Pseudo-Newtonian planar circular restricted 3-body problem

TL;DR

This paper investigates how incorporating first-order relativistic effects into the classical three-body problem affects the system's dynamics, revealing that pseudo-Newtonian models tend to be less stable and exhibit different chaotic behaviors.

Contribution

It introduces a pseudo-Newtonian approximation for the circular restricted three-body problem using a relativistic potential expansion, analyzing the resulting dynamics with new parameters.

Findings

Pseudo-Newtonian models show less stability than Newtonian ones.

Chaotic orbits can become regular when transitioning from Newtonian to pseudo-Newtonian regimes.

Most pseudo-Newtonian configurations are less stable than their Newtonian counterparts.

Abstract

We study the dynamics of the planar circular restricted three-body problem in the context of a pseudo-Newtonian approximation. By using the Fodor-Hoenselaers-Perj\'es procedure, we perform an expansion in the mass potential of a static massive spherical source up to the first non-Newtonian term, giving place to a gravitational potential that includes first-order general relativistic effects. With this result, we model a system composed by two pseudo-Newtonian primaries describing circular orbits around their common center of mass, and a test particle orbiting the system in the equatorial plane. The dynamics of the new system of equations is studied in terms of the Poincar\'e section method and the Lyapunov exponents, where the introduction of a new parameter $\epsilon$, allows us to observe the transition from the Newtonian to the pseudo-Newtonian regime. We show that when the Jacobian constant is fixed, a chaotic orbit in the Newtonian regime can be either chaotic or regular in the pseudo-Newtonian approach. As a general result, we find that most of the pseudo-Newtonian configurations are less stable than their Newtonian equivalent.