Gravitational Collapse and Equilibrium Conditions of a Toroidal Vortex with Thermal Pressure

We found equilibrium conditions for a self-gravitating toroidal vortex by taking into account thermal pressure. These conditions are shown to significantly differ from those for a disk and a sphere. The evolution of a thin vortex turns it into a compact vortex that loses mechanical stability for low masses at a polytropic index $\gamma <4\mathrm{/}3$ but retains stability for sufficiently high masses and densities determined by the velocity circulation in the vortex.