Lane-Emden stars, selfgravitating disks and the Sobolev inequality

TL;DR

This paper derives a lower bound on the minimal mass of self-gravitating polytropic disks using Sobolev inequality, connecting it to classical stellar mass formulas and verifying accuracy through numerical examples.

Contribution

It introduces a novel analytical bound for disk mass based on Sobolev inequality, extending concepts from stellar structure to disk configurations.

Findings

The minimal mass bound resembles the Lane-Emden star mass formula.

For n=3, the minimal mass is at least the Jeans mass.

The bound is more accurate for heavy disks than for light disks.

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.