lagrange points

The Lagrange P oin tsThere are /#0Cv e equilibrium p oin ts to be found in the vicinit y of t w o orbitingmasses/. They are called L agr ange Points in honour of the F renc h/-Italianmathematician Joseph Lagrange/, who disco v ered them while studing the re/-stricted three/-b o dy problem/. The term /#5Crestricted/" refers to the conditionthat t w o of the masses are v ery m uc h hea vier than the third/. T od a y w ekno w that the full three/-b o dy problem is c haotic/, and so cannot be solv edin closed form/. Therefore/, Lagrange had go o d reason to mak e some appro x/-imations/. Moreo v er/, there are man y examples in our solar system that canbe accurately describ ed b y the restricted three/-b o dy problem/. r1 r2M1 M2rmFigure /1/: The restricted three/-b o dy problemThe pro cedure for /#0Cnding the Lagrange po i n ts is fairly straigh tforw ard/:W e seek solutions to the equations of motion whic h main tain a constan tseparation b et w een the three b o dies/. If M/1 and M/2 are the t w o masses/, and/~ r/1 and /~ r/2 are their resp ectiv e p ositions/, then the total force exerted on athird mass m /, at a p osition /~ r /, will b e/~F /= /, GM/1 mj /~ r /, /~ r/1 j /3 /#28 /~ r /, /~ r/1 /#29 /, GM/2 mj /~ r /, /~ r/2 j /3 /#28 /~ r /, /~ r/2 /#29 /: /#28/1/#29The catc h is that bo t h /~ r/1 and /~ r/2 are functions of time since M/1 and M/2are orbiting eac h other/. Undaun ted/, one ma y pro ceed and insert the orbital/1 solution for /~ r/1 /#28 t /#29 and /~ r/2 /#28 t /#29 /#28obtained b y solving the t w o /- bod y problem forM/1 and M/2 /#29a n d lo ok solutions to the equation of motion/~F /#28 t /#29/= m d /2/~ r /#28 t /#29dt /2 /; /#28/2/#29that k eep the relativ e p ositions of the three b o dies /#0Cxed/. It is these stationarysolutions that are kno w as Lagrange p oin ts/.The easiest w a y to /#0Cnd the stationary solutions is to adopt a co/-rotatingframe of reference in whic h the t w o large masses hold /#0Cxed p ositions/. Thenew frame of reference has its origin at the cen ter of mass/, and an angularfrequency /#0A giv en b y Kepler/'s la w/:/#0A /2R /3/= G /#28 M/1 /+ M/2 /#29 /: /#28/3/#29Here R is the distance b et w een the t w o masses/. The only dra wbac k of usinga non/-inertial frame of reference is that w e ha v e to app end v arious pseudo/-forces to the equation of motion/. The e/#0Bectiv e force in a frame rotatingwith angular v elo cit y /~/#0A is related to the inertial force /~F according to thetransformation/~F/#0A /= /~F /, /2 m /#20/~/#0A /#02 d /~ rdt /!/, m /~/#0A /#02 /#28 /~/#0A /#02 /~ r /#29 /: /#28/4/#29The /#0Crst correction is the coriolis force and the second is the cen trifugal force/.The e/#0Bectiv e force can b e deriv ed from the generalised p oten tialU/#0A /= U /, /~ v /#01 /#28 /~/#0A /#02 /~ r /#29/+ /1/2 /#28 /~/#0A /#02 /~ r /#29 /#01 /#28 /~/#0A /#02 /~ r /#29 /; /#28/5/#29as the generalised gradien t/~F/#0A /= /,r/~ r U/#0A /+ ddt /#28 r/~ v U/#0A /#29 /: /#28/6/#29The v elo cit y dep enden t terms in the e/#0Bectiv e p oten tial do not in/#0Duence thep ositions of the equilibrium po i n ts/, but the are crucial in determining thedynamical stabilit y of motion ab out the equilibrium p oin ts/. A plot of U/#0Awith /~ v /=/0 /, M/1 /=/1 /0 M/2 /=/1 and R /=/1 /0 i s sho wn in Figure /2/. The extremaof the generalised po t e n tial are lab eled L/1 through L/5/./2 LL L 123 5L4LFigure /2/: A coun tour plot of the generalised p oten tial/.Cho osing a set of cartesian co ordinates originating from the cen ter ofmass with the z axis alined with the angular v elo cit y /, w e ha v e/~/#0A/= /#0A /^k/~ r /= x /#28 t /#29/^ /#10/+ y /#28 t /#29/^ /#11/~ r/1 /= /, /#0BR /^ /#10/~ r/2 /= /#0CR /^ /#10 /#28/7/#29where/#0B /= M/2M/1 /+ M/2 /; /#0C /= M/1M/1 /+ M/2 /: /#28/8/#29T o /#0Cnd the static equilibrium p oin ts w e set the v elo cit y /~ v /= d /~ r/= d t to zeroand seek solutions to the equation /~F/#0A /= /~/0/, where/~F/#0A /= /#0A /2 /#20x /, /#0C /#28 x /+ /#0BR /#29 R /3/#28/#28 x /+ /#0BR /#29 /2/+ y /2/#29 /3 /= /2 /, /#0B /#28 x /, /#0CR /#29 R /3/#28/#28 x /, /#0CR /#29 /2/+ y /2/#29 /3 /= /2 /!/^ /#10/#0A /2 /#20y /, /#0Cy R /3/#28/#28 x /+ /#0BR /#29 /2/+ y /2/#29 /3 /= /2 /, /#0By R /3/#28/#28 x /, /#0CR /#29 /2/+ y /2/#29 /3 /= /2 /!/^ /#11 /: /#28/9/#29/3 Here the mass m has b een set equal to unit y without loss of generalit y /. Thebrute/-force approac h for /#0Cnding the equilibrium p oin ts w ould be to set themagnitude of eac h force comp onen t to zero/, and solv e the resulting set ofcoupled/, fourteen th order equations for x and y /. A more promising approac his to think ab out the problem ph ysically /, and use the symmetries of thesystem to guide us to the answ er/.Since the system is re/#0Dection/-symmetric ab out the x /-axis/, the y comp o/-nen t of the force m ust v anish along this line/. Setting y /= /0 and writingx /= R /#28 u /+ /#0C /#29 /#28so that u measures the distance from M/2 in units of R /#29/, thecondition for the force to v anish along the x/-axis reduces to /#0Cnding solutionsto the three /#0Cfth/-order equationsu /2/#28/#28/1 /, s/1 /#29/+/3 u /+/3 u /2/+ u /3/#29/= /#0B /#28 s/0 /+/2 s/0 u /+/#28 /1/+ s/0 /, s/1 /#29 u /2/+/2 u /3/+ u /4/#29 /; /#28/1/0/#29where s/0 /= sign /#28 u /#29 and s/1 /= sign/#28 u /+ /1/#29/. The three cases w e need to solv eha v e /#28 s/0 /;s/1 /#29 equal to /#28 /, /1 /; /1/#29/, /#28/1 /; /1/#29 and /#28 /, /1 /; /, /1/#29/. The case /#28/1 /; /, /1/#29 cannoto ccur/. In eac h case there is one real ro ot to the quin tic equation/, givingus the p ositions of the /#0Crst three Lagrange po i n ts/. W e are unable to /#0Cndclosed/-form solutions to equation /#28/1/0/#29 for general v alues of /#0B /, so instead w eseek appro ximate solutions v alid in the limit /#0B /#1C /1/. T o lo w est order in /#0B /,w e /#0Cnd the /#0Crst three Lagrange p oin ts to b e p ositioned atL /1/: /#20R /"/1 /, /#12/#0B/3 /#13/1 /= /3 /#23/; /0 /!/;L /2/: /#20R /"/1/+ /#12/#0B/3 /#13/1 /= /3 /#23/; /0 /!/;L /3/: /#12/, R /#14/1/+ /5/1/2 /#0B /#15/; /0 /#13/: /#28/1/1/#29F or the earth/-sun system /#0B /#19 /3 /#02 /1/0 /, /6/, R /= /1A U /#19 /1 /: /5 /#02 /1/0 /8km/, andthe /#0Crst and second Lagrange po i n ts are lo cated appro ximately /1/./5 millionkilometers from the earth/. The third Lagrange p oin t /- home of the m ythicalplanet X /- orbits the sun just a fraction further out than the earth/.Iden tifying the remaining t w o Lagrange po i n ts requires a little morethough t/. W e need to balance the cen trifugal force/, whic h acts in a directionradially out w ard from the cen ter of mass/, with the gra vitational force exertedb y the t w o masses/. Clearly /, force balance in the direction p erp endicular to/4 cen trifugal force will only in v olv e gra vitational forces/. This suggests that w eshould resolv e the force in to directions parallel and p erp endicular to /~ r /. Theappropriate pro jection v ectors are x /^ /#10/+ y /^ /#11 and y /^ /#10 /, x /^ /#11/. The p erp endicularpro jection yieldsF /?/#0A /= /#0B/#0C y /#0A /2R /3 /#20/1/#28/#28 x /, R/#0C /#29 /2/+ y /2/#29 /3 /= /2 /, /1/#28/#28 x /+ R/#0B /#29 /2/+ y /2/#29 /3 /= /2 /!/: /#28/1/2/#29Setting F /?/#0A /= /0 and y /6/= /0 tells us that the equilibrium po i n ts m ust beequidistan t from the t w o masses/. Using this fact/, the parallel pro jectionsimpli/#0Ces to readF k/#0A /=/#0A /2 x /2/+ y /2R /#20/1R /3 /, /1/#28/#28 x /, R/#0C /#29 /2/+ y /2/#29 /3 /= /2 /!/: /#28/1/3/#29Demanding that the parallel comp onen t of the force v anish leads to thecondition that the equilibrium p oin ts are at a distance R from eac hm a s s /. Inother w ords/, L/4 is situated at the v ertex of an equilateral triangle/, with thet w o masses forming the other v ertices/. L/5 is obtained b y a mirror re/#0Dectionof L/4 ab out the x /-axis/. Explicitly /, the fourth and /#0Cfth Lagrange p oin ts ha v eco ordinatesL /4/: /#20R/2 /#12M/1 /, M/2M/1 /+ M/2 /#13/; p/3/2 R /!/;L /5/: /#20R/2 /#12M/1 /, M/2M/1 /+ M/2 /#13/; /, p/3/2 R /!/: /#28/1/4/#29Stabilit y AnalysisHa ving established that the restricted three/-b o dy problem admits equilib/-rium p oin ts/, our next task is to determine if they are stable/. Usually it isenough to lo ok at the shap e of the e/#0Bectiv e po t e n tial and see if the equilib/-rium p oin ts o ccur at hills/, v alleys or saddles/. Ho w ev er/, this simple criterionfails when w eh a v ea v elo cit y dep enden t p oten tial/. Instead/, w em ust p erforma linear stabilit y analysis ab out eac h Lagrange p oin t/. This en tails linearisingthe equation of motion ab out eac h equilibrium solution and solving for small/5 departures from equilibrium/. W ritingx /= xi /+ /#0Ex /; vx /= /#0Evx /;x /= yi /+ /#0Ey /; vy /= /#0Evy /; /#28/1/5/#29where /#28 xi /;yi /#29i st h e p osition of the i /-th Lagrange p oin t/, the linearised equa/-tions of motion b ecomeddt /0BB BBB B B BBB B/@ /#0Ex/#0Ey/#0Evx/#0Evy /1CC CCC C C CCC CA /= /0BB B B BBB B B BBB B B B/@ /0 /0 /1 /0/0 /0 /0 /1d /2U/#0Adx /2 d /2U/#0Adxdy /0 /2/#0Ad /2U/#0Ady dx d /2U/#0Ady /2 /, /2/#0A /0 /1CC C C CCC C C CCC C C CA /0BB BBB B B BBB B/@ /#0Ex/#0Ey/#0Evx/#0Evy /1CC CCC C C CCC CA /: /#28/1/6/#29Here the second deriv ativ es of U/#0A are ev aluated at /~ r /=/#28 xi /;yi /#29/.Stabilit y of L/1 and L/2The stabilit y of the /#0Crst and second Lagrange p oin ts of the earth/-sun systemis an imp ortan t consideration for some NASA missions/. Curren tly the solarobserv atory SOHO is park ed at L/1/, and NASA plans to send the Micro w a v eAnisotrop y Prob e /#28MAP/#29 out to L/2/. It has also b e suggested that the NextGeneration Space T elescop e /#28NGST/#29 should b e p ositioned at L/2/.The curv ature of the e/#0Bectiv e p oten tial near L/1 and L/2 rev eals them tobe saddle po i n ts/:d /2U/#0Adx /2 /= /#07 /9/#0A /2/; d /2U/#0Ady /2 /= /#06 /3/#0A /2/; d /2U/#0Adxdy /= d /2U/#0Ady dx /=/0 /: /#28/1/7/#29Solving for the eigen v alues of the linearised ev olution matrix w e /#0Cnd/#15/#06 /= /#06 /#0A q/1/+/2 p/7 and /#1B/#06 /= /#06 i /#0A q/2 p/7 /, /1 /: /#28/1/8/#29The presence of a p ositiv e/, real ro ot tells us that L/1 and L/2 are dynamicallyunstable/. Small departures from equilibrium will gro w exp onen tially with a/6 e/-folding time of/#1C /= /1/#15/+ /#19 /2/5/#0A /: /#28/1/9/#29F or the earth/-sun system /#0A/= /2 /#19 y ear /, /1and /#1C /#19 /2/3 da ys/. In other w ords/, asatellite park ed at L/1 or L/2 will w ander o/#0B after a few mon ths unless coursecorrections are made/.Stabilit y of L/3A p opular theme in early science /#0Cction stories w as in v asion b y creaturesfrom Planet X/. The requiremen t that Planet X remain hidden be h i n d thesun places it at the L/3 po i n t of the earth/-sun system/. Unfortunately forour w ould/-b e in v aders/, the L/3 p oin t is a w eak saddle po i n t of the e/#0Bectiv epo t e n tial with curv atured /2U/#0Adx /2 /= /, /3/#0A /2/; d /2U/#0Ady /2 /= /7 M/2/8 M/1 /#0A /2/; d /2U/#0Adxdy /= d /2U/#0Ady dx /=/0 /: /#28/2/0/#29T o leading order in M/2 /= M/1 /, the eigen v alues of the linearised ev olution matrixare/#15/#06 /= /#06 /#0A s/3 M/1/8 M/2 and /#1B/#06 /= /#06 i /#0A p/7 /: /#28/2/1/#29The real/, p ositiv e eigen v alue sp ells disaster for Planet X/. Its orbit is exp o/-nen tially unstable/, with an e/-folding time of roughly /#1C /= /1/5/0 y ears/. Whilethere can be no Planet X/, the long e/-folding time mak es L/3 a go o d place topark y our in v asion force while /#0Cnal preparations are made/././.Stabilit y of L/4 and L/5The stabilit y analysis around L/4 and L/5 yields something of a suprise/. Whilethese p oin ts corresp ond to lo cal maxima of the generalised p oten tial /- whic husually implies a state of unstable equilibrium /- they are in fact stable/. Theirstabilit y is due to the coriolis force/. Initially a mass situated near L/4 or L/5will tend to slide do wn the p oten tial/, but as it do es so it pic ks up sp eed andthe coriolis force kic ks in/, sending it in to an orbit around the Lagrange p oin t/.The e/#0Bect is analogous to ho wa h urricane forms on the surface of the earth/:as air rushes in to a lo w pressure system it b egins to rotate be c a u s e of the/7 coriolis force and a stable v ortex is formed/. Explicitly /, the curv ature of thepo t e n tial near L/4 is giv en b yd /2U/#0Adx /2 /= /3/4 /#0A /2/; d /2U/#0Ady /2 /= /9/4 /#0A /2/; d /2U/#0Adxdy /= d /2U/#0Ady dx /= /3 p/3/4 /#14 /#0A /2/; /#28/2/2/#29where /#14 /=/#28 M/1 /, M/2 /#29 /= /#28 M/1 /+ M/2 /#29/. The eigen v alues of the linearised ev olutionmatrix are found to equal/#15/#06 /= /#06 i /#0A/2 q/2 /, p/2/7 /#14 /2/, /2/3/#1B/#06 /= /#06 i /#0A/2 q/2/+ p/2/7 /#14 /2/, /2/3 /: /#28/2/3/#29The L/4 po i n t will be stable if the eigen v alues are pure imaginary /. This willbe true if/#14 /2/#15 /2/3/2/7 and p/2/7 /#14 /2/, /2/3 /#14 /2 /: /#28/2/4/#29The second condition is alw a ys satis/#0Ced/, while the /#0Crst requiresM/1 /#15 /2/5 M/2 /0/@ /1/+ q/1 /, /4 /= /6/2/5/2 /1A/: /#28/2/5/#29When the L/4 and L/5 p oin ts yield stable orbits they are refered to as T ro janpo i n ts after the three T ro jan asteroids/, Agamemnon/, Ac hilles and Hector/,found at the L/4 and L/5 p oin ts of Jupiter/'s orbit/. The mass ratios in theearth/-sun and earth/-mo on system are easily large enough for their L/4 andL/5 p oin ts to be h o m e to T ro jan satellites/, though none ha v e be e n found/.AuthorThese notes w ere written b y Neil J/. Cornish with input from Jerem y Go o d/-man/./8