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>
Solution Technique #2
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_n \rho_c^{1/n} . </math>
In terms of <math>\Theta_H</math>, then, the second-order ODE becomes,
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