User:Tohline/SSC/FreeEnergy/PowerPoint

Supporting Derivations for Free-Energy PowerPoint Presentation

The derivations presented here are an extension of our accompanying free-energy synopsis. These additional details proved to be helpful while developing an overarching PowerPoint presentation.

General Free-Energy Expression

We're considering a free-energy function of the following form:

<math>~\mathfrak{G}^*_\mathrm{type}</math>

<math>~=</math>

<math>~-ax^{-1} + b x^{-3/n} + c x^{-3/j} + \mathfrak{G}_0 \, ,</math>

where,

<math>~x \equiv \frac{R}{R_0} \, .</math>

As we have shown, setting,

<math>~\frac{\partial \mathfrak{G}^*_\mathrm{type}}{\partial x}</math>

<math>~=</math>

<math>~0 \, ,</math>

generates a mathematical statement of virial equilibrium, namely,

<math>~\frac{ b}{nc}\cdot x^{(n-3)/n }_\mathrm{eq} - \frac{a}{3c} + \frac{1}{j}\cdot x^{(j-3)/j}_\mathrm{eq} </math>

<math>~=</math>

<math>~ 0 \, .</math>

And equilibrium configurations for which the second (as well as first) derivative of the free energy is zero are found at "critical" radii given by the expression,

<math>~[x_\mathrm{eq}^{(j-3)/j}]_\mathrm{crit} </math>

<math>~=</math>

<math>~ \frac{a}{3^2c}\biggl[ \frac{j^2(n-3)}{n-j} \biggr] \, . </math>

Pressure-Truncated Polytropes

For pressure-truncated polytropes, set <math>~j=-1</math> and let <math>~n</math> be the chosen polytropic index. In this case, the statement of virial equilibrium is,

<math>~\frac{ b}{nc}\cdot x^{(n-3)/n }_\mathrm{eq} - \frac{a}{3c} - x^{4}_\mathrm{eq} </math>

<math>~=</math>

<math>~ 0 \, ;</math>

and the critical equilibrium configuration has,

<math>~[x_\mathrm{eq}]_\mathrm{crit} </math>

<math>~=</math>

<math>~ \biggl[ \frac{a(n-3)}{3^2c (n+1)}\biggr]^{1/4} \, . </math>

Case M

Set <math>~K</math> and <math>~M_\mathrm{tot}</math> constant and examine how the free-energy behaves as a function of the coordinates, <math>~(R,P_e)</math>. In this case (see, for example, here),

<math>~a</math>	<math>~\equiv</math>	<math>~\frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}\, , </math>
<math>~b</math>	<math>~\equiv</math>	<math>~n\biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, , </math>
<math>~c</math>	<math>~\equiv</math>	<math>~\frac{4\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, , </math>

where the structural form factors for pressure-truncated polytropes are precisely defined here. And (see, for example, here),

<math>~R_0 = R_\mathrm{norm}</math>	<math>~\equiv</math>	<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
<math>~P_\mathrm{norm}</math>	<math>~\equiv</math>	<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, .</math>

If we set all three structural form-factors to unity, we have,

<math>~\frac{a}{3c}</math>	<math>~=</math>	<math>~\frac{3}{2^2\cdot 5\pi}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, ,</math>
<math>~\frac{b}{nc}</math>	<math>~=</math>	<math>~\biggl(\frac{3}{4\pi} \biggr)^{(n+1)/n} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, ,</math>

so the statement of virial equilibrium becomes,

<math>~ x^{4}_\mathrm{eq} </math>	<math>~=</math>	<math>~\biggl[ \biggl(\frac{3}{4\pi} \biggr)^{(n+1)/n} x^{(n-3)/n }_\mathrm{eq} - \frac{3}{2^2\cdot 5\pi}\biggr]\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} </math>
	<math>~=</math>	<math>~\frac{3}{2^2\cdot 5\pi}\biggl[ 5\biggl(\frac{3}{4\pi} \biggr)^{1/n} x^{(n-3)/n }_\mathrm{eq} - 1\biggr]\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} </math>
<math>~ \Rightarrow ~~~ \frac{P_e}{P_\mathrm{norm}}</math>	<math>~=</math>	<math>~\frac{3}{2^2\cdot 5\pi x^{4}_\mathrm{eq} }\biggl[ 5\biggl(\frac{3}{4\pi} \biggr)^{1/n} x^{(n-3)/n }_\mathrm{eq} - 1\biggr] \, . </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