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 Poisson 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.

Usage of the Lane-Emden Equation: The physical Polytrope

The Lane-Emden equation has two boundary conditions, which are located at the center of the polytrope or at \xi =0:

begin{center}$\theta =1 , \frac{d\theta}{d\xi}=0.$\end{center

Since these requirements occur at the same point, the boundaries are in fact initial conditions. Thus, for every polytropic index value, there can only be one solution to the Lane-Emden equation.

The surface of a polytropic star is the value of \xiwhen \theta = 0, this will be defined as when \xi = \xi_{1}. Thus the surface is defined when the physical pressure and density go to zero.

There are only three analytic solutions: n=0. n=1 and n=5. The n=0 case corresponds to an incompressible fluid, i.e when \rho=\rho_{c}=const.. Thus the density is constant through out the star, but the pressure still goes to zero at the surface. This case is considered a crude approximation to the interior of our earth. The n=5 corresponds to the radius of this star being infinite. It can be shown that all indices greater than or equal to five will have infinite radii. The two cases that correspond to real stars are the n=1.5 and n=3 case, whose solutions are found numerically. The n=1.5 case is useful for approximating fully convective stars, and the n=3 case useful for approximation main sequence and relativistic degenerate cores of white dwarfs (though for the white dwarf case there is some special additions that need to made concerning K).

In order to make solutions of the Lane-Emden equation correspond to physical values, two input parameters are needed. These parameters can be K and central density, stellar mass and stellar radius, K and stellar mass etc. In the wiki:LE_Module, we require stellar mass and central density be given. We do not explicitly calculate K, but rather redefine a few terms as follows: begin{center}$\rho=\rho_{c} \theta^{n}$\end{center begin{center}$P=\frac{4 \pi G \alpha^{2} \rho_{c}^{2} \theta^{n+1}}{n+1}$\end{center where begin{center}$\alpha=[\frac{-M}{4 \pi \rho_{c} \xi_{1}^{2}} (\frac{d\theta}{d\xi})_{\xi_{1}}]^{1/2}$\end{center

Test case of n=3

Our initial parameters for this test case are: \rho_{c}= 76.13 g/cm^{3}, M= 1.989*10^{33}g ,density ratio and edge contrast both equal 10.

The code outputs: \xi_{1}= 6.8968, P_{c}= 1.25*10^{17} dynes/cm^{2} , andR_{outer} \propto 10^{10}cm .

Initial conditions:

2.5 sound crossing time: