New analytical planetary theories VSOP2010A and VSOP2010B

1. JOURNEES « SYSTEMES DE REFERENCE SPATIO-TEMPORELS » SEPTEMBER 2010, PARIS New analytical planetary theories VSOP2010A and VSOP2010B G. Francou, J.-L. Simon OBSERVATOIRE DE PARIS (SYRTE & IMCCE) CNRS UMR 8630 & 8028 – UPMC - GRGS The planetary theories VSOP are essentially issued from Bretagnon’s research works. The precision of the last version, VSOP2000 (Bretagnon & Moisson) fitted to DE403, was 10 times better than the previous solutions VSOP82 and VSOP87, both fitted to DE200. Subsequently, Bretagnon began to make some improvements : better accuracy for the computation of series, simultaneous determination of the 8 planets and the 5 big asteroids orbits, lunar perturbations taken into account for the motions of the planets. We took up again this work introducing various changes and complements : fits to the numerical integrations DE405 (JPL) and INPOP08A (Obs. Paris & Besançon) for VSOP2010A and VSOP2010B respectively, introduction of the perturbations coming from 298 asteroids, contribution of TOP theory (Simon), i.e., analytical theory of Pluto, perturbations due to Pluto, better accuracy for the orbits of Jupiter and Saturn.Over the time interval [1890,2000], the estimated precision is 3 to 10 times better than that of VSOP2000.Over the time interval [-4000,8000], the gain in precision is about 5 times better for the telluric planets and 10 to 50 times better for the outer planets in comparison with VSOP2000. Introduction The VSOP theories are analytical solutions for the motion of the planets of the solar system. They are essentially issued from Bretagnon's research works and they are based on the integration of Lagrange differential equations for the elliptic elements a, l, k, h, q, p where the time is expressed in TDB. • Main characteristics • For the theories VSOP2010A and VSOP2010B, we have kept a form similar to the solution VSOP2000, i.e., Poisson series where the arguments are : _ • the mean mean longitudes of the planets lp =l0p + n0p t where t is the time (TDB), l0p is the integration constant of the mean longitude l and n0p the integration constant of the mean motion n, • the Delaunay’s angles of the Moon D, F, l, • and the argument m(defined hereunder). Results In VSOP2010A and VSOP2010B, each element (a, l, k, h, q, p) of each planet has the form of Poisson series developed up to the power 12 of time (power 20 for a et l of Jupiter and Saturn); a is expressed in au, l in radian and the time in thousand of years. Table 3. Mean longitudes of the planets : gain in precision related to VSOP82 for VSOP2000, VSOP2010A and VSOP2010B over [1890;2000]. a : semi-major axis i : inclination l: mean longitude e : eccentricity k : ecos W: longitude of the ascending node h : esin w:argument of the perihelion q : sin i/2 cosW: longitude of the perihelion (W + w) p : sin i/2 sinW M : mean anomaly (l = M + ) n : mean motion (n2a3 = constant) Figure 1. Mean longitudes of the planets over [1890;2000] Differences VSOP2010A-DE405 (blue)and VSOP2010B-INPOP08A (red) Historical review VSOP82 : P. Bretagnon, A&A 1983, 114, 278. Perturbations developed up to the 3rd order of the masses and iterative method up to the power 5 of time in the Poisson series for the 4 outer planets; Perturbations of the Earth-Moon barycenter by the Moon; Relativity introduced by the Schwarzschild problem; Fitted to the numerical integration of the JPLDE200(E.M. Standish, A&A, 1981, 114, 297). VSOP87 : P. Bretagnon, G. Francou, A&A 1988, 202, 309. Extension of VSOP82 in several versions: rectangular or spheric variables, ecliptic reference frame J2000 or of the date, heliocentric or barycentric coordinates. VSOP2000 : X. Moisson, P. Bretagnon, Celest. Mech. 2001, 80, 205. Perturbations developed up to the 3rd order of the masses and iterative method up to the power 7 of time in the Poisson series for the 8 planets; Perturbations by the 5 big asteroids introduced during the iterations; Second order perturbations by the Moon on the telluric planets; Fitted to the numerical integration DE403(E.M. Standish et al, JPL IOM, 1995, 314, 10, 127). • Precision over one century [1980;2000] • The precisions of the VSOP2010A and VSOP2010B related to VSOP2000 can be evaluated by their maximum differences with, DE405,INPOP08A and DE403 respectively for the 6 elliptic elements (see Table 2). • In the case of VSOP2010B, the differences for the mean longitudes lp is smaller than that of VSOP2010A (Figure 1). That is due to our computation of the pertubations by the asteroids which is closer between VSOP2010B and INPOP08A than in the case of VSOP2010A and DE405. • The gain of precision of VSOP2000, VSOP2010A and VSOP2010B related to the first version VSOP82 for the mean longitudes of planets over [1890;2000] is given in Table 3. Figure 2. Heliocentric longitudes of the planets over [-3000;+3000] Differences VSOP2000-DE406 (green) and VSOP2010A-DE406 (blue) • We have computed the perturbations due to the J2 of the Sun on the telluric planets. • We have introduced the perturbations of asteroids (295 in the case of VSOP2010A, 298 in the case of VSOP2010B) under the form of Poisson series of m. (see hereunder). • We have improved the precision of the theories of Jupiter and Saturn over a large time-span starting from a new version of the solutions TOP(J.-L. Simon, private communication) where the motion of the outer planets and Pluto are computed under the form of Poisson series of only one angular argument m, linear function of time t given by : m = (n5-n6 ) t / 880 where n5 and n6 are the mean motions of Jupiter and Saturn respectively. • We have added the perturbations of Pluto over the outer planets at the first order of the masses • We have introduced a complete solution of the motion of Pluto issued from TOP (not yet finished for VSOP2010B). • Additional improvements : • Starting from VSOP2000, P. Bretagnon performed 15 additional iterations up to the power 12 of time in the Poisson series with the following improvements : • The numerical precision is 10 times better than VSOP82. • The eight planets and the five big asteroids have their orbits analytically computed during the same process. • The perturbations of the Moon on the eight planets are introduced during the iterations. • As for the solution VSOP2000, relativistic corrections have been included. Table1.Number of terms of the solution VSOP2010A for several levels of truncation in the series Table2. Maximum differences over [1890;2000] between VSOP2000 - DE403 VSOP2010A - DE405 VSOP2010B - INPOP08A Units : 10-10 au for Da/a, 10-10 radian for l, 10-10 for k, h, q, p. Precision over large time spans The comparison of VSOP2010A with VSOP2000 over 6000 years is illustrated by Figure 2 where DE406 is used as reference for the heliocentric longitudes of the planets. For estimating the precision of VSOP2010A we use also a comparison with an internal numerical integration over [-4000,+8000] (see Table 4). VSOP2010 solutions Unfortunately, P. Bretagnon did not have time to complete his work. We took up again his last improvements of VSOP2000, introduced various changes and complements and build two new solutions : VSOP2010A, fitted to the numerical integration DE405 (E.M. Standish, JPL IOM, 1998, 312.F, 98) which is the bases of The Astromical Almanach since 2003. VSOP2010B (not yet finished),fitted to the recent numerical integration INPOP08A(A. Fienga et al., A&A, 2009, 507, 1675) which is the new standard reference for the planetary ephemerides published by IMCCE. Table1 gives the number of terms for the solution VSOP2010A It would be quite the same for the solution VSOP2010B. Table 4. Maximum differences between VSOP2010A and an internal numerical integration over 12 thousands of years centered on J2000. Units: 10-7 au for Da/a, 10-7 rad for l, 10-7 for k, h, q, p. The numbers in the parenthesis give the maximum differences before introducing TOP corrections. Argument m : the choice of this argument ensures a much better convergence for the perturbations of the couple Jupiter-Saturn than in classical theories (Simon and al., A&A 1992, 265, 308). Also, it allows to build an analytical theory of the motion of Pluto in Poisson series of m (Simon, IMCCE, 2003, Note S081, 53). In conclusion, over large time spans we consider that the gain of precision of VSOP2010A and VSOP2010B is about 5 times better for the telluric planets and 10 to 50 times better for outer planets in comparaison with VSOP2000.