But for any polytrope, P c = W n G M 2 R 4 and ρ ρ c = K n ρ ρ avg . W n , K n are all constants independent of pressure and density, and the average density is defined as ρ ρ avg ≡ ≡ M 4 3 π π R 3 . Plugging this 2 equations into the equation for C, we have: C ∼ ∼ M 2 − − γ γ R 3 γ γ − − 4 where all multiplicative constants have been ignored. Recall that our original definition of C was: P 1 − − γ γ T γ γ = C