On the Evolution of Orbits in the Outer Version of the Restricted Elliptic Doubly Averaged Three-Body Problem

Abstract

The outer version of the restricted elliptic three-body problem, where a perturbed body of negligible mass is far from two bodies of finite masses (the main and perturbing ones) close to one another and its orbit is located entirely outside the orbit of the perturbing body, is considered. The terms up to the fourth order inclusive in the ratio of the orbital semimajor axes of the perturbing and perturbed bodies are retained in the expansion of the perturbing function for the problem. Explicit analytical expressions for the doubly averaged perturbing function and its derivatives with respect to the orbital elements on the right-hand sides of the evolution equations are presented. The integrable cases of the doubly averaged elliptic problem are investigated: planar and orthogonal apsidal orbits. Qualitative differences compared to the inner (satellite) version are pointed out. The evolution system is numerically integrated in the general (nonintegrable) case for a number of special initial conditions that hypothetically correspond to the orbital evolution of low-mass bodies (planetesimals, low-mass planets) in the exoplanetary system GJ 3512.