Rapidly Rotating Stars

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 (Math Z 36(1):481–562, 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 the supports of the stars in ${\mathcal{K}}$ become unbounded if we assume for instance an equation of state of the form ${p=\rho^\gamma,\ 4/3 < \gamma < 2}$. On the other hand, if ${6/5 < \gamma < 4/3}$, we prove that either the supports of the stars in ${\mathcal{K}}$ become unbounded or the density somewhere within the stars becomes unbounded. We consider two formulations, one where the angular velocity is prescribed and the other where the angular momentum per unit mass is prescribed.