On simultaneous binary collisions in the planar n-body problem

Abstract

We will introduce thegeneralized Levi-Civita transformation (§2.3(a)) near the simultaneous binary collision (SBC) singularity of the planar n-body problem and study the asymptotic behaviour ofSBC orbits. Then, we will show the following in these variables: (1) For eachSBC orbit there is a unique simultaneous binary ejection (SBE) orbit, together they form a C1 curve which we call a collision-ejection (C-E) orbit. (2) The collection of collision-ejection orbits form a Co submanifold which we call the collision-ejection (C-E) manifold. (3) There is a sectorS, made of line segments through theSBC singularity, such that near by solutions leaveS no matter how thinS is, and only theC-E manifold lies entirely inS.