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)