Exchange and Capture in the Planar Restricted Parabolic 3-Body Problem

Two attracting bodies m 1,m 2 move in parabolic orbits and a third massless body mo = 0 moves in the plane under the attraction of the primaries. We obtain the equations of motion of the massless particle in a rotating-pulsating coordinate system where the primaries remain fixed. Introducing an appropriate time scaling we obtain two invariant subsystems corresponding to final evolutions as time goes to±∞. We show that the set of initial conditions leading to parabolic escape of the infinitesimal mass is the union of invariant manifolds of dimension 3 and 4 and tend asymptotically to a central configuration. We also give a new criterion based in the Jacobi function analogous to the circular case, to guarantee elliptic capture. This criterion seems to be distinct from criteria introduced by Merman [1954] for the hyperbolic and parabolic restricted problems. We review Kocina’s example of exchange where the infinitesimal mass mo comes from infinity forming a bounded binary with m2 and escapes forming a bounded binary with m 1, and obtain new classes of orbits of symmetric exchange.