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Virial Equation
Free Energy Expression
Associated with any 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,
<math> \mathfrak{G} = W + U + T_\mathrm{rot} + P_e V + ... </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, Polytropic Sphere
For a uniform-density, 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> U = \frac{2}{3(\gamma - 1)} S </math> |
<math>=</math> |
<math> \frac{2}{3(\gamma - 1)} \biggl[ \frac{1}{2} a^2 M \biggr] \, ; </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> |
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© 2014 - 2021 by Joel E. Tohline |