Derivation of Capture Probabilities for the Corotation Eccentric Mean Motion Resonances

TL;DR

This paper derives the probabilities of a massless particle being captured into a corotation eccentric resonance in a three-body system, considering migration effects of both the particle and the perturber.

Contribution

It provides the first detailed derivation of capture probabilities into CER, including the effects of migration, which was not previously studied.

Findings

Capture occurs only with outward migration of the perturber.

Capture depends on the migration rate of the test particle and its radial gradient.

Results differ from Lindblad eccentric resonance capture conditions.

Abstract

We study in this paper the capture of a massless particle into an isolated, first order Corotation Eccentric Resonance (CER), in the framework of the Planar, Eccentric and Restricted Three-Body problem near a m+1:m mean motion commensurability (m integer). While capture into Lindblad Eccentric Resonances (where the perturber's orbit is circular) has been investigated years ago, capture into CER (where the perturber's orbit is elliptic) has not yet been investigated in detail. Here, we derive the generic equations of motion near a CER in the general case where both the perturber and the test particle migrate. We derive the probability of capture in that context, and we examine more closely two particular cases: (i) if only the perturber is migrating, capture is possible only if the migration is outward from the primary. Notably, the probability of capture is independent of the way the perturber migrates outward; (ii) if only the test particle is migrating, then capture is possible only if the algebraic value of its migration rate is a decreasing function of orbital radius. In this case, the probability of capture is proportional to the radial gradient of migration. These results differ from the capture into Lindblad Eccentric Resonance (LER), where it is necessary that the orbits of the perturber and the test particle converge for capture to be possible. Possible applications for planetary satellites are discussed.