User:Tohline/SSC/Structure/Polytropes/VirialSummary

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Virial Equilibrium of Pressure-Truncated Polytropes

Here we will draw heavily from an accompanying Free Energy Synopsis.

Whitworth's (1981) Isothermal Free-Energy Surface
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In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,

<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,

<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,

<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>


See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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