User:Tohline/SSC/Structure/Polytropes/VirialSummary
Virial Equilibrium of Pressure-Truncated Polytropes
Here we will draw heavily from an accompanying Free Energy Synopsis.
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Groundwork
Basic Relations
In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,
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<math>~\mathfrak{G}</math> |
<math>~=</math> |
<math>~W_\mathrm{grav} + U_\mathrm{int} + P_e V \, .</math> |
When rewritten in a suitably dimensionless form — see two useful alternatives, below — this expression becomes,
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<math>~\mathfrak{G}^*</math> |
<math>~=</math> |
<math>~- a x^{-1} + bx^{-3/n} + c x^3 \, ,</math> |
where <math>~x</math> is the configuration's dimensionless radius and <math>~a</math>, <math>~b</math>, and <math>~c</math> are constants. We therefore have,
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<math>~\frac{d\mathfrak{G}^*}{dx}</math> |
<math>~=</math> |
<math>~\frac{1}{x^2} \biggl[ a - \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 3c x^4 \biggr] \, ,</math> |
and,
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<math>~\frac{d^2\mathfrak{G}^*}{dx^2}</math> |
<math>~=</math> |
<math>~\frac{1}{x^3} \biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr] \, .</math> |
Virial equilibrium is obtained when <math>~d\mathfrak{G}^*/dx = 0</math>, that is, when
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<math>~\biggl( \frac{3b}{n} \biggr) x_\mathrm{eq}^{(n-3)/n} </math> |
<math>~=</math> |
<math>~ a + 3c x_\mathrm{eq}^4 \, .</math> |
And along an equilibrium sequence, the specific equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations — henceforth labeled as having the critical radius, <math>~x_\mathrm{crit}</math> — is identified by setting <math>~d^2\mathfrak{G}^*/dx^2 = 0</math>, that is, it is the configuration for which,
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<math>~0</math> |
<math>~=</math> |
<math>~\biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr]_{x = x_\mathrm{eq}}</math> |
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<math>~\Rightarrow ~~~ x_\mathrm{crit}^4 </math> |
<math>~=</math> |
<math>~ \frac{a}{3^2c}\biggl(\frac{n - 3}{n+1}\biggr) \, . </math> |
Inserting the adiabatic exponent in place of the polytropic index via the relation, <math>~n = (\gamma - 1)^{-1}</math>, we have equivalently,
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<math>~ x_\mathrm{crit}^4 </math> |
<math>~=</math> |
<math>~ \frac{a}{3^2c}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \, . </math> |
Useful Recognition
By comparing various terms in the first two algebraic Setup expressions, above, It is clear that,
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<math>~W^*_\mathrm{grav} = -ax^{-1}</math> |
and, |
<math>~U^*_\mathrm{int} = bx^{-3/n} \, .</math> |
Notice, then, that in every equilibrium configuration, we should find,
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<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{eq}</math> |
<math>~=</math> |
<math>~ \biggl(\frac{b}{a}\biggr) x_\mathrm{eq}^{(n-3)/n} = \frac{n}{3a} \biggl[ a + 3cx^4_\mathrm{eq} \biggr] </math> |
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<math>~=</math> |
<math>~ \frac{n}{3} \biggl[ 1 + \biggl(\frac{3c}{a}\biggr) x^4_\mathrm{eq} \biggr] \, . </math> |
And, specifically in the critical configuration we should find that,
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<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math> |
<math>~=</math> |
<math>~ \frac{1}{3(\gamma-1)} \biggl[ 1 + \frac{1}{3}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \biggr] = \frac{4}{3^2\gamma(\gamma-1)} </math> |
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<math>~\Rightarrow ~~~\frac{S^*_\mathrm{therm}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math> |
<math>~=</math> |
<math>~ -\frac{2}{3\gamma} \, . </math> |
The equivalent of this last expression also appears at the end of subsection ⑦ of an accompanying Tabular Overview.
See Also
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© 2014 - 2021 by Joel E. Tohline |