Steady States of Rotating Stars and Galaxies

A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. The existence of solutions is a very classical problem. Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed. The rotation is allowed to vary with the distance to the axis. A special case is when the equation of state is $p=\rho^\gamma,\ 6/5<\gamma<2,\ \gamma\ne4/3$, in contrast to previous variational methods which have required $4/3 < \gamma$. The continuum of particles may alternatively be modeled microscopically by the Vlasov-Poisson system. The kinetic density is a prescribed function. We prove an analogous theorem asserting the existence of a curve of solutions with constant mass. In this model the whole range $(6/5,2)$ is allowed, including $\gamma=4/3$.