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>

It will be useful to note as well that, for any polytropic gas, the three key state variables are always related to one another through the simple expression,

<math> P = H\rho</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 what is referred to in the astronomical literature as the,

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

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=0}^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 is,

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

Now, it is always true for polytropic structures — see, for example, expressions at the top of this page of discussion — that <math>~\rho</math> can be related to <math>~H</math> through the expression,

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

Hence, for the specific case of an <math>~n</math> = 0 polytrope, we deduce that

This means that an <math>~n</math> = 0 polytropic sphere is also a uniform-density sphere. It should come as no surprise to discover, therefore, that the functional behavior of <math>~H</math><math>(r)</math> we have derived for the <math>~n</math> = 0 polytrope is identical to the <math>~H</math><math>(r)</math> function that we have derived elsewhere for uniform-density spheres. All of the other summarized properties of uniform-density spheres can therefore also be assigned as properties of <math>~n</math> = 0 polytropes.

<math>~n</math> = 1 Polytrope

When the polytropic index, <math>~n</math>, is set equal to unity, the Lane-Emden equation takes the form of an inhomogeneous, <math>2^\mathrm{nd}</math>-order ODE that is linear in the unknown function, <math>\Theta_H</math>. Specifically, to derive the radial distribution of the Lane-Emden function <math>\Theta_H(r)</math> for an <math>~n</math> = 1 polytrope, we must solve,

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

subject to the above-specified boundary conditions. If we multiply this equation through by <math>\xi^2</math> and move all the terms to the left-hand-side, we see that the governing ODE takes the form,

<math>\xi^2 \frac{d^2\Theta_H}{d\xi^2} + 2\xi \frac{d\Theta_H}{d\xi} + \xi^2 \Theta_H</math> = 0 ,

which is a relatively familiar <math>2^\mathrm{nd}</math>-order ODE (the spherical Bessel differential equation) whose general solution involves a linear combination of the order zero spherical Bessel functions of the first and second kind, respectively,

and,

Given the boundary conditions that have been imposed on our astrophysical problem, we can rule out any contribution from the <math>y_0</math> function. The desired solution is,

<math> \Theta_H(\xi) = j_0(\xi) = \frac{\sin\xi}{\xi} . </math>

This function is also referred to as the (unnormalized) sinc function.

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