Emden–Chandrasekhar equation

In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden and Subrahmanyan Chandrasekhar. The equation was first introduced by Robert Emden in 1907. The equation reads

where ${\displaystyle \xi }$ is the dimensionless radius and ${\displaystyle \psi }$ is the related to the density of the gas sphere as ${\displaystyle \rho =\rho _{c}e^{-\psi }}$, where ${\displaystyle \rho _{c}}$ is the density of the gas at the centre. The equation has no known explicit solution. If a polytropic fluid is used instead of an isothermal fluid, one obtains the Lane–Emden equation. The isothermal assumption is usually modeled to describe the core of a star. The equation is solved with the initial conditions,

${\displaystyle \psi =0,\quad {\frac {d\psi }{d\xi }}=0\quad {\text{at}}\quad \xi =0.}$

The equation appears in other branches of physics as well, for example the same equation appears in the Frank-Kamenetskii explosion theory for a spherical vessel. The relativistic version of this spherically symmetric isothermal model was studied by Subrahmanyan Chandrasekhar in 1972.