Secular perturbations of quasi-elliptic orbits in the restricted three-body problem with variable masses

• Differential equations determining the secular perturbations of the orbital elements in the restricted problem of three bodies of variable masses may be integrable.
• Possible values of two integrals of motion, corresponding to the integrable cases, have been found.
• Analytical solutions of the evolutionary equations have been found in terms of elementary and elliptic functions.
• Points masses may vary isotropically with different rates while their sum reduces according to Meshcherskii law.

In this work we consider the satellite version of the restricted three-body problem when masses of the primary bodies P0, P1 vary isotropically with different rates, and their total mass changes according to the joint Meshcherskii law. Equations of motion of the body P2 of infinitesimal mass are obtained in terms of the osculating elements of the aperiodic quasi-conical motion about the body P0. Doubly averaging these equations and using the Hill approximation, we have obtained the differential equations determining the secular perturbations of the orbital elements and determined the domains of possible values of the system parameters for which their analytical solutions are expressed in terms of elementary or elliptic functions. The bodies P0, P1 mass variation laws for which the corresponding differential equations are integrable, have been found.