# Virial Equilibrium of Pressure-Truncated Polytropes

Here we will draw heavily from …

## Groundwork

### Basic Relation

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

 $~\mathfrak{G}$ $~=$ $~W_\mathrm{grav} + U_\mathrm{int} + P_eV$ $~=$ $~ - 3\mathcal{A} \biggl[\frac{GM^2}{R} \biggr] + n\mathcal{B} \biggl[ \frac{K_nM^{(n+1)/n}}{R^{3/n}} \biggr] + \frac{4\pi}{3} \cdot P_e R^3 \, ,$

where, when written in terms of the trio of structural form factors, $~\tilde{\mathfrak{f}}_M,$ $~\tilde{\mathfrak{f}}_M,$ and $~\tilde{\mathfrak{f}}_M,$ the pair of constants,

 $~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}$ and $\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, .$

### Often-Referenced Dimensionless Expressions

When rewritten in a suitably dimensionless form — see two useful alternatives, below — this expression becomes,

 $~\mathfrak{G}^*$ $~=$ $~- a x^{-1} + bx^{-3/n} + c x^3 \, ,$

where $~x$ is the configuration's dimensionless radius and $~a$, $~b$, and $~c$ are constants. We therefore have,

 $~\frac{d\mathfrak{G}^*}{dx}$ $~=$ $~\frac{1}{x^2} \biggl[ a - \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 3c x^4 \biggr] \, ,$

and,

 $~\frac{d^2\mathfrak{G}^*}{dx^2}$ $~=$ $~\frac{1}{x^3} \biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr] \, .$

Virial equilibrium is obtained when $~d\mathfrak{G}^*/dx = 0$, that is, when

 $~\biggl( \frac{3b}{n} \biggr) x_\mathrm{eq}^{(n-3)/n}$ $~=$ $~ a + 3c x_\mathrm{eq}^4 \, .$

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, $~x_\mathrm{crit}$ — is identified by setting $~d^2\mathfrak{G}^*/dx^2 = 0$, that is, it is the configuration for which,

 $~0$ $~=$ $~\biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr]_{x = x_\mathrm{eq}}$ $~\Rightarrow ~~~ x_\mathrm{crit}^4$ $~=$ $~ \frac{a}{3^2c}\biggl(\frac{n - 3}{n+1}\biggr) \, .$

Inserting the adiabatic exponent in place of the polytropic index via the relation, $~n = (\gamma - 1)^{-1}$, we have equivalently,

 $~ x_\mathrm{crit}^4$ $~=$ $~ \frac{a}{3^2c}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \, .$

### Useful Recognition

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

 $~W^*_\mathrm{grav} = -ax^{-1}$ and, $~U^*_\mathrm{int} = bx^{-3/n} \, .$

Notice, then, that in every equilibrium configuration, we should find,

 $~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{eq}$ $~=$ $~ \biggl(\frac{b}{a}\biggr) x_\mathrm{eq}^{(n-3)/n} = \frac{n}{3a} \biggl[ a + 3cx^4_\mathrm{eq} \biggr]$ $~=$ $~ \frac{n}{3} \biggl[ 1 + \biggl(\frac{3c}{a}\biggr) x^4_\mathrm{eq} \biggr] \, .$

And, specifically in the critical configuration we should find that,

 $~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}$ $~=$ $~ \frac{1}{3(\gamma-1)} \biggl[ 1 + \frac{1}{3}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \biggr] = \frac{4}{3^2\gamma(\gamma-1)}$ $~\Rightarrow ~~~\frac{S^*_\mathrm{therm}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}$ $~=$ $~ -\frac{2}{3\gamma} \, .$

The equivalent of this last expression also appears at the end of subsection of an accompanying Tabular Overview.

## Equilibrium Sequences

In all of the polytropic configurations being considered here, $~K_\mathrm{n}$ 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 $~R$ and $~P_e$ to vary while holding $~M$ fixed. In our accompanying, more detailed discussion, this is referred to as Case M. Adopting the normalizations,

 $~R_\mathrm{norm}$ $~\equiv$ $~ \biggl[ \biggl( \frac{G}{K_n} \biggr)^n M^{n-1} \biggr]^{1/(n-3)} \, ,$ $~P_\mathrm{norm}$ $~\equiv$ $~ \biggl[ \frac{K_n^{4n}}{G^{3(n+1)} M^{2(n+1)}} \biggr]^{1/(n-3)}$ and, $~E_\mathrm{norm}$ $~\equiv$ $~ P_\mathrm{norm} R^3_\mathrm{norm} \, ,$

— 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 coefficients in the above-presented Basic Relations become,

 $~a$ $~\equiv$ $~3\mathcal{A} \, ,$ $~b$ $~\equiv$ $~n\mathcal{B}$ and, $~c$ $~\equiv$ $~\frac{4\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .$

The relevant dimensionless free-energy surface is, then, given by the expression,

Case M Free-Energy Surface
 $~\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}}$ $~=$ $~ -3\mathcal{A} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, .$

### 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 $~R$ and $~M$ to vary while holding $~P_e$ fixed. In our accompanying, more detailed discussion, this is referred to as Case P. Motivated by Stahler's (1983) work, here we adopt the normalizations,

 $~R_\mathrm{SWS}$ $~\equiv$ $~\biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,$ $~M_\mathrm{SWS}$ $~\equiv$ $~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}$ and, $~E_\mathrm{SWS}$ $~\equiv$ $~ \biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}} = \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .$

the coefficients in the above-presented Basic Relations become,

 $~a$ $~\equiv$ $~3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, ,$ $~b$ $~\equiv$ $~n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}$ and, $~c$ $~\equiv$ $~\frac{4\pi}{3} \, ,$

where the pair of constants, $~\mathcal{A}$ and $~\mathcal{B}$, have the same definitions in terms of the structural form factors as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,

Case P:   Free-Energy Surfaces
 $~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}$ $~=$ $~- 3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} + n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} + \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \, .$

Across this Case P free-energy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virial-equilibrium condition is met, that is, wherever

Case P:   Virial Equilibrium Sequences
 $~0$ $~=$ $~ \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4 -\frac{3\mathcal{B}}{4\pi} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} + \frac{3 \mathcal{A}}{4\pi} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .$

An equilibrium model sequence is thereby defined for each chosen polytropic index in the range, $~0 \le n < \infty$.

Along each equilibrium sequence for which $~3 \le n < \infty$, there is one specific equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this critical configuration is identified by the relation,

 $~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4$ $~=$ $~ \frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .$

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, $~\mathcal{A}$ and $~\mathcal{B}$. Here, in an effort to provide a broad overview, we set,

$~\mathcal{A} = \frac{4\pi n}{3(n+1)}$       and       $~\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,$

and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,

 $~0$ $~=$ $~ \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4 - \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} + \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .$

(As has been demonstrated in an accompanying discussion, we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, $~M_\mathrm{mod}$ and $~R_\mathrm{mod}$ — instead of choosing these specific values of $~\mathcal{A}$ and $~\mathcal{B}$.)

Figure 1:   Virial-Analysis-Based Equilibrium Sequences

For comparison, the schematic diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of Stahler (1983). It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, $~\mathcal{A}$ and $~\mathcal{B}$. Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,

$~\mathcal{A} = \frac{1}{5}$       and       $~\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}$,

and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,

 $~0$ $~=$ $~ \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4 - \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}}\biggr]^{(n+1)/n} + ~\frac{3}{20\pi} \biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .$

The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an accompanying discussion, this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit $~M(R)$ or $~R(M)$ relation for the cases labeled, $~n = 1, 3,$ and $~\infty$; a generic root-finding technique has been used to generate points along each of the other depicted equilibrium sequences.

Figure 1:   Virial-Analysis-Based Equilibrium Sequences

For comparison, the schematic diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of Stahler (1983). It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.