A first-order secular theory for the post-Newtonian two-body problem with spin -- I: The restricted case

TL;DR

This paper develops a first-order secular relativistic theory for the two-body problem with spin, capturing key effects like precession and Lense-Thirring, and providing explicit solutions for long-term orbital evolution.

Contribution

It introduces a new explicit solution for the averaged equations of motion in the relativistic two-body problem with spin, accounting for complex interactions between effects.

Findings

Reproduces relativistic effects such as pericentre precession and Lense-Thirring.

Provides explicit solutions in terms of elliptic functions.

Identifies configurations with non-periodical behavior.

Abstract

We revisit the relativistic restricted two-body problem with spin employing a perturbation scheme based on Lie series. Starting from a post-Newtonian expansion of the field equations, we develop a first-order secular theory that reproduces well-known relativistic effects such as the precession of the pericentre and the Lense-Thirring and geodetic effects. Additionally, our theory takes into full account the complex interplay between the various relativistic effects, and provides a new explicit solution of the averaged equations of motion in terms of elliptic functions. Our analysis reveals the presence of particular configurations for which non-periodical behaviour can arise. The application of our results to real astrodynamical systems (such as Mercury-like and pulsar planets) highlights the contribution of relativistic effects to the long-term evolution of the spin and orbit of the secondary body.