Dividing both sides by r 2 yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with P = K ρ ρ c 1 + 1 n θ θ n + 1 and ρ ρ = ρ ρ c θ θ n , we have 1 r 2 d d r ( r 2 K ρ ρ c 1 n ( n + 1 ) d θ θ d r ) = − − 4 π π G ρ ρ c θ θ n