Long-Term Evolution of Coorbital Motion

Abstract

In these lectures the planar problem of three bodies with masses $m_0$, $m_1$, $m_2$ will be used as a model of coorbital motion, thus leaving the analysis of three-dimensional effects to later work. For theoretical as well as for numerical studies the choice of appropriate variables is essential. Here Jacobian coordinates and a rotating frame of reference will be used. The application of the Hamiltonian formalism in connection with complex notation will greatly simplify the differential equations of motion. The results obtained are partially of experimental nature, based on reliable numerical integration. Obviously, chaos plays an important role. An orderly behaviour occurs for small mass ratios $\epsilon : = (m_1+m_2)/m_0$; however, the typical phenomena persist even for mass ratios as large as $0.01$. In particular, proper coorbital motion seems to be chaotic, but stable for very long periods of time. The interaction of the satellites, as they approach each other, is qualitatively described by Hill's lunar problem. Temporary capture between independently revolving satellites is delicate and can only happen when close encounters are involved. It seems to be able to persist for very long times, though, even for mass ratios as large as $\epsilon = 0.1$.