A second-order theory for resonance capture in corotation centers

Abstract

In this paper we revisit the analytical determination of the corotation solutions of the restricted planar three-body problem under the effects of a Stokes drag. Previous calculations by different authors, and using different external dissipative forces, have always shown significant quantitative errors of the models with respect to the numerical values. Up to date no concrete explanation has been obtained for this discrepancy.We show that the origin of these errors lies not only in the modelization of the gravitational forces (i.e. disturbing function) but in the averaging process of the variational equations. A second-order averaging is developed based on a Lie transform method (Kamel, 1969). With it we can calculate new equilibrium solutions with a greatly improved accuracy.In a second part, this same perturbation method is applied to study the periodic orbits (i.e. limit cycles) which describe the motion of the particles around the averaged corotational solutions. We show how analytical approximations of these cycles can be obtained through the inverse transformation of the averaging process itself. In all cases, we present comparisons with numerical simulations of the exact equations.