Post-Newtonian corrections to Toomre's criterion

TL;DR

This paper derives a relativistic version of Toomre's stability criterion for self-gravitating, rotating gaseous disks using post-Newtonian theory, showing that relativistic effects generally enhance stability and align with neutron star disk simulations.

Contribution

The paper introduces the first post-Newtonian correction to Toomre's stability criterion, providing a relativistic framework for analyzing disk stability.

Findings

PN corrections increase the stability of rotating disks.

Relativistic effects destabilize non-rotating systems.

PN theory aligns with neutron star disk simulations, ruling out local fragmentations.

Abstract

The gravitational stability of a two-dimensional self-gravitating and differentially rotating gaseous disk in the context of post-Newtonian (hereafter PN) theory is studied. Using the perturbative method and applying the second iterated equations of PN approximation, the relativistic version of the dispersion relation for the propagation of small perturbations is found. We obtain the PN version of Toomre's local stability criterion by utilizing this PN dispersion relation. In other words, we find relativistic corrections to Toomre's criterion in the first PN approximation. Two stability parameters $\eta$ and $\mu$ related to gravity and pressure are introduced. We illustrate how these parameters determine the stability of the Newtonian and PN systems. Moreover, we show that, in general, the differentially rotating fluid disk is more stable in the context of PN theory relative to the Newtonian one. Also, we explicitly show that although the relativistic PN corrections destabilize non-rotating systems, they have the stabilizing role in the rotating thin disks. Finally, we apply the results to the relativistic disks around hypermassive neutron stars (HMNSs), and find that although Newtonian description predicts the occurrence of local fragmentations, PN theory remains in agreement with the relevant simulations, and rules out the existence of local fragmentations.