Some Peculiarities of the Evolution of Orbits in the Satellite Restricted Elliptic Doubly Averaged Three-Body Problem

Abstract

The inner (or satellite) version of the restricted elliptic three-body problem is considered. The terms up to the fourth order inclusive in small parameter are retained in the expansion of the perturbing function for the problem. The ratio of the orbital semimajor axes of perturbed and perturbing bodies is such a parameter, while their mean longitudes are the fastest variables. The Gauss scheme of independent double averaging over fast variables is used to analyze the orbital evolution of a body of negligible mass. Explicit analytical expressions for the doubly averaged perturbing function and its derivatives with respect to the elements on the right-hand sides of the evolution equations are presented. The integrable cases of the doubly averaged problem are studied in detail: planar and orthogonal apsidal orbits. The evolution system is numerically integrated in the general (nonintegrable) case for some special values of the problem parameters and initial conditions, in particular, for a set of orbital elements in which the so-called “flips”, i.e., transitions of the orbit from prograde to retrograde and vice versa, manifest themselves. In the Sun–Jupiter–asteroid model using some special asteroid orbits as an example, we show the influence of the retained fourth-order terms and the ellipticity of Jupiter’s orbit on their evolution.