Isothermal Lane-Emden equation (${\xi }^{-2}({\xi }^2{\Psi }^{\prime }{)}^{\prime }=e^{-\Psi }$)

For an isothermal sphere, put $\rho ={\rho }_c{e}^{-\Psi }$ and $r=a\xi$ with $a^2=K/(4\pi G{\rho }_c)$. The equilibrium equation becomes ${\xi }^{-2}({\xi }^2{\Psi }^{\prime }{)}^{\prime }=e^{-\Psi }$. Finite central density and smooth spherical symmetry require $\Psi (0)={\Psi }^{\prime }(0)=0$. The regular expansion is $\Psi ={\xi }^2/6-{\xi }^4/120+{\xi }^6/1890+O({\xi }^8)$. The enclosed mass is $M=4\pi {\rho }_c{a}^3{\xi }^2{\Psi }^{\prime }$, following by integrating the differential equation.