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> .

It is customary to replace <math>~H</math> and <math>~\rho</math> in this equation 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 mathematical relationship between <math>~H</math>/<math>H_c</math> and <math>~\rho</math>/<math>\rho_c</math> comes from the adopted barotropic (polytropic) relation identified above. To accomplish this, we replace <math>~H</math> with <math>H_c \Theta_H</math> on the left-hand-side of the governing differential equation and we replace <math>~\rho</math> with <math>\rho_c \Theta_H^n</math> on the right-hand-side, then gather the constant coefficients together on the left. The resulting ODE is,

<math>\biggl[ \frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr) \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{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\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. Our task is to solve this ODE to determine the behavior of the function <math>\Theta_H(\xi)</math> — and, from it in turn, determine the radial distribution of various dimensional physical variables — for various values of the polytropic index, <math>~n</math>.

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 <math>\xi = 0</math> (as required by one of the boundary conditions) to zero at <math>\xi = \xi_1 = \sqrt{6}</math> (by tradition, the subscript "1" is used to indicate that it is the "first" zero of the Lane-Emden function), then becomes negative for values of <math>\xi > \xi_1</math>.

The astrophysically interesting surface of this spherical configuration is identified with the first zero of the function, that is, where the dimensionless enthalpy first goes to zero. In other words, the dimensionless radius <math>\xi_1</math> should correspond with the dimensional radius of the configuration, <math>R</math>. From the definition of <math>\xi</math>, we therefore conclude that,

and

<math> \xi = \sqrt{6} \biggl(\frac{r}{R} \biggr) , </math>

Hence, the Lane-Emden function solution can also be written as,

<math> \Theta_H = \frac{H}{H_c} = 1 - \biggl(\frac{r}{R}\biggr)^2 . </math>

Since,

<math> a_n^2 = \frac{1}{4\pi G} \biggl(\frac{H_c}{\rho_c}\biggr) = \frac{R^2}{6} , </math>

we also conclude that,

This, combined with the Lane-Emden function solution, tells us that the run of enthalpy through the configuration as,

<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1 - \biggl(\frac{r}{R}\biggr)^2 \biggr]. </math>

Since, for an <math>~n</math> = 0 polytrope, <math>P/P_c = H/H_c</math>, we also deduce that the run of pressure through the equilibrium configuration is,

<math> \frac{P(r)}{P_c} = 1 - \biggl(\frac{r}{R}\biggr)^2 . </math>

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