Polytropes

The internal structure of a star can be approximated quite simply by the usage of polytropes. As opposed to manipulating the full rigorous solution's of all the equations of stellar structure, one can assume a power law relationship between density and pressure.

A derivation of the Lane-Emden Equation

So our three unknowns in this system are pressure, P, density, \rho, and mass as a function of radius, m(r).

We have two obvious equations right from the start that will do a good job with describing a spherical star: mass continuity and hydrostatic equilibrium. The two ordinary, first order differential equations are described as follows:

begin{center}$\frac{dP}{dr}=-\frac{G M_{r}}{r} \rho$ \end{center,

begin{center}$\frac{dM_{r}}{dr}=4 \pi r^{2} \rho$\end{center.

To be able to solve for our third unknown, we need an additional equation: thus, the power law relationship between pressure and density can be used,

begin{center}$P= K \rho^{\gamma} = K \rho^{(n+1)/n}$\end{center.

The adiabatic index \gamma, the parameter the characterizes the specific heat of a gas, can be redefined in terms of some n that we call the polytropic index. K will be assumed a constant. These three sets of equations can now be combined into a second order Poisson.

begin{center}$\frac{1}{r^{2}}\frac{d}{dr}(\frac{r^{2}}{\rho}\frac{dP}{dr})=-4 \pi G \rho$\end{center.

In order to make this equation less cumbersome, we will also introduce the following dimensionless variables:

begin{center}$\rho=\rho_{c} \theta^{n}, P=P_{c} \theta^{n+1}, r=\alpha \xi$\end{center,

where \theta is the dimensionless polytropic temperature, \xi is a dimensionless radial variable, \rho_{c} is the central density, P_{c} is the central pressure, and \alpha is a length constant defined as

begin{center}$\alpha^{2}=\frac{K (n+1) \rho_{c}^{\frac{1-n}{n}}}{4 \pi G}$\end{center.

Combining this with our polytropic pressure equation, we get a Possion equation in dimensionless variables which is known as the lane emden equation:

begin{center}$\frac{1}{\xi^{2}}\frac{d}{d \xi}(\xi^{2} \frac{d \theta}{d \xi}) = -\theta^{n}$\end{center.