Virial Equilibrium of Pressure-Truncated Polytropes

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

Groundwork

Basic Relations

In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,

<math>~\mathfrak{G}</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>

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>~\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>~\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>~\biggl( \frac{3b}{n} \biggr) x_\mathrm{eq}^{(n-3)/n} </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,

<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 equivalently,

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

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

Equilibrium Sequences

In all of the polytropic configurations being considered here, <math>~K_\mathrm{n}</math> is a constant — that is, the specific entropy of all fluid elements is assumed to be the same, both spatially and temporally.

Fix Mass While Varying External Pressure

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~P_e</math> to vary while holding <math>~M</math> fixed. In our accompanying, more detailed discussion, this is referred to as Case M. Adopting the normalizations,

<math>~R_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~ \biggl[ \biggl( \frac{G}{K} \biggr)^n M^{n-1} \biggr]^{1/(n-3)} \, , </math>

<math>~P_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~ \biggl[ \frac{K^{4n}}{G^{3(n+1)} M^{2(n+1)}} \biggr]^{1/(n-3)} </math>

and,

<math>~E_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~ P_\mathrm{norm} R^3_\mathrm{norm} </math>

— which, as is detailed in an accompanying discussion, are similar but not identical to the normalizations adopted by Horedt (1970) and by Whitworth (1981) — the constants in the above-presented Basic Relations become,

<math>~a</math>	<math>~\equiv</math>	<math>~3\mathcal{A} = \frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}\, , </math>
<math>~b</math>	<math>~\equiv</math>	<math>~n\mathcal{B} = n\biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, , </math>
<math>~c</math>	<math>~\equiv</math>	<math>~\frac{4\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, , </math>

where, <math>~\tilde{\mathfrak{f}}_M,</math> <math>~\tilde{\mathfrak{f}}_M,</math> and <math>~\tilde{\mathfrak{f}}_M,</math> are structural form factors. The relevant dimensionless free-energy surface is, then, given by the expression,

<math>~\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} </math>

<math>~ -3\mathcal{A} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, . </math>

Fix External Pressure While Varying Mass

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~M</math> to vary while holding <math>~P_e</math> fixed. In our accompanying, more detailed discussion, this is referred to as Case P.

User:Tohline/SSC/Structure/Polytropes/VirialSummary

Contents

Virial Equilibrium of Pressure-Truncated Polytropes

Groundwork

Basic Relations

Useful Recognition

Equilibrium Sequences

Fix Mass While Varying External Pressure

Fix External Pressure While Varying Mass

See Also

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