Langly Space Paper 1958

N7472579IIIIl ll IIIffIIIIlll NOTESONSPACETECHNCLOGY Compiled bythe FlightResearchDivision Langley AeronauticalLaboratory Langley Field,Va. February -May1958 (NASA-TM-X-69992) NOTESONSPACF IECiINCLOGY (NASA)-63b P 00/99N74-72579 Tll[_U U74-72596 Unclas J2270 REPRODUCED BY NATIONAL TECHNICAL INFORMATION SERVICE U.S.DEPARTMENT OFCOMMERCE SPRINGFIELD. VA.2216] https://ntrs.nasa.gov/search.jsp?R=19740074640 2020-02-24T15:56:13+00:00Z GENERAL DISCLAIMER Thisdocument maybeaffected byoneormoreofthefollowing statements •Thisdocument hasbeenreproduced fromthebestcopyfurnished by thesponsoring agency. Itisbeingreleased intheinterest ofmaking available asmuchinformation aspossible. Thisdocument maycontain datawhichexceeds thesheet parameters. Itwasfurnished inthiscondition bythesponsoring agency andisthebestcopyavailable. •Thisdocument maycontain tone-on-tone orcolorgraphs, charts and/or pictures whichhavebeenreproduced inblackandwhite. •Thisdocument ispaginated assubmitted bytheoriginal source. •Portions ofthisdocument arenotfullylegible duetothehistorical natureofsomeofthematerial. However, itisthebestreproduction available fromtheoriginal submission. PREFACE These notes arepart ofaspace technology course given attheFlight Research Division oftheNACA Langley Aeronautical Laboratory during the early part of1958. Thecourse wasconducted asaself-education program within theFlight Research Division andthevarious Sections ofthenotes were prepared forthemost part bymembers oftheDivision; however, four oftheseventeen Sections were prepared bypersonnel from thePilotless Aircraft Research Division andtheCompressibility Research Division who were very helpful inmaking theprogram more complete. Thenotes have been compiled onabrief time schedule anditwill be apparent tothereader that thepresent version isincomplete andtosome extent maylack uniformity inlength, type ofpresentation, andtechnical detail inthevarious Sections. Nevertheless, there hasbeen ademand for thenotes from those wbohave seen them, anditisthought that they might serve auseful purpose ifthey were made available onawider basis. Itis believed that forthesake ofexpediency this goal isbest acheived bymaking thematerial available nowinitspresent unedited form instead offollowing theusual NACA editing procedures. Thenotes arearranged under five broad headings. Thefirst four Sections areconcerned withSpace Mechanics; thenext four with Trajectories andGuidance; thenext twowith Propulsion; thenext three with Heating and Materials; andthefinal four with Space EnvironmentandRelated Problems. Since these notes have notbeen technically edited, they arenot suitable forreference inN_CA reports. I<Henry A.Pearson Course Coordinator ManeuverLoads Branch Flight Research Division / I /If /llII /INDEX Elementary Orbital Mechanics Satellite Time andPosition With Respect ToARotating EarthSurface TheMotionofASpace Vehicle Within TheEarth Moon System:TheRestricted Three Body Problem IVIOrbital Transfer/ VRe-entry With TwoDegrees ofFreedom / v_%l SixDegree ofFreedom Equations ofMotion andTrajectory Equations ofaRigid Fin Stabilized Missile with Variable Mass / J_ll Inertial Space Navigation VVlll Guidance andControl ofSpace Vehicles z JlXElements o£Rocket Propulsion _/X vIXICharacteristics ofModern Rockets andPropellants Aerodynamic Heating andHeat Transmission /XII Heat Protection / _III Properties ofHigh Temperature Materials // _IV TheSolar System / /UAppendix ontheEarth, sAtmosphere / XVCommunication andTracking / JXVI Some Dynamical Aspects oftheSpecial andGeneral Theories ofRelativity 4"VII Environmenta_ RequirementsW.B.Huston andJ.P.Mayer T.H.S_opinski J.P.Mayer A.P.Mayo A.P.Mayo J.J.Donegan D.C.Cheatham C.W.Mathews H.A.Hamer J.O.Thibodaux, Jr.andH.A.Hamer W.B.Huston W.S.Aiken, Jr. E.M.Fields C.R.Huss W.J.O'Sullivan andJ.L.Mitchell P.A.Gainer andR.L.Schot% D.Adamson W.A.McGowan 1-1 SECTIONI ELEMENTARY ORBITAL MECHANICS 1.1 Motion inaPlane. When aparticle moves inaplane (orspace) its acceleration satisfies the equation F=ma, but the quantities Fand ahave not only magnitude but also direction. Inorder todeal with magnitudes, wecan take components along aset ofcoordinate axes, say the rectan- gular XYaxes, thus ifthe component of Falong aline is Fx, and the component of aalong that line is ax, then, Fx=max sndsoon. Ifthe particle moves in aplane, and its coordinates are (x,y), the components of acceleration along the co- ordinate axes are O(1.1-i) If Fx, Fy are the components of Zhe force, the motion ofthe particle isdetermined bythe equations 3_ Q i-2 (1.1-_) m(i.I-3) Inmany cases itismore convenient touse polar co- ordinates. Aparticle at Pisrelated tothe origin bythe coordinates (r,@). Its velocity Visresolved into com- ponents along the line PQ normal toOP. These components have the values:vralong the line OP, and v@ V dr Vr =-- (l.l-h)dt d@v_=r-- (i.I-5)dt The acceleration also has components aralong OP and a@ along PQ. Projection along the appropriate axes and alittle manipulation will show that (i.i-6) (1.1-7) Ifthe force Facting onthe particle has the component Fralong 0P and Feperpendicular to OP, the motion of the particle canbeobtained bysolving the equations 5< i-3 1.2mar=Fr (1.1-8) ma@ =F@ (1.1-9) These equations are the starting point for studying the orbit ofanearth satellite. Gravitational Forces. Two bodies ofmasses mand Mare attracted bya force proportional tothe product ofthe masses, and inversely GmM proportional tothe distance between them, or _ • The inverse square concept, associated with the name of Newton (1687) which was also independently proposed byHooke (ofHooke's Law fame) has been called "the greatest dis- covery innature that ever was since the World's Creation". Ifthe origin istaken asthe center ofmass M, the point Pasthe center ofmass m, then the positive direction of risoutward, while the force isdirected back along rand _A_LA_ --z (i.2-I) Hence, the differential equations ofmotion from (1.1-8) and (1.1-9) are P(i.2-2) rm._=o r (1.2-3) Weare thus assuming that there are nodrag oraccelerat- ing forces atright angles tothe radius vector and affect- ing the rotational motion, that is, F9=O. Equation (1.2-3) canbeintegrated immediately, F. -de_/<, d% (1.2-k) This result, which isknown asKepler's Second Law is independent ofthe form ornature ofthe attracting force (in this case, gravity). The constant Khas aphysical meaning which can beseen from anexamination ofthe initial conditions. At t=0 VIinadirection Since rdg =Vcos_dtlet the radius be rl, the velocity relative tothe normal to r. the value of Kwill be (l.2-5) 1-5 The element ofarea dA swept out by through anangle d@ is dA= o) atrinmoving (1.2-6) from which follows Kepler's Second Law (derived originally from observation ofplanetary motion and published in1618), "each planet revolves sothat the line Joining ittothe sun sweeps over equal areas inequal intervals oftime". (It should benoted that Kepler's Constant isusually given in dA K the literature as d-_ or _.) Weknow have the set ofsimultaneous equations F/\d(._-] _"_GF_M (1.2-7) r"d0_K (1.2-8) dt Tofind the path orthe variation of rwith @wemust eliminate time between (1.2-7) and (1.2-8). Itisconven- ient tochange the variable from rto u=l/r, whence Thenc/r _U d-_=-Fdg 7_ I-6 With these substitutions equation (1.2-7) becomes d8_K_(1.2-9) The solution ofequation (1.2-9) can bedetermined by inspection as U--_ -l"A¢o5 _ (1.2-10) where Aisaconstant the significance ofwhich remains tobedetermined, and thus the path ofour mass mabout the center is _/_1=G.M+Aco_(9 (1.2-11)rK" Todetermine the significance of Aweneed to remember that there isaset ofcurves called conic sections defined bythe situation shown inthe figure onthis page. Given apoint Fcalled the Focus atafixed distance dfrom astraight llne D called the Dlrectrlx. Then Ifa point Patdistance rfrom F ismade tomove insuch away that for any value of @ distance of Pfrom distance of Pfrom stant orthe ratioof Ftothe Discon-P dD r PD=constant 1-7 then the curve defined bythe point Piscalled aCONIC SECTION. The constant inthe above ratio isacharacteristic constant ofthe curve called its eccentricity _which determines the nature ofthe curve. Since the distance PD is d-rcos @, the eccentricity is orr -rco)e (1.2-12) I ! I___,.r_+_cos0 (l.2-13) But this isJust the relation (1.2-ii) that came out ofour equations for themotion ofabody under aninverse square law ofattraction. What dothese curves look llke. = O circle O<6_I elllpse f --I--I parabola __ hyperbola These curves are illustrated infigures i-i and 1-2, plotted from equation (1.2-13) rewritten as _d r--I@"6¢o_8 (1.2-iJ4) and asrEe)_ r(o) 1+e o,e I-8 For anellipse there isacharacteristic dimension which is more useful than d. Itisthe length ofthe major axis, the longest distance from one side tothe other through the focus, from perigee toapogee. Usually weuse the semi- major axis a,which isrelated to dbythe formula gd = (1.2-15)l-&" Using thisrelation the orbit equation is 5e(1.2-16) The physical significance ofthe constants inequation (1.2-11) has thus been shown bycomparison with equations (1.2-13). The constant A=1/d while the gravitational constant G, the primary mass M, and Kepler's constant together determine the value _d, that isK The mass section, bedeferred pending astudy ofJust what itisthat fixes the values of_and d. For this purpose weneed toreconsider the motion interms ofthe potential and kinetic energy of the mass m.mmoves inapath about Mwhich isaconic Which conic section and why are questions which must io< 1-9 1.3 Total Energy. Wewrite our principal conclusion thus far bythe equation for the motion of mabout M. IGM I r-K"+Tcose (1.3-_) Inorder toestablish the factors which determine the eccentricity and the size ofthe orbit itisnecessary to examine the total energy. The kinetic energy is _/E --/Vz._.i__ r1_ (1.3-2) The potential energy isdefined for conservative forces by whereWdPE=-aW Isthe work defined by(1.3-3) _/=/rFdr (1.3-4) The potential energy is /_E=_/r/C-jr _/ GMF_dr= The total energy isthen /'.GMr_ U=T_V-- rGNI_ r(l.3-5) (1.3-6) 1-i0 Oneexplanation forthenegative signisthatthepotential energy atinfinity istakentobezero,andsincepotential energymustincrease asthedistance fromtheattracting center increases thenatalldistances lessthaninfinity the potential energymustbenegative. Ifwecancalculate thevalueof theorbit,weknowitforeverypoint. isaminimum and Q:0jUfor one point in Atperigee, where byequation (1.3-1), ...L_/_GM I rp-(1.3-7) Atthis point allofthe velocity isrotational, there isno radial component and dGV=rpdt But byKepler's Law, for this, e=K(or any) point inthe orbit orwith equation (1.3-7) (1.3-8) Inserting the values from equations (1.3-7) and (1.3-8) in(1.3-6) gives for the Total Energy 12< 1-II or, from equation (1.2-17) or _-_) (1.3-1o) Thus the orbit iselliptic, parabolic orhyperbolic, asthe total energy isnegative, zero, orpositive. This inturn de- pends upon the relative magnitudes ofthekinetic and potential energy. Hyperbolic orbits require the most kinetic energy (orspeed). For themost part weshall beinterested in elliptic orbits, (negative total energy). Such anorbit is specified byavalue ofeccentricity, _,equation (1.3-10) and bythe semi-major axis a. From equations (l.2-15) and (1.2-17) (1.3-II) From equations i(1.3-9) and (1.3-11) the total energy for anelliptical orbit is --_MnnU=2a (l.3-12) Thus itisseen that the total energy inanelliptical orbit isinversely proportional only tothe semi-major axis ofthe orbit. Thevelocity atany point inthe orbit can beobtained bysubstituting equation (1.3-12) into equation (1.3-6). IS< °'*_ 1-12 O=-GMm IV"2a 2-/'m --" r" (1.3-13) 1.4 gravity.Establishment ofanOrbit. Thus far wehave written equations for orbits interms ofgravitational constants, G, the mass ofthe principal corny M, auun_tant K _^ _-_4-_i TToll of_.h are hard tograsp physically. Inthis Section weshall convert the equations toaform containing constants which are more familiar tous. For example• abody ofmass mhas aweight mgoat the surface ofthe earth because ofthe attractive force of Thus wecan relate this weight tothe force as =oRz and wecould substitute for the term(z.h.-l) GM its equivalent ¢@ Similarly the constant Kisfixed, for arocket bycon- ditions atthe instant ofburnout, _e. its distance r1 from the origin, its Velocity V1and the angle _its path makes with the normal to rI. Atthis instant, from equation (1.2-5) 1-13 The total energy is V•U=__ =-z_r, Hence _U_ __ ° _ o Insertingthese values inthe expressions for l.B-lO) and a(eq. 1.3-11) wefind that(1.4-_) (1.4-5) E(eq. @ Itisworth looking atthis term Rgo. Ithas the dimensions ofvelocity squared. Suppose wewanted toestablish a circular orbit ofthe radius ofthe earth rI=Rwith zero eccentricity. The eccentricity can bezero only ifthe term onthe right under the radical isunity. If RI=R, If=O, Cos _=i, thenthis can betrue only if VI2 =Rgo. Hence Rgo isthe square ofavelocity Vo equal tothe velocity for acircular orbit ofradius R. Weshall write itthis way, i.e., V@=__@ (1.4-7) and with this substitution, £;Il¢/,e-r,Vo-,Ij°"y IS< -,t._,S(1.b.-8) 1-i_ Similarly leads toJD/&M _,_]A v.l(i.3-11) (l.h-9) or7,=z-vo/vj (l.h-lO) These last equations '""_ 11.1'4-9_, and 11,_h_!O ) determine uniquely the size and shape ofanorbit interms ofthe three parameters (rl, VI, _i)which characterize abody atthe instant itbecomes afree body, i.e. for a rocket -the instant ofburnout. 1.5 Orientation ofOrbits. Inaddition toknowing the size and shape ofanorbit itisnecessary toknow the orientation ofthe orbit in space, that isthe location ofthe major axis relative to the point ofburnout. Ifthe vehicle islaunched asafree body with the initial elevation angle (_i) equal tozero, burnout isthe perigee orapogee, depending onthe speed. IfY1 isnot zero than the position ofthe major axis in space isrotated with respect tothe position for zero angle ofelevation. Incalculating orbits with aninitial launch elevation angle weshall use thebasic orbit emuation (equation 1.2-16) with the origin of @atthe perigee and calculate the 1-15 initial orientation angle oflaunch (91) using the usual initial conditions atburnout (VI, rl, _i). The orbit equation is /":1+-(co._G (1.2-16) This can beexpressed as P (l.5-l)/"--/+Ecol_9 Atlaunch sothat-/-/-_co_@_ (l.5-2) (P/,",)-I (l.5-3) The eccentricity _could becalculated from equation (l.h-8); however, itwill befound more convenient touse aslightly different form. The angle (_) between the instantaneous direction ofthe velocity (V) and the normal tothe radius (r) can begiven asfollows:_f _anY=dr rda- 1-16 Differentiating equation (1.5-1) weobtain de--(i _e)_6_,_ But from equation (1.5-1) (/+__o_e)_=(_)" and thus Therefore,r.Ljr----E _l'nae,_(1.5-5) f,_7"=_-_.sJ_(1.5-6) (I._-,, _/.,@ (1.5-8) or "t_.]"=/+6 co._(9 __ _*_ _ (I.5-9) and _-- 51"n Substituting _from equation (1.5-9) intoequation (1.5-3) weobtain for the initial conditions: or_o_e--C_l_,-_)o;,,,o, ,P/r,"_,.v,(1.5-1o) (1.5-n) Inorder tocbtain the parameter p/rIinterms ofthe launch conditions, wenote from equation (1.3-11) that p=(_(l-e")-d_K'"(1.5-12) 18< 1-17 and from equations (l.h-2) -=0-_')-o._P But from equation (I._-3)(i.5-i3) (1.5-14) Therefore p= _ (l.5-15)_._ and intermsofcircular satellite velocity rIVs atradius (l.5-z6) Thus all the elements for calculating orbits have been found. These equations are listed below, and repeated inTable l-la inthe order usually found most convenient for computing orbits. Inaddition, formulas frequently used inorbit calculations are given inTable l-lb. 1-18 Step i. Step 2. Step 3. orC--P /-EcosO _nZ,(1.5-I) (I.5-16 (i.5-ii (i.5-3) (i.5-9) Since the angle (Y) atany point inthe orbit isoften required (i.e. for re-entry angles) equation (1.5-8) is also repeated: I+_.cos0 The semi-major axis isgiven interms of (I= /_EK The semi-minor axis is(i.5-8) pas (i.5-i8) (i.5-i9) i-19 The foregoing ecuations (1.5-17) arewell suited for the calculation ofthe elements oforbits including orientation when the initial velocity VI, initial elevation angle _i' and initial radius rIare known. Aconsistent set of numerical constants for use inorbit computations isgiven inTable 1-2. These eouatlons arewritten interms ofthe circular satellite velocity (Vs). Insome cases itmight beprefer- able touse the escape velocity VEorthe circular satellite velocity atthe surface ofthe earth Voasareference velocity. Therefore, the equations for these velocities are given: (1.5-20) (1.5-21) (1.5-22) The orbit equation isillustrated infigure 1-3 and the boundary conditions for the orientation angle @i are shown infigure 1-4. Infigure 1-3 the origin isatthe right focus and the angle @ismeasured inthe counter clockwise direction from 4 1-20 perigee. The vehicle isassumed tobelaunched asafree body at @=@I" The initial elevation angle _I is positive when the initial velocity vector VIisinclined outward from theperpendicular tothe initial radius rI. Boundaries separatin_ various values ofthe orientation angle @I are shown infigure 1-)4. presents the values of (VI/V s) and 90°or270°asdefined by V,The curve shown re- _i where @Iequals (1.5-23) This curve and the axis where _i=0divide the area into the four quadrants asshown infigure l-h and as indicated below: [email protected];_o.s ",'-,-,o<'i__-°"_=/.o o=l-'V,__°_'__zo 71"Q.oar,.% I zr _,_, -r" -: JW" Z2<0]KI[ 1-21 The eccentricity _isalways these definitions. The effect ofinitial velocitypositive when using (VI) onorbits is shown infigure 1-5 for zero initial elevation angle and in figure 1-6 for _i =I0°" Itshould benoted infigure 1-4 that when the initial elevation angle iszero that 91=O° for (VI/Vs)>i and that @i=180° for (VI/V s)<Isothat the eccentricity for _i=0is rl=o (1.5-2b) Infigure 1-6 itcanbeseen that when the initial elevation angle isnot zero the orientation angle @i is not zero but isafunction of (VI/Vs). The eccentricities are larger than those infigure 1-5 but the major axes (a) donot change from those given infigure 1-5 since the length ofthe major axis depends on VIsnd rIbut not The effect ofinitial launch constant initial launch velocity Itisseen that the effect ofchanging the initial launchangle _i onorbits for a VIisshown infigure 1-7. 2S< 1-22 elevation angle _i istochange the orientation ofthe orbit inspace and toincrease the eccentricity _• For example given itisseen that al0°elevation angle causes arotation ofthe major axis ofabout 32°. Inplotting orbits with different orientation angles the second and succeeding orbits must berotated through anangle A@: 1.6inorder tomake the launch points coincide with the launch point for the first orbit; orthe orientation can be accounted for bydefining the orbit as !where @ ismeasured counterclockwise from the initial radius rI. Time-Speed relationship inanOrbit. Inaddition tothe shape and the orientation oforbits itisalso important toknow the lapse time between points inthe orbit and the speed ateach point inthe orbit. The speed can beobtained using the relationship of equation (I._-3): _4 _. 1-23 or,-V_,,_7-,-'0V,_,,,Y, (1.6-1) orfrom equation (1.3-13): or V" _zI_-)_._:F_, o_(vlV)_ z__ -rpC__')(1.6-2) (1.6-3) (1.6-Lt) The time relationship isnot asobvious and will be derived inSECTION II. The equation derived is: _(e)Co_e_l_X:,_.;_*v,,t'*"t.. For parabolic and hyperbolic orbit_ (E_i) the equation becomes (i.6-5a) where Ccan begi.ven inthe following forms: (i.6-6a) (1.6-6b) •-5< ",_-. 2,ww,.(i.6-6C) I-2_ The time inequation (1.6-5) ismeasured from the origin of @which isthe perigee. Ifthe time from launch orfrom some other point is desired, equation (1.6-5) can beused toget the difference inthe time between any two points. The time from launch becomes : ,4t,=t(e)-Ce) From equation (1.6-5) the period ofthe orbit can be determined tobe(1.6-7) 1.73 (1.6-8) -- (1.6-9) Realizable Orbits. Thus far our principal results are embodied inthe equations which show the relationship between the three parameters, rl,Vl,_l, which define the position and path ofarocket atthe instant ofburnout, and the orbit which results ascharacterized byits semi-major axis a, its eccentricity _,and its orientation angle 91" It would benice tobeable tomake plots showing the relation- ship ofthese factors, but _parameters are hard toplot. 1-25 Since the elevation angle _ does not enter into the expression for a,wecan plot this asafunction of rI and VIasinfigure 1-8. The distances have been ex- pressed inearth radii R, the speed interms of Vo, the speed for acircular orbit ofradius R. The plot shows how aswegofarther and farther out, toestablish anorbit, we need toprovide less and less speed toestablish it. In fact, ifweprovide toomuch, we_will lose italtogether toaparabolic orhyperbolic orbit, ifatany time (1.7-1) aspeed which iscalled the escape _elocity. At rI=R, VE=36,695 ft/sec =25,019 mph For many purposes, itIsconvenient tonormalize the velocity not with Vo, but with aspeed Vs, which varies with altitude and isthe speed for acircular orbit ofradius thus the satellite velocity is V. ----{_/_ (1.7-2)rl; Onthis basis, figure 1-8 changes its form slightly, and becomes figure 1-9. Inorder toillustrate the relationship between eccentricity Cand launching conditions itissimplest G-7.<"7 1-26 first touse Vs, (1.4-8) becomesrather than Vo. Onthis basis, equation C= - cosy (1z-3) Itissimpler inthis case touse (VI/Vs) 2asthe inde- pendent variable, since for _=O, 2z_z which issimple and symmetrical toplot asshown infigure i-I0. There isonly one speed VI=Vs which atthe altitude rIwill produce acircular orbit, and this only if _=O. Any other speed orany departure from _=0will produce afinite value ofeccentricity. Thus circular orbits are exceedingly difficult toestablish asthe programming re- quirements onboth thrust and direction control are very stiff. Ifwewant toshow the same information asisgiven infigure I-I0 interms ofVI/Vo, rIand _the plot issomewhat more complicated, asshown infigure i-ii, which isanomogram for the solution ofequation (1.4-8). The variation of Vsand VEwith the distance from the center ofthe earth isshown infigure 1-12. The real question, ofcourse, inestablishing anorbit, isone ofwhether itwill clear the earth. Wecan launch a rocket atsome distance rl, and give itaspeed VIat 28< ........... i.J " 1-2? some angle _,but unless the perigee distance, the point ofminimum radius, isgreater than the radius ofthe earth, wewill not clear it, and will not even get one pass. This ignores the atmosphere, ofcourse, and inthe practical case, unless the perigee distance isatleast 50miles above the surface, the orbit isoflittle use. Now the orbit, interms of aand _may bewritten _SQC/- r=/+_co5_ (1,2-16) Hence perigee distance (9=O) isa(l- C). This distance does not seem tobeexpressible asasimple function of (rl,VI,_), but wecan essily write acondition for the limiting elevation angle _L which must not beexceeded if a(j-E)R (1.?-_) Combination ofeouations (l.h-6) and (l.h-8) with (1.7-4) yiel_the condition that (r,/g)CV,/Vo) =_, (I_ (1.7-5) 1-28 1.8relations which are plotted infigures 1-13 and 1-14, for various values of rl/R from 1to60. Note that for rI=R, _ must bezero,while ifwewish toestablish anelliptic orbit atsome othervalues of rIand VI, then the elevation angle must beless than the value of _L shown for each value of rI. Another limiting condition applies here_ VIcannot exceed the escape velocity ifan orbit istobeestablished. Minimum Altitude Orbits. The practical use offigures 1-13 and i-i_ islimited bythe existence ofthe atmosphere. The requirement for an orbit which does not approach the earth closer than some specified distance isshown infigure 1-15 where the ratio _R R+h, isplotted asafunction of VI/Vs for various values of For example, let usassume wehave asatellite vehicle which wewish tolaunc_ into acircular orbit at150 miles above the earth, and the maximum expected error inlaunch 1-29 angle (A_)is3degrees. Letusfurther assume that wedonotwantthevehicle todescend below90miles. In thiscase rmin = 01 I00 =0.985h r! From figure 1-15 wecan see that weneed aninitial velocity ofabout VI/V s=1.05 or5percent above circular satel- lite velocity toassure that the vehicle does not gobelow 90miles. Asmall section offigure 1-15 has been replotted in figure 1-16.for values of VI/V sinthe neighborhood of V1/V s=1.0. Also shown infigure 1-16, are the eccentricities and orientation angles associated with the orbits. Inorder toobtain amore direct measure oftheminimum altitude figure 1-17 ispresented tobeused inconJuction with figures 1-15 and 1-16. Shown plotted infigure 1-17 is4-1..^-_.,--,^ /......._j.-_.,.,.,.,.,rmin/ el p_o_ed against the minimum altitude min for constant values ofthe launch altitude. For example, ifthe vehicle were launched at150 miles altitude ataspeed ratio of V1/V s=1.05 and _l =3degrees the value of rmin/ rl from figure 1-15 is rmin/ ri=0.985. From figure 1-17 wesee that the m_nimum (perigee) altitude will beabout 90miles. Gi< o _ez_I-Io CONVENIENTFORMFORCALCULATINGCHARACTERISTICS'OF ORBITS FORTHEINITIALCONDITIONS rI,VI,TI _C r, Z.÷a.e,= .i_-o_._,[__-/] 4.t6.Y=_5in0 b="" 5 v-;.O 7T= I ",,I _)i ! "i _" I !II iU Ry 4.3 : " > YBh ae SSeo Xeow ~lsoeEsx">| gINGS Fale og Foy ‘ " vos Zz S$pee ESAT Gk ;>aleFZNu w«ofSSbye isgoe wo. aT ~~ ~ a & &. IN 3& wS3 s2§ .&§ ® o8 Qe ~ 18 “gs | ~3 v 3 > yoS it—, v A wE ° Seo 3 s|.¢aia 3 SSS |S cali as | ac fhv TS fe+ ofS =|= ° uwakeTs-lsYRS 3 SSfa N : iS §.ied{8=~——ooeasily Ss | a BS a|NE [ge[oeSPoe -g®2oY + T T 1se -le Asrssft ~~~~>_we 8°TT mo oy& & &www w 3a< Table 1-2 Consistent Set ofNumerical Constants for Use in Orbit Computations Various reference sources give values ofthe numerical constants (_,G,R,and go)used inorbit computations, which differ. Itwould bedesirable toadopt aconsistent set, that is,aset which satisfies the relation =GM-R2go The following set isaconsistent set. G=3.4_ x10-8ft4/Ib sec4 =6.66 x10-8dynes cm2 gm2 =I._077 x1016 ft3/sec 2 3.986 x1020 cm3/sec 2 W=6.59 x1021 short tons rL--_.V_ XIu_ slugs go=32.2 ft/sec 2 R=3960 milesuniversal gravitation constant weight ofearth mass ofearth radiusofearth F- F-/_I-1.-Conic sec÷lons_ellipse_ parat_ola, hyperbola FI91-2.- Chowac_erl_/lcs andd_rnen_ton_ o_v_rlous conic 5eC÷lOn_ ande/I/pse_. \ I P /+ECOS Z COB _'i rp/'] _an_,:_n_,LPlr'_/ £-cos_-I _,benr,-s,c-- -! iciE.I-3.-zTIlus_rotlOnoforbi_equolmn :-::i). _t72,: .....:_-7- H h 2L_ 2_7_- 2. _L-rH:_ i. ...._ "7-::ii:_ :-:T_.... :7;:_:T- -"L.=i:___ .... ;i _ i.L, -2__ T_-: 'N ::-:-----._,_:. :rT:=::7::: =71i!itll ',ii1!1 iiiill iTS1- e9< >,.:,_.:; Pals . 2 iN}aad Ww " sand $ S: > as} ‘ se : N Q 8 “ au 2 sINSey , @ > $ s N ¥O9 Saye aS 1Git akS34 % iS} x DGSI00%335 8iS~ ss @Solr SESE: a * SITUS - Seauy Gheds SSoesse gee SSseeiz uRSSos 40< @ : $ fpae Eggo ‘°s “¢NS = Boks ~~ 3833 —. 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Itisinteresting and often necessary toknow the satellite time ofpassage and its positionover arotat- ing earth surface. The object ofthis Section istouse the orbit equations presented inthe preceding Sections and develop time and position relationships that are a function ofthe angle inthe orbit plane. Abrief dis- cussion ofsome general features ofsatellite orbits will also bepresented. Time Relationship According tothe laws ofplanetary motion, the three initial orlaunch conditions V1, rl, and _l' determine the future path ofthe satellite neglecting drag or accelerating forces acting parallel toits path. From the preceding Sections weknow that the satellite will move inanelliptic path f mm6"J I÷6co5_ (2.1-i) or r= /_l-_C05@ (2.1-2) Wealso know that its motion will obey the second of Kepler's Laws that the radius vector sweeps out equal 53_ ., 2-2 areas inequal increments oftime or (2.1-3) The value of K angle described bythe radius vector rinatimeissimply twice the area ofany tri- dt and sweeping through anangle d@. Transposing and integrating (2.1-3) weobtain o K£=_rade(2.l-4a) @ Substituting equation (2.1-2) into equation (2.1-_b) we(2.1-kb) see that K,,(+6_°3_) z(2.1-5) Integrating and putting inthe limits weget the expres- sion for time from the perigee as (2.1-6) 2-3 Fromour previous notes weknow that p=a0-_U-K_/_R_ (2.1-7) Sothat equation (2.1-6) can bewritten as (2.1-8) Ifwelet K5 ii 9_R+(2.1-9} Now from our previous Sections wehave the following expressions K=,7,v,co5r,, P=C/!)-co_ r,_V5(1.h-3) (1.5-_6) (1.5-19) Thus the valueof Ccan bewritten inthe following forms I SS<(2.l-lOa) G--- _cos (2.1-10b ) (2.l-lOc ) The period issimply the value of t(@) for sothat from equations (2.1-8) and (2.1-10) -,,j-_T=0__),/, L_'aO T=(j__)_9=2_ (2.1-11a) (2.l-llb ) T_ (2.1-11c) where the semi-major axis The previous section on"Orientation ofOrbits" shows that perigee point and the launching point are afunction of @l assketched onpage 2-5. 2-5 r os=#mlh_ The angle between the perigee and the launching point isrelated tothelaunching angle _l by _p/r,I _=ne,=cant,LP/?,-_ The time from the perigee tothe launching point is simply the value of tfrom equation (2.1-8) for @=@l" Predicting Satellite Position One oftheobjectivesofthe IGY satellite program is todetermine the shape ofthe earth and its gravitational potential. Weknow that the earth isnot asphere, but 57< 2-6 anoblate shape. Consequently anartlfical earth satel- lite isattracted tothe earth byanet force that does not vary exactly asthe inverse square ofthe distPnce from the earth's center and the force isnot directed exactly toward the center ofthe planet. However, for locating manned satellites that may beinorbit for only afew hours orfor obtaining useful first approximations ofthe orbit elements ofunmanned satellites the assumption that the earth isauniform sphere should give results that are reasonably accurate. Two methods for obtaining the satellite position can beused both ofwhich produce identical relations provided the same reference positions are used. One method utilizes relationships existing inspherical triangles inlocating the orbital plane onthe earth surface while asecond method uses anequatorial system ofco-ordlnates used byastro- physical laboratories and other investigators (see refer- ence l,2and 3for example). Since the second method re- quires familiarization ofvector products which will be discussed inalater section ontrajectories, the position relations using spherical triangles will bedeveloped in this section. 2.2Reference Conditions Before weestablish the reference condition which will enable ustoorient ourselves onthe projected path ofthe 2-7 satellite, let's familiarize ourselves with how the satellite might move around the earth and let's review some relationships that exist inany spherical triangle. Ifthe earth istaken asasphere the plane ofthe satellite orbit remains fixed indirection and thenotation shoe infigure 2.1 ischosen totake advantage ofthis fact. The inclination angle iofthe orbit plane orthe maximum latitude reached and the point Natwhich the satellite crosses the equator from South toNorth specify the orientation ofthe orbital plane inspace. The follow- ing additional notations are presented for orientation in the equatorial and orbital planes. Z , 5_ Lv z\//f -_-E_uoJ'onoll plane X 59< 2-8 where 0-isthe center ofthe earth XOY- isthe plane ofthe earth's equator OX-isdirected towards the point where the satellite crosses the equator going North OZ-points tothe North pole N-point atwhich the satellite crosses the equator from South toNorth and iscalled the ascending node P-perigee location -angle NOP measured inthe orbit plane from where the satellite crosses the equator to theperigee LV-vertex ormaximum latitude reached bythe satellite (the point where the orbit cuts the plane OYZ) -angle inorbit plane between perigee and the apex ormaximum latitude reached bythe satellite (_+_=90°) S-any subsequent position ofthe satellite @V=angle inorbital plane from the apex to the subsequent position Softhe satel- lite @-angle inorbital plane measured .from the perigee (@=_+@V) (usually called the true anomaly angle) i-inclination angle ofthe orbit plane 60< 2-9 Toderive the relation between the latitude, longitude, and the angle @V weuse the following spherical tri- angle relations Where A,B,and Care three angles and a,b,and care the arc lengths ofthe opposite sides. From the sine law weknow that sin A=sin B=sin C sin a sin b sin c and from the cosine laws cos a=cos bcos c+sinbsin ccos A cos C=-cos Acos B+sin Asin Bcos c Ifweknow two sides and one angle ortwo angles and one side wecan solve for the unknown side orangle. The following additional notations are presented for defining the various additional elements 2-i0 Ap Av where the elements not defined previously are A-apex orvertex angle=90° -angle measured from the North tosatel-lite position atperigee 6_pV-difference inlongitude from perigee to vertex A_pv=)%p-_V A_VS- difference inlongitude from vertex to satellite A_VS =_v -)%S A_NS -difference inlongitude from nodal point Nand the satellite A_NS s_N -AS L-latitude measured North orSouth ofthe equator _- colatitude =(90 -L) -longitude measured EastorWestofthe prime meridian atGreenwich Subscripts refer toparticular position in orbit plane. 62< 2-II From the sine law sin A sin sin (90 -Lp) sin (90 -LV)_ sin sin (90 -i) Since the apex orvertex angle thenA=90° and sin (90 -L)=.cos L sin_ = cos iIcos Lp(2.2-1) From the cosine law for two sides and one angle cos (90 -Lp) =cos (90-i)cos +sin (90-i)sin _S cos 90° or andsin Lp=sin icos sinLp/ cos_= sin i(2.2-2) From the cosine law for two angles and one side (2.2-3) 63< 2-12 2.3 Position from Nodal Point For any other latitude and longitude location from the ascending node position Nonthe equator the follow- ing quantities for anon-rotatlng earth are derived: Latitude From the cosine law co.,(9o-L,)= Orcos(eo-z)co_ev 5//JLa=51,_i,co5@v COS_'V=5INL5 i 5mt., now from our previous notations weknow that @ (.,0+(3= 90(2.3-1) Latitude onthe earth surface isdirectly related to declination which isreckoned Indegrees North and Southofthe celestial equator. G4< 2-13 then so and_,,,=-[_o-c_*oU 5/wW=-cos(w+_)(2.3-2) (2.3-3) Substituting (2.3-2) into (2.3-1) weget the latitude of the satellite. (2.3-h) Longitude From the sine law szuCgo-z.,) .._o.j weget that cosL5(2.3-5) **2. Longitude onthe earth surface isrelated tothe right ascension, the hour angle and sidereal time inthe equatorial system ofcoordinates. 6S< 2-14 since or then (2.3-6) substituting (2.3-6) and (2.3-3) into (2.3-5) weget C0.5A"_Al_i=(2.3-7) Equation (2.3-7) can beput into amore convenient form bywriting itas (2.3-7a) Itcan very easily beshown bydotting aunit vector along the normal tothe orbital plane into aunit vector along a line drawn from the center ofthe earth tothe satellite position that L_ _oni. (2.3-8) 66_- 2-15 Substituting (2.3-8) and (2.3-_) into(2.3-7a) weget or .ani. Sothe longitude ofthe satellite for anon-rotatlng earth isgiven by (2.3-9) Since only the longitude position ofthe satellite is affected bythe earth's rotation the final expression for the longitude position from the nodal point is (2.3-lO) where We -rotational velocity ofthe earth equal to15°/hour or0.25°/min t(@) -time from perigee given byequation (2.1-8) The sign ofthe Wet(@) term isnegative for satel- lites launched East orwith the earth's rotation and positive for launchings tothe West. Since equations (2.3-k) and (2.3-i0) relate the latitude and longitude asafunction 67< 2-16 ofthe angle 9inthe orbit plane and time isa function of 9from equation (2.1-8) then equations (2.3-_) and (2.3-10) are obviously also afunction of time. Once the longitude position ofthe nodal point Nisestablished with reference toGreenwich then aplot of Ls(t) asthe ordinate and _(t) asthe abcissa is useful inlocating the satellite onaMercator projection. Each successive orbit isdisplaced inlongitude by where the period Tiscalculated from equation (2.1-11). Afew words about the earth's rotational velocity be- fore some typical results are presented. Weknow that the earth's rotational velocity atthe equator is =__2_R =1037.6 mi/hr equator 24 The rotational velocity atany latitude Labove or below the equator for agiven angle _ measured from the North is _L =1037.6 cos Lsin Then for agiven latitude you gain the most benefit from the earth'S rotation bylaunching due East (_ =90°) 2-17 Typical results Typical results arepresented toillustrate the application oftheabovedeveloped procedure. Theex- amplepresented isforthefollowing specified require- ments: I.Thesatellite height, hIabovethelaunching pointistobe150miles. Thelaunching velocity, V1shouldbegreatenoughtotolerate aninitial flight pathangle_1of_3°andaminimum radius overthelaunching radius rmin/rl ratio of0.986. 2.Thelaunching angleshouldbesuchthatthesatel- litecomesdirectly overthelaunchposition at CapeCanaveral, Florida onthesecond orbit around theearth. Theorbitcharacteristics fortheaboverequirements using equations (1.5-17) fromtheprevious Section aretabulated below Orbit Launching radius rI=_ll3 Launching velocity ratio, Initial flightpathangle,miles VI -1.05Vs _l=O Anglebetween perigee andlaunching point, Semi-latus rectum, p=h,535miles Eccentricity, _=0.1025@I=0 2-18 Perigee distance, rp=4,113 miles (hp =150 miles) = =1,090 miles)Apogee distance, ra 5,053 miles (ha Period T=105 minutes 4534.6 miles Radius inelliptic orbit, r= 1+0.1025 cos @ Because _I and 91 were zero the perigee point occurred over the launching point and consequently the launching height was also the height atthe perigee. Wenow solve for the difference inlongitude between the perigee and the vertex point knowing that due tothe earth's rotation =-O.Z,,_x 10,5' _1/_=-,26.°/..5 "' ('._-2&°/5' '/_L-'_P= •-80S2.@' 70< 2-19 then A}kFv=z3°7-5' Knowing Lp and A2%pv (2.2-2) and (2.2-3) forwesolve equations (2.2-1), and obtain _/ =83°39' From the same three equations then the inclination ofthe orbit plane orthe maximum latitude reached is i=Lv=29°6' and the angle inthe orbit plane between the perigee and the vertex is Since then=ii°31.5' +_S=90°, =78°28.5' Knowing the inclination angle iand the angle _in the orbit plane from where the satellite crosses the equator tothe perigee the declination orthe latitude of the satellite from equation (2.3-h) was simply and the incremental longitude from equation (2.3-10) is 71< when or then2-20 _=0 '_,,5(°)='_"_,,,,p=_-'_pv =-8°_2&'-7&°._2.S" _j_O) =--/Sg _5./ West ofGreenwich sothe final expression for the longitude orthe right ascession ofthe satellite asmeasured from Greenwich was or where the time from perigee from equation (2.1-8) was The Mercator projection ofthe initial orbit for the above example isplotted infigure 2.2. The surface of the earth covered did not include Europe orAsia. The only land areas crossed other than parts ofthe United States and Mexico included parts ofAfrica and Australia. 72< 2-21 2.5Aplot ofthe distance ofthe satellite from the center ofthe earth rasafunction ofthe latitude Lis shown infigure 2.3. The height onthe southern passage being greater than the northern passage for the same i_ latitude. Changes inlaunch angle _M1and _ havej been examined and the results indicate that change sin _/1 influenc_ the eccentricity 6 and the longitude ,while changes in _ effect the latitude L. Neither _l or _ effect the period T.iw_r_ k! Perturbation ofaSatellite. Without disturbing forces the orbital elements would not vary with time and the satellite would travel ina_ elliptical orbit described bythe elements. Inrea1'ity ,1 the continued action ofthe disturbing forces icauses the elements orcharacteristics_of the orbit tochan_e With time. These forces arise because asmentioned_p_reviously the net force that attracts the satellite tothe earth does not vary exactly asthe inverse square ofthe distance from the earth's center andisnot directed exactly toward the earth's center. The other main disturbing force tothe baslc elliptic orbit results from the atmosphere. Itshould beofinterest, therefore, tomention what some ofthe dis- turbing force effects are onthe elements appearing inour derived expressions for time and position. Because ofgravlty's more powerful effect atthe 73< 2-22 earth's bulging equator the satellite's orbit precesses orpivots, llke achild's top slowing down. The re- sultant effect does not decrease the inclination angle ofthe orbital plane but ifone revolution about anon- rotating earth carries the satellite over the equator at N(figure 2.1) the next circuit will cross atsome point west ofpoint N. The rate ofregression ofthe line ofnodes according toreference [_isafunction of the cosine ofinclination angle and is, therefore, the greatest for anequatorial orbit. For example, for a mean satellite radius of_,500 miles the rate ofregression iscalculated tobe-6._de_/day for anequatorial orbit and -_.61 deg/day for anorbit inclined 45°from the equator. Another effect ofthe earth's oblateness istorotate the orbital plane about the earth's axis inthe direction opposite tothemotion ofthe satellite. This rotation indegrees/day for agiven height is, according torefer- ence I,afunction ofthe cosine ofthe inclination angle iand is, therefore, the greatest for anequatorial orbit. Rate ofrotation ofSputnik IIorbital plane which was inclined 65°tothe NEwas between 2.69 and 2.88 deg/day. Itisalso found that the major axis ofthe ellipse rotates inthe orbital plane bysomany degrees per day for agiven height. This rotation isinthe same direction as the satellite motion ifthe inclination angle isless than 2-23 63.4°andintheopposite direction ifiisgreater than63.4.Forexample, fora200nautical mileorbital altitude themajoraxisrotates about16deg/day inthe samedirection asthesatellite foranear-equatorial orbitandatabout_deg/day intheopposite direction forapolarorbit. Bothoftheabovementioned disturb- ancescausetheperigee point Ptomovealongtheorbit sothat_theanglefromwherethesatellite crosses theequator totheperigee pointisnotconstant. Con- sequently thesatellite hasdifferent periods thanwould bepredicted bytheory. Alsotheheight ofthesatellite asitcrosses anylatitude goingNorthsay,maybediffer- entonsuccessive days. Themaineffect oftheatmosphere istoreducethe length ofthemajoraxismaking theorbitmorenearly circular. Thisisbecause mostoftheretardation dueto dragoccurs nearperigee; consequently thereisalossof altitude atthesubsequent apogee. Thereduction inthe length ofthemajoraxisshortens theorbital periodby somanyseconds perdayanditispossible toestimate theairdensity andthelife-time ofasatellite fromthe rateOfdecrease oftheperiod. 7S< 2-2[_ References i@ @King-Hele, D.G.and Gilmore, D.M.C.: "The Effect ofthe Earth's Oblateness onthe Orbit ofaNear Satellite." RAE Tech Note GWh75, October 1957. Ashbrook, J., Schililng, G.F.and Sterve, T.E.: "Glossary ofAstronautical Terms for the Description ofsatellite Orbits." Smlthsonlan Insitution Astro- physical Observatory Special Report _,Nov. 30, 1957. QKooy, J.M.J.: "On the Application ofthe Method of Variation ofElliptic Orbit Elements inCase ofa Satellite Vehicle." Astronautics Acts _(3)P179- 21 . 1957 Spltzer, Lyman, Jr.: "Perturbations ofaSatellite Orbit." British Interplanetary Society Journal v.9, May 1950, p131-136. 76< Z Z/\ _H SURFACE acid.2-._.--I_OTAT/OIq FOR SATELLITE OILER 5PNER/CAL EARTH, foot |TiN ee PPAica RTSCTE et{oS ee .poee ee : EOI agpesoeeelae FoRNerre ee CONNSIS o6k 2 oe eeeeeeA Tihse a ArersNsLE SE okeiyy Sean SEMeSe pew ikeTS Joa Soon,sche(toa Lebayh eeee a Baeymrr0KMMIGNSS7:A SEEepeee etreFresco eee ttt eyikl i7 r) (re eeaoriga erSmee aC,tome aeSa |SJeoYt RCC REL(COASECCORtet oN(OS eo PsPi te aHel R z =fa EeeaS a TG LSS Li{eee eA ti eeDS eee Naa9 as T'S iy iteee ‘SA APE VUEPTRDOPAT 0POONERS2PIFy % o8<e k.eq .p. t3 ta 1 ! e_44-._ON f 7_PnlVm7 7S_. SECTION III THEMOTION OFASPACE V_ICLE WITHIN THEEARTH-MOON SYSTEM INDEX 3.0 Introduction 3.1 Equations ofMotion (Inertial Reference Axes) 3.2 Equations ofMotion (Rotating Axes) 3.3 Jacobi,s Integral 3.4 Surfaces ofZero Relative Velocity 3.5 Points ofCoalescence 3.6 Velocities Near theSurface oftheEarth Corresponding toC2,C4,C6, andC8 3.7 Conversion ofRelative Velocity toVelocity inthe Earth Inertial _ystem 3.8 Effects ofNeglected Factors 3.9 Trajectories Near Minimum Velocities 3.10 UseofTwo-Body Ellipses toApproximate Lunar Orbits 3.11 Locus ofPoints ofEqual Attraction Between theEarth andtheMoon 3.12 Sphere ofInfluence 3.13 Approximate Method ofCalculating Trajectories of Lunar Vehicles 3.14 Characteristics ofApproach Trajectories 3.15 Trajectories toStrike theMoon 3.16 Circum-Lunar Trajectories 3.17 Allunar Trajectories 3.18 Periodic Trajectories 3.19 Establishing anArtificial Satellite oftheMoon 3.20 UseoftheMoon forAccelerating Space Vehicles 3.21 Propulsion Requirements forLunar Vehicles ReferencesPage 3-1 3-2 3-4 3-9 3-11 3-lh 3-18 3-20 3-23 3-25 3-28 3-31 3-34 3-39 3-51 3-53 3-59 3-62 3-63 3-65 3-69 3-72 3-73 80< SECTIONIIl THEMOTION OFASPACE VEHICLE WITHIN T_EEARTH-MOON SYSTEM _.O TheRestricted Three-Body Problem Sofarwehave been dealing with themotion ofavehicle under the -,ttr,_ction ofonelarge body (the Earth). If,however, thevehicle isto operate atlarge distances from theEarth (inthevicinity oftheMoon, forexample) then theorbit equations which have been developed areno _en,Eer valid andwemust take _nto account theforces duetothesecond 2"_rge body (the Moon). Ifthemass ofthespace vehicle wascomparable tothem_sses oftheMoon andEarth, wewould have toconsider the clssslcaS Three-_ody Problem. Inthis section wewill deal with the restricted Three-Body Problem inwhich themass ofoneofthebodies (the space vehicle) isinfinitesimal incomparison with theother two bodie_ (the Earth andtheMoon). TheThree-Body Problem isoneofthe _ssica] problems ofcelestial mechanics andthenames ofLagrange (1772), .Tmcobl (i_3), Hill (1_7_), andPoincare areclosely associated with the Freblem. Inthis section thedevelopment oftheequations ofmotion follows that ofMo_lton (1902) inreference 3-1. Thebasic development _ureference 3-], however, follows that ofHill (1_76) inreference 3-2. Thetrajectories shown andmany oftheresults given inthis section were taken from results ofstudies bytheRAND Corporation (Buchheim reference 5-_) andbytheHu._,_lans (Yegorov, reference J-h). (Itshould benoted that both references 5-3and3-4made wide us_ofthebook AnIntroduction toCelestial Mechan_c_ byW.R.Mou!ton; reference 3-1.) 3-2 3.]Equo.ti.?nsof_.!otlon (Inert!a] Ref'erene_ Axes) Thesystem of_es used istheinertial e_x_s systemsh.ovn inFigure 5-I and_.owhere theorigin ist_.ken atthecenter ofmass oftheE_rth-Meon system, _mdthepl._.ne,-frotation ofthemoon about theEarth isinthe Xo'ToPl_-ne"9: %o There on'et_oradial rr_nvltationslforccs acting onthevehicle "_b_reGmM]. Frl=maI r19 Fr2=m_2=.__ r22 = )2+(Yo )2+Zo2r].9(xo-xol .-YOl r22=(×o-xo2)2+(Yo"Yo2)2+_o2_.l-I }.]-..0 J/ Th_forcesdue tothee__rth inth:_. Xo,.Vo,and zodire:_t_,_n_ n,_:r.: FXl=Frl(:<o-xoI) r] (Yo-Y_$) 1rl ZoF=Fr -- zi 1rl3.1-} 2_/ng theforces duetotheearth andthemoon inthe Xo,Yo,sund zo directions wehave: ,,(Xo-Xoz) (Xo-Xo2) mxo=-GmMI -GmM2 r13 r23 (yo-yol) (yo-yo2) m@o""GmM1 -GmM2 rI3 r23 " ZO ZOmo - rl_ r2_3.1-4 or ,, (xo-Xol) (Xo-Xo_)XO=-GM1 -GM2 r15 r25 " (yo-Yol) (yo-Yo2) YO="GM1 -GM2 rl5 r25 " Z0 Z0 Zo=-GMI--- GM2r-_r153.1-5 83< 3-k 5.2 EquationsofMotion (Rotating Axes). Now, letusassume that theearth andthemoon revolve incircles about their con_non center ofmass. (Atleast forthetime itwould take avehicle tocomplete anorbit tothemoon.) Actually, theeccentricity is0.0_9 -notquite circular. _T__^_i_motion _v_referr _A__toan_......_y_t_m._ ofaxes rotatin_with_ the uniform angular velocity _,-themean angular velocity oftheearth-moon system. (See Figure 3-2.) Thecoordinates inthenewsystem aredefined bythefollowing transformations: O _Jo i fZoqK cos_t-sin_t0x =Isin_tL!0 c°s_tOli_O i_3.2-1 Xol YO_"m_COScot-sincot0x-a_i 0 0I:i_ , JLJ3.2-2 I!Xo y! iii =!slncot coscot0! J 0 0iii. JI! Yz"a_-_yi_3.2-3 Other useful transformations aregiven intable 3-1. Substitutin_ thevalues from equations (3.2-I), (3.2-2), and(3.2-3) into equ_tlons (_.!-5) weget: 3-5 II •N I I ÷ I g C.:. ¢) I I =NO rI + ,-.4 X O I ! i 0,_ + III ,,,,-4 % I I "N ÷ O I ÷0 0 r'----*_-i ÷ e--i t i I m i I ÷ x_ v .-'2 O L I! _,1OJ ! ! II0 o I I m 113 I U O 4_0 .,_ 0 I I ÷m o m_ O ..,-4! II4J'0 4) 4,_ 0 0 U u) bO I= ,,r-!I n 3-6 Substituting thevalues for A,B,C,andDwehave: and,, (x-xl) (x-x2) x-_-_2x=-GMZ G_% r13 r23 (Y-Yz) (y-y2) y+_._=_GM l G_ r15 r23 " Z Z Z=-GM1--- GM2-- rl3 r233.2-5 Yftheposition ofthe xrotating axis istaken through theearth-moon axis then Yl=0and Y2=0sothat x- =(_x-GMI(x-xl) (x-x2)G_ r13 r23 II y+_=o_2y-GM1y-GM2-X- r15 r253.2-6 ,, Z Z z--GMl G Note that rl2=(xo-xol)2+(Yo-Yol)2+Zo2 Then from thetransformation equations (5.2-i), (3.2-2), and(3.2-3) : r12=(x- xl)2+(y- yl)2+zZ which isindependent oft. 3-7 Thus these equations have theimportant property that theydonot involve explicitly theindependent variable tbecause thecoordinates of thefinite bodies have become constants because ofthemanner inwhich the axes arerotated. Thegeneral problem ofdetermining themotion ofasmall vehicle isof thesixth order; ifitmoves intheplane ofthemotion ofthefinite bodies, itbecomes fourth order. Theterms _ and 2_ aretheso-called Coriolis accelerations and _2x and_ arecentrifugal terms. Inthepractical case itisconvenient touseaunits system that does notrequire either very large orvery smallnumbers. Therefore ttheunits will bechanged to: Unit oftime unit ofdistance -iday ilunar unit, distance fromearth tomoon Inthis system: (Ref. 3-3) l =o.22_70_ rad/day D=1lunar unit GM=O.052_6587 (lunar unit) 3(day) "2 R=0.01655926 lunar units Xl=-0.01212_563 lunar units =- X2=0._7_7144 lunar units =i- M=M_ss ofearth plus mass ofmoon M1=Mass ofearth M2=Mass ofmoon Mo 1L_tt_.ng U=_"= (i-i_) M02._9M Equations (3.2-6) then becom, _: I! x-2___2_.oM(l-.)(X-Xl) r15GM_(x-x2) r25 I!y+,P_=2_,-oM(1-_) Y----oH. Y--- x r23 rl_ Zz=-GM(I-I_)z- GMU-- r15 r255.2-7 where r12=(x-Xl)2+y2+z2 r22=(x-x2)2+y2+z2 These aretheequations with which trajectories ofmoon rockets canbe computed. SS< _-9 3.5 Jacob!'s Integral L"_e only kno_rn zolution ofthe above eqtu_tion istheJacobi integcal. If w=!_#(×2÷y2) _GM(I- _)÷G__ 3.3-I 2 rI r2 th_n theequ,_tions ofmotion (3.2-6) canbewritten as " 6W _x If y__=__w 3.3-2 8v :.xj.nIp_e"8W _z _AW_1.,_?(.ox) GM(i-_) 2(x-x_) -GM#2(x-x2) 3x 2 r12 2rI - r222r2 ,:_2x -GM(i-_)(x-xl) (x-x2) GM# r15 r25 r,_itlplying equations (3._-2) by 2x,_,and 9zrespectively wehave: _1°-=+--"+ 8.'< I! _y I! 2;" =2_8w }-I0 _-_r, obt_in: adding these .,,ua,i,.ns we It _! tt Ox 6y Oz This equation canbeintegrated. Forexample3.5-4 ,--_ndxXdt--xdx=Ix)'-2 _w _w Ow )w _w dW=r--d_x+--dy+ dzox oy _z (_)2+r%_,_.2+_,_.2.2 +,-+z+c=o J\Ox oy5.3-5 or V2=2W-C 5.3-6 Substituting thevalue ofWfrom equation (5.5-1) into eqtu%tion (5.5-6) wehave: v2.;,?(x2+y2)+2o,_+2GM..a.c rl rfi3.3-7 This istheonly known integral fortheequations ofmotion. Ingeneral, solutions oftheequations ofmotion (equation (3.2-7)) must beobtained bystep bystep integration. 90< 5-II 5.5 Surfaces ofZero Relative V_loclty Jac_01's Yntezral (equation (3.3-7)) will give usagreat deal of information zbout themotion inanearth-moon system. When theconstant ofintegration hasbeen determined bytheinitial conditions, equation (3.3-7) determines the velocity intherotating plane atallpoints inspace. It isevident that V2must bepositive forreal motion intheearth-moon system. Thus theboundaries forpossible motion aregiven for V=0as: (l2_×2+. y2_+aaM_) rl+20M_--=C r2 rl2-(x-Xl)2+y2+z2 5.h-i 2 2y2 z2 r2=(x-x2)++ Curves ofzero relative velocity inthe XYplane areshown in Figure 3-B. (These curves areforillustration only andare nottoscale. Curves ofzero relative velocity inthe yzand yzplanes areshown in Figures 3-_and3-5.) Thesurfaces ofzero relative velocity, forlarge values ofC,mayberoughly described asconsisting ofaclosed fold about spherical inform about each ofthe large bodies, andofcurtains hanging from anasymptotic cylinder symmetrically with respect tothe XYplane. For smaller values ofCthefolds ofthe twobodies expand until they reach each other, andthen open upforming onesurface surrounding both earth andmoon. For still smaller values ofCthefolds andcurtains meet andopen up. Themotion isreal within thefolds oroutside ofthe curtains. 3-12 InFigure 3-3thevalues ofCarenumbered such thatCI>C2>C3.... _'crinitial conditions ofC=CIthebody canmove either inaclosed region about theearthorinaclosed region about themoon --itcannot travel from theearth tothemoon. Earth Satellites andballistic missiles will beinthis class. (Itcanalso beshown that iftheSunandEarth areconsidered the finite bodies andthemoon theinfinitesimal body, theconstant C, determinedbythemotion ofthemoon, issolarge that fold around the earth isclosed with themoon within it. Therefore, themoon cannot recede indefinitely from theearth within theassumptionsofthemethod.) For C=C5,thebody canmove within theclosed contour around the earth andmoon sothat travel tothemoon isnowpossible. C=C2represents thelimiting case separatin6regions inwhich itisnotpossible. For C=C5,thebody canescape theearth-moon system since theregion isopen behind themoon. C=C_isthelimiting case atwhich escape becomes possible. Inaddition tothese contours, within which motion ispossible, there are, forthesameCvalues, outer boundaries beyond which motion is possible. Forexample, abody (infinitesimal) fromouter space cannot approach anycloser than these boundaries (curtains). AtC-C4theinner andouter branches ofthesurfaces coalesce and for C<C4avehicle canenter theearth-moon system. AsCdecreases from C4theopening behind themoon widens.When C=C6thecontour also begins toopen behind theearth andwhe:A C=C7 only theinteriors ofthekidney-shaped regions cannothave real motion. 3"13 AsCdecreases furthertheareaprecluding motiondecreases until fina!lyatC=C6theregionsbecomeapoint.Thesepointsmake equilateral triangles withtheearthandthemoon.WhenC<C8no re@ioninthex-yplaneisexcluded. Thes:_fa_es ofzerorelative velocity mayalsobedescribed asthe _nve]opes ofallpossible orbitsforgiveninitialconditions. InFigure3-6 thezerorelative velocity contours corresponding toC2,C4,CO,andC6 aroshowndrawntoscalefortheearth-moon system. InFigure3-7are shownthecontours corresponding toavalueofCsimilartoCIof F_gure3-3, andinFigure 3-6thecontour about theearth isshown for awalue ofCcorresponding tothat ofanearth satellite. Fortheearth- moon system thecontours arealmost circles inthe x-y plane forvalues ofCnumerically greater than C2) (i.e. CI)since themass ofthe moon isonS_v 1/61.49 that oftheearth. 3-]_ 5.5 Points ofCoalescence From Figure 3-5_md5-6itcanbeseen that there erethrce points on the xsxls called points ofccalescence atwhich double solutions ar_ ob+slned and ther_zions ofpossible motion areenlarged. These points ccrrespond toC_,C5,and C6andare"indicated below: X_MC4"C'_0- Xc; X'cm Equation (5.5-7) intheXY plane iS F(x,y)=_(×2+y2)+2aM(1 -_) rl+2GMW---- C=0 r25.5-1 Th_ conditions fordouble points arc: _=_,?x-2oM(:-#) (x-x:.) bx rl5(x-x2) 2GM_ =O r25 _F_%v-2oM(1-_)-L-2oM_J---o = rl3 r235.0-2 Thedouble polnts onthe problem aregiven bytheconditions: _2x.GM(l-_)(x-Xl) $4<xaxis andthe straight line solutions tothe oM.(x-x2) [(x- =o3.5-3 3-15 This equation isalso thefirst equation of(3.2-7) with I! X-- y= 0 Moulton (ref 3-i) hassolved fortheroots ofthisequation andthey are: r=ci=(I) "_t3j "9 "....... r2cm 13J +_(3; "9 ÷....... r2ce=2-_-_ -23 +123.9-_ Itcanalso beshown that thedouble pointsnoto_thexaxis are rI=l,r2=l,such that thepoints form equilateral triangles with the finite bodies regardless oftheir masses. These arethepoints labeled C8inFigures 3-3and3-6. (Note that thevalues ofrareinlunarunits.) Itcanalso beshown that theparticular solutions onthe xaxis are unstable -ie., ifasmall body were d/splaced avery little from thepoint ofsolution itwould, ingeneral, deport toacomparatively great distance. Theequilateral trianglesolutions,ontheotherhand, arestable_ -abo_7 displaced alittle would oscillate about thepoint ofsolution. Itisofinterest tonote that theequilateral triangle soluti_s are known toexist intheSolar system. These arethewell known Trojan Asteroids between theSunandtheplanet Jupiter. There aretwelve known asteroids intheTrojan Group, seven ofwhich precedeJupiter inits revolution about theSunandfive ofwhich follow. These asteroids have anaverage diameter ofabout 80miles andoscillate near theequilateral triangle point. 3-16 Therealroots interms ofdistances along the x Xcl=I-#-r2cI =i-p+Xcm r2cm M2 1 Since p=- =- :M62._5Xce=i-_-r2ceaxis are: Xci =0.83702 lunar units rlci=0._919 lunar units Xcm =1.15560 lunar units rlcm=1.16773 lunar units Xce =-1.00505 lunar units rlce=-0.99292 lunar units Substitutlngthese values inJacobi's integral (equation 3.3-7) wecannowsolve for C2,C_,and C2=0.16_61 C4=0.16776 C6-0.159_ orC6.These values are: (lun unlts/ ) 2 .6000 ft/sec C_. _966 ft/sec -_32 ftl,ec3.3-5 3-17 Itmight beinteresting tcccmpare thevalue ofXci,thepoint between the earth andthemoon atwhich asmall body would remain relatively atrest, with thedistanceofthepoint ofequal attraction between theearth and themoon based onthestatic equations. Equating thetwofor:res ofattraction wehave: r12 r22 rr_l 2=_1.9\r2 rI+r2=D D1+iMl/v_ r1--=0.9D whereas r!ci=0.549. Thedifference, ofcourse, iscaused bythe inclusion ofthecentrifugal force term _2xinthecase ofthebody rotating with theearth-moon system. 3-18 3.6Velocities attheSurfaceoftheEarthCorresponding toC2,C_,C6,andC6 Thes_gnlflcance ofthesevaluesofC2,Ch,andC6canbeseen morereadilyifwecalculate thevelocity ofavehicle near theearth that corresponds toval_esofCequal toC._,C_,e.-_ C6; Weshall arbitrarily choose aposition 4,300 miles from thecenter oftheearth adjacent tothemoon. Atthis position theposition ofthevehicle is: Ys_ Zs= From equation (}.3-7) then: C=C2; V2,1 = C-c4; V4,m C=C6; V6,exs=o.oo_979 0 0lunar units 2.57_353 lunar unlts/day =5_,703._J ft/sec = 2.57_513 lunar unlts/day =5J_,706.h ft/sec = 2.577292 lunar units/day =_,752.4 ft/sec Thevalue of ThevelocityCbfrom theequilateral triangle solution is: C6=0.I_35 (lunar unlts/day)2 V_atthesurface oftheearth adjacent tothemoon is: V_= 2.577960 lunar units/day =_,756.5 ft/sec L D3-19 Thus theminimum relative velocity needed toreach themoon from the reference position is_,703.8 ft/sec andtheminimum velocity needed toescape theearth-moon system is_,706.4 ft/sec; adifferenceof only 2.6ft/sec indicating thesensitivity oftrajectoriesoninitial conditions. Inaddition, with avelocity greater thau _,736 ft/sec thevehicle could theoretically reach anypoint intheearth-moon system. Since V_,m isless than V6,e itiseasier toescape from the earth-moon system byprojecting toward themoon than itisbyprojecting away from themoon. 3-20 3-7ConversionofRelativeVelocity toVelocity intheEarth Inertial System Itshould beremembered that thevelociti_c _have been talking about arerelative velocities inarotating axes system andaredefined whereas=(£)2+(#)2+(;_.)2 5.7-i Ve2=(_e)2+(_,e)_+(_.e_2 3.7-2 Thevelocities relative totheearth arerelated tothevelocities inthe rotating axes system bythefoll_ing transformation: Ye Icosot-sin_t0 sin_t cos_t0 0 0 0(_-,_) +=(x-xz) 15.7-3 Therefore, =x2._+=_2+;,2._(x-x_)_+_2(x._1)2+;_2 Ve2=V2+m2[y2+(x-Xl)2]+2_[(x-x1)_r-:_'y]3.7-4 Writing this interms ofthepolar coordinatesofearthsystem (rl,r)we have inthe (Xe,Ye) plane: rl2=Xe2+ye2=(x-Xl)2+y2 5.7-5 iCO< 3-21 _nen Attime t=0;o_t=0: Ve2=V2+rl2e2+9mrl cos_-xsinq] IntheFig_;ce the_ngle ncrmal tothernz[ius r] and A]so Thereforebetween thevelocity vector _andthe from theearth is e=90+_'- _ =Vcos_' .7=Vsin_' (9cos_-_sinn)=-Vcos Ve2=_-2_rIcos_V+rl2_2 Them_xlm_m value ofVewill occur, then when cos_=-1andthe minimum valuew_l! occur when cos_=+1or3.7-6 3.7-7 3.7-8 5-20 Now Therefore,(v-'_rl) <ve_<(v+-'_l) =0.22997 rI=0.0179_6 _rI=0.00h!36 2.571197 _Ve2_2.379469 lu/day 34651 _Ve2_5k772 ft/sec3.7-9 2.371577 SVeh S2.3796&9 lu/day 3&65_ _Ve4 S54779 ft/sec Thus theminimum ve!ocity relative totheearth avehicle tothemoon is Ve=34651 ft/sec(Ve) required tosend Theminimum velocity needed toescape theearth-moon system is Ve=51_694 ft/sec Theescape velocity from thereference posi_ionbased onthetwobody equations is vEe= VE=2.4102919_ !u/day e or VE=35,214 ft/sec Thus thetwo-body velocity is560ft/sec more than thethree-body velocity. I0 < 3-23 5.8 Effects ofNeglected Factors Mean Distance from Moon Thedistance tothemoon used inreference 5-3of239,074 miles differs from themean distance 238,_57 miles determined byobservation. Thedistance used wasderived inreference 5-3asfollows: Theperiod isgiven by T=2_aI_ Themean angular velocity isgiven by3.8-1 But GMe-go_2_GM(1-_) •_:Also a=D,themean distance tothemoon D3=G__M _ED_= 3.8-33.8-2 Using appropriate values ofGo,R,N,and mthed_stanceD=2392074. Thedifference of217miles isduepartly totheaction ofthesumonthe moon andpartly dueto meglectimg theeccentricity ofthemoon's orbit. Some factors neglected intheThree Body Problem are: i.Thegravitational field oftheSun. 2.Eccentricity ofthemoon's orbit. 5.Inclination oforbit ofmoon. 103< 3-24 _.Oblateness oftheearth. 5.PressureofSolar radiation. Buchheim (ref. 3-3) hasinvestigated these effects andhisresults arelisted below. Theresults areshown asacorrection AVtothe initial velocity because oftheeffects ofthese assumptions. FACTOR I.Gravitational fieldofSun 2.Eccentricityofmoon's orbit 3.In_!ination oforbit ofmoon 4.Oblateness ofearth 5.Pressure ofSolar radiation&Vft/sec Percent io .o3 49 .13 2O .O6 20 .06 Thus youcanseethat theeffects aresmall butprobably should be included ascorrections toanymoon orbit calculations.V 104< 3-23 3.9 Trajectories Near MinimumVelocities InSection5.6and3.7therelative velocities corresponding tothe points ofcoalescence (C2,C_,C6,and C8)were determined forareference position 4300 miles from thecenter oftheEarth toward theMoon.In reference 3-_caAcu!ztions oftrajectories were made (three-body equations) using theminimum velocity corresponding toC2,butforareference positiom 200kilometers above thesurface oftheEarth, (reference radius 40_ miles) asindicated below: OM Thecorresponding minimum relative velocities areindicated below: rI=_300 mi. rI=40_ mi. /sec v2 34703.8 33372.2 10.8489o v4 34706.4 3337_._ 10.8_968 v6 34732.4 3_oo. 0lO.83738 V8 34736.3 3_603.8 10.898_4 Fortheinitial calculations, theinitial velocity corresponding to C2(V2)wasdirected perpendicular totheinitial geocentric (Earth) ICS< 3-26 radius andinthedirection oftheMoons rotation. From equation (3.7-b) itcanbeseen that this orientation will give themaximum value ofthe initial velocity relative totheEarth (Ve). Theresults ofthese calculations areshown inFigure 3-9from reference 3-4. (Note: This figure andmany ofthefigures tofollow aretaken from reference 3-2 anddistances aregiven inkilometers andvelocities inkilometers per second.) InFigure 3-9areshown thefirst five orbital revolutions plotted inthe xyrotating axes system. Also shown arethezero relative velocity boundaries corresponding toC2.Itcanbeseen that thevehicle does notreach theboundary infive orbital revolutions. Thetime elapsed during these five revolutions isabout 29days. Thesame orbits are shown intranslating Earth axes (Xe, Ye) inFigure 3-I0. Shown arethe first andfifth orbital revolutions. Itcanbeseen that theincrease in theapogee isnoticeable butsmall, andtheorbits appear tobevery close totwo-body ellipses, lbhasbeen estimated that about 200orbital revolutions wc,_id beneessary foravehicle toreach theboundary C2. (This would take abou_ 3years.) Calculations were also made ofthefirst orbital revolution when the initial launch angle waschanged from7=O°toY=160°. Trajectories areshown inFigure 3-11 forthefour cases indicated below. E 1C6< 3-27 From Figure 3-11 itcanbenoted that theinitial apogee i_greatest when since Vel•VelI•Velll •VelV. Inexamining Figures 3-9and3-11 itappears that the C2boundary night beapproachedmore closely ifthevehicle were launched behind the Earth _<90° sothat itnight come closer totheHoon onthefirst orbital revolution: Inreference 3-_calculations were alsomade using values ofthe initial velocity correspor_ing toC4,C6_and C8.Even when theinitial velocities corresponded tothevelocity forwhich allpoints inthe Earth-Hoon system could bereached (C8) thetrajectories didnotreach even theboundary C2onthefirst orbital revolution. Therefore itis apparent that theseminimum velocities arenotadequate forpractical considerations. 1G7< 3-28 3.10 UseofTwo-Body Ellipses toApproximate Lunar Orbits Since thetrajectories ofFigure 3-10 arealmost ellipses centered atthe Earth (geocentric) thetrajectory ofthefirst orbital revolution might be approximated byneglecting theeffect oftheMoon. Iftheinitial radius, initial velocity, andinitial launch elevation angle aregiven (rl, VI,71),andtheinitialvelocity isequal toor greater than tl_velocity necessary toreach theMoon theeccentricity andtheorientation angle 91canbecomputed from theequations of SectionI. Theangle e2isgiven by cose2,,_ -1 3.10-1 108< -29 where72isthedistance from theEarth totheMoon (72=D)and p--r P=rl(VI/VS) 2cos271 (SECTIONI)From thepreceeding sketch theangle )_between theoriginal Earth-Moon axis andtheprincipal axis ofthe orbit is: _'""+_F_t(o2)-t(el)]-e2 where t(B) isthetime toreach theangle Bmeasured from theperigee. (_esEC_o_I) Theminimum condition forlunar impact isfor r2=ra=D.For this case: 82=180° t(e2)=_/2 ),=_IT/2- t(el)] D+rp 2 Vp2=2goR2IrpD+rp3.10-2 3.10-3 (where thesubscripts aand prefer toapogee andperigee respectively.) _uL__== .,.-c_,.J.-_k._ uJ._._.,, miles theiL--Litial velocity _ur_perigee launch wouldbe vp=_2_/sec whereas theminimumvelocity toreach theMooncorresponding to (_=180°,seeequation 3.7-_)C2is Ve2=_772 ft/sec 109< 3-30 Theescape velocfty is VE=3521h ft/sec Thus thevelocity obtained from thetwobody approach is130ft/sec greater than thevelocity corresponding toC2,and312ft/sec less than the escape velocity. Inreference _-4trajectories were calculated using theabove equations andalsousing thethree body equations (equation 3.2-7) forvalues of 7from Oto160°.InFigure 3-12 theresults ofoneofthese calculations areshown for 71=0and h=124.3 mi(200 km). Thetrajectory labeled Iisthetwobody ellipse with theinitial velocity equal toVplequatlon (3.10-_)). Thetrajectory labeled IIIsanimpact trajectorywith Moon computed using thethree body equations (3.2-7). Thethree body trajectories areshown Inboth the(x,y)rotating axes system andthe(Xeo, Yeo) inertial axes system (taken through thecenter oftheEarth att=O). Itcanbeseen that thetrajectories arealmost thesame until they reach thevicinity oftheMoon. Itwasfound inreference 3-4that equation (3.10-3) canbeused fordetermining theminimum velocity necessary forstriking the Moon with aninitial velocity accuracy ofapproximately 0.02 meters persecond. Thus theinitial conditions canbecalculated disregarding theinfluenceof theMoon. Nevertheless, theactual orbits inthevicinity oftheMoon will vary considerably from thesimple elliptical orbit. 3-31 3.11LocusofPointsofEqualAttraction BetweentheEarthandtheMoon Considerable attention hasbeen given tothepoint ofequal gravitational attractionbetween theEarth andtheMoon. Ithasbeen suggested inmany sources that aspace vehicle needanly reach this point ofequal attraction toreach theMoon or"tofall into theMoon." Let rIbethedistance ofaspace vehicle from onecelestialbody and r2thedistance from another celestialbody. E ! Thelocusofpointsofequal attraction isthen given by r2 D8M+_M1/_2 -sin2eMCOS Mz/ -13.ii-i 111< 3-52 where MIand M2 Earth andtheMoon). radiusarethemass ofthetwobodies (for instance, the This locus canbeshown tobeequal toasphere of rgE___MM = DMI/_--I with itscenter located atD+ewhere5.11-2 e= i DMI/_-15.11-5 FortheEarth-Sun rgs----E= 0.00175177 R where inthis case Risthedistance between theEarth andtheSun; orinterms ofthedistance from theEarth totheMoon: FortheEarth-Moon: rgEM 0.1122 D Thesphere ofequal gravitational attraction (sometimes called "gravisphere") between theEarth andtheSunandbetween theEarth and theMoon isshown inFigure 3-13. Itistobenoted that theMoon is always attrac%ed more bytheSunthan bytheEarth. InFigure 3-14 is _hown the_ravltational attraction oftheEarth andoftheMoon along the 1_ne Joining theF.%rth endtheMoon. Ii2< 3-33 Inreference 3-4calculations were made ofamorbit which would reach thepoint ofequal gravitational attraction. Theresultsareshown inFigure 3-15. Thetrajectory labeledIistheellipse neglecting the presence oftheMoon_ andthecurve labeledIIistheresult ofthethree- body calculations. Theperturbation oftheorbit causedbytheMoon is verynoticeable inFigure 3-I_ anditcambeseen that although thepoint ofequal gravitational attraction isexceeded thevehicle wouldnot reach theMoon. Thus thebelief that thevehicleneed _ly reach the point ofequal gravitational attraction toreach theMoon isnottrue. 113< 3.12 Sphere ofInfluence There isanother space about attracting bodies called theSphere of Influence (references 3-1and3-_) which ismore important totrajectory studies than the"gravisphere". TheSphere ofInfluence isdefined as follows fortheSun-Earth system: Thelocation inspace about theEarth where theratio oftheforce with which theSunperturbs thegeocentric (about Earth) motion ofa vehicle (Ds) totheforce oftheEarth's attraction (AE) isequal tothe ratio oftheforce with which theEarth perturbs theheliocentric (about Sun) motion ofthevehicle (DE) totheattraction oftheSun(AS)iscalled theSphere ofInfluence oftheEarth. This definition ismore clearly stated with theequation where DSand DErepresent respectively and AS Earthrespectively. S3•12-i thedisturbing forceoftheSunandEarth and AErepresent theattraction oftheSunand Within theSphereofInfluence: DsDE AS # () E3.12-2 114" ' 3-33 Theratio oftheforce with which theSunperturbs thegeocentric motion ofaspace vehicle totheforce ofattractionoftheEarth isfound asfollows: From thepreceding theaccelerationofthespace vehicleduetothe Earth is Q_Am_=-- 3.m-3 r.2 Theacceleration ofthespace vehicleduetotheSunis ASM=_ 3._-4 TheaccelerationoftheEarthduetotheSunis o_ Theratio ofthedisturbingeffectoftheSuntotheattraction ofthe Earth isthen Inasimilar manner, theratio ofthedisturbingeffect with which the Earth perturbs theheliocentric motion ofavehicle totheattraction of theSunis As aZr.23.12-7 TheSphereofInfluence isdefined asthespace about theEarth inwhich D_s<_ 3.12-z A_AS 115<."-' 3-36 Equating thetwosidesofequation 3.12-2 thevalue fortheboundary oftheSphereofInfluence canbefound tobe 3.12-3 Thedisturbing effect isa_sximum_-hen r.ispositive (nearest theSun) andaminimum when r.isnegative (farthest from theSun). Nowsince MS>>ME,then R>>r.andtheradius oftheSphere of Influence c_nbe given approximately by r.- R \Ms/ r._o._7o9 RI_l2/9 FortheSun-Earth system equation (3.13-4) isanexcellent approximation since MS/ME=333,_3_. TheSphere ofInfluence given byequation (3.12-4) abaft theEarth is r,SE=902,000 miles orabout 2.1times thedistance between theEarth andtheMoon. Thus the Sphere ofInfluence oftheEarth includes theMoon. TheSphere ofInfluence oftheMoon found inthesame manner (equation (3.12-_)} is r.EM_ 35,800mi _0.1498 O This approximation fortheMoon isnotasexact asfortheEarth since ME/MM= 81.45. Equation (3.!2-4) caube used asafirst estimate, however, andifmore accurate values aredesired this value maybesubstituted in equation (3.12-3) andthen amore correct value found byiteration. 3-37 Thel_cation oftheSphere ofInfluence (onalinebetween theEarth and theMoon) found byiteration was32,200 miles infront oftheMoon8n_ 39,600 miles inbackoftheMoon. Theaverage ofthese twovalues is about thesameasgiven byequation (3.12_). InFigure 3-16theSphere ofInfluence fortheEarth andtheMoonis shown. TheSphere ofInfluence oftheMoonisalsoshown inFigure 3-13. Itwillbeofinterest tocompute theratio ofthedisturbance ofthe Sunonthegeocentric motion ofaspace vehicle totheattraction ofthe Earth. Within theSphere ofInfluence themagnitude oftheperturbing action oftheSunwillbeamaximum attheboundary. Thisvalue is: DS--=0.i04 AtadistanceoftheMoon'sorbit this value is: DS=0.011 AE andofcourse atdistances less than thedistance totheMoon from the Earth theeffect oftheSunismuch smaller. Thus theeffect oftheSun onthemotion ofaspace vehicle within theEarth-Moon system issmall (one percent orless)andprobably canbeneglected inpreliminary calculations forEarth-Moon vehicles. Theratio oftheperturbing effect oftheEarth ontheselenocentric (about Moon) motion ofaspace vehicle totheattraction oftheMoon attheboundary oftheSphere OfInfluence oftheMoon is --=0.702 7< .- 3-38 Therefore theperturbing effect oftheEarth isabout 7Cpercent of theattraction oftheMoon attheboundary oftheSphere ofInfluence. 3-39 3.13 Approximate _kthod ofCalculatingTrajectories ofLunar Vehicles Ifi_i_assumed that within theSphere ofInfluenceofaI_ that perturbations ofotherbodiesmaybeneglectedthenorbits canbe computed usingthetwobodyequationsofSECTION I.Thetrajectory canbedivided intotwoparts fortheEarth-Moon system: i.Motion toward ors_ay fromtheSphereofXufluence inwhich theeffectoftheMoon isneglected. 2.Motion withintheSphere ofInfluence inwhichtheeffect of theEarth isneglected. Themotion towardoraway fromtheSphereofInfluence (i)is calculated bymeansofthetwobo_yorbit equationsusingtheinitial conditions rl,VeI,71,andAIofthegeocentric coordinate system. Atthepoint where thevehicle enterstheSphereofInfluenceofthe Moon thecoordinates andtheentry velocityVe2 areconvertedtothe selenocentric (Moon) coordinate system. Theapproachtrajectory maybeanellipse_ aparsbola,ora hyperboladependingontheinitialvelocity.Inreference 3-_itwas shown that thepartofthetrajectory located withintheSphereof Influence oftheMoon isalw_vs ahyperbola inselenocentric coordinates. TheEscape velocityoftheMoononthebom_oftheSphereof Influence is 13 .6ft/sec 3-40 where GM= =_Moon =1.7283x1014 ft3/sec2 Since theentry selenocentrlc velocities arealways greaterthan the EscapeVelocity oftheM_on itisapparent that theMoon cannot "capture" thevehicle andanartificial satellite oftheMoon cannot beestablished without theuse ofretrogre_le rockets. 3.13 .iMethod Of,calculating lunar orbits using approximate method.-Theapproximate method isillustrated inthesketch onthe following page.Inthefollowingdescription thesmall rotation of the Earth about theMoon isneglected. Theeffects ofthis assumption arediscussed later. The initial conditions for theapproach trajectory asindicated in sketch aandbare Vel,rel,7el, andthe orientation angle _between themajor axis oftheapproach orbit andthe initial position ofthe Earth-Moon axis. Thepoint atwhich the approach trajectory intersects the Sphere ofInfluence isdetermined (either analytically orgraphically) andatthis point (point 2inthe sketch) the geocentric parameters r¢2,re2,and7e2areconverted tothe correspor_ling selenocentric parameters rm22 Vm2_and7m2" These valuesmaybefound inthefollowing manner. Referring tosketches (a)and (b), theradial velocity ofthe Moon (vJis isV_= _D 3.13-1 Theangle _2between theEarth-Moon axis Dandthe radius re2 q2=ee2÷X"_t2 "" I .0<3.13-2ql Thevelocity vectors may_found from thefollowi_ vector diagram: Note that thenumerals i,2,3refer totimes ofconsecutive positions oftheMoon. @ | éLR xe %o coc“\" @ Lr "" (b) 1e2<° Li\- .|we e ;<) 5-44 TheangleC2between Ve2 and V_is c2=.12-(m-.2+n2)3.13-3 but 7=_/2 -CL 3.13-4 Therefore C2=7e2"_23.]3.'.5 Theentry velocity inselenocentric coordinates isthen _=22=re22+V2-2%2v=co,c2I Theangle andthegeocentric radiusB,,c_-ween theinitial selenocentric radius Theangle A2between Ve2re2 attime t2isgiven by sinq2sinB= r./ D andV_isgiven by3.13-6 (r=2=r.) 3.]-3-7 V_sinC2 sin_2=vm-_3.1.3-3 Theangleam2 isthen equal to a_2=.-a_2-B*A23.13-9 1_4- 3_5 or 7m2-s-_-Te2 ,_nd, ofcourse, thefinalentry condition is3.15-10 I rm2=r. I 3.15-11 Wlth theae initial conditions (equations 3.13-6, 10,andII)the orbit within theSphere ofInfluence canbecalculatedbythemethods ofSECTIONI. Next wemust concern ourselveswith therelationships ofthe par_ters within theSphere ofInfluence. Referring tosketch (b) and(c)wenote thattheangle E2betweenr.andDisgiven by TheanglesinE2=re-_2sin_2 r. between theMoon xaxis and rm2=r.is _=Ee-_t23.13-12 3.13-13 Theangle SECTIONI:02insketch (c)lsthearien;atlon anglegiven as01in Theangle selenocentrlc orbit is Theangi_ O_02_01ofSECTION I Oobetween theMoon xaxis andthemajor axisofthe 0o=02-¢- botw#en themajor axis andrm3=r.is 05=2.-023.13-Ih 5.15-15 125< 3-46 andsince andrm2=rm3-r.: 7m3-.7m23-13-16 Vm3=Vm2 3.13-17 Theangle _between themajor axisoftheselenocentric orbit -_vm2_-,m3_-- =_-(am3+e3)3.13-18 orsince=m3=.Iz-7m3 ==/z+7m3-e33.13-19 Theangle _between theentryselenocentric velocity theexit selenocentric velocity (Vm3) is*--d 3.13-2o OF TheanEle""2(e37m_) I 3.13-21 between theMoonxaxis andtheexitvelocityVm3 is =eo+_ 3.13-22 Theconditions onleaving theSphereofInfluencearefound as follows (see sketch (b)and(c)): Theangle E3between Dand rm3-r.is E3=_-_t3+am33.13-23 I ,6< 3-_7 Therefore theexit geocentric radius is re32=r.2+D2-2r.DcosE33.13-24 .TheangleC3between 3.13-29 Therefore exit geocentric velocity is %32.%2,._2.2%%cosc33.13-26 Theangle _3between re3 andDisgiven by r_ sin_3"--sinE3_e33.13-27 Theangle A3between re3_vm3isgivenby V sinA3-_sinC33.13-28 Theangle %3 between thegeocentric radius vector Ve3 isthenre3andthevelocity Iae3="'A3 +_'wt3+_3 13.13-29 orsince 7--=" _-48 F_e3"_+c5""_I 3.15-_0! n3 Thus theexit geocentric conditions given byequations (5.15-24), (3.15-26), and (5.15-50) canbe.used w_th themethodsofSECTION Ito compute theorbit after leaving theSphere ofInfluence. 5.13.2 Motionwithin thelunar Sphere ofInfluence inGeocentric coordinates.- Themotionwithin theSphere ofInfluence canbegiven interms ofthe Earth coordinate system asfollows: \ \ \ \ D (d)\ \ \ \ \ 12.8< 3_9 Q__.n._le @Xm between t.he.Moonx radiusrmoftheselenocentric orbit isaxis andtheinstantaneous OXm=Om-003.i3-3i where 0mistheangle e(seeSECTION I)fortheselenocentric orbit. Thegeoc:_ntric radius isthen Ire2=rm2+I)2+2rmDcos(eXm+a_)I _neangle _isgiven by andtheangle @eissinn"rasin(exa+ re3.i3-32 3.i3-33 ee"mt-_l 3.13-M 5.13.3 Effect ofneglectin_ theEarth's revolution about theMoon.- Theapproximate method described intheprevious section wasbased onthe _ssumption that theMoon rotated about thecenter oftheEarth instead ofthecenter ofmass oftheEarth-Moon system.Themajoreffect of this assumption isthat theangle _between theEarth-Moonaxisand theradius from theEarth tothevehicle will beinerrorbyapproximately Ipercent orless. (<0.ideg) Incalculatir_ theapproach trajectoryusing thetwo-body relationships theerrors inthegeocentric parameters @eandreduetotherevolution oftheEarth about thecenter ofmass will beless than 2percent.This 1 9<.... isarelatively large effect butcanbecorrected bycalculating the initial approach trajectory about therevolving Earth using theequations ofSECTION Iinastep bystep procedure. Inaddition thetrigonometric proce dures ofSECTION 3.13.1 should beused w_th cautionsince these weredeveloped fortheparticular case illustrated lusketches (a), (b),and(c). Inother cases certain adjustmentsmayhave tobemadebased onthephysical characteristics of theorbitbeing calculated. Atrajectory calculated using theapproximate method (ref. 3-4) is shown inFigure 3-17. (Itshould benoted inFigure 3-17andother figures taken from reference 3-_that therotating yaxishasbeen shifted tothemidpointbetween theEarth andtheMoon andisdesignated asy'. Also inreference 3-hthe._bon waslocated at_t+_as compared totheresults presented inthis section, andthefigures taken from reference 3-4havenotbeen redrawn. ) _,i_Characteristics ofApproach Trajectories Inreference 3-4astudyofthecharacteristics ofapproach trajectories attheboundaryoftheSphereofY_luence wasmadeand theresults areindicated inFigures 3-18 and3-19 andinsketch (e). InFigure 3-18 thegeocentricentryangle _e2,thegeocentric entry velocityVe2,andtheinitial selenocentric velocityVm2ar cplotted against thedifferencebetween theinitial geocentric velocity andthe escapevelocity atthelaunching altitude (Vel -VeE).Curvesareshovn forentry radiiofD+r.andforinitial launch anglesof_I=+_/2 0 O(7"O,180 ).It,m_ybenoted that forinitial launching velocities greater +,h_n theescape velocity that thelocationofentry into the SphereofInfluence (re=D.+r.)doesnotmateriallyalter theangle _e2 ortheentry velocityVe2. Atlaunching speeds less than escape velocity andnear the:IninnJ_ velocity forreaching theMoon theeffect ofthelocationofentry ismare pronounced. Also itisindicated that theinitialselenocentric velocityV_ isnotchanged greatlybythedirection oflaunch. (c_i=_+_/2) Atlaunch speedsnear orgreater than escape veloci_/ theentry angle _2doesnotchangerapidly andisintherange below IOdegrees. Atlaunch speedsgreater than escape velocity anincrease of O.!_n/sec inthelaunching velocity 0.4km/sec intheentry velocityVe2 vel ityVelresults inanincreaseofabout andtheinitialselenocentric ISI< 3-52 Therange ofpossible orbits within theSphere ofInfluence is indicated inthesketch(e) L Theflight time required toreach theSphere ofInfluence ofthe Moc_ fromanaltitudeof200kilometersabove theEarth isshown in Figure 3-19 asafunctionofthelaunch velocity increment (Vel -VeE). (ref. 3-4)Itcanbeseen that theeffectofthelaunch elevation angle (a)onthetime isvery small.Theflight time totheSphereofInfluence varies fromabout 5d_vsnear theminum_n velocity toabout id_yat 0.5kilometer/second above escape velocity. 1 2< 3-93 ii m3.15Tra4ectories toStriketheMoon 3.15.1 Types ofIm_ct _r_ectories.- Inanystud_ onlunar flight thefirst trajectory onethinks ofisanimpact trajectory. Xmpact trajectories canbedivided into four classes (ref. _-_) asindicated insketch (f) (_Ce>0 _,<0 D•9_.t° -_X Xeo D A-A end,n9 Z)-Descendm (4:) Thetrajectories canbeclassified asstriklng theMoononan ascending arm Aoradescending arm D;andaslaunched inthe_Lirection oftheMoon's rotation (_>0)oropposite tothedirection oftheMoon's rotation (_<0). Atypical impact trajectory ofclass A,(_>O)isshown inFigure 5-20 ininertial xo-Yoaxes andinFigure 3-21 inrotating x-y axes. (Ref.3-9) InFigure 3-22 areshown theconditions atlaunch forimpact with theMoon calculated using thetwobody e_uatlons ofSECTION 3.10. The orientntion angle Abetween thcEarth-,_oon axis andtheinitial launch radius orep!ottcd against thelaunch velocity incrementVel-VeE Atalaunch angle of==9O°(7=0°)thevariation ofthelaunch orientation angle Aissmall and A_.85radian (48.7degrees). For alaunch angle of_=-x/2 which isopposite tothedirection of rotation oftheMoon _varies between iand-.5radian forthespeed rangeshown. Theerrors caused byusing two-body equations forFigure 3-Z2 :_ere found inreference 3-4toamount toadistance error of10-20 kilometers attheMoon near theminimum initial velocity anddecreasing to1kiAo::_i_-r atlaunch speeds above escape velocity. InFigure 3-23 (taken from reference 3-6) thelaunching conditions forstriking theMoon areshown forthree orientation angles @L" The orientation angle @Lisdefined by 154<- 3-55 0L-A+81 3.15-1 InFigure 3-22 theorbit orientation angle @i=O;therefore A"@L. InFigure 3-23 thelaunch velocityYlisplotted against thelaunch elevation angle7el atvalues of8L=45°,112.5 °,snd180°fora launch altitude ofaboul 3_0miles. Itcanbeseen from Figure 3-23 that alaunch velocity greaterthan 34,800 ft/sec isneeded tostrike theMoon foralaunch altitude of350 miles. Theescape velocity atthisaltitude isabout 35,165 ft/sec. From thepropulsionstan_ point itwould bedesirable toselect a velocity aslowasispractical,andfrom aguidance stand point we would like toselect asetofinitial conditions inaregion where the necessary launch elevation angle doesnotvary considera_ ly.aith velocity. Thus, from Figure 3-23 itisevident that theLaunch speed should beabove theminimumspeed andthat, forthis example 2thelaunch elevation angle7doesnotchange appreciably above 35,000 ft/sec. Therefore, inreference 3-6_ aspeed of35,000 ft/sec, alaunch orientation eL -^nangle =t_-, andaia_ch ................ _ ,,.,,._o.... selected fortheexample trajectory. Thetlmesrequlred toreach theMoon fortheconditions ofFigure 3-23 areasfollows: VI,ft/sec t,daysVatMoon 3l_,800 _ _9,000 ft/sec 35,000 2.5 " 35,500 1.5 " tS.S<.-- 3.15.2 Accuracy requirements forstriking _)on.- The accuracy requirements forlunar vehicles will depend onthe typeofguidance with which the vehicle isequipped. Ifthevehicle isequipped with both alaunch guidance system andaterminal guidance system the accuracy requirements will hedifferent than avehicle equipped with _. o1o,,,_ _,4_°_ =ystem. T,_°A,_÷.4,-_ ÷.h_,_,_._y reou_e_ment, will depend onthetype ofterminal guidance such asaninfold scanner oranoptical scanner.Inthis section weshall consider theaccuracy requirements ofavehicle which has launching guidance only andfollows afree body trajectory totheMoon.Inthis case wecanconsider only theerrors intheinitial launching velocity VIand launch elevation angle71. Inreference 3-6theaccuracy requirements tostrike some point ontheMoon were calculated fortheexample selected andtheresults are shown inFigures 3-24 and 3-25. InFigure 3-24 are shown the limiting conditions forimpacting ontheMoon forthe given setofinitial conditions. Ifthelaunchelevationan61e7wereexact then the velocity couldvarybyabout+45ft/second. Ifthe launchvelocitywee6 exact then the launchanglecould vary about+0.3degree.Itmaybe seen inFigure 3-24 that theaccuracy conditions arenot symmetrical, however_ andthat the tolerances are less inonedirection than another. Thetrajectories inthevicinity oftheMoon corresponding tothe limiting conditions ofFigure 3-24 areshown inFigure 3-25. Inreference 3-happroximate calculations were made oftheaccuracy necessary tostrike some point ontheMoon, andtheresults ofthese calculations areshown inFigure 3-26 foralaunching altitude of - 3-57 200kilometers. Themaximum errors inlaunch conditions areshowz, as afunctionof^the initial velocity totheescapevelocity (VI/VE). From this figure itisindicated that theoptimum condltion isnear the escape velocity. Atthis point errorsofabout 180ft/sec invelocity andabout O.3degree inangle canhetolerated. Atspeeds above and below theescapevelocity theallowable error invelocitydecreases. Although theallowable error inlaunch elevation angle increases rapidly below escape ve!ocity_ theallowable error inlaunch velocity decreases rapidly nullifying thebeneficial effect forlaunchangle. InFigure 3-27 trajectories areshown (ref. 3v_) foraninitial velocity close totheminimum necessary toreach themoon. Inthis case theallowable error ininitial velocity isaminimum. Forthe exact initial conditions impact occurs_ butforvelocity errors of 2meters persecond (6._ ft/sec) thevehiclemissestheMoon. Inreference 3-_itwasfound that errors intheinitial radiusof .+50km(31miles) were negligible.Themaxisn_n permissible error in theorientation angle Rwasabout idegree whichmeans rou_xlythat thetime oflaunch asafree bod_ must hecontrolled within several minutes. Theeffectoferrors intheplaneofthelaunch wasinvestigated inreference 3-4anditwasfound thatanimpact on.theMoonwould occur ifthevehicle werelaunched within 50km(31miles) oftheplane and withthevelocity inthezdirection lessthan50meters/second (16_ /sec). 1=7< _-'.8 Z ¥ Theerrors indicated above areforstriking theMoononthe ascending armofthetrajectory. Theaccuracies must be2-5times greater tostrike theMoon onthedescending arm.In_idition itwas found that theeffect oftheSundoesnotappreciably changethe accuracies stated above. 1:i,.8<q 3-59 3.16Circum-Ixmar Trajectories 3.16.1Tra_ectorie, withreturn toearth.-Thenextlunar trajectory ofinte.rest istheorbitwhich circles theMoon andthen returns tothevicinity oftheEarth. Trajectories ofthis typemay beuseful for study oftheback sideoftheM_x)n.There-are four types ofcircum-lunar trajectories which areindicated inthe sketchbelov. oC,>0 OC,<0 S,%I aD D A-Ascendln_ C rcurn/unarD _ D Thetrajectories canbeclassified astothedirectionoflaunch: inthedirection ofrotation ofMoon ,=>O,oropposite tothedirection ofrotation, =<O.Inaddition thetrajectories canbefurtherclassified astothetypeofapproach toandexit from the lineJoining themoon to 3-6o theEarth asindicated inthesketch fortherotatingx-y axis system. Theletter Arefers totheascending armofthetrajectory before apogee isreached andtheletterDrefers tothedescending armafter apogee isreached.Theupper letter inthesketch refers tothetype of trajectory before crossing theline Joining theMoon totheEarth, and thelower letter refers tothetype oftrajectory after crossing this line. Atypical circum-lunar trajectory from reference 3-9isshown in A Figure 3-28. This trajectory isofthetype _>O,A.Inthis type of trajectory thevehicle will either return tohittheEarth oritmaymiss theEarth andestablish anelliptic orbit about theEarth. Thellfe time ofsuch elliptical orbits would, ofcourse,depend ontheinitial conditions. Inreference 3-9itwasdetermined that theaccuracy requirement forthis type oftrajectory istheleast stringent ofanylunar trajectory; the allowable error ininitial velocity isi_0ft/sec andtheallowable error inelevation angle isi0degrees. Itwasfound thowevert that thetime ofthevehicle's return toEarth could vary asmuch as20days. Inaddition thedistance oftheclosest approach totheMoon will vary byabout 80,000 miles. Therefore, ifthepurpose ofthevehicle were tophotograph thefarside oftheMoon andrecover aninstrument package ontheEarth thetolerances wouldbegreatly reduced. Inreference 3-9itwasfound that foranuncertainty ofI000 miles inlocation oftheEarth reentry point thei_Itial velocity would have tobewithin 0.29 ft/secandthe initial elevation angle would have tobewithin 0.03 degree. 3-61 Inreference 3-4trajectories were calculated which would come within80OOmilesofthecenteroftheMoon.Theresults inthis case forcombinationsoferrors inVand7areasfollows: AV,ft/sec AT,deg. -30.6 +33 6 -336Remarks return toEarth return toEarth collide withMoon ordonotcircle moon Theaccuracyrequirementsdiminish rapidly with anincrease inthe distance that thevehicle comes from theMoon. 3.16.2Trajectories with return toEarth with a_raki_ellipse.- Ifnow, wewished tohaveavehicle circle theMoon andthenreturn to Earth inareentry orbit theaccuracy requirements wouldbevery stringent. Inthis case theresults ofreference 3-5indicate anaccuracyof1ft/sec intheinltla] ve_!ocity._an_Ia_es ___-_*-_ ..... obtainanelliptic reentry orbit about theEarth with anuncertainty of 50,000ftinperigee altitude. Anexample ofsuch atrajectory isshown inFigure 3-29. Inreference 3-_itwasfound thaterrors ininitial velocityas small as0.7ft/sec andangleerrors of0.3degreeproducederrors in altitude forreentry of525,000 feet and625,000 feet, respectively. Therefore, itisevident that areentry orbit would bevery difficult toobtain without corrections intheflight path. 3-62 3.17AllunarTrajectories Thenexttype oflunar trajectories istheallunar tra,Jectory; the_e pa_ infront oftheMoon butdonotpass behind theMoon. The four typos of_l].unar trajectories areindicated below (ref. 3-_)- oC,_O D D A-Ascendl-- Allu r_| -. A AI 9co Again thetrajectories areclassified astheclrcum-lunar trajectories except inthis case itispossible toapproach theMoon-Earth line ona descending trajectory andexit onanascending trajectory. This occurs when theattraction oftheMoon causesthevehicle toreverse itsdescending trajectory asindicated inthesketch. 3-63 3.18 Periodic Trajectories Oneoftheinteresting problemsoflunar trajectories isthe possibilityofestablishing aperiodic orbit about theEarth andthe Moon. Considerable workhasbeendoneonthisproblem, someofwhich isreported inreference 3-_. 3.18.1 Periodiccircum-lunar trajectories.- Themost interesting periodicorbit would beonewhich would circle both theMoon andtheEarth. J Thereexist such orbits which appearsomewhat llke thesketchabove. Inreference 3-_several periodic circumlunarorbits were calculated andsome characteristics ofthese arelisted inthefollowing table: _-64 re,ml rm,ml VI,ft/sec i. 4,o8_ 93 36,5o_ 2. 26,229 915 14,441 5. 91,475 952 10,446 4. 72,525 1,245 8,999 Itmaybeseen that theonly orbitwhich wouldnotstrike theMoon isnumber 4,andforthis orbit theminimum radius from theEarth isover 72,000 miles oralmost 20Earth radii.Such anorbitwould probably be oflittle use. Inaddition this type oforbit isunstable andperturbations would cause ittodiverge. Therefore, there seems tobelittle possibility ofestablishing acircumlunar periodic orbit about theEarth andMoon. 3.18.2 Periodic allunar trajectories.- Although there appears tobe onlyonefamilyofcirctunlunar trajectories there areanunlimitednumber ofallunar trajectories possible. There hasbeen considerable mathematical treatment ofsuch trajectories which arereferred toinreference 3-4. Several allunar trajectories computed inreference 5-4areshown in Figure 3-30. Theperiods ofthetrajectories shown vary from about 0.9 to1.9months. Althoughsuch orbits areofinterest itisdoubtful that such orbits couldbeestablished foravery long periodduetoperturbations oftheorbit. 144< o3-63 3.19 Establishing anArtificialSatellite oftheMoon Inaprevioussection itwasindicated that theMoon couldnot "capture" avehicle because theentry selenocentric velocity wasgreater than theescape velocity. Therefore inorder toestablish asatellite oftheMoon itisnecessary todecrease theselenocentric velocity below theescape velocity with aretrograde rocket.Inorder forthesatellite tostay _norbitabout theMoon indeflnitely itwouldbenecessary for thevelocity tobereduced considerably sothat thevehicle wouldnot leave theSphere ofInfluence. Themaximum velocity forthevehicle toremain inthevicinity ofthe Moon canbeobtained from Jacobi'sIntegral for C=C2(see section 3._ andFigure 3-3).Inreference 3-8astudy oflunar satellite orbits was made andthemaximum velocities were calculated.Themaximum allowable velocity inselenocentric coordinates isshown plotted against altitude above theMoon's surface inFigure 3-31.Besides thedanger ofrecapture bytheEarth there isalso thepossibility ofimpacting ontheMoon. corresponding tolunar impact using twobody equations (seeSECTION I) andtheseminimum velocities arealso plotted inFigue 3-31.Thus for establishing alunar satellite theselenocentric velocitiesmustbe kept approximately between thelimits shown inFigure 3-31.Inreference 3-8itwasfound that velocities slightly inexcess ofthemaximum could beused forretrograde orbits butthat velocities slightly below this maximum wouldhave tobeused fordirect (inthedirection oftheMoon's rotation)orbits. 145< 3-66 At_pical approach trajectory from theEarth isshown inFigure 5-32, andinFigure 3-33 circular lunar sate21ite orbits areshown asthey would appear InXo,Yoinertial axes. (Reference 3-8). InFigure 3-32 asatellite orbit about theMoon isshown inrotating x,ycoordinateg. InFigure 3-39 anorbit isshown inwhich theinitial selenocentric velocity wasabove themaximum allowable velocity andthevehicle is recaptured bytheEarth. Anear circular satellite orbit atadistance of20,000 miles from theMoon isshown inFigure 3-36 (ref. 3_8). InFigure 3-_ itisevident that thetwobody approach (SECTION 3.13) could notbeused formany orbits ofthelunar satellite since the perturbations become more noticeable after several orbital revolutions. Thedisturbing force oftheEarth isabout 70percent oftheattraction oftheMoon near theSphere ofInfluence. Therefore, formore than one ortworevolutions about theMoon thethree body equations must beussd. Inreference 3-8atypical satellite orbit about theMoon wascomputed andtheallowable errors inlaunching conditions were computed. This satellite orbit isshown inFigure 3-37. Thevehicle waslaunched as indicated below andasatellite orbit wasestablished at1,O00 miles from thesurface oftheMoon. E __ _ ./,_.2,.' V,: 146< 3-67 Theentryselenocentric velocityatthepointwhere theretrograderocket wasassumed tobefired was7707 ft/sec. Fora1,000 mile dlstamce the velocity mustbebelow5,338 ft/sec. Theminimum allowable velocity was 3,280 ft/sec. _ereforethe value _309 ft/sec wasselected forthe satellite orbit. This requires avelocity increment ofabout 3,400 ft/sec from theretrograde rocket. (For other satellite orbits velocity reductioms of2000 to6000 ft/sec arerequire_.) Theallowableerrors ininitial velocityVel an_launchelevation angle7eItoestablish this orbit were -33<ZXVel<77ft/sec -.3<A7el<.135 degree Inaddition itwasfound that anerror intheretrogradevelocity increment ofafewpercent would notsignificantly affect theallowable errors in velocity anddirection _iven above:. Fortheparticularsatellite orbit computed inreference 3-8, the __•_J. _ w_.s_.., a-awaa v_AJ.'_o t.*,,v¢:kul../t,,L_, "9LI%)tAJ.'i::J lIwllr"_ --r'm_reOl errors indicared above. Theabove values arerepresentative forretrograde satellite orbits; fordirect orbits theallowable errors invelocity areabout omehalf of those i_licated above, whereas theallowable errors inlaunch elevation angle areapproximately thesame asfortheretrograde satellite orbit. Ifwedesired that theinitial satellite orbit beestablished within I00miles ofthedesired altitude ofI000 miles, then accor_img to reference5-9itwasfound that theinitial velocity from theEarth must 3-68 beestablished within 4ft/sec andtheinitial launch angle must be within 0.09. degree. Intheabove discussion wehave tacitly assumed that th_retrograde rccket canbefired attheright time andintheright direction. The direction offlrlng m_ght becontrolled byspin stabilizing therocket inthecorrect attitude immediately after thepowered portion ofthe approach trajectory. Thetiming oftheretrograde rocket firing, however, might prove tobethemost difficult problem. Xnreference 3-8itis indicated that theuseofaclock totime thefiring oftheretrograde rocket might beimpractical because ofthefairly wide range oftimes ofarrival near theMoon. Other sources, however, have indicated that such atiming device might besatisfactory. 1, 8< 3-69 3.20UseoftheMoon forAcceleratingSpaceVehicles There hasbeen considerable interest inusing theMoonasameans foracceleratingaspace vehicleforinterplanetary travel.Since the Moon revolves about theEarthnear theplaneofother planetary orbits, atsome timeduringeachmonth the_vouldbeinapositionto accelerate avehicle tovard a_planet.TheuseoftheMoon foraccelerating space vehicles waFconsidered inreference 3-_. There arefour typesofmaximumacceleration trajectories as indicatedbelow GICo>0 I A _M|_"_*" DC£,_0 ,g'_eo ____x__eo -_Xe. 0 A-Ascen&n_ D-De_.cJ,.g Luf_cer ¢tccelero#'/_ froj¢c_'orles These four types areanalogous tothose forstriking theMoon (see SECTION 3.15.1).Inorder toobtain thegreatest accelerationthe I .9< 3-70 vehicle should pass very close tothesurface oftheMoon andpass outoftheSphere ofInfluence oftheMoon inadirection asclose as possible tothedirection oftheMoon's velocity about theEarth. Thus, thetrajectory formaximum acceleration (_V=Ve3-Ve2) passes around thmMoon inacounter-clockwise direction forapproach onanascending armandinaclockwise direction fordescending arms asindicated inthesketch. Theexit velocity Ve3 isalways greater than escape velocity andalmost independent ofinitial velocity Vel; however, theacceleration increment (&V=Ve3-Ve2) depends onthe Initial velocity V1andisgreatest near theminimum velocityV1 necessary toreach theMoon anddecreases astheinitial velocity V1 isincreased. Themaximum velocity increment near theminimum velocity is V=Ve3-Ve2-4920 ft/sec Themaximum velocity increment g_ven above isfortheturn around theMoon tobeattheradius oftheMoon. Because ofthepossibility of collidJng with theMoon thetrajectory must beraised from thesurface oftheMoon, thus reducing thegain invelocity. Inreference 3-4it isshown, foroneexample, that anerror of328ft/sec would cause an error inradius attheMoon of7_miles. Anerror inlaunch elevation angle of1degree would cause anerror inradiusof62miles. Inaddition toualng theMoon toaccelerate aspace vehicle it could beused todecelerate aspace vehicle. Themaximum deceleration would beobtained bypassing outoftheSphere ofYnfluence inadirection opposite totheMoon's rotation. ISO- 3-71 Thepracticality ofusing theMoontoaccelerate aspace vehicle depends onwhether itcosts mareinweight fortheadditional guidance accuracy sothatthevehicle could comeclose enough totheMoonto benefit fromitorwhether theextra weight could beputmoreefficiently intoalarger power plant. tS1< t 3-72 3.21 Propulsion Requirements forLunarVehicles AdiscussionofLunar trajectories wouldnotbecomplete without anindication ofthepropulsion requirements forsuch orbits. InFigure 3-38 thevelocity increments obtainable from several types ofpropulsion systems areshown asafunction oftheratio ofinitial weight topay- load weight.Itwill benoted that therange ofchemical propulsion shown indicates aconsiderablymore efficient system than that used with Vanguard type satellites. Thenumber ofrocket steps used also increases aseach curve increases,starting with 2steps atthelowestvelocity incrementsandincreasing to5stepsatthehighest velocity increments shown inFigure 3-38. Theapproximate velocity incrementsnecessary to perform various lunar orbits areindicated below: Trajectory i.MoonImpact 2.Circumlunar 3.Circumlunar with return to satellite orbit atEarth 4.Lunar satellite 5.Land/n6onMoonTotal AV,ft/sec 39,000 39,000 _7,00o 38,o00 41,000 Theadditionalvelocities fortrajectories 3,4,and_reflect theuse ofrockets inentering thesatellite orbits orlandln6 onMoon. IS < 3-73 REFERENCES 3-1 Moulton,F.R.:AnXntroductic_ toCelestial Mechanics,The MacMillanCom_,NewYork_ 191M. 3-2 Hill,G.W.:Researches intheLunarTheory, AmericanJournal ofMathematics,VolI,1878. 3-3 _chheim, R.W.:Motion ofaSmallBo_ inEarth-Moon Space, TheRAND Corporation,Research MemorandumRM-I_, JuBe 1956. 3-hYegorov,V.A.:Certain Problems ontheDynamicsofFlight to theMoon; SymposiumofSoviet Research onArtificial Earth Satellites andRelatedSubjects,Washington, D.C.,Oct. 1957; U.S.Joint PublicationsResearchService Rep.No.187; from Uspekhi FizicheskikhNauk (Progress inPhysicalSciences) Vol63,No.1-2, Sept. 1957. 3-5Lieske,W.A.:Accuracy Requirements forTrajectories inthe Earth-Moon System;TheRAND Corporation; Rep. P-I022,Feb. 1957. 3-6 Clement, GeorgeH.:TheM_on Rocket;TheRARD Corporation, RAND Rep.p-833, May1996. 3-7 Lieske,H.A.:Lunar Instrument Carrier-_raJectory Studies,The RAND Corporation, RAND RM- 1728, June 1956. 3-8 _chheim, R.W.:ArtificialSatellitesoftheMoon;TheRAND Corporation; RAND Rep. P-873, June 1956. Also: Proceedings oftheVIIInternational Astronautical Congress. Rome, Sept. 1956. 3-9 Klemperer, W.B.,andBenedikt, E.T.:Selenoid Satellites. Astronautica ActaVolIV,FASC i,1958. Some Flight Control ProblemsofaCircumnavigating WADCTechnical Note.58-82, MArch 1958. i53<3-10 Xenakis, George: Lunar Vehicle. 5-74 TABLE 5-i AXES TRANSFORMATIONS USE_;L INTHREE BODY PROBLEM (BasedonD=i) con_t-sin_t O" [A]=Isin_t cos_t 0 LJI I0 0 1 t YO=A z Yo Yo = Y zo o[A]+ tZo] Ye =A z 3-75 IX_ I III[A_Yo_-- I! .ZoJ"t! y2__ I! z .--[AJ _ej "o ?'°_}xe+mYe "_-76 ( Xeo=,x0-x1 Yeo=Yo Zeo =Z0 o--Xo =Yo Zeo=Zo Xm=X0-x2COSo_t Ym=Yo-x2sin _t zm=z0 x==xe-cos_t Ym=Ye"sin _t Zm=ze xe=Xo-Xlcos _t Ye=Yo"Xlsin _t Ze=Z0 156< 3-77 _=_.+_sin_t YmffiYe"mcosmt _= _e Xm=xo+_x2sin_t Ym=Yo"_x2cos_t /_',2 L)AJ <) f f X/-_B.900 ,,,_/' xz--z,36,/7o,,_/ LD-__2,3,9,070/.77/ /4=/v/e+Mm 14_//_=el.,¢5 R=8.96o /'72/ =O.22997r_o//o/_ /Tg.3-I.EurLh- Moon 158< X ,_jrjq /_g.3-2Cooro//nct_¢ _ffG_en?G 173._._Co#Laur_ of"zeroreluLzve yelociLy in(x,y)pl_xne. C Fg.3-4-ConLour_ (x,z)planeof"Z_f'O ve/oc,,t_ /r/ iz_9.3-5COHLOUFS (j,z}planeoPzcro rclalive _'eloc_L_i in 16,¢< Y Cz X fJ_.3-6 /n_,y)plunedr_n _oConLour_ oPzcrorc/o_/_e ve/oo/._y sooh.(ref_-_) x6j<-,_-,,_ e fig.3-7 Contours ofzero relate velocity inthe xyplane tortheearth. moon SYSLEM, Cz0.20 164<e t,=4.300tnlj A_3-aContour ofzororela_/vo >'e/oclty /z?lhe x,9'pla#e_or_hee_rlh-moo#sf/#_em. c--,.q,c,_.-_4.,_5 i65<'" X,lOaKm f_yJ-/o7;ajecLor_/ .J X Fgj-N_ujec_ory o<r<18o°{rJ.s-¢)7_l-/??l/?lmUm/n/t/dre%clf_ 167< "t"._ @ ‘ y Yen af F -). 0.Wgnn) a x7 a Fig3-12 Comparison ofimpact trajectory cakulated using two-body equations with that calculated using three-body equations (ret3-a) 188< @ "4..i,. k_3 ° _--_ | _o_ _ 169< .OZ 0eorthmoon •4 .6 ._ 1.0 1.2 i.4 170< reocb/t_3po;_lo/"agso/ EarLhond/_oon.(reP.a-g) _k.¢_.< S#hereofInF/uenceofEor_h Flq_re 3-/6.-ConTpar/son of-sphere o7c Influence ofEori-h w//-/7 respec_-I'oY-he _ul+h _he sphere _F/_iCtuenc_z o_ch?oon w_h _espec/-Iv_h_EorY'k).-_un 7.O Xeo \ H=200hrn ,_V,=O.OSzS56h,,_,,'sec FiGure 5-17.-Illustration ofapproximate method ofcatculatl_g Lunar frajectorles.(ref. 3-4) !'2:3< 2.0 1,0I., 0 o -o.t 0 _!1 02. 0.3 04 o.._ ^ re_=O+r, -- yea-- D-r, F'lGure _3-I8.- Cond,taonsafentr_, intofh_Lunar Sphere oF Z_flue_¢e. (ref.3-4) e iae ee PEC a 2GAPE TLEEEEEEEE EEetapa EEE MN tblzbalbe eae ctAIEEEECEEECEEE EEECLGSECECECEPEREPEELo EASE EEE EE *ettSRE ECEATsbere Lf alePPP ett tsleauge beceeesdcucagaseas=znaecPgscacaesceeGaceesseraa@esese eeET SES iePEEfinefatavaoueeeeaeceeeas: stata doll legeSuesueestevecicace audeeeaerages e 475< / I I I I I 0c_ _J c5 "_ o._x c_ c_ 17G< I" i°IJ,% 0 I tb __) _J e_ ! cr_ _J 4 3 2 0 -I_m'-0 I J I I ! I 0 O.I O,z _3 _4 O_ re,-Ve_.,_'/se: _,'l'h FiGure3-2P..-Vorloflo. of"_heorler_fafoo_ angle_,(re_,]'4._launch veloc_f_l {orimpocf w_'/'hthemoon. 178< I c_o 11 IIIi cO" Ii I_II Iq_ _k o_ ! 1.79< . 35_IO0 35,020 3_,98o 34,94Lo_L=17_.IP )_=14_.2" 3_,9oo I I I I /_ /+ 13 F_ure5-Z4.-Ll"mi_ing conditions Forimt_ct/n 9on theforward surface o_c_emoon. ref. _-G. S _80<" ToV,,_/sec 3-25.-Tra/ec (;ories correspondin Eirt_hevicini_ o{_he•._ moon _othehmlUn9condi_ions ref3-6 Si<. | i i I0 Q J tl u) 0 k) ta _2 I ,d 2 -300 1.14-I00 3o /_lgure 5-27.- Trnjectorles witherrors sn,_lhal veleclty close tofhernln/m_/m. '4-- /© C c5I I\ 184< / c_I I I I o 6 c_ o o ! i85< yn_ (b) _zm_ F_lure,5-30FCrl Oc/Ica//_nar tr_jec_ar,es. IS6< l13'11 mrll2 I_U.Oo.._ Xeo _Z../ 3-_0.- Periodic o/lunar _rajec_ortes concluded. 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Thefirst concern isgiven totheapogee kick transfer into there-entry orbit andthen theproblem ofakick atany arbitrary point inanorbit isconsidered. Even though theequations are directe_ toward there-entry problem they maybeadapted toavariety of other nonentry problems. 4.1 Apogee Kick Transfer Ifitisassumed that wehave avehicle insnorbit entirely outside themain atmoschere, oneofthesimplest andmore effective ways tocause thevehicle toenter theatmosphere isbyfiring aretarding rocket at theapogee. Theequations which pertain toanapogee kick re-entry are _erived asfollows: Thefollowing three equations were obtained from equations (1.2-17), (1.3-8), (1.4-2), and(1.I_-3) ofSECTION I. (h.l-1) (4.1-2) VpS-(4.1-3) [:-2 Thus from equation (h.l-2) v-=-_o_-_)_- andcombinin_ equation (h.l-l) and (h.l-h) /vo-Cl/v.. _.e which m_yber_c,:cedtc m O Other combinations ofequations (h.l-l), (h.l-2), and (h.l-3) yield: _oeV(h.l-h) (!_.1-5) (h._-6) (4.1-7) =/l+e(_.1-8) •_ and andafterIftheoriginal conditions atthe apogee areVao rao the retarding rockets are fired achange invelocity (AV) iseffectively instantaneously added, the newconditionsatthe apogee are now Val and - + The eccentricity, radius ofral where Val (Vao AV) and ral -rao. theperigee andvelocity attheperigee ategiven by 4-3 CI _,) v,-vo,(j+-(4.l-9) (4.1-io) (4.1-11) Also thenew semi majoraxis isestablished as _, 0,=Z. "Z /+6,(4.1-12) m (4.1-13) Theequation ofthenew ellipse is r-=_'("-'_') I+Eco_0 where _.isdefined by_q_:_tion (4,.1-9) J. Nowthat the neworbit hasbeen established, itisdesired toknow where thevehicle will enter theatmosphere andwhat will bethe re-entry angle, that is, theangle between theflight path and thehorizon atthe point ofre-entry. Itisassumed that AVissufficiently large tocause there-entry. Theupper limit oftheatmosphere isconsidered atre which isusually taken tobesomewhere about 50-70miles. The re-entry will occur attheangle0where theradius rofthe ellipse isequal to re.From equation (4.1-14) i<S<--(4.1-14) h-h r,OE,) (4.1-15) Solving forcos8e /"t' _coj _e:r(h.i-16) (h.1-17) andthere-entry positioneeisgiven by CO u.(h.l-18) There-entry angle isdefined astheangle between aline perpendicular torateeandthetangent totheorbit atthis point. Thegeneral equation forthis angle atallvalues of8wasgiven inSECTION Ias equation (1.5-8). tan7=/+Ecose (h.l-19) Thus there-entry angle7eat8eisgiven by /+6,co-_8e(4.1-2o) Forcounterclockwise vehicle motion, tan7"+indicates anexit, tan7"- indicates anentry. Thevelocity atre-entry isgiven from equation (1.6-2) ofSECTION Ias 159< v,-_+ o,)(4.1-21) Thevarious equations areadaptable tothefollowing problems. Problem Number I:Itisdesired tomake thevehicle re-enter atacertain _eand re. Thevaluesof 8eand rearesubsituted inequation (4.1-17) (a) Solvin_ for el: (b) r",,n__/e,co..,e,...=V_F_e, =_/c.-/_ __-o_e_-_ Thus theeccentricity required toeffect there-entry atee,re(c) (d) isdefined. Thevelocity attheapogee, Val,required togive this eccentricity isderived inthefollowing equations: From equation (4.1-9) (e) 4-6 Solving for Vayields: V"=(/-_)_"R=,, Q r_rd-_ ,-Vr-,tr_,,-.,& +r./(f) (g) This equation defines thevelocity requiredatapogee togive re-entry ateeand re. Thus Av=v.-v0 0 Co_ ..f.r(h) (i) Thus the AVrequired atapogee ratogive re-entry atee,reis defined. There-entry angle at7eisgiven by •-,f,¢=.t..an-IE,5,_e,., /+E, cos_£e.(J) 2,01< h-7 Thevelocity atre-entry isgiven byequation (h.l-21). Asol1:tionofa tzr_ical problem ofthis type isgiven infigure h-l. Problem Number II: Itisdesired tomake thevehicle re-enter atacertain •entry angle 7e and re. Substitution ofreinto the general equation ofthe orbit: r_(/__.)re= /+E-, co,s_(a) Solvin_ for el: r-_+_,_cos@_=r,.-Ce, (b) (c) Thus theeccentricity required toeffect the re-entry atany isdefined in(d). Theinterest is,however, inaparticular will give there-entry angle 7e" From equation (h.l-20) Z+_cos@ =,e._c,,_"r_II ee,re eewhich(d) (e) _,co_0-_,s,,_o_ / ,sin@e(f) (g) E_,,-_.. . h-8 This defines therequired eccentricity interms of@eand 7e" Equating thetwoequations fortherequired eccentricity gives i (h) (i) b11t co._6]--/l-.s,_" scthat (J) Squaring both sides, letting A 1(k) (z) or zA_5,,,_=-CA'-+_"9s,,.,"_.(m) 203< h-9 From which (n) (o) where Aisdefined byequation (k). This defines the eewhere the re-entry istooccur. Thus, the eccentricity ofthere-entry ellipse isgiven by El=recosE) +Fa Thevelocity required atapogee Val toobtain therequired 7e and reatre-entry isgiven by when equation (p)issubstituted for _i,equaticn (q)becomes and thus theincrement cfvelocity required atapogee togive re-entryat theprescribed 7mand rmisgiven by .(p) (q) (r) (s) h-lO where eeisdefined byequation (o). Thevelocity atre-entry isgiven by equation (4.1-21). Asolution toaproblem ofthis type isgiven infigure h-2. _.2UseofAKick AtAPoint InTheOrbit Other Than Apogee Theuseofthekick atapoint other than apogee isdirected toward the problem ofre-entry ataspecified angularposition 8e, altitude re, and re-entry angle7e. These arbitrary re-entry conditions cannot beobtained byakick atapogee. Thebasic equations given inSection h.lmaybe adapted tothis type ofproblem. Problem Number III: Itisrequired tore-enter atacertain @e, re,and Onesolution tothis problem istousethesolution given inSection h.1 todetermine are-entry orbit which will give theproper entry angle7e atreandthen torotate themajor axis oftheorbit togive theproper entry position ee.Thusare-entry orbit isobtained which satisfies the re-entry requirements. The£ransfer into this re-entry orbit ismade where theorbitofthevehicle intersects there-entry orbit. Theequation oftheorbit which gives theproper entry angle7eat reisgiven by r=,r,(/-e,) I+E,cose andtheequation oftheorbit ofthevehicle isgiven by r=Co(I-e.) i I+6co, 0 @(a) (b) 20S< h-ll Theellipseofequation (a)which gives theproper entry angle7e reisrotated counterclockwise through anangle A8togive theproper entry position 8esothat theequation ofthere-entry orbit becomes r-- I-/-_,co5(_-ae)at (c) Thepointsofintersection ofthevehicle's orbit (equation (b)) andthe re-entry orbit (equation (c)) where thetransfer maybemade arefound by equatin_ theequations forthetwoorbit equations. Thus, _,(/-&,) I+6,cos(@-zX@)=r..O-6.) (d) /+E'oco_8 Solving fortheO's oftheintersections. Inverting equation (d) f r',,°(/- Expandin_ equation (e) ]+E,<o.+(++_+e)= ;++°,:o.,,+ _,(/-,_=,) r+,0--+,) &(]-+:o) _o_/-¢o,) Let / --/ =A r+,(1-+,)_o¢"/-+.) Substitutin_ atrignometric ident{ty forcos(@-48)andcollecting in equation (f)(e) (f) 2C6< 4-12 _a 0-_)% Coilectinc inequation (i) ,4+_7T-__ _o()cosO=-,(,__sin0(_ _et:Andalso let Substituting equation (j)into equation (i)yields A+Bco5(_=C_,,_(J) (k) Andsquaring A_+2Aa_o_0+Bh-,,_'O=C_(/- _o,'e)(z) Collectin_ inequation (Y)yields: (A_-c_)÷2_s_o__+Cs'÷CO_o, "_=o L_:_=(,4_c9 ,¢'_=2AB where ABand Caredefined inequations (g)and(J). then:(m) (n) 207< 4-13 Thesolution forthe @ioftheintersections isthus: 8_=<o_-I(-_-*<_"-__,_ where El)K2_and K3aredefined inequation (m). Theredli attheintersections are(o) _o(/- ,.)_.-- _(p) Thevelocities ofihetwopaths attheintersection points aregiven by 2.(q) (r) Thea::vle between thevelocity vectors isgiven by nY--*T *- 0 +E,,_o,ed-to/ _)(s) -r....e,,x-J_.s uVwconsideration ofaproblem sketch. Thus thevelocity vector diagram isestablished byVI, Vo, and &_. The AVrequired totransfer from oneellipse toanother maybefound grom thecosine law AV"=(V,)"+.(V,,) _-Zv,V,,_o,(nr) (t) ,:.CS< h-lh andtheankle Dbetween AVand Vois f -iI )) 2-(av(V, Thus thevelocity increment required forthetransfer andtheangle ofapplicstion isdefined. Asolution toaproblem ofthis type is_iven infiFure h-3. ProblemNumber IV: acertain Be, re,Asintheabove problem itisrequired tore-enter at and7e. Thesolution tofollow ismore general than theone_iven above. Consider thefollowing sketch:(u) Point (I)isanarbitrarily specified point ontheoriginal orbit ofa vehicle. Itisdesired todetermine themaenitude anddirection ofthe velocity impulse required atpoint (i)which would cause thevehicle togo into are-entry orbit andenter ataspecified point (2)with re-entry angle _e" Thus re, rl, As,and Teareknown. Thefirst step isto determine thevelocity anddirection ofvelocity required inthere-entry orbit atpoint (I)to_ive thedesired entry atpoint (2). Then bysimple vector subtraction, thevelocity impulse required at(i)inarder tocause 2C9< 4-15 thevehicle toleave itsnormal orbit andgointo there-entry orbit will bedetermined. Theeccentricity andradius atapogeeofthere-entry orbit aregiven by (.) (I-_)(b) r,Ei+E<<>.,(_+,__J_7 _1-EJ Equating equation (b)toequation (c)andthen proceeding tosolve forthe unknown ee(c) (d) thus _+_ (e) Collecting (f) ,Eubstituting efrom equation (a)inequation (f) _..l_O<(g) 4-16 Tri_nometric identity: thus substitutin_ equation (h)into equation (g) tT r'_(i) Collecting inequation (i) andthus let Thusequation (k)becomes ACO.SO_4- _,sln_e =0 (,.) Zll< 4-17 From which A_B_,.,g_=o(n) or (o) where Aand Baredefined inequation ($). Equation (o)determines theangle 8ebetween theradius reand themajor axis ofthere-entry orbit. Thus theangle @eisdefined in terms oftheknowns. Using theknown ge, theeccentricity isfound from Using theknown ge, re, and zthesemimaJor axis isgiven by _,= (,/_(...) specified re-entryconditions isgiven b_ VI=-+og a, Using theknown A@, ge, ands,theangle 7atrIisgiven by .+_--I_s/,,(@_+_g),Y t.C,l¥l 1V-__o.sC_e+_@)(p) (q) (r) (s) ¢-"'_¢'%,e" 4-18 Theequation ofthere-entry orbit isgiven by e')-l-t-E @ Thevelocity anddirection ofthevelocity inthere-entry orbit at point (1)areestablished byequations (r)and(s). Thevelocity and direction ofvelocity oftheoriginal orbit atpoint (1)aregiven by equation (q)ofProblem Number IIIandequation (4.1-19). Theprocedure forobtaining therequired velocity impulse fortransfer anditsdirection maybeobtained byuseofequations (s)and(t)oftheforegoing Problem Number III. Thevelocity atre-entry maybeobtained bymeans ofequation (4.1-21). Asolution to•problem ofthis type isgiven infigure 4-4. Consideration uptonowhasbeen onthetransfer from anexisting orbit into are-entry orbit. Theadaptations oftheequations toother types oftransfers innon-entry considerations andthetreatment ofminimum energy transfers areleft forlater considerations.(t) 213< i -71220° 140_Specified: ee-._5o re-21.294240x106' ra=36.764638xi06' Vao•17133 '/S he=70ailes 250° IlO° 260° I00_ 60" 300• 310_ 3?.0" R h Z _J w i .__J240° 120°Specified: 7e=_2o re=21.294240xi06' r,=36.764638_i06' _ao •17133 '/S he•7OaLilesSolution: AvReq.=-430,/S Be=-9.49 ° Ve•28,980'/_ 130" 230" 120° 240° II0° 250" I00° 20& 90o 270_ 70° 290o 300° Figure 4-2,- Solution TypicalProblem Number2. I/ -t _z rIIr[lll_ 3300 J40" J#O° 0 I0" 20° 30"30" 20" I0" 350_ 340" 330" Earth surface Fi_ _between __ ure 4-3.-Solution Typical Problem Number 3.140" 220" 120" 240° I10" 250" I00" 260° 9O" 270" 280° IF"._A,._'G< L i220= 140° DO'I 1300 240° 120° 250° II0° 260° I00o 270° 90° 280' 80+ 290 70 3OO 6C 31C 5C 32( 4(Solu_on* AV-600,/S •r1•27.962 ° Vo•20226 '/S V1-19697 ,/S Ve=28,868'/S 120' 240" I10° 250° I00¢ 260° 90° 270° 80° 280° 70+ 2w 60° 300° _0• 310" Figure h-h.- SolutionTypical ProblemNumber ];. = "r330° J40e 350e 0 I0" 20" 30030" 20" I0' 350" 340" 330" 2 7< 5-1 SECTION V RE-ENTRY WITH TWO DEGREES OFFREEDOM @Introduction. Inthe previous SECTION the procedures for deter- mining the position, angle and velocity atre-entry were established. Once these quantities have been established, the complexity ofthe solution for the paths, velocities, and decelerations inthe atmosphere depends onthe complete- ness ofthe differential ecuation used. The equations dis- cussed here will beofthe simplest nature. The vehicle is toexperience lift, drag, gravitational and linear inertia forces only. Even the solution ofthe simple equations depen_on machine integration ofthe ecuations. However, there are some further restrictions concerning lift, drag, and air density that can beplaced onthese simpler equations and some strictly analytical results may beobtained. There are various paths that may betaken through the atmosphere. Ithas been proposed from aheatin_ standpoint touse askip trajectory where aplunge ismade, the vehicle heats up, returns toaltitude where radistion occurs and cools down for another plunge. The main problem for any path istoslow the vehicle down without excessive heating ondecelerations. Ithas also been shown that another some- °.,_< 5-2 what favorable heat path isaplunge tosay 200 - 250,000 feet where pressure drag isofafair magnitude and toremain atthat altitude until appreciable slow down occurs. The continuous high drag glide path has favorable deceleration aspects, but itisfairly un- favorable from aheating consideration. Only the skip and glide paths will bediscussed and then ina somewhat limited manner. A CD CL D g m r rc 1"o S S t V V _T ¥S W X Y _r5-3 SYMBOLS reference area for lift and drag evaluation, sq. ft. drag coefficient lift coefficient drag, lb. 2 acceleration due tcgravity, ft/sec. mass slugs distance from center ofEarth, ft. radius ofcurvature orflight path, ft. radius ofEarth, ft. distance along flight path, ft. surface area, sc. ft. time, sec. velocity, ft/sec. velocity divided bysatellite velocity at Earth's surface velocity ofsatellite atEarth's surface weight, lb. coordinate offixed axis system range coordinate vertical distance from surface ofEarth, ft. coordinate offixed axes system constant indensity altitude relation angle offlight path tohorizon, radians @ Panglebetween XaxisandV airdensity slugs/ft. _ anglebetween Zaxisandr alsoremaining range(F-_)radians (Section h.l); (Section h.3). Subscripts : b en exbody axis conditions atentrance toLarth's atmosphere conditions atexit from Earth's atmosphere 22.1< 5-5 S.I Development ofRe-entry Equations Involving Lift, Drag, Linear Inertia andGravitational Forces. The coordinate system showing the convention ofaxes and angles isshown infigure (5-1a) and anexploded view ofthe free body isshown infigure (5-1b). The equations ofmotion are derived (see reference 5-i) asfollows: Taking asummation ofthe forces inthe Xdirection: but _)_=Vco.se _m.d the rate ofchange ofvelocity inthe X dV=_o5o-eV5,_0 Xdirection is: and thus: /_jn@-_ -- f'2(V:o_o-V6._,,-,O)(5.1-I) Similarly summing theforces inthe Zdirection; (5.1-2) These two ecuations are the re-entry equations of motion where only lift, drag, linear inertia, and gravita- tional forces are involved. The equations are transferred 2.,22< 5-6 from the XZaxes tothebody XbZbaxes, thus pro- vidin Fforeasier handling oftheaerodynamic forces. Thetransfer ofaxes involves asimple rotation ofaxes bymeans ofthetrigometric formulas foraxes rotation. (5.1-3) (_.i-L_) Ifwesubstitute FXand Fzfrom equstions (5.1-1) and (5.1-2) into equations (5.1-3) and5.l-k) andcancel llke terms andcollect, ¢o y=VG(5.i-5) (5.1-6) Thus, substituting theexpressions forllft anddrag weight zWVV(5.1-7) (5.i-8) Also from figure (5-1) (5.i-9) _=Vcos'Y r =V_;_7-(5.1-Io) (5.i-ii) (5•1-12) Equations (5.1-7) through (5.1-12) provide the equations necessary for re-entry calculations _here only lift, inertia, gravitational, and drag forces are involved. The procedure for one means ofsolution, that isa step bystep ordigital integration ofthe e_uations is somewhat llke this• Weput the initial conditions Vo' $'o' _o' Po' to' CDo' CLo' inthe equations and obtain •_ " _,"_,_,r,,_,_a_'whichwhenmultiplied bythe increment oftime Atgives _i, Vl, hi, rl,_/i, Xl, and I.T.... .I."I" .= _ these qua_;tities back into _he equation with the appropriate CD, CL, and pand the process starts all over again toget the various quantities attime 2. It must beremembered that CLand CDare functions ofMach number and type offlow which must beknown for thebody be- fore any computations can bemade. Either CLor _may beprogrammed into the computations. Ifthere isnollft onthe vehicle, the lift term issimply dropped. Itmight •->4%"_,,,,,,A,"I"<_.-,.,.. 5.25-8 bementioned that there are various error reducing pro- cedures that could beused but which would complicate this simple integration technique. The above equations may beadapted tovarious problems. Various lift time histories, drag time histories, path angle histories may beprogrammed into the computations. For the skip and glide path some results may beobtained, with some further restrictions bysolely analytical means. The Skip Re-entry. Inthe skip re-entry the vehicle enters the atmosphere, negotiates aturn and isejecte0 into aballistic path. The equations ofmotion involving the forces perpen- dicular andparallel tothe flight path are respectively (see reference 5-2). Ci....._VL mV2" 7'_(5.2-1) (5.2-2) Wecan recognize these asequations (5.1-5) and (5.1-6) previously derived, but using the s_jmbol[sm offigure (5-3). Inthe top equation wehave the lift term, the weight term, and the centrifugal force due tothe path curvature. In the bottom equation wehave the drag term, the weight term, and the longitudinal acceleration term. 5-9 Ifinthere-entry turntheeffects ofgravity are assumed smallascompared tootherinertia andaerodynamic forces (gravity effects onthedownward patharealsosome- whatcancelled bygravity effects ontheupward path), then thetwoforceequations become: dV_- (5.2-I_)dtt _A-zCpfV = The atmospheric density isassumed tobegiven by Also from figure 5-3 itcan beseen that, -----,.Sl'#'i'_" Putting these quantities inequation (5-15) the V2and ds goout and C_foAe-#_d_=5,_ _nlch can beintegrated togiveYd_ (5.2-6) =cos"_-co_Y Cn (5.2-7) where pistaken as0atthe effective outer limits of the atzosphere. •.. 5-i0 Equation (F)shows that the magnitude ofthe angle inthe skip path isasingle valued function of y, that is, ithas only one ma_nltude atany particular altitude inall portions ofthe skip. Thus ifthe vehicle returns tothe outer atmosphere after each skip Ten = "_/ex for all skips. Dividing equation (5.2-3) byequation (5.2-h) weget or_6Y L_V_-_ D-aV at V.4T_ LaVas--5- at(_.2-3) (5.2-9) But and thusdV/,:IVL '-r Iav"=+ V_-_ Tas _ D Multiplying by ds and integrating for L/D =c VL= d7 V=Ven_-">/-b(5.2-10) (5.2-11) This defines the velocity inaskip. The _atvarious positions inthe skip !sdefined byequation (5-19). Thus, Z27< )5-ll the velocity, altitude and L/D =constant. any skip are defined for Sinceatvarious positions in of=e Ven(5.2-12) This defines the velocity loss inevery skip interms L/D since _enl= _en2 =_enh" Now weare interested inthe combined effect ofa series ofskips. Ithas been shown elsewhere that the range _, ofone ballistic phase isgiven by (5.2-13) where Vsissatellite velocity atthe Earth's surface. Itisassumed that the skipping process may beapproxi- mated byimpact problem considerations where the total range isthe sum ofthe ballistic trajectories. )Thus n (5.2-i£) 5.35-12 Glide Re-entry. The equations ofmotion (see reference 5-2) are asbefore : C" G.(5.3-1) The angles are assumed small inthe glide sothat sin _ _ ;cos __iand the atmosphere iscom- paratively thin sothat r ro_i. Itmay benoted that dV .idV'- "_--zas and from figure hitmay beseen that ---'q_--ds d__co_e as 7"I Thus equations (5.3-1) and(5.3-2) may bewritten as /_=-_V_d-_-K_+r_ dsmV" ------ (5.3-3)ro D__IdV" Z29< -13 Dividing equation (5.3-3) byequation (5.3-_) yields: ($.3-5) Itcan beshown that the terms may beneglected. The proof of this edhere. (See reference 5-2) So SinceZV_ -_--}--ZL-_=0 C5-5 equation (5.3-6) can beintegrated for constant give the velocity innon-dimensional form as [ /, IL/D gg' and V2dZ ds will not beconsider- (5.3-6) L/D to where Vo(5.3-7) isthe initial velocity divided bythe satellite velocity atEarth's surface. Visratio ofvelocity to satellite velocity atEarth's surface. Thus the velocity inthe glide path isdefined interms ofthe lift drag ratio and the range. ZCO<., ::. 5-14 From the original normal force equation COSL-,.__.co5_'= _ 1_ re_ro,)9_V" and divldin_ by weget Or Ifthe quantity for the llft isput in, there results (5.3-9) [1-- I.V*=(5.3-1o) Thus the vel_clty isexpressed intermsofthe density and thus indirectly interms ofthe altitude. Using the expression for density vs. altitude, velocity vs. density (5.3-10) and velocity vs. range (_.3-7) the flight path is defined. The foregoing equations onthe skip and _llde re-entry were obtained from hypersonic glider re-entry considerations. The velocity ofthe hypersonic glider may approach orbital Z&i< 5-15 velocity. Recently reference 3has been written which is also directed toward the re-entry problem under two degrees offreedom. Reference 3was not available soon enough for consideration when these notes were originally prepared. 5-16 REFERENCES 5-3.The Use ofLift For Re-entry From Satellite Trajectories. Antonio Ferri, Lewis Feldman, and Walter Daskin, Jet Propulsion, Nov. 1957. AComparative Analysis ofthe Performance ofLong- Range Hyper-velocity Vehicles. Alford J.Eggers, h.Julian Allen and Stanford Neice, NACA TN40h6. AnApproximate Analytical Method for Studying Entry Into Planetary Atmospheres. Chapman, Dean R., NACA TNk276. Figure 5-1a. Coordinate system forgeneral equations. Z Figure 5-1b. Free body forcediagram. Sketches from reference 5-1. I'I 0ff(f-.._tJ v_ fl _'_ __ <b _J.33_O11,,Y .,-3dt_J.Il T_fO3 m_ r_ _o_8 O_D _ 0RE .el m_.._o •.__ EO o _°m | o_r.Ja.o 0 •,-I_(2) (12: _0 (1) I._ I_, .elI 0 nO ¢o "I0 r_ _._(_- Figure 5-3. Skip re-entry geometry. "v',,r,.IY+_ Figure 5-h. Glide re-entry geometry. Sketch from TNhOb6. 6-1 SECTION VI SIXDEGREESOFFREEDOM EQUATIONSOFMOTION ANDTRAJECTORY EQUATIONS OFARIGID FINSTABILIZED MISSILEWITHVARIABLE MASS Thestudy ofspace mechanics thus farhasbeen restricted toatwo- dimensional analysis. Forma_yproblems however, itisnecessary touse athree-dimensional analysis. Someofthese problems are; theballistic missile with guidance whose trajectory isconstantlybeing changed tohit atarget; foraspace vehicle with guidance devices; foramanned vehicle reentry ororbit which iscapable ofbeing steered, forthelongitudinal andlateral stability analysis ofanyspace vehicle. Thus itbecomes necessary toexamine thecase ofmissilemotions andtrajectories for sixdegrees offreedom which aregeneral enough tocover allphases of missile motion, including thelaunch, exit from theatmosphere,space trajectory, reentry totheatmosphere addlanding phase. Thepurpose ofthis presentation will betoperform theclassical derivation oftheequations ofmotion ofamissile withvariable mass, andthen todevelop theequations ofthemissile trajectory referred to special sets ofaxes. Thederivation will beforthegeneral case of allsixdegrees offreedom. Inorder tomake thedevelopment more meaningful wewillbriefly review some elements ofvector mechanics andthematrix algebra oftransformation ofcoordlnates. Thepresentation will bedivided into four sections. 6.1. 6.2. 6.5. 6.4.Review ofvector mechanics. Reviewofmatrix algebra oftransformations. Development oftheequations ofmotionofamissile. Trajectory equations. 6-2 6.1.Review ofVector Mechanics Avector isaquantity which hasmagnitude anddirection. The analytical shorthand ofvector analysis hastheadvantage ofpermitting treatment ofdirected quantities such asforces without theneed ofreferring them toanarbitrary setofcoordinates. Inthefinal stages ofadevelop- ment, however, weusually define acoordinate system andresolve the vectors into components along theaxes. 6.1.1. Resolution ofvectors.- XZ k _y Avector canbewritten as: A=1Ax+JAy+KA_ 6.!-1 where i,Jand karebase vectors orunit v_ctors along the x,y,and zaxes respectively and Ax,Ay,andAzarethecomponents ofthevector Aalong the x,y,andzaxes, < 6-3 Itisseen that thema_Litude of_is A= 7_2 +Ay2+Az2 6.1-2 andthedirection cosCnes ofthevectorare =cosc_=cos(A,x)=Ax/A 6.1-3 m:cos_:cos(A,y)--A_IA6.1-_ n=cos7=cos(A,z)=Az/A 6.1-_ ft_ther Z2+m2+n2=i 6.1-6 More will besaid about direction cosines later. 6.1.2Addition andSubtraction ofvectors.- Addition andsubtraction ofvectors areperformed bythewell known force polygon graphicalmethod. 6.1.3 Multiplication ofvectors.- There aretwotypes ofvector scalar ordotproduct oftwovectors isascalar quantity andnota vector.Itisdefinedby A•B=ABcos(A,B) 6.1-7 orinterms ofcomponents A._:Ax_+__+AzBz6.1-8 6-_ Bymeans ofrelation (b.l-7) itisseen that thescalar products ofthe unitvectors are i.i=J. J=k. k=l}i.J=i.k=J-k=06.1-9 Thetime honored exampleoftheapplication ofthedotproduct is: W=_-X 6.1-10 m where Fisaforce wh4ch moves along avector distance XandWis work done. Thecross product orvector product oftwovectors isavector and isdenoted by u m m A×B-C 6.1-ii m where themagnitude ofCisgiven by C=ABsin(A,B) 6.1-12 andthedirection isgiven bytheright hand rule. ..... /17/ / / A 240< 6-9 orinterms ofcomponents m AxB=j 6.1-15 or i×E=(AyBz-__)i÷(__x-A_Bz)J÷(_By-__)k 6.1-14 Bymeans ofrelations (6.1-11) and(6.1-12) Itisseen that the vector productsoftheunit vectors are IxJ= Jxl k Jxk= -kxJ i k×i= ixk J6.1-19 Anexempleoftheapplicationofthecross products is T=RxF 6.1-16 m WhereFisaforce acting onaparticle whose radius orposition vector o isRand Tisthetorqueormoment ofFabout anaxis through the orig_n andperpendicular totheplane ofRand Fanddirected bythe right hand rule. #,_FJ-"- 6-6 Another example ofthevector cross product is: V=_x R 6.1-17 m where _isanangular velocity 3Risaposition vector and Visa linear velocity. Asindicated previously IVisgiven by pV-ijk Rx_Rz6.1-1b or orV=i(Rz%-_%)+J(__z-Rz%)+k(_%-_5) v-lVx+JVy+kVz6.1-19 6.1.4 Momentum.- Momentum ofaparticleofmass misdefined as: u-m_ 6.1-z) where Uisavector which hasthedirection ofthevelocity vector andtherefore from (6.1-17) itisseen that _42< 6-7 .=(ix_) 6.1-21 6.1.5 Momentofmomentumoran_ularmomentum.- Themoment of momentum orangularmomentumHissimply themomentofthemomentum vector about agiven point oraxis through that point. From equation (6.1-16) weseethat themoment ofavector about apoint issimply the radius vector Rofthat vector crossed into thevector U. Therefore: =RxU 6.1-22 andby(6.1-21) _-_x(7_x5)= 6.1-23 Forlater usewenote that 6.1-24 and x(;x_)==(_•_)m 6.l-2_ 6.1.6Vector operators: Gradientoperator:- TheoperatorV (the gradient ordel) isdefined by 6.1-26 Ifwehave ascalarfielddefinedby_=f(x,y,z),then Z43< 6-8 v_= i_-x+J by bz6.1-27 V_isthegradient @ordel Anexample ofanimportant scalar field isthegravitational field. Consider thepotential _duetoaconcentrated mass, m,intheearth's gravitational field,namely r where risthedistance from thecenter oftheearth tothemass. Then F,theresultant force, isgiven by F=V_-I_+O +k6.1-29 orthegravitational force isthegradient ofthegravitational potential. Toshow this wenote r=Sx2+y2+z2 6.1-3o and br,, x .x 6.1-51 bx Vx2+y2+z2r a=i ___=_._._r=.km .x 6.1-52 bx _r _xr2r 6.1-33 by _r byr2r 6-9 _____ 5__. b_Kr_km.z 6.1-_ bz()rbz r2r and F= _x/ r2 Th_, ifthemass misinthegravitational fieldofma_y other bodies klm k2m k_m whose potentials were defined asq_l=--;q)2=_---; _=--; etc. rl "2 r5 then thetotal potential would be_=q_i+q)2+q_5+............. _ndtheresultant gravitational force would be F=V_ Other vector operatorsnotused inthis development arementioned here merely forthesake ofcompleteness. Thecurl ofavector is: Q curl A=ijk bb_ bxby_z6.1-56 mdthedivergence ofavector is: _v7=_x+by_Tz6.1-57 _.5.5<..... 6-10 6.1.7 Time rate ofchange ofarotating vector.- m R l From thediagram itisseen that thevectordRisthevector sum ofacomponent @Ralong(R +_)and acomponent _to(R +_R) i m equivalent to(_RxR)dt oRI_ ÷ (_ XR)dt6.1-38 J_ I _'I_ ÷ (_ X I) 61]___ 9 dt_t This isthewell known transformation fortherate ofchange ofany - vector Rfrom fixed tomoving axes where ---istherate ofchange of_t measured with respect tomoving axes_and _RIsthea_velocity ofthemoving axes with respect tothefixed axes. 6-11 6.2MatrixAlgebraofTransformations Inthissection weshall deal only with rotational matrices. A rotation matrix performs anorthogonal transformation ons_aequantity such asavector orsetofvectors. Thecoordinate systems considered areCartesian andare"right-handed" systems. Orthogonal matrices, which perform only rotations (also called rotation matrices) have special properties. Wewill discuss some of these properties. 6.2.1 Single rotation.- Westart with asetofaxesXNYNZN asshown below. _, _'-YNcos, ZN,ZI Byrotating thesystem about the ZNaxis through anangle ,weobtain anewcoordinate system which weshall designate (XIY1ZI)"Thetrans- formationorrelation between theoriginalandnewsystem isobtainedby geometry andisseen tobe Y1 lScos,sin,0 l"ein* cos$0 0 00bXN Y.6.2.1 b-12 Thistransformation iscalledanorthogonal transformation. Theelements arecalleddirection cosines, sinceeachelementforinstanceIcos+ isthecosine oftheangle between XNandXI.Theelement sin_is thecos (90-_)theangle between YNand X1andtheelement -sin isthecog(90+_)theangle between XNandYI" Actually each element ofthematrix canbeconsidered thescalar product oftwounit vectors having thedirections oftheindicated axes. Ifwelet anddenote thetranspose of o,[_c,)]_,E'(*)]'_ obtainedbyreplacing (_)I.cos$sin$_I_T(,)_ = sin, cos, 0 0 anddefine thematrix isnoted !I:"c,)]-l-_c,)]": Stated inwords thetransposeof thematrixobtained byreplacing toequation (6.2-1) weget N- P.J6.2-2 anddenote theinverse IT(-_)_ asthat matrix then thefollowing property cos+-sin _il[.,:,oO.,o [_c,)] (,)by cos_-sin+0 sin_cos$0 0 0I • : .e.•6.2-3 isalso itsinverse andisalso (-$). Applying this principle IIYI Z]6.2-4 6-13 6.2.2 Tworotations.- Weoriginally startea with asetofaxes _NYNZNandrotatedabout the ZNaxis through anangle ,toobtain anewsetofcoordinates (XlYIZI)" Letusnowrotate theXlYIZI system about the YIaxis through anangle Gtothenewcoordinate system X2Y2Z2asshown below. X1 _5 Z_ From thegeometry oftheproblem thetransformation between the system andthe X2Y2Z2isseen tobe 2x21oInIIo Z2[sin80cos orX1 !YII rZI} zlXIYIZI 6.2-5 6.2-6 _.2Kg_AIA.':J[_"-• 6-14 Combiningequations (6.2-b)and(6.2-1) weget [_21 x{i1 U2J6.2-7 This equation implies therotation $isperformed firstonXNYNZN followed bytherotation 8todetermine thenewcoordinates X2Y2Z2" Inorder toreturn tothe XNYNZNcoordinates from the X2Y2Z2 coordinates, from aconsideration ofthegeometry wewould rotate first through a(-8)then througha(-,)or l'I,26.2.8 &. ,,_ Bymakinguseofequation(6.2-5),equation (6.2-8) could bewritten as zN6.2-9 Bytaking theinverse ofthetransformation matrix inequation(6.2-7) we canwrite 6.2-10 ..2S0< 6-15 Acomparison ofequation (6.2-I0) and(6.2-9) indicates Stated inwords equation (6.2-11) says theinverse ofaproduct oforthogonal transformation matrices istheproduct ofthe_inverses oftheindividual matrices taken inreversed order. This isthereversal property of orthogonal matrices. Forsubsequent usewenote that equation (6.2-5) maybestated LI!lllEc° e°slneol0h - X2 'Y2' Z26.2-12 6.2.3 Three rotations.-Webeganwith asetofaxes XNYNZNand rotated through anangle #toobtain thesetofcoordinates XIYIZI which werotated through anangle 8toobtain thesetofcoordinates X2Y2Z2" Letusnowrotate the X2Y2Z2system about the X2axis _,,__._-obtain =newcoordinate system which weshall designate (XbYbZb) asshown below _4 6-16 Fromthe.geometry wesee orqlooYb=0cos$sin zb |0-sin$cos 1n"(_)Y2 Combining equations (6.2-i), (6.2-5_ and(6.2-13)weget andinversely or z.jzb cos,-sln, 0 sin$cos$0 0 0ia cos80sin8 0 i0 sin ecos8. 00' 0cos_-sini Iko, sin_cos_I .L6.2-13 6.2-14 6.2-13 6.2-16 xb"_ I I ,Zb] 6.2-17 2.52.< r.i i x [ I m m 0 00 0 _._ 0 _I tJ .,-4 ._ m e2 0 .r4 +: mu_ f,.., 0 _ _ 0 0 _ °r| gl 0-,-t _ m _.. _ 0 0_ _ ,,4 m_m II! O! ,d A ! _ n o o o o i) "! OJ L I-----'-"--1 6-18 6.2._ direction cosinesandaredefined from equation (6.2.1_)as _ii=cos_cos8 _12"sin$cos 0 _13="sin8 m£1 =cos$sin 8sin_-sin$cos m12 =sin#sin 0sin_+cos$cos m13 =cos8sin n11-cos$sin0cos_+sin$sin n12=sin$sin8cos_-cos$sin n13=cos 8cosDirection cosines.- Theterms _iJ,mlj,nlj are called 6.2.21 Some interesting relations bet'_een thesedirection cosinesorbetween the 6.2-22directive cosinesofanyorthogonal transformation arethefollowing ZII2+mll2+rill2=i _122+m122+n122 =i _132+m132 +n132 =i _iI_12+mllm12+niln12"0 _12_13+m12 m13+n12n13"0 Z13_iI+m15mll+nl3nll"06.2-25 ZII2+ZI22+_132 =i mll2+m122+m132 =i rill2+n122 +n132-i6.2-2_ 54< 6-19 _iimll+ZI2m12+_15m15"0 rollnil+m12n12+m15n15=0 nll_Ii+n12ZI2+nl5Z15=06.2-25 and _iI ZI2ZI3 mllm12ml3 nlln12nl3=_ 6.2-26 Thethree rotations from XNYNZNthrough _,e,and $toXbybzb areindicatedonFigure i.Ifp,q,and raredefined astheangular velocities aboutthe XbYbZbaxes, then fromFigure 6-1itisa_parent that p=6-,sin8 q=8cos_+,sin_cos8 r=$cosecos_-8sin6.2-27 Inorder togettheweight component into theequations ofmotion (which wewill indicate later) thedirection cosines must becomputed. Onananalog type computer, however, thedirection cosines aregenerated byusing thederivative forms ofthedirection cosines which are _lJ=mljr-nljq "n!J=nljq"_lJr n!J=_Jq"mljrwhere J=I,2,5 6.2-28 [SS< 6-20 These relations arenotdeveloped here butmaybedemonstrated individually inthefollowing manner. Referring back toequation (6.2-21) itisseen that ZII,,cos$cos86.2-29 anddifferentiating weget @ Ill=-cos_sine8-sin_cos@ 6.2-50 From equation (6.2-28) itisseen _ll=mllr-nll q 6.2-31 Bysubstituting thevalues ofrandqfrom equation (6.2-27) andvalues ofroll and nll from equation (6.2-21) into (6.2-31) and performing theindicated multiplication andreduction weget =-COS$sin%8-sin$cos8 6.2-52 whichisidentical withequation (6.2-50). XN,YN,ZN _,8,q) Xb'Yb'Zb p,q,r UjV_Ig ib,Jb,kb iN'JN'kN V t_ Ve U UT m H RT6-21 List ofSymbols right-handCartesian coordinate system fixed innon- rotating earth, inertial axes (with ZNpositive down) angles used inspecifying missile attitude referred to asEuler angles (specified order ofrotation _,8,_). body axes -right-handed coordinate system fixed in missile body with theorigin attheinstantaneous center ofgravity. (See Figure 2.) angular velocities ofbody-axis system (xb, Yb,Zb) positive clockwise when looking inpositive direction ofaxes components ofresultant velocity Valong body axes Xb,Yb,Zb,respectively unit vectors along the xb,Yb,and unit vectors along the XN,YN,andzbaxes respectively ZNaxes respectively resultant velocity ofmissile center ofgravity resultant angular velocity ofmissile or(xb, Ybpzb) system _=ibp+Jbq+kbr Jetexit velocity relative tonozzle exit Momentum ofmissile U-mV momentum ofmissile andJet missile mass (slugs) angular momentum ofmissile with respect tothecenter ofgravity ormoment ofmomentum angular momentum ofmissile andJet : S7< 6-22 t T Fv F M Q D Pe re Cx,Cy,Cz Cz,Cm,Cn S LI,L3,time aftermissile separated fromla1_cher resultant Jetthrust resultant Jetthrust forthespecial case when the thrust isalong the xbaxis resultant Jetvane force resultant aerodynamic force resultant reaction control force resultant external force acting onmissile resultant Jetthrust moment about center ofgravity resultant Jetvane moment about center ofgravity resultant aerodynamic moment about center ofgravity resultant reaction controlmoment resultantexternal moment acting onmissile lift force onJetvane airdensity Jetpressure atnozzleexit (gage) nozzleexit area distance frommissile center ofgravity tonozzle exit aerodynamic force coefficients referred tobody axes aerodynamic moment coefficients referred tobody axes missile cross-sectional area representative missile length (dismeter, chord length_ lengthp etc.) distance ofJetvane lift forces behind center of gravity rl,r3 W A 7 X,Y,Z Xe,Ye,Ze Xg,YgtZg _e k L h Ro vw Ve va V Z,m,n 5a6-23 distances from xaxis toJetvane lift force missile weight, pound azimuth angle ofvelocity vector measured with respect tothe Xg,Yg,Zgaxes (+from south towest) elevation angle ofvelocity vector with respect to the Xg,Yg,Zgaxes (+up) angle ofattack angle ofsideslip right-hand Cartesian coordinate system fixed innon- rotating earth wlth Zpositive pointing north right-hand Cartesian coordinate system fixed inrotating earth Geographic axes angular velocity ofearth (rotational velocity) longitude latitude distance above earth's surface radius ofearth wind velocity with respect toinertial axis velocity component ofearth's atmosphere resultant aerodynamic velocity velocity ofmissile center ofgravity direction cosines aileron deflection rudder deflection 6-24 _e 51,82,55,54 Ix,_,Izelevator deflection Jetvane deflection missile moments ofinertia aboutXb,Yb,and respectively Ix=/(y2 +z2)dm ly=/(z 2+xe)am Iz=fC_2+_) _z, Ixz' Ixymissile products orxnertia about Xb,Yb,and Iyz=/yz dm Ixz=/Y.Z d.m Ixy=/Xy dm d (.)=--except thatdt'_= dmand m dtdI dtzbaxes zbaxes _60< .. 6.36-25 Derivation ofEquations ofMotion Thegeometry isshown inFigureb-l.T_o setsofCartesian coordinate axes areused indeveloping the_quations ofmotion. Onesetisfixed in thebody andisknown asbody axes andisshown inFigure _-2.The body axes wedesignate asXb,Yb,and zb.Theother setisfixed with respect to theearth andisknown asinertial axes which wedesignate asXN'YN'ZN" Theorientation ofthebody axes with respect totheinertial axes i_d_f_ned bythree angular coordinates @,e,and _which arecalled R_ler ang]e_. These anglesareshown inFigure 6-1.Therotations must be _-;ken inacertain specified order namely :,,e,and _. Furthnr definiticns ofpertinent quantities aregiven inthelist of symbols. Inappendix Asome oftheformulas pertaining tothedevelopment arelisted. Anattempt hasbeen made tousethestandard HACA symbols throughout although incertain cases this wasnotpossible. m Aschematic diagram ofthemissile isshown inFigure 6-3.The vector represents theresultant angular velocity ofthemissile or Xb,Yb,Zb_body system. Theequations ofmotion will bederived bywriting therate ofchange ofmomentum andrate ofchange ofangular momentum equations. 6.3.1 Force equations.- From Newton's lawwehave theequation: _dt_N 6!< 6-26 Therate ofchange with respect toinertial space ofmomentum ofthemissile andJetcanbewritten \-_-JN _ +_(_+_×re-_e)6.3-2 Since U angularvelocity ofchange ofisreferred tothe Xb,Yb,Zbsystem which isrotating with an (see Fig. G-_with respect toinertial space therate with respect toinertia space is dU I _ + _ X U dt6.3-3 asindicated previously insection 6.1. Bydifferentiating themomentmn U-m_ 6.3-4 anddefiningdm--- -mwegetdt mdU 6.3-5 Ifwesubstituteequations (6.3-3), (6.3-4), and(6.3-3) in(6.3-2) weget dt/Nm_-6.3-6 which reduces to dt/N m_ +6.3-7 Since dV a'_"G.ib+{,,Ib+"}kb 6.3-_ 6-27 and and_=Pib+qJb÷rkb = ib+reyJb+ kb _ereX rez6.5-9 6.5-io m mo_xV= I_JbkbIb qr vV=Ib(wq-=)÷Jb(=-vp)+kb(_-uq) 6.5-11 and X_e=ib(rez q"reyr)+Jb(rex r-rezP)+kb(rey p-rexq) 6.3-12 wehave (substituting equations (6.3-12), (6.3-11),and(6.3-6) in (6.3-7) d_T\ =_m(& vr) &(qrez-_ljN +wq- + -rrey)} Ib+ a(_r+ur wp)+_.(r rex-pez_Jb+ _(,+_-uq)÷_(_r_-qrex})kb-_We- 6.3-13 which istheforce equation. 6.5.2 Moment equation.- Themoment equation about themissile center ofgravity is =M N6.3-I_ ;.G3< 6-28 Therate ofchange withrespect toinertia space ofangular momentum ofthemissile with respect tothecenterofgravity including Jeteffect _S rex-N_N6.3-13 where d.H -- m-- -#-G_X H \dt/dtN6.3-10 From thedefinition ofangular momentum wecanwrite ff=/_x(T_x_)d,n 6.3-17 wherc R=xib+yJb+zkb6.3-18 Thetriple vector product asindicated previously (equation (6.1-25)) canbewrltten X (_ X R) l i_2 _J_. I_(I_ " _) 6.3-19 where andR2=(x2+y2+72) 6.3-20 "Pib+qJb+rkb 6.3-21 Substitutingequations (6.3-21), (6.3-20), (6.3-19) and(6.3-10) equatxon(6.3-!7) results in H'f{( x2+Y2+z2)(Pib+qJb+rk'°)"1 (xib+yJb+'zkb)(px+qy+rz)_ dm Jinto 6.3-22 _.64< 6-29 Byperforming theindicated multiplications equation (6.3-22) canbe expressed inthefollowing form m H={,,r<,,+.,>°-.,sx>..,=-rSx.°)-+ ,,f<_=+=,>o.,.f>,,<,=.,fx>. ,,,>+ 7s<-'°-,i.z°-,s,-°7,6.3-23 and_jintroducing theusual definitions ofmoment ofinertia _ndproduct ofinertia shown inthelist ofsymbols wefinally get m H=(Ixp-Ixyq-Ixzr)ib+(lyq-lyzr-Ixyp)Jb+ (Izr-Ixzp-Iyzq)kb 6.3-24 Itisusually conventional inmissile work tochoose thebody axes as theprincipal axes sothat Ixy=Iyz=Ixz=06.3-2.5 thus wehave thefinal expression forangular momentum m H=(Ixp)ib÷(Iyq)Jb+Izrk_6.3-26 Therate ofchange ofangular momentum isobtainedbydifferentiating (6.3-26) andbecomes (Ix_-Ixp)Ib+(Zyq-_yq)Jb*(Z_r{,.r)kb t m dt 6.3-27 T_Fpr--, 6-3o Theproduct _×His a_XH- Ixp_qIzr =(zz-_)rqib+(zx-z=)prJb÷(Zy-Zx)qpkb6.3-28 Similarly thethrust term becomes [-- We] rex(_Xre-_e)=mre2_-re(re•_)-rex6.3-29 =mI(rex2 +rey2+re2z)(Pib+qJb+rkb)- (rexib+ Jb+ kb)(rexp+ q+ r)_- rey rez rey rez Byperforming theindicated multiplications, equation (6.3-29) canbe expressed inthefollowing form [r - }-fo_ _ +rrez)t ib+ n:e×(_Xre-Ve) m(re +re)-rex(q rey r,:(rez+re)-rey(r rez "[ +q -m(rex2+re)-rez(p rex 6.3-30 266< 6-31 Finally bysubstituting equations (6.3-30), (6.5-2_), (6.3-27),(6.3-16), and(6.3-19) into equation (b.3@4)weget [/N=Ix_"(Iy-Iz)qr-IxP+ m(re+re2z)rex(q rey rez [__-(Iz-A)rp-_yq+ m + " + Jb(rez2 re2) rey(r rezPrex + Zz_-(Ix-zy)pq-{,r+ Ir + - +qrey)ll m'(re2xre_) rez(P rex kb- . m _ m mreXVe=M6.3-31 6.3.3 External force andmoment systems.- Theresultant external force Fisthesumoftheaerodynamic forces, thegravitational force, thecontrol forces acting onthemissile andtheforce caused bytheJetpressure at thenozzle exit or Theresultant moment Misduetoalltheforces listed above except thegravitational force which hasnomoment about thecenter ofgravity. m _ m m M-MA+MC+reXAePe 6.5-33 6.3.3.1 Aerodynamic forces andmoments.- Theaerodynamic forces andmoments along andabout theprincipal axes aredefined inthecon- ventional manner asfollows : _._.: 6-32 mqdFx=Cx_SVa2 0Fy.Cysv2 = RSVa2Fz Cz26.3-_ Mx=CZ_SVa2 0SMy=cm_Va2_ 6.3-35 0 =cn_sv2 where C_,Cm,and Cnarethemoment coefficients which give themoments about themissile center ofgravity. Wecanfurther specify that each aerodynamic coefficient isafunction ofthevariables 8,e,8,_,P,q,and rsuch that: Cx="Cxo"Cx__"Cx&"2V" "Cxq(2V) _ p_ r_ = Cyp(_) Cyr(_} Cy__+Cy_(_V)+ + Cz""Czo "Cz_ Cz_"2V" "Czq(2V) C_-C_lB8÷C_(W) +C_p(2) +CZr(2-V Cm=Cmo÷Cm_ +Cm_"2V" +Cmq(2-V) Cn-Cn__÷Cn_()÷Cnp(2P_V_)+Cnr(_-_) 41" f'_ '*_<6.5-56 6-33 Definitions ofa,_,sadVawill begiven later inthedevelopment. 6.3.3.2Gravitational force components.- Thec_ponentsofthe weight term which weincluded intheequations ofmotion arecomputed from thetransformation equationbetween inertial andbo_v axes (which was developed insection 2)andare % % _zbZII_12_151 =mllm12ml3 niln12nl3I°I I !) : • I IWLJ6.3-37 or _)2 w_=Zl3w=zl3_=_13mgoR_"+h Wyb=m13W=m13mg=m13mgo oh Wzb=nl3W= n13mg=n13mgo< Rn_2Ro+6.3-3,_ _oh)2= and go=32.2.where ggoRo+ Itshould benoted that although these weight components arethe ones usually used, they areapproximations based ontheflatearth concept, since they neglect thenonparallelism ofthegravitational attraction at different points oftheearth's surface. Thetrue gravitational force or weight components will bespecified later insection 6.4. 269< 6-_ 6.3.3.3 Control forces.- Thecontrol forces, ofcourse, depend on thetype ofcontrols used onthemissile. Forpurposes ofthis dis- cusslon weshall assume themissile isequipped with three types of controls. (SeeFigure b-2.) I.Conventional aerodynamic controls 2.Reaction controls 3.Jetvane controls Theresultant control force andmoment equations canthen beindicated as _c=F5 +FR +Fv 6.3-39 and Theconventional aerodynamic controls areconsidered tobeaileron 5a,rudder 8r,andelevator 5e.Thecomponents ofthese aerodynamic forces andmoments caused bythese controls areassumed tobethefollowing: F5x=0 F_-cy__sVa28r FSz-Cz5e_SVa25e6.3-41 M'/G-- D6-35 and P p %x=cz8a_sv$z6a+C%r_SVa2Z% PSVa_Z5e 6.5-_2 0 P _z-Cn8a_SVa2 Z6a+C_6r _SVaR Z_r Thereaction controls areass_ned tobeforce andmoment components along anlabout the._issile principal bodyaxes ofmagnitudes FRx,F_, %. 6.3.3.4 Jetvane controls.- The Jetvane forces andmoments are considered onthebasis ofthevanearrangement shown inFigures 6-4and 6-3. The components ofthetotal Jetvane force acting onthemissileare assumed tobethefoll_ing: FVx=-(total drag ofallJetvanes) =0 Fry=(Q3+%)=(_83+_6,+)=_d_(63+6_) aQ dQ,62)dQ(6z+62) Fv-(Q_.+_) =(_6j.+_ =z6.3-45 The componentsofthetotal Jetvane momentabout themissile centerofgravity ere _7i< 6-36 aQ dQ aQ53.r3dQ54MVX=rI_'_51-rI_'_52+r5_'_ My=LlaQ aQY _-_81+Io_--d552 d0,53, z_5_6.3.-_4 where LI,D2,L_,and 14arethedistancesoftheJetvane llft forces behind themissile center ofgravity. Other possible types ofmissile control notdetailed here arecontrol bymeans ofaswiveled orgimballed rocket andcontrol byuseofVernier engines. 6.3._ Equations ofmotion.- change ofmomentum "i_/_ becomes(whereTheequationofmotion duetorate of m =F 6.3.1 Veisadded toboth sidesof(6.3-I) 6.3-_5 where w 'Fj"AeP%+roVe =Jetthrust 6.3-46 Theequation ofmotion duetotherateofchangeofangular momentum 6.3-14 6-37 becomes (where 9.(re×VL))isadded tobothsides ofequation (6.5-i_) whereN6.3-h7 o m blj=re×(AePe+m_e)=Jetthrust moment about thec.g. 6.3-h_ Equation (6.5-h9) resolved into itscomponents thus generates thefollowing three algebraic equations. m(u+wq-vr)+&(qrez -rrey) =FAx+Wx+FOx+FJx 6.3-49 re(v+=+m(r -p)-FAy÷Wy ÷ +Fjy •rex rez FCy 6.3-50 m(w+vp-uq)+m(p -qrex) - +Wz+ +rey FAz FCzFJz 6.3-51 Equation (6.3-47) resolved into itscomponents generates thefollowing three algebraic equations F Ixp-(ly-Iz)qr-IxP+mlp(re_ +re_) +rex(qrey+rrez)i 6.3-52 LVJ< 6-3t_ M_*my+my 6.3-93 T.r-(Ix-ly)[q-izr+mfr(re_ =_;A+_z +MJzZ+re2)+re(Pre+qre_ y z x yJ 6.3-_ Inorder tosimplify theequations ofmotion the following assumptions _remade. Itisassumed the Jetexit velocity relative tothenozzle exit i__!cng thexbaxis_ that is ve=ixbv, er 6.3-95 slid then cr then furtherVex-re,Vey= vez=0 m Pe=ixbPe 6.3-96 r-e=ixbre 6.3-57 = -0 ray rez FJx"Ae.Pe+mVe=T Vjy-vjz-Mj-o With thes_ simplifications thesix_quations ofmotion become6.3-5b 6.3-99 •I"_J'4 6-39 m(V÷_-_)-_rer_Wy÷F_÷FCy Ixp-(zy-Iz)qr-Ix;-MA_+MCx Iyq- (Iz-Ix)rp-iyq+mqre2-MAy+MCy Izr-(Ix-Iy)pq-Izr+_rre2.MAz+MCy6.5-6o 6.3-61 6.3-6e 6.3-63 6.3-6_ 6.3-65 Bysubstituting thecomponents oftheaerodynamic force andmoments, theweight components andthecontrol force components derived earlier into equations (6.3-60) to6.3-65) inclusive wearrive atthefinal setof equations ofmotionofthemissile with variable massshown onthefollowing page. Theequations ofmotion forcoasting flightorthe(nothrust) con- ditions arealso indicated onthepage following. 6.3.9Remarks onsolution ofe_uatioms ofmotion.- Thesixequations ofmotion (6.1) and(_.b) contain sixunknowns u,v,w,p,q,and r Ifthesolution isperformed ontheanalog, three more equations areadded which generate thedirection cosinesneeded toinclude theweight term in theequations ofmotion• _13"m13r-n13q &13" n13P"ZI3r 6.3-66 n15=Z15q-ml_P __'IrT,r- U_ N "-tj,_ +k.) ++ ,¥_._.) .'_ %) q..4.,t- t_t_t_ _1+.I _ m II11II III _'-i,._..._._÷ •I-+ _""--i II _+-I- ++ •_. _.._ _____"£,_ IIIIIII|I iIIIII i.,. I I I III •_,_ + i •E.£ I.I',_IJ IIIIII I It U 0'_ ,_ II11, °_ II tl 6-_2 TheEulerangles _,8,and_which specify theinstantaneous missile attitudes canbedetermined from theequations (6A-2) given in appendix Awhich are • 1t=cose-- (qsin@+rcos_) = qcos_-rsin_ 6.3-67 -p+tane(qsin@+rcos_) 6.3.6 Remarks onchoice ofaxissystem.- Principal body axes have been used inthis development oftheequations ofmotion; however wecould equally aswell have chosen stability axes orwind axes. Each sethasits inherent advantages anddisadvantages. Choice ofbody axes offers theadvantage that themechanics ofthe problem aresimplifiedbytheelimination ofproductsofinertia andtheir rates. Thedisadvantageofprincipal body axes isthat theaerodynamic coefficients aresometimes difficult todetermine inthis coordinate system andthetrue weight orgravitational component isdifficult to incorporate intheequations.Theweight term isusuallyapproximated. Stability axes (see xsinFig. 6-6c) offer theadvantage that theaerodynamic coefficients areeasily specified. They have thedis- advantage that products ofinertia andtheir ratesmustbeknown andthe weight component isdifficult toincorporate intheequations. Wind axes areused frequentlysince theforce equations areeasily written forthis axis system andtheweight term iseasily incorporated. Themoment equations however have moments ofinertia andproducts ofinertia which varynotonly with thetimevarying mass, butalso with respect tothe body attitude inthewind axe_, 6.4 'AraJectc_, Equations Thepath ofthemissilecenterofgravity with respect toagiven setofcoordinates represents itstrajectory inthose coordinates. Inthe developmentofthemissileequationsofmotion, wedefined twosetsof axes (Xb, Yb,Zb)body axes and (YN,YN,ZN) inertial axes. 6.4.1 Inertial axes.- Thetrajectory ofthemissile center of gravity inthe (XN, YN,ZN) inertial axes canbecomputed inseveral ways, oneofwhich isthefollowing. From asolutionoftheequationsofmotion (6.5-60) to(6.5-65) time histories of(u,v,and w)areobtained. Using thetransformation defined inequation (6.2-19) we.canobtain thevelocity components alon@ theinertial axes from uli16.4-1 andanintegration ofVXN,VyN,and VZN,with theproper initial conditions yields thetrajectories XN,YN,and ZN.This operation isindicated by Integratlng 1 Matrix JvYN i vz l6.4-2 Since further informationofamissile trajectory maybedesired such aslatitude, longitude, andattitudeofthemissile with respect toa stable tableorstabilized platform wedefine some newsetsofaxes toyield this information. Ifweconsider theinertial axes(XN,YN,5N)fixed inthenon-rotating earth asshown inthefollowing sketch wenote that theearth would rotate about the(-ZN)axis using theright- hand rule. Wewould prefer tohave theearth rotate about the(+Z)axis; therefore, wedefine anewaxis system (X,Y,Z)such that the(+Z)axis goes through thenorth pole asshown inthefollowing sketch. Z ;y X The(X,Y,Z)axis isobtained from the(XN,YN,ZN)axis system bya rotation ofI_0°about theYNaxis.Thetransformation between the (xs,YN,z.)_a(x.Y,z)axesis 6-45 I°xI }Ooo This maybeobtained byletting e=180°inequation (6.2-5). (6.4-3) maybewritten as6.1_-3 Equation 6.4-_ 6._.2 Moving earth axes.- Theearth rotates atanangular velocity O_e. Ifwelettheangle H-o_t 6._-5 anddefine asetofaxes fixed intherotating earth as(Xe, Ye,Ze), then atagiven time the(Xe, Ye,Ze)axes would beoriented with respect to the(X,Y,Z)axes asshown inFigure 6-6a. A--hetransformation between thetwosets ofcoordinate systems becomes orICosSlnO= sinHcosH0 0 0 16._-6 6._-7 6..t_6 6._.3 Local geographical axes.- Aright-hand setofCartesian coordinates (Xg, Y8,Zg)isdefined asshown inFigure 6-6such that the Zgaxis is always pointing toward thecenter oftheearth andthe Xgaxis points south. Thelocal geographical axes canbeobtained from the(Xe, Ye,Ze) moving earth axes bythefollowing three rotations (i) Rotate (Xe, Ye,Ze)about Zethrough theangle _toobtain thenewcoordinate system (XI, YI,ZI)" (2) Rotate (XI, YI,ZI)about Ylthrou@h theangle of_toobtain thenewcoordinate system (X2, Y2,Z2)- (3) Rotate (X2, Y2,Z2)about X2through theangle 180°toobtain thenewsetofgeographical axes. (Xg, Yg,Zg) These three rotations aresimilar tothethree performed insection 2 where ,-k,0=_ and @=180°.Thetransformation which results from these three rotations on(Xet YetZe)toobtain (Xg_Yg,Zg)(which is similar tothat ofequation (6.2-18)) is Zel Xg ,Yg= Zgor_o,_,cos_)_in_,)(-_os_sin_) _in_cos_')(-cos _)_sin_,sin_) L(-sin_) o(-_os_) g (,in_) (-cos_) 0_l6._-8 Xel :_I 6.4-9 6-47 wecanwriteequation (6.4-9)as Yg= g6.4-1o inorder todefine thetransformation matrix [Tb]. 6.b.h Latitude tlongitude tandaltitude.- Time histories oflatitude, longitude, andaltitude arederived from time histories ofXe,Ye,andZe. Theangles used inspecifying theattitude oflocal geographical axes are k,thelongitude and2,thecolatitude. Ifthelatitude isdesignated byL,then L=9O-_ 6.4-n Thegeometry oftheproblem isillustrated inthefollowing sketch / xe__ >xe 6-48 From thegeometry oftheproblem thefollowing relations areobtained. (_o÷h)-_xe2+Ye2+ze2 Xe (RO+h)cosL=sin(9o-_)6.4-12 or Xe=(R0+h)cosLcosk Ye (RO+h)cosL-cos(9o-_)6.4-13 or Ye=(Ro+h)cosLsinA ze =sinL 6.4-14(Ro+h) or Ze=(Ro+h)sinL 6.4-15 From equations (6.4-10) and(6.4-12) wegettherelation forthelatitude L which is L=sin"I _e _Xe2 +Ye2+Ze26.4-16 Dividing equation (6.4-14) by(6.4-13), wegettherelation forlongitude which is A=tan"IYe 6.4-17 Thevelocityvectorofthecenter ofgravity isoriented inthe geographical axesbytheazimuth angle Aandtheelevation angle7. 6-h.9 Theazimuth angle Aistheangle inthe Xg,Ygplane measured inthe positive direction from south towest; inaphysical sense itisourcompass angle plus l_O°.Theelevation angle yismeasured +up. From the geometry itisseen VXg=Vcos7cosA Vyg=Vcos7sinA6.4-18 VZg=Vsin7 from these relations wegettwoexpressions for Aand y tanA=VYg Vxg6.4-19 and Time histories ofAand7sin7=--_ 6._-20V canbecomputed fromequations (6.4-19) and (6.4-20) iftime histories ofVXg,Vyg, and VZg areknown. Fromequations (b.h-1), (6.4-_), (6.4-7), and(6.4-10) wecanwrite gj wJ6.4-21 which permits thecomputation oftheneeded __VXg,Vv,andVZ. "_ g fromViscomputed v-"_/Vxg2+Vyg2+Vzg2 6._-22 285< 6-50 Insun_a_y, thegeographical setofaxes wasdefined inorder to si'eci_' theattitude ofthemissile with respect toastabilized platform. Theattitude isgiven bytime histories ofAand 7-Thespecification ofmoving earth axes Xe,Ye,Zepermits thecomputation oftJome histories ofaltitude, latitude, andlongitude. Thelocal geographical axesarethe same astheearth axes defined inASAYlO. 7-195_. b.h.9 True weight component.- From Figure 6-6c itcanbeseethat thetrue weight component Wacts along the Zgaxis. Thetrue weight components inbody axes maythus begiven by w_ZII112113 mllm12ml3 nlln12nl3-WcosLcos(h+H) WcosLsin(k-H) -WsinL6.4-23 where thedirection cosines llj,mlj,andnlj aredefined insection 6.2 equation (6.2-21). 6.h,6 Definition ofV,,_and _.- Thevelocity vector Vawhich isused todetermine aerodynamic forces andmoments istheresultant ofthree vectors Va=v-Ve-vw 6._-2_ n whereVisthevelocity vectorofthemissile centerofgravity referred toinertia axes,VWisanywindvelocity vector andVeisthevelocity component ofaparticle ofairduetotheearth's rotational velocity. 6-91 Ve=_XR= or Ifthecomponents of then Wecanexpress %'as andfromequation (6.4-26) thereforeijk 00_e x_xNZN ve--i_YN+J_x_ are =d m Va= iua+Jva+kwa V=iu+Jv+kw ve--i(_eY.)+J(_ex.) u_-u +_YN-"W va=v-_eXN-vw Wa=W -WW Theangleofattack aandsideslip6.#-29 6.#-26 W-aasshown inFigure 6-5 6.#-27 6.#-28 6.#-26 6.#-29 aredefined byFigure 6-9.It isassumed that themissile sideslips first then performs anangle ofattack. u_cos_= 6.4-50 vacos wasin_= 6.#-31 va_osFrom Figure 6-9itisseen "_S'7< 6-5z V sin_--_ 6.4-32 Va or =tan"Iw--%a 6._-33 ua =sln-Iv.__a 6.1_-_ Va 6._.7 An_ular velocityrates.-Ifwedesignate Pa,qaand ra astheangularvelocity terms which cause theaerodynamic moments about themissile center ofgravity andnote that m a_-iPa+Jqa+kra and _-ip+Jq+kr6.4-35 "then -r-6.4-36 where _X___ isacorrectionduetothecurvature oftheflight path. R2 This correction issmall andtherefore usually omitted. Thedensity 0isafunction ofaltitude handwould havetobe known toperformacomputation. Thecomputation ofinitial conditions tostart theproblem isamajor task andwillnotbediscussed here. 6-55 CONCLUDINOREMARKS This hasbeen abrief andrapid introduction ofthe equations of motion =.ndtrajectories ofarigid finstabilized missile with variablemass forthe genera_ case ofall sixdegrees offreedom. Iftime and space permitted, much more could besaid about reducing thegeneral sixdegrees of freedom totwoorthree degrees offreedom and simplifying byother techniques such asroll andyawstabilization. The equations ofmotion could also bepresented referred towind axesandstability axes, and compared with those usually found inthe literature. Itshould benoted further that there arealso spin stabilized missiles which entail other effects notdescribed here such asmagnus effects, gyroscopic effects, aerodynamic effectsduetospin and cross-spin, etc. This isanother subject initself, andisnotdealt with inthesenotes. AC)._ - APPENDIX 6-A Skm_uaryofFormulas Pertaining totheDevelopment From thegeometryofFigure 6-1_see theangular velocities about the body axes eregiven by p__-_sine q=ecos_+_sin_cos e r=_cosecos_-_sin6.A-I Equation (6.A-I) canalso bewritten as 1(qsin$+rcos$) e=qcos_-rsin 6.A-2 =p+tane(qsin$+rcos$) Theangular velocities about theinertial axes _XN._,.,=d'_NareXN,YN,ZN,denoted here by O_XN=_cos$cose-$sin$ _N=_sln Vcos e+_cos, O_N- ,-_sine6.A-3 Thetransformation from body axes Xb,Yb,Zbtoinertial axes XN,YN,ZN isgiven byequation (6.2-IG) which is 2SO< 6-33 XNIiimllnll Xb YN=_12m12n12 Yb ZN _13m13n13 Zh6.A-_ Thedirection cosines is where11J, mlJ, andnlJ aredefined inequation (6.2-21) Avectorquantity referred toXN,YN, ZNaxissystem denoted by(=) (=)=()XNiN+()YNJN+()ZNkN Avector quantity referred tothe xb,Yb, zb6.A-9 ()xN=zn()xb+mzz()Yb+nn()zb ()YN=zI2()Xb+m12 ()Yb+n12 ()zb 6.A-6 ()z_=h3()_+mz3()Yb+n(),_ axissystemdenoted by (')=()xbib+( )Yb4+()zbkb ()_=hz()xN+zz2()YN+h5()z_(-)is where ()Yb=mzz()xN+mz2()YN+ram3()z_ ()zb=nll ()XN+n12()ZN+n15()ZN Integration ofequation (6.A-2) yields6.A-7 6.A-8 _ot e=;0t =;0t_t÷_(o) dt÷e(o) 6.A=9 6-57 REFERERCES i.Sokolnikoff, I.S.,andSokolnikoff,E.S.:HigherMathematics for Engineers andPhysicists, McGraw-Hill Book Co., Inc., 1941. 2.Frazer, R.A.,Duncan,W.J.,andCollar, A.R.:Elementary Matrices TheMacMillan Co., 19_6. 3.Doolin, Brian F.:TheApplication ofMatrix Methods toCoordinate Transformations Occurring inSystems StudiesInvolving Large MotionofAircraft. NACA TN3968, 1997. 4.Perkins, Courtland D.,andHage, Robert E.: Aeroplane Performance Stability andControl. John Wiley andSons, Inc., 1949. 9.Charters, A.C.:TheLinearizedEquationsofMotion Underlying the DynamicStabilityofAircraft, Spinning Projectiles andSy_netrical Missiles, NACA TN3350, 1999. 6.Rankin, R.A.:TheMathematical Theory oftheMotion ofRotated andUnrotated Rockets, 19_9. Library Number N-_7_9 Phil. Trans. Series AVol. 241, No._37. 7.Chang, T.:General EquationsofMotion of"aRigid Missile. Cornell Aeronautical Laboratory, Inc., 1992. 8.Rosser, J.B.,Newton, R.R.,andGross, G.L.:Mathematical Theory ofRocket Flight, McGraw-Hill Book Co., Inc., 19_7. 9.Anonymous: AmericanStandary LetterSymbols forAeronautical Sciences, TheAmerican Society ofMechanical Engineers, 1954. ASAY10.7-195_. lO. Kooy, J.M.J.:OntheApplicationoftheMethod ofVariationof Elliptic Orbit Elements inCaseofaSatelliteVehicle. Astronautica Acta 3(3) P.179-21_, 1951. ZP Z Nol•••• e _--?coseco,¢),.-_s,,_¢ F#GURE {o-I.-OfllF-_TATION OF_OOY AxIs WITH RESPECT" NEWTOHIAN •oRINERTIAL :OR_PACE AXIS. T.!OROjR oFRo'rA'rloN _sY:pe,A.o(_."To \ \ \tu I-- ). v) ¢) Q (Jr) .,j o I-- o 0 uj I ! tU U. / // NOZZLE EXIT // -V_=,.TeTExITVELOc,TY RELATIVE TON.OZZLE E_',T 2_6(- \\\ ,TET I Zb_oz_l.E FIGURE Co-4.-'l'YrlcA LJETv__R_N__ LoOHIN C_FORWAKo XS COSOC = I TAN OC= sIN.J3 ="VV'Q V'o v. FIGURIE _-5,,-DEFINITION OFOC_NO # e Z X Ze X_ Z_ GEoGRAPHICAl" S_T$ oFAXES'. VARIouS 5_'S OFAxEs USED- v X9 Y, (c)9 g z Xgb _EC'GhAP_IcA/- AHo _OoY SETS o_AXE5 FIc,u_E(_-{o.-C _.cLuoeoXb 800< SECTIONVII INERTIAL SPACE NAVIGATION 7.0Introduction Inthepast fewyears increasing reference hasbeen made toanew type ofnavigation which hasbeen termed Inertial Navigation. Thepurpose ofthis section istogive abrief introduction anddiscussion ofsome ofthefundamentalsofinertial navigation. Thematerial presented is derived from thevarious references which arelisted attheendofthe text. Inmany cases thetext andtheFigures presented aretaken directly from these references. Although theconcepts oiInertial Navigation arerelatively newthe principles upon which theprocess isbased have been well known foryears andessentially itisasimple application ofapplied mechanics. Inorder tolearn something ofthesubject itwill beapproached from thestandpoint oftrying toanswer thequestions: A.What isit? B.Whydoweneed it? CoHowdoes itwork? D.Howwell does itwork? 7.iWhat isInertial Navigation? OnedefinitionofInertial Navigation hasbeen given as: Navigation without theuseofar_radiation aeither natural ormanmade. --inother words itisself contained. This definition isofarather negative nature that m_rely gives oneoftheattributes ofaninertial navigation system. Amore descriptive definition would be: Aprocess inwhich determination ofnavigational parameters with respect tothefixed stars ismade from 8,0!< 7-9- me_.surem_nts ofaccelleration actingonthebody. Ifthenavigation is concernedwith _osition with respect tothe Earth (ortoanother rotating pl%net) theprocess would also includeaconversion tonavigational param- eters referenced tothemoving earth. Aninertial navigation system is then onewhich isselfcontained and onethat byaprocess ofintegr_tion ofimposedaccelerations determineschanges invelocity and position. 7.2 WhV doweNeed Inert_a] Navigation? The present methods ofm%vigation areconsidered fairly accurate and wehave somsmy means ofnavigating that ttishard foreven the navigators tokeep upwith them. Airplanes cantm/_e offand flyhalf way around the Earth and lsnd onaparticular runway ataparticular airport. Bombers cmnflyover adesired target area with good precision even inconditions ofbadweather. Thecomparatively newdevelopments inOmni DME, Loran, r_Aar n__v_gation symte,_, etc. are successful totheextent that additional newmethods ofn_vigation must bequestioned astotheir usefulness. Inert%al n_vigation does promise tofill certsin needs that arenot adequately met byexisting systems. Theadvantages ofinertial n_vlgatlon systems canb_placed into two categories: military andcivil. Military advantages : (i) Does notdepend onground facilities forreference (2) Isnotsubject toJemming --Newton's l_ws ofmotion and gravitation aredifficult totamper with (5) F_titsnoradiation that canbedetected byanenemy (L) Byvirtue ofitsindependence ofoutside signals there isno limit astohowmany systems canb_utilized simultaneously. 802< 7-3 Civilapplication advantages: (i)Could possibly beused tosimplify theexisting complex network ofnavigational facilities (2) Would allow continued operation ofourcommercial transport fleet intimes where enen_y attack may bethreatened the existing navigational aids would have tobeturned off toprevent homing inbyenen_y /4!:tn_e roa3_n_: areinaddition tothe obvious advantage inapplica- tion toouter space flight where most ofthepresent navigation schemes have nomeaning. 7-3HowDoes Navigation Work? 7.3.i Basic principle.- Inertial navigation isarather fundamental application ofclassical mechanics whereby thepostion orchange inposition ofabody isdetermined from measurements ofaccelerations. Itisan application then ofNewton's Laws ofMotion. Inaddition there are other principles crfacts that have important bearing, such as: (I) Newton's Law ofgravitation which states that every mass particle attracts every. other mass particle with aforce proportional totheproduct oftheir mas_es andinversely proportional tothesquare ofthedistance between theparticles; (2)theprinciple ofequivalence inthe general theory of re!atlv_ty that says gravitational mass andinertial mass areequivalent; (3)thespatial direction oftheEarth's gravitational field atanypoint serves asaunique identification ofthat point. This latter point requires further discussion. Aswill bepointed out inSECTION XIV theshape oftheEarth isnotaperfect sphere, but itcan 7-4 beapproximated byHayford's Spheroid of1909 which hasanellipticity of 1/297. Because ofthis shape thenormal tothegravity field does not _!vays he,d directly tothecenter ofthe Earth. For example at&,5°latitude there _sabout llminutes ofarcbetween theplumb bobvertical and the true line tothe center oftheEarth. Shown onFigure 7-1isthe geoid zurface which isbydefinition anequipotential surface oftheEarth's gravity fie_.d. Thedirection ofthe groodient ofthe gravity potential atthe _n_fa_c r,fthegecid, theforce ofgravity, isdefined asthevertical. This Izdef_n-d byaplumb bobwith itsbase fixed with respect tothe zurf_ce ofthe Esrth. The specific force ofgravity _sthen avector addition ofthe _r_v!t_-tion sp_clflc force andthe centrifugal specific force assoc!_ted with daily rotation. Because thegeoid does nothave a smooth s_face, thevertical isnot ingeneral parallel tothenormal to thereference ellipsoid atthe same position. Theangular deviation, called the station error imgenerally less than onesecond offarc. 7.3.2 Coordinat_ s.v,_tems.- Theuniqueness ofthevertical atshy point isthebasis ofastronomical position. Theastronomical latitude is the complement between allne parallel totheEarth's polar axis andthe local gravity vector. The astrono_dcal longitude isthetingle about the Earth's polar axis between areference vertical (usually that atGreenwich) and thelocal vertical. Theastronomical set ofcoordinates arevery useful ininertial navigation but unfortun_.tely accelerometers measure with respect toinertial sp...ce sothat oth_rcoordinate systems arealso neeessery. Shown inF_gure 7-2areth_G-o_entrlc Inertial coordinate systemwhich iscentered intheearthwith the ax_s coincident with theEarth's polar ,gxis. Also 7-5 shown inFigure 7-2aretheGeocentrlc E_rth reference coordinates in which the Zaxis isagain coincident with thepolar axis butthe Xand Yaxis arefixed intheEarth andthus rotate around thepolar orZaxis attherate of19°perhour. Another useful coordinate system isshown inFigure 7-3andisreferred toasthelocal geographic reference coordinate. Inthis reference system there istheoption ofalining oneoftheaxes along agreat circle which could include thedeparture point anddestination and thus would simplify thenav_gation problem. Thesignificant difference between thegeographic coordinate ofFigure 7-5andthea_tronautical coordinate system isthat thelatter hasthe Zaxis alined with thelocal vertical rather than normal totheellipsoid. Knowing something ofthenature ofthegravity field about theEarth andthevarious coordinate systems which areuseful itiswell toreturn tothequestion ofhowdoes Inertial Navigation work. 7.3.5 Simplified example.- Asasimple example consider acart ona table top. Thecart isinitially atrest andaforce isapplied tomove it along thetable tog. Itsposition atarAytime canbedetermined bymeasureing anddoubly integrating theapplied acceleration. Mechanizing this simple problem brings outmany ofthesignificant features ofinertial navigation. First ofal]since acceleration isavector quantity accelerometers m_st bemcunted onthecart soastosense thecomponents oftheacceleration with respect tothecoordinate system within which themeasurement of position istobemade. Each acce]erometer must beaccurate endalso must be maS_ntalned inaprecisely kn_nn relationship tothecoordinate system this latter requirement g_Ives rise totheneed forastabilized mount for 7-6 theaccelerometers that will maintain thedesired orientation. Because an accelerometer cannot distinguish between acceleration andgravity either thestable mount must beoriented sothat gravity components arenotsensed orelse corrections must beadded toaccount forthis factor. Forsuch asimple example thecomponents ofaninertia] navigation system could besplit into three groups: (I)accelerometer package tosense accelerations wi_n respect totheaxis ofthechosen coordinate systems (2)astable platform (which normally uses integrating rate gyros asits primary instruments) that will maintain theorientation oftheaccelerometer package and(3)acomputer that will doubly integrate theoutputs ofthe accelerometers andputinsuitable corrections sothat position will be _nown. When therange over which thecart onthe"table top" travels becomes very large theproblem becomes more complicated because thecomponents of gravity which must beaccounted forbecome large. Inthesketch ofFigure 7-4which shows the Xcoordinate tangent tothesurface oftheEarth it canbeseen that asXbecomes large with respect totheradius ofthe Earth theforce ofgravity (which effectively lies along theline from the center oftheEarth totheposition ofthecart) tend tobecome alined along the Xaxis. Theacceleration sensed bythe Xaccelerometer is e. Ax=X+g x isthecomponentofgravitational force. This component maybewhere gx expressed gx=goa2x (a+h)3 3C6< 7m'_ • Ifiti_as_Ltm_d +hat there is.%neg!Igib]e _h_.nge inaltitude the e.vpr_s-ion may besimplified to X gx.=go- a an!_t!]! _rov_de anadequate approximation for gxforrelatively s_.!! _h_nge_ inX. 7.3.4 Schuler tuned pendulum.- Even though itis-_ithin the capabi!itv ofacomputer tocalculate gxforlarge changes inXthis operation places rather stringent requirements onthecomputer. Another, perhaps morereasonab_e, approach istor_intain theorientation ofthestable platform upon which the accelerometer ismounted sothat itwill benormal totheforce ofgravity an! th_ theaccelerometers would not sense the components ofgravity. This could bedone bymounting apendulum onthe stable table sothat as thependulum alined _tselfwith thegravity forces thetable would be alined perpendicular tothependulum. This problem was first approached byDr. Maxmilian Schuler, aGerman scientist andprofessor anditwas he who first pointed outthat what wasneededwas apendulum with vertical deterndnation characteristics independent ofvehicle movement. Asimple pendulum having areasonably long armwould besubject todisturbances away from the vertical, ifitsbasewere accelerated. Tobefree ofsuch errors the length ofthependulum armwould have tobeequal totheradius ofthe Earth. With such apendulum thepoint ofsuspension orbase ofthe pendulum could bemoved about over thesurface oftheEarth without dis- turblng thependulum mass andthus thependulum would always indicate the vertical. Such apendulum is, ofcourse, impossible tobuild andeven a distributed mass type ofpendulum having the same dynamic characteristics 7-8 would bevirtually impossible tobuild because oftheextremely small distances required between thecentroidofmassandthepivot point. Because thependulum isessentially asecond orderundamped system itispossible toconstruct aservosystem having thesamedynamic characteristics. Theperiod ofapendulum isdetermined from the formula T=2_-_ where Lis thelengthofthependulum andgthegravitational acceleration. TheSchuler pendulum hasaperiod of84.4minutes. Asimplified schematic oftheoperation ofaservo system SchulerPendulum is shown inthefollowingdiagram: X XT %T"angular rotation ofthetableupon which theaccelerometer ismounted 9x-angular arcovertheEarth's surface covered bythexmovement ofthetable -error inalinement oftable normal totruevertical 0 E 3C8< ?-9 Theacceler_aeter ismountedonatilting platform whose angle oftilt is deterz_ned byamotor which isdriven bythedoubly integrated output of theaccelerometer times theconstant K. Thetransfer function relating Xtotheapplied acceleration hasacharacteristic equation ofthe form p2+K=0which indicate anundamped second order system having a frequency ofoscillation equal to K. IfKwere adjusted toequal L theoscillatory characteristics would bethesame asaSchuler pendulum _dwould besaid tobe"Schuler tuned".Withsuch asystem the tilt angle oftheplatform becomes equal totheangular arcthat theplatform has t:'aversed over theEarth's surface. Thus ifthere were aws_ to"remember" theo[".ent.ation ofthevertical atthe starting point theposition ofthe :_'_,_r:_ couldbeobtained bymeasuring theangle between the instantaneous vertical andthevertical atthestart oftheproblem. This amgular position cou!d _asily berelated tolatitude andlongitude angles byalining the Xand Yaccelerometers along andperpendicular tothemeridian lines of theEarth. Thus there aremeans ofdetermining position both analytically bysuitable computer operations onthedouble integral ofacceleration and _eon_trically (orpartly geometrically andpartly analytically) bymeasuring the_::-_.,-ntation ofthevertical with respect tothereference coordinate syste. 7.3.5Hardware components.- Tofully understand themechanics of inertial navigation itisdesirable tounderstand theoperationofsome of +_he _,aslc component_ --the accelerometers, rate gyros, stable platforms, CO_IpU+eZ'S _etc. 7-10 Figure"_-5isaschematicdrawing oftheHIGtype, slngle-de_ree- offreedom gyroscopeoriginally developed bytheInstrumentation Laboratory, MIT. Basically, thegyro consists ofaspinning wheel driven byanelectric motorj mounted onpreloaded ball bearingsandcontained in ahermetically sealedcanorfloat with shaftextensions. Thefloatis completelysubmerged inaviscous fluid whichhas thesameaveragedensity asthefloat andshaft.This serves toreduce frictionabout theaxis defined bythepivotsdesign.Coaxial with theshaftisasignal generator which givesavoltage proportional totheangular displacement ofthe float relative tothecase_andatorque generatorwhichcanbeused toapply torque tothefloat. Thebasic principle ofoperation ofthegyro canbeexplainedin termsofthethree axes shown inFigure 7-5.The spinreference axis lies along theangular momentum vector(spinaxis) ofthewheel when thesignal 6eneratoroutput iszero.Theoutputaxisisnormal tothespinreference _w.is, andisthe axisaboutwhich thefloat isfree toturn.Th3 iny,:t _xis isnormal totheoutput axis andspin reference axis.Theinput quantity toth.- inqtrument isanangularmotion ofthecase, relative to inertialspace_about theinputaxis.The resulting output isamovement ofthefloatrelative tothecasewhich results inavoltage from the signal generator. Taeoperation isexplalned bythefamiliar physical fact that when atorqueisapplied toasplnnlng wheel soastochange thed/rection ofitsspin axis_ thespinaxis tends toalign itself with thetorquevector. Conversely, when theaxis ofaspinninE wheelisforcibly precessedorrotated_ thewheel through itsbearings exertsatorque aboutanaxis perpendicular to 7-ii theaxisofforcedrotation.IntheHIGgyro, movement ofthecaseabout theinputaxis causes aforceprecessionoftheg_TOwheel about this axis. Thegyrowheel thusexertsatorque onthefloatabout theoutputaxis. Initially_ this torqueaccelerates thefloat, butasthefloat gains angular velocity, theviscous she__torque reduces theacceleration tozero andthe float reaches asteadyangular velocity.Thegyro torque isthen balanced bytheviscous shear torqueandtheoutput angular rate isproportional to theinput an_,±_r rate.Thefact that theoutput signal isproportional totheinte6ra3oftheinput angular rate isthesource oftheterm " "Inte_'ating gy_'o".TheFjro thus serves asanattitude reference. HIGgyros act_u_precision angularmotion sensors, rather than sources oftorques toovercome friction orunbalances.Normally, they have an operating rangeofonlyaf_de_reesardtoprevent various cross coupling errors, inputa_les should bekept small.Therefore, forinertial guidance use, gyros areusually mountedonaplatform orbase which isservo-driven tomaintain thegyro outp, _J.__near anull. Theplatform thus remains fixed inorientation relative toinertial space, orisrotatedatarate _.t_r_ined bytorque generator input. Amuch simplified sk.'_tch oftheone degree offreedom stabilized platform issh_.-aiuFi_a_'e 7-6. Atypical three degreeoffreedom stabilized platform configuration isshown in Figure 7-7. Figure7-8isaschematic ofapractical type ofaccelerometer instrument based ontheHIGgyro construction. Theseismic mass exists in theform ofape_iu1_:: andtheforce generator isthetorque generator. The pickoff isthesignal gener___or. Flotation virtuallyeliminatesuncertainty 7-12 friction torques atthepivots. Because ofthependulous mass, acceleration oftheinstrument along theinput axi_ creates atorqueabout thefloat pivot axis. This torque causes rotation ofthefloat andaconsequent signal generator voltageproportional to8o.This voltage isused to generate acurrent which isappliedtothetorque generator togive a torque which "constrains" thependulumandkeeps 90small. Thecurrent,I, isthus proportional toacceleration along theinput axis. Thegain ofthe feedback systemmustbekept quite highsothatdeflection ofthependulum under high input acceleration issmall. Otherwise, a"cross talk"torque isdeveloped which isproportional totheproduct oftheacceleration along thependulous reference axis andthesine ofthedeflection angle 0o. Inztr _ents ofthis type, called force feedback pendul_ns orconstrained pendulums, areavailable commercially. They aremade with awide variety ofdynamic range andfrequency response. Velocity andposition arethequantities ofinterest innavigation, rather tl_an acceleration. Increased accuracyandreliability maysometimes beobtained byperforming integration intheaccelerometer. Basically, this isdonebymaking theforce acting ontheseismic mass proportional toa rate ofsomekind. Forinstance_ thecurrent fedtothetorque generator in Figure 7-8_Ightbeapplied inpulsesofconstant area butvariable rate. Pulse rate isthen proportional toacceleration andtotal number ofpulses tovelocity. Another usefuldevice _oneusedbytheGermans intheV-2 isth_pendulousgyro accelercmeter (PGA).This instrument i8apendulum inwhich theforce generator isagyroscopic element. Figure 7-9isa 7-13 schematic ofsuchanInstrument based ontheHIGgyro construction. The output ofthesignal generator isfedtotheservomotor which rotates the gyro case about theinput axis atarate sucL that thetorquedeveloped bythegyro ele_ent Justequals thepe_lulous tarque. Theangular rate ofthegyro case isthus proportional toacceleration andthetotal angle turned bythegyro case isproportional tovelocity. Aswith theinstru- ment ofFigure 7-8, thegain ofthefeedback loop must bekept high so that. 8isverysmall. Thetorque generator ofthegyro canbeused to apply additional torques tothegyro float which addtothepe_lulous torque. Inthis w_y, gravity maybea_ed tothethrustacceleration to give true acceleration andvelocity. 7.3.6 Typical configurations.- Asisusually thecasewhenengineers areallowed some freedom indevelopment oi"asystem todoaparticular Job there areanumber ofconfigurations ofinertialnavigation systems in operation orinthedesign stage. Figures 7-10, 7-11, and7-12 present three basic configurations ofinertialnavigation systems. These systems differaulte markedly _nthe__A+n__÷_D_,,+_ ..,_+ store÷_ reference coordinate systemanddetermine thedesirednavigational data. Figure 7-10 presents athree gimbal system inwhich theaccelerometers aremounted onthes_me platform astherate gyros which aretheheart of thestable table. Inthis case thetable isoperated alatheSchuler tun_ pendulum sothat oneaxis tracks thelocal vertical --thereference coordinate systems being store_ analytically inthecomputer. Thestable platform instr_,_nts theastronautic reference coordinate system by prec_sslng thegyros.Navigation data isobtained assignals representing 7-14 veloclty oraccelerationmeasurements relative tothegeocentric inertial coordinate system ofFigure 7-2andconverted into Earthdata bycomputer operations that convert thesignals into thegeocentric Earth coordinate ofFigure 7-2. Figure 7-11 shows afive gimbal system inwhich theplatform onwhich theacceleremeter aremounted isSchulertuned totrack thelocal vertical andinstruments theastronautical reference system. Thestable platform with therate gyros isstabilized ininertial spaceandthus instrument the geocentric inertial coordinate system. This configuration allowsdirect an_lar measures todetermine positions ontheEarth's surface relative totheastronautical coordinate system. Thecc_figuration shown inFigure 7-12 isalsoathree gimbal system where theaccelercmeters aremounted onthestable table which ismaintained ininertial space.Theplatform instruments only thegeocentric inertial coordinate system andnavigationdata isobtained from computer operations. 7.4HowWellDoes anInertialNavigationSystemWork Aninherent disadvantage ofaninertial navigationsystem isthat the errors intheincrease withtheproblem time.Determini_position by doubly inte_ratiltE anacceleration measurement causes anyerror inmeasure- menttoshowupasasecond power function oftime.Because ofthis time dependency oftheerrors theaccuracy requirementoftheaccelercmeters and therategyrosusedtoprovide stable table operation aremuch morestringent thanhasbeen required ofsuch instruments inprevious mechanisms. For 7-15 anIC_ typeofoperation theaccuracy requirementsofaccelerometers and rate gyros foreach i000 foot oftolerableerror aregiven asafunction ofrange inthegraph presented inFigure 7-13. Thefact that theoperation ofinertialnavigation systems (which operates independent ofoutside signals)arebaseduponnonvariantphenomena (Newton'slaws ofmotions,etc) isanadvantage inthat anyincrease in accuracy ofthesystem components results inincreased accuracyofthe overallsystem. Using theprinciples oftheSchulerpendulum enables thesystemerrors tobelimited toasomewhat restricted type ofoscillatory buildup --at least ontheaxeswith respect towhich thependulum canbeutilized.In this case theamplitude oftheerror oscillation would bedependentupona combinationofsuch factors asinitialmisalinement ofthependulum, accelerometer bias signals and"cross talk" between acceleration components along theotheraxes causedbymisalinement ofthestable table.Because theangulardriftofthestable table isafunction oftime theoscillating limits ofthecomputederrors ofeven aSchuler tuned system increase with time.Theprinciple ofaSchuler tuned systemhasnoapplication to thevertical axis (altitudeaxis) andthus theerrors inaltitude measurement would show aparabolic variationwithtime.These errors inaltitude measure- mentwouldalsoaffect themeasurements along theother axeswhere the Schuler tuned principle isapplied,because theerror inaltitudewould beanerror intheeffective length ofthePendulum andwould cause the period ofoscillation tobeinerror ascompared toaSchuler tuned system. This difference indynamicswould cause additional errors inthemeasurement ofacceleration duetosensing gravity components. 7-16 REFERENCES i.Wrigley,Walter,Woo_bury, Roger B.,an_Hovorka,John:Inertial Guidance. XASPreprint 698_ present_Jan.51,1997. 2.Slater_ J.M._an_Duncan_D.B.:InertialNavigation. Aeronautical Engineering Revlew_Vol15_No.i_Jan. 1956. 5.Klass, Phillip J.:Inertial Guidance. Aviation Week Special Report, Vol6_Nos. 1-4,Copyright 1996. 4.Russell_WilliamT.:Inertial Guidance forRocket-Propelle_ Missiles. JetPropulsion,Vol28No.1,Jan. 19_8. 5.Goetz,ErnestA.:ICBMInertialGuidance. Astronautlcs_ M_71998. 6.Duncan_D.B.:Analysis ofanInertial GuidanceSystem.JetPropulsion, Vol28,No.2,Feb. 1958. vl POLAR AXI| NORMAL TO ELrIPSOID OCEAN BASIN GRAVITY (NORMAL TOGEOID) ACTUAL SURFACE OFEARTH R,EFER_NCE ELLIPSOID OEOID MOUNTAINS Fin."YIP=l,.,tln,,,=k:,, ,-,1'ok...._,,I*.... t....... II:-.-:J 5.7< /£ ZIPOLAR AXIS E_JATOII NOTE:NIE EARTHAXESROTATE ONCE PERSIDEREAL DAYRELATIVE TOINERTIALAX_S•.". NOTE: ZE:ISALONG POLAR AXIS Xs,YEAREARBI- TRARILY ORIENTED INEQUATORIAL PLANE ANDFIXEDINTHE EARTH Geocentric Earthreference coordinate frame. (FIXED) ZE X__ER_IAN ATMAXIMUM LATITUDE ZCLOCU S iii::.' /_ Y¢_CIRCLE ,c_._........... ..::_:12/ "_ X_,'_" """-_-_'/_/ NOTE:REFERENCE XoISCOPLANAR WITH E_ ZGATMAXIMUM LATITUDE ANDZE XGISPARALLEL TOITSELF OVER GUIDANCE PATH;GENERATES CYLINDRICAL SURFACE. Fig._,Localgeographic reference frame,greatcirclegrid. // 1 V,%v_ 7.4 SPINMOTOR SKmmL .....GS_DtATOR OUTPUTmaS u_trf_os Fig.l.S HIGgyroscope _mplifier Commond inputs }ss :-Plotform "Geortrim /_"Bose GyrOcase "Servomotor HIGGyrostabilized- plotform STABILIZED platform (single axis)em- ploying anintegrating (HIG) g.v1"ol ,-/. YAW AJUS Fig._ StebiUzed plstform / _7.7 _.0< OUTPUT SIGNAL GENERATOR GENERATO_ Ig U_ Fig8Constrained pendulumacceIerometer JANUARY 1958 °Y'°_ •o _._ SYNC_., o REF. e_ FIfo 7._ Pendulousgyroaccelerometer (PGA) t_ ACCELEROMETERS GYROS ACCELEROMETER AND GYROPACKAGE; CONTROLLED MEMBER; INNERMOST GIMBAL ISOLATION GIMBALS BASE _INDICATED VERTICAL _¢e.7,IOAccelerometer pockage andgyropockage mounted together (3gimbals). INDICATED POLAR AXIS ACCELEROMETER PACKAGE \INDICATED VERTICAL. LOCAL EAST SIDEREAL DRIVE GREENWICH EASTAXIS (REFERENCE r/_// Accelerometer package indicates vertical. Gvropackage indicates inertial specc (polaraxis). ACCILIROMETERAHDOYllOPACKAGE BASE /_l_-_.'/,I_Accelerometer package fixedtogyropackage, whichisinertial. ACCURACY REQUIREMENTS FOR GUID- ANCE SYSTEM GYROS ANDACCELEROM- m 9o_ 0 °' 5ETERS Fo_logo'IT..RKo_ I | i I ! I000 2000 3000 4000 5000 RANGE (MILES) 8-1 SECTIONVlll GUIDANCE ANDCONTROL OFSPACEV_ICLES 8,0 Introduction Guidance asused herein isconcerned with theobtaining ofinput information required toachieve adesired trajectory (asrelated to space flight), andflight control isconcerned with thedetailed control oftheattitude andvelocity (both direction andmagnitude) ofthespace craft forthesame purpose. Guidance andflight control systems have re_ched astate ofdevelopment where they appear capable from component performance andaccuracy standpoints ofuseinmany types ofspace operations. Inspite ofthis apparent capability, much additional research anddevelopment effort will berequired inthis area before anys_t of routine space operations arepossible. This situation ischiefly aresult oftherather extreme complexity ofcurrent high performance systems in relation tothereliability ofcomponents andthelong operating times required inmost space missions. Therefore, inregard tothesystems tobedescribed, theproblems primarily relate toimprovements inreliability andflexibility andreductions incomplexity, size, weight, andp_er requirements. Such improvements mayoccur incomponents, system configu- rations, orintheinvention ofentirely newguidance andcontrol concepts. Asthebasis ofdiscussion ofguidance andcontrol systems, wewill break down space missions into their various phases such aslaunching, orbiting, space trajectories, andre-entry. Foreach mission phase, an attempt will bemade todescribe theoperational concepts; theguidance andcontrol equipment used, contemplated orneeded; andtheproblem areas. 8-2 8.i Launching Ascurrently envisioned theearthlaunchphaseofmostspace operations willcloselyresemble thelaunchoperation ofballistic missiles. Atypicallaunchtrajectory isshowninfigure8-1.Amulti-staged vehiclewillliftoffapproximately vertically andwillbestabilized in theattitude throughuseofanautomatic flightcontrol. Afteraperiod ofvertical fliehtthemissilewillperformanunguided gravityturnby programming themissiletostabilize attiltedattitudes maintaining nearzeroan_leofattack. Therealsomaybeperiodsofunguided coast- ingfollowing burnoutofthevariousstages. Active_uidance willtake placeinconnection withintermediate stagesandperhapsinconnection withthefinalstage,although insomecurrentsatellite vehicles the finalstagesareunguided andstabilized throughspinning. Themain purposeofthe_guidance istocontrolaccurately thedirection ofthe velocity vectoratburnoutofthelastguidedstage.Forunguided final stagesitisnecessary alsothattheattitude ofstagebecloselycontrolled atfirincandregulated duringburning. Anotherimportant guidance function istoaccurately controlthevelocity cutoffofthefinalstage.Asome- whatlesscritical guidance function isthecontrolofthespacialor _eogrschic position ofthevehicleatfinalstagecutoff. Anumberofdistinct typesof_uidance andflightcontrolequipment _recurrently usedintheperformance ofthefunctions Justdiscussed. Thecrimarytypesinvolveeitherinertial sensing_electromagnetic tracking. Practically alltheattitude systemsusedforlaunchcontrol incorporate inertial sensors(gyroscopes andlinearaccelerometers). 3Z6< 8-3 hblock diagram ofatypical attiutde control system ispresented in figure 8-2. Inanelementary system, theattitude stabilization may beobtained from three single degrees offreedom integrating rate gyros, orthogonally mounted onthebody. Ahigh gain control loop is used toaccurately slave themissile attitude tothegyro references in order toavoid gyro coupling effects. Infact, less than tendegrees of gyro gimbal freedom isusually provided. Thechief problem associated with these high gain attiutde control systems concerns theavoidance ofdynamic instabilities resulting from coupling between thecontrol system andthestructural orfuel sloshing molions. Inmost cases thebooster configuration iscontrolled byJet deflection andthecontrol _ains must besethigh enough tostabilize the system under maximum qconditions (where theconfiguration, saerodynamic instability isgreatest). This requirement results inanexcess in stability near lift offandtheshort period frequency under this condition isfairly high (say 1cycle persecond). Unfortunately, thefull fuel configuration results inthelowest frequency condition ofthestructural modes, andthefundamental fuselage bending mode mayhave afrequency ofjust afewcycles asecond. Anexample ofthebending modes andtheir frequency distribution issho_n infigure 8-3. Thefrequency ofthefuel motion inthemissile maybeeven lower, andthis motion also becomes critical some time after lift off. Inmost liquid fuel rockets ithas been necessary tousesome baffling inorder toprovide damping tcthe fuel motions. Iftheopen loop frequency response ofthecontrol system 8-4 issufficiently high itispossible toadddamping tothelowest frequency modes bycontrol action. This approach requires careful choice ofsensor location along thebody inorder toobtain proper phasing and/or requires theuseofelectronic shaping networks forthesame purpose. (See figure 8-2.) These techniquesarenotthecomplete solution totheproblem, however, because there arealways present higher frequency structural modes which maybeexcited bycontrol action. Infact, inonecase itwasnecessary toreduce thecontrol response bytheorder of1OOdecibels from its static value inorder toavoid trouble with amoderately high frequency mode (only afewoctaves above thefirst-bending mode). This reduction isordinarily accomplished byelectronic filters having avery sharp response cutoff orbytheuseofnotched filters ateach critical frequency. Oneproblem here isthat thefrequency andmode shape ofthese oscillations vary asthefuel isused. (See figure 8-2.) Deliberate useofcertain nonlinearities inthecontrol system issometimes helpful. Forexample, smallamounts offriction orhysteresis mayprevent thecontrol from responding tosmall-amplitude high-frequency signals generated within thestructure. Ofcourse thebasic control mode will have alimit cycle oscillation under such acondition, butthis oscillation maybesmall enough tobetolerable. Theuseoflarge solid propellants asfirst stage boosters would obviate thefuel sloshin_ probiem, andforthis andother reasons, they will undoubtedly beused. Thesesolid engines, however, also require 328< 8-5 solution ofcertain problems. These problems include thecontrol of thrust directionwithout high actuating forces (particularly friction) andtheaccurate control ofthrust cutoff. Arelated butsomewhatmore secondary problem isthat ofthrust modulation cfthese engines. Theattiutde control systems.just discussed (using body mounted gryos) canalso beused asaninertial guidance system forlaunching space vehicles. Theboosterswould bestabilized inroll andyawandthepitch gyro reference would beprogrammed sothat themissile attitudewould follow atrajectory ofthetype outlined previously. Ifthethrust also could be_ccurately controlled toadesired program this attitude systemwould beallthat isneeded toachieve adesired launch trajectory; however, because thethrust cannot beprogrammedwith therequired accuracy an integrating accelerometer ismountedonthebody tomeasure accelerations along thelongitudinal axis (mean thrust direction) andwith agravity correction established from thepitch attitude program thedirection or magnitude ofthevelocity vector canbedetermined. Guidance systems of this simple type have been used tolaunch satellites andcanbeused in other space operations where angle inaccuracies oftheorder ofone-half degree areallowed. Ifgreater precision isrequired from aninertial system, thethree gyros canbemounted onagimballed platform which isslaved tothe reference nulls ofthese gyros. Three orthogonally located integrating acceierometers c_nbemountedonthis platform toestablish thevelocitity. Thesuperior _ccurncy ofthis system results from thefact that the coordinates oftheplatform canbemaintainedwith greater accuracy _n 8-6 thoseofthemissile itself inaddition tothefact that allthree components ofaccelerations aremeasured. Thechief problems with inertial systems relate toinitialalign- ment (settin_ intheproper initial conditions), drift intheplatform orientation,andinaccuracies intheintegration oftheacceleration. Thelast twoproblems have received muchattention andcomponents have been developed with extremely _ood predicted precision. Theword "predicted" isused because oneofthere_l problems associated with this equipment isthat oftestin_ intheproper environment. Thehigh steady accelera- tions associated with thelaunch condition plus thevibration environment obviously introduce structural deflections, unbalance, etc. inexcess of those encountered inalaboratory environment. Ontheother hand, the extreme theoretical precision associated with thezerogenvironment of space flight cannot bechecked inaground based laboratory. Ashasbeen implied, theerrors ofaninertial guidance system increase with time offlight. Itispossible togetvelocity magnitude anddirection information from other types ofguidance systems wherein errors arepre- dominately afunction ofdistance. Into this category fall anumber of ground based equipment using thepropagation properties ofelectromagnetic waves. Perhaps thebest known deviceofthis class isthe conical scan tr_ckinz radar. Inthis system anindication oftheline ofsight tothe target isobtained asshown infigure 8-4.Thecenterofthelobes of transmitted energy ismade torotate inaconical pattern andthe direction tothetarget isdetermined bytherelative strengths ofthe < 8-7 return signal durin_ various portionsofthescan. Onthebasisofthis information thetracking head (antenna) isdriven tokeeppointing atthe target. Angle informationistaken right offthetracking head gimbals. Range information isobtained bydetermining thetime interval between thetransmitted pulses andtheir reflected return. Thechief problems with radar tracking systems areachievement of thedesired range capability andnoise intheangle tracking system. This noise problem isaggravatedinthelaunching problem because itis desired toobtain accurate velocity information andthis involves differentiation ofthebasic positional information. Totheradar the appearance ofthetarget somewhat resembles theappearanceofacrystal chandelier toeye.Slight motions cause thebright areas toshift around. Inthecase oftheradar thereisatendency totrack atrandom vsric_s o_rts ofthemissile. This condition isaggravated inthecase oftheconical scan radar bythefact that itdepends forangle information onthecomparison ofthereturn signalstrengths atslightly different times (scan positions). Ther2dsr, therefore, cannot distinguish between ageneral fading ofthereturn from adifference duetotheangle error. Fading atthescan frequency cancause theradar tomove entirelyoffthe target. Thefading effectorangle tracking canbeeliminated byusing a monopulse radar inwhich thereturn signalstrength comparison ismade between asingle pulse simultaneously lobed inslightly different directions. Thenoise duetoglint isstill present inthis system, however. Another 8-8 waytoimproveradar position information istousetwo(ormore) radars inconjunction with long andaccurately measured base lines• Inthis case thepositional information canbeobtained from therange measurements through useoftriangulation techniques. Forextremely large ranges the base lines must beproportionately larger tobeeffective. Forflights deep insolar space base lines between twoorthree 22,000 mile satellites have been suggested. Theproblem ofrange extension ofradars andother electromagnetic signalin_ devices islarge oneofincreasing thepeak transmitted power andincreasin_ thesignal gstherin_ capabilities ofthereceiver. The last requirement maydictate theuseofextremely large antennas (the size ofafootball field oreven larger). Range extension canalso be improved through useofabeacon (transponder) inthetarget. Useofa beacon mayalso improve theangle tracking accuracy. Another factor which mayaffect thepropogstion ofelectromagnetic radiationbetween theearth andvehicles inspace isthepresence ofthe inosphere. Itiswell known that this imnized layer reflects electromag- netic waves; however, itisthewriter's understanding that theionosphere will nothave much effect onhigh frequency waves• Refraction ofthe waves which result from this sourcewill produce smallangle tracking error s• Because during thelaunch phase theguidanceaccuracy ismost critical toVelocity errors, itisdesirable toobtain thevelocity information inamore direct manner than bythedifferentiation ofpositioK_ infcrmation. Doppler radar conaccomplish thi_ objective. Asitsnreimplie sits 8-9 operation dependsonameasurement ofthedoppler frequency shift between transmitted andreflected waves which results from thetargets velocity. The frequency shift asafraction ofthetransmitted frequency isdirectly proportioned tothetarget velocity radial tothetransmitter andinversely proportional tothevelocity ofpropogation oftheradiation. Because the velocity ofpropogation ofelectromagnetic radiation isextremely high (186,OOO miles per second) thefrequency shift must bemeasured with extreme accuracy. Ifthevelocity ofavehicle isdesired toonehundred feet per second thefrequency must bemeasured toroughly onepart inten million. Thus oneofthechief problemsassociated with the use ofdoppler systems for space flight istheachievement ofultra stable oscillators. Another problem connected with the useofdoppler techniques isthat only theradial component ofvelocity ismeasured@ This problem canbecir- cumvented through use ofmultiple installations onaccurate base lines. Thetotalvelocity canbedetermined from such anarrangement. Some ofthemore sophisticated methods oflaunch guidance donot depend ontheuse ofasingle type ofequipment butendeavor tomake use ofthebest features oftwoormore types. For example, atracking r_dar might beused forlong period information because itisnot subject tothe drift problems ofthe inertial system; however, aninertial system might beused atthe same time toprovide short period guidance. This arrangement wouldallow very heavy smoothing (filtering) tobeapplied totheradar data inorder toeliminate noise. Similarly adoppler system canbeused forvelocity information inconjunction with useof 333< 8-10 atracking radar system forposition information. This arrangement avoids the noise problems associated with data differentiation andalso avoids thedrift problems associated with data integretion. 8.2 Orbiting theEarth Once avehicle hesbeen established inorbit thecomplexionofthe guidance problem changes. Motions about thebody axis are essentially neutrally stable (noinherent modes ofmotion exist), andinmost cases, there arenosignificant external disturbances. Itisnecessary toapply only sm_ll control moments toreduce any initial angular velocities and tokeep thevehicle pointed inaparticular manner (ifdesired). Unless the vehicle islaunched into anorbit close tothe earth where theeffect ofaerodynamic drag maybesignificant thetrajectory isstable and predictable anddesired minor modification tothe orbit canbemade by relatively small thrust kicks intheappropriate direction and atthe appropriate point. Ontheother hand theorbital phase oftheoperation may extend over very long periods oftime andcover large distances over the surface ofthe earth. Thefirst factor mentioned dictates extreme emphasis on simplicity, reliability, drift andlowpower consumotion oftheguidance andcontrol systems tobeused. The second factor implies that on-board guidance andcontrol systems wculd bedesirable. Theground based systems described inconnection with thelaunch operation (and other types of ground systems) canand arebeing used inconnection with orbital operations buttheir use islimited bytheextreme size ofthenetwork req_[red forcomplete _rfacecoverage. 8-II Mostadvanced satellite missions require somesortofattitude control. Examples ofsuch missions areearth reconnaissance missions and stellar observation missions. Some ofthemost elementaryattitude control systems include thesensing ofgravitation orcentrifugal force gradients bycontrol ofthevehicle geometry (for earth observation) and theuse ofsolar photon sails (for solar observation). Theuse ofadrag device for stabilization ispossible innear earth short-term orbits. These schemes are characterized byextremely small restoring moments and their useisdependent upon theresponse times achievable ascompared to those required insparticular operation. Their practicality isalso depen4ent onthemagnitude ofdistrubances enc_mtered both externally endinternally. Another type ofsimplealtitude system which have been used issimply spinnin_ the entire Vehicle. Theattitude stabilization inthis c_se iswith respect tofixed space. Optical systems appear well suited fororbit attitude control and other guidance functions inspace. These systems are capable ofvery high resolutionandarewell adapted tousebythehuman. This combination isfelt bymany toadduptohighly reliable system. Attitude stabilization with respect totheearth canbeaccomplished bya360°scan ofthehorizon. Stabilization with respect tothesun, planets, orstars canbeobtained through use ofanastro tracker. Systems ofthis type require ameans forconversion ofthe sensed quantities into control moments. This conversion involves ahuman pilot orautopilot andsuitable moment producing devices. Such devices arenowenvisioned asinertia wheels orsmall reaction 8-12 jets. Problems associated with such systems arelargely intheareas ofdesign information ordevelopment. More information isneeded on disturbances (particularly internal), onpower requirements (novel power sources need tobedeveloped), onthedesign ofsmall controllable rockets orhiCh reliability, andonthereliability ofcomponents and subsystems ingeneral. Inorder toperform such functionsasorbit transfer ortore-enter from anorbit, information onsuch parameters asgeocentric radius (altitude), velocity,andposition. These measurements could bemade using inertial platforms ofthetype already described. Theinertial system hasanadvantage inthis andother space flight applications inthat itisaself-contained on-board system. Itdoes appear, however, that some backup system mayberequired inallbutshort-term operations because ofthelong-period drifts associated with theinertial systems. Theground-based systems previously described could also beused in orbit determination. Inaddition, another type ofground-based system is usedforsucE measurements. This system involves thedetermination of th_direction ofarrival ofawave front transmitted from thetarget. Thedirection tothetarget isalong aline normal tothis front. A phase comparison between thesignal received atstations alon_ known base lines isused toestablish thedirection ofarrival ofthewave front. This type ofequipment hastheadvantage ofbeing basically D_sive inn_ture. Itiscapable ofobtaining position information onanytarter forwhichatransmitted signal canbedetected. Itdoes, however, require af_ir]y elaborate ground range communication and 35.6< 8-13 computing system. Attitude ofssatellite with respect toaground station canalso beobtained using such equipment provided thed6,ice for measuring angle ofarrival isaboard the satellite. Optical systems might also beused toadvantage inobtaining velocity andposition information asillustrated infigure 8-5. Forasatellite maintaining afixed orientation with respect tothe earth (as, forexample, bymeans ofahorizon scanner) thespacial velocity parallel totheearth,s surface canbeobtained from rate gyros providing thegeometric radius isknown (V=R_). This radius measurement might beobtained bystadia- metric methods using thehorizon scanner orbyaradar altimeter. Similarly, instantaneous velocity andposition with respect toapoint ontheearth could beobtained optically foranattitude stabilized satellite by topographic identification anddrift measurement. Itshould bementioned that theclose inphases ofmerging oforbital position oftwovehicles might very possible beaccomplished byhoming techniques. The type ofguidance system might bemuch the same asthose developed forair-to-air orsurface-to-air missiles andcould employ o_tical, radar orinfra redtracking. 8.3 Space Travel The guidance andcontrol devices already described forthe orbiting vehicle canalso beapplied toavehicle traveling incislunar space or ateven greater distances. There islikely tobeonedifference, however. _lthou_h guidance ofvehicles ininitial orbiting near theearth may often beaccomplished with afairly elementary computation routine, thetrajectories 8-14 inspace will dictate theuseoffairly complex computers inconnection with theguidance function. Coupled with long travel time, this computer presents aserious reliability problem. Anexample isgiven inapaper byXenakis inwhich acircumnavigating lunar trip ending inare-entry totheearth's atmosphere wasconsidered. Thetrip time wasroughly ten days. Thesystem consisted ofanastro tracking system, rate gryo control, inertia wheels, andacomputer. Thecomputer wasthedominating influence inlowering theprobability ofsuccess. Using failure rate data fora modern airborne fire control computer butonly considering one-tenth thecomputer capacity, theprobability ofsuccess ofthis mission was 0.22. Replacing thecomputer byahuman pilot raised theprobability ofsuccess toO.70; however, theprobability ofsuccess ofthecomputer equipped system could beraised to0.64 through useofintermittent operation. Even with this intermittent system, theprobability ofsuccess ofaonewaytrip toMars would below(less than oneintwenty) because ofthelong travel time assumed (2400 hours). Guidance accuracy requirements forsuch missions aslunar impact, lunar circumnavigation, orlarge radius lunar orbits arenotgreat. Angle accuracies oftheorder ofone-half degree would berequired, however, ifreturn tothesurface oftheearth isdesired much higher precision isrequired. Ifabraking ellipse re-entry isused, afirst pass perigee altitude precision ofabout 20,000 feet appears toberequired. At20,000 miles from perigee theangle accuracy would have tobeabout 0.02 degrees andthevelocity accuracy would have tobeabout lOfpsto 35S< 8-15: meet this requirement. Thus aneed isindicated forsome means oflimited corrective control (thrust control) astheperigee point iuapproached. 8.h Atmosphere Re-entry There-entry into theatmosphereofaspace vehicle will bemade at ashallow angle inorder tolimit aerodynamic heating rates toacceptable values. Ingeneral regulation ofre-entry angle andposition will be accomplished bysmall amounts ofretro-thrust. Theconfigurations considered forre-entry vehicle vary widely andinclude balloons, parachutes, ballistic re-entry capsules, andseveral varieties ofwinged vehicles. Thewinged vehicles will probably beoperated athigh angles ofattack (thirty to ninety degrees) inorder tolimit theheating rates and/or thetotal heat input. Ballistic re-entry bodies will behigh drag configurations, butultimately might utilize small amounts oflift formaneuvering. Thesignificant addition totheguidance andcontrol problem here as compared toother phases ofthemission istheimportance ofaerodynamic lift, drag, stability, andcontrol. Blunt (approximately flat faced) .................... wu_o orwiz_edv_:_Ac,es atextreme angles ofattack (approaching ninety degrees) arecharacterized byhigh drag, negative lift-curve slopes, andvery smallamounts ofstatic stabi]ity. _ctually stangles ofattack approaching ninety degrees theresultant force coefficient shows little variation with angle ofattack both as toitmmagnitude andastoitsdirection with respect tothebody. Theaccelermtion time history during re-entry, therefore iscontrolled ;iLg< 8-16 bycontrolling thetrajectory through theatmosphere. Because noimmediate effect ontheaccelerations results from control action piloting problems arelike]y tooccur. Adding tothis difficulty isthe effect ofthe necative lift-curve slope. Thevehicle must bepitched down inorder toapply lift inanupward direction (pull out). Another problem common toallre-entry configurations islow damping oftheshort-period modes. Infact, the negative lift-curve slope associ- ated with theblunt configurations maymake thedampir_ slightly negative inthis case. Ingeneral, however, theamplitude ofthis oscillatory motion tends todecrease during there-entry because oftheincreasing density. Thedecrease inamplitude isapproximately proportional tothe decrease inperiod because intheabsence ofcontinued disturbances the energy intheoscillation will remain about constant. These conditions result inthemaximum angular accelerations increasing inproportion to theincrease inperiod. Inthecase oftheblunt body, these motions sho_]d not produce significant transverse accelerations nor oscillations inthetotal accelerations forthereasons given previously. This result would notexist forwin_ vehicles re-entry atangles ofattack of35or ]_5de,Fees. The winged vehicles operating intherange ofh5degreesangle of attack may _iso encounter problems associ*ted with strong aerodynamic and inertial coupling ofthemoments about oneaxis duetomotionsabout another. This problem isparticularly difficult iftheangle ofattack must bevaried. Although the effects might becompensated inthe design (orbyautomatic means for oneangle ofattack, such compensaticn would ,iO< 8-17 notbepossible over arange ofangles ofattack. Control deflection coupling hasbeen found p_rticularly bothersome. Incertain instances themoN_ents produced about thecontrol axis hasbeen smaller than the moments produced about another axis. Added totheperhaps more subtle problems previously mentioned, there aretheproblems relating totherapid changes incontrol effectiveness, static stability, trim, andresponse associated with therapid variations in_sch number anddynamic pressure during these re-entry maneuvers. These characteristics will require considerable adaptive capabi]itieson the part ofthehuman pilot orthe_utopilot. nw yoBsS30 $ra 8Rss x 8s 8~ || S |7é 4|| h 3 NSQ ~ S C) s + % 8~~ >» x 3 N Vvw % a! x é >v 22 ‘o ;s4 W uv 1 ~~ 3 ao 342< cy ei, Re . . § v Ww n > = ye 28 ~88 3 as n Egs §ss 6rd Bs 8iN 3a8 ~ e wD eas = BRE 8gtsASR 45 =sk ay Bs Md ve =RAK <a> ' KL& IN] . gy *x NG vgs 3 DS & B43< QJ b 0"70- OJ \_0- _0. 0J f fJ /J / /JI f f J J / / mode_. s_,4r"8,i4- \ q_ _J _J t,. ! 0Qc_ _3 I IIt.) N £ 0 I u_ 3,_6< 9-1 SECTIONIX ENEMENTS OFROCKET PROPULSION 9.1History ofRockets Thehistory ofrockets will bebriefly discussed. Only those few n_mes anddates which areconsidered sufficient forabrief outline ofthe subject will bementioned. Thefirst recorded useoftherocket occurred in1232 A.D. when the Mongol, Ogadai, thethird sonofGenghis Khan, began hisattack onKaifeng, capital ofHonan Province inChina. Rockets were used bythedefenders of Kaifeng andwere described asfire arrows. They were nothing more than rockets attached toarrows toincrease their range. Itisprobable that therocket principle wasknown totheGreeks several centuries earlier, possiblyasfarback asthe8thcentury. Going back still farther theChinese have been credited bysome with theuseof _Jnpowder rockets asearly asseveral centuries B.C. Ofcourse, all propellants used inthose times were solid. Infact, gunpowder orits variations wastheonly known type ofrocket propellant until early inthe 2Oth century. Assuming the13th century asthestarting point inrocket history, themost rapidadvances were made intheOrient during thenext 500years. TheIndian soldiers used large numbers ofrockets with telling effect against British troops during theIndian campaigns. Their success aroused theinterest ofmany inEngland, oneinparticular being SirWilliam Congreve. Hebegan earnest study in1804. Hisrockets were used against Napoleon 9-2 andalsoagainstAmerican troopsintheWarof1812....whose redglare ismentioned intheStar-Spangled Banner. Thelong, trailing stick ontherocket which wasused forstability waseliminated around themiddle ofthe19th century. Itwasreplaced by small curved vanes built into thepath ofthejetwhich were able tospin- stabilize therocket. This idea ofspin stability wasalso applied to artillery shells byrifling thebores. Bytheendofthe19th century therocket wasreplaced byartillery because ofthis increased accuracy due torifling andbecause ofrapid dewelopments made whereby therange ofthe artillery wasgreatly increased. In1903 Ziolkovsky ofRussia made adefinite proposal foraliquid- propellant rocket unit inanarticle dealing with thepossibility ofrocket space travel. Nevertheless, credit forrebirth ofrockets inthe20th century isgenerally given toDr.Robert Goddard ofMassachusetts. Although thebasic idea ofutilizing therocket principle toattain extreme altitudes hadbeen conceived inthemiddle ofthe18th century, Dr.Goddard conducted thefirst scientific experiments along these lines. Hebecame interested inthesubject in1909 andbegan experiments asearly as1915 with solid propellants. Financial backing forhisexperiments wasprovided bythe S_ithsonian Institution whopublished hisreport, "AMethod ofReaching Extreme Altitudes" in1919, thefirst publication onthis subject. Init, Dr.Goddard claimed itwastheoretically possible tosend arocket tothe moon. In1920, Goddard sawtheneed forliquid-propelled rockets inorder toobtain therequired speeds, altitudes, andendurance. In1926 hesucceeded 9-3 inmaking thefirst flight with aliquid-propelled rocket, andin1932, thefirst flight ofarocket stabilized byasmall gyroscope. Inthemeantime, shortly after Goddard hadrenewed theinterest in rockets others from all overtheworld followed with work onthesubject, notable among these being Hermann Oberth inGermany. Except forGoddard, thebulk ofrocket research inthefirst half ofthe2Oth century was centered inGermany. Therocket wasused inWorld WarI,butnotasanoffensive weapon. In_brld WarIImany different uses oftherocketasamajor weapon were made byeach ofthepowers. Since World WarII,theapplication ofrocket Fower hasbeen rapidlyaccelerated asexemplified bythedevelopment of intercontinental ballistic missiles andthelaunching oftheartificial satellites oftheearth. 9.2Rocket Principles Anattempt will bemade toderive andexplain afewofthemore common quantities associated with rocket motors. 9.2.1 Equation fortheAcceleration ofaRocket. Thepropelling action ofarocket motor isderived from thegeneration oflarge quantities ofgases bythechemical reaction ofsuitable proepllants within therocket motor. Theforce which isproduced bythe generation ofthegases iscalled thrust andisthebasic performance parameter ofarocket since itdetermines thespeed anddistance which canbeobtained. 9-4 Theequation fortheacceleration ofarocket canbederived from a special principle ofthemomentum theoremwhich takes into account the fact that aburning rocket iscontinuously losing part ofitsmass. This principle states that forasystem ofSparticles, thevector sumofall theexterior forces acting onSisequal tothetime rate ofchange ofthe total momentum ofSplus therate atwhich momentum isbeing transferred outof$bytheparticles that areleaving S. Inderiving theequation fortherocket acceleration consider the rocket shown below l_-----Nozzle exit i where therocket system consists ofthemetal parts oftherocket, the unburnt fuel, andthegasinside thegeometric surface. Themomentum of the_-stem isequal tothemass ofthesystem times itsvelocity, that is, My. Therate atwhich thedeparting particles remove momentum from the system isJust therate atwhich momentum crosses theexit surface. Ifthe velocity oftheg_srelative totherocket attheexit ofthenozzle is re,then thegascrossing theexit surface hasavelocity ofw-ve.If istherate atwhich gasisstreaming through theexit surface, then the Jetistaking momentum from thesystem attherate ofm(v-re),and, by theabove principle 9-5 _-Fexter.al (t)=a-_(My)+m(V-re) Therate ofgasstreamin_ through theexit _which istherate atwhich thefuel isbein_ burnt must equal therate ofchange inmass ofthesystem. Therefore _EFexter.al(_)=MG-_v =MG-_ve•&(v-ve) or MG:_va÷_Fexternaj (t) 9.2,2 Derivation oftheThrust ofaRocket, Among theexternal forces acting onthesystem, asshown inequation (9-1), arethegaspressures over thesurface ofthesystem. Thegas pressures consist ofatmospheric pressure over theoutside oftherocket andtheJetpressure over theexit surface. Theforce duetothejetpressure isPeAe where PeistheJet pressure attheexit surface and Aeisthearea oftheexit plane. The etmospheric pressure iscomposed ofthestatic atmospheric pressure plus theaerodynamic forces duetomotion through theair. Theforce dueto thestatic atmospheric pressure over theoutside oftherocket isjust the(9-1) Si< 9-6 negativeoftheforce thatwould beproduced bystatic atmospheric pressure over theexit surface. (Thisisobtained bysumming upalltheforces due toatmospheric pressure acting ontherocket exterior excluding thenozzle exit surface.) Therefore, inten-ms oftheJet, theforce duetostatic atmospheric pressure over theoutsideoftherocket is-PaAe where Pa isthestatic atmospheric pressure. Aspreviously stated theremaining forces duetoatmospheric pressure areduetomotion through theairandareconsidered asaerodynamic forces Fa• Collecting theabove andother external forces thetotal external force is _:Fexter.al('-,.)=peAa- p_Ae-Fa-Mclx-F (9-2) Theterm Mgx istheforce duetogravity andtheterm Fincludes all other exterior forces, forinstance, iftherocket istowing outaline, etc. From equation (9-1) then M_/=rove+(Pe"Pdi)Ae-Fa-McJx-F(9-3) Nowconsider arocket held motionless byatest stand. Theterm M_ becomes zero since v-O.Also, there arenoaerodynamic forces onthe stationary rocket, sothat Fa=O.Iftherocket isheld inahorizontal position, then MgxaO.Thus, theterm Finequation (9-3) becomes the thrast, _smeasured bythethrust gage• 9-7 or F-- +(Pe-P,)Ae(9-4) (9-5) where &istheweight rateofpropellant flow. Thethrust iscomposedof twoterms. The first term isknown asthemomentum thrust. Thesecond term isknown astheoressnre thrust andconsists oftheproduct ofthe cross-sectional area oftheexhaust _ietandthe difference between the exhaust oressure an_i theatmospheric pressure. From equation (9-4) itis evident that arocket exhaust nozzle isusually designed sothat the exhaust _essure isequal toorsli_htl7 hi,hem than theatmospheric pressure. Ifthe atmospheric pressure isequal tothe exhaust pressure, thethrust is F:("_Ig)Ve and_iwes amaxim_nm thrnlst for agivAn eh_mbe_ pre_su_ Therocket no_z!e design which permitsthe expansion ofthepropellant products tothesame pressure ofthe surrounding fluid isreferred toastherocket nozzle with o_timum expansion retio. For underexpsnsion (that is,when theexhaust gaspressure ishigher thanatmospheric pressure) aportion ofthe energy ofthe_ases isnot converte_ into kinetic energy andislostasfar asthrust development is concerned. 9-8 Ifamotor istooperate atdifferent altitudes, thearea ratio (Aexit/Athroat) must beselected sothat itisthebest compromise for theoperating conditions. Usually, the nozzle isdesigned sothat itwill operate with aslightunderexpansion atthe most significant operating altitude. Inthis case the thrust atsealevel would bereduced. TheGerman V-2 isatypical example anditsthrust variation with altitude isshown below. _6 .Q mo_,2 0 o L. I-- i I S4- I I 0 2o ,To Go 80 I00 Altitude, ioooft 9.2.3 Effective ExhaustVelocity. Inorder toexplain themeaning ofeffective exhaust velocity reference canbemade toequation (9-5) which is F:(W/9)ve+(Pc" Pa)Ae The term effective exhaust velocity isintroduced asaconvenient way of definin_ thethrust intheabove equation only interms ofthepropellant weight rate offlow andanexhaust velocity which iscalled theeffective exhaustvelocity.Theeffective exhaust velocity cisdefined as ..F" I'Pe-Pa)Ae9(9-6) 9-9 (_en velocity ofthepropellant Eases velocity, thenPe=Pa' theeffective exhaust velocity cisequal totheexhaust With thisdefinition ofexhaustVe.) F-cC '/9)(9-7) T?%_ical values forarocket might be Te-6,200 ft/sec c-7,300 ft/sec Effective exhaust velocity depends ontheatmospheric pressure (see equation (9-6)) ;therefcre, when values ofeffective exhaustvelocity arestated thecorresponding atmospheric pressure isalso stated. Thus theuseof effective exhaust velccity relates theperformance orthrust ofarocket onastandard basis andonerocket canbeeasily compared with another rocket. Intheactual testin_ ofarocket motor todetermine itsperformance thev_lue oftheeffective exhaust velocity canbeeasily determined from m__s_r_, _no_ ofthethrust andpropellant flow, asisshown below. Thetctal weight ofthepropellants consumed is Wp:_/At Then from equation (9-7_ (9-8) cr c:Ig (9-9) 9-10 where Iiscalled theimpulse inpound-seconds and isthe integralofthe thrust vs. time duration curve. (See below.) -a e_ 4J L r-/ k_ l< Duration, At Theuse ofimpulse ishandy, especially for solid propellant rockets where itisoften difficult tomeasure thepropellant flow rate accurately. Theeffective exhaust velocity isoneofthemost important criteria ofrocket-motor performance andacontinual effort isbeing made toincrease itsvalue. Ifthethermodynamic properties ofthepropellant gases are known, the effective exhaust velocity canbecalculated from the following equation. C -- (9-1o) 9-11 where Pcisthecombustion chamber pressure kisthespecific heat ratio Risthegasconstant perunit weight andisequal totheuniversal gasconstant (1,544 ft-lb/mole o_)divided bythemolecular weight (ib/mole) Tcisthechamber temperature, OR P_2istheexpansion ratio Pc _ctual!y, equation (9-10) gives theeffective exhaust velocity forthe optimum expansion ratio, that is,Pe=Pa" Intheabove equation the specific-he_t ratio andexpansion ratio canexert only aminor influence onthemagnitude oftheeffective exhaust velocity. Therefore, Forgases _enersted bytheliquid propellants currently being used, thevalues ofRvary from 60to80. Combustion chamber temperatures range from 3500°R to6000OR. Atypical chamber pressure is300psia, while some are ashigh ss500psia. Rockets aregenerally designed foraconstant chamber pressure. 9.2.4 _pecific Impulse. Rocket motor performance isfrequency expressed interms ofspecific impulse ofthepropellants. This istheimpulse delivered perunit weight ofpropellant consumption. or9-12 is=F-T-- W (9-12) Interms oftheeffective exhaust velocity (equation (9-6)) which is thespecific impulse is .C mIs- s Therefore, itisseen that specific impulse ismerely another wayof expressing rocket performance andisconvenient tousewhen itispossible tomeasure thepropellant flow rate andthrust (usually inliquid-propellant rockets). Inthefigure below isatypical calculated performance ofaliquid oxygen-gasoline propellant system.(9-13) (D000 3000 4000 ,_3000 "_40 oE__zo :J "_200 v) IiIIII 1.5Z.0 Z.._ 3.0 Mixture ratio,0 J.z"_ I,I_ z, zog "3 u tu ox_dizer wt. _elwt. 9-13 Itisseen that onedefinite mixture ratio (oxidizer weight tofuel weight) _ivesanoptimum rocket performance. This optimum mixture ratio occurs usually atavalue which isricher infuel than the stoichiometric mixture ratio, atwhich al! thefuel istheoretically completely oxidized andthe flame temperature isamaximum. Ingeneral, themixture ratio oftherocket issoselected astobevery close tothe optimum value. Inliquid pro- pellant rockets this ratio determines theproportions ofthepropellant tank volumes andinsolid propellant rockets itdetermines theproportions of oxidizer andfuel tobeused during propellant preparation. 9.2.5 Propulsive Efficiency. There arevarious efficiencies defined inconnection with rockets. These arenot commonly used indesigning rockets; however, they permit anunderstanding ofthe energy balance ofarocket. Oneefficiency, _own asthe propulsive efficiency, will nowbebriefly discussed. The propulsive efficiency determineshowmuch ofthe kinetic energy oftheexhaust jet isuseful forpropellin_avehicle. Itisdefined as vehicle energy _P=vehicle energy +residual kinetic jetenergy and sinceFv I v (9-14) 9-14 where visthevehicle velocity and cistheeffective exhaust velocity. The term (c-v) isthe absolute exit velocity ofthepropellant gases. The following figure shows the instantaneous propulsive efficiency ofthe rocket jet asafunction ofthevelocity ratio !. C _,_I00 _.80 ,-_60 o40 e_ _20 9. 0• 0 0. 0--.... I.o 2.O Velocity rat;o,¢3.0 The propulsiveefficiency isamaximum when theforward vehicle velocity isequal tothe exhaust velocity, that is,when the absolute velocity of thejet iszero. This means that thetotal available kinetic energy is bein_ used topropel thevehicle. Since the object ofarocket isusually toattain ashigh aflight speed aspossible the speed isconstantly increasing, thus thepropulsive efficiency varies during the flight ofa rocket. Itisapparent from thefigure shown above thattherocket isan inefficient propulsion system atspeeds comparable tothose ofourpresent- d_yairplanes. &GO< 9-15 9.2.6 IdeslVelocityofaRocket. Ifthe external forces _Fexternal (t) then MY=inequation (9-1)areneglected, (9-15) From this very useful expression theideal velocity oftherocket canbe obtained; that is,the speed therocket would attain neglecting theeffect ofdraz orgravity. Theeffective exhaust velocity cisassumed tobe ccnstsnt. Theideal velocity isdetermined from equation (9-15) byfirst rememberin_ that sothat dMcM_:-aT Multiclyin_ through by dt, then orMdv:-caM AM dv:-c-----:-M Both sides oftheequation canbeintegrated over the period ofburning as shown below _f _( dMav:-c v_:-c(L.M;-I.,Mo) =CLnMo Mo:v._ or" e'_ Mf Mf(9-16) C.G:i< 9-16 where vf isthefinal velocity and Moand Mfarethe masses ofthe rocket (plus fuel) atthebeginning and endofburning, respectively. It should benoted that forequation (9-16) thevelocity isgiven interms of thefinal velocity reached forarocket started initially from rest. If therocket isatsome speedatthebeginning offiring, then the vf term becomes incremental velocity Av. _quation (9-16) canbeexpressed interms ofthepropellant mass ratio S. VforAV=-cL.(I-_) (9-17) where _isthe ratio ofthemass ofthepropellant tothe gross mass ofthe rocket Mp Mo Since theendresult ofarocket isthat ofdelivering auseful load orpayload itisconvenient toexpress equation (9-17) interms ofa parameter known asthepay-load ratio __Ms where MZ isthe mass ofthe Mo payload• Byintroducing anacditional parameter e,called the structural factor, andbysubstituting this term along with thepay-load ratio into equation (9-17) theexpression for thevelocity becomes or=cL.[¢(x), x] where kisthepay-load ratio M_ Mo and Mf M; M_+M.p Mo-Mz(9-i8) 9-17 such that =(,-x)(,-¢) Itshould benoted that Mf does notincludethemasso£thepay ]oad. Although these results forideal conditions cannot beapplied directly, they give anapproximation ofthemaximum speed which canbeexpected for aRiven rocket. Thedifference between the "ideal" andactual performance varies forevery rocket. Asmall fireworks rocket mayactually onlyattain 2or3percent ofthis "ideal" performance. TheGerman V-2 attained some 70oercent ofitsideal performance. The larger therocket thesmaller the difference between thetwovalues; arocket capable ofovercoming thewhole gravitational field ofthe earth would probably attain better than 95 percent of"ideal" performance. 9.2.7 Multiple-Step Rockets. ItisobTiousfrom equation (9-17) that very high final velocities cannot beobtained with asingle rocket because thepropellant mass ratio Mp M_ that would berequired becomes physically impossible using present-day propellants. Thedisadvantage ofasingle rocket inregard toattaining high velocities lies inthe fact that theentire empty mass ofthe rocket must becontinually accelerated even after themajor portion ofthat empty moss isnolonFer useful. Forthisreason thestep principle was conceived inth_i arocket consists ofseveral steps, each ofwhich operates independently. Asthe propellant isexhausted ineach step, thestep isseparated from the rccket andpropu/sion fortheremaining unit isprovided bythenext step. 9-18 MoX IThis principle isbriefly discussed below bytaking athree-step rocket as anexample. Inorder toavoid confusion, theindividual rockets aredefined hereinasthestepsandthecombinations ofthesteps arereferred toas sub-rockets. propellant mass empt_mass (excl.ain 9payload) pay-loadratiostructuraJ _actor propellant massPatio Monz X==-- MoxM_j Mp, Mr,-I-Mp, Mox _ Mfz _:_MPz (:= Mfz+ Mp2 M:= £'- Mfa _J.r:"_ Mfat-Mp3 Moo :564< D I9-19 Inthecaseofastep rocket theideal velocities ofthesub-rockets maybeadded, sothat thetotal ideal velocity ofthethree-step rocket shown above is -c,,,_.['_=(,-_.=)+x=-i Itisassumed that each step starts tofire immediately after thepreceding step hasceased firing, andisdiscarded immediately after ithasstopped firing. Ifitisassumed that allthesteps aredesigned with thesame structural factoreandthat thepay-load ratios forallthesub-rockets areequal (itisbelieved that these assumptionsarenotfarfrom thetruth foranactual multiple-step rocket) then thevelocity oftherocket atthe endofburning ofthethirdstep is v3=-_,,l"[_(',-,.)*x] _-_o) where caistheaverage value ofeffective exhaust velocityover the entire firing period. Incase themass ofthepayload ofthefinal step M_andthemass oftheentire rocket Moyaregiven then thepay-load ratio kin equation (9-20) foreach ofthesub-rockets is Hoz(9-21) C6S< 9-20 Inpractice, each step multiplies thetotal weight byafactor upto IO, sothat for reasons ofshear bulk, rockets ofmore than 5steps are not often contemplated. Inequations (9-16) through (9-20) theeffects ofdrag and gravity have been neglected.Actually, even under conditions where flight is made intheatmosphere atlowaltitude the effect ofdrag onthefinal _Iccity isprectically negligible (about 5percent) forlarge, slender rockets. The deceleration due togravity canbeincluded inthese equations bysimply adding the term -gtp, where tpisthe duration of powered flight inseconds. The acceleration due togravity atsealevel is used here forthevalue ofgsince the error introduced byassuming a constant _will bevery small iftheburning stopsatadistance above the earth which issmall compared with theradius ofthe earth. Itis apparent that theshorter theburning time, the less theeffect ofgravity isonthefinal velocity ofarocket. 566< 9-21 REFERENCES I. Sutton, George P.:Rocket Propulsion Elements,1956. 2.Zucrow, M.J.: JetPropulsion andGasTurbines, 1948. 3. Rosser, J.P.,Newton, R.R.,and Gross, G.L.: Mathematical Theory ofRocket Flight, 19h7. )_.icy, Willy: Rockets, Missiles, and _epsce Travel. Revised Edition, 1957. 5. Seifert, H.S.,Mills, M.M.,and Summerfield, M.: The Physics of Rockets. American Jour. ofPhysics, Jan.-Feb., Mar.-Apr., May-June, 1947. 6.Malina, F.J., andSummerfield, M.: TheProblem ofEscape from the Earth byRocket. Journal oftheAeronautical Sciences. V.14, no.8, August 1947. I0-i SECTIONX CHAP_CT_,ISTICS OFMODERNROCKETSANDPROPELLAi_S i0.i GeneralComments Afewbriefremarkswillbemadeaboutrocketsingeneralsoas toprovideanoverallpictureofthevarieddetailsinvolved inrocket engineering. Therearevastdifferences betweenliquid-andsolid-propellant rocketsinregardtotheirdesignandoperation. Ithasbeenapparent thatliquid-propellant rocketshavebeenmuchmorecomplicated in relation tosolid-propellant rockets. Sincethesolid-propellant rockets aremoreconvenient andeasiertomaintain -andthuscheaper-theyhave beenusedexclusively whereshortduration andrelatively smalltD_usts arerequired° Themagnitude ofthethrustofsolidrocketshasbeen limitedbecausethesizeofthepropellant islimitedbythemanufacturing process. Nowadays, therearethosewhoareconvinced thatsolidswillcompletely _.___._-_ _=_±_ systez,s___:_z-±u_u_'e. Othersgive thesmallmissilefieldtosolids,equalbilling_themediumthrustrange, butprobably lessimport&nce tosolidsinlargemotorssincesolidsincrease incomclexityandhsndlin_ difficulty toapoint where they lose any adventage they once had over theliquids. Nomatter which isbetter, there will undoubtedly bemuch toseeandhear about both until higher energy =y_tems aredeveloped Onefact worth mentioning about these advanced _ystem¢ isthat some may still beintheform ofliquid-propellant _¢stems. For instance, itwill bepossible touse theheat generated by 568< 10-2 anatomic pile tovaporize aliquid (which canbecalled thefuel) and heat thevapor toanydesired temperature stahigh pressure. Thus a fuel such aswater could beused. Ifhydrogen were heated to4,OOO °C atapressure of300psiandallowed toexpand down toatr_ospheric pressure ajetexhaust velocity ofnearly 30,000 ft/sec could theoreti- cally beobtained. This isabout three orfour times more than is beingachieved stpresent. Asimple definition ofarocket isajetreaction motor that carries itsownworking fluid. Thepropulsion unit iscalled amotor since it hasnomoving parts. Inarocket, thermal energy isconverted into kinetic energy producing animculse which isathrust foraperiod of time. Fuels av_il_ble forpresent orfuture usearechemical, nuclear, sol,r, andionormagnetic (sometimes called plasma). Acomparison of theenergy ofvarious fuels canbemade with theuseoftheterm Jet horsepower which isobtained by thrust xenhaust velocityjethorsepower =55o Typical values ofjethorsepower forseveral rockets aswell asfora jetengine areshown inthefollowing table. Each ofthejet-horsepower values given correspond toa30pound thrust. 10-3 For30pound thrust Exhaust velocity Mass i_ow ft/sec Ib/sec Jethorsepower Turb ojet 2,700 O.3 150 Chemical rocket (present) 7,5OO .12 400 Chemical rccket (advanced) 15,OOO .07 800 Nucl earrocket 45,OO0 .023 2,500 Ionrocket "_ 3,000,000 uptothe Fhoton rocket _ velocity oflight .00032 160,OOO andup InSECTION IXitwas shown that theenergy ofachemical rocket is limited b_temperature andmolecular weight (and toalesser extent, by specific heat). F--------- exhaust velocity _V_ c- where Tisthe specific heat ratio Tisthe temperature inthecombustion chomher C _isthemolecular weight Ahi[b performance canbeobtained byhigh temperature orlow molecular weight. Thehi_her +emper_tures forbetter performance areinturn limited duetocooling difficu.lties. Thus, high performance depends onhigh efficiency inthat aslittle thermal energy 8spossible must belost fromtheworking fluid tothechamber andnozzle walls.Nuclear systems must have high efficiencies orelse thevery high temperatures will damage therocket unit. Theenergy inanionrocket isproportional to --where the Visthevoltage, and MisthemolecularQis charge, lO-h weight. Here,forhighperformance thefuelmusthaveahighcharge, ahighvoltag,e,oralowmolecular weight. Inre_ardtochemical propellants, theycanbeclassified intothree _roups: (I)liquid,(2)solid,and(3)hybrid,whichcanbeacombination ofaliquidfuelandsolidoxidizer oracombination ofasolidfueland l_q_[doxidizer. Furtherdiscussion ofhybridfuelswillnotbemade sincetheyhavenotbeenusedwithmuchsuccess. Thewordpropellant referstoboththefuelandoxidizer whicharecombined toproduce b_Jrn[n_. Fuelcanbereferred toasthereducin_ a_entandtheoxidizing a_entis_hatcomFourJd thatincreases theproportion ofoxygenoracid formin_elements orradicals whencombined withanothercompound. Liquid propellants requireamixin_apparatus (injector) andinmostcasesmust havepu,ps.Solidpropellants aremixedduringthemanufacturing process and,ofcourse,neednopumping. 10.2 LiquidRocketsandPropellants Incontrast tothesolidrocketthereseemtobemanymoreitems tobecoveredwhendiscussing themorecomplicated liquid-propellant rocket. Itshouldbenotedthatthestatements madehereinaregeneral _nn_ture. Actually, allthevariousrocketunitsnowinexistence wi]]differfromeachotherinmanyrespects sinceeachisdesigned for• specific task. Inre_ardtothesizeofthecombustion chamber, itmustbelarge ,,no_1_htoallowforcomplete combustion ofthepropellants. Aspherical 10-5 shape isdesired since foragiven volume this shape provides theminimum wall-surface area, therefore, theleast weight. Also, manufacturing con- siderations indicate apreference foraspherical orcylinderical shape. Inaddition, thecooling jackets aremore difficult todesign andbuild forthemore complicated shapes. Asforfeeding thepropellants into thecombustion chamber, itis important torealize that they have tobeforced into thechamber under considerable pressure since thehigh temperature gases inthechamber arealready athigh pressure. Ofthemany ways offeeding thepropellants tothechamber twoaremost widely used. Thefirst employs agaspressure feed system andisused onlow-thrust, short-duration units. Inthis system thepropellants areforced outofthetanks byreplacing them with high pressure gas. This system isnotused onlarge rocket units because large tanks would berequired. Since these have tobemade towithstand pressures greater than thecombustion chamber pressure, their weight would become excessive. Fort_ehigh-thrust, long-duration rocket useismade oftheturbopump system. Here thepropellants arepressurized bypumps which, inturn, areusually driven byturbines. Ordinarily aseries of centrifugal pumps would beused togenerate thehigh pressures, butdue totheweight limitations inarocket, only asinglepump islikely tobe used. This will result inthepump being runatanabnormal speed. The power todrive theturbines canbeobtained inmany ways, butnormally is obtained from agasgenerator either having itsownpropellant supply orusing thesame propellants astherocket combustion chamber. Other means bywhich turbines arepowered arebybleeding gasdirectly from lO-6 thecombustion chamber orbyusing gasgiven offfrom solid-propellant charges burning ataslow rate. Twoimportant features ofthecombustion chamber aretheinjector andtheigniter. Theinjector hasadual Jobinthat itmust spray both thefuel andoxidizer into thechamber asfinely divided mists andatthe same time properly mixthem inthecorrect proportion. There aremany kinds ofinjectors inuse; some spray injets, some inthin sheets, while others useaconical spray. Different propellants require different t}_pes ofinjectors, with thebest injector usually being found by testing. Anigniter isnotalways necessary, assome propellants ignite spontaneously oncontact. Where itisneeded, theigniter itself maybe avery small liquid-propellant rocket, which canbeignited byaspark plug. This produces aDencil offlame which canbeused tolight offthe main stream ofpropellants. This arrangement allcws therocket motor to beswitched onandoff. Itisoften claimed that thestarting isthemost difficult operation inrunning acombustion chamber. Thestart must beaccurately controlled toproduce asmooth andev,_n combustion. Care must betaken toavoid forming anunlgnited explosive mixture ofpropellants. Theinitial propellant flow isusually regulated tobeless than full flow andthestarting mixture ratio ofthepropellants isdifferent from that used during normal operation. Frequently, amore reliable ignition isassured when oneof thepropellants isintentionally made toreach thecombustion chamber first. Ordinarily, forafuel-rich mixture thefuel isadmitted first andvice versa foranoxidizer-rich mixture. L73< I0-7 Itseems almostunnecessary tomention thefact that theproblem ofcooling aliquid-propellant rocket motor isalarge one: Asfor solid-propellant rockets, they arealmostexclusively uncooled; however, liquid rockets areusually cooled,especiallywhen used foralong duration. Inthese rockets some orallofthemetal parts incontactwith thehot _ases must becooled -such asthechamberwalls, nozzle walls,injector faces,etc. Thecooling jacket,which provides forthecirculation ofthe coolant, isdesigned sothat thecoolant velocity isthehighest atthe critical regions andthefresh, cold coolantenters thejacket ator near these regions. Helical coolin_ passages arefrequencyused, especially atthecritical regionswhere high velocities areneeded. Regenerative cooling isused bythemajority offlying liquid-propelled rockets. This method involves circulatingoneofthepropellants through thecooling Jacket onitswaytothecombustion chamber. Itisof interest tonote that theinitial energy content ofthepropellantis thus augmented prior toinjection increasing theexhaust velocity upto 1-1/2 percent.Water isoften circulated incooling jackets andisused extensively instatic test stand firings. Another methodofcooling is known asfilm cooling. Here, athin fluid film covers andprotects the exposed _al] _urfaces from excessive heat transfer. Thef_im isintroduced byinjecting small quantities Offuel,oxidizer, orinert fluid atvery lowvelocity inalarge number ofplaces alon_ theexposed surfaces. This method iseffective inthat itforms arelatively cool boundary layer andthecoolant isable toabsorb aconsiderable amount ofheat byevaporation. Aspecial type offilm cooling which hasbeen tried is iO-8 called sweat ortranspiration cooling. This method uses aporouswall material which admits coolant through pores over the surface. However, difficulty has been encountered inmaking thecoolant distributions uniform due tovariations inthepressure dropacross thecombustion chamber, particularly inthenozzle. Also, difficulty arises inthe manufacturing forthis process. The German V-2used acombination of film andregenerative cooling with thefuel being thecoolant. Considerable research effort hasbeen expended tofind materials tomeetrocket requirements. Inbrief, thewall materials have to withstand relatively high temperatures, high local gasvelocities, the chemicalaction ofcorrosion andoxidation, high stresses, and initial heat shock. Inaddition, thewalls have topermit high heat transfer rates and thermal expansion. These severe conditions are somewhat alleviated bythecomparatively short operational duration ofthe rocket. High thermal conductivity andhighstrength atelevated temperatures, two important properties required inthe inner walls ofacooled rocket combustion chamber, arenotusually found inthe same material. For instance, stainless steel withstands high stress athigh temperature, butitswalls have tobemade thin forgood heat transfer. Thus, material ofthis type would notbestrong enough tobeused inalarge combustion chamber. Onthe other hand, high conductivity materials such ascopper _ndaluminum have lowstrengthatthehigher temperatures and, therefore, reqnire heavy and thick chambers.Some materials that have been used successfully are aluminum alloys, low carbon steel, alloy steels, and stainless steels. LTS< 10-9 There aremany chemical compounds that areused forthefuel and oxidizer inliquid propellants. Some ofthemore common oxidizing agents andliquid fuels arelisted below along with afewpertinent remarks about each. Oxidizing Agents: (i) Hydrogen peroxide (H202).- This compound,which isliquid atroom temperature, does nothave avery high energy content. Itisdiffi- cult tohandle inthat itisquite unstable, decomposing readily into H20 and 02.This reaction, which isoften spontaneous, occurs with most organic material such asgrease andoil(explosive) orskin (burns). Inaddition, itispoisonous andvery corrosive. (2) Nitric acid +nitrogen tetraoxide (HNO3+N2Oh).- This isliquid atroom temperature, easy tohandle, butvery corrosive. (3)Liquid oxygen (02).- This abundantly available element isliquid only atvery lowtemperatures. Itisdifficult tohandle because ofthesevere cold andmust bekept away from oilbecause ofthe fire hazard. (h) Liquid fluorine (F12).- Fluorine isliquid atvery lowtemperatures, only, andisvery toxic andspontaneously reactable with many materials andmetals. (1)Liquid Fuels (Reducing Agents): Gasoline (CsH18).- This wasone ofthefirst fuels used andisstill inwide usetoday. I0-I0 (2)Alcohol (C2H5OH).- Alcoholisreadily available and, therefore, used inmany oftheliquid-propellant rockets. (3) Aniline (C6H5NH2).- Aniline isahydrocarbon that ignites spon- taneously with nitric acid; therefore, this propellant combination does notrequire anignition system. Spontaneously ignitable propellants areoften termed hypergolic. (4) Ammonia (NH3).- This isavery stable compound; itcanbemixed with other compounds without dan_er ofexplosion. Ammonia canbe made hypergolic with nitric acid byaddition oflithium. (5)Methane (CH4).- Methane isnon-corrosive andstable, butpresents ahigh fire hazard. Inaddition, itisanasphyxiating agent. (6) Hydrazine (N2H4).- This hasaloosely bonded chemical structure, therefore, high energy. Itisafire hazard andonly slightly corrosive. (7) Unsymmetrical dimethyl hydrazine [N2H2(CH3)2].- This hasgood storability; that is,theliquid canbestored inordinary tanks over long periods andatmany temperatures without decomposition or change ofstate. (8)Liquid hhdrogen (H2).- Liquid hydrogen isvery cold and, therefore, itscontact with metals causes severe brittleness. Because ofits lowspecific _ravity large tanks arerequired resulting inaheavier tank weight. Performance ofarocket isexpressed interms oftheeffective exhaust velocity corthespecific impulse c/g. Specific impulse isthethrust delivered perunit weight ofpropellant consumption, itsunits being 77< I0-II / Ib/Ib/sec orjust sec. propellant combinations aregiven below: Oxidizer andFuel Hydrogen peroxide -gasoline Hydrogen peroxide -hydrazine Nitric acid -ammonia Nitric acid -aniline Liquid oxygen -gasoline Liquid oxygen -alcohol Liquid oxygen-hydrazine Liquid oxygen-liquid hydrogenCalculated values ofspecific impulse forafew Specific Impulse (sec) 25o 26o 24o 24o 260 260 28O 360 Theabove values areforachamber pressure of500psiandexpanded at sealevel pressure; thechamber temperatures arefrom 4500 to5500°F. Incombination with most fuels, liquid fluorine affords higher values ofperformance than most other oxidizers. Several examples are shown below forachamber pressure of500psiexpanded tosealevel presser@ : Oxidizer andFuel Liquid fluorine -ammonia 300 Liquid fluorine -hydrazine 320 Liquid fluorine -liquid hydrogen 380Specific Impulse (sec) Chamber Temperature (_) 7200 79OO 51OO Expansion athigher altitudes with alarger nozzle mayincrease the specific impulse byasmuch as70to80percent. Inactual practice, three totenpercent less than the8bove values isobtained. D78< 10-12 Combustion temperatures above 6500 °Fareusually not feasible since themolecules will become unstable anddissociat%consuming some ofthe energy ofthe gases. Fluorine rockets canberun athigher temperatures than most other chemical systems since fluorine ismuch more stable at these temperatures. Forafluorine-hydrogen rocket half asmuch hydrogen would have tobecarried than foranoxygen-hydrogen rocket since only one H2molecule isrequired for each FI2molecule whereas two are required foreach 02molecule. This isabig advantage since thelow density ofhydrogen requires bulky fuel tanks. Inregard tothehighest energy content which canbeexpected there arenoknown chemical propellants which have anenergy content twice that ofthose nowincommon use. Fuel utilization isanimportant problem with aliquid fuel system. The optimum case would bethat allfuel andoxidizer areused upatburn- out. This means very close control must bemaintained over the amounts offuel andoxidizer carried andover thecombining ofthese twocompounds intheproper ratio forthedesired thrust. Any oxidizer orfuel left over atburnout means areduction intheperformance oftherocket ..... 8scanbeseen byreferring toequation (9-16) inSECTION IX. where effective exhaust velocity, atthebeginning, and Mfvf=cSnwM° Mf vf isthedragand gravity free velocity atburnout, cisthe Moisthemass oftherocket (plus fuel) isthemassatburnout. 4 10-13 I0.3 Solid Rockets andPropellants Solid-propellant rockets arestorable andthus donotrequire long delays after adecision ismade forfiring. Since solid fuel ismore dense than liquid fuel, asolid-propellant rocket will besmaller than liquid rocket foragiven impulse. Duetothehigh temperatures at whichssolid fuel burns andsince solid-propellant rockets arenot cooled, they canbeoperated foronly ashort time; otherwise, thecase would bemelted orweakened. Thesolid rocket hasnomoving parts such asvalves orpumps; itconsists ofthefour parts shown inthesimple sketch below: f/A // r/A i/ f/A -- //A -z/AV/_ fJA -- f/A r/A , -/_ // _._ /.- "-,,/ _-_ NozzleIgniter Propellant charge Case Essentially, theentire solid-propellant rocket ispressurized, asthe solid fuel comprises most oftherocket. Theigniter emits ahotgas over theentire exposed surface ofthepropellant. Solid propellants can beignited almost instantly, theshortest time being 10or12milliseconds. :SO< 10-14 Thedesign andmanufacture ofthepropellantorgrain, which isthename given todescribing thephysical body ofthepropellant, allows thethrust tobepredetermined andprogrammed. Recent developments have been made where thesolid rocket canbeturned off andreignited. There aretwoprincipal types ofsolid propellants. The first is referred toasdouble base propellant type because thepropellant is mainly acombination oftwochemical compounds, nitrocellulose [C6H702(ON02)3] andnitroglycerin [C3H5(ONO2)3]. These areunstable compounds whicharecapable ofcombustion without the addition ofan oxidizing material. These double-base compositions areeither extruded orcast. The specific impulse ofextruded Ballistite (JPN), which is51.5 percent nitrocellulose and43.O percentnitroglycerin is246 seconds atachamber pressure ofiOOO psiand239 seconds atachamber pressure of500psi. Inthefuture, the specific impulse ofsolid propellants will probably reach avalue ashigh as300 seconds. The other type ofsolid propellant iscalled composite propellant type andcontains twoprincipal ingredients, afuel and anoxidizer, neither ofwhich will burn satisfactorily without the presence ofthe other. Thepropellant usually consists ofafinely ground oxidizer mixed with afuel (while thefuel isintheliquid state andoften atanelevated temperature) and isusually cast. Typical chemicals used asoxidizers areammonium perchlorate (NHhC!O4) and ammonium nitrate (NHhNO3) andare used with such fuels asThiokol, polyurethanes, polyesters, andbutyl rubbers. Usually, 60to80percent ofthese mixtures isthe oxidizing LSi< lO-15 agent. The ammonium perchlorate produces anexhaust gaswhich istoxic andhighly corrosive tomany materials. Inboth types ofsolid propellants, other chemicals areincluded, •inaddition totheprincipal ingredients, tocontrol thephysical and chemical properties ofthepropellant ..... such asserving asacatalyst toaccelerate ordecelerate theburning rate. Forasolid propellant rocket the highest efficiency isobtained when therocket isoperated ataconstant pressure. When this happens then themass ofthefuel burned equals themass dischsrged. Now where and where Then-RSpp Ristheburning rate orthevelocity atwhich asolid propellant isconsumed Sisthe propellant burning area ppisthepropellant density _d=CdPcAt Cdisthedischarge coefficient Pcisthechamber pressure Atisthenozzle throat area RSpp -CdPcAt The burnin_ rate isafunction Ofthechamber pressure, n R=apc 10-16 whereaandnareconstants. (Theconstant adependsuponthe temperature ofthepropellant grainpriortocombustion.) Thus oraPcnSPp-CdPcAt 1 1 (i- where theconstant Kisdetermined bythechoiceofthepropellant. The ratio oftheburning area tothenozzle throat area isanimportant quantity inthedesign ofsolid propellant rockets. For example, the relation permits anevaluationofthevariation necessary inthethroat area ifthe chamber pressure (and therefore the thrust) istobechanged. From theforegoing development itappears that theproblem ingrain design istomake theshape ofthegrain such that itsburning will main- tain aconstant pressure. This isdone bytrying todesign the grain sothat theburning surface will always beaconstant area during burning. Some examples ofgrain design where itispossible toobtain aconstant burning area areshown below: (Arrows indicate burning surface.) (a) (b) (c) LSS< i0-17 Inthecase ofaninternal burning design, asillustrated by(b)or(c), none oftheheat ofcombustion gets tothecase; therefore, thecase can bemade lighter giving arocket ofbetter performance. ll.lll-1 SECTION XI AERODYNAMIC HEATING ANDHEATTRANSMISSION General Physical Principles, Temperature, HeatandEnergy Weareallfamiliar withthekinetic energy ofabody, duetoitsmassinmotion, oritspotential energy, dueto itsposition inagravitational field. Manyengineers do notfrequently workquantitatively withheatenergy; the purpose ofthesenotesis,therefore, toreview someofthe physical principles, theconcepts, andthequantities which areusedinthisfield. Weallhaveaninstinctive ideaofwhatwemeanby temperature andheat.Wesaythatheatisthatwhichcauses abodytoriseintemperature, andtemperature isaquanti- tative measure ofthedegreeofhotness orcoldness ofabody. Itisratherlikeapotential, andweordinarily adoptsome arbitrary scale, andarbitrary reference. Theicepointand boiling pointofwateraretwostandard points, givenvalues of0to100degrees ontheCentigrade scale, 32degrees and 212degrees ontheFahrenheit scale. Thesevalues areold standards, newphysics hasassociated withheatthevibration- alenergyofthemolecules thatmakeupthesubstances. This energy becomes zeroatatemperature ofzeroontheabsolute scaleoftemuerature, 0degrees Kelvin forthecgssystem, ii-2 0degrees Ranklne intheFahrenheit scale. adopted relations areThegenerally T,(°K}=T(C°)+273.18 T(°R}=T(°F)+h59.72(ii.I-I) (ii.I-2) Nowitwasoneofthegreatilluminating principles ofscience thatenergy initsvarious formscanbemeasured, andcompared onacommonbasis, byreducing eachtothe amount ofmechanical workitcando.Thispointhasnot alwaysbeenasobvious asitseemsnow,infactitwasnot believed generally true,especially inthebiological sciences untilHelmholtz, asrecently as18h7published his famous paper, "OntheConservation ofEnergy". Sincethekinetic energy ofamassmisi/2mV2, theworkdoneinlifting aonepoundweight onefootagainst theforceofgravity willgivethatsameweight avelocity V=_2=_2x32.2_8 ft/sec. Conversely, aweight of onepoundmoving at8ft/sec, would, incoming torest,do oneft-lbofwork. Somesharpfellow, probably ablacksmith, noticed thatiftheweight wasahammer thathewasswinging withallhismightagainst ananvil, sothatitstopped dead, thenthehammer, andalsotheanvilgotwarm. Thiswasa greatmystery atfirst; it'snotsolongagothatitwas finally demonstrated thatthesameamount ofworkalways produced thesameamount ofheat,andwewereledtotheidea ofthemechanical equivalent ofheat, JfortheBritish physicist Joulewhomeasured itin18_3. Themodern value LS6< ll-3 isthat Jis778.26 ft-lb per BTU; that is778.26 ft-lb ofwork done will generate sufficient heat toraise the temperature ofone pound ofwater byone degree Fahrenheit. The BTU, British Thermal Unit, isthe quantitative measure ofheat, the amount ofheat required toraise the temperature ofone pound ofwater one degree Fahrenheit. Inanearlier SECTION onthe Orbits ofSatellites we _iscussed the concept ofthe total energy ofasatellite, 2 U=_mY2-- _ "Itisinstructive toconsider r" this energy, not only from the standpoint ofthe usual measure, ft-lb, but also as U/J, orthe heat capacity. For acircular orbit ofradius rthe speed ofthe satellite is _=VR_Z@ and its total energy is r U= r Since thevalue oftotal energy uniauely fixes theseml- _n_ _Y4_ _ _n_ f.h11_ th_._4_e Ofanye,ll]nt]aal orb_t_.... _,.......................... j. wecan regard this value oftotal energy asUathe total energy ofany elliptical orbit ofsemi-major axis a, regardless ofeccentricity and write /mRe f4 Ii-5_ -o.s For the same satellite atrest onthe earth (V=O,a=R) U =_/ m These relations are illustrated infigure l,where the total energy isplotted asafunction of a/R. The abscissa isdiscontinuous tocover awide range. The ordinate on the left isanorbital energy scale, where the zero applies toabody atrest atinfinity. The scale onthe right isthe total energy ofamass of1pound inanorbit at a/R, expressed inBTU/lb. The zero for this scale istaken as atrest onthe ground, lh,000 BTU/lb are required to establish anorbit ofradius R,very little more would move the orbit out tobOO miles (a/R =1.1), twice would put it atinfinity. ll_,000 BTU/lb isalot ofheat, when itisrecalled that 180 BTU will raise apound ofwater from the ice point tothe boiling point. Anadditional 970 BTU will vaporize apound ofwater into steam, lh3 BTU will melt aone pound block ofice at32°F. Thus aone pound satellite has enough energy toturn anllpound block ofice into steam. Converse- ly, this much energy must bedissipated ifasatellite in orbit istobereturned toearth. Much ofthis energy can bedissipated asdrag, some will appear asheat, which must inturn either bedisposed ofbyradiation orbeconducted to the interior and absorbed. The following sections will review 11.2 11.3Ii-5 theprinciples ofradiation, conduction andconvection. Symbols anddefinitions ofquantities generally usedin thisdiscussion aredefined inthetable. ModesofHeatTransmission Heatismovedfromplacetoplaceinthreeways,by radiation, byconduction andbyconvection. Thelawsgovern- ingthesemodesoftransmission areofgreatengineering importance, andhavebeenstudied foralongtime,bymany ofthegreatnamesofscience. Fourier (1822)invented Fourier series todealwiththeproblem oftheflowofheat insolids, Newton statedthelawofheattransfer froma solidtoafluidin1701. Radiation Everybodyemitsradiant energy inalldirections. Whenthisenergy strikes another bodysomemaybeabsorbed, somereflected, andsometransmitted through thebody. The partabsorbed istransformed intoheat. Thelawofthermal radiation wasdiscovered empirically (1879)thenderived theoretically (188h). Thequantity ofheatradiated per second fromasurface ofareaAatatemperature Tis dQ:A dt (11.3-1) This law aswritten applies toaso-called "black body", anideal radiator, orwhat isthe same thing, aperfect absorber, atall wavelengths. Most substances reflect some part ofthe illumination that falls onthem and are thus not L89< Ii-6 black bodies. They are characterized byaquantity called the emissivity, which isdefined asthe ratio of the total energy radiated byabody attemperature Tto the energy radiated byablack body atthe same temperature, i.e. (dQ/dt) = surface (ii.3-2) (dQ/dt)black body Polished metallic surfaces atroom temperatures have low values ofemissivity, the lowest being gold witm avalue of 0.018. Most polished surfaces darken asthe temperature increases, due tooxidation ofthe surface; under these conditions the emissivity may range uptovalues between 0.8 to0.98. Extensive tables ofemissivities have been published. The constant cinequation (11.3-1) isthe Stephan- Boltzmann radiation constant, the value ofwhich has been determined as0.47594 xi0"12 BTU/(Sq ft)(Sec)(°R) 4. Thus for practical surfaces the total energy per second emitted asradiation may bewritten as 4Q d--'_'- /000 /(ii.3-3) This radiant flux isdistributed continuously over arange ofwavelengths inthe spectrum ofelectromagnetic radiation which includes radio waves, and light, ondown tothevery short such asX-rays. Some representative points and bands inthis spectrum are tabulated below, where the wave lengths 90< Ii-7 are expressed inAngstrom Units; one AUis10-8centimeter. Spectrum ofElectromagnetic Radiation Radi ation X-rays Ultra Violet, less than Visible spectrum Infra red, greater than Hertzian waves, greater than Radar WVEC -TV WVECWave Length AU io-15o 4,000 4,000 -7,000 7,000 2.2 x106 109 6.26 x109 2.012 x1012 The energy per unit area per unit time emitted bya black body inthe wave length range/_ to)_+ d_ isexpressed byPlanck's Spectral Distribution Law as Whereec /RT_/ c=velocity oflight h=Planck's constant k=Boltzmann's constant c2=1.4387 cm°K =2.5897 cm°R(n.3-4) ii-8 The total area under this function, i.e., integration between O(__m_ leads toequation (11.3-3). The shape ofthe spectral distribution was first determined experiment- ally but classical physics was unable toexplain it, orpro- vide aformula for it. Itremained apuzzle until Planck proposed the quantum theory and in1901 derived his dis- tribution function which agreed with the hitherto inexplicable curve. The spectral distribution has asingle peak atawave- length _p which varies inversely with the absolute temperature inarelation known asWien's Displacement Law, Or(ii.3-5) The movement ofthis peak from the red end ofthe spectrum tothe blue with increasing temperature isreflected inthe increasing whiteness ofluminous sources such aslight bulb filaments with increasing temperature. Practical use is made ofthis property inuse ofthe color scale oftemperature. For many practical pruposes, radiant energy has its source onhot walls ofasolid, and the temperature ofa body can beestimated visually from its color byuse ofthe following table. _'_,, ii-9 Color Scale ofTemperature Color Incipient Red Dark Red Bright Red Yellowish Red Incipient White WhiteTemp. OR 1390 -lh80 1660 -18hO 2020 -2200 2380 -2560 2740 -2920 3100 -3280 Gases like nitrogen and oxygen donot radiate to any extent, nor dothey absorb. Carbon dioxide and water vapor onthe other hand, like many other gaseous compounds have complicated absorption bands especially inthe infra- red and thus can emit acertain amount ofheat. The net effect ofall atmospheric components has been correlated onthe basis ofanemperical relation which treats the atmosphere asaradiator atthe ambient atmospheric temper- ature TAwith anem_q_v_ty....... a__,_,,_,, ....,_^- linearily withthe square root ofthe atmospheric pressure. The relation forthis sky radiation factor may bewritten as (11.3-7) andthus anexpression forthe radiant energy from space and the outer atmosphere absorbed onasurface is :A rA (11.3-8)dt :.93< 11.4ii-i0 This relation, based principally onground observations, should beapplicable atmoderate altitudes, its suitability for use inthe upper fringes ofthe atmosphere isnot known, since here some atmospheric com- ponents are ionized oratomic rather than molecular. The energy exchanges associated with ionization and dissocia- tion ofgases are frequently good sources ofradiant energy, the sun for example. Asasource ofradiation the sun is not ablack body, but the radiant flux from the sun has a value known asthe solar constant Cand taken as0.1192 BTU/(sq ft)(sec). This value istaken asapplying atthe top ofthe atmosphere, atthe mean distance between the sun and the earth. Thus the heat absorbed from the sun can be written J_ = (li.3-9) where _isused instead ofthe absorptivity for solar radiation, aquantity which for some materials differs to some degree from the absorptivity for black body radiation. Convection Newton, in1701, defined the relstlon for the heat transfer between fluid atatemperature Tand asurface atatemperature TS. The relation may bewritten as J-gQ: A(r- (ndt Equation (iI._-i) isnot aphysical law but anexpression of experimental observations which isapplicable tomany sur- faces and fluids; the latter may beliquids orgases. The < ii-Ii convective heat transfer coefficient hisnot acal- culable Quantity but theoretical considerations indicate that itshould beafunction ofboth the flow conditions and the thermal properties ofthe fluid. Flow conditions can becorrelated bymeans ofthe dimensionless ratlop the Reynold's Number Re, fluid properties interms ofthe Prandtl Number Pr; the convective heat transfer coeffi- cient isexpressed indimensionless terms bythe Nusselt Number, Nu. Theory does not provide anindication ofthe functional nature ofthe relationship between these three dimensionless ratios, but experimental evidence tends to confirm the general indications oftheory. Anumber of experimental correlations have been published, one fre- quently used isofthe form or where Aissome constant and dimension and the exponent m about 0.31. the formLissome characteristic appears tohave avalue For spheres inair equation (ll._-3) takes _D _0.37(eVD )0.6T - x For useofequation (Ii._-_) temperature T fluid isrequired. Inaeronautical applicationsfor the Twill Ii-12 not ordinarily bethe ambient temperature ofthe atmos- phere because ofthe phenomenon ofaerodynamic heating. To discuss this subject several special temperatures must be defined. Stagnation temperature TT, adiabatic wall temperature Taw,and surface temperature TS. Sta_natlon temperature: Consider abody moving with avelocity Vthrough air atrest atatemperature TA. Atastagnation point onthe body the air isatrest relative tothebody and has thus also acquired aspeed V relative tothe ambient air. Work has thus been done on this air, increasing both temperature and pressure, in- creasing its total energy. For aperfect gas theory ex- presses the stagnation temperature interms of TA, the Mach number, and the adiabatic exponent _ which isthe ratio ofthe specific heat ofthe gas atconstant pressure tothe specific heat atconstant volume. This relation is (ll.&-5) Although useful for values ofMach number upto2or3, equation (11.4-5) isstrictly applicable only for the limited range oftemperatures and pressures over which the specific heat ofair issubstantially constant. More accur- ate values ofstagnation temperature are obtained from enthalpy tables. Enthalpy isameasure ofthe total energy ofagas, Air atrest and anambient temperature TAhas L96< n-13 avalue ofenthalpy per unit mass hTA given inTable 2. Anincrease inspeed toavalue Vprovides anenthalpy increase Ah-V2/2gJ. The stagnation temperature isthen that temperature corresponding tothe enthalpy hTT where V_ h_ =_7A+29J" (ii._-6) Acomparison ofvalues ofstagnation temperature computed byequations (ll.h-5) and (ll.h-6) isgiven below for TA=518.h orsea level inthe standard atmosphere. TT TT M_ _"=I._ enthalpy basis 1 622 622 2 933 929 3 1_51 lh18 h 2177 2060 5 3110 28h4 6 4251 3770 7 5599 4830 8 715_ 6o_o Adiabatic Wall Temperature: Atany point ona moving object which isnot astagnation point, the air is also atrest, but through the boundary layer there isa transition tofree stream conditions. The nature ofthis transition issuch that the surface may assume anelevated temperature near but less than the stagnation temperature. Ifthe heating process isanadiabatic one, free from the ii-i_ effects ofconduction and radiation, the temperature reached istermed the adiabatic wall temperature Taw. Data on Taw are correlated byatemperature recovery factor Kwhich isindicative ofthe proportion ofthe total temperature rise TT-TAachieved, that is _ (n.E-7) The actual value of Kvaries with the boundary layer, being smaller for laminar than for turbulent boundary layers. Theoretical studies byPolhausen indicate that for flat plates parallel tothe stream Kisafunction ofthe Prandtl number, based onaPrandtl number for air atordinary temperatures of0.72, these studies indicate Laminar boundary layer, Turbulent boundary layer, values which are ingeneral confirmed byexperiment. For blunt bodies, the recovery factor will, ingeneral, vary with the shape ranging upward from the flat plate valuesK=(Pr){=0.85 K=(Pr_ =.895 toward unity. Surface Temperature: The temperature actually assumed byasurface under the action ofconvective heating depends not only onthe adiabatic wall temperature and convective heat transfer coefficient, but also onany exchange ofenergy -q%Q< n-15 byradiation, and byconduction ofheat tosome other parts. Under steady conditions asurface initially at atemperature TSwould reach atemperature Taw under the action ofaerodynamic heating alone, but radiation to other bodies, the earth and outer space can reduce this value considerable. Equilibrium Temperature: Under the combined actions ofaerodynamic heating, and raSiative processes, with or without solar heating abody will ultimately reach an equilibrium temperature Tewhich isordinarily less than the value of Taw appropriate tothe operating conditions. At Tethe net rate ofheat transfer iszero. The time taken toreach this equilibrium temperature depends toa large extent upon the magnitude ofthe heat transfer co- efficient. Consider aconducting flat plate initially ata temperature TS, one face ofwhich with area A/2 is total aero@ynamic heating input would be, byequation (ll._-l) hA(Taw -TS),the solar input would be C_ The radiation exchange with the earth and space would be expressed by£o'A(_7AA4- _J .Atequilibrium where TS-_ Te, the sum ofthese terms iszero and With due attention tothe areas involved, and the presence orabsence ofsolar heating, similar expressions can be written for other shapes. These expressions are all ofthe 3..gg< ll-16 fo= _re÷_r_-_-o,thepositive _eal root isthe equilibrium temperature. When speeds are ofthe order ofsatellite speeds, Taw Isofadifferent order ofmagnitude than TA, for such speeds more Isknown atpresent about the product hTaw appearlng Inequation (ll.[_-8) than about either hor Taw alone. Under such conditions, a frequently used correlation isthat byRomlg, which gives the heat transfer rate atthe stagnation point ofahemis- pherical nose ofradius Rninterms ofthe Mach number M and free stream pressure pas 11.5I__o0/45Ma'__ oft.,-" _ (ll.h-9) Conduction Inanopaque solid, conduction Isthe only mechanism ofheat flow. Under the influence ofatemperature gradient, kinetic energy istransferred from amolecule toadjacent molecules. The flow ofheat through amedium isthus In many ways analogous tothe flow ofelectricity. Heat flows from aregion ofhigh potential (temperature) toaregion oflow potential; asubstance Ischaracterized byits conductivity k, asort ofreciprocal resistivity. Just aselectricity can bestored byincreasing the charge Ina condenser, heat isstored Inasubstance byincreasing the temperature. These two properties offlow and storage, are characterized bytwo equations which are fundamental toany 4C0< II-17 consideration atthe conduction ofheat. The flow equation: JQ --_AJr dt -dx (II.5-i) expresses the quantity ofheat passing aboundary inthe medium interms ofthe temperature gradient dT/dx atthat boundary. The storage equation: = wf,__drc,,. (n.5-2)dZ relates the quantity ofheat stored inavolume ofarea Aand thickness Ax, tothe rate oftemperature increase and the heat capacity ofthe medium. The heat capacity, the quantity ofheat required toraise one cubic foot of the substance one degree fahrenheit, isthe product cw, ofthe specific heat c(BTU per pound) and the specific weight w(pounds per cubic foot). Values of cand w for some representative materials are given inTable 2. The application ofthese two equations tohest con- duction problems isillustrated bythe classical Fourier Heat Conductionequation for one-dlmenslonal heat insteady heat flow. Consider ahomogeneous medium, either arod of area Aan4 length Linsulated sothat heat can flow only inthe xdirection, orasmall section ofalarge slab ofthickness Lunlfolnnly heated sothat heat flow inthe yand zdirections canbeignored. Ifatsome point x inthe medium weconsider aboundary across which heat is flowing, the quantity ofheat which flows perunit time by 401< ii-18 equation (11.5-1) is f<,,a/ kd_Jx:-kA(ax)x Between the boundaries at xand x+Ax there lles aslab ofthickness Ax, and heat capacity cw. The difference between the heat which flows into the slab at x, and that which flows out at x+Ax, is ,ax X÷AX (11.5-3) This isthe heat which remains inthe slab an4 which by the storage equation produces arate oftemperature increase dT/dt specified bythe relation The temperature gradient at x+Ax isrelated tothat at xbythe relation :+ax t_--7._-]_x (ll.5-5) where the second derivative isevaluated atsome point between xand x+Ax. For vanlshingly small values of Ax, then thebracketed term inequation (II.5-_) becomes _,-.,= tdx,/x(ll.5-6) 402< II-19 hence, the equation for one dimensional unsteady heat flow, ord_ m/_* (11.5-7) cwA_L. -k d_ ax-'r (ll.5-S) where the partial derivative isused since Tisafunction ofboth time tand the coordinate x. Inheat transmission problems, equation (11.5-8) is generally written as z (n.5-9) where the quantity =k/cw termed the diffusivity, and which has the dimension (length squared) isauseful measure ofthe heat conducting properties ofamaterial, the meaning ofwhich can perhaps bebest illustrated byasolution of equation (11.5-9). Asapartial differential equation, there are nosimple direct methods for the solution ofequation (11.5-9). Solutions toanumber ofspecial problems areknown, however, (see for example reference 1). The nature ofeach solution isdetermined bythe boundary conditions, that isthe temper- atures atparticular times atparticular points inthe medium. Asanexample ofsome practical interest, consider aplate ofthickness L, initially atauniform temperature TO, "JC_"- ll-20 one edge ofwhich, (that atx=0)isinstantaneously changed toatemperature T1. Such aproblem represents anidealization ofathick skin, one face ofwhich issudden- lysubjected toahigh temperature asinaJet, i.e., where the heat transfer rate istaken asinfinite. For this pro- blem the initial, orboundary condltlons, are, for all values of t (I)T,-T,:,_x-O (2) -_ =Oo_x=L Q"K This second boundary condition follows from the assumed idealization that noheat can flow across the unheated face at x=L. The solution toequation (11.5-9) isthen found bythe method ofseparation ofvariables, inwhich itis assumed that the functional dependence oftemperature onboth xand tcanbeexpressed bythe product oftwo functions, here denoted @and Xwhere @isafunction of and Xisafunction of xalone, that istalone, Often itiseasier tosolve for the unachieved temper- ature rise Thus if7":7-_,_-).- _ (ll.5-1o) 7: X(4 4C4< ii-21 and<)2._ --_E)_ (ii.5-12) Jx_'d,(" substitution ofthese relations inequation (11.5-9) gives X_ _/_ (ii.5-i3) ordividing both sides by T=_"X Bd'-_X(/x" (ii.5-i4) This relationship can apply only ifboth sides ofequation (ll.5-1h) are equal toaconstant, which can bewritten as -b2 Then d@, .÷6__8---C) de (11.5-15) J'_X÷b_ =0 (ii.5-i6)TV Itcan beverified that E_=ge isasolution toequation (11.5-15), also that X--Aco5bx+B_,_bx(ii.5-i7) isasolution toequation (11.5-16) and thus that Tis given bythe product ofthese two relations, or ;lOS_ Ii-22 where the constant Gisadded asaconstant ofintegration. The boundary conditions are now used todetermine the values of d, F, Gand b. From the first condition T=T1at T= 1. Thus D=0,since e G 1. Hence T/V-_- _a_ = _INb×x=0,therefore, isnot zero, and dT dx=F6 - From the condition that that F#0,and b=)_L ---.__7-:_0 a_X=L; itfollows 7)_ -;7=/_3j-_---- and thus - 5/Nng_K 2L n=I,3,_----- There are anumber oftermsF_IN _K each of 2L which represents asolution ofthe equation, hence their sum represents asolution, and thus 7-= /+_F__-_a_'/_" where _==S/N 2_ (ii5-18)ZL (n.5=19) At t=0, T=TOand _p= 0for 0< x___L, hence the Fncan beestablished from the Fourier series relation for aunit step, 4C6": ii-23 Hence,.sIN =0 £x-o3 w,- 2L -2_-_/_c J(ii.5-20) + isthe solutiontothe problem. Thetemperature distribution expressed byequation (11.5-20) isplotted infigure 11-2 for _arious values ofthe ratio t/to.The initial temperature step to T1atthe face x=0penetrates farther and farther into the slab with the passage oftime. There is essentially nowarming ofthe inner face at x=Luntil atime tslightly inexcess of O.lt chas passed after which the temperature increase israpid. The time interval (iI._-20) isacharacteristic time fixed bythe thickness L and the diffusivity _which isthus seen tobethe square ofacharacteristic dimension ofthe substance. This dimension isameasure ofthe distance into the material which aninitial surface temperature increase will penetrate inagiven time. The two values ofdiffusivit_/ used for illustration infigure ii- 2are representative ofiron and carbon, the solutions are for the unheated surface ofone inch slabs ofthese two materials. 4%7< n- Carbon, with its higher diffuslvlty, experiences amuch more rapid temperature increase than does iron. Equation (ll._-lS) converges rapidly, and is, therefore, easy touse for calculations. Inorder toobtain it, how- ever, simplifying assumptions were necessary, infinite heat transfer attheheated surface, and noradiation. These assumptions would generally betoo unrealistic for any practical aeronautical applications. The assumption of infinite heat transfer isnot anecessary one, the solution isknown (reference l)for afinite value ofheat transfer coefficient from aconstant temperature medium, but itdoes not converge rapidly, and isawkward touse. Ifradiation must beaccounted for, exact solutions ofthe differential equation ofheat conduction have not been found, and recourse must behad toapproximate ortonumerical methods, some of which will bediscussed under Heat Protection, SECTION XII. ii-25 Reference I.Carslaw, H.S., and Jaeger, J.C.: Conduction of Heat inSolids. The Clarendon Press (Oxford), 1947. .McAdams, William H.: Heat Transmission. McGraw- Hill Book Company, Inc., Third Edition, 1954. C C Cp Cv g h hT J K k ? P q q Re T t V W Nu Pr c:_ £ "p (JIi-26 S_BOLS solar constant, 0.1192 (BTU)/(Sq ft)(sec) specific heat ofmaterial (BTU)/(lb)(°F) specific heat atconstant pressure (BTU)/(lb)(°F) specific heat atconstant volume (BTU)/(lb)(°F) acceleration due togravity, 32.17k0 ft/sec 2 convective heat transfer coefficient (BTU)/(sec)(sq ft)(°F) enthalpy per unit mass ofair corresponding totemperature T,(BTU)/(Ib) mechanical equivalent ofheat (778.26 ft-lb/BTU) temperature recover factor thermal conductivity BTU/(sec)(sq ft) deg F/ft characteristic length atmospheric pressure (ib)/(ft) 2 quantity ofheat, BTU rate ofheat flow per unit area BTU/(ft)2sec Reynolds number, (V_p/_) temperature °Fand °R time, seconds velocity, (ft/sec) specific weight ofmaterial, (ib)/(cubic ft) Nusselt number, (hq/k) Prandtl number, (Cp_g/k) diffusivlty, k/cw emissivity Adiabatic exponent, Cp/C v density (slu_s)/(cubic ft) Stefan-Boltzmann radiation constant O.h759h xlO-12 BTU/(sq ft)(sec)(OR) _ coefficient ofviscosity (lb)(sec)/(sq ft) 410< IH H _0_-+_-__.__-__._"u_ I.__D_-_0_(M _dddddddd'___gd_ 0D'-C_0_0r-ir-I 0r--I_O_1_ ---f__D'..-0____0C____0 I 0 0 • I H,-I 0_O00"_COr'4__0 0_O0 _ P-__Od:_._0"_D-.D0_m('_00_"I 0 * 0 • I ,--t r-t [',-D--c_0__100 c_('_ cO I I___.__DO__-I__D OJ000H I "L____-J"_OOq_)_O--.--t"__ I O " O *I•I I_.1C_q9C__-I_"_O 0D'--D--D'-OO I C_COCO__0_0_D- 0____0_._ C_0_0"_0C'__;._00COD'-_O_(_)__.-J- 0 I r__-_I'4 C_JL_COr___- D'--O_['.-_O_-_('_D- _._.,D_'X_.-_J'X(M 0 • * 0 _-0__-__o___,_o__.._,_ _-,_-_-,___oo__-,_0__o o _o4,____g4,_,,'___-o_._o__o_oc_o o 0 0 Qr-I ! _D0 ('_ I 0 0_0 Ag 0d ! OO0d t'-.- P,I A (_ O Og_ g._4 o_O _ t'-- OO OO ,D ('..-Oc_J .-I_-I A 00._D _'_cO g Ox OO OO coO_.O I-4 0 0 0 OIO O O O O 0O_ O_ (M ,4:) g O_ 'L_ O O ;OCOO4 OO_. -.__D OO OO OO (_O CXJ cO O _O A O_ O_. O O O _-t-o _0(_- _0 c',l,_ i--4 i.-4 O O OO O12) /._,,DO m O_ 'x_0 E "0 I1) II 4: .1< e Ir-4OC 0 b- 0 ,.-I_/x 04 O_oD.J ,..-I 4Du", I---- 00 /_ .-_ I i--It,_ _ o,J:0 ,--I_ * .O C.- :0 0_-t"Lr_ I._ d 0 0 od _ _'- icd _t;X _D _0. r-_od 0t'- a: _ _ _ lI(__ C'_ _•0 I 0 0 ,'-4 _g ei'--I _ m 0 m e cO _m _0 m_ •t_4-_ _"_ Nr_o_ •I-_I_ D'- 0u__.,-I I_ @c_ A •,-I__Or.4 _f_OcO cO _,.0 .D !l/x_-.0 0",0_] I - _.__o__._._-_etl ,-4e-4 e 04 _ L_-q_ L_- 0_"CO _._:._.._,,,_ ll-28 THERMAL PROPERTIES OFINSULATORS (from reference 2) InsulatorW Temp. °R ib/ft 3k BTU/sec ft2 _F/ft Magnesite Diatomaceous earth (moulded and fixed) Kaolin Aluminum foil sandwich858 1661 2651 858 2059 852 1860 56o 811158 h2•3 19 .26.1 xi0-h _.h5 3.05 0.390 .528 •139 .31h •069 .106 I I I i 1 \ II II kG "x3 i_ . _:_ 4:;..4< _o I I _J _x }-/.0 .6 6 .4 .2 0 .2 .4 ._ .8AO [- I I "4/,0 .6 .6 .4 .2 II 0-4- o<[=_0×lO 7/- / /oL=lINch, ,_=IxlO 20 .70 40 _0 "__..v_-c Figure 11-2 Temperature variation with position and time for se_,i-lnfi:_ite slab subjected tostep te,_perature input 8t face x=O. 12.112-1 SECTION XII HEAT PROTECTION Introduction Inthe discussion ofbasic heat concepts inSECTION XIitwas pointed out that energy equivalent tol_,O00 BTU/lb must beexpended toreturn objects toearth from satellite speeds. Some ofthis energy isdissipated in the form ofdrag, that is, velocity changes, while the remainder must beabsorbed asheat orreradlated. The energy values which enter into the heat protection problems tobeconsidered here are only asmall fraction ofthe l_,O00 BTU/lb figure. For example, toreturn awinged vehicle similar tothat proposed bythe Flight Research Division from orbit requires the disposal ofonly lSO BTU/lb. This amount ofheat, for example, for a_,000 pound airplane could beabsorbed by600 pounds ofberyllium with atemperature rise of2,000°F. This, ofcourse, may not bethe most efficient way tohandle the heating aswill bedemonstrated later. There are available structural materials which retain sufficient strength tobeused attemperatures asnigh as 2000°F. There isconsiderable question atthe pressnt time, however, whether practical structures canbeoperated at 12-2 2000°F. Oneimportant feature sometimes glossed overis theeffect oftemperature gradients whichwouldbeassoci- atedwithhotstructures undertransient heatin_ conditions. Forexample, aRenekl(nickel alloy) structure couldsupport temperature gradients ofonlyabout350°Fbefore approaching failure frombuckling considerations. Several meansareavailable tolimitstructural temper- atureto2000°Foranydesired lowervalue(practical limits arise, ofcourse, inthesizeandweight oftheinsulating or shielding material). maybeclassified as a. b. C. d. e.The various heat protection schemes Radiators Insulators Heat sinks Surface evaporators Internal coolers Every heat disposal scheme contains elements ofmost ofthe others. First let usconsider the methods just listed to see what physical processes are involved ineach. a. Radiators.- The ideal radiator isablack body and most materials radiate heat from the surface only but itis still impossible toraise the surface temperature ofa radiator toapoint for efficient radiation without conduction ofsome heat tothe in%erior ofthematerial. 417< i 12-3 b. Insulators.- The ideal insulator has, ofcours_ a low conductivity but large bulk even with low density is usually associated with insulating materials. c. Heat sinks.- The process here istoattempt to retain insome non-structural item the heat input associ- ated with high-speed flight. Materials with high specific heat are desired and ifhigh specific heat can beassoci- ated with low density somuch the better. d. Surface evaporators.- Under this classific_tlon wecan consider the melting, evaporation, sublimation or ablation ofvarious metals and orsynthetic compounds such asteflon. Inthese processes the latent heat offusion or vaporization isused tolower the surface temperature. In anablation process achange ofstate brought about bya chemical reaction and relatively high temperatures may be required for efficient heat absorption. e. Internal cooling.- With this system ofheat pro- tection complications arise from the plumbing requirements and the difficulties involved inforced convection heat transfer calculations. The major portion ofthese notes will bedirected to- ward presentation ofequations useful inthe evaluation of heat transfer problems for Radiators, Insulators, Heat Sinks and Ihope combinations ofthese three schemes. The basic laws ofheat transfer have been covered in SECTION XIbut the actual solution of most heat transfer problems isdifficult compared tothe apparent simplicity 12-4 ofthebasicprinciples. Fortheremainder ofthisdis- cussion simplifying assumptions willbemadesothatsome answers andsomephysical pictures oftheprocesses in- volved alongwithnumerical answers maybeobtained. Thefirstsimplifying assumption willbethattheheat inputtoourstructure isindependent ofthestructural temperature andhasbeenspecified bysomeindependent calculation. Thuswewillbediscussin_ timehistories of heatinputs suchasareshowninfigure 12-1. Fi___ou_eio_-ia istheheatinputinBTU/ft2sec. ascalculated forthe_,e- entryofalargeflatobject at90°angleofattackf:_'_ orbitatanaltitude ofabout70miles. Figure 12-1bhas aboutthesametotalheatinputoverashorter period of timesimilar toapulsewhichwouldcorrespond morenearly totheheating ratetimehistory ofacapsule vehicle __rin_ re-entry. 12.2Radiation Ifalltheheatinoutcouldber_<_iated back,th_ temperature timehistory oftheskinofthestructure c,::_uld becomouted fromtheeoustion forradiant heatinterc__-_Te _-2-+Jk_-i ifbody_(theatmosphere) beconsidered ablackbody withT2h_0then 4i9< 12-5 _A_T¢ I I-,,'z I I.J.._+I.J.__I EII :g,E,a---r,4(12.2-1) Where and Note:qisthe rate ofheat radiated Athe area athe Stefan Boltzman constant the emissivity Tthe absolute surface temperature can range from 0.018 for gold to.98 for flat black lacquer; for metals _increases with temperature until they become molten. For most nonmetallic substances crease intemperature. negative gradient ofdecreases with in- Quartz has alarge with Tand isone Agenerally used value for emissivity is0.8.ofthe few substances whose thickness effects the radiation properties. Thus equation (12.2-1) becomes 3_=._8OlI-_-ooo)eA(12.2-2) where qisinBTU/sec Ainft2 TIindegrees Rankine 42L;0< 12.312- 6 The surface temperature time histories using equation (12.2-2) for the time histories ofheating rates infigure 12-I are shown infigure 12-2. Heat Shield Case The forgoing example is, ofcourse, entirely hypothet- ical since all the heat has been re-radiated and none con- ducted orabsorbed bythe structure. Aheat shield or_'-:k may beused toreduce thesurface temperature toalevez _c which structural elements retain some strength. Aheat shield ismost efficiently constructed ofmaterials ,_lich have ahigh specific heat. Two substances inliquid form, water and lithium have high specific heats iand 1.;4, how- ever, aside from the diffioulty ofcontaining these fluids asliquids the operatlng temperatures ofconcern here are such that both would become vaporized. For both, ofcourse, vaporization would absorb large amounts ofi_leat but then they would not strictly speaking, beheat sinks. The material which seems most suited for aheated shield isthemetal beryllium which has as_ecific heat rangln_ from 0.[, atO°F toabout 0.7_ atl_O0°F, the variation isnot linear. Another metal whose soeciflc :eat isofinterest isnickel wlth aspecific heat ofC.l at O°F and 0.15 at1600OF. The following m2terlal will cover the calculation ofthe temperature rise due tothe absorption ofhe_t for the :_:_t in- 12-7 putsoffigures 12-1aand12-1bforberyllium andnickel. Theappropriate formula foruseinheatabsorption calcu- lations is 4Q-VwcdT Inthis equation the volume inft3, specific heat, and R.W T (Asused herein)(12.3-1) Qisthe quantity ofheat inBTU, V the density inIbs/ft 3, c the the absolute temperature indegrees Inthe derivation ofequation (12.3-1) f_see reference I)the assumption was made that the material was thin with ahigh conductivity and thus notemperature gradient existed from the front toback face. Equation (12.3-1) may also bewritten as Tt Q= c J (12.3-2) when the specific heat varies with tenperature. Since Wthe weight =wV equation (12.)-2) is k,-_._.-o ..)_j I where andTOisthe temperature attime O. Infigure 12-3 the specific heat curves for beryllium and nickel asafunction oftemoerature are shown while inj.'_ figure 12-k the variation of C_ asafunction of Tare given for aninitial @To=500°R. Equation (12.3-3) isthen used tocalculate the temperatures which nickel and beryllium will reach when subjected tothe 12.h12-8 heating ratecurvesoffigure12-1aand12-lb. There- sultsareshowninfigure12-_forunitweights (W/A)of nickel =12lbs/ft2andberyllium equalto3lb/ft2. Different weights werenecessarily usedotherwise the nickelwouldmeltbefore absorbing thetotalheatinput. Theresults showninfigure 12-_werequantitatively obvious fromfigure 12-5whereitcouldbeseenthatroughly timesasmuchnickel asberyllium wouldberequired to absorb theheatinputequivalent toabout2300°R. Theefficiency ofberyllium asaheatshield is demonstrated bytheseexamples, however, intheactual physical casewhereradiation enters thepicture therewill occurinstances wheretheheatshield onlyaddsuseless weight. Radiation PlusHeatShielding Eouation 12.2-2) applies forthecaseofsingle sur- faceradiation. Thethinmaterials wearespeaking ofhere mayradiate frombothsurfaces atthesametemperature, thus equation (12.2-2) maybewritten as whereSiseither 1or2. Integration ofequation (12.|_-I) A )kloool t-ogives(12.[!-1) (12.k-2) 12-9 Ifheat_sbothradiated andabsorbed combination of equations (12.3-3) and(12.h-2) gives -- To (12.k-3) Thesolution ofequation (12.k-3) forTIcanprobably behandled bysomeveryhighgrademathematics butthe approach detailed inthefollowing produces answers ina reasonably shortperiod oftimebytriala_derror. The valueofCisverysmallfortheexamples tobeusedand willbediscarded. Thetimeinterval chosen isAtand trapezoidal inte=ration willbeemployed, :t-- \I_°°;n+,3801 5 ¢ .%8o15\i01] _\ioool_-, To (12.1 -4) ,_o'-aoion (.4___ --_ i_.c-_s ma_, besolved bytPial _lu err_ u_by guess until the left hand side equals the right hand side. Asanexample let ususe 2ib/ft 2ofberyllium with At double surface r_diation, -_and the heat input2 curve offigure 12-1b. Equation (12.h-4)becomes 4T, ,,t(.-O 0 4a4< 12-i0 The time chosen for evaluation isi0_ec., thus "g°56r',,oooj 4, .roedr--- 2.&a7jogoo°% =.5(6o) -o-1.9o (.5) =30.00 -0-.12 =29.88n=l a try T1=800 then d_ from figure 12-_ :ik5 and the lefthand side becomes 1.903 (.8)£+145 =.78 +lh5/ 29.88 From this itisseen that the temperature inthis temperature range isprimarily determined bythe heat capacityso from figure 12-4, (TI)est =560°R 1.903 (.56)h +29.5 =.19 +29.5 =29.69_29.88 This isabout asaccurately asthe temperature (560 °)can beobtainea from the scales used infigure 12-h. The temperature time histories for various unit weights ofberyllium when both radiation and absorption effects are considered are shown infigure 12-6 for the heat input of figure 12-lb. Increasing the unit weight from 1lb/ft 2to klb/ft 2lowers the heat shield temperature 1020°F orin other words, anadditional heat shield of3lb/ft 2of beryllium lowers the temperature 1020°F. q_r D 12.5 12.612-ll Comparison ofRadiation and Absorption Effects for Pulse Type Input Arehash ofsome ofthe previously presented data for the pulse type ofheat input offigure 12-1b demonstrates the effectiveness ofthe heat shield asindicated infigure 12-1. For the case where all the heat isassumed tobe radiated the surface temperature follows the heat rate in- put curve and amaximum temperature ofabout 3500°R is reached. _2nen the temperature iscalculated using the absorption formula (and W/A =k.0 lb/ft 2)the surface temperature time history has the shape ofthe integral of the heat rate input curve with amaximum temoerature of about 1700 degrees. The combination ofboth effects which isquite close tothe actual physical case reaches amaximum temperature ofabout 1600 deg. after which heat islost by radiation. For this type ofheat input with the relatively large amount ofberyllium used, radiation has arelatively _ effect on_lelaaxlmum temperature. Asthe unit weight ofthe heat absorbing material isreduced the time history will tend toapproach the shape ofthe radiation case curve asindicated infigure 12-6. Low Heating Rate Re-entry Inthis section the effects ofsingle and double sur- face radiation incombination with the heat absorption capabilities ofberyllium andnickel for the low heat rate data offigure 12-1a are presented. 6 12-12 Equation (12.4-3) isused for the following cases although the actual calculations were made bymeans of equation (12.h-k) orminor modifications of(12.[_-k). The cases calculated for illustration are shown infigure 12-8 and listed inthe followin< table: Material W/A Type Radiation Beryll ium 2.0 sing le Beryllium 2.0 double Beryllium 1.0 double Nickel 1.0 double Nickel and 1.0 for double Beryllium eachCurve No. a b C d e Several conclusions may immediately bedrawn. a. The maximum temperature reached for single sur- face radiation ascompared todouble surface radiation (curves 12-8a and 12-8b) isonly about 200°higher for beryllium. b. AW/A =1for beryllium, curve 12-8c re8ches a 150°hi_her than the W/A =2 temperature only for beryllium, curve (b), suggesting that an efficient way ofnan@ling theheat input isby radiation. c. The observation of(b] above isborn out bycon- sideratlcn ofcurves (d) and (e) where curve (d) can beconsidered anickel structure with skin wei_ht ofiib/ft 2the tempereture ofwhich isnot 12-13 materially effected bythe addition ofa 1.0 ib/ft 2beryllium heat shield asshown incurve (e). Arapid temperature rise for the nickel skin occurs early inthe time history allowing the structure toradiate alarge portion ofthe heat input. 12.7 Heat Shield vsRadiator The decision astothe most efficient type ofheat dis- sipation touse tokeep the all upaircraft orcapsule weight within reasonable bounds depends, ofcourse, onthe type of heat input aspreviously demonstrated. One more figure will bepresented tobelabor this point for the last time. In figure 12-9 the effectiveness ofheat shielding for the two types ofheat input used isshown. The solid curves correspond toanunprotected nickel structure while the dash- edcurves have a1lb/ft 2heat shield ofberyllium added. figure 12-1b datathe heat shield reduces the maximum temper- ature about 350°F, but for the slow heating rate data of figure 12-1a thetemperature drop isonly about 70°F and the structural weight has been unnecessarily penalized. Of course, the maximum allowable structural temperature must also beconsidered indeciding uponthe most efficient form ofheat protection includingthe use ofinsulating orablat- ing materials, 4 8< 12-14 12.8 Insulators and Conductors The solution ofheat protection problems involving insulators isaconsiderably more complex problem than any- thing discussed sofar. The general differential ecua[c'. which must besolved is 3 %)tJ3ti Ifthe conductivity kisassumed tobeconstant, an,_.:_ substance ishomogeneous and isotropic, the equation for aninfinite slab i,e., one directional heat flow becomes We _Xz .__ (12.8-2) Numerical methods have been developed which make the application ofequation (12.8-2) topractical cases some- what easier. The following isfrom reference 1and isbased onthe "General Numerical Method ofDusinberre". Figure 12-10 shows the cross section ofalarge slab ofthickness xdivided into 4equal slices of Ax each. The heat balance for the cross hatched zone abcdiswritten as k(n--r,)_k(-r,-u)= --r,) A'X z_ _4: (12.8-3) byusing and 12.912-15 and lapseoftimeAt. Equation (12.8-3) maybesolvedfor .To Misthe new temperature atplane Iafter a where(12.8-4) Ibelieve that quite accurate solutions may beobtained using equations ofthe type of(12.8-_) if Miskept large byusing small values of At. Constant Heating Rate Ifthe conductor orinsulator inthis case isassumed tohave aspecific heat ofzero, zero emissivity, and a constant heating rate input the heat balance for the conductor may bewritten as _:kA(To-T,) X (ze.9-1)A x For aspecified thickness ofinsulation xthe temperature gradient (To-Tl)which can bemaintained isonly afunction ofthe heating rate factor q/A " Equation (12.8-k) ismost useful for home insulation purposes. 4_0< 12.1012-16 Cases with Heat Transfer ataSurface The steady conduction case occurs only infrequently inaerodynamic problems andthe Dusinberre numberical method ofsolution previously given has tobemodified toaccount for avariable heat input and changes inthe heat balance equation due toradiation. Tohandle this case figure 12-10 may again bereferred toand aheat balance onthe half sliceO_o iswritten __=,,_<_,_0-;-rD+k(To-_,)+_,(T_')'_ 6"X (12.10-i) where q/A the right is transmitted and the third term the heat radiated. Asanapproximation the termisthe variable heat input, the first term on the heat absorbed, the second term the heat ITp-Tv--To-To (12.10-2)Thus eauation (12.10-1) becomes ____: _A(_'-To)+ k(T_-T,) +E_'(T_')* AA_ _x Equation (12.10-2) issolved for T1bythe equation ___+ToC:P-Q) +-'F,Q=PT;+e'_('To')+ A(12.10-3) 12-17 Asanillustrative example use aninsulator 3inches thick with 3slices = . I _ BTU i . °F/ t c:.iz13T0, Ibde_lF then Az:t =.I'_Y,12. equation (12.10-3) becomes &-+(.018-.ooo43_)'Fo't-.ooo4_ _3x.Iz 20 =.00043_.018 =.olSTd +._8o7(Jo11@ _,JO00 01" _f._ff__+•97_%t.oz4TI A=+21.15"o ,o Iooo(12.10-k) Eouation (12.10-2) serves for the determination ofthe surface temperature ToI the time ofinterest and for the previous time. Todetermine the temperature made ofthe Dusinberre equation (12.7-h.).using the heating rate q/A at Toand TIwhich were determined TIdirect use may be For the example chosen 411-_1< 12-18 11_×BI5 _t_x.lZBTUxI, _. Ib°F".j_B'Tu h,"B =._.T&?x3(=oo= 17- and equation (12.7-3} becomes _'=To+81,o7_ +v_ _.©7 (12.10-5) similarly T21 becomes =-r, 3,o7(12.10-6] The temperature T3must beestablished onthe basis ofaheat balance equation similar toequation (12.10-2). Ifthe rear surface isassumed toradiate heat atits temper- ature T3the heat balance for the last slice may bewritten (12.10-7) Using the same properties ofthe insulator aswere used previously (12.10-8 now all necessary equations have been written and the temper- atures Tol, T1l, T2l, and T31 may besolved for ina 433< 12-19 step bystep process using equations (12.10-4), (12.10-_), (12.10-6)and(12.10-8). Anillustrative example ofre- sults ofthe numerical procedure isshown infigure 12-11 where the initial heat input isthat offigure 12-1b and the time interval chosen isl0sec. Changing the number oftemperature calculating stations has aneffect onthe calculations which should beinvesti- gated before the procedure isadopted. The surface temper- ature gradually approeches the equilibrium radiation temper- ature and the rear face temperature decreases asthe number ofstations increases. The insulator isavery effective way tohandle heat inputs but there isaweight penalty involved. The insulator used inthe present example which keeps the rear surface temperature toless than 600°R weighs 9Ibs/ft 2. Ecuations have been presented which may beused to obtain temperature time histories for any combination of heat shields and insulators which include the effects of transient conduction, radiation and variable heat inputs. The only restrictions tothe equations are that they are one-aimensional and numerical procedures must beused thus producing some variation inresults dependent upon the size ofthe time intervals used and the number ofintermediate temperatures determined. Arecent report, reference 2,presents amethod of calculating the temperature distribution ofthick walls which uses time series and the response toaunit triangle 434< 12-20 variation ofsurface temperature. Details ofuse of the method for cases involving surface radiation are not given for the case where the surface temperature isunknown. 4&S< 12-21 i. .References McAdams, William H.: Heat Transmission. McGraw- Hill Book Company, Inc., Third Edition, 195_. Hill, P.R.: AMethod ofComputing the Transient Temperature ofThick Walls From Arbitrary Variation ofAdiabatic-Wall-Temperature andHeat-Transfer Co- efficient. ,156< 9 . x AS .x 8 s ay 8 ~ 3 BN BN: 8.3 . anN N gk »Ss yegs e BN8 Nan NSKS NiS N N NY 8 &x 8 N x gs NN x Vox my iS} aay g ** & 9&SoD SsgraN®%SAYge XN SSSR . e 457< e 3x NS Nay §X 3 NaS Na 3 esRzSAR8 RRRES 98NONE eBese ; emes 8 8 8 \ es & Re Ngy S s8 vy Fe eee EES wyee e -458< % .S \ i} \ |N : \ | \x \ \ NN \ S n + Ne NNNEN N> ‘8 e Ns ay QN Ss ? ra >8 aN 9 ¥8BS vs & rs Nyyae8 es FS © & yy & BY N" N oO% e 429< x .X % X N ® NS f\jfN Ng 8 g N gsN BRR ry 8Q N gN 8 2 8 XK y x N * bx 8 BNY »§§FEFePeRew § 2g iN Yrsg‘io,f? . . e 450< Ns f SYHV nv|eyby aN' ayff .; |\| ay 3: \ ai :iN 3| \ ; 5 \\ 3| . :! So3; \ . a|\ ut| ghHfaN VES eq! s\\ : glo.3 \ N 8 ia! |3 ;x g 2 alo & , R| | aN || : \ 3‘ ; \SSON 5: Se any 9N Pugs eS$8§88 Ss & ;x‘.ayS40Z 441< ; %¥ 8 g&i 8 S3.X88 Xx!% NaNRS XNs, § aN e gk8NN es Ny y VY & N g “®R MSy N § s8 8 iN yap % _3 "k _o\ %.q /_3 _3 % % N % _,% 443< \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6ap_"_o ! 4'_4< / / / \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % @ Q ‘ 2 3 4|fo, | | V a YG5, y75a °ot|_| Laatobeseoleae ohareore Aigure 12-0.-Dusinberre Wagram @ ai6< _3 I!II % '_.o 0 Q I 447< 13.13-1 SECTION XIII PROPERTIES OFHIGH-TEMPERATURE MATERIALS General. Inflight through the atmosphere, compression ofthe air layer around the body heats the air and the body. The temperature ofthe heated air increases roughly asthe square ofthe increase inspeed, and the heat transferred from the hot air tothebody becomes aproblem above M_2. For aperfect gas, the temperature ofthe hot-air layer a- round the body (boundary layer) isgiven approximately by TB.L. _Tatmosphere (1+1/5 M2) and even athigh altitudes Tatmosphere _500°F sothat at M=5the boundary layer temperature is>2,000°R and atM=lOthe boundary layer temperature is_7,000 -8,000°R. The actual boundary layer temperatures are somewhat less, due toanumber offactors, but flight atM_6 will melt _n_ _ _i_+ ............ _ at M_I2 will melt allknown substances, neglecting body radiation. This does not mean that itisimpossible tofly atM=6or12, since other factors such asduration offlightp shape ofnoses and wings, artlflcal cooling, andbeneficial body radiation enter the picture. Itdoes mean, however, that disaster lurks continually for the unwary and that prodigious effort and ingenuity are required for hypersonic (M_5) flight inthe atmosphere. Everyone isfamiliar with the fiery trails of meteors andwith the fact that many ofthe meteors are 13.113-2 metallic. Themeteors arefierybecause theyareburned upbyairfriction. Meteor velocities onentering the atmosphere rangefromM_25-30upward andthesespeeds areonlyabouttwicetheM_12-15speeds thatcanbe achieved bypilotless vehicles atpresent. Theforegoing serves topointoutthatvehicle speeds whichcanbeachieved atpresent aregreatenough tocause rapiddestruction ofthevehicle andoccupants duetoheat- ingfromairfriction. Sincesustained flight atM_ willmeltmagnesium andaluminum andatM_6willmeltsteel itisclearthatspeedsofM_lO-15willrequire more exoticmaterials having ahighmelting point, plusprobably artifical cooling, plusinsomecasespossibly controlled destruction ofpartsofthevehicle totakeadvantage ofthe beneficial cooling resulting frommaterial destruction. The restofthediscussion willdealonlywithsomeofthehigh- temperature materials, someeffects oftemperature onmaterials properties, andsomeusesofmaterials inhigh-speed vehicles. Materials. Theexistence ofhightemperature materials hasbeenknown formanyyears. Thediscovery ofcarbon occurred inpre- historic times. Tungsten, platinum, andmolybdenum were discovered inthe1700's. Tantalum, Niobium, Beryllium, and Rhodium werediscovered inthe1800's. Rhenium wasdiscover- edin1925.Unfortunately. mostofthehightemperature 13-3 materials are cuite hard and brittle and are thus unwork- able. Inaddition, most ofthem are expensive toproduce and the lack ofacommercial demand has prevented the lengthy research and development that will benecessary to produce more usable forms ofthese materials. Asaresult, most ofthese materials have remained laboratory curiosities. However, within the past 3orhyears extensive and intensive research and development efforts have been applied tothe field ofhigh temperature materials and alloys, and itis booed that this ccotinuing effort will pay off with new materials ormore-workable forms oftheknown materials. 13.1.1 Melting Temperatures. The majority ofthe elements are shown infigure i, with meltinz temperature plotted against molecular weight. The elements are divided into the periodic groups and the meltin5 temperature has been plotted against increasing rnnl_l _...._+ "_*_- ^ach group. _isimmediately apparent that themelting temperature varies atrandom with molecular weight andwith periodic group. Figure 13-1 serves toillustrate that there doesn't seem tobeareason- able relation between the melting temperature and the prop- erties ofthe elements. However, figure 13-1(a) does show that the high-meltin_z elements appear tobegrouped roughly around the molecular weights of12, h8, 96, and 192; the significance ofthis grouping isnot apparent. Asfar as can bedetermined, there does not seem tobeany correlation r 13.1.2 13.1.313-4 between melting temperature and physical orchemical properties ofthe elements. Compound s. Few ofthe elements are hsed intheir pure form, and itisofsome interest tolistthe elements and their corresponding refractory compounds that may have high- temperatures uses. The majority ofthe elements are listed inTable 13-1 inorder ofdecreasing melting temperatures, and the meltin_ temperature ofanumber oftheir known com- pounds that melt above 2,500°F are listed. There are three main items ofinterest inTable 13-1. (i) Two ofthe carbides are the highest-melting two-element compounds known and their melting-temper- atures are several hundred degrees higher than either ofthe constituent elements. (2) Over half ofthe compounds have higher melt- ing temperatures than either ofthe constituent elements. (3) The carbides and nitrides tend tohave the highest melting points, averaging about i,700°F. Some Oxides. Some ofthe oxides that melt above 4,000°F are listed inTable 13-2(a) indecreasing order ofmelting temperature, and qualitative information ontheir oxidation and thermal 4SI< 13-5 shock resistance are included. Itshould bestressed that the oxidation resistance inparticular isqualitative, since the oxides asaclass aregenerally considered tohave in- different orpoor resistance tooxidation because most of the oxides tend tobesomewhat porous. Abody made entirely ofanoxide would, ofcourse, becompletely oxidation re- sistant with the exception ofthe few oxides that change to higher forms under heat and agenerous supply ofoxygen. Most refractories are too weak and brittle for load-carrylng applications, sothat their main use isasaprotective coat- ing over astrong, but oxidation-prone strength member. For coating aoplicetions, then, the porosity ofthe protective coating isofparamount importance since aporous coat allows the free oxygen ofthe air toattack the load-carrying "pro- tected" member. The thermal shock resistance isroughly tied tothe thermal conductivity and expansion characteristics ofa material. Thermal shock isadescriptive phrase used to indicate the break-up ofamaterial from uneven temperature distributions, andbroadly speaking comes about because of the material's inability toconduct heat away from ahot spot coupled with atendency toexpand when hot. Thus, a material with bad thermal shock properties, when heated on one face while the other face remains unheated except for conduction from the hot face, expands atthe hot face and breaks upthe cooler portions because the relatively poor 13-O conductivity keepstheheatfromequalizing throughout thematerial. ItcanbeseenfromTable13-2thattheoxides tendto breakupundertransient heating. Theoxides, then,asa classforgeneral application arenottoohighly regarded forhightemperature protective coatings orashomogeneous bodies fortransient-heating applications. 13.1.4 SomeCarbides. Listed inTable13-2(b)aresomecarbides thatmelt aboveh,000°F, andsomeinformation ontheirqualitative re- sistance tooxidation andthermal shockisincluded. The oxidation andthermal shockproperties arenotwellknown forthecarbides. AsnotedinTable13-1,thehighest- melting compound known, Hafnium carbide, isfoundamongthe carbides. Frommelting considerations only,am_ssile made entirely ofHfCcouldflyindefinitely atM=12. Thecarbides arecharacterized byextreme hardness, andtheyhavebeenusedforyearsasabrasive gritson grinding wheels andascutting tooltipsandmasonry drill tips. Tungsten carbide isperhaps thebestknownofthe carbides fortheseapplications. Theuseofcarbides forbodies andprotective coating hasnotmetwithunqualified success. Thecarbides arevery hardandalmost impossible toshapeorwork,andgenerally arequitebrittle; thesequalities tendtodiscourage their 453< , ....• 13-7 13.1.5use asrefractory bodies. Their brittleness makes them less desirable asprotective coatings, but itisfelt that there isroom for considerable progress indeveloping car- bides with good high-temperature characteristics. The PARD Vapor-Desposition Laboratory plans tospend some time in basic research into the carbides tobeused asprotective coatings. There isatendency tothink ofacarbide asacoat- ing oncarbon, but carbides may beformed onmany materials such astantalum, ziconium, titanium, tungsten, etc. Some Borides. Several borides are listed inTable 13-2 (c), and the most outstanding characteristic ofthe borides isthe lack ofknowledge concerning their properties. All ofthe borides contain boron, and the literature indicates that there is little commercial use for the pure boron; most boron uses are ascompounds inmild antiseptics, washing powers, and enamels and glasses for covering refrigerators and the llke. Several ofthe borides have attractive melting temper- atures (ranging upto5,500°F) and more intensive research may reveal other useful high-temperature properties. Asof the present, however, the borldes aremore alaboratory curiosity than useful products. The borides aregenerally quite hard andbrittle, and one investigator postulates that some oftheborides may become superconductive (electri_ally) athigh temperature 4S4< 13.1.6 13.1.713-8 and may thus beuseful asatemperature-sensing element. Some Nitrides. Several nitrides that melt above 4_,O00°F are listed inTable 13-2 (d). The general information available in- dicates that several ofthe nitrides have good thermal shock properties, but their oxidation resistance iseither generally unknown orthought tobepoor. The higher-melting nitrides are metallic inappearance and range incolor from gold (ZrN) togray (TAN). The nitrides tend tobequite brittle and hard, although one source reports that TaN athigh temperature isabout as soft ascopper isatroom temperature. There isascarcity ofdata inthe literature concern- ing the use ofnitrides asrefractory bodies orcoatings. This lack ofinterest may bedue tothe inherent hardness and brittlesness ofthe nitrides. Some Other Refractories. Table 13-2 (e) lists several other refractories thought tohave good oxidation resistance. The lowest-melt- ing refractory, MoSi 2,has been included because the author has some knowledge ofthe behavior of MoSi 2from personal experience. MoSi 2atabout 3,000°F has good oxidation re- sistance. Amolybdenum model coated with MoSi 2was tested inthe PARD hot air Jet and lasted for nearly lOminutes while .455< 13.2 13.2.113-9 the uncoated model lasted about 6seconds. Temperature Effects. Ingeneral, itmay bestated that things get worse as the temperature increases. There are certain exceptions to this rule, such asthe increasing e.m.f, with increasing temperature for thermocouples and the increasing emissivity wish increasing temperature characteristic ofmany materials. Effect ofTemperature onStrength. The strength ofamaterial usually goes topot when the material gets hot. The effects oftemperature onthe strength ofafew "high-temperature" materials are shown in figure 13-2. These "high-temperature" materials arebasic- ally Ni-Cr-Fe alloys except for the 0.5% Ti-Mo which is almost pure molybdenum. Itcan beseen infigure 13-2 (a) tha= most ofthe materials have lost about half their ultimate strength at1,600°F orbelow, and ingeneral this isthe story ofmaterial commonly used for load-carrying members inaircraft andmissiles; airframes have tobe "beefed up" and far over-strength atordinary temperatures sothey will bestrong enough when heated upbyhigh flight speeds. One ofthe significant things tobeseen infigure 13-2 (a) isthe strength ofthemolybdenum alloyed with 0.5% titanium, even athigh temperatures. Molybden_nn un- fortunately oxidizes badly above 1,300 -1,_O0°F and catastrophically at1,800 -2,000°F and this one character- istic has been (and still is) amajor reason for not being dSG< 13-lO able toutilize thehigh-temperature strengthofmolybdenum. Ahlgh-temperature protective coating that istightly bonded, oxidation resistant, self-heallng, and ductile for molybdenum would permit the utilization ofits great hlgh-temperature strength. Figure 13-2 (b) shows the effects oftemperature onthe lO0-hour-rupture strength ofstainless steel, several super- strength alloys, and molybdenum. Itisobvious that high temperatures drastically reduce the allowable loading. Again, the alloyed molybdenum shows aconsiderable advantage, assuming ofcourse that some means ofprotecting the molybden- umfrom oxidation can befound. Ifthe highest temperature that astructure will reach isnot more than 1,100 -1,200°F, figure 13-3 shows that beryllium isanattractive material. The upper part of figure 13-3 shows the relative weight ofathln-wall structure plotted against atomic number ofthe material ofconstruction for room temperature. The weight ofthe structure when made ofaluminum isthe base for comparison. The lower part of figure 13-3 isthe same type ofplot, except that the temper- ature is1,200°F and asteel structure isthe base for com- paris on. Itcan beseen that atroom temperature several materials arebetter than aluminum purely onastrength basis. At 1,200°F many ofthe materials are nolonger usable. Itis interesting tonote that beryllium appears tobeanexcellent material atboth temperatures. These comparisons are ona 457< 13.2.213-II strength basis alone and donot consider cost, workability, availability, and oxidation effects. Effect ofTemperature onThermal Conductivity Figure 13-h (a)shows that ingeneral the thermal conductivity generally tends todecrease with increasing temperature. There are enough exceptions tothis togenerate plenty ofargument. Adecreasing conductivity with increas- ing temperature isadvantageous from aninsulation viewpoint but disadvantageous from athermal shock viewpoint unless the expansion rate also decreases. Itmay benoted from figure 13-_ (a) that copper and silver are _,000 -6,000 times more conductive (thermally) than some ofthe better insulators. This might indicate that ofcourse copper and silver would beworthless asin- sulators and indeed they are ifthe generally-accepted de- finition ofinsulation isused. However, there are some special cases where copper isused asan"insulator" and this will bediscussed inthe Heat Sink section ofthis discussion. Itisseen infigure 13-4 (a) that most ofthe usual airframe materials and the refractories have about the same order ofmagnitude ofconductivity asindicated bythe cross- hatched areas atthe bottom ofthe figure, although there are exceptions tothis. Figure 13-4 (b)shows the effect oftemperature onthe thermal conductivity ofseveral refractories. ZrO2(zirconia) has arather low conductivity and Be0 (beryllia) has ahigh 8.I...I 13-12 conductivity which decreases rapidly with increasing temperature. These refractories show analmost consistent decrease ofconductivity with increasing temperature.* Some ofthese refractories such asZrO2and A1203 have low enough conductivity tomake them attractive asinsulators incertain flight applications. Ingeneral, however, the refractories are fairly dense and are not thought ofas efficient insulators because oftheir weight; itisusually more efficient touse the so-called insulators which are lightweight and have relatively low conductivity. Figure 13-4 (c) shows the effect oftemperature onthe thermal conductivity ofseveral insulators that are suitable for high temperature application, and the conductivity curve for air isincluded for comparison. Incontrast tothe refractories shown infigure 13-4 (b), the insulators show anincreasing conductivity with increasing temperature. All ofthe insulators shown (except air) are made offibrous materials surrounded byair spaces, and this construction is typical ofthe llght-weight insulators. Asanitem of interest, the thermoflex isactually arefractory (A1203 • SiO2)infibrous form. •Some sketchy information indicates that porous refractories may tend toshow anupward trend ofthermal conductivity with temperature beginning atabout 2,000°F, due to radiation across the internal air spaces. 4S9< 13-13 Atlow temperatures, air isagood insulator and a vacuum isconsidered tobealmost aperfect insulator. How- ever, attemperatures above 1,%00 -2,000°F, neither air nor avacuum are good insulators because ofthe radiation. At 2,000°F, for example, the heat conducted across a1foot thick airspace would beabout 0.0_ Btu/ft2-sec but the heat radiated from the ho_ surface tothe cool surface could be asmuch as1%or16Btu/ft2-sec. Ifthe space were avacuum the conduction would beabsent but theradiation would still be1%or16Btu/ft2-sec. Onthe other hand, afibrous type insulation such asquartz fibers would have atotal conducti- vity of.0% or.06Btu/ft2-sec since the solid materials effectively block radiation. The selection ofacertain type ofinsulation for a particular application isnot astraight forward cut-and- dried process. Assuming that the insulation will take the temperature, there must beaconsideration ofsuch things am conductivity, difficulty ofholding the insulation inplace, tendency ofthe insulation tovitrefy and compact, and the weight ofthe entire insulation assembly. Insome cases, such asfor missile noses and wing leading edges, itmay bebetter touse one ofthe relatively heavy refractory seml-lnsulators such asZr02. The difficulty ofinsulating ahigh speed flight vehicle iscompounded toastaggering degree bythe weight restrictions / J 460< 13.2.3 13.2.413-14 imposed. Atthe present time noone can bepositive about the allowable weight ofinsulation, but there seems tobe asort oftacit agreement that about 2-3lb/ft 2isnear upper limit. At3lb/ft2? a3-foot-diameter vehiclethe 25feet long would require 500 -600 lbs ofinsulation on the body alone. Effect ofTemperature onSpecific Heat. The specific heat isthe ability ofamaterSal tostore heat, and generally ahigh value isdesirable because ahigh specific heat means arelatively low temperature rise for a given heat input. Figure 13-5 shows that most materials have ahigher specific heat athigh temperatures than atlow temperatures and that the curves tend tokeep rising with temperature. Graphite has adifferent trend, with ahighest value at about 2,000°F. Ascanbeseen bycomparing figure 13-5 with figure 13-4 (a), there isnot such awide variation ofspecific heat between materials aswas shown for thermal conductivity. Effect ofTemperature onEmissivity. The emissivity ofamaterial isameasure ofits ability toradiate heat. Aperfect radiator iscalled a"black body" and has anemissivity of1.0 and abody that does not radiate any energy has anemissivity ofzero; thus, all materials 461< -15 have emissivities between 1.0 and 0.0. The desirability ofhaving ahigh orlow emissivity depends onthe function ofthe material. Ingeneral, materials considered for high speed missile skins should have ahigh emissivity so that when the skin heats upitwill radiate tospace and tend to"cool" the skin. For example, askin having a heat input of8Btu/ft2-sec will come toequilibrium at 1,600°F for _=1and at2,600°F for C=0.2. There is noknown perfect radiator (_=l)but graphite comes close .75(__ .95 -.96 according tosome investigators; -.78 according toothers). Figure 13-6 (a) shows the effect oftemperature onthe emissivity ofafew selected materials. Itcanbeseen that ingeneral the emissivity tends toincrease with temperature increases. Figure 13-6 (b) shows the effect oftemperature onthe emissivity ofInconel which has had various previous heat treatments. Since emissivity isessentially asurface phenomenon, changing the surface conditions (roughness) of amaterial will generally change the emissivity. Ingeneral, surfaces that appear dark tothe eye under ordinary light have ahigh value of _because absorbtivity and emissivity are the same thing. Conversely, surfaces that appear bright tothe eye under ordinary lighting are poor radiators (but good reflectors)• 462< 13-16 13.2.5 Effect ofTemperature onOxidation. There appears tobeaconsiderable store ofknowledge concerning the oxidation ofmany materials invarious atmospheres near room temperature. However, oxidation be- havior athigh temperatures particularily inthe presence ofahigh speed airstream islittle understood and sparsely documented. The oxidation problem isaserious one for high speed vehicles and isattracting considerable interest. The reason that oxidation isimportant isthat practically all materials (except the oxides) are prone tooxidation and heat isalmost always given off inthe oxidation process. This heat liberated tends toheat the material tohigher temperature where the oxidation reaction isaccelerated. In some cases ahigh speed oxidation reaction may set inthat isself-regenerative and violent enough todestroy the material inamatter ofseconds. Afamiliar example of oxidation isthe operation ofthe oxy-acetylene cutting torch which works onthe principle ofheating the material tobecut to1500OF and then impinging onthe hot metal aJet ofoxygen. Steel plate 12" toh8" thick can becut at2"to6"per minute bythis method. Flame machining isalso done, using the principle ofmaterial removal bysurface oxidation; metals may beoxidized inthis process atthe rate oflOlb-l_lb per minute. 46S'- 13-17 The oxidation ofmaterials inhigh speed flight in air isnot well understood orpredictable. Itisaserious problem, asevidenced bythe fact that molybdenum (M.P. h700°F) will break into flaming destruction at2800°F ina supersonic air jet and steel (M.P. 2800°F) has been made to flame like atorch ina600°F supersonic air jetwhen the steel was preheated with atorch tored heat. Figure 13-7 (a) shows some effects oftemperature on the oxidation ofSiC powder and some effects ofadding water vapor tothe air when the temperature isheld at about 2000°F. Itisseen that higher temperatures aggravate the oxidation problem for SiC powder, and this isgenerally true for other materials. The magnitude ofdarbon lost in the upper part offigure 13-7 (a) isnot ofparamount importance from afllght-vehicle standpoint since missile skins will not bemade ofpowder. The size ofthe powder ......... o............................. rata; everyone isfamiliar with the explosions ofdust-laden air. Figure 13-7 (b) shows the effects oftemperature onthe oxidation rate ofmolybdenum which has amelting temperature ofabout h700°F. Itisinteresting tonote the large Jump inoxidation rate ingoing from1350°F to1500°F. This is due tothe behavior of MoO3which isformed asasolid oxide below 1350°F but which melts atabout 1350°F and in- creases the oxidation rate manyfold ofthe remaining molybdenum. 13.3 13.3.113-18 Therearemanyformsofoxides formed onmaterials such astitanium, steel, aluminum, molybdenum, tungsten, etc. Someoftheoxides arerelatively hardandimpervious and tendtoretardfurther oxidation onceathinoxidelayerhas formed, aluminum beinganexample ofsuchprotective oxidation. Otheroxides aretoofragile toprovide anyprotection, such astheironoxides. Uses and Arrangements. Noattempt will bemade tocover all ofthe uses and arrangements ofmaterials for high temperature use inhigh speed vehicles. The protection ofmissiles from high temper- atures byuse ofinsulation and "heat sinks" will bedis- cussed briefly asillustrations. Insulation. Ininsulating anobject, alayeroflow-conductlvity material isplaced between the hot gases and the object. Any object canbekept atnear room temperature for hours even when surrounded bygases at5000°F ifthere are no restrictions concerning the cost orweight ofthe insulation; for example, analuminum plate 1/lO inch thick protected from 5000°F gases by150 feet offirebrick will rise from 32°F to room temperature inabout _hours. The effect ofinsulation istodelay the temperature rise ofsuch vehicle components asthe load-carrying members, electronic instruments, and the inhabitants. Insulation 165< k • .. 13-19 keeps the inside cool only inarelative sense, inthat regardless ofhow much insulation isused the entire vehicle will finally come toanequilibrium temperature unless some sort ofcooling scheme isused. This equilibrium temperature may be2000°F orhigher, and itisobvious that insulation isused only todelay the temperature rise ofthe interior ofthe vehicle for areasonable time sothat the vehicle can perform its intended function. Figure 13-8 shows thequalitative effect ofinsulation onthe flight time ofahypothetical missile inthe atmosphere. Itcanbeseen that athicker insulation extends the flight time, afact that would beintuitively clear. The obvious conclusion todraw from this figure isthat the flight time could beextended atwill byusing thicker and thicker in- sulation, and this istrue toavery limited extent. However, the one factor that makes the insulation problem sodifficult isthat theweight ofthe insulation and insulation-support- ing structure quickly mount toseveral pounds per square foot and such added weights are intolerable. Itseems generally agreed that the weight ofthe insulating structure for high speed flight vehicles will have tobeless than about 2to3 ib/ft 2and this weight immediately translates into insulation thicknesses ofthe order ofafew inches and flight times of afew minutes. The problem ofinsulation isaback-breaker and anumber ofpeople are atthis minute racking their brains trying tosolve it. 13-20 13.3.2Heat Sink. Another method ofusing materials toprotectagainst high temperatures isthe so-called "heat sink" approach which might also begenerally thought ofasinsulation. In theheat-sink arrangement, the material's ability tostore heat isofprimary importance. The theory oftheheat sink isasfollows: thehigh-heat inputareas ofahigh speed vehicleerecovered onthe outside withamaterial which hasahighheat capacity per degree temperature riseand m conductivity high enough tocarry theheat tothe interior oftheheat sink material. Theheat input tothe skin is absorbed inraising the temperature oftheheat-sink material instead ofbeing conducted tothe interior ofthe vehicle. Itisdesirable that theheat-sink materialhave aconduc- tivity Justhigh enough toprevent surface melting, and this necessary conductivity ofcourse varies with theheat input rate. Figure 13-9 (a) shows acomparison ofseveral convention- almaterials onthe basis oftime tostart melting atvario;Is heat input rates for aninfinitely-thick slab. For agiven heat input, the time tostart melting isafunction ofthe material heat capacity, conductivity, and melting tempera- ture.Itisinteresting tonote that copper ranks quite high although its melting temperature isrelatively low (about 1900°F); the reason itranks high isbecause ofits .467< 13-21 13.4heat capacity and conductivity. Molybdenum isone ofthe highest-ranking materials but isless attractive than copper because ofbrittleness and oxidation characteristics. Inusing amaterial asaheat sink, itwould seem that the thicker the material the longer itwould take tostart melting. This istrue Only uptoacertain point, aswill beshown byfigure 13-9 (b). Inthis figure thin plates of Inconel are considered, and itissurprising tonote that there islittle reason for using thicknesses greater than i/_" to1/2" except for very low heating rates. The reason for this isthat Inconel conductivity isrelatively low and atmoderate tohigh heating rates the surface heat cannot be conducted away fast enough tokeep the surface from melting. Figure 13-9 (c) shows the same type ofplot for copper. The main point from this figure isthat the greater conduc- tivity ofcopper allows more surface heat tobeconducted Aw_v _ _h_ m_xlmi_m thickness ofcoooer would beofthe order of2"to3". Ablation. The process ofremoval ofsurface material from a vehicle inflight, through heating and the scrubbing action ofthe boundary layer, iscalled ablation. The process of ablation thus includes melting, boiling, sublimation, and the removal ofmaterial inthe form ofdiscrete particles. Ablation isthe controlled destruction ofavehicle surface 4 S< 13-22 soastouse upheat supplied bythe boundary layer and thus cool the vehicle. Agood ablating material must have alow thermal conductivity sothat the surface will remain much hotter than the interior and thus the ablation process will be confined tothe surface. Afurther requirement isthat this material have good thermal shock properties; i.e., the material will resist breakup due tolarge thermal gradients. Itgoes without saying_ ofcourse, that such amaterial will not oxidize exothermically and supply additional heat tothe unablated portion. Surprisingly enough, anumber ofmaterials meet the above requirements tosome degree. Even mo_e surprising, some good ablation materials have alow melting temperature. Nylon, for example, with asoftening temperature ofhOO-500°F isattractive from anablation viewpoint. The cooling achieved bythe ablation process comes mainly from the following: (1) Raising the material surface temperature tothe melting orsubliming point. This isgenerally avery small part ofthe total heat carried away bythe ablation process. (2) Changing the material state, from asolid toa gas orasolid toaliquid and toagas, without a change intemperature. Inmost cases, this isnot a L169< 13-23 large part ofthe total heat carried away by the ablation process. (3) Mixing ofthe relatively cool gases from the ablating material with the hot boundary layer. This results inacooler boundary layer next tothe ablating surface, and the heat transferred tothe surface islowered. This process inmany cases accounts for alarge part ofthe total heat carried away bythe ablation process. The desired effect ofthe ablation process istokeep the underlying structure from getting too hot. The under- lying structure receives heat byconduction through the ablating material and the heat conducted isalinear function ofthe outside surface temperature ofthe ablation material. Agood ablation material will have anessentially-constant surface temperature for awide range ofboundary layer temper- tures and heat transfer rates, since anincrease inheat transfer rate tothe outside ofthe ablating surface will re- sult inmore Ibs/sec ofablating material being boiled away rather than anincrease insurface temperature. Suggested reading: (I) "Sublimation," Aircraft and Missile Engineering, Feb., 1958; (2)Aviation Week, pp52et. seq., May 12, 1958 Figure 13-10 shows the calculated effectiveness of beryllium oxide and plexiglass used asablating materials. Melts Below 2500°F xx Not GivenTABLE 13-1 This compound melts athigher temp. than either constituent NAME SYMBOL MELT TEMP.°FOXIDE BORIDE CARBIDE NITRIDE SILICIDE SULFIDE Carbon C 6700 Tungsten W 6170 2683 5288 5184 -- 3956 -- Rhenium Re 5740< ...... _3092 -- Tantalum Ta 5425 3434 3632 _ 5396 3992 -- Osmium Os 4900 .......... Molybdenum Mo 4760 < 3956 4874 3686 • Ruthenium Ru 4500 .......... Iridium Ir5450 ...... _A_A _ "- Columb ium Cb 4380 3222 )3632 6332 3707 -- Boron B 4200 ( -- 4442 _ "- Rhodium Rh 3570 ............ Chromium Cr 3430 _ 3632 3434 -- 2795 2822 Thulium Tm 3400 xx Titanium Ti 3300 _ _ 5684 _ 2804 Zirconium Zr 3200 _ 6386 2768 Platinum Pt .3230 ............ Lutecium Lu 3100 xx Vanadium V 3150 .... Iron Fe 2800 2822 3002 .... Palladium Pd 2830 ........ 2552 -- Yttrium Y 2700 4370 ........ 3497 Ytterbium Yb 2700 xx Cobalt Co 2720 _ ...... _ Erbium Er 2650 xx Dysprosium Dy 2600 xx Holmium Ho 2650 Nickel Ni 2650 _ _ .... _ "" Silicon Si 2600 -- 3812 -- _ Gadolinium Gd 2500 Beryllium Be 2340 _ -- 3902 _ .... Samarium Sm 2370 .......... Scandium Sc 2190 ...... _ .... Manganese Mn 2270 _ -- 2768 .... Europium Eu ,2100 xx Copper Cu 1980 xx Gold Au1945 xx Silver Ag 1760 xx Germanium Ge 1760 xx Praseodymium Pr 1700 xx TABLE13-1CONTINUED MeltsBelow2500°FxxNotGiven[k-k-k-_This compound melts athigher temp. than either constituent NAME SYMBOL MELT TEMP. OFOXIDE BORIDE CARBIDE NITRIDE SILICIDE SULFIDE Calcium Ca 1560 _)3812 4172 ...... Neodymium Nd 15hO Cerium Ce 1500 35_2_}3812 ...... Strontium Sr lh20 _3812 _3501 .... > _arium Ba 1300 )3812 _3236 .... Magnesium Mg 1202 _ < .... Aluminum A1 1220 < 5072 _ -- Antimony Sb 1170 xx Lanthanum La 1519 ___ _3812 ........ Zinc Zn 787 XX Tellurium Te 8h0 XX Cadmium Cd 609 xx Terbium Tb 621 xx Lead Pb 621 xx Thallium TI 572 xx Tin Sn h_9 xx Bismuth Bi 520 xx Selenium Se h28 XX Li_hi_m Li 367 xx Indium In 313 _ .......... Sodium Na 207 xx Sulfur S 246 xx Iodine I 237 xx Potassium K I_5 xx Rubidium Rb 102 xx Gallium Ga 85 _ ........ < Phosphorus P iii xx Cesium Cs 82 xx Bromine Br 19 xx ."_7..< o r._ o I--4 I--t oI- 13::;00000000000 00000000000 t_I_I_1_1:___I_I_I_I:_ o0ooooooooo ooooooooooo oJ ! ,-Iz H o_a_ 0_ _o 1--i o_ o r.D00o000o0o80 OdC_I C_I_1 0o_0 0000000Od0Od ___o____"_o o__ oo -_,C 473< v c_ ! ,--4 _3 < E_oo_ O H X O "il1--1-00 00 _'.e-0e-0e-¢_ _-_0e-0e- 00 _4 _'.c-.0e- 0e- _¢-.¢-*¢-.e-6-- OOOOOOOOOOOO OOOOOOOOOOOO o UI tO o_ Z_ o_ H o_ o I e_ _o o oo000000 000000 475< "0 (_J ! c_ro _Z E-G Z_ Or..) H_ E-_ _r_ H H00 O _" _"0O _- _. _-. _. 0 000000000 000000000 e,.,l0_OJ(3"OD000 O_ O_ ___0o__® r,.) o Z o H i--t oc 0 rS0 rd 0 o _.oo o_ o oo o¢-._-'oo 0.) ° o E-4o_ r_go°g_gg .= o= c_ L',,I I_O ,_ol.o_. m? 2" ',1__,_,ctI Ik_A/.7,/ II;C _'78< e soi - ran y,Al g eae, |es j AA « S 3s 2 FO ‘ iD pe oyBi yes a3eerie 83 & 3ey g 3 a e 479< _5 _orr_ _r__!I I--"°) i___ _ t. I 0 _0< CONDUCTIVITIES OF SOME SELECTED MBTERIAL S . Thos Figure 13Uusteative Only Cd * ®‘Ay co a e <2gz4 y S93 N g 3 2¢) ——s 8 megu=a & Sein SC,Mee, Alaes sasulaPSSSSIOSS OS oreETA OPaOOeeneal OOOO LOIS °mee os sone TEMPERATURE, OF (4)Overall comparison FigurelS-4Electofbemperature onthermal Conduativity 481< CO,,vbu c2"/r/;,')loF_o,_,_G i.0.6" i ,_4 .03 .o2 .ol Deoa_ J_o 7"_,,tn,_, af _,_,rG/1':.q¢,,nc/,,,¢_' 4S3< 5DGCePlC_"j_,4T_/_T'___F_c_r_ b ii C .I "YZooo /J ,6 ,2fC _U 7"f_p/ ep ,4 ,Z16m/._2_o 486< SlLiCo_ C'AnIP_.,. _qo"F _ / __°'° 0_o 6o qo I_l,o 15"o 7¢Jb,.t'/eGm'F SILl RI_III_ 1 totArJ_r/mm IbooISs_mC911A/r) ! 4** \ 0 _NI/_ Iv@B¢oeq ¢ ,_SS< ,_P._._-.. 7"o_:/_vs_..,_;,'to_/' %/ / / , / / /4/_ /N&/.M.AI_YON I IINarW I_TuI.A_T_O_'_?,,_E ---_I I ,,_",,2gse/,.,;'-B" e'((¢"=_ o//.,su/q,_io,_ on-/'/;gh-/ ,_,;,w_". _om s4t_INPu• M /o 2o 7",,P,,'IC Fo,3T,_R;" ,'_,_r/,_r4 ,_coN_rJO 490< 2e00 (Neonat ASHEAT SUK Inconee Hear apr | see My : " x “ Npee e 8 g z N : seo inpunreny rie we wos" ary = 7 Ss Tus FOR MELTING. [Oo START, secouDt (b)Thin slab of/ncone!. Figurel3-9 continued e 491< ZO JO /'_tTureIS-9 co_c/_,d_d. _.__, \ 2 g = & ry S3 ek Boe as (OWS °5s * at z2€-19 Lar == oe 4H eyrm Sno «6 agsy alg$ gs3 ° JagEy Sbeg= BEL. 3 £28For ese BSae yzsak r BS228eog2eelow2836)CERSE: : itat "geodle & ye3z: 32% sil,- Ql9gI £ ‘2§22y35zi7 \ ae Wra 3 8 8 3 8 3 $ 3 $ 8 8 s to Fil 483< SECTION 14 SOLAR SYSTEM As_u_: o!"space technology would notbecomplete without some reference berne m:uLe t_.the.ue.mbers ofourownsolar system. Since there arecomplete books dealing with thevarious aspects ofthis subject thefollowing notes aremore inthecategory ofammmmry oftheavailable information covering thephysical make-up ofoursolar system. Fordetails and/or amore complete understanding oCaparticular point oneofthereferences listed attheend ofthese notes maybeconsulted. Thenotes essentially consist oftwoparts; thefirst part belng more inthenature ofageneral discussion ofthemembers ofthesolar system; •_h[ie thesecond part contains tabulations ofnumerical data aswell asa hrlef discussion ofterms andcoordinate systems used inAstronomy. Theorigin ofthesolar system isstill amatter ofconjecture. H_¢ever, ithasbeen fairly well established that oursolar system is merely as_mll part ofatremendous galaxy made upofcountlessother bodies andmatter. Thegalaxy ofwhich oursolar system isapart isbelieved to beofthespiral type andoursystem islocated about twothirds oftheway outinoneofthearms. Itisestimated that thegalaxy is6xlO17or600,000,000,000,000,000 miles indiameter or98,000 light years. Thegalactic system rotates ina clockwise direction asviewed from itsnorth pole (located atthehour angle =I'__4_m,anddeclination8=2_°). Thegalactic rotational velocity inthevicinity ofourqunis178.3 mi/sec. Thiswasdetermined in195h. 4S4< NowtheSunwhich isthecenter ofoursolar system is33,000 light years fromthegalactic center andhasaperiod ofrevolution about the galoctic center of22_000,000 years. Oursolar system ismoving withavelocity of12.9mi/sec toward an apexnear _-18h,_-30°which isnotfarfrom thedirectionofthe star MHercules (theendpointoftheconstellation Hercules).The diameter ofoursolar system isabout 7,_O_O00s000 miles or 0.0000000734 x1017 miles ascompared to6x1017 forthegalaxy. Th_members ofthesolar system aregenerally considered tobethe Sunandthenine planets with their satellites plus many other smaller bodies called asteroidsorplanetoids. Also contained inthesolar system aresuch thlngs ascomets, meteor showers andother socalled cosmic matter. Thesenotes will only beconcerned with theSun, planets with their satellites andtheasteroids.Thesketch onthenext pa@e gives some idea ofthe relative sizes oftheprincipal members ofthesolar system. Itisapparent that theSunisbyfarthelarges object ofoursolar system. Theplanets areusually grouped astheterrestlal planets (Mercury_Venus, Earth, and Mars) andthemajor planets (Jupiter,SaturnlUranus_andNeptuDe).The planet Pluto issoremote thatnotmuch isknownabout itandofsuch a size that itissometimes classed asaplanetoid. Thegeneral discussion ofthesolar systemtofollow will firsttake theSunandthen each oftheplanets inorder accordingtotheirdistance from theSun. 14-3 0 _° o_o eo o.o 1 •' =o 496< Sun TheSunistherulero1"oursolarsystem. Ztccutrols themotions of theplanets, comets, meteoric bodies, andothersatellites. TheSunisa star. Xtisnotthelargest orbrightest starbutitisthenearest one tous.Thenextnearest staris275_000 times asfarmra_. Radiatiom of theSunissolesource ofpower, warmth_ acttvity_ andlifeonEarthtwith theexceptionoftidesandvolcanic action. Thediameter oftheSunisal_proximately _,000miles.It's volume is1,300,000 that ofE_rthanditsmass is333,000that ofEarth.Its density is1.4times thatofwater,andgravityatitssurface is3.6 times thatattheEarth's surface.Itssurface temperature isabout 6,000°C.anditsinternal temperature isestimated tobe20,000,000°C. TheSunrotates inthesamedirection astheEarthonanaxis inclined 83°totheplaneoftheecliptic (equator inclined 7°toeclipticplane). ThepolesoftheSunaredirected tovardapointabout hal_a_between the stars Polaris andVega.Thepoints coordinates are=.18h_m, 5=+_o. TheSunisofagaseousnature anditsequator turns faster than the poles.Thesiderealrotational period isabout25da_sattheequator, 27.5d_ysatlat.-+45°andabout 33c_ysat.+80°lat. TherateoftheSun's outpour ofenergM isexpressed asasolar constant ofradiation. Thesolar constant ofradiation isdefined asthenumber ofcalories which would bereceived fr_ntheSuneachminute UlX_asurface onecentimeter square, ifthesurface wereexposed perpendicularly tothe Sun's r_soutside theEarth'satmosphere_attheEArth's meandistance from theSun.Thevalue is1.94 _5percent.Onecalorie istheamountofenergy 497< 14-5 required toraise thetemperature ofonegram ofpure water at15°C.to 16°C.Itisequal to4.18x107ergs.Thesolar constantofradiation istherefore equalto1.35×106ergs persquare centimeter persecond. TheSun's mass is2x1053grams.This isequivalent to1.8xiO_ ergs ofenergy.TheSungives offenergy attherate of1.2x1041ergs peryear anditsmass isdiminishingattherate of1.3x1020 grams or 1.7x10I_tons peryear(4,200,000 tons asecomd).Ifweassume that alltheenergyoftheSunisduetodestruction ofmatter, itwillcontinue togiveoffenergyatthis samelevelandrate for1.5x1013years (19million million). Taking theintensity oftheSun'slightandheatattheEarth's distanceasunity,wehave thefollowlng valuesoftheintensity atthe meandistancesofthevarious planets. Mercury 6.7 SaturnO.Ol Venus i.9 Uranus O.003 EarthI.O NeptuneO.001 M._ nh3 p_--_- ^ JupiterO.Ok Light from theSuntakes _98.6seconds or8.31minutes toreach theEarth. •""4S8< Mercury Mercury istheplanet nearest theSun. Withtl_eexception ofPluto andsomeoftheasteroids ithasthemosthighl_ inclined andmosteccentric orbit andtheleast diameter andmassofanyobject inoursolar system. Mercury seem tohavesomesortofatmosphere (atleast according to someobservers). Following areccmnents byAntonlodi, oneofthemost ablevisual observers, canpiled in19_using a33inchrefractor telescope. (a)ThehazeonMercury iswhitish; itoccurs morefrequently and isdenser than thatonMars. (b) Itisrare that adarksurfacemarkingonMercury retains its normal intensity foraslong asseveral weeks. (c)Thehaze onMercury presents alldegreesofconcentration, from very tenuous toadensity sufficient toobliterate thedarkest surface markings. (d) Thehaze isusually invisible inthecentral portionsofthe disc butappears chiefly toward thelimb (aresultofperspective) and lessoftennear theterminator; near thelimb itmayextendover anarc of oookm(3ooomi s). (e)Thechanges observed inthehazema_beveryrapid; 'inasingle da_darksurface _rkings, 3500kminsize, m_7disappear completely, while theopposite ma_alsooccur. (f) _einvisibilityofsurface markingsm_Vlast _d_vs. (g)Lighthazem_ycoveraregion forweeks, with improved and diminished visibility alternating. (h)Thenorth pole region, usuall_ clear,onceshowed abright haze forsix o 14-7 (i) Oneparticular dark region,ontheequator andat60°longitude from thesub-polar point (totheright intheimage, with Southontop), ismuchmore often covered byhaze than anyother surfacemarking; two regions atthesame longitude butat.+40olatitude respectively are covered lessoften than theequatorial regionmentioned, while thewhole left half ofthevisible hemisphere iscovered still less frequently. Ithasbeen sucgested that thehaze described byAntoniodlmightbe carbon monoxide plus other inert _ses andcosmic dust duetoimpacts (possibly replenished bythese impacts). Thedust particles being the reflectors t_t m_kc thehaze visible. Atanyrate theatmosphereof _rcury _,.st nothave much depth since othermeansofdetection have not yielded _os_tive results. Asfarasweknow thepl__r.et rotates onitsownaxis with thesame period asitrcvo!_2s about theC-mo-nd, therefore, thesideofMercury C_ein_ the_unhasatemperat-_re of68_°K.atperihelion and_0°K.at aphelion. T"nedark hemisphere isintensely cold (I0°to-20°K.)andtends +__'__,_ outallbutthcmost vo!otile compoundsoftheatmosphere. _tro.n_it ofa_÷ occurs when one_!anet passes between theSunand another planet. AsfarastheEarth isconcerned theonly planets that can _ve transit_ areMercury andVenus. Transits areimportant astronomically forobservational reasons. Thetransits ofMercury occur asfollows Date TimeC_ DateTimeG_ 1957 _4_y9 13 15_6 Nov.1216 1960 Nov. 7 9 l_X_3 Nov.516 1970 _y8 2O 1999 Nov. 159 1973 Nov.9-°3 •2003 Nov.6 19 .500< i_-8 Venus Venus hasthemost circularorbit oftheplanets andhasabout the same diameter astheEarth.Ithashigh reflectivityandiseasily visible however, notagreatdeal isknown about itssurface.This lackof knowledge about itssurface isduetothedense l_yersofcloudsanddust intheatmosphereofVenue. Visual observations andspectrographic analysisoftheatmosphereof Venus have resulted intheconclusion that carbon dioxide makes upabout 90percent oftheatmosphere.There isundoubtedly somenitrogen, argon, andpossiblyCree oxygen. Notraceofwater hasbeen found. Theatmosphere isestimated tobe6000mile deep with aCO2cloud layer atleast i_miles deepnear thesurface.There isahaze is.ver about _000 feet thick near thebottom oftheatmosphere witheither theplanets qurface oranopaque is_verbeneath. Itisbelieved that theatmosphere isverydusty (both from cosmicdust andfrom wind erosion ofthesurface). Theperiod ofrotation ofVenus onitsownaxis isopen toquestion. Itwasoriginally thought that itsperiod wasthesame asitsperiod of revolution around theSun, however, thelatestopinion isthat itsperiod ofrotation isabout 20-30 d_s.This ispartlybasedonthefact that thetemperature onthebright side (500-60 °C.)isnottoomuchdifferent thanthatonthedarkside(-32°C.). Thefact that thetemperature onthedarkside doesnotapproach -973°C.canbepartly attributed toatmospheric circulation. However, this cannot account forallthedifference, thus theplanetmust rotate slowly onitsuwnaxis. 11_-9 Transits ofVenus occuratintervals of8,121-1/2, 8,105-1/2,8, 121-1/2, 8,years etc.Some datesoftransits ofinterest tousare Date 1874Dec.9 1882Dec.6 aO04Jume8 20].2June6 IS02< 14-10 Earth TheEarth isameandistance of92.9 x106miles from theSun.Itis anoblate spheroid havingadiameter of7,927miles attheequator and 7,900 miles atthepoles. TheEarth is29percent land and71percent water. Ittravelsaround theSunatavelocityof18.5 miles persecond. Themass oftheEarth is6x1021 metric tansor6.593x1021short tons. Themean density is5.52 times that ofwater anditrotates once in24sidereal hours. Themagnetic pole isabout 20°from thegeographic pole. Theattraction ofgravity oftheEarth isabout 1/190 ofitself less a_theequator than atthepoles. Onehundred ninety pounds atthepole would only weigh 189pounds attheequator onaspringbalance. Onepound outof289ofthis difference isduetocentrifugal force, andonepound outof555isduetotheEarth's shape. From Hs_fo_ "Spheroid of1909" wegetthefollowingdimension for Earth: F_uatorial radius 3_963._ ,_les Polar radius 3,949.99miles Oblateness 3963. _-39_9.99 =_=O.00_ 3963. _ 297 Latitude Length ofonedegree ofarc o miles o 68.7o8 13 68.757 3o 68.882 45 69.056 60 69.231 SO < 69. o7 i_-ii Note: Onestatute mileequals 1.1516times anautical mile. Perturbations. oftheEarth.-TheEarth experiencesthreedetectable perturbations; precession,nutation, andvariationoflatitude. Precession.- Pole moves inacircle.Onerevolutiontakes about 25,800 years. Discovered in120B.C.byHipperchus. Therateofrotation ofpoles is50'.'26 peryear.This precessiondisplacesequinoxbyabout 30°in2,150 years.That iswhythefirst pointofAries isnowinthe constellation Pisces.Starnearest pole _,000yearsagowas _Draconis. Itisnov =UrsaeMinoris. In12,000years itwillbeVega. Nutations.- Nutations aredepartures ofthepoles from perfect circular motion. Theyarecaused mainly bySunandMoon. _elargest effect istheMoon.This lunar effecthasaperiod of18.6years andan amplitude of9"2inlatitude. Latitude variation.- Twosmalleffects -o_ewith a14-monthperiod andtheother with aI-year period withamplitudes ofabout 0:'2-are called latitude variations.The14-montheffect isduetotheelasticity oftheEarth. Theannual effect isduetoseasonaldisplacementofmatter over theEarth. Both discovered byChandler in1891 andexplained by Newcomb. _e --.7Cause peric__ of_-tb_'srotatio_ toslowdown by2to5milliseconds. o× wf ,e #0 .4.Z. 0-.Z III_t _ __ IIIIII .4" .2"0-2" +XII If•"_i_ sketch shows thc variation oflatitude from 191_ to1918. ,_$" 504< i_-12 Perturbatiou oftheF_th' sorbit: 1.Thelineofapsides isrevolving eastward ataratethat, if continued, would carry itentirel_around inabout108,000 years) itwillnotcontinuealways attheseinerate. 2.Theeccentricityoftheorbit which isnow0.016 isdiminishing andwill continue todosoforabout 24,000yearsatwhich time itwillbeabout0.003. Itwillthen increase forsome 40,000years butwillnever exceed.0.07. 3.Theplaneoftheorbit isslowly changingposltiou.Thevalue ofobliquity isn_23027 'andisdiminishi_ attherateof 0_5ayear.Thisdecrease will continue forabout 15,000years, after which theobliquity will increase. Itoscillates inthis mannerabout 195o_eitherside ofthemean. Earth isatperihelion aboutJanuary 3andataphelion aboutJul_5.The Earth andtheMoourotate about accmnon ayparent center ofmass. This center isabout 2_0 miles fromthecenter oftheEarth orabout 1,000 miles within thesurface oftheEarth. TheEarth hasoneknovn satellite. TheMoon TheMoon isabout anaverage of60F__-thradii from theEarth.Its nearestdistance is222,000 miles anditsfurthestdistance isabout 253,000 miles from theEarth.TheMoon'sdiameterwhich lies inallne with theEarthisabout 2,163 miles.Theequatorialdiameter (atright an_es toabove) 1mileshorter.isabout 1/7mile shorterandthepolar diameter isabout TheMoon'smass is1/81.5 thatofEarthanditsmeandensity 1_-13 is3.39 that ofwater. Thesurface gravity is1/6that ofthesurface gravity ofEarth. Itreflects about 7percent ofthelight that itreceives. Itssidereal period around theEarth is2_7h4_nmaditsperiod ofrotation about itsaxis isthesame asitsrotationaround theFArth. TheMoonaxis istilted 6.5°toitsorbit additsorbit istilted 2Oto theplane oftheecliptic. Itrotatesandrevolves West toEast. The eccentricity ofitsorbit is0.056. Thetemperature isgreater than I00°C.onsunlight side andless than -i_0°C.onunlight side. Thetemperaturedrops rapidly asEarth shadow falls onit. Material ofMoon, therefore, isalowconductorof heatandhasalowspecificheat. _eonly substancethat weknowof having suchproperties areloosely packeddust, ash_ coarse pc_er,etc. TheMoon hasnoatmosphere aswethink ofanatmosphere andnowater. There issome chance that theheavier gasesmaybefound insome ofthe craters andpossibly enough moisture tosustain some lowtypes ofmoss. Polarization studies indicate that thechemical compositionofthe Moon's surface issimilar tothatoftheEarth's crust. ThesurfaceoftheMoon ismade upofmaria,mountains, craters_ rills andrays. Some ofthese canbeseen inthefollowing pictures. Themaria were originally thought tobeseasbutactuallythey are comparatively flat areasmarkedbysmall cratersp hills,andcracks. Someofthemountaims arevery high, above 25_000 feet andaremostly imchains orgroups. Rills aremarrow crevices -ternto300miles long am_less tham two miles wide. 506< i_-14 Rays arenarrc_streaks, lighter incolor than theirsurroundlngs, radiatingoutfr_n prominent craters.Theyextendhundreds ofmiles across thefaceofthemoon. Cratersarebyfarthemostn_rous lunar formations.They vary insize from 150miles indiameter toi/I0 mile indiameter. Slopes ontheMoon inexcessof49°arequiterare.There areno great clefts orfaultswhich aretremendouslydeep. Thefaults ofthe Moon have slopesofabout 45°. TheMoon's surface hasnotchangednoticeably since observations of itwere first recorded.There issome speculation that this isnotexactly true. Someobservers claim tohavedetected some chants butthese have notbeen verified. SC7< _,_LTS Marshasadiameter of4,200miles. It'sdayis2_hours, 37minutes - about 37minutes longer thanours. Marsisameandistance of1_2._ m_ll_on m41es fromtheSunandrevolves about theSunancein687d_s. Theclosest Mars comes toEarth is35millionmiles anditsgreatestdistanceatOl_O- sition is61million miles.Theseasons onMkrsareabout twiceaslong 8_3ours. Mars andEarth areinopposition about every 2-1/8 Earth years. Opposition iswhen Eazth, Mars,andSunareinstraight line.Theclosest oppositions occur every 15to17years.Thelast onewasonSeptember i0, 1956.Thenext opposition then willbeonOctober 25,1958, (approx.). Indications arethatMars isalivingplanet.Faintatmospheric belts havebeendetectedacross thefaceoftheplanet -andclouds havebeen detected arou_i thenorthern icecapinfall andwlnter.Some- times atmosphere isclear andatother timesopaque (calledbluehaze). Itchanges rapidly forunknown reasons. Whatever atmosphere there is contains very littleoxygen.On_ heavier g_sesremain(nitrogen, C02, argon, andH20).This isprobablyduetosmallmassofplanet (mall gravitation attraction). _eatmospheric pressure isnotknownbutit isestimated tobeabout i/lO thatofEarthatthesurfaceofMars. Otherestimatesplace thepressureat1.16 ib/in. 2.Atmospheric circu- lationonMars issimilar tothatonEarth.However,weather ismore regularduetoKeographyofitssurface. Thetemperature attheequatorduring thed_yisabout 70°to80°F. andatnight about -95°F.Atpoles itisfarbelow zerobothd8_andnight. SO@< i_-16 These temperatures weremeasured with avacuum thermocouple which can measure the heatofacandle _0miles ormore _V- Radiometricmeasurements perndtestimation ofsurfacetemperatures over areas assmall as200miles inradius.This allowsdelineation of the @eneraltemperature field.Such adescription ispresented inthe following sketch. S ?0 @o 4o ! ,,o,'_s_ 2_4 I,- E_-z. 'I •zf__,.'_ aos,d"_.__ -:---_ •I'L"",,v"J ¢ -fro 1 0 Go I_o JSOC/""t3-I,@ ,IAA -- •'"e-.e_I/,--_ __"/-__.,._,.__?.0_.< /1.I"_.,4 _/4__.__ 9o4_'o 2L0 ,,fo, j&o NT.'c eO _o 'W'_0 LO .°00 -110 -10 -40 Thedistribution oftemperature (oC.)onMars inNorthern Hemisphere winter.ValuesmarkedQarequestionable. Measurements made in1926. Streamline nml_ canbedrawn cc_sistant with thetemperature distri- butions. Suchachart isshown below. SES< l_-17 (oO 4d Jo Eo iio 40f,,% \ &OL-- 9 lao 18'00 J_o •40 3oo A/W Aschematic stresmline mapforMers inNortherHemisphere winter.Thearrows represent observed clouddriftdirections. Cloud drift observationsmade in1894. Themoststriking aspectofthese twosketches istheirresemblance toterrestrial weather maps.This indicates aswasmentioned earlier that Mars hasanatmospheric circulatianvery similartoEarth's. Thepoles ofMars arecovered byicecaps butthese caps areprobably only afewinches toacouple offeet thick. Thefamous "canals" onMars arereal.Theyseem tobethelines that thewater from themelting polar icecaps follow.Vegetation growsalong these canals insunder. Some peoplebelievethey areartificial since they aresostraight andintersect -oneruns 1500 miles.Sc_e believe they maybefault lines. These fault lines have been caused bycollision with meteorites orasteroids. Othersseem tothink that they maybe wind rows ofvolcanicdust blown into patterns that weseeandthat they areclearer inspring because thewinds increase.Theonly sure thing 5£0< 14-18 isthat thereissome sort ofnetwork ofso-called canals that appear to lead thewater from themelting polar caps totheequator. M_rs hastwoti_ satellites. They areprobably less than 20miles indiameter. They revolve incircular orbits intheplane oftheplanet's equator. _nenearer, Phobos, isonly _,800 males from thecenter ofthe planet.Itrevolves inthedirection oftheplanet's rotation andits rerlod is7h40m. Theother satellite, Deimos, revolves atadistance of14,600 miles from thecenter oftheplanet anditsperiod is30h18m. • • v ... 14-19 TheAsteroids FirstdiscoveredbyPiazzl in1801(based onBode's law).Between MarsandJupiter there arema_7asteroids orminor planets; 1_00havebeen cataloged andithasbeen estimated that there areasmany as30,000. Ofthefirst900cataloged, themainbo_7 begins atadistance of 2.1astronomicalunitsandcontinues to3.5; then there isagapanda group of6at3.9units (Hilda group); then anisolated one,Thule, at 4.5units; andfinally, theTrojan group(6)at5.2units. Theeccentricities vary: 209haveeccentricity ofOto0.087; 375 from 0.C_7 to0.174; 2_fr_n0.174 to0.299; _9from 0.259 to0.3_2; 7 from 0.342 to0.423 and4stragglers, Albert, Alinda_ _de, andHidalyo that have eccentricity greater than 0.90. Theinclination oftheir orbits totheecliptic alsovaries -222being inclined0°to50;29"(from5°toI0°; 222fromi0°to15°;98from15°to 200; 35from 20°to250; lhfrom 29°to30°adn3above 30°. There isatendency forhigh eccentricity andinclination togo together.Thediametersalso vary -there are195thathavediameters greater than61miles;502between61miles and25miles;193between 25 milesandi0miles and22less thani0miles. Some ofthebigger onesare Name Dia.(miles) Al_lo Ceres _88 .06 Pa11 .07 Vesta 248 .26 Juno 118 .12Inclination of orbit toecliptic 1OO37' 43' 14-20 Eros isanother importantminor planetoid.Itisonly17miles in diameter.Ithasaperiodofrevolution of643d_ys, aneccentricity of 0.222.Itisimportantbecause itcomes towithinI_,000,000miles of Earth andi.13astronomical units from theSun.Itsorbit isgreatly affected bythemassoftheEarthandSun. Observationofthesepertur- bations willaidindetermlning themass oftheEarthandSun.However, itonly comesnearest totheEarth aboutevery 40years.Itsorbit is greatly inclined totheEarth's orbit. Arecently discovered asteroid calledGeog_'aphos isoneoffew asteroids whose orbit isinside that ofourEarth's. Itsplane or revolution isinclined 13°toours. Itsperiod ofrevolution about the Sunis17months.TheEarth andGeographos willbehmillion miles apart durinK August 1969, theclosest forthis century. S:[3< i_-21 Jupiter Jupiter isthelargest planet andthe second brightest. Itsequatorial diameter is88,800miles; itspolardimeter82,0OO miles. OneJupiter yearequals11.86 Earth years. OneJupiterd_y isabout 9hoursand55 minutes. Itsrotational speed atequator isabout 28,800 miles perhour. Thetemperature atthetop oftheatmosphere isbetween -130°C.to-180°C. Itsbulk isequal to1,300 Earths. Rotational speed ofJupiter plus its gaseous atmosphere causes the planet's "atmosphere" toappear tohave analternately dark s_ light band Bike structure. This rapid rotation causes rapid heating andcooling oftheatmosphere which inturn probably causes wind andelectricalstorms far surpassingarA7onEarth.Thedarkbands arecalledbeltsandthelight bandsare zones.Thebands areblueandyellow incolor. Atmosphere probablyconsistsofclouds offrozenammonia floating in aseaofmethane_also probablysomehydrogenandhelium.The atmosphere isestimated tobelO00'sofmiles deep.Colorsandsepctrums canbe reproducedbyfrozen-freeradicals ofammonia andmethane intheLaboratory. Freeradicals aremade byexposingmolecules toproperwavelength oflight andproper temperature. They arestableatlowtemperatures (order of -200°C.).Whenheated theycombine toform moleculesandheat.Couldbe apossiblesource offuel.The circulationofJupiter's atmosphere is similar tothatofEarth. Theyearsofgreatestsunspotactivity aretheyears ofmaxlmtlm spottedness ofJupiter. 5:}.4< i_-22 Thegreat redspot ofJupiter isofunknown cca_sition. Theopinion isthat itfloats intheatmosphere andthere isnodoubtbutthat it affects theatmospheric flow near it. Jupiter has12satellites. Theseven outer ones aresubject to enormous perturbations bytheSun. Their plane oforbit ishighly inclined totheplaneoftheequator. Theinner five about coincides with Jupiter's equatorial plane. Jupiter VIII, IX,XI,andXIIrevolve intheretrograde direction•Jupiter Vissometimes called Amalthea. 4_ 112,500 261,000 _19,000 664,000 1,167,000 7,200,000 7,300,000 .+ 7,500,000 .+ 15,000,000 14,000,000 i_,600,000t 14,900,000_no ................ i iih57m 3d13b 7__h 16d17h 26oa12', 266dt + oh 692dlehto I007 2200 1925 5350 2O 8oT 29T 18 zTT 19Y........ r O 5°6!9 3°6!7 3°9_8 3°2_3 290 _8o z_8o z56°JupiterV Io Europa Ganymede Callisto JupiterX Jupiter VI Jupiter VII Jupiter XII Jupiter XX Jupiter VIII JupiterIX!°i o.ooz8 ...... 0 2•7 .69 •o0o52.6 •76 .oo191•5 ._9 •oo79 l•O.z6 .195o ...... •2O7O ...... •210 ...... .250o ...... S .S< 14-23 s.s. Temperate zone Se zone S.Tropical zone Equatorial zone N.Tropical zone N.Temperate zone N.N. Temperate zone NS.Polarregion S.S.Temperatebelt S.Temperate belt Equatorial belt band Equatorialbelt N.Temperate belt N.N. Temperate belt N.Polar regions 9h55m5s 9h55m29s 55m NS.Polar current S.S.Temperate current S.A"_erate current Tropical current PartS.Equatorial current Great redspot Great equatorial current N.Tropical current N.Temperate current NoN.Te_rate current N.Polar current Zones, belts, currents, an_rotationaltimes ofJupiter 26< Saturn Onlyknown planet withrings around it. withicesurrounded byagaseous mantle ofmethane. Itsmeandistance fromSunis886,000,000 miles. L_).46 Earth years. Itrotates in10hours 14minutes.Theplanet isprobably coated Oneyea_equals Itsequatorial diameter is75,000 miles.Itsrotational speed differsatdifferent latitudesasinthecaseofJupiter andtheSun. Itstemperature isabout -i_0°C.Saturn ismore flattened atthepoles than Jupiter. Itspolar diameter is67,000 miles compared to75,000 miles fortheequator. Its surface(llkeJupiter) ismarkedbydusky beltswithlight intermediate zo_le8, Theoutstanding feature ofSaturn isitsrings. _ese rings are translucent andarecomposed ofti_ hig_reflective solidparticles ormoonlets. _eringsareintheplaneoftheequatorofSaturn andare inclined about 27°totheecliptic. RingsofSaturn Radiusofouter limitofrlngsystem Widthofouter ring (called A) WidthofCassini's division (1675) widthofrl.g(B) Width ofCrepering(C) Distance ofinner edge ofring Ctosurface ofSatura ThicknessofringsMiles 86,300 ii,I00 2,200 18,000 11,000 6,000 less thani00 l_oba_y 20to_.0 1_-25 Another feature ofSaturn hasbeentheperiodic observation ofa white spotonitsdisk. Thisspotwasobserved in1876, 1903, and1933. Thespothasvaried insizefromabout 1/5theplanet diameter to2/3 theplanetdiameter. _he1933 spotstarted with anEast-Westdiameter equal to1/5oftheplanets equatorialdiameter.Itgrew to2/3of theplanets equatorialdiameterandfinallydissolved intoawhite band 1/2theplanets equatorialdiameter insize. Asecondspotemerged from thecenter ofthefirsto_eandmerged into thepreviously existing bright zone changing itinappearance.Therotational period ofi0h16m decreased toI0h13m._]_ere also hasbeen observedasemlregular fluctuation inthereflectivity ofSaturn.This fluctuation hasaperiod ofi0years forSaturn. Saturnhasnine satellites. Theorbits ofthefive innerones are circular andlieintheplaneoftheplanet'sequator andrings.Titan andHyperion also revolve nearly intheplaneoftheringsbutJapetus isinclined about i0°.Phoebe revolves intheretrograde dlrecticn in theplane oftheorbit ofSaturn. Hyperion hasanorbital retrograde motionof18°40'annually. D _-z6 Satellites ofSaturn Encela_us Tethys Dione Rhea Titan Hyperion Japetus Phoebe117,000 157,000 186,000 238,000 332,000 771,000 9_,000 2,225_ 000 8,000,000crj0 d.h 022.6 18.9 i21.3 217.7 412.4 15_.7 21 6.6 79 7.9 550 10.60 0 26 z6 26 26 26 26 26 16 174.742_.7 _.7 A_.7 44.7 4_.9 7.i 0 18.1O.0190 600+ w .o001 8oot .o0oo 1200t .o020 11o0t .0oo9 150ot .o289 3o8o-+ ._o_3 500+. .02842O00t .1659 2oo±1789 rr89 16_ 16_ 1672 16_ 1_8 1671 1898 _-Z7 Uranus Uranus wasaccidentally discovered byHerschel in]7_1. Uranus appears asapale green oblate spheroid. Itseems tohavebands orbelts and streaksacross itssurface. Ithasarotational periodofabout 10-2/3 hours. Theequator isinclined 02°toplane oforbit.Its rotation isretrograde. Itstemperature isabout -170°C. Whereas the outeratmospheres ofJupiter andSaturn consistmainly ofhydrogen andhelium twith methane thenextmostcce_o_ element;neon and other inert gases with lower boiling pointsm_ybemore abundant on Uranus andNeptune. Asimi-regular long period fluctuation inthereflectlvity of Uranus ofO._years hasbeen observed. Uranus hasfive satellites w_ich revolve insame plane asplanet's equator. Their motion isretrograde. Ns/De Ariel Umbriel Titania Oberon MirandaSatellitesofUranus Distance from e_n_ter of Uranus, miles 120,000 107,000 273,000 363,000 75,000Period sidereal zaz2.5h 8dl_._ 13dll._ z_zo._Diameter _AJA_A_A_.I r_les _0 _00 i000 _0 5£0< 14-2t_ Neptune Diameter ofNeptune(discovered initS6) isabout20,000 miles.Its period ofrotation isabout IGhours.Ithasadenseatmosphere. Asemi- regular fluctuation inthereflectivity of_.4yrs. hasbeen observed. PredictedbyAdams andLeverrier.Neptune hastwosattelites which are called Triton andNereid. Tritons motion isretrograde. Other characteristics ofTriton are: Diameter -2,000 miles Distance from center ofNeptune -222,000 miles Sidereal period-5ds_ys,21hours Inclination oforbit toecliptic -35° Discovered ini_6 Thecharacteristics ofNereid are- Diameter -200miles Distance from center ofNeptune -3,500,000 miles Sidereal period-559dOh0m Pluto There isnothingmuch known about Pluto.Itsdiameter andmass are smaller than that ofearth.Ithasayellowishappearance. Predictedby Prof. Lowell in1915.Heworked from1905 to1915.Hisprediction was notvery accurate, placing theplanet inoaeoftwoopposite regionsofthe Zodiac. After Lowell's death in1916, theplanet wasfinally discovered Ln1930 byTombaugh. Some claim that thediscrepancies oftheorbits ofUranus andNeptune cannot becaused byPluto because ofitssmall mass. Pluto, because ofits size, probably belongs intheclass ofsmaller planets mreven asteroids. SZI< -29 Havingdisposed ofthegeneral discussionofthesolarsystemandthe major elements therein, theremainin6part ofthenotes isdevoted tothe presentation ofnumericaldataanddefinitionsofterms. Thetables present thenumericaldata fortheSunstheplanets, the satellites oftheplanets andforSome oftheasteroids. Aparticular valuepresented inthetables isacompilation ofvalues obtained from manysources andthereforemaynotbeexactly consistant with other values inthetables, however, itisbelieved that thevalues presented arethe best that areavailable, atleast totheknowledge oftheauthor.Values notincluded ormarked withquestion marks aresubject toquestion, either because they cannot bemeasured accurately orbecause thevarioussources presented widely differentvalues. a o C d G h i L M m P s T yr. 5 u) nSymbols semi-n_or axis semi-minor axis distance between center ofellipse andfoct time ind_s (sometimes used assuperscript) eccentricity time inhours (sometimes used assuperscript) inclinationoforbit totheecliptic mean heliocentric longitude mean anomaly ofaplanetataspecific epoch. (The same a_le as@used inprevious lectures.) time inminutes (sometimes used assuperscript) sidereal period time inseconds (sometimes used assuperscript) time ofperihelionpassage time inyears right ascension declination longitude ofperihelion meandaily motion argumentofthelatitudeofperihelion longitudeofascendingnode 52S< 14-31 _o be I0. Ii.Definition ofTermsUsed inAstronomicalLiterature i.Siderealda_-Theperiod ofonerotation oftheEarth relative tothe stars. 2.Sidereal month-Theinterval between twosuccessive arrivals ofthe moon atagiven apparentplace smongthestars. 3.Sidereal year -Theinterval between twosuccessive arrivals ofthe Sunata81yen apparent place amongthestars. Solar ds_v-Theinterval between twosuccessive meridian passages of Sun. 5-Synodic month -Thetime ofarevolution oftheMoon with respect to theapparent place oftheSun-that isfrom conjunctiontoconjunction. Nodical month -Thetime oftheMoon's revolutien with respect to eithernode. 7.Tropical year -Theinterval between successive arrivals oftheSun attheVernal equinox. o.AnomAlistic year -Theintervalbetween twosuccessive arrivals of theSunan_Earth atthesame true anomoly. o___._o_a!. _+_--_,,_ -"_-.,,_hotu-an@leoftheVernalequinox ortheright ascension ofthemeridian. Solar time-Thehourangle oftheSun.Itdiffers fromsidereal time bytherightascensic_ oftheSun. Astronomlclatitude-Theangle between theplane oftheequatorand thedirectionofgravityatthelocation inquestion. Geocentriclatitude-Theangle between theplane oftheequator and astraight line passing from thelocatio_ inquestien tothecenter 5.7. 4< 14-32 oftheearth. Itdiffers from astronomic latitudeduetothe oblatenessoftheEarth. 13.Geographic latitude -Theanglebetween theplane oftheequator end anormal tothestandard spheroid. Itdiffers from astranomic latitude only bytheeffectsoflocal deviations ofthedirection ofgravity. 14.Geocentric longitude -Thearcoftheecliptic_measured Eastward from theVernal equinox totheapparent positien oftheSunasseen fromtheEarth. 15. Heliocentric longitude -Samedefinitic_asGeocentric longitude except that itisasseen from theSun.ItisI_0°opposite Geocentric longitude. 16. Geocentric -Asseen from orreferred totheEarth. 17. Heliocentric -Asseen from orreferred totheSun. i0. Conjunction -Atime atwhich either thebod_ isbetween theSunend theEarth (inferiorconjunction) orwhen theSunisbetween thebody andtheEarth (superior conjunction). 19. Opposition -Atime when theEarth isbetween theSunandthebod_. 20. Synodic period-Thetime between twosuccessive oppositionsor successive superior conjunctions. 21. Transits-Anoccurrence inwhich oneplanet passes directlybetween anotherplanet andtheSun_ thus appearingasablack dot_ the photosphere oftheSun. 22.F_uinox -points ofintersection oftheapparent pathoftheSun(the ecliptic) andtheEarth's equator. _5< 14-33 23.Vernalequinox-TheequinoxwheretheSuncrosses fl-anSouth toNorth oftheequator.(Occurs inthespring.) 24. Line ofapsides -Aline ofinfinite lengthpassing through theapes (perihelion andaphelion) andthrough theloci oftheellipticorbit. Themajoraxis oftheellipse isasegment. 25. Solar parralax -Theapparentsemi-diameter oftheEarth asSunfrom theSun. 26.Nutation -Asmall oscillation oftheEarth'spoles ofrotationdueto theregression oftheMoon'snodes. 27. Aberration -Theeffect oftheorbital motion oftheEarthupon the apparentdirection ofthelight that comes tousfrom astar. 2_. Albedo -Ameasure ofreflective power. Theratiooflight reflected tolight received. 29.Dicrnalmotion -Theapparent revolution ofalltheheaven_vbodies a around theEarth. 30.Mean Anomaly -Theangle theradius vectorsweeps through asthe planet revolves around theSunmeasured fromperihelion. 31. Hourannie -Thearcofthecelestial equator included between the meridian andthestar's hour circle. Themeridianandhour circle aredefined later under Coordinate SystemsUsed inAstronomy. .- Elements ofaPlanet' sOrbit There areseven terms whichdefine co=plete_7 theorbit ofaplanet. They are i. 2. 3. 5. 6. eThelon6itu_eoftheascendingnode, _. Theinclination totheecliptic, i. Thelongitudeofperihelion, ,orthe"the argumentofthelatitude ofperihellon", _. Thesemi-major axis,a. Theeccentricity, ¢. Themean helicentric longitude,L,orthemean anomaly M,ofthe planet ataspecified epoch;orthetime ofperihelion passage, Thesidereal period, P,ormean daily motion, _. Theexplanation ofthese terms isbest understood bytheuseofthe following figure. R JJTI 0N| _# )rbit ofplanet // a Ac: KOrb: of % n L II_-39 Plane _represents theplane oftheEarth's _rbit (ortheecliptic) andORBItheplane oftheorbit ofanother planet. Thelineofintersection_ IQI'which passes through thesunisthelineofnodes. _eplanet _asses fromtheSouth totheNorth sideoftheecliptic atthepoint nswhich isthebeginning oftheasceDdi_ node. Theline S_risdream fromthecenteroftheSuntoward theposition ofthevernal equinox onthecelestial sphere.Theangle betweenS and NN'measured toward theascendingnode (Eastward) istheIcMgitude oftheascendingnode _.Theanglebetween thetwoplanes isi, theinclination. These twoelements (_and i)define theposition oftheorbit plane. Theorientationoftheorbit within itsplanema_bedescribedbythe angle _,measured intheplaneoftheorbit andinthedirectionofthe body'smotion (eastward forplanetsbutwestward formany comets)between SNandSP,Pbeing theperihelion. Fortheprincipal planets, however, itiscustomary tosubstitute for _,thelongitudeofperihelion _which isthesumof_andft. Thesemi-majoraxis,a,ormean distanceoftheplanet from theSun, defines thesizeoftheorbit. Itisusual_expressed inastronomical units, which isbased onthesemi-major axisoftheEarth'sorbit, (92,900,000 milesori_9,500,OO0kilometers). Theeccentricity, _,defines theshapeoftheorbit. Xtisthe ratioofcover a,cbeing thedistanceoftheloci orSunfrom the centeroftheorbitandabei_ thesemi-majoraxis. S< Todetermine thepositionoftheplanet ata_time wemustknow theposition ataspecific time (atEpoch) andthetimeofrevolutio_ orthemean dail_ motica_; _;which issi=pl_ 360°dividedbythenumber ofd_vs inP. Elements ofanElliptic OrBit A F SP2 D Fand SareFoci. APisthelongestdiameter andiscalled themajor axis. BDistheshortest diameter andiscalled theminor axis. adenotes thesemi-major axis. b'denotes thesemi-mlnor axis. cdenotes thedistance from thecenter Ctothefoci ForS. Theeccentricity is¢andequals candisnever greater than i. a Thesmaller theeccentricity themorenearly circular theellipse. IfSisthefoci about which theplanet isrevolving then Pis called theperihelion and AIscalled thee_helion.Theline SEis called theradius vector andtheangle PSE iscalled thetrue anomaly. ,-4 .... ,_....... A I_"_"__',iR__f_,A., _,R,_,RIIA"AAI ,_o_§_§, _(B "•",-i••x_W-•$o r"4!@,Q @,,, ®_ ,__o.__' _ "_•mo°. __9_ _x__ 0lh-_Tm 53O< r-4_'_ ,____i¢'"" _.__ . e'l e_ e_ _.o.o. _ _,,, 0_._i_,,_,___ ,,.__'' ___ II__ il__ ,_.ii_14-37b S_.,i< -M C ___ r-_ I qP r-g m,_._ m•-_____ I I_i_ ,-I I I14-37c 532< F_ 0 0 r-I e-4° _o,,-I '__,___ I_o,,-4 0C_ 0000 ! I I I I I "i S_3<z_-37d CD l q_ ,-411)4_ ; oIIIII'_,_,,III I IIIsM,,__,I!II ,_ I,_ ,___, IIII IIIII"!''''_ _,,, __, ''_ _',, _,= ,. ! _.i"" 0.,-I0 -_ -I I I''_I , ,__ ,,_,_,I I I_ I '__''_a_ ' ._" C_lC_IN__-f (__ (%1 _ W ,__' ,._-_.,"_-I__ ,M{_ .._iL-37e 0 00 0 014-38a 0 o&_ 0 5._5< e 8 14-38 vag 888 82 7 Pie , 2 gzadeR H Ray. & xX MA 85 i 3 eSS ZA -4 \ Roy a4 SO ANSSy ;SINS //||E % e 5c6< ,-I ,-I ,-II,..4 i | i eA u_ _°_ +;+_._.-I 0 ,M 0 .la o X) !I I m!! !i e.-e_ _o 00 e,. m14-39a XXXx____,IIIIIIII____o,,,,,,,, e-_ e.-@* e-. e-@* e-. i... A o,_ _, 0 • I 00 0 0 0 ________•. 0 --iiiii- )o ,.j| .o ..-,., ii 5XX e..x x !____x___,,_:iiii! :•,••••,•• =G_.=O : !"U .t' ,._e© ,,4 0_,,4 _o_ I II.... __i-,i-,i.-i..e* - '__.__ _o "_ °_._ _ii_:_ __'_.I,4 llli -'!_"_•i _._!, ) 14-40a v o _e U r-4 [email protected] ,-4 .... _•_........ V S&J,_.4_.......... ,-4 5._i9< Ib-bOb "0 0p 'I0 o 0 I0 r_ I,,,iG) •4_r..) 00 0 $ #_e iii| ,-4 lh-hl 4oa;0 _o_ _o m0.8II A A A4_ ,4-- S,ll< • .... i_-4Z Dimensions oftheTerrestrial Spheroid (Fr_n Hayford' sSpheroid of1909) Equatorial radiusa-637_._ Km= 3963._ ml Polar radiusb=63_6.909 Km=3949.99 mi Mean semidiameter, i/3(2a+h)=6.37123 xlO_cm=6371.23 km =39.5o._9 mi Oblateness a-b/a =1/297 IOlatitude, _.(instatute miles) =69.0569 -.3_ cos2_+.0007 cos4_ i°longitude (_nstatute miles) =69.2316 cos_-.0_ cos3_+.0001 cos Thedistanceoftheseahorizon inmiles isequal tothesquare root of3/2oftheobserver's height infeet. Thedipofthehorizon inminutes ofarcisequal tothesquare root oftheobserver's height infeet. AstronomicalConstants Lengthofday: Sidereal =23h_=6m4s.091 ofmean solar time. Mean solar =2_h3m56s555ofsidereal time. Length ofyear (inmean solar units), 1900 -Newcomb Tropical- 365_.24219_79=365d5h_m_5.s98 Sidereal =36_.29656042 =56_ 6h9m9.s_ =3.15_ X107sec Anomalistic =36_b'.2.._:_.13k •=365d6h13m53.a01 Lengthofmonth (inmeansolar units) (according toBrown) s_i_al- e_.53o_- 29d12h4_m2_.8 Sidereal =27_.521661 =27d7h43mllS5 .odi_az=2_.2_2_=2_5h5'_35._ 14-43 Obliquity oftheecliptic =23°27'8"26-0'.'46t_(t-1900)_ Generalprecession =50"2564+"000222 (t-1900)N_comb -% Constant ofnutation -9'.'21 Aiopte_ forZphemerls purposes Constant ofaberration =20"47 Paris comference, 1911 Solar parallax m_"80 Velocity oflight =299,776 km/sec -I_6,273 ml/sec (mrge 19_I) Constant ofgravitation,G-(6.670f.005) xi0-8C.G.S. units (Birge 1941) Acceleration ofgravity, g,(inmeters) =9._ -.0260 Cos2_-2h/R g (from Helment), inwhich histheelevation above sealevel inmeters andlogR=6.t_16. Earth's weight =(5.975 .+.00_) x1027 grams =6.59 x1021 short tons Sun's weight -1.992 ×1035 gra_s Sun's mean radius =6.965×I0I0cm Oneastronomical unit (A.U.) =1.496_ ×108_=9.3×107mi Onelight year -6.322×104A.U.-9.260×1012 km=5.t_×1012 mi Oneparsec =3.263 light years -2.06265 x105A.U. =3.087 x1013 km =1.92 ×lOIS ml SiS< Kepler's LawsofPlanetary Motion i.Each planet moves inanellipse which hastheSunatoneofitsloci. 2.Theradius vectorofeach planet passes over equal areas inequal intervals oftime (isv ofareas). 3.Thecubes ofthemeandistance ofanytwoplanets from theSunareto each other asthesquares oftheir periodic times ora13:a_3:= :PI2:P2 2. IMxle'sYaw Theapproximate mean distance oftheplanets from theSunmaybe convlently reme_red byarelation first pointed outbyTitus butnow commonly known asBode's law.Ifwewrite aseries of_'sandaddtothem thenumber O;3×I=3;3×2=6;6×2=12;12×2=2_;etc. thus 4 4 4 4 4 _ercury Venus Earth Mars AsteroidsJupiterSaturnUranusNeptune wegetaseries ofnumbers that areapproximately tentimes themean distances oftheplanets inastronomical units. There isnoknown reason ..'L.__ .X.'t.../............... 1--- W,LL_t_AJ.L_ _I,J[A_::UV'_.A" WUJ.'I"_3e Units Astronomical units areusually used intalking about distances in thesolar system. Fordistances greater than those ofoursolarsystem theastronomical unit istoosmall since itisbased onthesemidiameter oftheearth' sorbit. 544< 14-_5 TheEnglish have invented anewunit called aparsec. Theparsec isbased cmthedistance astar isfrom theEarth that hasaparalla_ of i". Thedistance ofastar whose parallax isPseconds isthensimply I/P parsecs. Onelight yr. Oneparsec Distance of proxi,naApproximate Ecuivalents Centi- Light yrs. Parsecs Ast. unitsMillions ofmi.meters 1.00 3.°_6 *.200.32 1.00 1.2865,000 206,265 26_,_006 2O 2_i0I_ 3.lOl_ .i0I_ Prcxima isthenearest star tous. CoordinatesSystem Used inAstronom_ Inastroncmy theskyisconsidered acelestial sphere with theearth atthecenter. Theapparent positionofthestars isdescribedbylocating itsprojection upon thecelestial sphere. There arefour coordinate systems used tolocate theobjects onthecelestial sphere andthey allhave common elements. I.Fundamental circle -great circleofthesphere. Poles are located90°tothis circle. 2.Secondary great circle -these circles pass through poles (ecrrespond tocircles oflongitude onEarth). 3.Parallels -smaller circles parallel tofundamental circles. Thefour systems areesfollows: S S< Horizon System 1.Fundamental circle -itisthehorizon. Thepoles arecalled thezenith (overhead) andNadir. Their position isdefined bythedirection of gravity. 2._condary circles -theyarecalled Vertical circles. Thevertical circle going through north andsouth iscalled themeridian andthe onegoing through eastandwestiscalled thepr_ne vertical. _.Parallels -Almucanters. Coordinates ofastarareazimuth and altitude. Azimuth isthearcofthehorizon measured intheclockwise direction fromthesouth point topoint ofthestar's vertical circle. Itisexpressed indegrees (0°to560°).Altitude ofastaristhe arcofavertical circle included between thestarandthehorizon in degrees. Thecomplementary angle ofthealtitude isthezenith distance. Equator System i.Fundamenta] circle -itiscalled thecelestialequator. Poles are points intheskywhich have nodiurnal motion. These arepoints where theearth's axis intersect thecelestial sphere. 2.Secondary circle -hour circle. 3.Pp_ral!els -Parallels ofdeclination. Thedeclination ofastar isthearcofanhour circle included between thestar andthecelestial equator. Reckoned indegrees, +if above equetor. Symbol is5. Thehour angle isthearcofthecelestial equator included between the meridianandthestar's hour circle. Itisusually measured westward and isexpre,qsed inhours. Itismeasured from theobserver's meridian. $16< Rightascension ofastar isthearcofthecelestial equatorincluded between thevernal equinoxandthestar'shourcircle. Xtisreckoned eastwsrdfromvernalequinoxandisexpresse_ inhours.Itssymbol is=. Onecomplete revolution ofthecelestialsphereiscalledasidereal ds_7.Itisabouthminutesshorter than solar day.Siderealnoon occurs when thevernalequinox isonthemeridian. Sidereal time atanymoment isthehour angle ofthevernal equinox orright ascention ofthemeridian. _]quinoctial colure Sidereal time 8 _o__P Eastward _auator Starhour angle =sidereal time -rightascension EclipticSystem i.Fundamental circle-path ofsunamong thestars iscalled theecliptic. Angleatwhich itintersects equator (23_-i/2) istheobliquity and thepoints ofintersectionoftheequinox.90°totheequinox isthe solstice.Hour circle thatpasses throu6htheequinoxesandsoltices ereknown astheequinoctical andsolstitial colures. 2.Secondary circles -theyarecalled secondaries totheecliptic.They aregreat circlespassing through thenorthandsouthpoles. 3.Parallels -they arecalledparallels tolatitude.They aresmaller circles parallel totheecliptic. 1 .-48 Celestial longitude ofastar isthearcoftheeclipticmeasured eastward from thevernal equinox tothesecondsrycirclethat passes through thestar. Celestial latitude isthearcofthesecondarybetween the star andtheecliptic.Itis÷ifthestarliesnorthoftheecliptic. Galactic System I.Fundamental circle -Galactic circle.Itisthecenterline ofthe milky way.Itisinclined 62°tocelestial equator.North pole isatright ascension 12h_manddeclination+27°. 2.Secondary circles -secondaries togalactic circle.Same assecondaries totheecliptic. 3.Parallels -parallelsofgalactic latitude.Same asparallelsoflatitude intheecliptic system. Galacticlongitude isreckonedfrom theintersection ofthegalactic circle with thecelestial equator at==18h_hm.Galactic latitude and longitude arerelated tothegalactic circle exactly ascelestial latitude andlongitude arerelated totheecliptic. GeneralRemarks onCoordinate Systems Thehorizon systemmoves with theobserver.Theequatorsystem is based ontheearth's rotation about itsaxis.Itisthesame foreverybody onearth. Theecliptic system isbasedonearth's revolution around the sun(earth's orbit).This system isalso _nesame foreverybo_7 onearth. Thegalactic system isbasedonstructureofvisable universeofstars. Itwould hold forallplanetsofoursolarsystem andnearbyneighboring stars. 14-49 =0 g ro 0 ¢f3 O to r,.)¢J ¢J ,,-4 r_ _o o= N _o ¢,J 4am 8 0 B _o !",-4 °_ 0 OCH I_0o_=.,-,o'=__ '_ ,.< °__°_d o=o#._ _ 0 .,.,,-4__®0 0 •,,.40 i. 14-50 REFERENCES io e _e e 5e e 7. _e QQ / !O. 11. 14.Rice,F.O.:TheChemistry ofJupiter.Scientific American. V.194,no.6,pp119-120, 122, 124, 126.June 1956. Slipher, E.C.,Hess, Seymour L.,Blackodar, Alfred K._Gidos, H.L., Shapiro, R.,Lorenz, E.N.,Gifford, F.A.,Mintz,Y.,andJohnson, H.L. TheStudy ofPlanetaryAtmospheres. (Final Rept.) (Sponsoredby AirForce Cambridge Research Center). Lowell Observatory. September 30,1952. Menzel, D.H.:Harvard College Observatory. Exploring OurNeighbor World, TheMoon. TheNational Geographic Magazine. Page 277-296. February 19_. S!ipher, E.C.:NewLight ontheChangingFace ofMars. TheNational Geographic Magazine. Pp427-436. September 1955. Bowen, I.S.:S_Survey Charts theUniverse. _heNational Geographic Magazine. Pp.7_0-790. December 1956. Duncan, J.C.:Astronomy. _zL_per andBrothers Publishers. Astronom_-I-TheSolar System. Russell, Dugan, andStewart. Ginn andCo. _omeAspects oftheP_teorologyofMars.Journal ofMeteorology. 7:1-13 (195o) Compendium ofMeteorology, American Meteorology Society, 1951. p391-397. Reports onProgress inPhysics.VolXIII. 1950. _atel!ite .6_.mogpheres. p.__........ _.Planetary a_ TheAtmosphere ofVenus.S.H.Dole.TheRand Corporation. October 12,1956. Encyclopedia Britannica. Interim Report onSearch forSmall Earth Satellites forPeriod 1953-1956. C.W.Tombaugh, Physical Science Laboratory, State College, NewMexico. TheAtmospheresoftheEarth andPlanets. RevisedEdition, KT TheUniversity ofChicago Press, Chicago ILl. Edited byG.P.Kuiper. 15. TheAmerican Ephemeris 8ndNautical _Imsnac,1958. 550< FJJ.-_I_I 'SATHOSI_ERE W.J.O'Sullivan_ Jr.andJ.L.Mitchell Upuntil about 1_6theatn_phere hadbeenexplored indetail - pressure_ temperature, density, andcomposition -ataltitude upto amaxim_ ofabout 1_0,000 feetbymeans ofsotw_tng balloon_ i.e.I radiosonde techniques. Onthebasis ofanaverage ofthese measure- ments theInternational Civil Aviation Organization_ ICAO, agreed to theadoption ofastandard atmosphere foraltitudes to65s800 feet. ThisICAOstandard atmosphere ispublished asNACATR1235_ reference 1. After about 1_6, thesounding rocket cameintouseasapowerful research toolfarupper atmospheric research. Theinitial sounding rocket research wasconducted under theguidance ofapanel ofupper atmospheric scientists offlciall_r designated astheV-2panel. The V-2panel later changed itsnametoUpper Atmosphere Rocket Research PanelandisnowcalledtheRocket andSatellitePanel.Underthis panel(whatever itsname) wascoordinated theupperatmospheric researchdaneprior totheInternational GeophysicalTear,using the V-2andlater theAerohee, .Viking andNlke-Deacon orCajunsoundin8 rockets. These sounding rockets haveextended thedirect measurements oftheatmosphereuptoamaximum height ofabout 700,000 feet. Asa result, uptoabout 400_000 feetwenowknowagreat dealabout the detailed pressure,temperature,densltytandcomposition oftheair. SSi< 14-52 Theresultsofthesedirectsounding rocketdataavailableuptoabout 1955, along with indirectmeteor andaurora data anddeducticus from theory, wereused tocompile amodel atmosphere upto118_0,870 feet. This atmosphere waspublishedbytheAirForce Cambridge Research Center as"The ARDC Model Atmosphere 19_6", reference 2. Atthepresent time,F_y1958, first results fromSputnikIand Explorer Isatelliteshavegivenvaluesofatmosphericdensity considerably higher than the"ARDC Atmosphere" ataltitudesofabout 720,000 feet and1,200,000 feet, respectively. Onthebasisofthese measurements, Sterne, Folkert, andSchillingoftheSmithsc_ian Institution Astrophysical Observatory,reference 3,have suggested arevision ofthe"ARDC Atmosphere". Figure ishowsthetwosatellite points, the"ARDC Atmospher" andtheproposed revision.Explanations forsome ofthechanges inthevarious curves aregivenonthefigure. Sincetheconcept ofgeopotentialaltitude isofinterest_ an explanation regarding therelationship of_eopotentialandgeometric altitude isinorder. Adetaileddiscussion willbefound intheARDC ModelAtmosphere Report.Thebasic definition of@eopotential isas follows:Thegeopotential ofapoint isdefined astheincrease in potential energy perunitmass lifted fran mean sealevel tothat point against theforce ofgravity. Nowtheincrease inpotential energypermxltmassofabod_ lifted against theforceofgravity,from sealevel,throt_h avertical distance toagiven point is: SS2< 14-53 _-=gdz-.O.P. AE=increase inpotential energy, ft-lb m=massofbodyinslugs g-acceleration ofgravity atpoint zf/sec2 z=geometricaltitudeabovemean sealevel_ft G.P.-geopotentlal inft-lb/slugs(z) Nowifwedefineageol_tentialaltitude through which themassmustberaisedataconstant standard value of accelerationofgravity, i.e.,Gtogetthesame change inpotential energM:Htobethatvalueofaltitude theno.P.-m_ (2) _oZs--_z (3) furtherusing theinversesquare lawforthevariationofgwithz,i.e., g<p=sealevelvalueofgatlatitude of r_=radiusoftheEarth atlatitude _ft H=G/° r?,r_+ dzofthepoint,f/s2(_) (9) 55S< 1_-54 integrating gives or(6) Z8r_H g-_r_-wG(7) ThedefinitionofGissuch that atlatitude 45°32'40", _/G=i. Atthis latitude r=20,855,531 feet, so H Zi+ 20,85.5, 531(8) Thesignificance andusefulness ofthegeopotentialaltitude is that theaverage atmospheric propertiesonanonspheroidal Earth are invariant with geopotential altitude rather thangeometric altitude. Theequation ofhydrostatic equilibrium forinstance is dp= -ogdz (9) where gdepends onlatitude andaltitude byequation(_). Differentiating equation(3)weget GdH= gdz (zo) sothat independentoflatitude dp=-GodH (n) Equation(Ii) along with theperfect gasis_, themolecular weight of theari, aspecified variationoftemperature with height (usually 554< i -55 geopotential height), andasealevel value ofpressurehavebeenused tocalculate thestandard atmospheres inthereports referred to previously. Abrief history ofstandard atmospheres anddetailed anal_sis of thetheory andcalculation procedures aregiven inreference 2.Asummry ofradiosondetemperature measurements isgiven inreference 4.Reference 5presents asummaryofsounding rocket measurements toJanuary 19_. Sounding rocket techniques arediscussed inreference 6.Reference 7 isanextensivestudyoftheatmosphere andcontainsawealthofbasic references foa"further stud_. 14-56 i.TR1235(NACA), Standard Atmosphere-Tables andData forAltitudes to65,800 Feet (1959). 2.Minzner, R.A.,andRipley, W.S.:TheARDC Model Atmosphere, 1996. AirForceSurveys inGeophysics,No.86,December1956. 3.Smithsonian Institution Astrophysical Observatory. Special Report No.7,IGYProject No.30.i0,AnXnterim Model Atmosphere Fitted toPreliminary Densitie_ InferredFromUSSR Satellltes. _.Tolefson,HaroldB.:A5kwmmkryofRadiosondeTemperature Observations forAltitudes uptoI00,000 Feet Over Several Geographical Areas. _ACAT_4169. 5.Physical Review,volume 8_,no.9,pages1027-1032, Decemberi,1952. 6.Boyd, R.L.F._andSeaton,M.J.:RocketExploratian oftheUpper Atmosphere. 7.Kuiper_ Gerard P.:TheEarth asaPlanet. 556< 1800 1700 s_ 1400 t MII00_- '!f 5O0 200 lO-0o io-0O10-_oUNITED STATES 5_7< ATMOSPHERE iT1 __,1700 ,/xlO3 1500 ................... / -. ................ --•-- 1400 .............. t3oo emDerature 1200 /i, i i _I Molecularweight ratio -/H- 1100ightemperature heretheorized ............ 7_- duetotheextension ofthe - solar corona beyond earth's orbit I000 .......... 7z.................. 900_ ......... N2Dissociation ........................ _k_......................... 700 5OO -_oioi÷c,_'ion i___i--_i2-_i_--400 /-Sound waveconcept notusedabove here /500 L ozone _ .................... 0 200 400 600 800 I000 1200 1400 1600 1800 2000 2200 2400 2600 2800 Temperature, °R 660 700 740 780 820 860 900 940 980 1020 1060 I100 1140 1180 Speed ofsound, ft/sec 0.4 05 06 0.7 0.8 09 I0 I.I 1.2 1.5 1.4 15 1.6 1.7 Molecular weight ratio,M/Mo=M/(28966) Viscosity rotio,#/Fo=F/(&7579 x10-7)ARDC 1956 Extension Speculative Tentative __L f Standard ICAOt 558< i5-i S_L_ION XV Communications andTracking For many years tocome, the major purpose ofeach vehicle sent into space willbetogather information both for application tothe design offuture vehicles and for the further- smce ofgeneral scientific knowledge. Asystem ofcommunicating this information toearth, assoon aspossible after itIscollect- ed, isanessential part ofany foreseeable space vehicle. The ability totrack aspace vehicle isessential also, for the position and velocity ofthe vehicle are required for guidance and for observing possible disturbances inflight path. Since electromagnetic waves form the only known practical medium for communication through space, itwill bewell tocon- sider some ofthe factors which influence the transmission and reception ofthese waves and the relative merits ofsome methods ofimpressing information signals upon them. For the present, visible wavelengths will not beconsidered. The first consideration indesigning atransmitter will bechoosing the frequency ofthe carrier wave. Several factors enter here. The earth's atmosphere, particularly the leyer called the ionosphere, attenuates, refracts orreflects radio waves de- pending onthe frequency ofthe radio waves and the angle of incidence. Commercial, military and amateur radio transmissions pretty well cover the practical wavelengths, tosay nothing of interference from radio stars. Asakind ofoptimum, U.S. satellite designers have chosen 108 megacycles per second.This frequency occurs atthe upper edge ofthe commercial F.M. band, 15-2 andmosthomeF.M.setscanbemodified easily toreceive thestronger U.S.satellite signals. Another matter forconsideration isthemanner in whichtheradiocarrier waveistobemodulated bytheinformation tobetransmitted. Thetwomostcommonmethods arefrequency modulation, wherethecarrier frequency ischanged inproportion totheinformation signal, andamplitude modulation, whereby the strength ofthecarrier waveismadeproportional totheir_or- mation signal whileitsfrequency isheldconstant. Ofthese twomethods, theA.M.hastheadvantage thatavcry selective receiver maybeused,sincethemajorfrequency content ofthe transmitted signal isatthecarrier frequency. Highreceiver selectivity isverydesirable foreliminating noiseinterference. Itisnotdifficult toseethatthefewerthenoisefrequencies allowed intobeamplified alongwiththeuseful signal, the morethesignalwillstandout.Everyone hashadexperience withtheinterference ofonecommercial radiobroadcast with another atsomenearfrequency. Acomplete lackofselectivity wouldallowallthestations upanddownthedialtobereceived atthesametime. A.M.isnotsuitedfordatatransmission ifitisusedin theusualmanner, thatis,wheretheamplitude ofthecarrier varies directly withtheinformation signal. Toomanythings, besides theinformation signal, caninfluence thereceived amplitude, asyouwellknow. Thesolution istomodulate the carrier amplitude withanother carrier frequency, whichmaybe !" 560< h 15-3 inthe audible range (up toabout 15Kcps.). The frequency of this subcarrier isthen varied according tothe magnitude of the measured quantity tobetransmitted. The actual amplitude ofthe received signal now has nosignificance whatever with regard tothe information content. Inthis manner the best characteristics ofboth F.M. and A.M. are combined. The power required ofthe space vehicle's transmitter will depend onthe sensitivity ofavailable earth-based receiving equipment, distance, atmospheric attenuation and noise levels. Receivers have been designed which are sensitive to10"18 watts power input. With this sensitivity, however, information could only besent atthe slow rate ofone "bit" per second. The major stumbling block toincreasing sensitivity appears tobe the noise generated inthe receiver circuitry. Ifyou turn up the volume onahigh gain radio receiver, you may hear the effects ofr_dom motions ofelectrons inthe vacuum tubes and even in the wires and other components ofthe set, These noises ar_ in large part duetothermal agitation ofmolecules and atoms. Some reduction inreceiver noise level can begained bycooling parts ofthe receiver tovery low temperatures. Noise which originates outside the receiver can bere- duced bythe same methods used byowners ofTVsets, though much refined. Maximum receiver selectivity restricts the noise toa narrow range offrequencies, while highly directional antennas will pick uponly the noise which comes from the same direction i," I-% _ asthe signal. The TVset owner increases selectivity by installing high pass, low pass orband-pass filters inthe antenna system, whereas high selectivity (usually variable) isbuilt into communications receivers. Directional TV antennas with remote controlled rotators are not uncommon. Many steerable parabolic reflectors oflarge diameter, 80to 250 ft., have been built for use inradio astronomy and have been brought into play for tracking and receiving telemetered data from the U.S. and Russian satellites. The reason for all the concern over receiver sensitivity, antenna gain and soonisthat transmitter power isexpensive when itmust becarried onasatellite orother space vehicle. Atthe present state ofthe craft, batteries "cost" about 2.5 pounds per watt-hr. The extra rocket fuel required toget one more watt-hr, into space istremendous. For this reason, U.S. satellite transmitters have allbeen less than .iwatt. (The Russians used a1watt transmitter.) WVEC-AH pours out 250 watts, WGH-AM uses 5,000, Just for comparison. Inorder toguide avehicle properly during the launching phase, some accurate system oftracking isrequired. This system will later beused toobserve the trajectory ofthe vehicle. Any deviation (orfor that matter, lack ofdeviation) from the predicted path isofgreat interest. Tracking systems are oftwo main types, the radar system which uses power transmitted from the ground and reflected back from the vehicle, and the directional receiver system which uses 15-5 the power tramsmitted from thevehicle. Mostofusare familiar with the principles ofradar. The most accurate ofthe other systems isknown asthe radio interferometer. Basically, it consists oftwo antennas located some distance apart and feeding into acommon receiver. Asignal source moving across the field ofthis antenna system will produce signals inthe two antennas which alternately cancel and reinforce. Asusual, between the basic system and something workable, there isawide discrepancy. With the aid ofafew blocks, Iwill try toexplain some ofthe details ofanactual system built byagroup ofradio amateurs out inCalifornia. This system isasimplified version ofthe Microlock tracking and communications system developed bythe Jet Propulsion Laboratory ofthe California Institute of Technology. Most ofthe quite complicated circuitry shown in figure Iisused tocorrect for frequency shifts inthe incoming sign_l due toDoppler effect. Forthis purpose, asingle ref- erence isused. This _f_n_ _nt_o _oo_......;_^ I_ ..................... _._v _o_* _u_- _'_:_iv the telemetered information. Beginningwith the reference antenna, the signal from the satellite ispicked upand sent toapre- amplifier, (I). The output ofthe preamp., still atthe satellite frequency of108 mc., isfed into amixer, (2), where itismixed with (added to) a127 mc. signal from avoltage controlled oscillator, block 6.The output from the mixer consists of the sum and difference ofthe 108 and 127 mc. signals. Block 3 isanordinary communications type receiver tuned toreceive only the difference frequency, 19mc. This receiver has itsown local 563< oscillator which istuned to19.455 mc., producing anoutput of455 KC. (19._55-19) when mixed with the 19mc. input. Now, ifthere isashift ininput frequency due tomotion ofthe satellite, the input toreceiver (3) will nolonger be19mc., but some other frequency. The output will also nolonger be455 KC. One method ofkeeping the output ofreceiver (3) ata constant 455 KCistovary the frequency ofthe oscillator, (6), the required amount tokeep aconstant 19mc. difference from the satellite signal. This can bedone automatically asfollows. The output ofreceiver (3) iscompared with the output ofa crystal controlled 455 KC. reference oscillator,(7), bythe phase detector inblock (h)- This phase detector isadevice which pro- duces avoltage proportional toeither the sine ofthe phase angle orthe frequency difference between two inputs. The in- stant theoutputofreceiver (3) begins todiffer from the 455 KC. reference oscillator, either infrequency orphase, acorrect- ing voltage from the phase detector issent through the filter (5) tothe voltage controlled oscillator (6) which changes fre- quency insuch amanner astobring the output ofreceiver(3) back toexactly 455 KC. The filter (5) limits the response of the correcting system sothat nochange infrequency faster than lOcycles per second will becorrected. Thus any telemetered data inthe form offrequency modulation faster than lOcps will not bedestroyed. Without filter (5), the system could not dis- tinguish between the relatively slow changes infrequency due to 5 4< 15-7 Doppler affect and the rapid changes infrequency due toF.M., and would cancel out both types offrequency change. T_e output ofreceiver (3) isalso fed toatelemeter detector and recorder. Thus far, the only accomplishment has been the establish- ment ofaconstant h55 KC. output frequency and the recording of the telemeter data. The interferometer channel isalso control- led byoscillator (6) .Aseparate receiver isused, block ll, and the local oscillator (19._55 mc.) ofthis receiver must be synchronized with that ofreceiver (3) toavoid undoing the work ofthe frequency correction network. (The local oscillator in receiver _3) may drift, but such changes infrequency would be corrected asifthey were changes insatellite frequency. Re- ceiver (11) isnot inthe feedback loop ofthe correction system, and drifting ofits oscillator would not becorrected. Therefore, receivers (ll) and (3)must, effectively, use the same local oscillator. This isindicated bythe dashed line Joining (3) and (ll) .) Signals from the satellite reach the receiver _ll) through mixers (9) and (10). Since the antennas Eand W(East and West) are separated bysome distance, the two signals fed Into(9) will differ inphase according tothe relative positions ofthe satel- lite and the two antennas. The output ofC9) can thus becon- sidered aphase modulated signal of108 mc. frequency. This is reduced to19inc. inmixer lO and again to_55 KC. inreceiver (ll), retaining the phase modulation. Phase detector (12) detects the difference inphase between the satellite signal and the $6S< 15-8 _55 KC. reference oscillator, converts itinto signal strength variations which are recorded onanink-recorder,(13_. The resultant record looks like this, asasatellite crosses the antenna field. T_me Byproper interpretation ofthe recorded time history, the angular position ofthe satellite with respect tothe antenna base line atany time can bedetermined very accurately. Time correlation ofall satellite signals isvery important, sothat records from several interferometer stations can beused topredict the path ofthe vehicle. Radio station WWV, operated bythe Bureau ofStandards onseveral frequencies, brcadcasts extremely accurate time signals which can easily be used for the necessary correlation. The frequency shift due toDoppler effect, which was so carefully avoided inmeasuring angular position and receiving telemetered data, isused inother apparatus tomeasure the range rate (radial velocity) ofthe space vehicle. The stability ofthe satellite transmitter issufficient toallow accurate measurements ofthis sort. For asatellite coming toward orgoing away from anobserved at18,000 mph., the Doppler shift isabout +-2KC. from 108 mc. The most sophisticated circuitry may detect ashift S 6< 15-9 ofafraction ofacycle per second. The actual range ofthe vehicle may beobtained from one station byradar orfrom two ormore correlated stations bytriangulation using interfero- meter data. Asthe range increases tointerplanetary distances, neither ofthese methods will besufficient. Asolution might betoestablish abase onthe Moon for triangulation purposes, with the other base onearth. Arecent discovery byDr. John D.Kraus ofOhio State U., resulting from some efforts toreceive the radio signals from Sputnik I,may beofaid indetecting ICBM'B approximately one minute earlier than canbeaccomplished byordinary radar methods. The discovery came about inthis manner: itwas noticed atareceiving station incentral Ohio, that the 20mc. signal from Sputnik Iwas not received clearly until the satel- lite had passed over. Furthermore, the 20mc. signal from station WWV, which isnot normally heard atnight incentral Ohio, was received inthi_ A_ _ _° +_ "_^_*_- pas_ed over, and reception began before the satellite itself came up over the radio horizon. The indications were that acloud of ionized particles was being pushed ahead ofthe satellite. This cloud reflected the satellite transmitter's signal away from the receiving station while itreflected the WWV signal toward the receiver. The short wavelength signals from radar equipment pass through such acloud, and thus cannot detect it. ,567< 1540 Since anICBM would betravelling through the ionosphere atapproximately satellite speeds, itisexpected that alow frequency radar-type installation could beused todetect the concentration ofionized particles ahead ofthemissile. This cloud, according tocalculations, canbeasmuch as300 to_00 miles inlength, giving anextra minuteofwarning. When it isconsidered that radar would only give atmost nine minutes warning, the extra minute canbeseen tobevery important. This phenomenon had been observed earlier asmeteors entered the ionosphere, but atthat time itwas attributed to the meteor trail. Now, let usexamine the data telemetering system ofa U.S. satellite. The quantity tobemeasured acts onatrans- ducing element tovary one ofits electrical properties. For instance, changes intemperature might act onadevice called a thermistor tovary its electrical resistance. Micrometeorltes might erode away parts ofathin metal film, increasing the edge- to-edge resistance ofthe film. These are techniques with which most ofusare familiar. Inthe Vanguard satellite which didn't quite make itonApril 28, 12quantities were tohave been measur- ed. Totransmit all this data atonce would require alarge bandwidth oftransmitted frequencies. Since most ofthe quantities are only slowly changing, apulse type oftransmission was tohave been used. Inthis system, each pulse consists ofasine wave superposed onthe 108 mc. carrier frequency bycarrier amplitude modulation. The frequency ofthe impressed sine wave isproportion- altoone ofthemeasured quantities, the duration ofthe pulse 568< i5-ii (uptoabout 30milliseconds) isproportional toasecond quantity, and the interval between the first pulse and the second represents athird quantity. The frequency ofthe second pulse can represent afourth quantity, and soonuntil each ofthemeasured quantities has been transmitted asa frequency, pulse duration orinterval together with any desired calibration pulses (similar inprinciple tothe calibrate de- flections and galvanometer zeros used onoscillograph records). Atthe end ofeach sequence ofpulses, orframe, asitiscalled, apulse orinterval ofrecognizable duration will appear asa marker, and another cycle ofpulses begins. Atthe earth station, these pulses can berecorded onmagnetic tape together with a time signal from WWV. The transmission rate isabout 3to4 frames per second. Probably the most complicated piece ofequipment inthe satellite isthe encoder, whlcn determines whether each quantity istoberepresented byapulse frequency, pulse duration or interval between pulses, and the sequency inwhich the data is tobetransmitted. Italso must determine the frequency and duration ofeach pulse, and the duration ofeach interval be- tween pulses inaccordance with pre-set calibrations. This re- markable gadget iscontained inadisc 3/5" tIL%ck, 5-1/2" in diam. weighing 3-1/2 oz. and consuming 8toi0milllwatts of power. There are nomoving parts. 569< 15-z Transistor a_nplifiers are used inplace ofvaccuum tubes inthe U.S. satellites. Signals can only bereceived from the satellite when it isinthe line ofsight. Inview ofthis fact, some advantage istobegained bystoring the recorded data for readout only once each orbit asthe satellite passes near one oftile receiving t_is stations. Atape recorder has been incorporated for^purpose in one ofthe'_Explorers"whlch isnow inorbit. The data are record- edatarather slow tape speed. Onreceiving acode signal from one ofthe earth stations, the tape isplayed back athigh speed and the data transmitted toearth. Atthe same time, the tape iserased for use onthe next orbit. The power requirements for satellite transmitters are very small. 100 milllwatts can give avery good signal from adistance ofseveral thousand miles. The power required varies directly wlth the square ofthe distance tobecovered. If l0row., for instance, are the minimum requirement for adistance of2.500 miles, then 100. watts would give the same signal atthe distance ofthe moon, 2%0,000 miles. However, present satellites are radiating power inall directions. Ifsome method can be found for orienting anantenna inthe direction ofthe earth at all times, the radio power can beconcentrated into asmall cone. Acone ofabout 2°apex angle would permit the l0milllwatt trans- mitter tocover the distance between the Earth and Moon. However, even with the reflecting antenna, the power required still In- creases with the square ofthe distance. When the vehicle reaches 570< 15-13 25,000,000 miles (not quite out tothe nearest point onthe orbit ofMars) the power required isagain lO0 watts with the 2_beam. The extent towhich the beam may benarrowed depends upon the wave length ofthe radio waves and the diameter ofthe reflector. Out ininterplanetary space, avery large reflector could becarried because its strength requirements would be small. For instance, one ofthe large plastic balloons could beused asareflector byaluminizing only half ofit. Such antennas have actually been used pnearth inradar installations, where they could beshielded from wind forces. Iftrips tothe outer edges ofthe solar system are to besuccessful, itwill probably benecessary toestablish some method ofrelaying messages between the earth and space vehicles. Itmay bepossible toestablish automatic repeater stations on each planet visited, gradually working outward. These stations might beatomic powered, orthey might bepowered bynatural movements ofthe planet's atmosphere orground fluids. In addition tofurnishing arelay station for messages, each would furnish abeacon, identified bysome code, for navigational purposes. Another fact which may have tobetaken into account is that there will beno"urgent" messages asweknow them. The radio operator inthe neighborhood ofPluto who says, "Rush this message through toEarth" must reconcile himself tothe fact that hewill have towait atleast l0hours for areply. The velocity ofpropagation ofelectromagnetic waves isone thing wehaven' t been able tospeed upyet. ....571< 15-1n Many phases ofcommunications theory and practice could not betouched upon inthis presentation. Itishoped that these objectives have been obtained: toshow how, inageneral way, itispossible toget back enough information toJustify a space vehicle; toshow how the path ofthis vehicle may be followed from the ground, and tostate some ofthe obstacles tocommunications over very long distances. References : QST (magazine) Dec. 1957 Radio and TVNews March 19_8 and May 19%8 Science andMechanics May 1958 Missiles and Rockets February 19%8 Westinghouse Engineer May 1958 SV2 e 15-15 aty getzsFa §_& 8Ets, 8H8 ss gige 2“ow>-F@S < red <Zé gee as)#8 | ese 3ve 2 ! % zgli £Z| = Boe:5 se | g s e 1gy& 21| 2 o| cs} De}|| : i a) \ oy =333!ifBs 5iit lio | # 5>3 HOOTRo 3 =z6:$ 3Z 2dvd = NS e va 573< 16.1 16.1116-1 SECTION XVI SOME DYNAMICAL ASPECTS OFTHE SPECIAL AND GENERAL THEORIES OFRELATIVITY Special Theory ofRelativity Evolution ofSpecial Theory ofRelativity Towards the close ofthe last century Michelson and Morley performed anexperiment. Atthe outset itappeared Just another routine experiment. Noone expected itto turn out, asindeed itdid, tobeone ofthe most signifi- cant experiments inthewhole ofscientific history. The experiment had asits objective the determination ofthe velocity ofthe earth' sdrift through the all prevading ether. Atthat time itwas generally supposed that light and electromagnetic phenomena ingeneral were propagated through the ether with constant velocity. Ifalight Wave were tomeet the earth head onasthe earth drifted through the ether, the velocity ofthe light relative tothe earth would besomewhat augmented. Onthe other hand, ifthe light signal were toovertake the earth the velocity would be correspondingly reduced. The apparatus (figure 16-1) con- slated inprinciple ofalight source Slocated atthe intersection oftwo mutually perpendicular arms SP and SQ, both ofequal length, atthe extremities ofwhich were mounted two mlrrors. ,574< 16-2 Mirrors (Light source) S Figure 16-1 Bycomparing the times taken for alight pulse initiated at Stotraverse the respective arms andback again the velocity ofthe earth's drift could becalculated. The apparatus was sufficiently sensitive todetect avelocity ofdrift of1mile per sec. Asissofrequently the case, itwas the unexpected that happened; they got anull result. Ofcourse, there was apossibility that the earth was drift- ing inadirection bisecting the angle PSQ. This was elim- inated byrotating their apparatus through _5°and repeating the experiment and again getting anull result. There re- mained afurther possibility, anunlikely one albeit, that the velocity ofthe earth relstive tothe ether was zero, (oratany rate less than 1mile per sec). This possibility was eliminated when they repeated their experiment after an elapse ofsix months (the earth's velocity having inthe meantime changed byabout 38miles per sec.) and again obtained anull result. Something was clearly wrong with the current concept of propagation oflight through space. Nor would ithave helped torevert tothe corpuscular theory for this demanded that the velocity oflight bedependant onthe velocity ofthe SVS< 16-3 emitting source -aresult known tobeatvariance with experimental results. Einstein was the first toovercome this impasse, only, however, byabandoning the intuitive concept ofspace and time. 16.12 Kinemetics ofSpecial Theory ofRelativity Einstein took the experimental result atits face value viz the velocity oflight isconstant relative toall / observers. Consider two observers 0and O' figure 16-2 z°I I I I/Y' / I/ I/ X I/ X' O' )V(Velocity of0'Relative to0) Figure 16-2 and let ussuppose alight pulse isinitiated at I)and let itbesubsequently received atB Let observer 0 coordinates :assign x y z t x+dx y+dy z+dz t+dtA(event (event II). (event I) (event II) Similarly let observer 0' assign coordinates: x' y' x'+dx' y'+dy'z' t' (event I) z'+dz' t'+dt' (event II) 576< 16-4 SinSe byhypothesis the velocity oflight isthe same for both observers (say equal toc) •then 0 =dx2 + dy2 + and0 =dx'2 +dy'2 +dz2 .c2 dz'2 .c2dt2 dt'2 i.eo t dx2+dy2+dz2-c2dt2=dx'2÷dy'2+dz'22-cdt'2 (16.12-i) This differential relationship isassumed tohold for all contiguous pairs ofevents. Clearly itimplies some form offunctional relationship between x' y' z' t' onthe one hand and xyztonthe other. tegrated form ofequation (16.12-1). cussion let usdisregard the yandLet usseek the in- Tosimplify the dis- zcoordinates. dx2+(icdt) 2=dx,2+(icdt,) 2 Compare this with dx2÷dy2=dx,2+dy,2 This latter defines simply arotation of xy some angle @asdepicted infigure 16-3.(16.12-2) axes through Y \ \ \ \x' s s . _,I X Figure 16-3 577< 16-5 x'=xcos @+ysin @ y'=-xsin @+ycos @ Clearly then the integration of(16.12-2) leads to x'=xcos @+ict sin @ ict' =-xsin @+ict cos @ (16.12-3) or Since purely imaginary, say cos(ict' =ixsin @+ctcos @ (where @isthe constant ofintegration) x,tand x',t' are all real this implies isin (ii@ )=cosh a )=sinh¢@is Thus the equationS (16.12-3) take the form x'=xcosh _-ctsinh G ct' =-xsinh a+ctcosh a ccan beexpressed interms of V. ofO'(x w=O) relative to 0isV. 0=Xcosh¢X -ctsinh _;cc-r-"V C Hence 1V C(16.1a-_) Thus since velocity tanhu, Substituting in(16.12-4) weobtain finally $7S< 16-6 x-Vt _-c2 t,-C2 (16.12-5) These are theLorentz Transformations. 16.121 Dilatation ofTime This ismost readily obtained from the differential expression _c2ds2mdx2+dy2+dz2_c2dt 2 (dsdefines the absolute interval between two events: -The spacial and temporal separations ofthe events are simply components which vary from observer toobserver). Let ussuppose 0observes aparticle Pinmotion and let usfurther suppose aclock iscarried on P. (Track ofPin x_t frame ofO) X FigUre16-4 Assume light blips are initiated at Aand B. $7S (These 16-? events are designated Iand II respectively.) 0assigns coordinate differences dx dy dz tothe interval separating these two events. clock carried by P ofseparation ds.dt Howeverp the Nowwill actually mark off the true interval 2 =dx2dy2 -cds2 + +dz2-c2dt2 Time between blips as registered bymoving clock Time between blips as registered byobservers clock i.e., clock inmotion appears toberunning slow bya factor i_-_(16.12-6) 16.122 Lorentz-Fitzgerald Contractiono Toobtain this wemust appeal tothe integrated forms ofthe equationsZ - I I I I I I (I)v/ / ;(2)/ / Figure 16-5 Assume O' iscarrying ameasuring scale which extends from his origin xI'=0to x2'=i. Omeasures the length of Ibydetermining the coordinates ofits •tI extremities atthe same instant^_ccordlng tohis own clock). SSO< ! xI=0= ! x2=i= 816-8 xI-VtI x2-VtI x2-xI (16.122-1) Thus the measuring rule inmotion appears tobereduced in length byafactor _I-_ (This is thec2 socalled Lorentz-Fitzgerald Contraction). Thus ineffect what Einstein has accomplished inhis special theory ofrelativity isthe welding together of space and time. Weare not saying that time isofthe same intrinsic nature asspace. Quite the contrary. The mathematican recognizes the difference byattaching the identifying label itothe time coordinate. The physicist recognizes the distinction byusing clocks tomeasure time like separations andmeasuring rules tomeasure space like separations. We ourselves intuitively recognize adifference between them. Ofthe two time isamore mysterious entity than space, It israther interesting tospeculate onthe reason for this. Weasindividuals are somewhat ofthe nature offour dimen- sional worms -relatively extended inthe time dimension. Thus weare much more intimately associated with time than SSi< 16.1316-9 with space and this may conceivably explain why weregard itassomysterious.* Dynamics ofthe Special Theory ofRelativity Itisclearly impossible todetect uniform motion. This implies that the laws ofmotion must have the same form relative toall inertial observers,c-_ i.e., the equations ofmotion ifcorrectly formulated must beinvariant with re- spect tothe Lorentz transformation. Toput itanother way, the terms appearing inthe equation must transform according tothe same rules. (They must bevectors ortensors inthe revised 4dimensional sense.) Clearly wewill beassisted inthis reformulation bythe fact that our modified equations must tend tothe Newtonian form as V ,0. Look atthe equations ofmotion asformulated byNewton. d__(mvi) =X.dt --1 where (i=1,2,3) dxi Vi=--dt (16.13-1) *Bythe same token wefind material thin_3s much less mysterious than our own psyche. **Newton concurred inthis and his equations ofmotion are indeed invariant with respect toGalilean transformation. His error lay inassuming that the Galilean transformion accurately described the transformation from one inertial frame toanother. • 5S < 16-i0 Consider the velocity vector vIv2v3.This isnot avector inthe extended 4dimensional sense. Let usfor amoment look atthe situation ofaparticle moving with relatively low velocity relative toour own x_t reference frame. dts X Figure 16-6 Its world track will departonly slightly from the time axis. Consider asegmentofits track ds. Atlow velocit- ies itwouldhardly besurprising ifds were confused with dt since the two are sonearly equal. dxI dxIdxIdtVl Inplace of vI=-- introduceUl=-= _i dt ds dtds -_2 dx2 dx2dx2dt v2 v2-dt u2 ds dtUs _l- _2 v3 dt u= dt. dsdtds _i_2 dxhdx4dt ic u4 dsdtds $83< 16-ii (N.B. uIu2u3u4isaproper vector inour 4space. Moreover its spatial components uIu2u3tend respective- ly vIv2v3asvelocity tends tozero thus itsatisfies all the requirements imposed inthis reformulation). Bythe same token inthe left hand side of(16.13-1) wereplace differentiation with respect to tbydifferen- tiation with respect to s. Onmaking these substitutions wefind the left hand side Isnow truly invariant with re- spect toLorentz transformations. Wemust similarly doctor the right hand side. Here weneed toreplace the force vector XIX2X3byanappropriate 4vector. Electro- magnetic theory tells ushow tomake this generalization. XIX2X3are replaced by KIK2K3K4 XI X2 X} iF"V where KI= ;K2- ;K3= ;K4_ (KIK2K3K4isaproper vector andmoreover its three spatial components tend to X1X2X3atlow velocities.) Thus inplace ofNewton's equations wehave d__(mui) =_ds(i=1,2, 3,h) Writing thisout interms of xyz coordinates(16.13-2) 16-12 x= -- or m --.=X ,(..,°_ iY dsc,,,_=.iY 2 (16.13-3) The fourth equation deserves special note. d -F.V dt However, mc2=mc2+_1mv2+.... Atallbut the highestvelocities itsuffices toretain only two terms ofthe above expansion" Note the similarity with the classical equation -.% d__(K.E.) =F•V dt Consider thecollision between two lead masses (assuming no rebound). 585< 16-13 _'_mc2+ =EF"V=0 imv2__constant Atimpact theK.E. isentirely destroyed since there is assumed tobenorebound. Since citself isconstant the disappearance ofacertain amount ofenergy isassociat- edwith anincrease inmass. AM=AE-_ (16.13-[_) C Inplace ofthe two classical laws IConservation ofMass II Conservation ofEnergy wenow have only single law ofconservation ofmass +energy, The lawsofmotion (16.13-3) between them express the con- servation ofmomentum and conservation ofmass and energy, Weshallfind how these results are further generalized interms ofgeneral theory ofrelativity, f 16.2 16.2116-i£ GeneralTheory ofRelativity Evolution ofGeneral TheoryofRelativity. Newtonhad recognized the impossibility ofdevising anexperiment todetect uniform motion (which initself implies the invarience ofform ofthe equations ofmotion relative toall sets ofinertial axes see footnote page 16-9) Onthe otherhand,however,hebelieved itpossible todetect absolute rotation (his criterion: simply hold abucket full ofwater inyourhand -ifthe surface ofthe water iscurved you are rotating, ifitisflat you are atrest).* Bishop Berkeley held acontraryviewpoint onphilosphic grounds. A hundred years later Ernest Mach discerned anillogicality in the concept ofabsolute rest inarotating sense. Mach argued thusly; consider apail rotatinguniformily inauniverse otherwise atrest. Clearlythere isnothing todistinguish this situation from one inwhich the pail isatrest and the remaining masses oftheuniverse rotating about it.Thus if bysome means the masses oftheuniverse could beset into a rotation about apail ofwater atrest inthis instance also the surface ofthe waterwould becurved.Thus the curvature ofthe surface i.e., centrigual and centripetal forces etc., *The possibility ofdetecting acondition ofabsolute rest implies that inarotating frame indeed inaccelerated frames ingeneral the laws ofmotion are not invariant inform - this isreflected bythe presence ofadditional terms i.e., Centrifugal Force Terms, etc. $87< 16-15 are inthe last resort attributable tothe influence of the remaining masses ofthe universe. They aregravitational inorigin. Mach's hypothesis, ifaccepted, explains the pro- portionality ofinertial mass and gravitational mass, a circumstance which Newton and other scientists who followed him had been ataloss toexplain. Inertial mass entered the scheme ofClassical Mechanics via Newton's law ofmotion, gravitational mass onthe other hand entered via Newton's law ofgravitation. Now inview ofthere being nodiscernable tie upbetween these two laws the strict proportionality between the twokinds ofmasses was somewhat perplexing. Bessel and Eotvos had carried out delicate experiments with the aim of detecting slight differences between them ifthey existed. Their experiments consisted ofmeasuring the direction ofpull onmasses ofvarious sizes suspended atthe earth's surface. Two forces inthis instance are operative, acentrifugal force due toearth's rotation (proportional toinertial mass)and gravitation (proportional togravitational mass). Ifthe inertial and gravitational masses were notstrictly proportional this would show upinachange ofdirection ofthe resultant pull asthe mass changed. Nosuch effect was observed. Follow- ing Mach, Friedman attemped todetect "centrifugal" force on aparticle atrest near the center ofarotating spherical mass. This experiment too yielded anull result. Weknow now that the effect for which hewas looking was much too small tobe 16-16 detected. Einstein beingastudent ofMachs was exposed toand acceptedhis radical viewpoint. ItistoEinstein that weowe the rather convincing example ofthe "man inthe l.ift", anexample which further supported the essential equivalence ofinertial and gravitational mass. Einstein argued thusly -imagine alift being acclerated upwards at 32ft/sec. 2inagravity free field. All experiments per- formed inthis lift will yield the same result aswould be obtained ifthe experiment were performed inalaboratory on the surface ofthe earth. Thus projected particles will in both cases describe parabolic paths. Note,however, that the man inthe lift interprets mass asinertial mass whereas the man inthe earth bound laboratory interprets mass asgravi- tational mass. Having convincedhimself ofthe equivalence ofinertial and gravitational mass itremained forhim toexplain the latter. The following story isattributed toEddington. The gist ofitisthis: Uptothe time ofthe middle ages it was popularly believed that the earth was flat. Letus suppose this beliefhad persisted down tothe present day. Amercators chart would thushave been regarded asgiving an accurate description ofthe disposition ofplaces onthe earth's surface.Ifour mercators chart were centered onthe USA, the chart would serve adequately solong asour Journeys were SSS< 16-17 limited tothe confines ofthe USA. Consider, however, the case oftravelers toGreenland. They would find that they apparently covered immense distances with relatively little expenditure ofeffort. Scientists would doubltless explain the phenomenon byinventing ademon who helped travellers to Greenland ontheir Journeys. Being scientists they wouldn't choose touse the word "Demon" but would probably use aword ofGreco-Latin derivation such as"gravity". Consider now our scientific colleagues inGreenland. They would use a mercators chart centered intheir own country and they would see the demon operative inthe USA. The demon isnever where weare always where the other fellow is. Ifyou'll reflect amoment you'll see the situation isquite analogous tothe situation with respect togravity. Just asthe demon owes his existence tothe circumstance that weare_trying toforce aspherical surface into aflat surface sogravitational forces result from our regarding what isinreality atwisted space time continuum asbeing flat; (P_rtic!es instead ofdescrib= ing curved paths inflat space actually follow straight or geodesic paths incurved space). Einstein having come this far the rest was relatively straight forwardo Inreformulating his law ofgravitation Einstein instead ofsaying something about aforce had now tosay something about curvature. Fortunately there are relatively few things wecan say about curvature. Thus inthe case ofatwisted two dimensional surface in3-space there isonly asingle SSO< 16-18 intrinsic measure ofcurvature- the socalled Gaussian or spherical curvature.* Passing from two dimensions tofour dimensions wefind the latter bycomparison relatively ingen- ious indevising new contortions -this isexemplified bythe fact that ittakes aEuclidian space ofI0dimensions to accommodate atwisted four dimensional domain. Inplace of asingle intrinsic measure ofcurvature a4-dimensional domain has twenty -ten ofwhich are pincipal and ten ofwhich are secondary.** What was Einstein tosay about these curvatures? Ifhe'd set them all zero he'd have been back where hestarted with flat space time. Ifhe'd left them all entirely arbi- trary this would have implied absence ofany constraint or law. Hechose amiddle ofthe road course and set the ten principal measures ofcurvature equal tozero. -X XXX G_v= X XX =0 XX X (G_v isasymmetric tensor and has ten components as indicated) This still left himwith ten degrees offreedom which was sufficiently flexible toenable him toadopt his law toany *There are, ofcourse,other measures ofcurvature, however, these are only meaningful tobeings external tothe sur- face. Wemust clearly limit ourselves tomeasures of intrinsic curvature. **Tounderstand the rather subtle distinction between principal and secondary curvatures one has todelve rather deeply into Riemannian Geometry -See Eisenhart: Riemannian Geometry. Sgi'- 16.2216-19 particula_ physical situation, Kinematics ofGeneral Theory ofRelativity What ismeant bythe curvature ofspace time. Clearly there isnopoint tobeserved byattempting tolook atthe thing from aconceptual standpoint. The best wecan dois toformulate the mathematical equations and draw our impli- cations from them. Itisarelatively easy matter toeval- uate the gravitational field associated with asingle mass particle. This istantamount tosolving the gravitational equation G_v =0imposing the condition ofspherical symmetry. Infigure 16-7 Misanisolated mass the presence of which results inadeformation ofspace time. "A" isan ! observer atadistance ro (asmeasured byits own measuring rods). ! ro M A B Figure 16-7 Relative totheobserver Athe interval separating two @ contiguous events isgiven bythe expressionS- *The unitshavebeen selected tomake the velocityof light infree space unity. $92< 16-20 ds2=-.+_'r--° r2_r' dr'2_r'2de'2-r'2sin2e'd_'2 \ r° /i 2m' 2m'_ ,2 +_+ro' r'jdt (16.22-l) m' isameasure ofmass masdetermined at"A". r' isthe distance tothe two adjacent events. dr' d@' d_' dt' are the coordinate differences relative to A, separating the two events. Inthe special case ofanobserver atinfinity (de- signated becomes dr2 ds2_-_ (I-2-m) Z'Binfigure 16-7) the expression for interval -r2de2-r2sin2ed_22m÷(i-7)dr2 (16.22-2) Let ussuppose theobservers Aand B same pair ofevents. The interval ds same for all observers.** Hencebothobserve the separating isthe *Note mass mappearing in(16.22-2) istobedistinguished from the mass m' appearing in(16.22-I) since itistobe expected that different observers will assign different measures tothe mass ofthe gravitating body. **ds provides anabsolute measure ofseparation (See Special Theory). 5_q< 16-21 f_÷2m, / i1" 22 2ds2 -1"-2_, "2m'i '2d8' -r'sin28 - dt' \ +ro ro' dr2-r2de 2-r2sin2ed_ 2+(I- r2-_m)dt2 16.221(16.22-3 ) This differential relationship isstrictly analogous tothe relationship c2ds2=_dx2_dy2_dz2+c2dt2=_dx,2_dy!2_dz,2+c2dt,2 which appeared inthe Special TheoryofRelativity. Inthe Special Theory wewere able tointegrate the expression quite simply. With regard to(16.22-3) noone tomyknow- ledge has been clever enough toexpress the relationship in integrated form. Wedonot therefore have inthe caseofthe gravitational field the equivalentofthe Lorenz equations ofSpecial Theory. Fortunately weare able todeduce quite alot from the relationship initsdifferential form. Variation ofthe VelocityofLight The velocity oflight isobtained bysetting Thus inSpecial Theory dx2.dX2+dz2 v2=c20=1- or c2dt2 i.e., the velocity oflight isuniformly equal to c. Wederive anexpression for the velocity oflight in the gravitational case byagain setting ds=O. S 4<ds=Oo 16-22 Thus for observer A I_+2m' 2m,l2m' 1 /+r'2sin2e,_ ,2=i+2m'I ro2m' rI N.B. The velocityoflight varies with direction. inthe case oftransverse propagation (@' varying and _'constant) I 2m'2m'_1/2c=1+_-_- r,/0 whereas inthe radial direction 2m' 2m'_I+-- r9'-r'/ c-_2_,\z12l+r' I Restricting our attention toradial propagation asThus rt rI becomes smaller i.e., the more closely weapproach the gravitating mass the more sluggish does light become. On theotherhand as rI-----+co 12m--_'_z/2c---_÷r'/>I Bearing inmind that interms ofpresent units the velocity oflight infree space isunity. Thus for the observer A the velocity oflight canbeeither less than orgreater than its velocity ingravity free space. For the observer atinfinity (16.221-I) This equation atfirst sighthas •rather startling impli cations. i.e., ifwemake mlarge enough and r 16-23 smallenough ccanbemadezero orindeednegative. This clearly demands further scrutiny. Letustobegin byseeking the radius ofamass (having uniform density equal tothat water) such that atits boundary and hence c=0. Let the radius ofthe mass be Rxl06Kms Mass ofsphere = Mass ofthe sun = m= Therefore_(RlO6)3x(105)3 X gms.3 __(0.695 x106)3x(105) 3x1.41 gms. 4R3x x x1.5" l018 io15 kms.2m --= 1 r s0.6953 x10°8x1015 x1.413 2m_ 3R3 R Rx106x0.6953 x1.41 =1 R20.69_ x1.4_= x1063 R=395. i.e., the radius ofthe mass inquestion =395,000,000 kms. Ifwedelve mnre d_.A_Iv I_th_ _=I +_a_ _ _ thatthis mass results inaclosing inofspace upon itself, Thus the presence ofmass induces acurvature ofspace time. Adding more and more mass producesever greater distortion and ifweadd enough itwill cause space tocurve back on itself and become closed. Ifwedirect alight signal to- The mappearing inomr equations for interval though proportional tomass isnot measured in gms. Asa result ofour choice ofunits ithas dimensions oflength. Expressed inappropriate units the sun's mass is1.5 kms. ....- 596< 16.22216- 24 wards sueh amass asitapproaches its boundary the light signal will slow down tozero and will never actually attain it. Amaterial particle ifprojected towards itcan therefore never reach it. Thus there isnopossibility ofour adding more water from the outside. Can water beadded from the in- side? Ifwelook atthe relevant equations wefind the velocity oflight iszero atall interior points and that clocks stand still. There canbenoactivity whatever, there- fore, onthe inside. Itisimpossible tosend asignal from aninterior point toanexterior point and vice versa. If such masses exist there canbenomeans ofour knowing it. Let usreturn torather more mundane matters. Time Dilatation inGravitational Field. IntheSpecial Theory wewere able toevaluate time dilitation effects onthe basis ofthe differential expression and wecan dolikewise here. Consider inthe first place a clock atrest and let uscompare its timekeeping with our reference clock atinfinity. Since the clock isassumed tobeatrest dr=d8=d_=0 ds2=(I-p)dt2 Reference clock(Distance asmeasured by observer atinfinity_ Q• --4J. (_ Clock under observot|on dse The clock underobservation marksoff true interval SS7< 16-25 dsIclock under observation) =i-2m dt (clock atinfinity) r (16.222-1) Toour observer atinfinity the clock under observation 2m appears toberunning slow byafactor 1----.i.e.r • themore closely the clock approaches thegravitating mass the more sluggish itappears tobecome. Table Itabulates the degree towhich clocks locatedatthe surface ofvarious gravitating bodies are sloweddown relative toclock at infinity. (Effects ofrotation ofbodies are ignored.) Sun Typical red giant (OScorpii A).TABLE I Extent towhich clocks run slow referenced to m(kms) Radius (kms) clock atinfinity 1.5 2parts in106 4_5.0 1/8 parts in106 Typical whit_ d_.e 1I,I, (QCan.MaJ. B).695,000 480x 695,000 _I,vev_ X 695,000>.q parts inIU# (1/2 hr. inayear approximate ) Wenote from this table that the most dense agglomerations of matter weknow ofi.e., the white dwarfs, result inonly aslight wrinkling ofspace time. Turn next tothe slowing down ofclocks carried bybodies moving ingravitational fields. $9S< 16- 26 dr2 r2 r2 2m ds2=-_- d82- sin28d_ 2+(I- _)___oat2 r (i2m( ) mNow P.E. =-- r(P.E. isreferenced toinfinity) (16.222-2) 22 2 •2r2r2;2÷r sin 8_ (16.222-31 K.E. =r÷ + Inthe unit system wehave adopted the expressions ofP.E. 2andK.E. are their values incustomary units divided by c. Both are therefore small quantities and canbetreated as such. Toafirst order therefore: ds_2=l*2(PE) -2(K.E.)dt/ ds(clock inmotion) dt(reference clock at infinity )=1+(P.E.)-(K.E.) (16.222-/$) Thus because oftheir smallness the effectsofgravitating field and themotion ofthe body inquestion can besuper- imposed. Apply (16.222-4) toaclock carried bythe earth (effects ofearth's rotation about its axis are quite negligible). The influence ofthe gravitational field results inaslowing down of1part in108. Inaddition since the earth is following toallintents and purposes acircular path the K.E. ofits motion isequal toabout 1/2 (P.E.) and hence, the motion results inafurther slowing down ofabout 599< 16-27 1/2 part in108.Hence aggregate slowing down amounts to about 1-1/2 parts in108.Inthe case ofNeptune the aggre- gate slowing down amounts toabout 1/3 part in109. Consider next the case ofspaceship following the elliptic orbits designated Iand IIinfigure 16-8. --Neptune orbit Figure 16-8 Orbit I. The spaceship approaches close tothe Sun and bycomparison with earth's clocks the clocks carried bythe spaceship will berunning slow. Inthis instance the space traveller will age somewhat less than his earthbound counter- part. Orbit II. Asignificant portion ofthe time isspent inrelative close proximity toNeptune orbit where clocks run fast relative tothe earth. Onsuch avoyage thisspace •,• 600< 16-28 traveller will besomewhatolderonreturn than his earth- bound counterpart. Bear inmind, however, that the extent ofrelative aging issoslight inall interplanetary flights astobe utterly insignificant from the point ofview ofchanging man' slifespan.The only way inwhichhis lifespan could bematerially extended isbyprovidinghim with ameans of attaining velocities comparative inmagnitude with speed oflight. Afurther example ofsomewhat current interest isthe possibility ofverifying the general theory bycomparing the timekeeping ofaclock carried byasatellite with an identical clock atthe earth's surface. For aconsideration ofthis problem itsuffices todis- regard all masses save that ofthe earth. Let our observer atinfinity make the comparison for us. The clock onthe earth's surface isslowed down asthe result ofitbeing inthe earth's gravitational field (the additional slowing resulting from the earth's rotation is quite negligible). Satellite orbit Figure 16-9 6G1< 16.2316-29 The clockofthe satellite isslowed down: (a) Asaresultofthe earth's gravitational field. (b) Asaresult ofits motion inits orbit. Ifwesuppose the satellite tobeinarelatively tight orbit around theearth thegravitional affects are about the same onboth clocks. _nd the difference intheir re- spective rates isattributable almost wholly tothe motion ofthe satellite inits orbito Thus experimental comparison oftheir rates oftimekeeping _uld provide uswith atest ofthe Special Theory ofRelativity rather than the General Theory. Atest ofGeneral Theory could bemade byplacing the second clock onthe moon' ssurface for then gravitational effects are significantly different. Even here, however, if aquantitive check onGeneral Theory were tobemade an accuracy intime measurement ofthe order ofIpart inI0I0 would bedemanded. Somuch for the kinematics ofGeneral Theory. Let usproceed now tothe discussion ofRelativistic Dynamic s• Dynamics ofGeneral Theory ofRelativity Aswehave had occassion tomention previously there isnomeans ofour detecting absolute rotation orindeed acceleration inany shape orform. Clearly then this implies that our equations ofmotion must assume identically the same form inall frames ofreference. For otherwise ifthe form ofthe equation changed aswepassed from one accelerated frame toanother this fact could initself beutilized to 802 16-30 distinguish one accelerated frame from another and hence, define zero rotation etc.* All frames ofreference are then equivalent and the equations ofmotion ifcorrectly formulated must therefore beinvariant with respect toquite arbitrary transformations ofspace time. This invariance will beachieved ifwecan express our equations intensorial form (i.e., all terms having the same tensorial characteristics and hence, subject tothe same transformation rules). InGalilean frame ofreference the classical equations ofmotion assume the form:- 8_2p+8(pvi) =0 (equation of 8t 8xi continuity) (16.23-1 ) Dv. Fe ÷ Fi (equation ofPDt momentum) (16.23 -2) (external (internal forces) forces) With reference toexternal forces (i.e., gravity and inertial forces) these have been shown tobefictitious (they result from our mistakenly regarding what isinreality atwisted space time continuum asbeing flat) and can therefore, bedis- *Inthe classical set-up the eouations ofmotion did assume characteristic form when referred torotating axes, i.e., Centrifugal and Centripetal forces appeared. Byensuring that these inertial force terms vanished wewere able to define acondition ofabsolute rest inarotational sense. 6 3< 16-31 regarded. Wecannot soeasily dismiss the forces since these are atomic inorigin. Wecan rewrite (16.23-2) intheform 8X _A=--AA (j-l, 2,3) p8vj+pvi 8xi8t BtFi (16.23-3) ormaking use of(16.23-i) a(_vj) a +- (PViV j-Xij) =0 8t 8xi(16.23-_) Following the example ofSpecial Theory, wereplace it by x4•Ifwethen define Tij=Tji=PViVj_Xij (i,j =l,2,3) Ti4=T4i=ipvi (16.23-5) Then the classical equations ofmotion (16.23-1) and (16.23-4) canbeexpressed inthe symmetric form: - (j=l,2,3,4)• Thus setting J=i,2,3. axl ax2 ax3 ax4 a_ a(ipvi) .dt__ 8xi(PViVj -Xij)+at dx4=0. ...... 604< 16-32 8 a (p_j)+_(pv_j-x_j).o Setting J= a(pvl) ÷ia ._ apdt=0 iaxI _(pv2)+iax3(pv3)- atdx4 ap÷a___ ataxi(PVi) =o. However, Tij isnot atensor since vivjare not even vectors inthe _dimensional sense nor isstress component Xij atensor ina4dimensional sense. Weproceed todoctor the equations inmuch thesame way aswedid inSpecial Theory. Thus inplace ofthe velocity components: dx1 Vl dt weintroducedx1 Ul=d--s-=Vldtds dx2 dx2 dt v2 dt u2 ds v2 v3 dt u3ds .-_A.i dt u4 ds ds d__.5_IN.B. ds atlow speed and the spatial components ofour hvector--mb uwill tend toequality with our classical velocity vector -_. Inaddition weseek anew stress tensorXiJ(i,J "1,2,3,4) such that atlow speeds the spatial components of Xlj..--_ xij (iJ=i,2,3). 6CS< 16-BS and atthe same time Having proceeded thus far redefineandX_-----._0 Tij byreplacinE vIv2v3byuIu2u3uhand Xij byXij Yielding Tij--Tji=Puiuj-Xij (i,J =1,2,3,4) (16.23-6) TiJ asdefined by(16.23-6) isatensor which approximates closely to(16.23-5) atlow speeds. Inour Galilean frame ofreference itseems reasonable therefore, toreplace the classical equations bythe equation 8Tij -- =O 8xi or ]_ivT=0 ij This equation being intensoral form itisimmediately applicable toall framesofreference (inwhich divergence isappropriately interpreted) Div TiJ =0orafter wecan write italter- natively as (16.23-7 ) Let uslook attheequation ofmotion alittle more closely mv(zj)_o (16.23-8) Whenever the divergence .ofanentityvanishes this implies per- manence ofthe entity inquestion, i.e., the entity inthis instance, the stress-energy tensor, isconserved. 606< 16- Thus bymaking our transition from classical mechanics tospecial relativity wereplaced the two laws ofconservation ofmass and conservation ofenergy bythe single law of conservation ofmass ÷energyo Wehave now generalized even further andshown that what isinreality conserved imthe stress energy tensor which inaddition toembracing mass and energy, has momentum and internal stress asadditional facets. Itisrather interesting toapply the equation ofmotion Inthis (16.23-8) tothe case ofthe single mass particle. instance the equation ofmotion assumes the form dxG-_ =0 d2 +(c_,_ _-sds(16.23-9) This isthe equation ofageodesic i.e., particles describe geodesic (or straight) paths incurved space time rather than curved paths inflat space time; this latter represent- ing the classical viewpoint. Wecan rewrite (16.23-9) intheform dsds Compare this with theequation ofmotion asderived onthe basis ofSpecial Theory ofRelativity d(mds-_) =KI_ Bearing inmind that the only forces wehave taken into consideration are the gravitational forces (and, ofcourse, 6C7< le-35 inertial forces) wenote that inthe formulation ofgeneral relativity inplace offorce wehave the expression _mdsds which defines atwisting ofour reference (4, _} isintimately tied upwith ourframe.* Since choice ofreference frame which canbechanged quite arbi- trarily itisfor this reason weregard forces and the associ- ated concept ofpotential energy asquite fictitious. Thus wecan deliberately choose atwisted reference frame (i.e., accelerated frame) inflat space time. Inthis instance the forces (asmanifested intheterm _m(_, _dxcds dxBds ) which appear inour equations ofmotion are ofreducible nature and can beremoved atwill. (i.e., centrifugal and centripetal forces are example_) Onthe other hand ifspace time isitself twisted there does not exist atransformation the Christoffel three symbols (c_, _which makes vanish and inthis case the force field isirreducible (i.e., non-uniform _" .....iAeld of_gravitatir_ _,i_s. Itmay beshown that Div(Gj* ½g_G)=O. Itwould benatural thePefore tomake the identification GU 1uu ÷-g_G--2 (16.23-1o) This relationship isofsignificance inthe following *No_ beitnoted atwisting inspace time -for the symbol (a_,_) isnot atensor andhence, does not define anything intrinsic inspace tim_. 608< 16-36 l/ respect. The left hand side involves G_ which the law ofgravitation isformulated The right hand side involves law ofmotion isformulated (16.23-2) which ties G_interms of 0). TJ inter,ms ofwhich the ((TJ) v=0). Byvirtue of and T_ together we would deduce that the law ifmotion isdeducible from the law ofgravitation and vice versa. ItistoEddington weowe the following rather graphic explanation ofthe nature of this tie up. Figure 16-10 Consider two material particles inspace time. Along their world lines space time iscreased (figure 16-10). The law ofgravitation states G_u=0 inthe intervening space, inother words, nature isarather fastidious tailor who will not tolerate ripples infree space. Such being the case the creases must run inprescribed directions ifripples are tobe avoided. Thus the relative motion ofthe particles isthe direct outcome ofthe law ofgravitation. 629< 17.17-I SECTION XVII ENVIRONMENTAL REQUIREMENTS Introduction Prior toastudyofthe problems confronting man in space travel itperhaps would beproper tolist some ofthe reasons given asto_why itwould bedesirous tohave manned space vehicles. The most sublime reason given isthat man's expansion into new lands isnecessary for his survival and therefore, because ofthe limitations ofearth, the survival ofthe human race depends upon space travel; there being anesti- mated i00,000 planets inthe known universe capable of sustaining life asweknow it. The reason most often given for manned vehicles isthat generally, man has amuch greater tolerance tovibration, more versatile than the most elaborate electrical brain or man-made man. Finally, the proposed "National Aeronautics and Space Act of1958" inpart declares that "adequate provisions ... bemade for the development ... of... satellites and other space vehicles, manned and unmanned .." Initial steps taken toobtain recruits for the space vehicles would betoscreen applicants toassure that they " 610< 17-2 stand upunder the physical and mental stresses commonly experienced byflying personnel such asthose induced by hazards, combat, authority relations, space living con- ditions, and separation from family. Arecruit ofmature Judgement and emotional stability isdesired. The manned vehicle will besubjected tohigh linear accelerations and possibly some stabilizing rotation during launch. While inorbit orwhile traveling within the solar system the vehicle, equipment, and occupants will experience weightlessness, cold, heat, darkness, brilliant sunlight, vacuum, relatively unknown and unnatural atmospheres of other planets, cosmic radiation and meteorites. The final stages oftravel, re-entry into earth's atmosphere orentry into any planets atmosphere, will probably mean the space vehicle will experience high aerodynamic heating and the occupants will feel the effects ofrapid deacceleration. Man has, rather definite, but most certainly, limita- tions, astowhat hecan withstand physically and still be able toperform prescribed tasks. Man's physical makeup has remained effectively constant over the years and is expected toremain sointhe foreseeable future. These limitations will necessarily beconsidered then inthe design ofanymanned vehicle. 611< 17-3 17.1 Launching ofVehicle Presently reasonable linear accelerations and time- durations needed toplace athree stage rocket inanorbit about the earth have been calculated tobeofthe following order: 5 0 0 -27,_e Xec. S300 17-4 Also the time-duration ofcontinuous accelerations required for avehicle toreach the escape velocity ofthe earth have been calculated tobe: Acceleration (g) Time endured 3 9min 31sec 4 6 21 5 4 45 6 3 _8 7 3 i0 8 2 _o 9 2 2O i0 2 6 Humans have withstood these accelerations inacentrifuge for the listed times without blacking out. Infact the subjects riding the three-stage rocket-launch cycle felt good enough tohave the test repeated several times in succession. One man withstood 17g's for aminute with the acceleration acting inadirection nearly atright angles tothe spine. The subjects, withstanding i0g's, during aperiod of 2minutes and 6seconds were able toconverse inmono- syllables. Their vision was clear, and they were mentally alert and able torespond tovisual and auditory signals. They also retained relatively unimpaired control oftheir hands, wrists, and ankles. Generally speaking the individuals tolerance oflinear 61S_P 17- acceleration actingfromheadtofootisafunction of thecolumn ofarterial blood,between theheartandbrain, thatisapproximately 12inches inheight. Theheart normally pumpsbloodupthiscolumn atapressure of120mm Hg.Atanacceleration ofabout5g'sthedownward force actingonthebloodintheartery aboutequals theupward force. Thebloodcannotthenflowtothebrainandeyes andalsothebloodinthelowerbodyandextremities can notreturn totheheart. Theresult isthattheeye,the firsttonotice anoxygen-carrylng-blood lack,greysout between 3-5and5g'sandblacks outbetween _to5.5g's. Whenthegforceisincreased to_._to6g'sandmaintain- edforthreetofiveseconds unconsciousness willresult. Wehavementioned previously, however, thathuman subjects havewithstood ashighas17g'swithout losing consciousness. Itwasfoundthatiftheacceleration occurs fromthefronttothebackofthebody,thesubject beingin thesupineposition, orfromthebacktothefrontofthe body,theproneposition, humantolerance toacceleration is increased considerably. Tolerances oftheaverage manto various linear accelerations foragivenlength oftimeare showninfigure 17-1. Infigure 17-2thetolerances offivehumansubjects tolinear accelerations atvarious _degrees ofsupination are given. Inanupright position thesubjects blacked outat abouthg,s,however, whensupinated 85°thebloodwasnot subjected topooling andtheyallwithstood 15g'swithout 17-6 blackout. When subjected tominus g,acceleration acting from feet tohead, the blood rushes tothe head causing an increase ofpressure inthebrain. The practical limit of tolerance isabout aminus 3g's for l0to15seconds. The so-called red out, occuring atminus gaccelerations, is apparently due tothe lower eye lid acting asared curtain over the eye. Ifthe space vehicle isrotated ortumbles during flight and ifthe occupants rotate ortumble with the vehicle another physical problem isencountered ofwhich the human body has but acertain endurance. When thebody isrotated the blood tends toaccumulate atthe extremities. In figure 17-3 are presented the results oftests ofhumans being rotated about ax@s through the heart and pelvis region. The tests were run atdifferent revolutions per minute and the duration oftime spent ateach rpm was limited bythe occur- ance ofpain and ocular hemorrhage. The greatest endurance occurred when thebody was rotated about anaxis through the heart. Ag-suit will increase tolerance ofplus acceleration about 2g's. What isprobably anultimate ina"g-suit" is obtained byimmersing the body inacapsule ofwater. As the capsule issubjected toincreasing g's the increasing water pressure onall parts ofthe body prevents pooling of the blood and g-tolerance isgreatly extended. Anadditional 6 5< 17.217-7 advantage ofthis type "g-suit" would berelative freedom tomove the limbs. Adequate provisions should bemade toprotect the orbiting vehicle and contents from the heat and noise generated bythe propulsion units. Somewhat ofaguide tonoise level restrictions may bederived byknowing that the ear usually feels uncom- fortable atanoise level of120 decibels and feels a strong tickling sensation at130 decibels and deep pain atI_0 decibels and above. Men have withstood 115 decibels for a56hour period. This noise level, however, caused atemporary hearing loss ofapproximately 50decibels which cleared upin_days. Ahearing loss of50decibels is considered aserious hearing impairment and makes it difficult tounderstand even loud speech. Orbiting and Travel inSpace. While inspace the temperatures tobeencountered have the aweinspiring range offrom near absolute zero (-273°C) inthe shade ofsome planets tonear 6000°C near the surface ofthe sun. Although itmay bedifficult, temperature can bere- gulated within aspace vehicle byestablishing abalance between reflectors andabsorbers onthe vehicle surface to reflect and absorb radiant heat, primarily from the sun. 61.'_6< 17-8 Afew measurements made inthe nose section ofthe satellite, Explorer I,while inorbit, are presented in figure 17-_ along with the calculated temperatures that were expected tobemeasuredo Human tolerance toheat and cold depends upon the body environment, health, activity, and type and amount ofclothing worn. Safe heat and cold exposure times to air over arange oftemperatures for normal healty men at rest, clothed and partly orwholly exposed, inablack- walled room free offorced draft and radiation are presented infigure 17-5. Some pain may beexperienced bythe human skin ifitis heated toh_°C and at45°C the pain becomes unbearable. If the skin isheld atatemperature of55°C for more than i0 seconds burns will occur. Time-tolerance tocold exposure for humans, sitting and doing nomore than light manual work, for several types of clothing are given infigure 17-6. Additional experience gained bythe personnel winter- ing atthe South Pole indicates that man, properly clothed, can withstand -lO0°F for several hours. They also found out that the pain felt atplus _O°F, ifimproperly dressed, isas much ascan befelt at-_O°F. The reason given isthat the nerves are limited tothe amount ofpain that they can feel due tocold. 6 7< 17-9 Combinations ofvehicle interior air temperature, interior surface temperature, and relative humidity can cause interior frosting. Frosted instruments, control knobs and observation ports may beuncomfortable and a hazard tothe safety ofthe vehicle. Curves similar to those shown infigure 17-7 may beofconvenience tothe designer. The effect that the strange andmysterious weightless- ness orzero gstate may have onman over prolonged periods will probably not beresolved until man isput into orbit. Weightlessness can befelt for afew seconds during adive into aswimming pool orwhen Jumping down tothe ground from above until the resistance ofthe air becomes appreci- able. Inexperimental work with aF-9_c airplane weightless- ness has been experienced for periods ofh3seconds byfly- ing inaprescribed parabolic arc. (figure 17-8) Subjects experiencing zero ginthese tests had varied reactions. Most, however, enjoyed the weightless state. Asobering thought onweightlessness isprovided, however, byanexperienced _est pilot Major Chuck Yeager, who, after 8-I0seconds atzero g,felt his head grow thick and he got the impression that hewas spinning around slowly inno particularly defined direction. Here onearth, many persons have chewed, swallowed and begun todigest food while upside down, oreffectively at minus one gand hence, could prQbably eat with little orno 17-I0 difficulty atzero g. Drinking, though,while ina weightless state may cause drowning; for liquids would float freely and could flow into the nose. However, squeeze tubes could beused toforce fluid into the mouth where muscular action would move the fluid down the throat. Manorientates himselfinanenvironment byusing three different bodysystems: the eye; the semicircular canals and otolith organs oftheinner ear; and bythekinesthetic system. Atzero gthe eye will beareliable system for orientation. However, ithas been determined that two of the three systems are required for positive orientation. The three semicircular canals, aligned inthe three planesofspace, and theotollth organs oftheinner ear are responsive tothe rateofrotation ofthe head and linear accelerations alongone particular direction re- spectively. These organs evolvedinastateofone gand adjustment tothe zero state will have tobemade. The kinesthetic system gives usasenseoforientation through the nervesinthe skin, muscles, and connecting tissues ofthebody. Here onearth the body can besub- Jected toaone ggravity field atany one time inone of six directions (to the right, left, back, front, and up and down)without undo discomfort. ThereforeitiSexpect- edthat the kinesthetic systemwill adjust readily tothe zero gravity state. 6.(9< -'7 17-11 17.3 Air Equivalent Theoxygen partial pressure atsea level is3psi.; atthisoxygen pressure the blood isnormally saturated to 95% and thebodyworks fine. At15,000 feet the oxygen pressure isbut 1.5 psi. and oxygen saturation ofthe blood drops to70%. The person used tobreathing air atsea level ifsuddenly placed inanatmospherewith but 1.5 psi. oxygen pressure for about two hourswould experience fatigue, drow- siness, headache, and poor Judgment. Due toapractically total lack ofoxygen inspace the airequivalent needed by the occupants ofany space vehiclewill have tobetaken along. Ithas been determined thatone cubic ft/hr/manof oxygen (probably carried inliquid state) isrequired when light exercise orwork isperformed and 2.26 cubic ft/hr/man under moderate exercise conditions. Experiments have been carried out during the pastseveral years todevelope aneconomicalsystem tofurnish oxygen and food inaspace vehicle from cropsofalgae. The system can bedescribed briefly. The algae takes light from the sun or other source, carbon dioxide from the air, and nitrogen, water, and other foods from the soil. These are synthesized into carbohydrates, proteins and fats. Man could then eat a percentage ofthe plants andbreathe the excess oxygen pro- duced bythe plants during photosenthesis. Humanwastes, (including the carbon dioxide exhaled from the lungs) con- taining almost exactly the_ec@ssary foods topromote 6 0< 17-12 vigorous algae growth, would bereturned tothe soll to serve asfood for the algae and tostart the cycle anew. Aninternal cabin pressure aslow aspractical would bedeslrable from the vehicle structural designers stand- point. The differential between internal and external vehicle pressures would then beatits lowest while orbiting inthe near vacuum ofspace. However, the vehicle would probably bepressurized while onthe ground Just prior to launch. The most practical differential pressure value then from astructural consideration would bethe value half way between the sea level pressure (if launched at sea level) and the vacuum orzero pressure level. Under these asstLmptlons the internal cabin pressure would be 7.35 lbs/In 2,equivalent toanaltitude ofabout 18,000 feet. One disadvantage offlying around inspace enjoying acabin pressure of7.35 ibs/in 2isthat itmight slowly orsuddenly vanish. Machine failure orpuncture ofthe space vehicles skin could cause loss ofpressure. Suddenly losing the atmosphere oreven apartial loss ofpressure will subject the traveler todecompression. Atthe time ofdecompression, ifthe breath isheld,or swallowing istaking place, the rapid exit ofair from the lungs will beprohibited orslowed down. The trapped 6 11-'- ]-7-13 gases inthe lungs will then expand causing chest pains, blurred vision, nausea, andheadaches. Other body organs that are sensitive topressure changes are the ears and sinuses. Infigure 17-9 are pressure differentials causing middle ear pain. Inaddition tothe effects ofrapid decompression even aslow reduction ofpressure onthebody may produce ill effects. Man breathes nitrogen innormal air and some ofit isdissolved inthe tissues ofthebody. Ifthe pressures onthe body are reduced to615 lb/ft 2the nitrogen will be released from the tissues toform bubles inthe Joints, called "bends", and inthe pulmonary mechanisms ofthe body, called "chokes". Still further reduction inair pressure toabout 130 lb/ft 2will cause "boiling" ofthe body fluids. This is called ebullism and isundoubtly the lowest pressure limit aman could stand. Itwas mentioned before that carbon dioxide isexhaled inthebreath. Ifthe carbon dioxide content ofthe air is allowed tobuild uptowhere it's more than 0.3% byvolume (ten times the normal content ofair) itwill cause labored breathing ,headaches, and ifgreatly exceeded even death. Figure 17-lOshows the tolerance ofman toasudden exposure tovarying amounts ofcarbon dioxide. The standard technique ofabsorbing carbon dioxide in aclosed compartment isbyusing anoxide ofanalkalin 17-iI+ 17.4earth metal. Anexample being Lithium which requires 325 grams/man/day toremove the carbon dioxide that is exhaled from the lungs atthe rate of1kg/man/day. Radiation When man removes himself from the protective covering ofthe earth's atmosphere hewill besubjected tointense solar radiation and primary cosmic rays. Serious sunburn hazards will exist and the viewing ofobjects will beanything but comfortable athigh altitudes. Inside acabin, with sunlit patches adjacent todeep shadows, viewing will bemade even more uncomfortable, and itisre- commended that sun glasses beworn. The earth's atmosphere isequivalent toalead shield about three feet thick and when this isleft behind the space vehicle will bebombarded bythe primary cosmic rays. When aprimary cosmic-ray particle enters matter, such asaspace vehicle, ithas two main processes bywhich it may give upits great kinetic energy: itmay gradually give upits energy byionizing the atoms ofthe material itenters, or, itmay collide with anatomic nucleus, producing aviolent nuclear reaction, called a"star". The energy ofthe ray is distributed among the fragments ofthe colliding nuclei. It's common tospecify the radiation dosage received interms ofintensity, ornumbers ofroentgens per hour and the time ofduration. Aroentgen isthe sLmount of 17.517- 15 irradiation which will produce 2.1 x109 pairs ofions inacubic centimeter ofair atO°C and 760 mmHgpressure. According tothe Bureau ofStandards, the maximum permissible dose ofionization based onyear-round exposure for male hum_s is300 milliroentgens per week for X-, _-, and _-rays and 15mrper week for Z-rays. UptoI0,000 mrcan beaccumulated below theage of30 and upto50,000 mrbelow the age of[_0years without damage tooffspring orshortening ofllfe. Meteorites The meteorite Isthe particle which causes the atmos- pheric effect wesee and call ameteor. Innavigating inspace itwould probably bebest to avoid the fundamental plane ofthe Solar system, especially between the planets Mars and Jupiter, where there are high concentrations ofmeteorites. Calculations have been made onthe probability ofa apace vehicle being hlt and penetrated byameteorite. The calculations were based onthe number ofobservations made ofmeteors over acertain period. However, meteors are only visible from amaximum ofabout 80miles from the earth and meteorites may bemore abundant inspace. Chances ofaspace vehicle being penetrated byameteorite, according tothese calculations, are about 1in2000 over a 6?4< 17-16 17.62[_hour period. Data collected from orbiting unmanned vehicles will undoubtly beofgreat help inthis respect. Re-entry The foremost problems tosolve, ordesign for, during re-entry are deacceleration and aerodynamic heating. Human tolerance tovarious magnitudes ofvehicle de- acceleration can beconsidered the same astoacceleration ifthe occupants face aft during the re-entry. Aerodynamic heating during re-entry was considered inprevious papers.f 67.S< 17-17 Glossary Aeroembolism -decompression sickness Astrobiognosis -the study ofthe support oflife inspace Astronautics -design, production, and operation ofspace craft Bends -pain inand about the Joints due tothe formation ofnitrogen bubbles Blackout -loss ofsight due tolack ofblood inthe eyes Bioastronautics -human factors envolved inastronautics Chokes-symptonsofaeroembolism referable tothe thorax- blockage ofpulmonary vessels bybubbles. Clo -amountofclothing aseated man needs tobecomfortable in70°F air with arelative humidity of50% and air movement of20feet per minute Dysbarism -the effects ofreduced barometric pressure on the body Ebullism -"boiling" ofbody fluids Ecosphere ofthe sun -limited area orbelt inthe planetary system within which life (as weknow it) isconceivable Greyout -vision becomes cloudy due tolack ofblood inthe eyes Hyperventilation -breathing three afour times the normal rate, thereby upsetting nature's balance ofoxygen and carbon dioxide, causing dizziness and loss ofcoordination Hypoxia -lack ofoxygen inthe blood lliac crest -inthe region ofthe dorsal and upper end of the three bones composing either lateral half ofthe pelvis Kinesthetic apparatus -special receptors situated inmuscles , tendons, connective tissues, skin, etc. and their nervous connections with the brain 626< 17-18 Milieu- environment Otolith organ -cavity ofear responsive tolinear acceleration Roentgen -anamoun_ ofradiation aswould have produced 2.1 xI0vpairs ofions inacubic centimeter ofair at0°C, 760 mmHgpressure Squeeze -fluid andblood being squeezed into the lungs Supine -lying onthe back -opposed toprone Trauma- aninjury, wound, shock Vestibular system -paired small balance organs and their nervous connections with the cerebral cortex asthe center ofperception 6L7<9 . I0. II. 12.17-19 Test Facilities doing Work onHuman Tolerances I. Naval Research Laboratory, Washington, D.C. 2. U.S.Naval School ofAviation Medicine, NAS, Pensacola, Fla. 3. Naval Medical Research Institute, Bethesda, Maryland 4. U.S.Navy Aeronautical Medical Equipment Laboratory, Philadelphia, Pa. 5. Naval Medical Research Laboratory, New London, Conn. 6. Aviation Medical Acceleration Laboratory, Johnsville, Pennsylvania. 7. USAF School ofAviation Medicine, Randolph Field, Texas. 8. Aero Medical Laboratory WADC, Wright-Patterson A.F.B., Dayton, Ohio. Holloman A.F.B., New Mexico Gunter A.F.B., Montgomery, Alabama. Army Medical Research Laboratory, Fort Knox, Kentucky. The Lovelace Foundation for Medical Education and ReseArch; Albuquerque, New Mexico. I. 2. 3. 5. . . 8. 9. I0. ii, 12. 13. 14. 15.17-20 References Realities ofSpace Travel, Edited byL.J.Carter. Your Body inFlight, Department ofthe Air Force manual 51-7. Rockets, Missiles, end Space Travel, Willy Ley. Physics and Medicine oftheUpper Atmosphere. C$_4h,te_O.OSe-_o,?Jr Missiles and Rockets, Dec. '56, Jan '57, Feb. '58 and Apr. '58. The Journal ofAviation Medicine, Aug. 'Sh, Dec. '5h and Feb. '58. Scientific American, Feb. '56 and Jan. '57. Aeronautical Engineering Review, Mar. 1958 and Apr. '58. Air Force Magazine, Mar. 1958. Aviation Age, March 1958. Aviation Week, Aug. '57, March '58and Jan. '58. The National Geographic Magazine, Aug. 55and Apr. '58. Spaceflight, Apr. '57. Journal British Interplanetary Society, 1954. Summary Session Astronautics Symposium, Feb. 18-20, 1957, under sponsorship ofUSAF Office ofScientific Research.! 619< ..... J+ -....................................................... ° ...... /00 I0 I u .01_-+-i.+-_ _+ •:'-""5.;_.i: _'L'Tf 2HTTN= ;27::z:_::_. N N N ,..:2 N N li:i!l 111111 630< 16 i,4- iZ 4. 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