Stellar Structure and the Lane-Emden Function:
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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
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.
The boundary conditions of Equation 2 require that the integration
Interestingly enough, because of the singularity at ,
the imposition of the first boundary condition fixes both
the resulting values also solve the second boundary condition.
Here is the solution.
For this case, .
Here the boundary conditions of Equation 2 require that the integration
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, .
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 .
Numerical Solution >>