User:Tohline/SSC/FreeEnergy/PolytropesEmbedded

Free-Energy of Truncated Polytropes

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

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

where, as derived elsewhere,

Structural Form Factors for Isolated Polytropes

Structural Form Factors for Pressure-Truncated Polytropes

<math>~\tilde\mathfrak{f}_M</math> <math>~=</math> <math>~ \biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math> <math>\tilde\mathfrak{f}_W </math> <math>~=</math> <math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math> <math>\tilde\mathfrak{f}_A </math> <math>~=</math> <math> \frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1} </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>

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

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

<math>~=</math>

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

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

The M₁ Free-Energy Surface

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

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

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

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

<math>~\equiv</math>

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

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

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

where the constants,

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