Rapidly Rotating Stars

TL;DR

This paper proves the existence of rapidly rotating star solutions modeled by the Euler-Poisson system, showing their properties depend continuously on rotation speed and identifying potential unbounded behaviors as rotation increases.

Contribution

It establishes the first existence theorem for rapidly rotating stars with continuous dependence on rotation speed, solving a long-standing problem since 1933, using global continuation theory.

Findings

Solutions form a connected set in function space.

Either the star's support or density becomes unbounded at high rotation speeds.

Applicable to a range of equations of state with specific gamma values.

Abstract

A rotating star may be modeled as a continuous system of particles attracted to each other by gravity and with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. We prove an existence theorem for such stars that are rapidly rotating, depending continuously on the speed of rotation. This solves a problem that has been open since Lichtenstein's work in 1933. The key tool is global continuation theory, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. As the speed of rotation increases, we prove that {\it either the supports of the stars in $\mathcal K$ become unbounded or the density somewhere within the stars becomes unbounded}. We permit any equation of state of the form $p=\rho^\gamma,\ 6/5<\gamma<2$, so long as $\gamma\ne4/3$. We consider two formulations, one where the angular velocity is prescribed and the other where the angular momentum per unit mass is prescribed.