Dynamics in the Schwarzschild isosceles three body problem

TL;DR

This paper investigates the qualitative dynamics of a three-body system under Schwarzschild potential, revealing unique behaviors such as triple collisions at non-zero angular momentum and non-homothetic collision orbits, contrasting with Newtonian predictions.

Contribution

It provides a detailed analysis of the Schwarzschild three-body problem, including equilibria, collision behavior, and the impact of the potential on orbit types, highlighting phenomena absent in Newtonian models.

Findings

Triple collision can occur at non-zero angular momentum.

Existence of non-homothetic, non-homographic triple collision orbits.

Positive measure of initial conditions leading to triple collision.

Abstract

The Schwarzschild potential, defined as U(r)=-A/r-B/r^3, where r is the distance between two mass points and A,B>0, models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy-momentum diagram. We further employ appropriate regularization transformations to analyse the behaviour of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for non-zero total angular momenta (e.g., when two of the mass points spin infinitely many times around the centre of mass). This phenomenon is known in celestial mechanics as the "black-hole effect" and it is understood as an analogue in the classical context of the behaviour near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits which are neither homothetic, nor homographic.