User:Tohline/SSC/FreeEnergy/PolytropesEmbedded
Free-Energy of Truncated Polytropes
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In this case, the Gibbs-like free energy is given by the sum of three separate energies,
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<math>~\mathfrak{G}</math> |
<math>~=</math> |
<math>~W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_eV</math> |
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<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,
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<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,
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Structural Form Factors for Pressure-Truncated Polytropes <math>~(n \ne 5)</math> |
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As we have shown separately, for the singular case of <math>~n = 5</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,
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<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,
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<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> |
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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> |
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,
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<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,
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<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> |
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<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,
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<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,
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<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,
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<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,
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<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>.
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<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,
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
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<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> |
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<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,
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<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> |
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<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,
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<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,
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<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,
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
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<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.
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<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> |
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<math>~=</math> |
<math>~- 2\mathfrak{a}_c \biggl(\frac{\nu}{q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math> |
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<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> |
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<math>~=</math> |
<math>~- 2\mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math> |
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<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> |
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<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,
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<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,
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<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,
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<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:
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<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,
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<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> |
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<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> |
|||
|---|---|---|---|
|
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> |
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<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,
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<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,
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<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
From Detailed Force-Balance Models
Equilibrium Radius
In an accompanying chapter we find,
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<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,
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
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<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> |
Gravitational Potential Energy
Also from our accompanying discussion, we have,
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<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}} </math> |
<math>~=</math> |
<math> - \Chi^{-1} \biggl( \frac{3}{5}\biggr) \chi_\mathrm{eq}\biggl(\frac{\nu^2}{q} \biggr) f(\nu,q) </math> |
|
|
<math>~=</math> |
<math> - \Chi^{-1} \biggl( \frac{3}{5}\biggr) \biggl\{ \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] \biggr\}^{n_c/(n_c-3)}\biggl(\frac{\nu}{q^3} \biggr)^2 q^5 f(\nu,q) </math> |
|
|
<math>~=</math> |
<math> - \Chi^{-1} \biggl( \frac{3}{5}\biggr) 2^{-n_c/(n_c-3)} \biggl(\frac{4\pi}{3} \biggr)^{1/(n_c-3)} \biggl( \frac{\nu}{q^3}\biggr)^{(5-n_c)/(n_c-3)} \biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr]^{n_c/(n_c-3)}q^5 f(\nu,q) </math> |
Internal Energy Components
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,
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<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,
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<math>~\biggl(\frac{P_i }{P_0} \biggr)_\mathrm{eq}</math> |
<math>~=</math> |
<math>~1 - \biggl\{\frac{2}{5}q^3 f + \biggl[1 - \frac{2}{5} q^3( 1+\mathfrak{F} ) \biggr]\biggr\}^{-1} \, .</math> |
So, copying from our accompanying detailed derivation, we have,
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<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> |
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<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> |
See Also
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© 2014 - 2021 by Joel E. Tohline |