This little pdf file covers some interesting properties of polytropes – but what’s a polytrope? Basically it’s a sphere of gas, or some other matter, governed by a particular equation of how the pressure and the density are related.
P = K.(rho)1+1/n
…(rho) being density, P is pressure, K is a constant, and n is the polytropic index.
This equation is used to create an expression for the structure of the sphere that is converted into a differential equation, the Lane-Embden equation, which then can be integrated. Polytropes of n = 3/2 are used to model brown dwarfs and planets, for example, while polytropes of n = 3 are used to model stars like the Sun. Both need to be computed numerically as closed form solutions only exist for n = 0, 1 & 5.
The paper referenced above derives an expression for the gravitational binding energy of a polytrope of arbitary index. And it’s surprisingly easy…
(Omega) = -[3/(5-n)]*GM2/R
…thus a sphere of constant density (n=0) is -(3/5)*GM2/R,
n=3/2 case is -(6/7)*GM2/R,
and n=3 case is -(3/2)*GM2/R. What that means is that the Sun has squeezed into it about 5/2 times the potential energy that you’d expect from the Kelvin-Helmholtz solar model. If its energy derived from gravitational contraction then it has about 50 million years stored up inside it in its current configuration.
A puzzle of stellar structure, prior to the breakthrough that was relativity and quantum mechanics, was what was stopping a star from collapsing forever? Nothing seemed strong enough to hold back the inexorable squeeze of a star’s own gravity.