Resonant returns to close approaches: Analytical theory

We extend Opikʹs theory of close encounters of a small body (either an asteroid or a comet) by explicitly introducing the nodal distance and a time coordinate. Assuming that the heliocentric motion between consecutive close encounters is Keplerian, or given by an explicit propagator, we can compute the initial conditions for an encounter as functions of the outcomes of a previous one; in this way it is possible to obtain a completely analytical theory of resonant returns. It is found that the initial conditions of a close encounter that lead to a resonant return must lie close to easily computable circles on the b-plane of the first encounter. By further assuming that the nodal distance varies uniformly with time, due to secular perturbations, and considering the derivatives of the coordinates on the bplane of the second encounter with respect to those on the b-plane of the first encounter, we compute in the latter the location, shape and size of collision keyholes.