# Maximally Rotating Supermassive Stars at the Onset of Collapse: Effects of Gas Pressure

## TL;DR

This paper investigates how gas pressure influences the critical configurations of maximally rotating supermassive stars at collapse, revealing significant effects even at high masses and evaluating approximation methods against exact calculations.

## Contribution

It provides a fully nonlinear analysis of gas pressure effects on supermassive stars nearing collapse, extending previous perturbative studies across a wide mass range.

## Key findings

- Gas pressure significantly affects critical configurations at masses around 10^6 solar masses.
- One common approximation deviates increasingly from exact results as mass decreases.
- Another approximation remains robust for masses above 10^4 solar masses.

## Abstract

The "direct collapse" scenario has emerged as a promising evolutionary track for the formation of supermassive black holes early in the Universe. In an idealized version of such a scenario, a uniformly rotating supermassive star spinning at the mass-shedding (Keplerian) limit collapses gravitationally after it reaches a critical configuration. Under the assumption that the gas is dominated by radiation pressure, this critical configuration is characterized by unique values of the dimensionless parameters $J/M^2$ and $R_p/M$, where $J$ is the angular momentum, $R_p$ the polar radius, and $M$ the mass. Motivated by a previous perturbative treatment we adopt a fully nonlinear approach to evaluate the effects of gas pressure on these dimensionless parameters for a large range of masses. We find that gas pressure has a significant effect on the critical configuration even for stellar masses as large as $M \simeq 10^6 M_{\odot}$. We also calibrate two approximate treatments of the gas pressure perturbation in a comparison with the exact treatment, and find that one commonly used approximation in particular results in increasing deviations from the exact treatment as the mass decreases, and the effects of gas pressure increase. The other approximation, however, proves to be quite robust for all masses $M \gtrsim 10^4 M_{\odot}$.