Asymptotic expansion of the angle between successive maximal distances for a 2-body system with potential −1/r1+ε

Based on an example of Arnold (1989, page 37) we perturb the classical Newtonian 2-body problem by considering the potential −1/r1+ε with ε≪1. Restricting to orbits close to circular motion we compute an asymptotic expansion in ε of the angle between the successive positions where the distance between the bodies is maximal.