2 Dynamic models of planetary motion

The influence of dynamical models on the determination of parameters has been revealed by the use of various planetary ephemerides as the bases for the adjustment to the same set of observations. The JPL ephemerides - DE200 (Standish 1990), DE403 (Standish et al. 1995), DE405 (Standish 1998) and the similar ephemerides EPM98, EPM2000 (Pitjeva 2001), constructed at the Institute of Applied Astronomy of Russian Academy of Sciences, have been used with the same set of observational data.

Common to all these dynamical models is a simultaneous numerical integration of the equations of motion of the nine planets, the Sun, the Moon and lunar physical libration, performed in the Parameterized Post-Newtonian metric for the harmonic coordinates ($\alpha=0$) and General Relativity values ( $\beta = \gamma = 1$). The Ephemerides of Planets and the Moon (EPM2000) took into account the perturbations from 300 asteroids by numerical integration of the equations of motion of all objects, but including the mutual perturbations of only the planets, moon, sun, and the 5 most significant asteroids. Previously, EPM98 had accounted for the asteroid perturbations in a manner similar to that used for DE403.

There are slight differences among the models of the various ephemerides: the modelling of the lunar libration is different, but this has no significant effect on the parameters considered in this paper; the solar oblateness J₂= 2.0   10^-7, obtained from some astrophysical estimations (Duvall et al. 1984; Brown 1989), has been taken into account; and the constructed ephemerides differ only slightly by the accuracy of interpolation for planetary coordinates by a set of Chebyshev polynomials. DE200 differs significantly from the other ephemerides in the modelling of the perturbations from asteroids: only the three most massive asteroids were taken into account in DE200, while the other ephemerides were supplemented with the modelling of perturbations from 300 asteroids upon the orbits of the planets.

The astronomical parameters have been determined by computing for each ephemeris residuals of the same observational data and then by re-computing them by adjusting a number of ephemeris parameters. The data included radar ranging of Mercury and Venus and spacecraft ranging and doppler of the Viking and Pathfinder martian landers (observations marked by a * in Table 2). Note that the uncertainties, given in this paper, are formal standard deviations; realistic error bounds may be an order of magnitude larger. The formal standard deviations are given in Table 1 for the orbital elements of Mercury, Venus, Earth and Mars: a_i, sini_icos$\Omega_i$, sini_isin$\Omega_i$, e_icos$\pi_i$, e_isin$\pi_i$, $\lambda_i$, where a - the semi-major axis, i - the inclination of the orbit, $\Omega$ - the ascending node, e - the eccentricity, $\pi$ - the longitude of perihelion, $\lambda$ - the mean longitude, index _i = 1, 2, 3, 4 - planets; also given in Table 1 are the corrections to the value of the astronomical unit ( ${\rm AU}=149\,597\,870\,691.0$ m) and to the Mars precession ( $\dot\Omega_q =-7\hbox{$.\!\!^{\prime\prime}$ }576$/y) with their uncertainties determined by using the various ephemerides.

From Table 1 it is seen that, first, the dynamical models of the ephemerides DE403, DE405, EPM98, EPM2000 are nearly equivalent and yield virtually identical accuracy for the determination of the parameters. Secondly, the formal standard deviations of the solution parameters are improved by 30-50$\%$ using DE403, DE405, EPM98, EPM2000 ephemerides instead of DE200.

In addition, it should be noted that the DE200 ephemerides were created before many spacecraft-based determinations of the relevant parameters existed. Thus, many of the fitting parameters (e.g., planet masses, station locations) were less accurately determined for DE200 than they are known today. Consequently, part of the higher uncertainties of DE200 must be attributed to the embedded effect of these parameters. The numerical experiments have shown that the accuracy of adjusted parameters essentially depended on values of planet masses and asteroid modeling.

 

 

Table 1: The formal standard deviations of parameters, readjusted to the same set of observations, using a selected set of parameters and DE200, DE403, EPM98, DE405, EPM2000

parameters	DE200	DE403	EPM98	DE405	EPM2000
a1 [m]	0.430	0.306	0.309	0.305	0.308
sini1cos$\Omega_1$ [mas]	5.451	3.886	3.917	3.874	3.909
sini1sin$\Omega_1$ [mas]	6.010	4.284	4.318	4.272	4.310
e1cos$\pi_1$ [mas]	0.537	0.383	0.386	0.382	0.385
e1sin$\pi_1$ [mas]	0.460	0.328	0.331	0.327	0.330
$\lambda_1$ [mas]	1.708	1.218	1.227	1.214	1.225
a2 [m]	1.463	1.042	1.050	1.038	1.048
sini2cos$\Omega_2$ [mas]	2.322	1.655	1.668	1.650	1.665
sini2sin$\Omega_2$ [mas]	2.244	1.600	1.612	1.595	1.609
e2cos$\pi_2$ [mas]	0.127	0.090	0.091	0.090	0.091
e2sin$\pi_2$ [mas]	0.119	0.085	0.086	0.085	0.085
$\lambda_2$ [mas]	0.990	0.706	0.711	0.704	0.710
a3 [m]	0.180	0.109	0.104	0.103	0.103
e3cos$\pi_3$ [mas]	0.001	0.001	0.001	0.001	0.001
e3sin$\pi_3$ [mas]	0.001	0.001	0.001	0.001	0.001
a4 [m]	0.518	0.291	0.271	0.270	0.270
sini4cos$\Omega_4$ [mas]	0.033	0.024	0.024	0.024	0.024
sini4sin$\Omega_4$ [mas]	0.040	0.026	0.026	0.026	0.026
e4cos$\pi_4$ [mas]	0.002	0.002	0.002	0.002	0.002
e4sin$\pi_4$ [mas]	0.003	0.002	0.002	0.002	0.002
$\lambda_4$ [mas]	0.013	0.008	0.008	0.008	0.008
scale factor [m/au]	21.354	1.408	0.310	2.010	0.206
	$\pm$0.288	$\pm$0.190	$\pm$0.186	$\pm$0.184	$\pm$0.184
$\dot \Omega_q$ [mas/yr]	-10.98	2.35	4.61	-3.42	-7.36
	$\pm$7.28	$\pm$5.17	$\pm$5.20	$\pm$5.15	$\pm$5.17