User:Tohline/StabilityVariationalPrincipal
Free-Energy Stability Analysis
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Most General Case
Consider a free-energy function of the form,
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<math>~\mathcal{G}</math> |
<math>~=</math> |
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math> |
where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,
<math>~\chi \equiv \frac{R}{R_0} \, ,</math>
is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.
Virial Equilibrium
The first variation (first derivative) of this function with respect to the configuration's radius is,
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<math>~\frac{d\mathcal{G}}{d\chi}</math> |
<math>~=</math> |
<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math> |
According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,
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<math>~0</math> |
<math>~=</math> |
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math> |
Stability
Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,
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<math>~\mathcal{G}^'</math> |
<math>~=</math> |
<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math> |
The first variation (first derivative) of this function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,
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<math>~\frac{d\mathcal{G}^'}{d\chi}</math> |
<math>~=</math> |
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math> |
If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.
Pressure-Truncated Configurations
For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,
| Free-energy expression: |
<math>~\mathcal{G}</math> |
<math>~=</math> |
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math> |
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| Virial equlibrium: |
<math>~0</math> |
<math>~=</math> |
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math> |
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| Stability indicator: |
<math>~\frac{d\mathcal{G}^'}{d\chi}</math> |
<math>~=</math> |
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math> |
Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,
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<math>~6g \chi_\mathrm{eq}^4 </math> |
<math>~=</math> |
<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math> |
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<math>~=</math> |
<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math> |
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<math>~\Rightarrow ~~~ 6g \chi_\mathrm{eq}^4 +\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4 </math> |
<math>~=</math> |
<math>~ 2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr] </math> |
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<math>~\Rightarrow ~~~ \biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4 </math> |
<math>~=</math> |
<math>~ 2e - \biggl[\frac{af(n+3)}{nb} \biggr] </math> |
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<math>~\Rightarrow ~~~ \chi_\mathrm{eq}^4\biggr|_\mathrm{crit} </math> |
<math>~=</math> |
<math>~ \biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, . </math> |
Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,
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<math>~ \chi_\mathrm{eq}^4\biggr|_\mathrm{crit} </math> |
<math>~=</math> |
<math>~ \biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr] </math> |
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<math>~=</math> |
<math>~ \frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, . </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |