The integral of the squared density is ∫ ∫ 0 R max 4 π π r 2 ρ ρ ( r ) 2 d r = 4 π π 3 R s 3 ρ ρ 0 2 [ 1 − − R s 3 ( R s + R max ) 3 ] so that the mean squared density inside of R max is ⟨ ⟨ ρ ρ 2 ⟩ ⟩ R max = R s 3 ρ ρ 0 2 R max 3 [ 1 − − R s 3 ( R s + R max ) 3 ] which for the virial radius simplifies to ⟨ ⟨ ρ ρ 2 ⟩ ⟩ R v i r = ρ ρ 0 2 c 3 [ 1 − − 1 ( 1 + c ) 3 ] ≈ ≈ ρ ρ 0 2 c 3 and the mean squared density inside the scale radius is simply ⟨ ⟨ ρ ρ 2 ⟩ ⟩ R s = 7 8 ρ ρ 0 2