Advanced Astrophysics (2) - Solution

For a uniform sphere with a polytropic index n = 0 for uniform density, show that V = -6/5 (GM²/R). Take the potential to be:

W=  3/ (n – 5) (GM²/ R )

For any polytropic gas sphere.

Then we may simplify using n = 0 to get:

W=  3/ (0 – 5) (GM²/ R ) = -3/5 (GM²/ R )

We have already shown that: V = 2W  so:

V = 2W  = 2 [-3/5 (GM²/ R )] = -6/5 (GM²/ R)