MPP01, a New Solution for Planetary Perturbations in the Orbital Motion of the Moon

A&A 366, 351–358 (2001) Astronomy DOI: 10.1051/0004-6361:20000097 & c ESO 2001 Astrophysics

MPP01, a new solution for planetary perturbations in the orbital motion of the Moon

P. Bidart

DANOF – UMR 8630, Observatoire de Paris, 61 avenue de l’Observatoire, 75014 Paris, France e-mail: [email protected]

Received 18 May 2000 / Accepted 3 November 2000

Abstract. We present in this paper MPP01, a new solution for the planetary perturbations in the orbital motion of the Moon. Comparisons of the semi-analytical solution ELP2000-82B to numerical integrations of the Jet Propulsion Laboratory and to LLR observations have shown deﬁciencies resulting mainly from planetary perturbations series established twenty years ago. The publication of a new planetary solution VSOP2000 and the progresses in numerical tools have contributed to improve the precision of the MPP01 solution. Comparisons between MPP01 and the ELP2000-82B solution for planetary perturbations show diﬀerences which are mainly due to the long periodic terms of the solution.

Key words. celestial mechanics, stellar dynamics – Moon – solar system: general

1. Introduction 2. Statement of the problem

Since 1996, the Lunar Laser Ranging at CERGA 2.1. Description of ELP theory (Centre d’Etudes´ et de Recherche en Géodynamique et The ELP lunar solution comes from the theory elaborated Astronomie) provides lunar observations with a precision by (Chapront-Touzé & Chapront 1980). They took their under the centimeter level in the distance. Using a com- inspiration from Brown’s theory. The construction of this plete model including the ELP2000-82B solution for the theory is divided in two fundamental steps. lunar orbital motion, the residuals are estimated roughly In a first step, one considers the system restricted to to 3 cm in rms. the three point masses, the Sun (S), the Earth (E) and Nevertheless the main limitation in the precision in the Moon (M) called “main problem”, taking into account ELP solution results from planetary perturbations whose that the Sun’s orbit around the Earth-Moon barycenter precision is around few meters in rms. Hence, numerical is Keplerian. The solution of this problem provides us complements are introduced to improve the solution and an intermediate solution called main solution (Chapront- to allow comparisons with observations. The aim of this Touzé 1980). work is to improve the accuracy of the planetary pertur- In a second step, all the other effects are regarded as bations component in order to reduce the contribution pertubations to the main solution: Earth and Moon fig- of these complements and to enlarge the timescale valid- ures, planetary perturbations, relativity and tides. In this ity of the solution providing series expanded up to the paper, we limit ourselves to the computation of the plan- fifth power of the time. It has been possible thanks to etary perturbations in the orbital motion. the publication of a recent planetary solution VSOP2000 by (Moisson 2000). Besides, progresses in numerical tools 2.2. Main problem solution allow us to handle large series with a great precision. We refer to Chapront-Touzé (1980) in particular for the In the first part of this paper, we desbribe the main definition of the constants used. The series provided solv- features of the ELP planetary perturbations theory, and ing the main problem’s equations in spherical ecliptic co- in a second part, we present the algorithms used for the ordinates (V, U, R) take the general form: numerical computations and the methods to handle large series necessary to ephemerides computation. Finally, we X X6 w δ + (A + Bj δz0) compare our series to the ELP ones and we exhibit the 1 V i1,...,ip i1,...,ip j largest differences arising from long periodic terms, as we i1,...,ip j=1 × ··· expected using a new planetary solution. sin(i1λ1 + + ipλp +Φi1,...,ip )(1)

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20000097 352 P. Bidart: MPP01, a new solution for planetary perturbations

Moon has been built using the IERS92 set of masses; the con- W1 stants of integration are ﬁtted on the numerical integra-

l F tion DE403 of the Jet Propulsion Laboratory. It contains

D perturbations developed up to the eighth order of plane- Earth W2 tary masses. The accuracy of VSOP2000 with respect to VSOP82 has been improved roughly by a factor of ten to l’ Te -180° Sun twenty for inner planets and one hundred for the outer Origin of longitudes ones. Here we have neglected the effects of asteroids and J ω_ ’0 Pluto on the planets, and truncated the VSOP2000 series W3 γ ’ to the fifth power of the time. The planetary series take then the form: Fig. 1. Delaunay arguments with respect to wj X5 X tn C(n) sin(i λ + ···+ i λ +Φ(n) )(3) i1,...,ip 1 1 p p i1,...,ip where δV = 1 for the longitude V and δV = 0 otherwise. n=0 i1,...,ip Ai1,...,ip are numerical coefficients and Φi1,...,ip are numer- j ical phases. B are the first derivatives of Ai ,...,i (n) i1,...,ip 1 p where, as in the main problem series, C are numeri- with respect to the constants used. A first set of constants i1 ,...,ip cal coefficients and Φ(n) are numerical phases. The λ includes the arbitrary constants of the main problem: i1,...,ip j are the linear combination of the mean longitudes of the 0 n 0 0 z0 = ,z0 =Γ,z0 = E (2) eight planets: λj = Nj t+λj . Table 1 shows the integration 1 ν 2 3 constants of the planetary solution that we have used in where ν and n0 are respectively the sideral mean motion of the computation of planetary perturbations of the Moon. the Moon and of the Sun, Γ the half coefficient of the sin l term in longitude, and E the half coefficient of the sin F term in latitude. The others are e0, the solar eccentricity, 3. Equations of motion α, the ratio of the semi-major axis of the Moon over the 3.1. Reference frame Sun’s one, and the masses of the Moon mM and the Earth 0 R mE through the ratio µ = mM/(mE + mM).δzj are the We consider the reference frame 0 defined with the in- literal linear variations of the three arbitrary constants. ertial mean ecliptic J2000 and the inertial mean equinox I The component λj in the argument stands for T , γ2000. It is a fixed frame. The ELP moving reference frame the heliocentric mean longitude of the Earth-Moon R is defined as follows: 0 barycenter, $0, the longitude of the perihelion of the Earth-Moon barycenter restricted to the constant part – the reference plane is the inertial mean ecliptic of date; 1 0 – the origin of longitudes points towards the point γ0 and the three angular variables wj = wj t + wj : 2000 defined such that: w1, the mean longitude of the Moon; w2, the secular part of the longitude of the lunar Nγ0 = NγI perigee; 2000 2000 w3, the secular part of the longitude of the lunar where N is the ascending node of the inertial mean ecliptic ascending node. 0 of date on the inertial mean ecliptic J2000 (Fig. 2). γ2000 is the non-rotating origin that Brown used in his theory All these arguments are polynomials expanded up to the named departure point. The coordinates of the Moon, ex- fourth power of the time. The phases w0 are three other j pressed in the fixed reference frame are deduced from coor- arbitrary constants of the theory. Practically, in the for- dinates expressed in the moving reference frame by adding mulation of coordinates, the λ are described with the four j the accumulated precession in longitude p between J2000 Delaunay’s arguments D, F, l, l0. They are derived from A and the date. The numerical value of the precession p w , T and $0 by the relationships (Fig. 1): A j 0 that we used in our computations is given in Simon et al. ◦ D = w1 − T + 180 (1994). Each coordinate will hereafter be expressed in the frame R. F = w1 − w3

l = w1 − w2 0 − 0 3.2. Equations in R l = T $0. Let us consider the four body problem and their masses: 0 Earth (E, mE), Moon (M, mM), Sun (S, m ) and a planet 2.3. Planetary solution (P, mP). G is the Earth-Moon barycenter. We refer to the Fig. 3 for the deﬁnition of distances D and ∆. µ is the We used in our computations the new planetary solution masses ratio: µ = mM . The equations of motion in mE+mM VSOP2000 established by (Moisson 2000). This solution the Hamiltonian formulation can be written in the system P. Bidart: MPP01, a new solution for planetary perturbations 353

Table 1. Integration constants of the VSOP2000 solution ﬁtted to DE403 in (Moisson 1999). Units are radians per thousand of julian years for N 0 and radians for λ0

Planet N 0 λ0 k0 h0 q0 p0 Mercury 26087.9031405997 4.40260863422 .04466063577 .20072330414 .04061565385 .04563549294 Venus 10213.2855473855 3.17613445715 −.00449281998 .00506684832 .00682411318 .02882282404 EMB 6283.0758504457 1.75346994632 −.00374081857 .01628448821 .00000000000 .00000000000 Mars 3340.6124347175 6.20349959869 .08536559272 −.03789970867 .01047042990 .01228448748 Jupiter 529.6909721118 .59954667809 .04698584885 .01200370563 −.00206544402 .01118380996 Saturn 213.2990797783 .87401678345 −.00296006971 .05542959194 −.00871728519 .01989130917 Uranus 74.7816656905 5.48122762581 −.04595338320 .00564804585 .00185927537 .00648605543 Neptune 38.1329181312 5.31189410499 .00599810784 .00669103056 −.01029144683 .01151685345

S inertial mean ecliptic of date ∆ 1 inertial mean ecliptique J2000

M ∆ p ∆ A γ ’ 2 2000 mean equator J2000 γ I D d γ I 1 2000 D G P N D E 2

Fig. 2. ELP reference frame Fig. 3. Schema of the four bodies system dx of rectangular coordinates x ; u = i : i i dt is the force function giving rise to direct perturbations, which are the perturbations induced by a planet act- 1 X H = u2 −F−G ing directly on the Moon. Then, the expression of the 2 i i Hamiltonian H becomes: G is the force function induced by the rotation of the ref- H = H −F −F −G (7) erence frame, and F is: c I D 1 P k(mE + mM) 0 1 1 1 1 2 −F F = + km + with Hc = i ui c(xi,t) describing the main r µ ∆ 1 − µ ∆ 2 2 1 problem. The integration method used to solve this 1 1 1 1 Hamiltonian system is the variation of arbitrary constants. + kmP + · µ D2 1 − µ D1 F It includes the main problem force function. We divide 4. Integration method into several components to distinguish the effects that we will take into account: 4.1. Integration of the Hamiltonian k(m + m ) 1 1 1 1 F = E M + km0 + (4) The method that we used has been described in c r µ ∆ 1 − µ ∆ 2 1 c (Chapront-Touzé & Chapront 1980). We briefly summa- is the force function of the main problem. Index c meaning rize the main features of this method. In rectangular co- that the Sun’s motion around the center of mass G is ordinates, the equations of motion from the main problem Keplerian. are written: k(m + m ) 1 1 1 1 F E M 0 −F dxi ∂Hc dui ∂Hc I = + km + − c (5) = , = − · (8) r µ ∆2 1 µ ∆1 dt ∂ui dt ∂xi is the force function giving rise to indirect perturbations. dx These perturbations are induced by the whole set of plan- In the main problem, u = i . Let (x∗,u∗)bethesolu- i dt i i ets on the Moon through the Earth-Moon barycenter. tion of the system (8) expressed in Fourier series like (1) 0 1 1 1 1 whose coefficients are functions of zj and arguments are FD = kmP + (6) µ D2 1 − µ D1 linear combinations of wj, as we saw in (2.2). Then, we 354 P. Bidart: MPP01, a new solution for planetary perturbations have: or if α ≥ 2 " ZZ # ∗ X3 (1) x = x (z0; w ; t) ∂w ∂P i i j j w(α) = j C−1 dt ∗ j 0 0 ∂z ∂wl ui = ui (zj ; wj ; t). i=1 i i α "Z # t 0 0 ∂P By the means of three functions zj = zj (ck) we define a − [C−1]T · (14) canonical change of coordinates: ∂z0 i j tα −→ ··· α (xi,ui) (wk,ck). The brackets [ ]tα mean the coefficient of the term in t . Then we consider a new Hamiltonian H deduced from Hc by: X 4.2. Constants adjustment H −F −F −G −P 1 2 − F P = Hc I D = Hc = ui ( c + ) 0 2 The introduction of new arbitrary constants δzj allows us i to force the solution to be fitted on the set of constants P is the disturbing function to be added to the main prob- defined for the main problem (ν, Γ,E). Therefore, we lem. The system (8) becomes: evaluate the new arbitrary constants taking into account P that the main problem’s constants must be unchanged: dwj (1) −1 T ∂ = w +[C ] (1) (1) j 0 w = w = ν dt ∂zi 1 1,c dz0 ∂P according to (2), we have: j = C−1 (9) dt ∂w (1) (1) i ∂w ν ∂w 1 = − and 1 =0 if i =2, 3 (15) × 0 0 0 0 C is a 3 3 matrix function of zj which is computed with ∂z1 z1 ∂zi the main problem’s values. The order of magnitude of the the relation (13) with j = 1 provides: function P is small enough to apply an iterative process 0 z0 ∂P for the integration. The numerical values for zj and wj δz0 = − 1 [C−1]T · (16) 0 1,c 0 given by the main solution, that we note (zj,c,wj,c), are ν ∂z1 1 t0 substituted in the function P before integration. After in- 0 To obtain the remainder δzi , we express the longitude of tegration of the three last equations of the system (9) we the Moon: obtain: Z V = V +∆V ∂P c 0 0 0 −1 X ∂V X ∂V zj = zj,c + δzj,c + C dt (10) 0 − 0 − ∂wi j = Vc + 0 (zj zj,c)+ (wj wj,c). (17) ∂zj ∂wj 0 0 j j = zj,c +∆zj,c The coefficients of the sin l-terms in V and Vc must be where δz0 are the new integration constants introduced equal, hence we have: j,c   for each disturbing function. Substituting these results " # Z X X P into the three first equations of the system (9) and con- ∂V 0  ∂V −1 ∂  (1) 0 0 δzj + 0 C dt sidering the increments of w with respect to ∆z ,we ∂z ∂z ∂wi j k j j sin l j j j get:   sin l X X3 (1)  ∂V  dw ∂w + δwj · j (1) j 0 ∂w = wj,c + 0 δzk,c j j dt ∂z sin l k=1 k Z The adjustment of the sin F -term in the latitude pro- X3 ∂w(1) P j −1 ∂ vides another linear equation. Solving this linear system + 0 C dt ∂zk ∂wl k and taking into account the relation (16) one determines k=1 P the δz0. − −1 T ∂ j [C ] 0 (11) ∂zi j 5. Algorithm for the left member computation thus after integration: 5.1. Direct perturbations X5 0 (α) α wj = wj + δwj + wj t (12) As shown in Fig. 3, D1 and D2 occurring in the expression α=1 of FD (6) can be developed (Chapront & Abu el Ata 1977) 1 where if α =1 in Legendre polynomials of the variable θP = TL·GP : rD " # X3 (1) X n ∂w ∂P kmp r (1) (1) j 0 − −1 T F = c P (θ ) (18) wj = wj,c + 0 δzi,c [C ] 0 (13) D n n P ∂z ∂z D ≥ D i=1 i i j t1 n 2 P. Bidart: MPP01, a new solution for planetary perturbations 355

L(i) where scalars cn are functions of masses: 0i+1 The quantities a i are dimensionless and of degree i n−1 n n−1 a0 cn =(1− µ) +(−1) µ . (19) with respect to the monomials xj/a0. Whatever the planet concerned, even for indirect perturbations, lunar coordi- The Legendre polynomials are expanded with an anal- 0 ogous approach to Brown’s. One makes a separation of nates and their partials with respect to wj and zj are the monomials depending on lunar coordinates solely from same. Therefore we construct the above quantities once polynomials referring to planetary elements. for all. Let (x1,x2,x3)and(y1,y2,y3) be respectively the coordinates of the vector TL and of the vector GP .We 5.2. Indirect perturbations have for θP: Coming back to the disturbing function FI in (5), the x y + x y + x y θ = p 1 1 2 p2 3 3 Sun’s orbit is no longer Keplerian but contains the plan- P 2 2 2 · 2 2 2 x1 + x2 + x3 y1 + y2 + y3 etary perturbations up to the eighth order of the masses. However, we have to remove the terms which are al- θ is substituted in (18); we obtain for F : P D ready computed in the main problem. Hence we com- X pute with a procedure analogous to the direct case, the F = km L(i). (20) D P polynomials in term of the planetary coordinates from i≥2 VSOP2000 and from their Keplerian expressions. With The index i indicates that the expression is homogeneous the VSOP2000 integration constants of the Earth-Moon by a degree i with respect to xj .Thuswehave: barycenter (Table 1), X 00 00 00 00 00 00 L(i) (i) n1 n2 n3 N ,λ ,k ,h ,q ,p (24) = L x1 x2 x3 (21) n1+n2+n3=i we construct the rectangular coordinates from the for- where L(i) are polynomials (X )ofy and odd powers of mulae of Keplerian motion solving Kepler’s equation j i F the inverse of the distance D: (Chapront et al. 1975). As in the direct case (18), I is X developed in Legendre polynomials: (i) 1 2n−2 0 X L = Xj · y . (22) km r n D2n+1 j F = c P (θ0) (25) n≤i I r0 n r0 n n≥2 This separation of coordinates suggested by Brown is ex- 1 tremely important for the integration of the Hamiltonian where θ0 = TL · GS,andc given in (19). Let rr0 n system. As a matter of fact, the multiplication of series 0 0 0 (x1,x2,x3) be the coordinates of the Sun. The separation whose arguments are different increases considerably the 0 between the lunar coordinates (xj ) and the solar ones (x ) number of terms. On the other hand, during the integra- j leads to an expression similar to FD: tion, most of the short periodic terms vanish whereas small X 0 0(i) divisors increase the amplitude of long periodic terms. A FI = km L ≥ drastic truncation level in the multiplication would pre- Xi 2 X 0 0 vent us from obtaining all the significant long periodic = km L (i)xn1 xn2 xn3 (26) terms. The separation of coordinates allows us to accom- 1 2 3 i≥2 n1+n2+n3=i plish simultaneously the multiplication and the integration avoiding truncation in multiplications alone. L0(i) are homogenous polynomials of degree i with respect 0 0(i) In the equations of system (9) the elements of the to the coordinates xj.WehaveforL : −1 2 C matrix are proportional to 1/νa0. Here a0 is the 0 0 2 3 0(i) (i) − (i) lunar semi-major axis tied to ν by: ν a0 = k(mE + L = LVSOP Lkep P F mM). Replacing in (9) by D from (18), we get the 0(i) 0(i) coefficient: where LVSOP (resp. Lkep) have the same expression as in (22) substituting D and y by r0 and x0 (resp. r0(kep) and km m n02a03 j j P = P β (kep) 2 0 2 xj ). νa0 m νa0 m0 with β = 0 . Then we have, for instance, the 5.3. Perturbations induced by the rotating frame m + mE + mM partials of FD with respect to the angles wj: The expression of the function G providing the pertur- F bations induced by the rotating frame has been given in 2 −1 ∂ D νa0 C 2 = (Chapront-Touzé & Chapront 1980). We have, in rectan- ∂wj νa0   gular coordinates:   n02 m ∂ X a i−2 L(i) G − − 2 −1 P 0 0i+1 · = θ1(x2u3 x3u2)+θ2(x3u1 x1u3) νa0 C β 0 0 a i (23) ν m ∂wj  a a  i≥2 0 +θ3(x1u2 − x2u1) (27) 356 P. Bidart: MPP01, a new solution for planetary perturbations

θi are the components of the instantaneous vector of ro- Delaunay arguments. We can divide those arguments in tation of R with respect to R0. One can express the θi in three categories: terms of the polynomial developments of p0 and q0 of the Earth-Moon barycenter and their derivatives with respect – Lunar short periodic terms to the time: Combinations of planetary and lunar arguments whose frequency is close to ν. The precision of those terms is 0 0 2 0 0 dp 02 dq the truncation precision η; θ1 = p p q +(1− p ) 1 − p02 − q02 dt dt – Planetary long periodic terms 0 0 combinations of planetary arguments whose frequency p 2 − 02 dp 0 0 dq θ2 = (1 q ) + p q (28) is small compared to ν. The precision of those terms 1 − p02 − q02 dt dt 0 0 coming from the simple integration is the product of 0 dp 0 dq precision in the planetary with the factor ν/ε; θ3 = −2 q − p · dt dt – Lunar long periodic terms combinations of lunar and planetary arguments whose With the aid of (28), (27) we integrate the system (9) frequency is small compared to ν. The precision of considering that the coordinates xi and ui are those of those terms arising from the double integration is the the main problem. product of main problem’s precision with ν/ε2.

6. Results and comparisons 6.3. Comparisons to ELP2000-82B 6.1. Timescale and long periodic terms Before any comparisons to the ELP2000-82B solution, one has to take into account the differences in the construc- Developments as in (23) have been undertaken up to the tion of the previous solution. For indirect perturbations, power i = 4 for each planet except Venus and Mars. Taylor developments using increments of the coordinates That means we have neglected terms containing the fac- with respect to the elliptic elements have been used. Here 0 3 ∼ −8 tor (a0a ) 1.610 . The analogous developments for we subtract the Keplerian contribution to the complete Venus, Mars, and indirect perturbations have been un- expression of the coordinates of the Earth-Moon barycen- dertaken up to the power i = 5 neglecting terms with ter. ELP2000-82B has been computed with VSOP82 plan- 0 4 ∼ −11 (a0/a ) 410 in factor. etary solution and developed up to the square of the time. Thanks to the separation of coordinates, we can per- In that case, the perturbations include only the secular form the multiplication of terms weighted by the di- variations due to solar eccentricity. visors appearing when integrating the equations. Let ε Differences in longitude (∆V ) between the ELP2000- be the divisor associated to the argument θ of a term 82B solution for planetary perturbations and MPP01 re- A cos θ + B sin θ resulting from a multiplication. During stricted to Fourier terms produce the results that we ex- the integration, we keep all the terms with a magnitude pected (Table 3). The largest differences are provided by | | | | j | | | | 2 ( A + B )/ε>ηfor z0 and ( A + B )/ε >ηfor wj , 00 lunar and planetary-type long periodic terms resulting −7 where η is the truncation level. We set η ∼ 410 in lon- from the computation of δw1 (17). This effect is mainly 00 gitude, η ∼ 210−7 in latitude and η ∼ 410−7 kilometers explained by the improvement of the planetary solution in distance. especially for outer planets whose precision has been im- Terms whose frequency ε is of the same order as the proved by two orders of magnitude (Moisson 1999). The Moon’s sideral mean motion ν have a truncation η.The numerical tools are more powerful. They allowed us to integration of long periodic terms provides small divisors handle larger series, so that the precision of computation ε ν and even ε2 ν in the double integration that im- has been increased. The amplitude of such long periodic ν ν terms after integration is then better defined. plies the degradation of the precision by a factor . ε ε2 The terms () do not appear in MPP01 whereas the Thus, the precision of computation of long periodic terms new terms (∗) did not appear in ELP2000-82B. The argu- depends on the period, and whether it comes from the ments of terms (†) are resulting from combination of long simple or double integration. In that respect, we neglect periods. The short periods in the lunar motion should not terms whose period exceeds 5000 years. appear in the difference at this level of precision. But the terms (?) come from the product of long periods described 6.2. Results above with Delaunay arguments in ∂V/∂w1δw1 (17). The differences in latitude (∆U) and distance (∆R)showes- The contribution of planetary perturbations in the final sentially short periods (Table 4). Concerning the Poisson solution occurs through the secular contributions in angu- terms for each coordinates, the magnitude of differences lar variables wj (12) and through Poisson series formally does not exceed those of Fourier terms over the time span 0 expressed by δwj (12) and ∆zj (10). In the series of plane- of two centuries. tary perturbations, the arguments are linear combinations In Table 5, we show the main differences in plane- of the eight planetary mean longitudes and of the four tary perturbations through the secular contribution in the P. Bidart: MPP01, a new solution for planetary perturbations 357 Table 2. Summary of main differences in the construction of ELP2000-82B and MPP01

ELP2000-82B MPP01 Indirect perturbations method increaments with respect to complete motion for EMB elements minus Keplerian contribution Poisson Series development up to t2 (so- development up to t5 lar eccentricity perturbations only) Planetary solution VSOP82 VSOP2000 Set of masses IAU76 IERS92 00 00 Precision of computation 10−5 4.10−7 Size of series 10 000 terms 100 000 terms

Table 3. Largest differences in longitude between ELP2000- Table 4. Largest differences in latitude and distance between 82B solution for planetary perturbations and MPP01 restricted ELP2000-82B solution for planetary perturbations and MPP01 to Fourier terms. Periods are in julian years, ∆A are the dif- restricted to Fourier terms. ∆A for distance are the differences ferences in magnitude expressed in 10−3 arcsec of degree in magnitude expressed in meters

argument period ∆A latitude ω (year) (mas) argument period ∆A 34Te − 41Ma + 2Ju − D 1237.472 58.65 ω (year) (mas) 15Ve − 9Te − 4Ma − l 2810.039 37.53 34Te − 41Ma + 2Ju − D − F 0.0745 2.61 10Ve − 3Te − l 1911.772 36.15 34Te − 41Ma + 2Ju − D + F 0.0745 2.61 2Ju − 5Sa 883.279 25.66 15Ve − 9Te − 4Ma − F − l 0.0745 1.66 4Te − 8Ma + 3Ju 1783.181 25.57 15Ve − 9Te − 4Ma + F − l 0.0745 1.66 4Te − 8Ma + 3Ju + l? 0.075 10.35 10Ve − 3Te − F − l 0.0745 1.65 4Te − 8Ma + 3Ju − l? 0.075 10.32 10Ve − 3Te + F − l 0.0745 1.63 Te + Ju − Sa + D − F 293.300 6.22 Te + Ju − Sa + D 0.0745 1.50 18Ve − 12Te − 8Ma + 3Ju − l∗† 322.414 4.95 2Ju − 5Sa + F 0.0745 1.02 19Ve − 24Te + 10Ju − D + F − l 4852.856 3.75 2Ju − 5Sa − F 0.0745 1.02 34Te − 41Ma + 2Ju − D − l? 0.075 3.26 distance 34Te − 41Ma + 2Ju − D + l? 0.075 3.26 argument period ∆A 20Ve − 16Te + 7Ma − 8Ju + 6Sa − l 272.845 3.06 ω (year) (meter) 20Ve − 12Te − 15Ma − D − F + l 3296.767 3.01 4Te − 8Ma + 3Ju + l 0.0754 9.49 20Ve − 19Te − 3Ju − l 556.916 2.89 4Te − 8Ma + 3Ju − l 0.0754 9.47 22Ve − 28Te + 7Ma − 2D + l 640.069 2.87 34Te − 41Ma + 2Ju − D + l 0.0754 3.01 2Ju − 5Sa − l? 0.075 2.82 34Te − 41Ma + 2Ju − D − l 0.0754 3.01 2Ju − 5Sa + l? 0.075 2.82 2Ju − 5Sa − l 0.0754 2.69 4Te − 8Ma + 3Ju − 2D + l? 0.087 2.80 2Ju − 5Sa + l 0.0754 2.69 4Te − 8Ma + 3Ju + 2D − l? 0.087 2.78 4Te − 8Ma + 3Ju − 2D + l 0.0871 2.17 18Ve − 16Te − l 273.045 2.66 4Te − 8Ma + 3Ju + 2D − l 0.0871 2.13 4Me − 5Ve + 3Te − 2D + l 1010.584 2.54 10Ve − 3Te 0.0754 2.02 Ur − 2Ne 4233.468 2.28 10Ve − 3Te − 2l 0.0754 1.97 10Ve − 3Te? 0.075 2.20 15Ve − 9Te − 4Ma − 2l 0.0754 1.84 10Ve − 3Te − 2l? 0.075 2.13 15Ve − 9Te − 4Ma 0.0754 1.83 14Ve − 9Te + 2Ju − 2F + l 1807.486 2.03 10Te − 19Ma + 3Sa + 2D − l 0.0871 1.63 Me + 15Ve − 16Te − D − F + l 1771.543 1.99 10Te − 19Ma + 3Sa − 2D + l 0.0871 1.63 15Ve − 9Te − 4Ma? 0.075 1.99 8Te − 16Ma + 6Ju + l 0.0754 1.27 15Ve − 9Te − 4Ma − 2l? 0.075 1.99 8Te − 16Ma + 6Ju − l 0.0754 1.27 10Te − 19Ma + 3Sa + 2D 0.0404 1.23 10Te − 19Ma + 3Sa − 2D 0.0404 1.22 angular variables. They are mainly due to the constant ad- Te + Ju − Sa + D − F + l 0.0755 1.03 − − − justment and the indirect perturbations. The modification Te + Ju Sa + D F l 0.0755 1.03 in VSOP2000 with respect to VSOP87 of planetary constants such as the masses or the mean motions described in (Moisson 1999), explains the small differences observed motion in the complete solution of the Earth-Moon in the constants adjustment. The difference in the contri- barycenter. This method requires a high precision in the bution of indirect planetary perturbations is larger than computation of Keplerian coordinates. The small contri- what we expected. It is a consequence of the method bution of the neglected effects in the fitting of VSOP2000 used to compute the indirect perturbations. Our method to the numerical integration DE403 can explain the differ- consists in subtracting the contribution of Keplerian ences observed. The results concerning the t3 (and higher 358 P. Bidart: MPP01, a new solution for planetary perturbations

Table 5. Diﬀerences of the secular contribution in angular variables. Units are arcseconds per century to the power α

(1) 00 (1) 00 Parameter w2 ( /cy) w3 ( /cy) Direct planetary perturbations 0.0036 −0.0013 Indirect planetary perturbations −0.0498 −0.0159 Moon on EMB 0.0032 0.0002 Relativity −0.0002 0.0001 Constants adjustment −0.0121 0.0009 Total −0.0553 −0.0160 (2) 00 2 (2) 00 2 (2) 00 2 Parameter w1 ( /cy ) w2 ( /cy ) w3 ( /cy ) Total 0.00605 −0.00135 −0.00038

Table 6. Comparison of long periodic terms determined by Moshier with those present in MPP01. Periods are in julian years, amplitudes are in arcseconds per century square, and phases are in degrees

terms in t2 sin Moshier MPP01 ω period (y) A (00/cy2)Φ(◦) A (00/cy2)Φ(◦) 18Ve − 16Te − l 273.045 −230.310−5 0 −228.010−5 +23.6 8Ve − 13Te 238.922 −17.710−5 +22.5 −4.110−5 +58.9 10Ve − 3Te − l 1911.772 −30.110−5 −60.8 −25.410−5 −17.3 powers) Poisson terms are not definite. They do not ap- 7. Conclusion pear in (Table 5). In this paper, we have presented a new solution for planetary perturbations of the Moon. This solution has been built taking advantage of the improvements of the plane- In the last part of our comparison, we consider the tary theory VSOP2000, and with a greater accuracy with results presented by (Moshier 1992) on a comparison of respect to the ELP2000-82B solution for Moon’s plane- ELP2000-82B with a lunar ephemeris over a 7000 years tary perturbations. The comparisons revealed differences time span. To make consistant ELP with the numerical in- with long periodic terms and with their associated short tegration on a long time, Moshier adds a quadratic expan- periodic terms as expected. The adjustment of constants sion of three long periodic terms and secular contributions and orbital parameters to numerical integrations of the in lunar arguments. The effect of these adjustments is to Jet Propulsion Laboratory DE403 and DE406 are under reduce the residuals of the difference in longitude to about construction. The improvement of the analytical solution 1.500 rms. The extremum is below 1000 over the 7000 years for the planetary perturbations should provide a lunar time span. The comparisons of the long periodic terms of ephemeris obtained with a greater precision. The numer- MPP01 with the results obtained by Moshier are shown in ical complements remain still necessary for the compar- Tables 5 and 6. We observe a good agreement concerning isons to Lunar Laser Ranging observations. the amplitude of the term 18Ve − 16Te − l. The differences in the amplitudes of the terms 10Ve − 3Te − l and Acknowledgements. The author would like to thank 8Ve−13Te are reduced when we consider the development X. Moisson, M. Chapront-Touzé and especially J. Chapront of these terms in higher powers of time and also the other for their many useful pieces of information, and their great long periodic terms (2Ju − 5Sa and 4Te − 8Ma + 3Ju) help in numerical computations. which were not mentioned in (Moshier 1992). The comparisons are also in good agreement with a shift of 25◦ References in phase observed by Moshier concerning terms involving Chapront, J., Bretagnon, P., & Mehl, M. 1975, Celest. Mech., l, and produced by the limitation of lunar arguments to 11, 379 their linear development. Chapront, J., & Abu el Ata, N. 1977, A&A, 55, 83 Chapront-Touzé, M., & Chapront, J. 1980, A&A, 91, 233 Chapront-Touzé, M. 1980, A&A, 83, 86 Finally, the difference with arguments corrected with Moisson, X. 1999, A&A, 341, 318 the shift in phase does not exceed 100 over the 7000 years Moisson, X. 2000, private communication Moshier, S. L. 1992, A&A, 262, 613 time span and is therefore non significant with respect to Simon, J. L., Bretagnon, P., Chapront, J., et al. 1994, A&A, the background noise appearing in the comparisons with 282, 663 numerical integrations.