Analytical theory of the motion of Phobos: A comparison with numerical integration

Abstract

A new theory of the motion of Phobos - a moon of Mars - has been presented. The theory is based on the problem of two fixed gravitational centers. The analytical functions appearing in this model are expressed as a series of the third order with respect to the J2 and J 3 J2 zonal harmonics of Mars. The interaction between Phobos and Mars is described by a potential consisting of very many elements. The zonal harmonics of order ≤12 and tesseral harmonics of order ≤6 are taken into account. Interactions of Phobos with the Sun, Jupiter, Deimos - another moon of Mars, and the tidal potential of the Sun are also taken into account. In the next two decades one should expect several missions to the vicinity of Mars aimed mainly at a preparation for the land- ing of humans on this planet. Therefore a precise knowledge of the phenomena that may influence the motion of all objects in the proximity of Mars is of particular interest and impor- tance. In particular, a good analytical model of the motion of the Martian satellites may be used as a starting point for a pre- cise description of other objects, e.g. of the artificial satellites of Mars that will be launched in the future. Recently a new analytical model of the motion of Phobos has been developed by the present author (Wa ¸u z 1999a). The theory is based on the problem of two fixed gravitational centers. The Hamiltonian, apart from the integrable part describing the problem of two fixed centers, contains perturbative terms which include pertur- bations due to the Sun as well as perturbations connected with zonal harmonics j4 − j12 and with sectorial-tesseral harmonics of order 6. Also the influences of the tidal interactions of the Sun and of Jupiter and Deimos on the motion of Phobos have been taken into account. In the present paper the results obtained in the process of the analytical integration of the Hamilton equations of motion using the previously defined Hamiltonian (Wa ¸u z 1999b) are de- scribed. The formulae describing the secular and periodic per- turbations in Euler's orbit of Phobos are also presented. To es- timate the precision of the analytical formulae, the final analyt- ical solution has been compared with the results of numerical integration.