Polytropic Spheres (structure)

Here we will supplement the simplified set of principal governing equations with a polytropic equation of state, as defined in our overview of supplemental relations for time-independent problems. Specifically, we will assume that <math>~\rho</math> is related to <math>~H</math> through the relation,

<math>~\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>

Governing Relations

Lane-Emden Equation

Adopting solution technique #2, we need to solve the following second-order ODE relating the two unknown functions, <math>~\rho</math> and <math>~H</math>:

<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .

Inserting the polytropic <math>~\rho</math>(<math>~H</math>) function, shown above, into the right-hand-side of this ODE, we obtain,

<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n</math> .

Our task is to solve this ODE to determine <math>~H</math>(<math>r</math>),for various values of the polytropic index, <math>~n</math>.

It is customary to replace <math>~H</math> everywhere by a dimensionless polytropic enthalpy, <math>\Theta_H</math>, such that,

<math> \Theta_H \equiv \frac{H}{H_c} = \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} , </math>

where the central value of the enthalpy, <math>H_c</math>, is related to the central density, <math>\rho_c</math>, through the expression,

<math> H_c = (n+1)K_\mathrm{n} \rho_c^{1/n} . </math>

In terms of <math>\Theta_H</math>, then, the governing relation becomes,

<math>\biggl[ \frac{(n+1)K_\mathrm{n}}{4\pi G}~ \rho_c^{1/n - 1} \biggr] \frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\Theta_H}{dr} \biggr) = - \Theta_H^n</math> .

The term inside the square brackets on the left-hand-side has dimensions of length-squared, so it is also customary to define a dimensionless radius,

<math> \xi \equiv \frac{r}{a_\mathrm{n}} , </math>

where,

<math> a_\mathrm{n} \equiv \biggl[ \frac{(n+1)K_\mathrm{n}}{4\pi G}~ \rho_c^{1/n - 1} \biggr]^{1/2} , </math>

in which case our governing ODE becomes,

<math>\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math> .

In the astronomical literature, this is referred to as the Lane-Emden equation.

Boundary Conditions

Given that it is a <math>2^\mathrm{nd}</math>-order ODE, a solution of the Lane-Emden equation will require specification of two boundary conditions. Based on our definition of the variable <math>\Theta_H</math>, one obvious boundary condition is to demand that <math>\Theta_H = 1</math> at the center (<math>\xi=0</math>) of the configuration. In astrophysically interesting structures, we also expect the first derivative of many physical variables to go smoothly to zero at the center of the configuration — see, for example, the radial behavior that was derived for <math>~P</math>, <math>~H</math>, and <math>~\Phi</math> in a uniform-density sphere. Hence, we will seek solutions to the Lane-Emden equation where <math>d\Theta_H /d\xi = 0</math> at <math>\xi=0</math> as well.

Known Analytic Solutions

While the Lane-Emden equation has been studied for over 100 years, to date, analytic solutions to the equation (subject to the above specified boundary conditions) have been found only for three values of the polytropic index, <math>~n</math>. We will review these three solutions here.

<math>~n</math> = 0 Polytrope

When the polytropic index, <math>~n</math>, is set equal to zero, the right-hand-side of the Lane-Emden equation becomes a constant (<math>-1</math>), so the equation can be straightforwardly integrated, twice, to obtain the desired solution for <math>\Theta_H(\xi)</math>. Specifically, the first integration along with enforcement of the boundary condition on <math>d\Theta_H/d\xi</math> at the center gives,

<math> \xi^2 \frac{d\Theta_H}{d\xi} = - \frac{1}{3}\xi^3 . </math>

Then the second integration along with enforcement of the boundary condition on <math>\Theta_H</math> at the center gives,

<math> \Theta_H = 1 - \frac{1}{6}\xi^2 . </math>

This function varies smoothly from unity at the center of the configuration (as required by one of the boundary conditions) to zero at <math>\xi = \sqrt{6}</math>, then becomes negative for larger values of <math>\xi</math>.

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