Stellar Structure and the Lane-Emden Function: Introduction

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Introduction

In Chapter IV of "An Introduction to the Study of Stellar Structure" (Dover, New York, 1958), S. Chandrasekhar discusses the equilibrium configuration of polytropic and isothermal gas spheres. There he derives, in terms of a rescaled temperature, , and rescaled radial coordinate, , an equation governing . He refers to this as the Lane-Emden equation. It has the form,

Physical constraints at the origin of coordinates require the boundary conditions,

		(2)

Chandrasekhar explicitly solves the Lane-Emden equation for special values of : 0, 1, and 5. To do this he invokes a number of clever transformations of Equation 1. For all other cases he indicates that numerical methods must be used.

An important physical parameter associated with the Lane-Emden function is the location of its first positive real zero, : . Because the density of the star is related to the temperature via , the position of the first zero of corresponds to the surface of the star, and thus gives the radius of the star.

Let's look at the explicit solutions.

For :

The boundary conditions of Equation 2 require that the integration constants and 1. Interestingly enough, because of the singularity at , the imposition of the first boundary condition fixes both and : the resulting values also solve the second boundary condition.

Here is the solution.

For this case, .

For :

Here the boundary conditions of Equation 2 require that the integration constants and . As in the case, the first boundary condition fixes both constants and those values also satisfy the second boundary condition.

Here is the solution

Here is the second boundary condition .

For this case, .

For :

In this case a number of transformations must be made before the equation yields up its solution. We will not exhibit these here, but you can consult Chandrasekhar's book and use Mathematica to reproduce and verify the manipulations.

For this case has the form   and it follows that .

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