Lane-Emden stars, selfgravitating disks and the Sobolev inequality
arXiv:1007.1652 · doi:10.1111/j.1365-2966.2010.17764.x
Abstract
We estimate the minimal mass of selfgravitating polytropic disks using the famous Sobolev inequality. This bound resembles the well known mass formula for Lane-Emden stars. For ideal gas with the polytropic index n = 3 the minimal mass is not smaller than the Jeans mass. The accuracy of the estimate is verified in a number of numerical examples. The bound works well for heavy selfgravitating disks and is less useful for light disks.
9 pages, 2 figures