Lane-Emden equation

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden. The equation reads

where ${\displaystyle \xi }$ is a dimensionless radius and ${\displaystyle \theta }$ is related to the density (and thus the pressure) by ${\displaystyle \rho =\rho _{c}\theta ^{n}}$ for central density ${\displaystyle \rho _{c}}$. The index ${\displaystyle n}$ is the polytropic index that appears in the polytropic equation of state,

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