User:Tohline/SSC/Structure/Polytropes/VirialSummary

Virial Equilibrium of Pressure-Truncated Polytropes

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

Setup

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>

and,

<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

<math>~0</math>	<math>~=</math>	<math>~a - \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 3c x^4</math>
<math>~\Rightarrow ~~~ \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 — henceforth labeled as having the critical radius, <math>~x_\mathrm{crit}</math> — that marks a transition from dynamically stable to dynamically unstable configurations is identified by setting <math>~d^2\mathfrak{G}^*/dx^2 = 0</math>, that is, it is the configuration for which,

<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>
<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 alternatively,

<math>~ x_\mathrm{crit}^4 </math>

<math>~=</math>

<math>~ \frac{a}{3^2c}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \, . </math>

First Recognition

By comparing various terms in the first two algebraic Setup expressions, above, It is clear that,

<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,

<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} </math>
	<math>~=</math>	<math>~ \frac{n}{3a} \biggl[ a + 3cx^4_\mathrm{eq} \biggr] </math>
	<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,

<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] </math>
	<math>~=</math>	<math>~ \frac{4}{3^2\gamma(\gamma-1)} \, . </math>

See Also

• Index to a Variety of Free-Energy and/or Virial Analyses

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation