Some estimates regarding a finite time stability problem of the Sun-Earth-Jupiter system

The spatial three-body problem of the Sun, Earth, Jupiter is studied over a finite time interval comparable with the age of the solar system. Some basic concepts of Nekhoroshev's theory are adopted; however, because of the finite time horizon considered, the canonical transformation scheme can be stopped with a third-order remainder. The overall effect of Jupiter is estimated from its maximal gravitational forces acting on the orbital elements of the Earth. This is done both analytically and numerically. The conservation of energy and angular momentum are rigorously taken into account. As part of a rather extensive programme which has not yet been completed, the effects of the first-order resonances and of a typical third-order rest term are estimated. Both contributions are found to be at their largest when the two osculating ellipses are coplanar. The third order force examined is zero at opposition by time reversal symmetry, but its maximal value lies close to this constellation. It can give rise to a 5% fluctuation of the semimajor axis of the Earth, not before 12 billion years, provided the eccentricity and inclination of the Earth are confined to 0e0.2 and 0i3/4, respectively. The self-consistent check of the two latter conditions is left to a future study. The results of this paper rely on a certain adiabatic approximation.