Stability in the Full Two-Body Problem

Abstract

Stability conditions are established in the problem of two gravitationally interacting rigid bodies, designated here as the full two-body problem. The stability conditions are derived using basic principles from the N-body problem which can be carried over to the full two-body problem. Sufficient conditions for Hill stability and instability, and for stability against impact are derived. The analysis is applicable to binary small-body systems such as have been found recently for asteroids and Kuiper belt objects. In this paper, some classical results from the N-body problem are applied to the problem of two interacting rigid bodies, each with an arbitrary gravity field. Such a problem can serve as a model for the dynamics of a binary small-body system, such as an asteroid or a Kuiper belt object, especially during the early stages of its evolution following a disruptive impact or planetary flyby. The specific interest expressed in this paper concerns the long term stability of the binary against disrup- tion (escape) or impact. Stability against disruption for this problem is essentially Hill stability, and we find sufficient conditions for this stability and sufficient con- ditions for violation of this stability. Necessary conditions are more difficult, and this is explained. In the N-body problem, stability against impact is often related to Lagrange stability which restricts both the positions and velocities of the bodies to be bounded. For interacting point masses, such a restriction guarantees that impact will not occur. However, for rigid bodies with distributed mass, impacts can occur with finite velocity; thus we introduce a definition called stability against impact (SAI). We find sufficient conditions for SAI in the full two-body problem.