Kinetic Theory of Collisionless Self-Gravitating Gases: Post-Newtonian Polytropes

TL;DR

This paper derives a corrected collisionless Boltzmann equation in the 1PN approximation for self-gravitating gases, addressing inaccuracies in previous formulations, and applies it to construct post-Newtonian polytropes relevant for high-density astrophysical systems.

Contribution

It provides a consistent 1PN collisionless Boltzmann equation correcting previous inaccuracies and applies it to develop post-Newtonian polytropes.

Findings

Corrected the 1PN collisionless Boltzmann equation.

Ensured the equation's consistency with 1PN conserved quantities.

Constructed post-Newtonian polytropes for astrophysical applications.

Abstract

In this paper we study the kinetic theory of many-particle astrophysical systems and we present a consistent version of the collisionless Boltzmann equation in the 1PN approximation. We argue that the equation presented by Rezania and Sobouti in A&A 354 1110 (2000) is not the correct expression to describe the evolution of a collisionless self-gravitating gas. One of the reasons that account for the previous statement is that the energy of a free-falling test particle, obeying the 1PN equations of motion for static gravitational fields, is not a static solution of the mentioned equation. The same statement holds for the angular momentum, in the case of spherical systems. We provide the necessary corrections and obtain an equation that is consistent with the corresponding equations of motion and the 1PN conserved quantities. We suggest some potential relevance for the study of high density astrophysical systems and as an application we construct the corrected version of the post-Newtonian polytropes.