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Virial Equation
Free Energy Expression
Associated with any isolated, self-gravitating, gaseous configuration we can identify a total "Gibbs-like" free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<math> \mathfrak{G} = W + U + T_\mathrm{rot} + P_e V + \cdots </math>
Here, we have explicitly included the gravitational potential energy, <math>W</math>, the total internal energy, <math>U</math>, the rotational kinetic energy, <math>T_\mathrm{rot}</math>, and a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e</math>.
Uniform-density, Uniformly Rotating Sphere
For a uniform-density, uniformly rotating, spherically symmetric configuration of mass <math>M</math> and radius <math>R</math>,
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<math> W </math> |
<math>=</math> |
<math> - \frac{3}{5} \frac{GM^2}{R} \, , </math> |
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<math> T_\mathrm{rot} </math> |
<math>=</math> |
<math> \frac{1}{2} I \omega^2 = \frac{J^2}{2I} = \frac{5}{4} \frac{J^2}{MR^2} \, , </math> |
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<math> V </math> |
<math>=</math> |
<math> \frac{4}{3} \pi R^3 \, , </math> |
where, <math>~G</math> is the gravitational constant, <math>I=(2/5)MR^2</math> is the moment of inertia, <math>\omega</math> is the angular frequency of rotation, and <math>J=I\omega</math> is the total angular momentum.
Adiabatic
If, upon compression or expansion, the gaseous configuration behaves adiabatically, in which case the pressure will vary with density as,
<math>P = K \rho^{\gamma_g} \, ,</math>
where, <math>K</math> specifies the specific entropy of the gas and <math>~\gamma_\mathrm{g}</math> is the ratio of specific heats, then
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<math> U = \frac{2}{3(\gamma_g - 1)} S </math> |
<math>=</math> |
<math> \frac{2}{3(\gamma_g - 1)} \biggl[ \frac{1}{2} a_s^2 M \biggr] \, , </math> |
where <math>S</math> is the total thermal energy, and the square of the (adiabatic) sound speed,
<math>a_s^2 \equiv \frac{\partial P}{\partial\rho} = \gamma_g \frac{P}{\rho} = \gamma_g K \rho^{\gamma_g-1} \, .</math>
Appreciating that <math>\rho = M/V</math> for the case being considered here (i.e., for a uniform-density sphere), the free energy can be written as,
<math> \mathfrak{G} = -AR^{-1} +BR^{3-3\gamma_g} + C R^{-2} + DR^3 \, , </math>
where,
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<math>A</math> |
<math>\equiv</math> |
<math>\frac{3}{5} GM^2 \, ,</math> |
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<math>B</math> |
<math>\equiv</math> |
<math> \biggl[ \frac{\gamma_g K}{3(\gamma_g-1)} \biggl( \frac{3}{4\pi} \biggr)^{\gamma_g - 1} \biggr] M^{\gamma_g} \, , </math> |
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<math>C</math> |
<math>\equiv</math> |
<math> \frac{5J^2}{4M} \, , </math> |
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<math>D</math> |
<math>\equiv</math> |
<math> \frac{3MP_e}{4\pi} \, . </math> |
Isothermal
If, upon compression or expansion, the configuration remains isothermal — in which case <math>\gamma_g =1</math> — then both the (isothermal) sound speed, <math>c_s</math>, and the total thermal energy, <math>S=(1/2)c_s^2 M</math>, are constant. But as pointed out, for example, in Appendix A of Stahler (1983, ApJ, 268, 16), the total internal energy will vary according to the relation,
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<math> U </math> |
<math>=</math> |
<math> \frac{2}{3} S \ln\rho \, . </math> |
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© 2014 - 2021 by Joel E. Tohline |