Williams torque on oblate

1994AJ 108. . 71 IW THE ASTRONOMICAL JOURNAL VOLUME 108, NUMBER 2 AUGUST 1994 CONTRIBUTIONS TO THE EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION James G. Williams Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 Electronic mail: LOGOS::JGW, [email protected] Received 1993 December 21; revised 1994 April 13 ABSTRACT The precession and nutation of the Earth’s equator arise from solar, lunar, and planetary torques on the oblate Earth. The mean lunar orbit plane is nearly coincident with the ecliptic plane. A small tilt out of the ecliptic is caused by planetary perturbations and the Earth’s gravitational harmonic 72- These planetary perturbations on the lunar orbit result in torques on the oblate Earth which contribute to precession, obliquity rate, and nutation while the J2 perturbations contribute to precession and nutation. Small additional contributions to the secular rates arise from tidal effects and planetary torques on the Earth’s bulge. The total correction to the obliquity rate is —0.0247century, it is an observable motion in space (the much larger conventional obliquity rate is wholly from the motion of the ecliptic, not the equator), and it is not present in the IAU-adopted expressions for the orientation of the Earth’s equator. The72 effects have generally been allowed for in past nutation theories and some precession theories. For the planetary effect, the contributions to the 18.6 yr nutation are —0.03 mas (milliarcseconds) for the in-phase At// plus out-of-phase contributions of 0.14 mas in Aif/ and —0.03 mas in Ae. The latter terms demonstrate that out-of-phase contributions can arise by means other than dissipation. The sum of the contributions to the precession rate is considered and the inferred value of the moment of inertia combination (C—A)/C, which is used to scale the coefficients in the nutation series, is evaluated. Using an updated value for the precession rate, the rigid body (C-A)/C=0.003 273 763 4 which, in combination with a satellite-derived J2, gives a normalized polar moment of inertia Cl MR2=0.330 700 7. The planetary contributions to the precession and obliquity rates are not constant for long times causing accelerations in both quantities. Acceleration in precession also arises from tides and changing J2. Contributions from the improved theory, masses, ecliptic motion, and measured values of the precession rate and obliquity are combined to give expressions (polynomials in time) for precession, obliquity, and Greenwich Mean Sidereal Time. 1. INTRODUCTION Torques on the oblate Earth due to the gravitational attrac- tion of the Sun and Moon cause the Earth’s equator to pre- cess and nutate. The precession is retrograde and its rate is 507yr, roughly 1/3 of it due to the Sun and 2/3 from the Moon. The rate depends on the lunar and solar masses and distances, the orbital eccentricities and inclinations, and the obliquity angle between the Earth’s equator and ecliptic planes. Recent decades have seen impressive advances in the ac- curacies of techniques measuring positions of artificial satel- lites, the Moon, and radio sources. Accurate theories for the motion of the Earth’s equator in space are needed. This paper examines several theoretical contributions to precession, obliquity change, and nutation. The orbit of the Earth-Moon system about the Sun de- fines the ecliptic plane. The lunar orbit is inclined 5° to the ecliptic plane and the strong solar torques drive the preces- sion of the lunar orbit plane along the ecliptic with an 18.6 yr period. But several influences cause a slight tilt of the mean plane of orbital precession with respect to the ecliptic. The Earth’s oblateness contributes a small torque which attempts to precess the lunar orbit along the equator. The net result of these two torques is a lunar orbit precession along a plane tilted 8" with respect to the ecliptic and this plane intersects the ecliptic at the dynamical equinox, the intersection of the ecliptic and equator planes. This small influence of the Earth’s oblateness on the lunar orbit in turn causes a small change in the precession of the Earth’s equator. The orbit planes of the planets have small inclinations with respect to the ecliptic plane. As a consequence of the planetary attractions, the ecliptic plane moves. The Moon’s mean plane of orbital precession follows the moving ecliptic closely, but not perfectly. This motion causes a 1.4" tilt of the plane of orbital precession to the ecliptic. There are also direct planetary torques on the lunar orbit which contribute a smaller displacement. These two influences on the lunar orbit result in torques on the oblate Earth which modify its orien- tation. In addition, the planets directly torque the Earth. The torques from these three planetary influences are not aligned with respect to the dynamical equinox. Consequently, they contribute to both the precession of the equator and the obliquity rate. While the precession rate must be a measured quantity, the obliquity rate is not a free parameter of the dynamics. These planetary influences are not included in the IAU-adopted theory of precession and obliquity change (Lieske et al. 1977). Neither have all of the consequences of the planetary tilts on the lunar orbit been included in recent nutation theories. The above sources of precession and obliquity rate also cause accelerations. Acceleration corrections also arise from tidal effects and the Earth’s changing J2- The above outlined corrections to the motion of the 711 Astron. J. 108 (2), August 1994 0004-6256/94/108(2)/711/14/$0.90 © 1994 Am. Astron. Soc. 711 © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 712 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 712 Earth’s equator are developed in the following sections. To these corrections are added precession corrections developed by Kinoshita & Souchay (1990) due to the Earth’s J4 and second-order corrections due to nutation. From the revised theory are developed new polynomial expressions for the motion of the Earth’s equator and revised values of the Earth’s fractional moment of inertia, (C—A)/C, and the nor- malized polar moment, C/MR2. 2. FUNDAMENTALS This section sets up the fundamental equations for calcu- lating the motion of the Earth’s equator (or pole) in space. As the computations of the subsequent two sections are limited to small effects, it is reasonable to introduce simplifications. The Earth will be treated as a rigid body without oceans and without the influence of a liquid core. Small differences in the directions of the axes of angular momentum, instanta- neous spin, and figure (equivalent to celestial ephemeris pole for a rigid body) are ignored. The equations will be written for the angular momentum axis, but strictly speaking it is the motion of the figure axis of the rigid Earth which is desired. Also ignored are second-order effects due to the change of the Earth’s orientation, e.g., precession and nutation modify- ing the computation of precession and nutation. The oblate, rigid Earth is torqued by an external body. The attracting body has a geocentric distance r and a product of the gravitational constant and mass Gm. The Earth has moments of inertia A, A, C with A<C, and mass M. The z axis is aligned with the Earth’s principal axis corresponding to the maximum moment C, and the x axis points toward the intersection of the ecliptic and equator planes, the dynamical equinox. The potential energy of the external body in the gravity field of the oblate Earth is V=Gm[M/r-(C-A)(3 sin2 5-l)/2r3], (1) where the declination of the attracting body is S and the right ascension is a. Equivalently, the vector r has components (x,y,z). The torque T on the Earth is T=-rXVV, (2) 3Gm(C-A)sin S cos ô sin a — cos a 0 T= 3Gm(C —A) (3) The rate of change of the vector angular momentum L is governed by dL/dt=T. Given the orbit of the external at- tracting body, the resulting precession and nutation of the Earth can be calculated. The analytical theories for the Sun, planets, and Moon are referred to the ecliptic plane. Consequently, the conversion from geocentric ecliptic coordinates (X,Y,Z) to equatorial coordinates (x,y,z) requires a rotation about the x axis by the obliquity e Y cos €—Z sin e Z cos e+Y sin e (4) In the torque vector the products of equatorial coordinate components become (l/2)(72-Z2)sin 2e+VZ cos 2e —XZ cos e—XY sin 6 0 (5) The ecliptic coordinates (X,Y,Z) of the attracting body can be written in terms of the geocentric distance r and the geo- centric ecliptic longitude \ and latitude ß I X\ j COS yö COS X\ Y = r cos y0 sin X . (6) \z) \ sin ß ! Because the Earth’s path about the Sun is well approxi- mated by an elliptical orbit in the ecliptic plane, the solar torque may be computed with good accuracy with little ef- fort. Averaged over an integral number of revolutions the average x component of torque is Tx = 3Gm(C-A)ún ecos e/2a3(l—e2)3/2, where a is the semimajor axis, e the orbital eccentricity, and e the obliquity. The x component of the torque gives rise to a retrograde precession along the ecliptic with rate dif//dt = Tx/C(x)z sin e, where a)z is the major component of the Earth’s angular velocity and Ca)z approximates the total angular momentum of the Earth’s spin. dt('/dt=3Gm(C-A)cos e/2a3(l-e2)3/2Cwz. (7) G/a3 may be replaced with the square of the mean motion divided by the sum of the masses (Sun+Earth+Moon) using Kepler’s third law. The analogous precession from the Moon includes an inclination factor of 1-1.5 sin2 i. The other two torque components have zero average, but of course the first two components have time variations which contribute peri- odic nutation terms. The elliptical approximation above works well for the solar-induced precession of the Earth’s equator along the ecliptic, but it is a coarser approximation for the lunar effect because the lunar orbit is strongly perturbed by the Sun. These difficulties in the major precession and nutation effects have been dealt with by Kinoshita & Souchay (1990). Their computation for the solar precession is only larger by 2X10-6 so Eq. (7) is a very good approximation for the Sun. The lunar orbit is highly perturbed and the equivalent equa- tion for the lunar-induced precession, including the inclina- tion factor, is less precise. The computation of many small corrections in this paper can use the elliptical approximation. 3. EFFECTS DUE TO THE TILTED LUNAR MEAN PLANE The lunar orbit precesses along a plane which is tilted slightly with respect to the ecliptic plane. The Earth’s J2 causes an 8" tilt and planetary effects cause a 1.5" tilt. As a consequence of these small sizes, expansions will be used. © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 713 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 713 The lunar latitude arises from the 5.15° inclination of the orbit to the ecliptic i, and smaller perturbations Aß so that sin yß^sin i sin F + Aß, (8) where F is the mean argument of latitude. Similarly the lunar longitude \ may be written in terms of its mean longitude L, mean anomaly /, eccentricity e, and smaller contributions AL k=L + 2e sin l + AL. (9) The perturbing terms most important for precession and nutation are selected from Chapront-Touzé & Chapront (1988, 1991). For J2 perturbations Aß=-8.045" sin L + 0.326" sin(L-2F), AL =7.063" sin 0 + 0.361" sin(L+F), (10) where is the lunar node (Í1=L—F). The important planetary-induced terms are A/3= 1.510" sin(L + 96.68°), AL=—0.289" sin(0 + 95.13°) -0.062" sin(L + F+95.13°). The J2 and planetary effects also cause radial perturbations, but compared with the longitude and latitude perturbations they are relatively ineffective in modifying precession and nutation. For the purposes of expansions, the above perturbations in ecliptic longitude and latitude will be represented symboli- cally as Aß = B sin(L+ 0)+£' sin(L-2F), (12) AL=E sin(il +<p)+ F' sin(L+F+cp). The ecliptic plane is rotating about a line which is dis- placed from the dynamical equinox by <p—90o=5.13°. The phase <f) is different from cp because the first term in latitude combines both the direct effect of the planets with the indi- rect effect of the ecliptic motion. After introducing these per- turbations into the differential equations of the previous sec- tion and carrying out the expansions through first degree in e and sin i there are contributions to both rate and periodic terms in ÿ and e. The rate terms are dip 3Gm{C—A)cos 2e dt Aa3Co)z sin e X[2B cos <£ + (£'—F)sin i cos <p], de —3Gm(C—A)cos e dt Aa?>Co)z X\2B sin <f) + (Ef -E)sm i sin <p]. A contribution to the obliquity rate requires phase shifts. The planetary effects contribute a —0.254 mas/yr (mas =milliarcsecond) correction to the obliquity rate while J2 perturbations contribute nothing. To the precession, J2 per- turbations contribute -2.630 mas/yr while planetary effects contribute -0.056 mas/yr for a total of -2.686 mas/yr. The largest of the nutation corrections has the 18.6 yr period of the lunar node (rate Ù). While the rigid-body nu- tation caused by the main lunar theory only contains in-phase terms (sines of the arguments for Aip and cosines for Ae), the phase shifts with the planetary effects also induce out-of- phase terms (cosines for Aip and sines for Ae) 3Gm(C —A)sin e cos e sin eAil/= ^ {T-7 sin i B cos ó 4a3Co)zil + 2E' cos 9 + 6 sin i F'jsin Í1 + [5 sin i B sin </> —2F' sin 9]cos Í1} (14) 3Gm(C—A)sin e A6= : {[sin i B cos ó 4a3Cct)2fi — 2E' cos 9]cos íl + [sin i B sin </> — 2E' sin 9] sin Í1}. Nutation terms at half of the nodal period must also be con- sidered. The contributions to the nutation terms with argu- ment twice the lunar node are 3Gm(C —A)cos 2e sin eAiI/= ; ^ [(sin i E cos 9 8a3Ccu2il — 2F')sin 2íl + (sin i E sin ip)cos 2Í1] (15) 3Gm(C—A)cos e Ae = ; : [ — (sin i E cos 9 8a3C(x)zQ — 22?')cos 2íl + (sin i E sin 9)sin 2Í2]. Finally, there are small corrections to terms with argument 2L (rate 2L) 3Gm(C-A)cos 2e sin eAif/= ; : [-B cos ó sin 2L 4a3C(ozL — B sin cp cos 2L], (16) 3Gm{C—A)cos e A 6= ; ^ [Æ cos 4> cos2L 4a3CoL>zL — B sin cp sin 2L] Using the numerical values of the coefficients and phases for the J2 and planetary effects from Eqs. (10) and (11), the above contributions to the nutation have been calculated. They are presented in Table 1 (units mas). The major contri- bution is to the 18.6 yr A^ term with a lesser contribution to the 18.6 yr A£ term. Both of these contributions increase the magnitude of the conventional 18.6 yr terms. The contribu- tions to the 9.3 yr nutation are small. The two contributions in Eq. (15) from the J2 effect nearly cancel and the values in the Table 1 are effectively zero. The higher frequency of the 2L( = 2F + 2il) term prevents those half-month nutation corrections from being large. Out-of-phase terms in the nutation theory will arise from dissipative processes in the oceans (Wahr & Sasao 1981; Zhu © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 714 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 714 Table 1. Nutation terms due to J2 and planetary tilt effects of lunar orbit. Lunar mean longitude is L = il + F. Ae Argument sin mas cos mas sm mas cos mas J2 Tilt Í1 20 2L Planetary n 20 2L -1.4782 0.0049 0.0151 -0.0301 -0.0005 0.0003 0.0000 0.0000 0.0000 0.1366 0.0060 -0.0028 0.0000 0.0000 0.0000 -0.0277 0.0032 -0.0015 0.1557 -0.0026 -0.0081 0.0029 0.0003 -0.0002 et al 1990) and interior of the Earth (Wahr & Bergen 1986; Dehant 1988, 1990). The out-of-phase terms in Table 1 arise from the phase shifts in the planetary effects which in turn arise because the orbit planes of the planets other than Earth have no special alignment with the ecliptic plane or dynami- cal equinox. There are still smaller corrections with argu- ments of 2L +Í1, 2L - il, 2m—íl, and 3Í1 which are not given. Woolard (1953) was aware that out-of-phase terms in nu- tation theory could arise from planetary perturbations on the lunar orbit. The out-of-phase 18.6 yr term for nutation in longitude occurs in his Table 24 (it is marked with a ? and a footnote, but matches the value in Table 1 of this paper), but the obliquity term was too small for his cutoff limit. Woolard also calculated the obliquity rate contribution, showing -0.256 mas/yr in his Table 24 at the year 1900. In the text (p. 127) he also comments that the planetary-induced lunar terms contribute to precession and to the acceleration of obliquity. Kinoshita (1975, 1977) considered the obliquity rate contribution to be due to an error in Woolard’s equations of motion. This assertion will be discussed further in Sec. 6. Kinoshita’s Mj correction to precession is -2.68 mas/yr and it appears to correspond to the sum of the J2 and in-phase planetary effects computed in this paper. In Kinoshita & Souchay (1990) a more elaborate “second-order” correction to precession replaces the earlier correction. It contains a -2.60 mas/yr correction to precession due to the J2 effects, but the —0.056 mas/yr planetary effect is missing. Presum- ably, their nutations contain the corresponding contribution from J2, but not from the planetary tilt. 4. RATES DUE TO DIRECT TORQUES OF PLANETS ON EARTH The torques from the Sun and Moon dominate the preces- sion of the Earth’s equator. There are small additional torques from the planets which contribute to the precession. The inclination of the planetary orbits to the ecliptic will also cause a small obliquity rate. A calculation of the precession contribution was given by Kinoshita & Souchay (1990), but not the obliquity rate. A brief derivation of both rates is given below. Note that the effect of these direct planetary torques on the Earth is distinct from the tilt effect due to the direct planetary perturbations on the lunar orbit. In order to compute the geocentric coordinates of the at- tracting planet, it is necessary to difference the heliocentric coordinates of the planet and the Earth. Primes will be used for the planet’s variables, no primes for the Earth. The effects are small; so to keep the derivation from becoming unwieldy, two approximations will be introduced. The heliocentric or- bits will be taken as circles and the planetary inclinations will only be carried to first degree (cos Then the plan- et’s geocentric ecliptic coordinates are af cos(w' + n') —a cos(M + fT) af sin(w'Tí!') —a sin(H + iy) a ' sin / ' sin M ' (17) where a and a ' denote the semimajor axes, i ' the inclination to the ecliptic, Í1' the node on the ecliptic, and u and «' the arguments of latitude measured from that same node for both the Earth and attracting planet. The geocentric distance r is given by r2 = a2 + a'2-2aa' cos(u — ur). (18) The ecliptic coordinates are rotated into equatorial coor- dinates following Eq. (4) and the products of coordinates of Eq. (5) are formed for substitution into Eq. (3) for the torque. There result expressions involving products of sines and co- sines of u and «'.In order to isolate the secular rates from the periodic terms Gauss’ method of averaging over « and « ' is used. Denoting the average with (), an example is ((YZ/r5)) = f i(YZ/r5)dudur I4tt2 with both integrals evaluated from 0 to 277. To winnow out terms which will disappear during the double integration a mathematical de- vice is useful. The transformations (sin w, sin «')—►(—sin «, -sin«'), (cosm, cos «')—►(—cos «, -cos«'), and both taken together leave unchanged the distance r which appears in the denominators of the integrals. Any component of the numerators Y2-Z2, YZ, XZ, or XY which reverses sign un- der any of the three transformations will average to zero. Also, since only « — « ' appears in the denominator, changing variables of integration to « —«' and « + « ' makes it clear that additional components average to zero. Finally one gets ((Y2 —Z2)/r5) = (l/2)(l/r3), (YZ/r5) = (a'/2)sin i' cos il'(«'(l/r5) — <z(cos(m —w')/r5)), (19) (XZ/r5)= -(«72)sin /' sin il'(«'(l/r5) -«(cos(w-w')/r5)), (XY/r5) = 0. The three different averages on the right-hand sides above are only functions of « —«' and they may be evaluated in terms of complete elliptic integrals of the first and second kind, K(k) and E(k), respectively, © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 715 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 715 Table 2. Precession and obliquity rates from direct planetary torques on the Earth’s bulge. Planet if/ rate mas/yr e rate mas/yr Mercury Venus Mars Jupiter Saturn Uranus Neptune Total 0.003 651 0.187 273 0.005 393 0.116 665 0.005 177 0.000 100 0.000 029 0.318 287 -0.000 090 -0.017 372 0.000 255 0.002 782 0.000 217 0.000 001 0.000 001 -0.014 207 (l/r3) = 2£(&)/[7r(fl + fl')3(l -£2)L {llr5) = 2[-K{k) + 2E{k){2-k2)l{\-k1)y [37r(a + a')5(l —&2)], (20) {cos(u — u,)lr5) = 2\_ — K(k){2 — k2) + 2E(k){l — k2 + kA)¡{ \ —k2)y [37r(a + a')5/:2(l — A:2)]. The modulus k is the geometric mean of the two semimajor axes divided by the arithmetic mean or A:2 = 4flfl7(ö + ß')2. (21) The rates induced by the direct planetary torques are dil/ldt = [Gm' {C—A)l ir{a + ar){a — a'ŸCoiz] X[3 cos e E{k) — G cos 2e sin ï cos il'/sin e], de/dt^lGm’ (C-A)l Tr(a + a'){a- a'ŸCoiz] X[-G cos e sin V sin ilf], (22) G = (a2 + la,2)E(k)/2(a2-ar2) — (a — a')K(k)/2(a + a'), where m[ is the attracting planet’s mass. In the precession rate the larger term involves cos eE(k). It contributes to precession as though the attracting body were in the ecliptic plane and it has the same cos e dependence as the dominant precession due to the Sun and Moon. For both rates the com- binations sin sin Í!' and sin/'cosí!', which are two of the coordinates of the planet’s orbit pole direction, allow for small contributions due to the tilt of the planet’s orbit plane with respect to the ecliptic. The numerical results for the precession and obliquity rate contributions from the direct planetary torques on the Earth are given in Table 2. The 0.3183 mas/yr precession rate re- sults from 0.3269 mas/yr due to the cos eE(k) term and —0.0086 mas/yr from the planetary inclinations. The com- parison of precession rate with Kinoshita & Souchay’s (1990) computations for Venus through Saturn shows differ- ences of 3% for Venus and 1% for Jupiter. In Table 2 the largest values of the modulus k occur for Venus and Mars, 0.987 and 0.978, respectively. The obliquity rate contribution of —0.014 mas/yr combines with the larger contribution of —0.254 mas/yr from planetary effects through the lunar orbit (Sec. 3) to give -0.268 mas/yr. Tidal torques contribute ah additional 0.024 mas/yr to obliquity rate; that derivation is interconnected with nonlinear contributions and will be de- ferred (Sec. 7) until after the summarizing of the rates. The total obliquity rate with respect to space is —0.244 mas/yr. This correction to the obliquity rate is not included in the expressions accompanying the IAU-adopted precession theory. 5. TOTAL PRECESSION AND OBLIQUITY RATES This section summarizes the various contributions to pre- cession and obliquity rates, gives the total values, and dis- cusses the implications. The precession and nutation of the Earth’s pole in space depend on the dynamical flattening (C—A)/C. Since the precession rate was measured with a smaller relative error than the nutation coefficients, the rate of precession was chosen as a primary 1976 IAU constant and recent nutation series have been computed from the de- rived value of (C-A)/C (or proportional quantities for the Sun and Moon called ks and kM). Knowledge of the precession rate and obliquity has im- proved since the adoption of the 1976 IAU constants. The value of (C—A)/C appropriate to the IAU constants, but with the theoretical modifications of this paper and updated ecliptic motion, is 0.003 273 978 26. The featured computa- tions will use improved values of the precession rate, ob- liquity, masses, mean motions, and ecliptic motion. A —3 mas/yr correction to the IAU-adopted value of the precession constant has been indicated by several lines of evidence: lu- nar laser ranging (Williams et al 1991, 1993), very long baseline interferometry (Herring et al 1991; Herring 1991; McCarthy & Luzum 1991; Steppe et al 1993), the two com- bined (Chariot et al 1991), and systematic proper motions in star catalogues (Miyamoto & Soma 1993). Several recent fits have given corrections near -3.2 to —3.3 mas/yr and a gen- eral precession rate of 5028.777century has been chosen for this paper. The change from the IAU general precession rate is -3.266 mas/yr and the change in the luni-solar precession rate is —3.219 mas/yr (the two do not match because the ecliptic motion is different from the IAU paper). For the obliquity at J2000 (e0), the value of 84 381.409"=23° 26'21.409" is based on analyses of lunar and planetary ob- servations. This obliquity and the mass ratios Earth/Moon =81.300 59 and Sun/(Earth+Moon)=328 900.560 are from the recent ephemeris DE 245 (Newhall et al 1993). See Standish (1982) for the technique of extracting the obliquity from an ephemeris. The corresponding {C—A)/C is 0.003 273 763 40, £5=3475.197397century, and =7546.737 00"/century (or 7567.305 75"/century with the 1/^2 factor). The various contributors to precession and obliquity rates are summarized in Table 3. Taken from Kinoshita & Souchay (1990) are the first-order equations for the computation of the lunar- and solar-induced precession (the values were computed from the equations), the value of the second-order © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 716 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 716 Table 3. Contributions to precession and obliquity rates. {C—A)!C =0.003 273 763 4 and obliquity 23°26'21.409" at J2000. Contribution Free, rate arcsec/yr € rate arcsec/yr Sun first order Moon first order Second order 74 Tilt effects Direct planetary Tidal Geodesic precession Total space motion Ecliptic motion General motion 15.948 870 34.457 698 -0.000 468 0.000 026 -0.002 686 0.000 318 -0.019 194 50.384 565 -0.096 865 50.287 700 -0.000 254 -0.000 014 0.000 024 -0.000 244 -0.468 096 -0.468 340 lunar plus solar effects (excluding the J2 orbit effects), and the small value for the precession induced by the Earth’s J4 gravitational harmonic. The contribution to precession and obliquity rate due to the lunar orbit tilt comes from Sec. 3 of this paper and the planetary contribution due to direct torques on the Earth’s oblateness comes from Table 2 in Sec. 4. The relativistic precession, variously called the geodesic, geodetic, and de Sitter-Folker precession, is computed from the following equation based on that in Barker & O’Connell (1970, 1975): Pg=3(na/c)2n/2(l-ez), (23) where c is the speed of light and n and a are the mean motion and semimajor axis of the orbit of the Earth-Moon system about the Sun. The convention of measuring the pre- cession constant in a left-handed sense (retrograde) results in a negative sign for the geodesic precession in Table 2. The tidal influence on obliquity rate is taken from Sec. 7. The sum of all of the above contributions gives the precession and obliquity rate with respect to space for the stated value of (C—A)/C. Conventionally the precession along the fixed ecliptic with respect to space is referred to as the “luni-solar precession” (which includes contributions from the planets as well). Clearly, it would be inappropriate to refer to the companion —0.244 mas/yr obliquity rate as luni-solar obliq- uity rate since most of it ultimately comes from planetary influences. To get the precession and obliquity rate for the moving equator with respect to the moving ecliptic plane it is neces- sary to subtract off the motion of the ecliptic plane. This is done in the last two lines of Table 3. The values for the ecliptic motion have been improved upon since the IAU theory (Lieske et al. 1977). Improved ecliptic motion and its influence on the precession expressions has been considered by Bretagnon (1982), Bretagnon & Chapront (1981), Laskar (1986), and Simon et ai (1994) and the improved motion from Simon et ai, including the correction for mass changes, has been used in Table 3. There is a problem with the no- menclature of the past. What have been called “planetary precession” and obliquity rate (Woolard uses “precession in obliquity”) come from the motion of the ecliptic. We now have two planetary contributions to each of precession and obliquity rate which are motions in space, not ecliptic mo- tion. It is conventional to refer to the (mean-of-date) motion of the dynamical equinox along the moving ecliptic plane as “general precession in longitude.” Consequently, the final line has been labeled general motion and by extension the obliquity rate might be called general obliquity rate. Both very long baseline interferometry (VLBI) and lunar laser ranging (LLR) are capable of measuring the motion of the equator with respect to space rather than the moving ecliptic. Thus both measure the luni-solar precession rate, not the general precession rate (the IAU primary constant), and have the potential to measure the —0.244 mas/yr obliquity rate with respect to space. There is weak evidence for the latter in the VLBI results (Herring et al. 1991; Steppe et al. 1993). Better precession and obliquity rate measurements can be anticipated as the VLBI and LLR data spans lengthen and separation of rates and 18.6 yr nutation becomes stron- ger. The nutation theory of Kinoshita & Souchay (1990) is a significant improvement on previous theories and it is impor- tant to understand the corresponding values of (C-A)/C, kM, and £5. In part due to misprints, three different values of (C-A)/C and two of kM have been published and it is im- portant to resolve the discrepancy. From the 1976 IAU con- stants and Kinoshita & Souchay’s numerical values and equations are calculated (C-A)/C =0.003 273 967 71, £5=3475.414 267century, £^=7547.199 697century (or 7567.769 707century with the l/Fl factor). The set of their values which has internal consistency is (C—A)/C =0.003 273 967 (Souchay & Kinoshita 1991), £s=3475.41357century, and £^=7547.198l7century (7567.768l7century with factor). The relative difference is 2.2X10-7, but is small enough to only influence the nutation at the few microarcsecond level. In Kinoshita & Souchay the (Earth+Moon)/Sun mass ratio is incorrectly labeled Earth/ Sun ratio in two places and correctly labeled in a third, but any discrepancy is too small to explain the difference. To adjust their theory to this paper’s updated precession rate and (C-A)/C with all of the corrections, multiply their nutation series by the factor 0.999 937 82. It is instructive to consider several contributions to the above factor and the proportionate (C—A)/C. The largest is the correction to the IAU precession rate causing a relative change of — 6.48X10-5. Updating the mass ratios and ob- liquity causes -8X10“7. The change in the ecliptic motion causes 9X10-7. Theoretical differences due mainly to the planetary tilt induced precession (absent in Kinoshita & Souchay) and a somewhat different geodesic precession ac- count for 2.2X10-6. By comparison, the relative uncertainty due to the present precession rate determination is about 7X10“6. The ratio of J2 = (C—A)/MR2, where R is the Earth’s equatorial radius, and (C-A)/C gives C/MR2 the normal- ized polar moment of inertia. Combining J2 from the GEM-T2 solution of Marsh et al. (1990), including a suitable addition for the model’s permanent tide, with the precession derived (C—A)/C from above yields a rigid body C/MR2=0.330 700 7. With the mean moment /=(C + 2A)/ 3, then 7/MR2=0.329 978 9. © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 717 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 717 6. THE PHASE OF THE TILT TERMS There are two reasons to consider the seemingly prosaic subject of the phase of the tilt terms used in Sec. 3. (1) It has been stated (Kinoshita 1975, 1977) that these terms do not give rise to an obliquity rate and the resolution of the differ- ence between that claim, Woolard (1953), and this paper hinges on the origin of the phase. (2) Time variations of the phase will give rise to higher derivatives of the precession and obliquity. The /2~induced tilt terms in Sec. 3 have zero phase so long as L and ÎÎ are referred to the moving equinox. They do not give rise to an obliquity rate and do not need to be considered further in this section. The planetary-induced tilt terms in the lunar orbit arise in two ways. The direct terms arise from the forces of the planets on the lunar orbit. The indirect terms arise from the force of the Sun on the lunar orbit, coupled with the motion of the ecliptic plane due to the forces of the planets changing the heliocentric orbit. The tilt from the direct effect is an order of magnitude smaller than the indirect effect. The two components have been combined in Sec. 3. The indirect contribution will dominate the follow- ing discussion. We wish to express the secular motion of the lunar orbit plane acted upon by the Sun. The coordinates of the pole of the variable lunar orbit plane are {Pv,-Qv,cos iv) where Py^siniysinily and ßy=sincos fly. The analogous time-varying variables for the ecliptic pole are Pf and Q'. Using an inertial frame aligned with the ecliptic and equinox at the initial time, e.g., at J2000 P' = g' = 0, a good approxi- mation for the differential equations for the secular motion is (see chapters 12 and 16 of Brouwer & Clemence 1961) dPy/dt = Ù0(Qv-Q'), dQy/dt=-Ù0(Py-Pf). (24) In the first approximation il0 is a quantity which is propor- tional to the mass ratio Sun/(Earth+Moon), the lunar mean motion, and the cube of the ratio of the lunar to heliocentric semimajor axes (a/a')3. When the ratio of ecliptic motion to il0 is small, it is 1CT5 for the Moon, a good approximation for the solution of the differential equations is Pv=P + P'-Q'/Qo, Qv=Q + Q,+P,/Úq. (25) P=sin i sin il0 and g=sin i cos no represent a uniformly processing lunar orbit plane with rate ÍÍ0 (retrograde 18.6 yr period) and fixed inclination i, which the additional solar perturbation terms modify. Py—P' and Qy—Qr are good approximations for the motion of the lunar orbit pole with respect to the moving ecliptic pole (Pf and ß' are functions of time). In the above solution the orbit is precessing along a plane tilted slightly with respect to the moving ecliptic with the tilt given by the last terms on the right-hand side. At J2000 ß7ii0= 1-386" and P’/Ù0=-0.124", so the indirect term causes a 1.39" tilt with an orientation governed by the node about which the ecliptic plane is rotating (II=174.87° at J2000). The tilt of the lunar orbit plane to the moving ecliptic can be expressed as a perturbation in the lunar latitude Aß=(ß7ii0)cos L0 + (P7iio)sin L0, (26) where the mean longitude L0 is given in terms of the mean argument of latitude and node (L0 = F-fiî0). Note that the differential Eqs. (24) and their solution (25) are written in an inertial coordinate system. While the differences Pv-P' and Qy—Q' are useful for seeing that the lunar orbit nearly fol- lows the ecliptic, the node fl0 is a quantity referenced to the equinox at the initial time (J2000) and its retrograde rate is with respect to inertial space. The subscript zero has been used to distinguish L0 and fl0 from L and il which in con- ventional lunar theory are measured from the moving (mean of date) equinox along the moving (mean) ecliptic. The A/3 equation can be put in the form of an amplitude times sin(L04-phase) where the phase is given by tan-1(ß7P')> which at J2000 is 95.13° or 270°-n. At J2000 this phase rate measured along the moving ecliptic plane is -17.36 7yr, twice the rate of II measured along the fixed J2000 ecliptic. To put A/3 in the form of Eq. (11) with sin(L +phase), with L measured from the moving (mean of date) equinox along the moving ecliptic, then for compatibility the phase must be measured from the moving equinox, along the moving ecliptic, to the node of rotation of the ecliptic on the moving ecliptic. In the notation of the IAU precession paper this phase is 210°—I[{T,0). The phase rate is then the general precession rate minus 17.36 or 32.93"/yr (this rate is wrong in Brown’s lunar theory). The smaller direct contribution has its own phase which depends on the difference between the moving equinox and the planetary nodes, so in Eq. (11) the phase of the combined direct and indirect terms is slightly larger than 95.13° and the amplitude is slightly larger than 1.39". Woolard (1953) earlier computed the obliquity rate con- tribution from the planetary-induced tilt in the lunar orbit. When explaining it (pp. 127-128) he used the (direct plus indirect) A/3 contribution from Brown’s lunar theory which is equivalent to Eq. (11). Broken into sine and cosine com- ponents of F + il=L, only the cosine component [he gave it as 1.53" cos(F+fl)] was displayed because the sine does not give rise to a secular obliquity rate. If the argument was measured with respect to the moving equinox, as Í1 and L conventionally are in lunar theory, and if the coefficient was constant, then there would be no obliquity rate. For this rea- son Kinoshita (1975, 1977) argued against Woolard’s ob- liquity rate. For the indirect contribution (there is a parallel argument for the direct contribution), Eq. (26) and its discus- sion show that the A/S expression could have been written in terms of cos[L —IliTjO)] or in the form of Eq. (26) using L0. To write it using L requires introducing a function of IliTjO), and II(r,0) depends on the moving equinox. Wool- ard was only computing the linear time term (his discussion shows that he was aware of the higher powers of time). His notation was numerically suitable for the epoch [like Eq. (11) of this paper], but if taken literally it is functionally mislead- ing because the equinox dependence is hidden in the numeri- cal coefficient [in Eq. (11) it is in the phase]. The discussion in the appendix of Kinoshita (1977) was unaware that A/3 © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 718 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 718 had a functional dependence on the equinox which was un- displayed in Woolard. Kinoshita’s discussion depended on a finite partial derivative of À/3 with respect to the equinox, but the equinox dependence given above causes the partial to be zero. It is concluded here that the obliquity rate from the indirect (and direct) tilt is real and that Woolard’s numerical value was reasonable. There is an anomaly that I do not understand. The com- parison by Souchay & Kinoshita (1991) of their theory with a numerical integration showed as discrepancies neither the obliquity rate term nor the out-of-phase 18.6 yr nutation terms which arise from the same source. Since the indirect tilt terms depend on the time-varying P ' and Q ' and the two direct planetary effects (direct tilt and direct torque on the Earth) depend on the time-varying plan- etary orbits, it should be understood that all three of these contributions to the obliquity rate are not constant. Over long time scales (>10 000 yr) the obliquity exhibits quasiperiodic variations and the rate will show both signs. By contrast, the small obliquity rate due to tidal dissipation (Sec. 7) is always positive. Both the precession and obliquity rates arising from the indirect tilt will vary with time. The time dependence of the P' and Q' derivatives in Eq. (26) can be used with Eqs. (12) and (13) to compute the precession and obliquity rates as a power series in time. For use in Sec. 8 it is convenient to express these rates in a coordinate system moving with the ecliptic and equinox. In the two-time (7V) power series of Lieske et al (1977) or Simon et al (1994) differentiate P' and Q' with respect to t, set i=0, and use T as the time variable. The series for the indirect contribution then be- comes —0.003 92+0.000 703 T for precession rate (units " and centuries) and — 0.0233+3 X10-6 T2 for obliquity rate. The computation of the nonlinear contributions is similar for the direct planetary torques on the Earth and the direct plan- etary tilt on the lunar orbit. In Eqs. (22) the dependence of the direct torque effects on the planetary P’s and g’s is ex- plicit and power series for the P’s and g’s (Laskar 1986; Simon et al. 1994) can be used. Venus dominates the accel- eration and the result is -17X10-6 "/centuries2 for preces- sion and 2X10~6 "/centuries2 for obliquity. The acceleration due to the direct tilt terms is more difficult, but is estimated to be about 40% larger than for the direct torque effect. Be- cause the polynomials for the planets used a fixed equinox, to these figures must be added the larger accelerations which result from transforming from a fixed to a moving equinox: 411X10-6 "/centuries2 for precession and -48X10-6 "/centuries2 for obliquity. The total of the preceding direct and indirect tilt terms and the direct torques on Earth are listed under “planetary tilt and direct torque” in Table 4. Note that part (0.032 69 "/century) of the direct torque effect for precession in Eq. (22) does not depend on planetary Ps and Qs, contributes no accelerations, and is included with the entry for luni-solar precession in the table. The coefficients of the planetary-tilt-induced nutation (Table 1) will also have secular changes. Assuming that the secular changes in the AL coefficients scale in proportion to those of the latitude coefficient, then the in-phase contribu- tions are (-0.0301+0.0050L) sind to the longitude nuta- Table 4. Time and obliquity dependence of precession and obliquity rates ("/century) which are needed to calculate the evolution of precession and obliquity with time. Rate in "/century e Dependence Luni-solar, direct planetary torque Geodesic precession Second order (M3) Second order J4 precession J2 tilt Planetary tilt and direct torque 1 ides on lunar orbit Fides on spin and moments ' 2 rate Precession P0 cos £0 - 0.003395 t - éxlO“6 t2 -1.919362 + 2.7x1o-61 -0.03310 -0.01368 +0.00260 -0.2630 -0.00643 + 0.0010741 -0.0001021 -0.000133 t -0.0140 t COS £ 1 6 cos2 e - 1 3 cos2 £ - 1 cos £ (4-7 sin2 e) cos 2e / sin £ cos 2e / sin £ cos2 £ cos3e COS £ Planetary tilt and direct torque Tides Obliquity -0.0268 - 0.0000441 + 3x 10“612 cos £ +0.0024 sinE cose tion and (0.0029-0.0005L) cos Í1 to the latitude nutation (units mas and centuries). The relative changes of the out-of- phase coefficients is about 10_3/century and is ignorable. 7. TIDAL AND NONLINEAR EFFECTS This section considers effects which cause accelerations and higher derivatives in the accumulated precession (inte- gral of precession rate), and another contribution to obliquity rate. There are effects due to the change in the eccentricity of the orbit of the Earth-Moon system about the Sun which have been considered in previous theories, plus tidal effects in the lunar orbit and Earth rotation, and possible changes in (C-A)fC. Many of the results of this section can be derived from Eq. (7) and its lunar counterpart. Moving toward poly- nomial expressions for the precession quantities as in Lieske et al. (1977), the units of that paper are now adopted (arc- seconds and centuries). The results of this section are sum- marized in Table 4. That table also gives the e dependence since the change of obliquity contributes additional accelera- tions and higher derivatives which will be utilized while solving the differential equations for orientation in the next section. As seen from Eq. (7), the eccentricity of the heliocentric orbit enters into the solar-induced precession of the equator and changes in the eccentricity will affect the derivatives of the precession rate. The evaluation of the eccentricity- induced acceleration (first derivative of the precession rate) in the accumulated precession in the IAU theory dates to de Sitter & Brouwer (1938). It is re-evaluated here at J2000. Using the eccentricity polynomials in Laskar (1986) or Simon et al. (1994), ¿/e/¿/í = -42.0Xl0_6/centuries. This causes the solar-induced precession rate to have a first de- rivative of -3362X10-6 "/centuries2. The eccentricity of the heliocentric orbit also enters into the geodesic precession Eq. (23). The derivative of that rate (in retrograde sense) is 2.7XlO-6 "/centuries2. The lunar orbit includes perturbations by the Sun and those which depend on the heliocentric ec- centricity contribute accelerations. These periodic terms in the lunar latitude and distance weakly influence the lunar- © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 719 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 719 induced precession through their squares. The fractional in- fluence on the precession rate is 1.97X10-6 by these radial terms and -5X10-8 by these latitude terms, yielding -33.3X10-6 "/centuries2 or 1.0% of the solar-induced accel- eration. The luni-solar acceleration is — 3395X10-6 "/centuries2 exclusive of the contribution of the geodesic pre- cession. There is also a small (higher derivative) contribution due to the second derivative of e2. The t and t2 terms in the precession rate (without geodesic precession) due to helio- centric eccentricity changes are -3395X10-6 "/centuries2 Í-6X10-6 "/centuries3 t2. To convert the t coefficient to the Px parameter of the I AU theory divide by cos 60 to get -0.003 70 "/centuries2. The agreement with the -0.003 69 "/centuries2 used in the IAU theory is excellent, aided by the small t2 term. Tides are raised on the Earth by the Moon and Sun and their energy dissipation causes the Moon to recede and the Earth’s rotation to slow. Lunar laser analyses indicate a secu- lar acceleration of —26.0 "/centuries2 and a tidal (semimajor axis) recession of 3.84 cm/yr (Williams et al 1993) so da/di/a = 1.00X10~8/centuries. The l/<z3 dependence of pre- cession in Eq. (7) implies the precession changes by — 103X10-6 "/centuries2. The tidal changes in the lunar orbit eccentricity e and inclination i are small (Chapront-Touzé & Chapront 1983, 1988) and lead to only 0.9 and 0.2X10-6 "/centuries2, respectively, in the precession. Equation (7) de- pends on the obliquity e, the angular momentum Ccùz, and the moment difference (C—A) which exhibit secular changes due to tidal effects. An angular momentum balance between the Earth’s spin and the lunar orbit for long time scales gives an estimate for secular changes in these quanti- ties. Writing the angular momentum components perpendicu- lar and parallel to the ecliptic plane Hz = C(oz cos €+Mm[G(M+m)a(l-e2)]1/2 Xcos //(M+m), Hy = Co)z sin 6, (27) where m and M are the masses of the Moon and Earth and G the gravitational constant. Differentiating both equations for secular changes, conserving angular momentum, and com- bining gives d€/dt = [m/(M+m)](n/(i)z)(MR2/C)(a/R)2 sin e X(1 — e2)1/2 cos i[da/dt/2a-ede/dt/(l-e2) — tan i di/dt] (28) d(Cü)2)/dt/Cù>z= —cot e deldt, where n is the lunar mean motion and R the Earth’s radius. Evaluating with the tidal changes in the lunar orbit (domi- nated by da/dt) gives de/df = 19.6X10-4 "/centuries and d(Ccu2)A/f/Ccu7=—2.20X10-8/centuries. The latter causes 110X10-6 "/centuries2 in the precession. Both C—A and the deviation of C from the mean moment depend on the square of the Earth’s rotation rate. Thus the value of dtojdt! ct)2:=: 2.19X 10_8/centuries, 0.44% less than d{Cù)z)ldtlC(x)z. The change in C—A causes a precession change of —220X10“6 "/centuries2. This tidal despinning of the Earth by the Moon causes changes in both lunar- and solar-induced precession. The solar tides also act to despin the Earth. The solar torque is much less well known than the lunar. It is a common approximation to assume that the ratio of solar to lunar torques is proportional to the square of the ratio of tide heights (0.46), though there is some evidence for a smaller torque ratio (Brosche & Wunsch 1990). Here the factor 1.21 is used to amend the lunar cal- culations for the solar contribution: the tidal obliquity rate is 24X10“4 "/centuries and the tidal precession change is (-102-1.21 X HO) X10-6 "/centuries2= -235 X 10~6 "/centuries2. A related, but not identical, calculation of the obliquity rate by Kaula (1964), when adjusted for recent secular acceleration measurements and the solar contribu- tion, gives an obliquity rate of 17X10~4 "/centuries. There are a host of nontidal processes which change the spin rate of the Earth by exchanging angular momentum be- tween the liquid core, solid mantle plus crust, oceans, and atmosphere, but these leave Ca)z unaffected. However, some of these processes do affect the precession through changes in C—A. The Earth’s gravitational harmonic J2 is propor- tional to C—A and exhibits a small secular decrease which has been detected from the analyses of ranges to the Lageos and Starlette satellites (Yoder et al 1983; Rubincam 1984; Cheng et al. 1989; Gegout & Cazenave 1991; Watkins & Eanes 1993; Nerem et al 1993). Consequently the preces- sion rate should also exhibit a decrease. The J2 rates from these studies lie in the range of (-2.5 to —3.6)XlO~9/centuries; they induce a sizable precession rate change in the range of (—11.6 to — 16.8)X10-3 "/centuries2. This is about 0.7% of the —2 "/centuries2 classical accelera- tion induced by ecliptic motion (next section) and two orders of magnitude larger than tidally induced accelerations. Though seeming to vary on thousand year time scales, the nontidal acceleration of the Earth’s spin (Stephenson & Mor- rison 1984, 1985) appears to be in accord with the reported J2 rates. While the J2 rate is clearly visible in satellite track- ing data taken since 1976, that rate is imperfectly separated from the 74 rate and there appear to be rate irregularities and questions about the separation of A8.6 yr tidal signatures which limit knowledge of the long-time average (Watkins & Eanes 1993). For Table 4 a J2 rate °f —3XlO_9/centuries has been adopted; this yields —0.014 "/centuries2 in precession. This choice will give a precession acceleration valid since 1976, but future extrapolation is less certain. The J2 rate uncertainty is the largest recognized uncertainty in the accel- eration of precession. The precession is only sensitive to long-time changes in J2 and the appropriate contribution to the precession will depend upon further monitoring of J2 changes with artificial satellites. 8. POLYNOMIALS The IAU precession paper (Lieske et al 1977) gives poly- nomials in time for orienting the Earth based on the IAU- adopted general precession rate, obliquity, and masses. Equivalent matrix formulations have also been published (Lieske 1979; Fabri 1980). Improved ecliptic motion led to polynomial updates by Bretagnon & Chapront (1981), © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 720 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 720 Fig. 1. Relation between the fixed equator (mean equator) and fixed ecliptic of J2000 and the moving (mean of date) equator and ecliptic. The arc from the moving equinox to the node of the moving ecliptic on the fixed ecliptic is = FQ = o-¿ + . The subscript A is not used with symbols in the figure. Laskar (1986), and Simon et al. (1994). The latter paper in- cludes improved precession rate, obliquity, and masses as well. In this paper theoretical contributions to precession and obliquity rates and higher derivatives have been identified and computed. In this section the theoretical improvements plus updated values for precession rate, obliquity, masses, and ecliptic motion will be used to generate revised polyno- mial expressions. The notation of the IAU paper is used in this section except that the tilde has been dropped. The subscript A (for accumulated) denotes an angle. Thus ijjA and pA are accumu- lated (integrated) luni-solar and general precession rates, re- spectively. Rates will be indicated with derivatives. The polynomial expressions will be derived for a single time ar- gument for use with the J2000 epoch. The fixed ecliptic and equator planes of J2000 and the moving ecliptic and equator of date constitute the basic geometry. See the IAU paper and Fig. 1 for the definition of the variables. The basic differential equations were given in the IAU paper but they require additional terms due to the obliquity rate contributions with respect to inertial space. The obliq- uity and precession rates of Table 4 use a coordinate frame which is moving with the equinox. The total contribution of the obliquity rate (no ecliptic rate) from Table 4 is denoted R € and the total contribution to the precession rate multiplied by sin eA is denoted These two components of the equa- tor’s rotation vector are in the plane of the moving equator. Two of the differential equations are just the projections of these two rates through the angle (“planetary preces- sion”) between the moving equinox and the intersection of the fixed ecliptic and moving equator. da)A/dt = cos Y/^e+sin XaRij,, (29) sin a)Adi//A/dt = cos sin XaR€- The differential equation for the obliquity rate with respect to the moving ecliptic involves both the motion of the ecliptic and Re d€A/dt —cos pAdQ'/dt —sin pAdPr/dt + (1 —cos 7rA)cos(IlA+pA)d7rA/dt + R€, (30) where P'=sin tta sin UA and g'=sin irA cos UA describe the ecliptic pole with UA and irA being the node and inclina- tion, respectively, of the moving ecliptic on the fixed ecliptic. In addition to P' and Q ', which are input functions for the ecliptic motion, the right-hand sides of the differential equa- tions are functions of Xa and Pa • Two geometrical equations are needed to link these latter two variables with the others sin Y^ = sin tta sin(IIi4+/?y4)/sin oja , (31) COS(V\A+pA) = COS ^ COS(nA + >llA) + sin xa sin(IIA+ i/^cos • The three differential equations and the two geometrical equations must be solved simultaneously for (x)A, if/A, eA, Xa , and pA . The simultaneous solution was performed with a numeri- cal technique. The variables are represented with polynomi- als of time. The five equations are evaluated at equal time intervals, the polynomial coefficients are fit, and the proce- dure is iterated to convergence. Input quantities are the J2000 obliquity (e0) and general precession rate plus the polynomi- als for P ' and Q '. The constant P0 in Table 4 is determined from the initial obliquity and precession rate and the other precession rates in the table. At J2000 the rates of general and luni-solar precession are linked through the rate of the planetary precession xa projected on the ecliptic plane. The iterative solution for the polynomials was done in extended precision on a microcomputer. Parameters such as the commonly used ÇA, 0A, and zA are derived geometrically from the above set of variables. Nu- merical polynomial fits were also used. Two points are noted for generating the polynomials for ÇA and zA . The polyno- mial for o)A is carried to one higher degree than those for ÇA and zA. It is necessary to include a constant in ÇA and zA when the zero coefficient of t in a)A is finite. The constant is d(oA/dt/sin €0di//A/dt evaluated at J2000 and has opposite signs for and zA . The computer program was tested against the IAU expres- sions. For the IAU theory, three terms are used which have the form, but different numerical values, of those on the first two lines for precession rate in Table 4: the P0 and t terms on the first line and a constant geodesic precession rate. With the input values used in the IAU theory, the program was able to successfully reproduce the polynomials in the IAU paper with a deviation of no more than 1 in that paper’s last digit except for IIA . As also experienced by Fabri (1980), in the IAU paper the number of digits given for fl^ exceeds those given for P' and Q’ and apparently additional digits in P ' and Q ' were used there. © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 721 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 721 Table 5. Polynomial expressions for orientation of the Earth’s equator (arcsec). Time t in Julian centuries from J2000 (JD 2451545.0). Greenwich mean sidereal time (s) at 0 h UT1. Time in UT1 centuries from 12 h UT1, JD 2451545. Angle Constant t2 t* P' Q’ ka nA Pa Va C0A Xa £a Ça zA eA Ça £â ru GMS TO 0 0 0 629543 0 0 84381 0 84381 2 -2 0 0 84381 0 24110 000000 000000 000000 967373 000000 000000 409000 000000 409000 511180 511180 000000 000000 409000 000000 4, -46. 46. -867. 5028. 5038. -0. 10. -46. 2306. 2306. 2004. 10. -46. 5038. .54841 8640184 199610 809560 997570 919986 770000 456501 024400 557700 833960 071060 065079 182023 557700 809560 456501 7928613 0.193971 0.051043 -0.033506 0.153382 1.105407 -1.078977 0.051268 -2.381366 -0.000174 0.299027 1.092516 -0.429466 0.493164 0.051142 1.558353 0.0927695 -0.000223 0.000522 .000124 .000026 .000076 .001141 .007727 ,001208 .002000 0.018017 0.018265 -0.041822 -0.000309 0.000531 -0.000186 -0.0000003 -0.000001 -0.000001 0.000000 -0.000004 -0.000024 0.000133 0.000000 0.000170 -0.000001 -0.000005 -0.000029 -0.000007 -0.000003 0.000000 -0.000027 -0.0000020 The expression for Greenwich Mean Sidereal Time (GMST) relates UT1 to the angle between the mean equinox of date and the Greenwich meridian. This paper will follow the convention that the GMST expression is a solution of the dynamical equations for rotation. The IAU-adopted polyno- mial expression for GMST given by Aoki et ai (1982) is specific to the IAU precession theory. As pointed out by Williams & Melbourne (1982) and Zhu & Mueller (1983), changing the precession expressions without changing the GMST expression would alter the determination of UT1 from observations. Consequently, an additional equation has been evaluated numerically. The fundamental parameter is the rotation rate of a rigid Earth with respect to inertial space about its symmetry axis ¿/(GMST+^)/<ii —cos o)Adil/Aldt (32) with due consideration for the units. The nonlinear parts of the GMST expression at zero hour UT, GMST=GMST0 -bUTl, come from cos (x>Adil/A/dt dt — XA (33) dividing arcseconds by 15 to convert to seconds. The coeffi- cients of the constant and linear terms were set by imposing the condition that at J2000 there would not be a discontinuity of UT1 (the constant coefficient matches Aoki et al), its rate, or the rotation rate of the Earth in space (there are small ambiguities at the level of truncated digits). In Aoki et al the constant and linear coefficients were picked for continuity of UT1 determined from optical astrometric measurements of catalogue stars rather than continuity with respect to an in- ertial frame. Inertially referenced techniques now dominate the determination of UT1 so there is no counterpart to the catalogue equinox drift. It is conventional to derive the small nonlinear terms of GMST0 using a linear time scale for the independent time, but to evaluate the entire GMST0 expres- sion using a UT1 time scale. Since the IAU theory for precession appeared, there have been improvements in the computation of the motion of the ecliptic due to theoretical advances and improved planetary masses, better measured values for precession rate and ob- liquity, and the theoretical computations of Kinoshita & Souchay (1990) and this paper. To illustrate the resulting changes, revised expressions are presented here. The input values match those used to generate Table 3 (Sec. 5). The ecliptic motion is taken from Simon et al (1994) including planetary mass corrections. The theoretical contributions of Table 4 have been used. The resulting expressions are given in Table 5. The units are arcseconds and Julian centuries measured from J2000 [i = (JD-2451545.0)/36525], except for GMST which uses seconds and centuries of UT1 measured from JD 2451545.0 UT1 = 12 h. UT1 on January 1, 2000. While these expressions can serve those who need the high- est accuracy now, it should be anticipated that there will be future improvements: some theoretical, certainly in the mea- surement of the precession constant and obliquity, and hope- fully in the predictive knowledge of the J2 rate. The polynomial expressions in Table 5 can be used for times extending out to a few millennia, but are not suitable for longer times. The polynomials are equivalent to expan- sions of expressions appropriate for longer times: an average precession rate and obliquity plus long-periodic, or at least quasiperiodic, terms with periods exceeding 10 000 yr (Berger 1976; Laskar et al 1993). For millions of years the small tidal acceleration is inexorable and modifies the pre- cession and obliquity behavior for ancient times (Berger et al 1992). Nontidal J2 change must be transient. Most of © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 722 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 722 the Earth’s oblateness is caused by its spin and fluctuations of the active Earth’s J2 from the spin-controlled equilibrium will damp down. 9. ROTATIONS Considered in this section are sequences of rotations suit- able for the various sets of precession parameters in Table 5. When combining the rotations for precession and nutation, the number of rotations can be minimized. Finally, an expe- dient procedure is given which is suitable for introducing the most important (linear) corrections to precession and ob- liquity without undertaking the more extensive and complete modifications. Consider the sequence of rotations which can be used to orient the precessing and nutating Earth. The standard pro- cedure is to precess from the mean equator and equinox of J2000 to the mean equator and equinox of date using the angles £4, , and zA and then to nutate to the true equator and equinox of date by rotating into the mean ecliptic of date, applying nutation in longitude to reach the true equinox of date, and then to rotate to the true equatorial plane includ- ing nutation in obliquity. The sequence of six rotations (Æ, is the rotation matrix around axis Ï) is X/?^)/^0-^). (34) An alternative is to precess by moving along the fixed eclip- tic to the intersection with the mean equator of date, then rotate along that equator to the mean equinox of date, and then to nutate as before. The seven rotations are Æ ! ( - - A e)i?3( - A 1 ( (Xa 1 ( - "a ) X7?3(-,AA)«i(eo)- (35) A second alternative is to precess by moving along the fixed ecliptic to the intersection with the mean ecliptic of date, rotate back along the mean ecliptic, and then nutate flli-eA-AiO^-nA-pA-A^l^A) xR3(Ha)Ri(€q). (36) The third procedure requires only five rotations. If one is to nutate as well as precess, it is possible to bypass the mean equator of date and combine the precession and nutation along the moving ecliptic. Five rotations is not the minimum for combining precession and nutation. With four rotations one can move along the fixed equator to the intersection with the moving ecliptic (angle £4), rotate into the moving ecliptic (64), combine the rotation along the ecliptic of date (r}A to the mean equinox of date) and nutation in longitude, and rotate to the true equator Rl{-eA-^e)Ri(<-7)A-^ili)Rl{e'A)R3{^A). (37) The angles £4, eA, and r]A have not been given with con- ventional precession expressions in the past. They are illus- trated in Fig. 1 and the expressions are given in Table 5. Note that for the changes in precession and obliquity polynomials not due to ecliptic motion, only the last two (left) rotations in the last two sequences above would be changed; for the first two sequences the precession and obliquity changes are distributed over multiple rotations. The last two sequences make it clear that a change to the preces- sion and obliquity rates could be added into the correspond- ing nutation parameters as an alternative way to introduce them (a similar conclusion was reached by Folkner et al 1994 and VLBI fits to observations have often included lin- ear terms in their “nutation” corrections). Adding -0.02447centuries t to Ae and -0.32197centuries t to A'F (t in centuries from J2000) is an expedient way to incorpo- rate the most important corrections without reprogramming the precession and GMST expressions. For this expedient approach the equation of equinoxes associated with A^ will automatically satisfy the concerns of Williams & Melbourne (1982) and Zhu & Mueller (1983) about precession modifi- cations changing UT1, so that the GMSTO polynomial does not require revision. Neither does the J20Ö0 value of the obliquity need changing for the expedient approach since the difference tends to cancel between the rotations. Other theo- retical contributions, such as the nonlinear corrections, could also be added as nutation corrections. The expedient proce- dure does not work for geometrical revisions to ecliptic mo- tion (purely P’, g', IÏ4, and 774 but also parts of other parameters) since they appear in multiple rotations and tend to cancel, but it could be applied to the dynamical conse- quences of those revisions. However, it is observed that if an expedient procedure becomes too complicated then it is not expedient. It is best suited to easily inserting the linear cor- rections to precession and obliquity while more thoroughgo- ing revisions can use Table 5’s polynomial expressions for precession and GMSTO. 10. NUTATION CORRECTIONS, SCALING, AND COMPARISONS The 1980 IAU nutation series (Seidelmann 1982) was a combination of the rigid-body series of Kinoshita (1977) and the elastic and structured-Earth corrections due to Wahr (1979, 1981). Since the 1980 IAU nutation working group paper there have been two revisions of rigid-body nutation (Zhu & Groten 1989; Kinoshita & Souchay 1990). The nu- tation theories allowing for the Earth’s elasticity and core are based on the rigid-body theories and it is well to compare and understand those rigid-body theories. Zhu & Groten utilized the earlier work of Kinoshita (1977) extending the nutation series to smaller terms, and adding both second-order terms and corrections for the Earth’s J3. It has served as the basis for nonrigid body treat- ments by Zhu et ai (1990), and the several ZMOA series of Herring (1990), Mathews et al. (1991), and Herring et al. (1991). Kinoshita & Souchay also extended the series to smaller terms, and added second-order terms, J3 effects, and planetary terms involving planetary arguments. In addition they added small solar terms due to the offset of the Earth from the center of mass of the Earth-Moon system and re- vised the expression for the (C-A)/C scaling of the nuta- tion series from the precession constant. In Kinoshita (1977) and Kinoshita & Souchay the J2 tilt effects on the scaling and 18.6 yr terms are present. The in-phase 18.6 yr nutation © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 723 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 723 coefficients and the —0.00567centuries precession due to the planetary tilt effect is present in the former, but not the latter work. Kinoshita & Souchay also add — 0.0147centuries second-order contributions to the precession and make small revisions to the first-order contributions which are not present in the earlier works. Thus there are small differences in the scaling of Kinoshita & Souchay, Zhu & Groten, and this paper which are addressed below. Only Kinoshita & Souchay have the small center-of-mass offset corrections, three of which have amplitudes of 0.02 mas in longitude. Comparison of the J3 contributions in the two papers shows poor agreement; the 3231 day obliquity coefficients have dif- ferent signs and differ by 0.12 mas. In addition, the 6164 day coefficients disagree by a factor of two. See Souchay (1993) for further comparisons. The coefficients of Kinoshita & Souchay’s rigid-body nu- tation theory would have to be multiplied by 0.999 937 82 (Sec. 5) to match the precession rate and other changes of this paper. This would cause the 18.6 yr coefficient to increase by 1.075 mas and the 18.6 yr At coefficient to change by —0.574 mas. These corrections are in addition to those of Table 1, and taken together the in-phase corrections to the 18.6 yr coefficients are 1.045 mas in A(/f (giving -17.280 76") and -0.571 mas in A* (9.228 00"). To match the constants of this paper the coefficients of Zhu & Groten’s rigid-body nutation series need to be multi- plied by 0.999 930 8 for the lunar terms and 0.999 929 7 for the solar terms. For the in-phase 18.6 yr coefficients this gives —17.280 75" in A^ and 9.227 92" in At. Thus after correction to a common (C—A)IC and compensation for the planetary tilt effect, the 18.6 yr terms of Zhu & Groten and Kinoshita & Souchay differ by only 0.01 mas in longitude and 0.08 mas in obliquity. To all nutation series since Woolard (1953) the out-of- phase planetary tilt contributions of Table 1 need to be added. To all nutation series since Woolard, a —0.15 mas annual term from the yearly variation of the geodesic preces- sion needs to be added to the nutation in longitude (Voinov 1988; Gill et al 1989; Fukushima 1991). For highest accu- racy, nutation terms with planetary arguments, such as those of Kinoshita & Souchay, should also be included. While it causes minor changes in the resulting nutation series evalu- ation, the arguments of the 1980 IAU series and other series can be improved upon by using the values of Simon et al (1994). The annual argument (/') differs by 5" at J2000, but the values of V and L ' depend very much upon which long- period terms are being carried when these arguments are fit to the time-varying heliocentric orbit. It should be compat- ible with the formulation used to generate the nutations with planetary arguments. 11. SUMMARY Improvements in the accuracy of the observed motion of the Earth’s equator plane and the wish to use these observa- tions to infer the Earth’s properties make improvements in the theories of precession, obliquity rate, and nutation desir- able. The rate terms computed in this paper come from lunar orbit perturbations due to the planets and the Earth’s J2 plus direct planetary torques on the Earth and tidal effects. The corrections to the obliquity rate are due to direct plan- etary torques on the Earth (see Sec. 4 and Table 2), torques due to planetary perturbations on the lunar orbit (Sec. 3), and tidal influences (Sec. 7, Table 4). Together these corrections are —0.244 mas/yr. This correction is a motion in space; the conventional —0.468"/yr obliquity rate is due solely to eclip- tic motion, not to changes in the Earth’s orientation. The LAU-adopted theory of precession and obliquity changes re- quires correction for this contribution to the obliquity rate. The largest contribution to the obliquity rate in space was earlier computed by Woolard (1953), but its reality was ques- tioned by Kinoshita (1977). Section 6 discusses the reason for this discordant interpretation and concludes that the rate is real. The obliquity motion in space should be observable by the very long baseline interferometry and lunar laser rang- ing techniques. In addition to the obliquity rate amendments, there are small contributions to the precession rate due to direct plan- etary torques and lunar orbit effects (Secs. 3 and 4, Table 2). The sum of the various contributions to obliquity and preces- sion rates is given in Table 3 (Sec. 5). Based on recent mea- surements a general precession rate of 50.2877"/yr at J2000 was adopted. For a rigid Earth this corresponds to the moment-of-inertia combination (C—A)/C =0.003 273 763 4. Combined with a satellite-determined J2 this gives a normalized polar moment of inertia C/MR2=0.330 700 7 and a normalized mean moment //Mi?2=0.329 978 9, with R the equatorial radius. The contributions to obliquity and precession rates are not constant with time and the higher derivatives from these and other sources are computed in Sec. 7. Table 4 summarizes both linear and nonlinear (in time) contributions. The largest nonlinear correction arises from the Earth’s J2 rate. The theory for orienting the Earth (precession, obliquity changes, and Greenwich Mean Sidereal Time) is considered in Sec. 8 and revised polynomial expressions are presented (Table 5). In addition to the theoretical corrections of this paper, these expressions use improved values of the obliquity, precession rate, masses, and ecliptic motion. Matrix rotations which combine precession and nutation are considered in Sec. 9. The conventional rotation scheme is not optimized for the number of rotations. A sequence of four rotations is given which incorporates both precession and nutation. The torques, due to lunar orbit perturbations from the planets, also give rise to nutation contributions (Sec. 3 and Table 1). The largest contributions are to the 18.6 yr nuta- tion: -0.030 sin Í1+0.137 cos Í1 to A*/' (in mas) and -0.028 sin Í1+0.003 cos Í1 to Ae. The small out-of-phase corrections arise because the perturbing planets’ nodes on the ecliptic are not aligned with the dynamical equinox. Out-of- phase nutations are conventionally considered to arise from energy dissipation in the Earth and oceans, but these are exceptions. The torques which cause precession and nutation depend on {C—A)!C so that an accurate determination of the pre- cession rate sets the scale of the nutation series. This scaling © American Astronomical Society • Provided by the NASA Astrophysics Data System 1994AJ 108. . 71 IW 724 J. G. WILLIAMS: EARTH’S OBLIQUITY RATE, PRECESSION, AND NUTATION 724 of the two most recent rigid-body theories is discussed in Sec. 10. Also discussed are the additions appropriate to each of these nutation theories. Since the IAU expressions for precession and nutation were adopted, both theoretical improvements and refined measurements have become available. The theoretical contri- butions of this paper may be added to revised computations of ecliptic motion, rigid-body nutation, dissipative effects in the Earth’s interior and oceans, and relativistic effects. Im- proved measurements of the precession constant and indi- vidual nutation terms are available. 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