User:Tohline/SSC/FreeEnergy/PolytropesEmbedded

Free-Energy of Truncated Polytropes

In this case, the Gibbs-like free energy is given by the sum of three separate energies,

<math>~\mathfrak{G}</math>	<math>~=</math>	<math>~W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_eV</math>
	<math>~=</math>	<math>~ - \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} - \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}} + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>

where, as derived elsewhere,

Structural Form Factors for Pressure-Truncated Polytropes <math>~(n \ne 5)</math>
<math>~\tilde\mathfrak{f}_M</math> <math>~=</math> <math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math> <math>\tilde\mathfrak{f}_W</math> <math>~=</math> <math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] </math> <math>~ \tilde\mathfrak{f}_A </math> <math>~=</math> <math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1) \biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} </math>
As we have shown separately, for the singular case of <math>~n = 5</math>, <math>~\mathfrak{f}_M</math> <math>~=</math> <math>~ ( 1 + \ell^2 )^{-3/2} </math> <math>~\mathfrak{f}_W</math> <math>~=</math> <math>~ \frac{5}{2^4} \cdot \ell^{-5} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] </math> <math>~\mathfrak{f}_A</math> <math>~=</math> <math>~ \frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] </math> where, <math>~\ell \equiv \tilde\xi/\sqrt{3} </math>

In general, then, the warped free-energy surface drapes across a four-dimensional parameter "plane" such that,

<math>~\mathfrak{G}</math>

<math>~=</math>

<math>~\mathfrak{G}(R, K, M, P_e) \, .</math>

In order to effectively visualize the structure of this free-energy surface, we will reduce the parameter space from four to two, in two separate ways: First, we will hold constant the parameter pair, <math>~(K,M)</math>; adopting Kimura's (1981b) nomenclature, we will refer to the resulting function, <math>~\mathfrak{G}_{K,M}(R,P_e)</math>, as an "M₁ Free-Energy Surface" because the mass is being held constant. Second, we will hold constant the parameter pair, <math>~(K,P_e)</math>, and examine the resulting "P₁ Free-Energy Surface," <math>~\mathfrak{G}_{K,P_e}(R,M)</math>.

The M₁ Free-Energy Surface

It is useful to rewrite the free-energy function in terms of dimensionless parameters. Here we need to pick normalizations for energy, radius, and pressure that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~M</math>. We have chosen to use,

<math>~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>

which, as is detailed in an accompanying discussion, are similar but not identical to the normalizations used by Horedt (1970) and by Whitworth (1981). The self-consistent energy normalization is,

<math>~E_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~P_\mathrm{norm} R^3_\mathrm{norm} \, .</math>

As we have demonstrated elsewhere, after implementing these normalizations, the expression that describes the M₁ Free-Energy surface is,

<math> \mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} = -3A\biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} -~ nB \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, , </math>

where the constants,

<math>~A</math>	<math>~\equiv</math>	<math>\frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \, ,</math>
<math>~B</math>	<math>~\equiv</math>	<math>~ \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, . </math>

Given the polytropic index, <math>~n</math>, we expect to obtain a different M₁ free-energy surface for each choice of the dimensionless truncation radius, <math>~\tilde\xi</math>; this choice will imply corresponding values for <math>~\tilde\theta</math> and <math>~\tilde\theta^'</math> and, hence also, corresponding (constant) values of the coefficients, <math>~A</math> and <math>~B</math>.

The P₁ Free-Energy Surface

Again, it is useful to rewrite the free-energy function in terms of dimensionless parameters. But here we need to pick normalizations for energy, radius, and mass that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~P_e</math>. As is detailed in an accompanying discussion, we have chosen to use the normalizations defined by Stahler (1983), namely,

<math>~R_\mathrm{SWS}</math>	<math>~\equiv</math>	<math>~\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
<math>~M_\mathrm{SWS}</math>	<math>~\equiv</math>	<math>~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>

The self-consistent energy normalization is,

<math>~E_\mathrm{SWS} \equiv \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}</math>

<math>~=</math>

<math>~ \biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>

After implementing these normalizations, the expression that describes the P₁ Free-Energy surface is,

<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>	<math>~=</math>	<math>~ \biggl\{\biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr\}^{-1} </math>
		<math>~ \times \biggl\{- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} - \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}} + \frac{4\pi}{3} \cdot P_e R^3 \biggr\} </math>
	<math>~=</math>	<math>~- \biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} </math>
		<math>~ \times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} + \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}} - \frac{4\pi}{3} \cdot P_e R^3 \biggr\} </math>
	<math>~=</math>	<math>~- \biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} </math>
		<math>~ \times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \frac{R_\mathrm{SWS}}{R} \biggl[G M_\mathrm{SWS}^2 R_\mathrm{SWS}^{-1} \biggr] </math>
		<math>~ + \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n} \biggl[K M_\mathrm{SWS}^{(n+1)/n} R_\mathrm{SWS}^{-3/n} \biggr] </math>
		<math>~ - \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl[ P_e R_\mathrm{SWS}^3 \biggr] \biggr\} </math>
	<math>~=</math>	<math>~- \biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} </math>
		<math>~ \times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R_\mathrm{SWS}}{R}\biggr) G \biggl[\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^2 \biggl[\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^{-1} </math>
		<math>~ + \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n} K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^{(n+1)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^{-3/n} </math>
		<math>~ - \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 P_e \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^3 \biggr\} </math>

<math>~\mathfrak{G}_{K,P_e}^*</math>	<math>~=</math>	<math>~- \biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} </math>
		<math>~ \times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R_\mathrm{SWS}}{R}\biggr) G \biggl[\biggl( \frac{n+1}{nG} \biggr)^{3} K^{4n/(n+1)} P_\mathrm{e}^{(3-n)/[(n+1)]} \biggr] \biggl[\biggl( \frac{n+1}{nG} \biggr)^{-1/2} K^{-n/(n+1)} P_\mathrm{e}^{(n-1)/[2(n+1)]} \biggr] </math>
		<math>~ + \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n} K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3(n+1)/(2n)} K^{2} P_\mathrm{e}^{(3-n)/(2n)} \biggr] \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{-3/2n} K^{-3/(n+1)} P_\mathrm{e}^{3(n-1)/[2n(n+1)]} \biggr] </math>
		<math>~ - \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 P_e \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{3n/(n+1)} P_\mathrm{e}^{3(1-n)/[2(n+1)]} \biggr] \biggr\} </math>
	<math>~=</math>	<math>~- \biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} </math>
		<math>~ \times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R_\mathrm{SWS}}{R}\biggr) \biggl[\biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr] </math>
		<math>~ + \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n} \biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr] </math>
		<math>~ - \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr] \biggr\} </math>
	<math>~=</math>	<math>~ \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)\biggr] </math>
		<math>~ - \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n} \biggl( \frac{n}{n+1} \biggr) + \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{n}{n+1} \biggr) </math>

See Also

• Index to a Variety of Free-Energy and/or Virial Analyses

© 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