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,

where ${\displaystyle P}$ and ${\displaystyle \rho }$ are the pressure and density, respectively, and ${\displaystyle K}$ is a constant of proportionality. The standard boundary conditions are ${\displaystyle \theta (0)=1}$ and ${\displaystyle \theta '(0)=0}$. Solutions thus describe the run of pressure and density with radius and are known as polytropes of index ${\displaystyle n}$. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.