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.
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General Free-Energy Expression
We're considering a free-energy function of the following form:
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<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,
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<math>~\frac{\partial \mathfrak{G}^*_\mathrm{type}}{\partial x}</math> |
<math>~=</math> |
<math>~0 \, ,</math> |
generates a mathematical statement of virial equilibrium, namely,
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<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,
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<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,
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<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,
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<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),
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<math>~a</math> |
<math>~\equiv</math> |
<math>~\frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}\, , </math> |
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<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> |
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<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),
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<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> |
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<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,
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
And, along the equilibrium sequence that is generated by this virial expression, the transition from stable to unstable configurations occurs at,
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<math>~[x_\mathrm{eq}]^4_\mathrm{crit} </math> |
<math>~=</math> |
<math>~ \biggl[ \frac{(n-3)}{3(n+1)}\biggr] \frac{3}{2^2\cdot 5\pi}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} </math> |
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<math>~\Rightarrow ~~~ \frac{2^2\cdot 5\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)[x_\mathrm{eq}]^4_\mathrm{crit} </math> |
<math>~=</math> |
<math>~ \biggl[ \frac{(n-3)}{3(n+1)}\biggr] </math> |
which, in combination with the virial equilibrium condition gives,
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<math>~5\biggl(\frac{3}{4\pi} \biggr)^{1/n} [x_\mathrm{eq}]^{(n-3)/n }_\mathrm{crit} -1</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{(n-3)}{3(n+1)}\biggr] </math> |
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© 2014 - 2021 by Joel E. Tohline |