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>~ - 3\mathcal{A} \biggl[\frac{GM^2}{R} \biggr] + n\mathcal{B} \biggl[ \frac{KM^{(n+1)/n}}{R^{3/n}} \biggr] + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>

where the constants,

<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>

and

<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, ,</math>

and, 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>; giving a nod to Kimura's (1981b) nomenclature, we will refer to the resulting function, <math>~\mathfrak{G}_{K,M}(R,P_e)</math>, as a "Case M" 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 "Case P" free-energy surface, <math>~\mathfrak{G}_{K,P_e}(R,M)</math>.

Case M 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 "Case M" free-energy surface is,

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

Given the polytropic index, <math>~n</math>, we expect to obtain a different "Case M" 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>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.

Case P 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 \biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}</math>

<math>~=</math>

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

After implementing these normalizations — see our accompanying analysis for details — the expression that describes the "Case P" free-energy surface is,

<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>

<math>~=</math>

<math>~- 3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} + n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} + \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \, . </math>

Given the polytropic index, <math>~n</math>, we expect to obtain a different "Case P" 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>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.

Free-Energy of Bipolytropes

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

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

<math>~=</math>

<math>~ \biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{core} + \biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{env} \, . </math>

In addition to specifying (generally) separate polytropic indexes for the core, <math>~n_c</math>, and envelope, <math>~n_e</math>, and an envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>, we will assume that the system is fully defined via specification of the following five physical parameters:

• Total mass, <math>~M_\mathrm{tot}</math>;
• Total radius, <math>~R</math>;
• Interface radius, <math>~R_i</math>, and associated dimensionless interface marker, <math>~q \equiv R_i/R</math>;
• Core mass, <math>~M_c</math>, and associated dimensionless mass fraction, <math>~\nu \equiv M_c/M_\mathrm{tot}</math>;
• Polytropic constant in the core, <math>~K_c</math>.

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

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

<math>~=</math>

<math>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, q, \nu) \, .</math>

Order of Magnitude Derivation

Let's begin by providing very rough, approximate expressions for each of these four terms, assuming that <math>~n_c = 5</math> and <math>~n_e = 1</math>.

<math>~W_\mathrm{grav}\biggr|_\mathrm{core}</math>	<math>~\approx</math>	<math>~- \mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot} M_c}{(R_i/2)} \biggr] = - 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] \, ;</math>
<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>	<math>~\approx</math>	<math>~- \mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot} M_e}{(R_i+R)/2} \biggr] = - 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] \, ;</math>
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{core} = U_\mathrm{int}\biggr|_\mathrm{core} </math>	<math>~\approx</math>	<math>~\mathfrak{b}_c \cdot n_cK_c M_c ({\bar\rho}_c)^{1/n_c} = 5\mathfrak{b}_c \cdot K_c M_\mathrm{tot}\nu \biggl[ \frac{3M_c}{4\pi R_i^3} \biggr]^{1/5} </math>
	<math>~=</math>	<math>~\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5} \, ;</math>
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>	<math>~\approx</math>	<math>~\mathfrak{b}_e \cdot n_eK_e M_\mathrm{env} ({\bar\rho}_e)^{1/n_e} = \mathfrak{b}_e \cdot K_e M_\mathrm{tot}(1-\nu) \biggl[ \frac{3M_\mathrm{env}}{4\pi (R^3-R_i^3)} \biggr] </math>
	<math>~=</math>	<math>~ \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) K_e [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} \, . </math>

In writing this last expression, it has been necessary to (temporarily) introduce a sixth physical parameter, namely, the polytropic constant that characterizes the envelope material, <math>~K_e</math>. But this constant can be expressed in terms of <math>~K_c</math> via a relation that ensures continuity of pressure across the interface while taking into account the drop in mean molecular weight across the interface, that is,

<math>~K_e ({\bar\rho}_e)^{(n_e+1)/n_e}</math>	<math>~\approx</math>	<math>~K_c ({\bar\rho}_c)^{(n_c+1)/n_c}</math>
<math>~\Rightarrow ~~~~ K_e \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr) {\bar\rho}_c\biggr]^{2}</math>	<math>~\approx</math>	<math>~K_c ({\bar\rho}_c)^{6/5}</math>
<math>~\Rightarrow ~~~~ \frac{K_e}{K_c} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}</math>	<math>~\approx</math>	<math>~\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5} \, .</math>

Hence, the fourth energy term may be rewritten in the form,

<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>	<math>~\approx</math>	<math>~ \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} K_c\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5} [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} </math>
	<math>~=</math>	<math>~ \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5} \frac{(1-\nu)^2}{(1-q^3)} \, . </math>

Putting all the terms together gives,

<math>~\mathfrak{G}</math>	<math>~\approx</math>	<math>~ - 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] - 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] + \mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5} </math>
		<math>~ + \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5} \frac{(1-\nu)^2}{(1-q^3)} </math>
	<math>~=</math>	<math>~ - 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] + \mathcal{B}_\mathrm{biP} K_c \biggl[\frac{(\nu M_\mathrm{tot})^{2}}{ qR} \biggr]^{3/5} </math>
<math>~\Rightarrow ~~~~ \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~ - 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} + \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} K_c \biggl[\frac{M_\mathrm{tot}^{2}}{ R} \biggr]^{3/5}\biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} </math>
	<math>~=</math>	<math>~ - 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{R_\mathrm{norm}}{R} \biggr] + \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggl[\frac{R_\mathrm{norm}}{ R} \biggr]^{3/5} \, , </math>

where,

<math>~\mathcal{A}_\mathrm{biP}</math>	<math>~\equiv</math>	<math>~\biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr) + \mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] \, ,</math>
<math>~\mathcal{B}_\mathrm{biP}</math>	<math>~\equiv</math>	<math>~\biggl( \frac{3}{2^2\pi} \biggr)^{1/5} \biggl[5\mathfrak{b}_c + \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] \, .</math>

Equilibrium Radius

Order of Magnitude Estimate

This means that,

<math>~\frac{\partial\mathfrak{G}}{\partial R}</math>

<math>~=</math>

<math>~ + 2 \mathcal{A}_\mathrm{biP}\biggl[ \frac{GM_\mathrm{tot}^2 }{R^2} \biggr] - \frac{3}{5} \mathcal{B}_\mathrm{biP} K_c \biggl[\frac{\nu^{2}}{ q} \biggr]^{3/5} M_\mathrm{tot}^{6/5} R^{-8/5} \, . </math>

Hence, because equilibrium radii are identified by setting <math>~\partial\mathfrak{G}/\partial R = 0</math>, we have,

<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>

<math>~=</math>

<math>~\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl[\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}\biggr]^{5/2} \biggl(\frac{ q} {\nu^{2}}\biggr)^{3/2} \, . </math>

Reconcile With Known Analytic Expression

From our earlier derivations, it appears as though,

<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>	<math>~=</math>	<math>~\biggl( \frac{3^8}{2^5\pi} \biggr)^{-1/2} \biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} </math>
	<math>~=</math>	<math>~\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl(\frac{q}{\nu^2} \biggr)^{3/2} \biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2} \frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr] \, . </math>

This implies that,

<math>~\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}</math>	<math>~\approx</math>	<math>~ \biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2} \frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr]^{2/5} </math>
	<math>~=</math>	<math>~\biggl(\frac{\nu^2}{q} \biggr) \biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/5} \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} </math>
<math>~\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr) + \mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] </math>	<math>~\approx</math>	<math>~\frac{1}{2^2\cdot 5}\biggl(\frac{\nu^2}{q} \biggr) \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c + \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] </math>
<math>~\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c + \mathfrak{a}_e \cdot \frac{q(1-\nu)}{\nu(1+q)} \biggr] </math>	<math>~\approx</math>	<math>~\frac{\nu}{2^2\cdot 5} \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c + \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] </math>

Focus on Five-One Free-Energy Expression

Approximate Expressions

Let's plug this equilibrium radius back into each term of the free-energy expression.

<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr|_\mathrm{core}</math>	<math>~\approx</math>	<math>~- 2\mathfrak{a}_c \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} \biggl[ \frac{GM_\mathrm{tot}^2 }{R_\mathrm{eq}} \biggl(\frac{\nu}{q}\biggr) \biggr] </math>
	<math>~=</math>	<math>~- 2\mathfrak{a}_c \biggl(\frac{\nu}{q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math>
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr|_\mathrm{env}</math>	<math>~\approx</math>	<math>~- 2\mathfrak{a}_e \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} \biggl[ \frac{GM_\mathrm{tot}^2 }{R_\mathrm{eq}} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] </math>
	<math>~=</math>	<math>~- 2\mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math>
<math>~\frac{S_\mathrm{core}}{E_\mathrm{norm}} = \biggl[\frac{3(\gamma_c-1)}{2}\biggr] \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr|_\mathrm{core} </math>	<math>~\approx</math>	<math>~\biggl[\frac{3}{2\cdot 5}\biggr]\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} K_c (M_\mathrm{tot}\nu)^{6/5} (R_\mathrm{eq}q)^{-3/5} </math>
	<math>~=</math>	<math>~ \biggl[\frac{3}{2\cdot 5}\biggr]\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)^{3/5} \, ;</math>
<math>~\frac{S_\mathrm{env}}{E_\mathrm{norm}} = \biggl[\frac{3(\gamma_e-1)}{2}\biggr] \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr|_\mathrm{env} </math>	<math>~\approx</math>	<math>~\biggl[\frac{3}{2}\biggr] \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} K_c M_\mathrm{tot}^{6/5}R_\mathrm{eq}^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5} \frac{(1-\nu)^2}{(1-q^3)} </math>
	<math>~=</math>	<math>~\biggl[\frac{3}{2}\biggr] \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[ \frac{q^3}{\nu} \biggr]^{4/5} \frac{(1-\nu)^2}{(1-q^3)} \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)^{3/5} \, . </math>

From Detailed Force-Balance Models

In the following derivations, we will use the expression,

<math>~\chi_\mathrm{eq} \equiv \frac{ R_\mathrm{eq}}{R_\mathrm{norm}}</math>

<math>~=</math>

<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr)^3 \biggl( \frac{\pi}{2^3} \biggr)^{1/2} \frac{1}{A^2\eta_s} = \biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} \, .</math>

Keep in mind, as well — as derived in an accompanying discussion — that,

<math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>

<math>~=</math>

<math>~ (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, ,</math>

where,

<math>m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>

From the accompanying Table 1 parameter values, we also can write,

<math>~q</math>	<math>~=</math>	<math>~\frac{\eta_i}{\eta_s} = \eta_i \biggl\{\frac{\pi}{2} + \eta_i + \tan^{-1}\biggl[ \frac{1}{\eta_i} - \ell_i \biggr] \biggr\}^{-1}</math>
	<math>~=</math>	<math>~ \eta_i \biggl\{\eta_i + \cot^{-1}\biggl[ \ell_i - \frac{1}{\eta_i} \biggr] \biggr\}^{-1} \, , </math>

where,

<math>~\eta_i</math>

<math>~=</math>

<math>~m_3 \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] \, .</math>

Let's also define the following shorthand notation:

<math>~\mathfrak{L}_i</math>	<math>~\equiv</math>	<math>~\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i \, ;</math>
<math>~\mathfrak{K}_i</math>	<math>~\equiv</math>	<math>~\frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] + \frac{\Lambda_i}{\eta_i} \, .</math>

Gravitational Potential Energy of the Core

Pulling from our detailed derivations,

<math>~\biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr] \, .</math>
<math>~\Rightarrow ~~~~ -\chi_\mathrm{eq} \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr] \biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} </math>
	<math>~=~</math>	<math>~ \biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^5} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] </math>

Out of equilibrium, then, we should expect,

<math>~\frac{W_\mathrm{core}}{E_\mathrm{norm}} </math>	<math>~=~</math>	<math>~ - \chi^{-1} \biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^5} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] </math>
	<math>~=~</math>	<math>~ - \chi^{-1} \biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2} \biggl[ \mathfrak{L}_i - \frac{8}{3} \biggr] \, , </math>

which, in comparison with our above approximate expression, implies,

<math>~\mathfrak{a}_c </math>

<math>~=~</math>

<math>~ \biggl( \frac{3}{2^5} \biggr) \frac{\nu}{\ell_i^5} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] \, . </math>

Thermal Energy of the Core

Again, pulling from our detailed derivations,

<math>~\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr] </math>
<math>~\Rightarrow ~~~~ \chi_\mathrm{eq}^{3} \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]^5_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \frac{1}{2^5} \biggl( \frac{3^8}{2^5\pi} \biggr)^{5/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]^5 \biggl[\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}\biggr]^{3} </math>
	<math>~=~</math>	<math>~ \frac{1}{\pi}\biggl(\frac{3}{2^{2}}\biggr)^{11} \biggl(\frac{\nu^2}{q}\biggr)^{3} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]^5 \biggl[\frac{(1+\ell_i^2)^9}{\ell_i^{15}}\biggr] \, . </math>

Out of equilibrium, we should then expect,

<math>~\frac{S_\mathrm{core}}{E_\mathrm{norm}}</math>

<math>~=~</math>

<math>~ \biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl[ \chi^{-1} \biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} \biggl(\frac{3}{2^{2}}\biggr)^{2}\mathfrak{L}_i \, . </math>

In comparison with our above approximate expression, we therefore have,

<math>~ \biggl[ \biggl(\frac{3}{2\cdot 5}\biggr)\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggr]^5</math>	<math>~=~</math>	<math>~ \frac{1}{\pi}\biggl(\frac{3}{2^{2}}\biggr)^{11} \biggl(\frac{\nu^2}{q}\biggr)^{3} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]^5 \biggl[\frac{(1+\ell_i^2)^9}{\ell_i^{15}}\biggr] </math>
<math>~\Rightarrow~~~~ \mathfrak{b}_c </math>	<math>~=~</math>	<math>~\frac{ 3 }{2^3\ell_i^{3}(1+\ell_i^2)^{6/5}} \biggl[ \ell_i (\ell_i^4 - 1 ) + (1+\ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] \, . </math>

Gravitational Potential Energy of the Envelope

Again, pulling from our detailed derivations and appreciating, in particular, that (see, for example, our notes on equilibrium conditions),

<math>~A</math>	<math>~=~</math>	<math>~\frac{\eta_i}{\sin(\eta_i - B)} \, ,</math>
<math>~(\eta_s - B)</math>	<math>~=~</math>	<math>~\pi \, ,</math>
<math>~\eta_i - B</math>	<math>~=~</math>	<math>~\frac{\pi}{2} - \tan^{-1}(\Lambda_i)\, ,</math>

<math>~\Rightarrow ~~~ \sin(\eta_i -B) = (1+\Lambda_i^2)^{-1/2}</math>

and

<math>~\sin[2(\eta_i-B)] = 2\Lambda_i(1 + \Lambda_i^2)^{-1} \ ,</math>

we have,

<math>~\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 \biggl\{ \biggl[6(\eta_s-B) - 3\sin[2(\eta_s - B)] -4\eta_s\sin^2(\eta_s-B) + 4B\biggr] </math>
		<math>~ - \biggl[6(\eta_i-B) - 3\sin[2(\eta_i - B)] -4\eta_i\sin^2(\eta_i-B) + 4B \biggr]\biggr\} </math>
	<math>~=~</math>	<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl[\frac{\eta_i}{\sin(\eta_i - B)} \biggr]^2 \biggl\{ 6\pi - \biggl[6(\eta_i-B) - 3\sin[2(\eta_i - B)] -4\eta_i\sin^2(\eta_i-B) \biggr]\biggr\} </math>
	<math>~=~</math>	<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2(1+\Lambda_i^2) \biggl\{ 6\pi - 6\biggl[\frac{\pi}{2} - \tan^{-1}(\Lambda_i)\biggr] + 6\biggl[ \frac{\Lambda_i}{(1 + \Lambda_i^2)} \biggr] + 4\eta_i \biggl[ \frac{1}{(1+\Lambda_i^2)} \biggr] \biggr\} </math>
	<math>~=~</math>	<math>~ -\biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} \, . </math>

So, in equilibrium we can write,

<math>~-\chi_\mathrm{eq}\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} \biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} </math>
	<math>~=~</math>	<math>~ \frac{3}{2^2} \biggl(\frac{\eta_i}{m_3}\biggr)^3 \biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{\ell_i^5} </math>
	<math>~=~</math>	<math>~ \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} \biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} \, . </math>

And out of equilibrium,

<math>~\frac{W_\mathrm{env}}{E_\mathrm{norm}}</math>

<math>~=~</math>

<math>~ -\chi^{-1}\cdot \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} \biggl[\mathfrak{K}_i+ \frac{2}{3} \biggr] \, . </math>

This, in turn, implies that both in and out of equilibrium,

<math>~\mathfrak{a}_e </math>

<math>~=~</math>

<math>~ \frac{3}{2^3} \biggl[\frac{\nu^2(1+q)}{q(1-\nu)} \biggr] \frac{1}{\ell_i^2} \biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} \, . </math>

Thermal Energy of the Envelope

Again, pulling from our detailed derivations,

<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 \biggl\{ \biggl[6(\eta_s - B) - 3\sin[2(\eta_s-B)] \biggr] - \biggl[6(\eta_i - B) - 3\sin[2(\eta_i-B)] \biggr] \biggr\}</math>
	<math>~=~</math>	<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl[\frac{\eta_i}{\sin(\eta_i - B)} \biggr]^2 \biggl\{ 6\pi - 6(\eta_i - B) + 3\sin[2(\eta_i-B)] \biggr\}</math>
	<math>~=~</math>	<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 (1 + \Lambda_i^2) \biggl\{ 6\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + 6\biggl[\Lambda_i(1 + \Lambda_i^2)^{-1} \biggr] \biggr\}</math>
	<math>~=~</math>	<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} \, .</math>

So, in equilibrium we can write,

<math>~\chi_\mathrm{eq}^{3}\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} \biggl[\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}\biggr]^{3} </math>
	<math>~=~</math>	<math>~ ~\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{3^2\pi^2}{2^{12}} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^3 \biggl\{ \frac{(1 + \Lambda_i^2)}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \frac{\Lambda_i}{\eta_i} \biggr\} \biggl[\frac{(1+\ell_i^2)^9}{3^9\ell_i^{15}}\biggr] </math>
	<math>~=~</math>	<math>~ ~\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr) \biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr] \biggl\{ \frac{(1 + \Lambda_i^2)}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \frac{\Lambda_i}{\eta_i} \biggr\} \, . </math>

And, out of equilibrium,

<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>

<math>~=~</math>

<math>~ ~ \chi^{-3}\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr) \biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr]\mathfrak{K} \, . </math>

Combined in Equilibrium

Notice that, in combination,

<math>~\biggl[\frac{2S_\mathrm{env} + W_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=</math>	<math>~ - \frac{2}{3}\biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^3 </math>
	<math>~=</math>	<math>~ - \frac{2}{3}\biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl[3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \ell_i \biggl( 1 + \ell_i^2 \biggr)^{-1}\biggr]^3 </math>
	<math>~=</math>	<math>~ - \biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[\frac{\ell_i^3}{( 1 + \ell_i^2)^3}\biggr] \, . </math>

Also, from above,

<math>~\biggl[ \frac{2S_\mathrm{core}+W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \ell_i \biggl(- \frac{8}{3} \ell_i^2 \biggr) (1 + \ell_i^2)^{-3} \biggr] </math>
	<math>~=~</math>	<math>~ + \biggl( \frac{2\cdot 3^6}{\pi } \biggr)^{1/2} \biggl[ \frac{\ell_i^3}{(1 + \ell_i^2)^{3}} \biggr] \, .</math>

So, in equilibrium, these terms from the core and envelope sum to zero, as they should.

Out of Equilibrium

And now, in combination out of equilibrium,

<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>

<math>~=</math>

<math>~ \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} + \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\} +\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{2n_c}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} +\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2n_e}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \, . </math>

Hence, quite generally out of equilibrium,

<math>~\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] </math>

<math>~=</math>

<math>~ -\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} + \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\} -\frac{3}{5}\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{10}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} -3\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \, . </math>

Let's see what the value of this derivative is if the dimensionless radius, <math>~\chi</math>, is set to the value that has been determined, via a detailed force-balanced analysis, to be the equilibrium radius, namely, <math>~\chi = \chi_\mathrm{eq}</math>. In this case, we have,

<math>~\biggl\{\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] \biggr\}_\mathrm{\chi \rightarrow \chi_\mathrm{eq}}</math>

<math>~=</math>

<math>~ -\chi_\mathrm{eq}^{-1}\biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} + \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} +2\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} +2\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \biggr\} \, . </math>

But, according to the virial theorem — and, as we have just demonstrated — the four terms inside the curly braces sum to zero. So this demonstrates that the derivative of our out-of-equilibrium free-energy expression does go to zero at the equilibrium radius, as it should!

Summary

In summary, the desired out of equilibrium free-energy expression is,

<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>	<math>~=</math>	<math>~ \frac{W_\mathrm{core}}{E_\mathrm{norm}} + \frac{W_\mathrm{env}}{E_\mathrm{norm}} +\biggl(\frac{2n_c}{3}\biggr)\frac{S_\mathrm{core}}{E_\mathrm{norm}} +\biggl(\frac{2n_e}{3}\biggr)\frac{S_\mathrm{env}}{E_\mathrm{norm}} </math>
	<math>~=</math>	<math>~ - \chi^{-1} \biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2} \biggl[ \mathfrak{L}_i - \frac{8}{3} \biggr] -\chi^{-1}\cdot \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} \biggl[\mathfrak{K}_i+ \frac{2}{3} \biggr] </math>
		<math>~ + \biggl(\frac{2\cdot 5}{3}\biggr) \biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl[ \chi^{-1} \biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} \biggl(\frac{3}{2^{2}}\biggr)^{2}\mathfrak{L}_i +\biggl(\frac{2}{3}\biggr) \chi^{-3}\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr) \biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr]\mathfrak{K} </math>
	<math>~=</math>	<math>~ - \biggl( \frac{3}{2^4} \biggr) \biggl[\chi^{-1}\frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2}\biggr] \biggl[ \mathfrak{L}_i + 4\mathfrak{K}_i \biggr] + \biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl(\frac{3\cdot 5}{2^3}\biggr) \biggl[ \chi^{-1} \biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} \mathfrak{L}_i </math>
		<math>~ + \biggl( \frac{\pi}{2^{5}\cdot 3^6} \biggr) \biggl[\chi^{-1}\biggl(\frac{\nu^2}{q} \biggr) \frac{(1+\ell_i^2)^2}{\ell_i^{4}}\biggr]^3\mathfrak{K} \, . </math>

Or, in terms of the ratio,

<math>\Chi \equiv \frac{\chi}{\chi_\mathrm{eq}} \, ,</math>

and pulling from the above expressions,

<math>~\biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr] </math>
	<math>~=~</math>	<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3} \biggl[ \mathfrak{L}_i - \frac{8}{3}\biggr] </math>
<math>~\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ -\biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} </math>
	<math>~=~</math>	<math>~ -\biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3} \biggl[4\mathfrak{K}_i + \frac{8}{3} \biggr] </math>
<math>~\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr] </math>
	<math>~=~</math>	<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3}\mathfrak{L}_i </math>
<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>	<math>~=~</math>	<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 \biggl\{ (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} </math>
	<math>~=~</math>	<math>~ ~\frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3} (4\mathfrak{K}_i) \, , </math>

we have the streamlined,

Out-of-Equilibrium, Free-Energy Expression for BiPolytropes with <math>~(n_c, n_e) = (5, 1)</math>
<math>~\biggl( \frac{2^5\pi}{3^6} \biggr)^{1/2} \biggl[ \frac{(1+\ell_i^2)}{\ell_i} \biggr]^{3} \biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math> <math>~=</math> <math>~ +\Chi^{-3/5} (5 \mathfrak{L}_i) +\Chi^{-3} (4\mathfrak{K}_i) -\Chi^{-1} (3\mathfrak{L}_i +12\mathfrak{K}_i ) </math>

where,

<math>~\mathfrak{L}_i</math>	<math>~\equiv</math>	<math>~\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i \, ,</math>
<math>~\mathfrak{K}_i</math>	<math>~\equiv</math>	<math>~\frac{\Lambda_i}{\eta_i} + \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] \, ,</math>
<math>~\Lambda_i</math>	<math>~\equiv</math>	<math>~\frac{1}{\eta_i} - \ell_i \, ,</math>
<math>~\eta_i</math>	<math>~=</math>	<math>~3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] \, .</math>

From the accompanying Table 1 parameter values, we also can write,

<math>~\frac{1}{q}</math>	<math>~=</math>	<math>~\frac{\eta_s}{\eta_i} = 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr] \, ,</math>
<math>~\nu</math>	<math>~=</math>	<math>~ \frac{\ell_i q}{(1+\Lambda_i^2)^{1/2}} \, . </math>

Consistent with our generic discussion of the stability of bipolytropes and the specific discussion of the stability of bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,

<math>~\chi = \chi_\mathrm{eq}</math>

<math>~=</math>

<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} \, .</math>

Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, it is unstable whenever,

<math>~\frac{ \mathfrak{L}_i}{\mathfrak{K}_i}</math>

<math>~></math>

<math>~20 \, .</math>

Table 1 of an accompanying chapter — and the red-dashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.

Focus on Zero-Zero Free-Energy Expression

Here, we will draw heavily from the following accompanying chapters:

• Analytic Detailed Force Balance Models
• Free-Energy Analysis

From Detailed Force-Balance Models

Equilibrium Radius

First View

In an accompanying chapter we find,

<math>~ \frac{P_0 R_\mathrm{eq}^4}{G M_\mathrm{tot}^2 } </math>

<math>~=</math>

<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] </math>

where,

<math>~f</math>	<math>~\equiv</math>	<math> 1+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] \, , </math>
<math>~\mathfrak{F} </math>	<math>~\equiv</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (-2q^2 + 3q^3 - q^5) + \frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr] \, , </math>
<math>~\frac{\rho_e}{\rho_c} </math>	<math>~=</math>	<math>~ \frac{q^3(1-\nu)}{\nu(1-q^3)} \, . </math>

Here, we prefer to normalize the equilibrium radius to <math>~R_\mathrm{norm}</math>. So, let's replace the central pressure with its expression in terms of <math>~K_c</math>. Specifically,

<math>~P_0</math>	<math>~=</math>	<math>~ K_c \rho_c^{\gamma_c} = K_c \biggl[ \frac{3M_\mathrm{core}}{4\pi R_i^3} \biggr]^{\gamma_c} = K_c \biggl[ \frac{3\nu M_\mathrm{tot}}{4\pi q^3 R_\mathrm{eq}^3} \biggr]^{(n_c+1)/n_c} ~~~\Rightarrow~~~ \frac{P_0}{P_\mathrm{norm}} = \biggl[ \frac{3}{4\pi}\biggl(\frac{\nu}{q^3}\biggr) \frac{1}{\chi_\mathrm{eq}^3}\biggr]^{(n_c+1)/n_c} </math>
<math>~\Rightarrow~~~K_c \biggl[ \frac{3\nu M_\mathrm{tot}}{4\pi q^3 R_\mathrm{eq}^3} \biggr]^{(n_c+1)/n_c} \frac{R_\mathrm{eq}^4}{G M_\mathrm{tot}^2 } </math>	<math>~=</math>	<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] </math>
<math>~\Rightarrow~~~R_\mathrm{eq}^{(n_c-3)/n_c} </math>	<math>~=</math>	<math>~ \biggl(\frac{G}{K_c}\biggr) M_\mathrm{tot}^{(n_c-1)/n_c} \biggl[ \frac{3\nu }{4\pi q^3 } \biggr]^{-(n_c+1)/n_c} \biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] </math>
<math>~\Rightarrow~~~\chi_\mathrm{eq}^{(n_c-3)/n_c} \equiv \biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]^{(n_c-3)/n_c}</math>	<math>~=</math>	<math>~ \frac{1}{2}\biggl(\frac{4\pi}{3} \biggr)^{1/n_c} \biggl( \frac{\nu}{q^3}\biggr)^{(n_c-1)/n_c} \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \, . </math>

Or, in terms of <math>~\gamma_c</math>,

<math>~\chi_\mathrm{eq}^{4-3\gamma_c} </math>

<math>~=</math>

<math>~ \frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c} \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \, . </math>

Second View

Alternatively, from our derivation and discussion of analytic detailed force-balance models,

<math> \biggl[ \frac{R^4}{GM_\mathrm{tot}^2} \biggr] P_0</math>

<math>~=</math>

<math>\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} \, ,</math>

where,

<math>~[g(\nu,q)]^2</math>

<math>~\equiv</math>

<math> 1 + \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \, . </math>

In order to show that this expression is the same as the other one, above, we need to show that,

<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] </math>	<math>~=</math>	<math>\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} </math>
<math>~\Rightarrow~~~ f - 1-\mathfrak{F} </math>	<math>~=</math>	<math>~\frac{5}{2q^3} \biggl[g^2-1\biggr]</math>
	<math>~=</math>	<math>~\frac{5}{2q^3} \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math>
	<math>~=</math>	<math>~\frac{5}{2q^5} \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl\{ 2 ( q^2 - q^3 ) + \frac{\rho_e}{\rho_0}\biggl[ 1 - 3q^2+ 2q^3 \biggr] \biggr\} \, .</math>

Let's see …

<math>~ f - 1-\mathfrak{F} </math>	<math>~=</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] - \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (-2q^2 + 3q^3 - q^5) + \frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr] </math>
	<math>~=</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2}-1 \biggr) - \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (-2q^2 + 3q^3 - q^5) \biggr] </math>
		<math>~ - \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ \frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr] +\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] </math>
	<math>~=</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl\{ (q^3- q^5 ) + (2q^2 - 3q^3 + q^5) \biggr\} </math>
		<math>~ + \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \frac{1}{q^5} \biggl[ 3 (1 -5q^2 + 5q^3 - q^5) \biggr] +\frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \frac{1}{q^5} \biggl[ 2 - 2q^5 + 5\biggl( q^5-q^3\biggr)\biggr] </math>
	<math>~=</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (q^3- q^5 ) + (2q^2 - 3q^3 + q^5) \biggr] + \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \frac{1}{q^5} \biggl[ 3 (1 -5q^2 + 5q^3 - q^5)+2 - 2q^5 + 5( q^5-q^3) \biggr] </math>
	<math>~=</math>	<math>~ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ 2q^2 - 2q^3 \biggr] + \frac{5}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ 1 - 3q^2 + 2q^3 \biggr] \, . </math>

Q.E.D.

Hence, the equilibrium radius can also be written as,

<math>~\chi_\mathrm{eq}^{4-3\gamma_c} </math>

<math>~=</math>

<math>~ \frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c} q^2 g^2 \, ; </math>

or, in terms of the polytropic index,

<math>~\chi_\mathrm{eq}^{n_c-3} </math>

<math>~=</math>

<math>~ \frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \, . </math>

Gravitational Potential Energy

Also from our accompanying discussion, we have,

<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}} </math>	<math>~=</math>	<math> - \Chi^{-1} \biggl( \frac{3}{5}\biggr) \biggl(\frac{\nu}{q^3} \biggr)^2 q^5 \biggl[ \frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \biggr]^{-1/(n_c-3)} f(\nu,q) </math>
	<math>~=</math>	<math> - \Chi^{-1} \biggl( \frac{6}{5}\biggr) q^5 f \biggl[ 2^{n_c-(n_c-3)} \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{(1-n_c)+2(n_c-3)} b_\xi^{n_c} \biggr]^{1/(n_c-3)} </math>
	<math>~=</math>	<math> - \Chi^{-1} \biggl( \frac{6}{5}\biggr) q^5 f \biggl[ \biggl(\frac{6}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{n_c} \biggr]^{1/(n_c-3)} \, . </math>

Internal Energy Components

First View

Before writing out the expressions for the internal energy of the core and of the envelope, we note from our separate detailed derivation that, in either case,

<math>~\biggl[\frac{P_i \chi^{3\gamma}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma}</math>	<math>~=</math>	<math>~\biggl[\biggl(\frac{P_i }{P_0} \biggr) \biggl(\frac{P_0 }{P_\mathrm{norm}} \biggr)\chi^{3}\biggr]_\mathrm{eq} \biggl[\frac{\chi}{\chi_\mathrm{eq}}\biggr]^{3-3\gamma}</math>
	<math>~=</math>	<math>~\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma} \, ,</math>

where, in equilibrium,

<math>~\biggl(\frac{P_i }{P_0} \biggr)_\mathrm{eq}</math>	<math>~=</math>	<math>~1 - b_\xi q^2</math>
<math>~b_\xi q^2</math>	<math>~=</math>	<math>~\biggl\{\frac{2}{5}q^3 f + \biggl[1 - \frac{2}{5} q^3( 1+\mathfrak{F} ) \biggr]\biggr\}^{-1} </math>
	<math>~=</math>	<math>~\biggl[1 + \frac{2}{5}q^3 (f - 1-\mathfrak{F} ) \biggr]^{-1} </math>

So, copying from our accompanying detailed derivation, we have,

<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>	<math>~=</math>	<math> \frac{4\pi/3 }{({\gamma_c}-1)} \biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_c} \biggl\{ \biggl( \frac{P_0}{P_{ic}} \biggr) \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] \biggr\} </math>
	<math>~=</math>	<math> \frac{1 }{({\gamma_c}-1)} \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_c} q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, , </math>
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>	<math>~=</math>	<math> \frac{4\pi/3 }{({\gamma_e}-1)} \biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_e} \biggl\{ (1-q^3) + b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\} </math>
	<math>~=</math>	<math> \frac{1}{({\gamma_e}-1)} \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_e} \biggl(\frac{P_i }{P_0} \biggr) \biggl\{ (1-q^3) + b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\} </math>
	<math>~=</math>	<math> \frac{1}{({\gamma_e}-1)} \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_e} \biggl\{ (1-b_\xi q^2)(1-q^3) + b_\xi \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\} </math>
	<math>~=</math>	<math> \frac{1}{({\gamma_e}-1)} \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_e} (1-q^3) \biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2 \biggr\} \, . </math>

Furthermore,

<math>~\biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]</math>	<math>~=</math>	<math>~ \biggl(\frac{3}{4\pi}\biggr)^{\gamma_c - 1} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c} \biggl\{\chi_\mathrm{eq}^{4-3\gamma_c}\biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)} </math>
	<math>~=</math>	<math>~ \biggl(\frac{3}{4\pi}\biggr)^{\gamma_c - 1} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c} \biggl\{\frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c} \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)} </math>
	<math>~=</math>	<math>~ \biggl(\frac{3}{4\pi}\biggr)^{(\gamma_c - 1)/(4-3\gamma_c)} \biggl( \frac{\nu}{q^3} \biggr)^{(6-5\gamma_c)(4-3\gamma_c)} \biggl\{\frac{q^2}{2} \biggl[ 1 + \frac{2}{5} q^3( f - 1-\mathfrak{F} )\biggr] \biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)} </math>
	<math>~=</math>	<math>~ \biggl(\frac{3}{4\pi}\biggr)^{1/(n_c-3)} \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)(n_c-3)} \biggl\{\frac{q^2}{2} \biggl[ 1 + \frac{2}{5} q^3( f - 1-\mathfrak{F} )\biggr] \biggr\}^{-3/(n_c-3)} </math>
	<math>~=</math>	<math>~ \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} \, . </math>

Hence, we have,

<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>	<math>~=</math>	<math> n_c \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c} \biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{-3/n_c} q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] </math>
	<math>~=</math>	<math> n_c \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} \Chi^{-3/n_c} q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, , </math>
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>	<math>~=</math>	<math> n_e \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} \Chi^{-3/n_e} (1-q^3) \biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2 \biggr\} \, . </math>

Second View

In our accompanying discussion of energies associated with detailed force balance models, we used the notation,

<math>~\Pi</math>

<math>~\equiv</math>

<math>~ \biggl(\frac{3}{2^3\pi}\biggr) \frac{GM_\mathrm{tot}^2}{R^4} \biggl(\frac{\nu}{q^3}\biggr)^2 = P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2 \, , </math>

which allows us to rewrite the above quoted relationship between the central pressure and the radius of the bipolytrope as,

<math>~P_0 = \Pi (qg)^2 \, .</math>

We also showed that, in equilibrium, the relationship between the central pressure and the interface pressure is,

<math>~P_0 =P_i + \Pi_\mathrm{eq} q^2 \, .</math>

This means that, in equilibrium, the ratio of the interface pressure to the central pressure is,

<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>

<math>~=</math>

<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0} = 1- \frac{1}{g^2} \, , </math>

or given that (see above),

<math>~ \frac{5}{2q^3} \biggl[g^2-1\biggr] </math>	<math>~=</math>	<math>~ f - 1-\mathfrak{F} </math>
<math>~\Rightarrow~~~~ g^2 </math>	<math>~=</math>	<math>~ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} ) \, , </math>

we have,

<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>

<math>~=</math>

<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0} = 1- \biggl[ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} ) \biggr]^{-1} \, . </math>

This is exactly the pressure-ratio expression presented in our "first view" and unveils the notation association,

<math>~b_\xi q^2</math>

<math>~\leftrightarrow~</math>

<math> \frac{1}{g^2} \, . </math>

From our separate derivation, we have, in equilibrium,

<math>~\mathfrak{G}_\mathrm{core} = \biggl(\frac{2n_c}{3}\biggr) S_\mathrm{core}</math>	<math>~=~</math>	<math>\biggl(\frac{2n_c}{3}\biggr) \biggl( \frac{4\pi}{5} \biggr) R_\mathrm{eq}^3 q^5 \biggl (\frac{5P_i}{2q^2} + \Pi \biggr)_\mathrm{eq} </math>
	<math>~=~</math>	<math>\biggl( \frac{ q^5n_c}{5} \biggr) R_\mathrm{eq}^3 \biggl( \frac{2^3\pi}{3} \biggr) \Pi_\mathrm{eq} \biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
	<math>~=~</math>	<math>\biggl( \frac{ n_c}{5} \biggr) \biggl[ R_\mathrm{norm}^3 P_\mathrm{norm} \biggr] \chi_\mathrm{eq}^{-1} \biggl(\frac{\nu^2}{q}\biggr) \biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{P_0} \biggr)_\mathrm{eq}\biggl( \frac{P_0}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
<math>~\Rightarrow ~~~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>	<math>~=~</math>	<math>~\biggl( \frac{ n_c}{5} \biggr) \biggl(\frac{\nu^2}{q}\biggr) \biggl[\frac{5}{2q^2} \biggl( 1-\frac{1}{g^2} \biggr)\biggl( q^2g^2\biggr) + 1 \biggr] \chi_\mathrm{eq}^{-1} </math>
	<math>~=~</math>	<math>~\biggl( \frac{ n_c}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr) \biggl[ g^2-\frac{3}{5} \biggr] \biggl\{\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \biggr\}^{-1/(n_c-3)}</math>
	<math>~=~</math>	<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2} \biggr] \biggl( \frac{ 1}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr) g^2 \biggl\{2^{n_c}\biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{1-n_c} (q g)^{-2n_c} \biggr\}^{1/(n_c-3)}</math>
	<math>~=~</math>	<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2} \biggr] \biggl\{2^{n_c}\cdot 2^{(3-n_c)}\biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{1-n_c} \biggl(\frac{\nu}{q^3}\biggr)^{2(n_c-3)} q^{5(n_c-3)} q^{-2n_c} g^{-2n_c} g^{2(n_c-3)} \biggr\}^{1/(n_c-3)}</math>
	<math>~=~</math>	<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2} \biggr] \biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15} g^{-6} \biggr\}^{1/(n_c-3)} \, .</math>

Finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>	<math>~=~</math>	<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2 \biggr] \biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15} b_\xi^3 q^{6} \biggr\}^{1/(n_c-3)} </math>
	<math>~=~</math>	<math>~n_c q^3 \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2 \biggr] \biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^3 \biggr\}^{1/(n_c-3)} \, ,</math>

which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression. Also from our previous derivation, we can write,

<math>~\mathfrak{G}_\mathrm{env} = \biggl(\frac{2n_e}{3}\biggr) S_\mathrm{env}</math>	<math>~=~</math>	<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr) R_\mathrm{eq}^3 \Pi_\mathrm{eq} \biggl\{ (1-q^3) \biggl(\frac{P_i }{\Pi}\biggr)_\mathrm{eq} + \biggl( \frac{\rho_e}{\rho_0} \biggr)\biggl[ (-2q^2 + 3q^3 - q^5 ) + \frac{3}{5} \biggl( \frac{\rho_e}{\rho_0} \biggr) ( -1 + 5q^2 -5q^3 + q^5 )\biggr]\biggr\} </math>
	<math>~=~</math>	<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr) R_\mathrm{eq}^3 \biggl[ P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2\biggr]_\mathrm{eq} \biggl\{ (1-q^3) q^2(g^2-1) + \biggl(\frac{2}{5}\biggr) q^5 \mathfrak{F} \biggr\} </math>
	<math>~=~</math>	<math>~ \biggl[ P_\mathrm{norm} R_\mathrm{norm}^3 \biggr] \frac{n_e}{2} \biggl(\frac{\nu^2}{q^4}\biggr)(1-q^3) \biggl\{ (g^2-1) + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \chi^{-1}_\mathrm{eq} </math>
<math>~\Rightarrow~~~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>	<math>~=~</math>	<math>~ n_e (1-q^3) \biggl\{ (g^2-1) + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \frac{q^2}{2}\biggl(\frac{\nu}{q^3}\biggr)^2 \biggl[\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c}\biggr]^{-1/(n_c-3)} </math>
	<math>~=~</math>	<math>~ n_e (1-q^3) \biggl\{ (g^2-1) + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \biggl[2^{[n_c-(n_c-3)]} \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{(1-n_c)+2(n_c-3)} q^{2(n_c-3)-2n_c} g^{-2n_c} \biggr]^{1/(n_c-3)} </math>
	<math>~=~</math>	<math>~ n_e (1-q^3) \biggl\{ (g^2-1) + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} g^{-2n_c} \biggr]^{1/(n_c-3)} \, . </math>

And, finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>	<math>~=~</math>	<math>~ n_e (1-q^3) (b_\xi q^2)^{-1} \biggl\{ 1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} \biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} (b_\xi q^2)^{n_c} \biggr]^{1/(n_c-3)} </math>
	<math>~=~</math>	<math>~ n_e (1-q^3) \biggl\{ 1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} \biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6-2(n_c-3)+2n_c} b_\xi^{3-n_c+n_c} \biggr]^{1/(n_c-3)} </math>
	<math>~=~</math>	<math>~ n_e\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{3} \biggr]^{1/(n_c-3)} (1-q^3) \biggl\{ 1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} \, , </math>

which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression.

Summary

In summary, the desired out of equilibrium free-energy expression is,

<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>

<math>~=</math>

<math>~ A\Chi^{-3/n_c} + B\Chi^{-3/n_e} - C\Chi^{-1} </math>

where,

<math>~A \equiv \biggl( \frac{\mathfrak{S}_\mathrm{core}}{E_\mathrm{norm}} \biggr)_\mathrm{eq}</math>	<math>~=</math>	<math> \frac{n_c}{b_\xi} \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^{n_c}\biggr]^{1/(n_c-3)} q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, , </math>
<math>~B\equiv \biggl( \frac{\mathfrak{S}_\mathrm{env}}{E_\mathrm{norm}} \biggr)_\mathrm{eq}</math>	<math>~=</math>	<math> \frac{n_e}{b_\xi} \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^{n_c} \biggr]^{1/(n_c-3)} \biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2 \biggr\} b_\xi^{(n_c-3)/(n_c-3)} \cdot b_\xi^{-1} \, , </math>
<math>~C \equiv \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{eq} </math>	<math>~=</math>	<math> \biggl( \frac{2\cdot 3}{5}\biggr) q^5 f \biggl[ \biggl(\frac{6}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{n_c} \biggr]^{1/(n_c-3)} \, . </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