= Isothermal Lane-Emden equation
{title2=$\xi^{-2}(\xi^2\Psi')'=e^{-\Psi}$}
For an <isothermal sphere>, put $\rho=\rho_ce^{-\Psi}$ and $r=a\xi$ with $a^2=K/(4\pi G\rho_c)$. The equilibrium equation becomes $\xi^{-2}(\xi^2\Psi')'=e^{-\Psi}$. Finite central density and smooth <spherical symmetry> require $\Psi(0)=\Psi'(0)=0$. The regular expansion is $\Psi=\xi^2/6-\xi^4/120+\xi^6/1890+O(\xi^8)$. The enclosed mass is $M=4\pi\rho_ca^3\xi^2\Psi'$, following by integrating the differential equation.
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