Polytrope of index five
= Polytrope of index five
{title2=$\theta=(1+\xi^2/3)^{-1/2}$}
The regular $n=5$ <Lane-Emden equation> solution has no finite first zero. Its <Lane-Emden mass formula> gives $m(\xi)=4\pi\rho_c\alpha^3\xi^3/[3(1+\xi^2/3)^{3/2}]$, so its infinite-radius total <mass> is $4\pi\sqrt3\rho_c\alpha^3$. Its enclosed mean <mass density> is $\rho_c(1+\xi^2/3)^{-3/2}$ and tends to zero. A finite externally confined core instead requires a specified truncation pressure.