The binary collision in the N-body problem

Abstract

The binary collision in the N-body problem of celestial mechanics and its regularization are studied from the point of view of analytic differential equations, i.e., for complex values of the time and of the coordinates as independent and dependent variables. Based on a suitable definition of a binary collision, a proof is given for the classical result that a binary collision corresponds to an algebraic branch point of the first or second order. The convergence of Sundman's integral and the regularization of the binary collision are briefly discussed also for more general problems not necessarily possessing an energy integral; this situation is present, e.g., in the reduced three-body problem.