Lane–Emden equation

In astrophysics, the Lane–Emden equation is Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. It is named after the astrophysicists Jonathan Homer Lane and Robert Emden. Its solution provides the run of pressure and density with radius r in terms of a re-scaled radial variable \xi and a re-scaled density variable \theta:

\frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) %2B \theta^n = 0

where

\xi = r \left(\frac{4 \pi G \rho_c^2}{(n%2B1)P_c}\right)^{\frac{1}{2}}

and

\rho = \rho_c \theta^n \,

where the subscripts "c" refer to the values of pressure and density at the center of the sphere. Here n is the polytropic index in which the pressure and density of the gas are related by the polytropic equation

P = K \rho^{1 %2B \frac{1}{n}}\,

Note that solutions to the Lane–Emden equation for a given polytropic index n are known as polytropes of index n . Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. It should be clear then if we know nothing about the gas other than the way pressure and density vary with respect to one another, we can reach a solution, in principle. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct, resulting in the Lane–Emden equation. This is a useful "zeroth order" solution for self-gravitating gaseous spheres such as stars. It is still a useful approximation in certain situations, but typically it is a rather limiting assumption.

Solutions of equation

This equation can be solved analytically when n = 0, 1 or 5:

n = 0 1 5
\theta = 1 - \frac {\xi^2}{6} \frac{\sin\xi}{\xi} \left(1%2B \frac{\xi^2}{3}\right)^{-\frac{1}{2}}
\xi_1 = \sqrt 6 \pi ∞

Here \xi_1 is the first root of \theta , corresponding to the re-scaled radius of the sphere.The equation reduces to a Spherical Bessel differential equation when n = 1 which gives a sinc function.

References

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