Instabilities in Bounded and Composite Polytropes

Unbounded, Complete Polytropes

Free-Energy Function and Its Derivatives

The free-energy function that is relevant to a discussion of the structure and stability of unbounded configurations having polytropic index, <math>~n</math>, has the form,

<math>~\mathcal{G}(x)</math>

<math> -ax^{-1} +b x^{-3/n} + \mathcal{G}_0 \, , </math>

where <math>~x \equiv R/R_\mathrm{SWS}</math> identifies the radius of the configuration and <math>\mathcal{G}_0</math> is an arbitrary constant. If the coefficients, <math>~a, b</math>, and <math>~c</math>, are held constant while varying the configuration's size, we see that,

<math>~\frac{d\mathcal{G}}{dx}</math>

<math> x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} \biggr] \, , </math>

and,

<math>~\frac{d^2\mathcal{G}}{dx^2}</math>

<math> x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} \biggr] \, . </math>

In terms of the system's mass and its structural form factors, <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_A</math>, and <math>~\mathfrak{f}_W</math>, the two relevant coefficients are,

<math>~a</math>	<math>~=</math>	<math>~\frac{3}{5}\biggl(\frac{n+1}{n}\biggr) \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^2 \mathfrak{f}_W \, ,</math>
<math>~b</math>	<math>~=</math>	<math>~n \biggl( \frac{3}{4\pi}\biggr)^{1/n} \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n} \mathfrak{f}_A \, . </math>

Equilibrium Radius

A configuration's equilibrium radius, <math>~x_\mathrm{eq}</math>, can be determined one of two ways:

Extrema in the Free Energy

Equilibria are identified by extrema in the free-energy function. Setting <math>d\mathcal{G}/dx = 0</math>, we find,

<math>~x_\mathrm{eq}</math>	<math>~=</math>	<math> \biggl(\frac{an}{3b} \biggr)^{n/(n-3)} </math>
<math>~\Rightarrow ~~~~~ R_\mathrm{eq}^{n-3}</math>	<math>~=</math>	<math>\frac{R_\mathrm{SWS}^{n-3}}{M_\mathrm{SWS}^{n-1}} \biggl( \frac{4\pi}{3\cdot 5^n}\biggr) \biggl(\frac{n+1}{n}\biggr)^n \biggl[ \frac{\mathfrak{f}_W^n \mathfrak{f}_M^{1-n}}{\mathfrak{f}_A^n} \biggr] M^{n-1} </math>
	<math>~=</math>	<math>\biggl( \frac{4\pi}{3\cdot 5^n}\biggr) \biggl[ \frac{\mathfrak{f}_W^n \mathfrak{f}_M^{1-n}}{\mathfrak{f}_A^n} \biggr] G^n K^{-n} M^{n-1} \, . </math>

If one assumes that the equilibrium system has no internal structure — that is, that the interior density and temperature are uniform throughout — then <math>~\mathfrak{f}_M = \mathfrak{f}_A = \mathfrak{f}_W = 1</math> and this derived expression gives a good estimate of the equilibrium radius, given any choice of the pair of parameters, <math>~M</math> and <math>~K</math>.

Detailed Force Balance

Alternatively, a solution of 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>

gives information regarding the detailed interior structure of the equilibrium system via knowledge of the properties of the Lane-Emden function, <math>~\Theta_H(\xi)</math>, as well as an exact expression for the equilibrium radius, namely,

<math>~R_\mathrm{eq}^{n-3} = \frac{3^n}{(n+1)^n} \biggl(\frac{4\pi}{3}\biggr) G^n K^{-n} M^{n-1} \cdot \frac{\mathfrak{f}_M^{1-n}}{\xi_1^{2n}} \, , </math>

where,

<math>~\mathfrak{f}_M \equiv \frac{\bar\rho}{\rho_c} = \biggl(- \frac{3\Theta_H^'}{\xi} \biggr)_{\xi_1} \, .</math>

Together

Now, once the Lane-Emden function, <math>~\Theta_H</math>, is known from a detailed force-balance model, the other two structural form factors also can be straightforwardly determined. For unbounded, complete polytropes, the relevant expressions are,

<math>~\mathfrak{f}_W</math>	<math>~=</math>	<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta_H^'}{\xi} \biggr]^2_{\xi_1} \, ,</math>
<math>~\mathfrak{f}_A</math>	<math>~=</math>	<math>~\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta_H^' \biggr]^2_{\xi_1} \, .</math>

Plugging the appropriate ratio of these two functions, namely,

<math>~\biggl[\frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n</math>

<math>~\biggl( \frac{3\cdot 5}{n+1} \biggr)^n \biggl[ \frac{1}{\xi} \biggr]^{2n}_{\xi_1} \, ,</math>

into the expression for the equilibrium radius obtained from the free-energy analysis gives precisely the same answer as was obtained from the detailed force-balance analysis. Using either method of determination we conclude, therefore, that,

<math>~\frac{G R_\mathrm{eq}^{n-3}}{K^n M^{n-1}} = \frac{4\pi}{(n+1)^n} \cdot \xi_1^{-(n+1)} (-\Theta_H^')^{1-n}_{\xi_1} \, . </math>

Related Discussions

Wikipedia introduction to the Lane-Emden equation
Wikipedia introduction to Polytropes

User:Tohline/SSC/Stability BoundedCompositePolytropes

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