User:Tohline/SSC/Structure/Polytropes

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_\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> .

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation