On the Mars rotation under the action of gravity torque from the Sun, Jupiter and Earth

P. S. Krasil'nikov ; R. N. Amelin

Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 329-342

Résumé

The Mars rotation under the action of gravity torque from the Sun, Jupiter, Earth is considered. It is assumed that Mars is the axially symmetric rigid body ($A=B$), the orbits of Mars, Earth and Jupiter are Kepler ellipses. Elliptical mean motions of Earth and Jupiter are the independent small parameters. The averaged Hamiltonian of problem and integrals of evolution equations are obtained. By assumption that the equatorial plane of unit sphere parallel to the plane of Jupiter orbit, the set of trajectories for angular momentum vector of Mars $\mathbf{I}_2$ is drawn. It is well known that «classic» equilibriums of vector $\mathbf{I}_2$ belong to the normal to the Mars orbit plane. It is shown that they are saved by the action of gravitational torque of Jupiter and Earth. Besides that there are two new stationary points of $\mathbf{I}_2$ on the normal to the Jupiter orbit plane. These equilibriums are unstable, homoclinic trajectories pass through them. In addition, there are a pair of unstable equilibriums on the great circle that is parallel to the Mars orbit plane. Four heteroclinic curves pass through these equilibriums. There are two stable equilibriums of $\mathbf{I}_2$ between pairs of these curves.

Détail

Keywords: four body restricted problem, the track of the angular momentum vector, method of averaging.
Mots-clés : Deprit – Andoyer variables