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>~- \biggl[\frac{3}{5}\cdot \frac{\tilde{f}_A}{\tilde{f}_M^2} \biggr] \frac{GM^2}{R} - \biggl[\biggl(\frac{3}{4\pi}\biggr)^{\gamma-1} \frac{\tilde{f}_A}{\tilde{f}_M^\gamma} \biggr] \frac{KM^\gamma}{R^{3(\gamma-1)}} + \frac{4\pi}{3} \cdot P_e R^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>. 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,
which are closely related to the normalizations used by Hoerdt and by Whitworth. (Following Stahler's lead, we define, )
See Also
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