User:Tohline/SSC/Structure/Polytropes
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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.
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