SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2

Virial Equilibrium of Adiabatic Spheres (Summary)

Part I:   Force Balance, Free Energy, & Virial

Part II:  Mass-Radius Relation

III:  Discussion & Other Model Sequences

Mass-Radius Relation

Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes. In effect, we have viewed the problem through the same lens as did 📚 Gp. Horedt (1970, MNRAS, Vol. 151, pp. 81 - 86) and, separately, 📚 A. Whitworth (1981, MNRAS, Vol. 195, pp. 967 - 977), defining variable normalizations in terms of the polytropic constant, $K$, and the configuration mass, $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$, which were both assumed to be held fixed throughout the analysis. Here we switch to the approach championed by 📚 S. W. Stahler (1983, ApJ, Vol. 268, pp. 165 - 184), defining variable normalizations in terms of $K$ and $P_e$, and examining the mass-radius relationship of pressure-truncated polytropes.

Detailed Force-Balanced Solution

As has been summarized in our accompanying review of detailed force-balanced models of pressure-truncated polytropes, 📚 Stahler (1983) — hereafter, SWS — found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:

$\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	$=$	$(\frac{n^3}{4\pi }{)}^{1/2}{\tilde{\theta }}^{(n-3)/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}),$
$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	$=$	$(\frac{n}{4\pi }{)}^{1/2}\tilde{\xi }{\tilde{\theta }}^{(n-1)/2},$

where,

$M_{\mathrm{S}\mathrm{W}\mathrm{S}}\equiv (\frac{n+1}{nG}{)}^{3/2}{K}^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]},$

$R_{\mathrm{S}\mathrm{W}\mathrm{S}}\equiv (\frac{n+1}{nG}{)}^{1/2}{K}^{n/(n+1)}{P}_{\mathrm{e}}^{(1-n)/[2(n+1)]}.$

Mapping from Above Discussion

Deriving Concise Virial Theorem Mass-Radius Relation

Looking back on the definitions of ${\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}$ and ${\mathrm{X}}_{\mathrm{a}\mathrm{d}}$ that we introduced in connection with our initial concise algebraic expression of the virial theorem, we can write,

$P_e$	$=$	$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{3}{4\pi }\right){\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}{]}^{1/(n-3)}$
	$=$	$\left(\frac{3}{4\pi }\right){\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}\left[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}{]}^{1/(n-3)}\right[\frac{K^{4n}}{G^{3(n+1)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(n+1)}}{]}^{1/(n-3)},$
$R_{\mathrm{e}\mathrm{q}}$	$=$	$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}{\mathrm{X}}_{\mathrm{a}\mathrm{d}}[\frac{{𝒜}}{{ℬ}}{]}^{n/(n-3)}$
	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}\left[\frac{{𝒜}}{{ℬ}}{]}^{n/(n-3)}\right[(\frac{G}{K}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}{]}^{1/(n-3)}.$

The first of these two expressions can be flipped around to give an expression for $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ in terms of $P_e$ and, then, as normalized to $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$. Specifically,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(n+1)}$	$=$	$\left(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{n-3}\right[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}\left]\right[\frac{K^{4n}}{G^{3(n+1)}{P}_e^{n-3}}]$
	$=$	$M_{\mathrm{S}\mathrm{W}\mathrm{S}}^{2(n+1)}\left(\frac{n}{n+1}{)}^{3(n+1)}\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{n-3}\left[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}\right]$
$\Rightarrow \frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	$=$	$\left(\frac{n}{n+1}{)}^{3/2}\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-3)/[2(n+1)]}\left[\frac{{{ℬ}}^{2n/(n+1)}}{{{𝒜}}^{3/2}}\right].$

This means, as well, that we can rewrite the equilibrium radius as,

$R_{\mathrm{e}\mathrm{q}}^{n-3}$	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\left[\frac{{𝒜}}{{ℬ}}{]}^n\right(\frac{G}{K}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}$
	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\left[\frac{{𝒜}}{{ℬ}}{]}^n\right(\frac{G}{K}{)}^n\left\{\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{n-3}\left[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}\right]\left[\frac{K^{4n}}{G^{3(n+1)}{P}_e^{n-3}}\right]{\}}^{(n-1)/[2(n+1)]}$
	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\left[\frac{{𝒜}}{{ℬ}}{]}^n\right[\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}{]}^{(n-1)/[2(n+1)]}\left(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-3)(n-1)/[2(n+1)]}\right(\frac{G}{K}{)}^n\left\{\right[\frac{K^{4n}}{G^{3(n+1)}{P}_e^{n-3}}]{\}}^{(n-1)/[2(n+1)]}$
	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\left\{\right[\frac{{𝒜}}{{ℬ}}{]}^{2n(n+1)}\left[\frac{{{ℬ}}^{4n(n-1)}}{{{𝒜}}^{3(n+1)(n-1)}}\right]{\}}^{1/[2(n+1)]}\left(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-3)(n-1)/[2(n+1)]}\right\{\left(\frac{G}{K}{)}^{2n(n+1)}\right[\frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}{P}_e^{(n-3)(n-1)}}]{\}}^{1/[2(n+1)]}$
	$=$	${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\left(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-3)(n-1)/[2(n+1)]}\right[{{𝒜}}^{-(n+1)(n-3)}{{ℬ}}^{2n(n-3)}{]}^{1/[2(n+1)]}[G^{(3-n)(n+1)}{K}^{2n(n-3)}{P}_e^{(n-3)(1-n)}{]}^{1/[2(n+1)]}$
	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}^{n-3}\left(\frac{n}{n+1}{)}^{(n-3)/2}{\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-3)(n-1)/[2(n+1)]}[{{𝒜}}^{-(n+1)(n-3)}{{ℬ}}^{2n(n-3)}{]}^{1/[2(n+1)]}$
$\Rightarrow \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	$=$	$\left(\frac{n}{n+1}{)}^{1/2}{\mathrm{X}}_{\mathrm{a}\mathrm{d}}\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(n-1)/[2(n+1)]}\left[\frac{{{ℬ}}^{n/(n+1)}}{{{𝒜}}^{1/2}}\right].$

Flipping both of these expressions around, we see that,

${\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}$

$=$

$\frac{4\pi }{3}\left\{\right(\frac{n+1}{n}{)}^{3(n+1)}\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)}\right[\frac{{{𝒜}}^{3(n+1)}}{{{ℬ}}^{4n}}]{\}}^{1/(n-3)},$

and,

${\mathrm{X}}_{\mathrm{a}\mathrm{d}}$	$=$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left(\frac{n+1}{n}{)}^{1/2}\right[\frac{{{𝒜}}^{1/2}}{{{ℬ}}^{n/(n+1)}}\left]\right(\frac{3{\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}}{4\pi }{)}^{(1-n)/[2(n+1)]}$
	$=$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left(\frac{n+1}{n}{)}^{1/2}\right[\frac{{{𝒜}}^{1/2}}{{{ℬ}}^{n/(n+1)}}\left]\right\{\left(\frac{n+1}{n}{)}^{3(n+1)}\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)}\left[\frac{{{𝒜}}^{3(n+1)}}{{{ℬ}}^{4n}}\right]{\}}^{(1-n)/[2(n+1)(n-3)]}$
	$=$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(1-n)/(n-3)}\right(\frac{n}{n+1}{)}^{n/(n-3)}[\frac{{ℬ}}{{𝒜}}{]}^{n/(n-3)}.$

Hence, our earlier derived compact expression for the virial theorem becomes,

$1$	$=$	$\left\{\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(1-n)/(n-3)}\left(\frac{n}{n+1}{)}^{n/(n-3)}\right[\frac{{ℬ}}{{𝒜}}{]}^{n/(n-3)}{\}}^{(n-3)/n}$
		$-\frac{4\pi }{3}\left\{\right(\frac{n+1}{n}{)}^{3(n+1)}\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)}\right[\frac{{{𝒜}}^{3(n+1)}}{{{ℬ}}^{4n}}\left]{\}}^{1/(n-3)}\right\{\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(1-n)/(n-3)}\right(\frac{n}{n+1}{)}^{n/(n-3)}[\frac{{ℬ}}{{𝒜}}{]}^{n/(n-3)}{\}}^4$
	$=$	$\left(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n-3)/n}\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(1-n)/n}\left(\frac{n}{n+1}\right)\left[\frac{{ℬ}}{{𝒜}}\right]-\frac{4\pi }{3}\left(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-2}\left(\frac{n}{n+1}\right)\frac{1}{{𝒜}}.$

Or, rearranged,

$\frac{4\pi }{3}(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4-{ℬ}(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n-3)/n}(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n+1)/n}+{𝒜}(\frac{n+1}{n}\left)\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2=0.$

After adopting the modified coefficient definitions,

${{𝒜}}_{M_{\ell }}$	$\equiv$	${𝒜}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-2}=\frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2},$
${{ℬ}}_{M_{\ell }}$	$\equiv$	${ℬ}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-(n+1)/n}=(\frac{3}{4\pi }{)}^{1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}},$

as well as the modified length- and mass-normalizations, $R_{\mathrm{m}\mathrm{o}\mathrm{d}}$ and $M_{\mathrm{m}\mathrm{o}\mathrm{d}}$, such that,

$\frac{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}$	$\equiv$	$\left(\frac{4\pi }{3}{)}^{2n/(n+1)}\right[\frac{3(n+1)}{4\pi n}{]}^{3/2}\frac{{{𝒜}}_{M_{\ell }}^{3/2}}{{{ℬ}}_{M_{\ell }}^{2n/(n+1)}},$
$\frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}$	$\equiv$	$\left(\frac{4\pi }{3}{)}^{n/(n+1)}\right[\frac{3(n+1)}{4\pi n}{]}^{1/2}\frac{{{𝒜}}_{M_{\ell }}^{1/2}}{{{ℬ}}_{M_{\ell }}^{n/(n+1)}},$

we obtain the

Virial Theorem in terms of Mass and Radius

$(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^4-(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{(n-3)/n}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{(n+1)/n}+(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^2=0.$

For later use we note as well that, with these modified coefficient definitions, we can write,

${\mathrm{\Pi }}_{\mathrm{a}\mathrm{d}}^{n-3}$	$=$	$\left[\right(\frac{4\pi }{3}{)}^{n-3}\left(\frac{n+1}{n}{)}^{3(n+1)}\frac{{{𝒜}}_{M_{\ell }}^{3(n+1)}}{{{ℬ}}_{M_{\ell }}^{4n}}\right]{{𝒴}}^{2(n+1)},$
${\mathrm{X}}_{\mathrm{a}\mathrm{d}}^{n-3}$	$=$	$\left[\frac{n}{n+1}\right(\frac{{{ℬ}}_{M_{\ell }}}{{{𝒜}}_{M_{\ell }}}){]}^n{{𝒳}}^{n-3}{{𝒴}}^{1-n},$

where ${𝒳}$ and ${𝒴}$ are defined immediately below.

Corresponding Concise Free-Energy Expression

Let's also rewrite the algebraic free-energy function in terms of Stahler's normalized mass and radius variables. Expressed in terms of the polytropic index, the free-energy function is,

${\mathfrak{𝔊}}^{*}=-3{𝒜}{\chi }^{-1}+n{ℬ}{\chi }^{-3/n}+{𝒟}{\chi }^3.$

First, we recognize that,

$\chi \equiv \frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\right)\frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}.$

From the definition of $R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ — reprinted, for example, here — we can write,

$(\frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{n-3}$	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}^{n-3}\left[G^{-n}{K}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{1-n}\right]$
	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}^{n-3}{M}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{1-n}\left[\right(\frac{K}{G}{)}^n\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{1-n}\right];$

and from the definitions of $R_{\mathrm{S}\mathrm{W}\mathrm{S}}$ and $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$ — reprinted, for example, here — we have,

$(\frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{n-3}$	$=$	$\left[\right(\frac{K}{G}{)}^n\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{1-n}\right]\left\{\right(\frac{n+1}{nG}{)}^{1/2}{K}^{n/(n+1)}{P}_{\mathrm{e}}^{(1-n)/[2(n+1)]}{\}}^{n-3}\left\{\right(\frac{n+1}{nG}{)}^{3/2}{K}^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]}{\}}^{1-n}$
	$=$	$\left[\right(\frac{K}{G}{)}^n\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{1-n}\right](\frac{n+1}{n}{)}^{[(n-3)+3(1-n)]/2}{G}^{[(3-n)+3(n-1)]/2}{K}^{n[(n-3)+2(1-n)]/(n+1)}$
	$=$	$\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{1-n}\right(\frac{n}{n+1}{)}^n.$

Hence, in each term in the free-energy expression we can make the substitution,

$\chi$

$\to$

$\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\right)\left\{\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{1-n}(\frac{n}{n+1}{)}^n{\}}^{1/(n-3)}=(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left)\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(1-n)/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{n-1}(\frac{n}{n+1}{)}^n{\}}^{1/(n-3)}.$

Next, drawing on the definition of $P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ — reprinted, for example, here — along with the definition of $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$, we recognize that,

${𝒟}\equiv \frac{4\pi }{3}\cdot \frac{P_e}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$\frac{4\pi }{3}\cdot P_e[K^{-4n}{G}^{3(n+1)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(n+1)}{]}^{1/(n-3)}$
	$=$	$\frac{4\pi }{3}\cdot \left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)/(n-3)}{P}_e\right[K^{-4n}{G}^{3(n+1)}{]}^{1/(n-3)}{M}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{2(n+1)/(n-3)}$
	$=$	$\frac{4\pi }{3}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)/(n-3)}\right\{\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-2}\right(\frac{n+1}{n}{)}^3{\}}^{(n+1)/(n-3)}.$

After making these substitutions into the free-energy function, as well as replacing ${𝒜}$ and ${ℬ}$ with ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$, respectively, we have,

${\mathfrak{𝔊}}^{*}$	$=$	$-3{{𝒜}}_{M_{\ell }}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^2\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-(1-n)/(n-3)}\right\{\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{n-1}\right(\frac{n}{n+1}{)}^n{\}}^{-1/(n-3)}$
		$+n{{ℬ}}_{M_{\ell }}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n+1)/n}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/n}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3(1-n)/[n(n-3)]}\right\{\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{n-1}\right(\frac{n}{n+1}{)}^n{\}}^{-3/[n(n-3)]}$
		$+\frac{4\pi }{3}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{2(n+1)/(n-3)}\right\{\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-2}\right(\frac{n+1}{n}{)}^3{\}}^{(n+1)/(n-3)}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{3(1-n)/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{n-1}(\frac{n}{n+1}{)}^n{\}}^{3/(n-3)}$
	$=$	$-3{{𝒜}}_{M_{\ell }}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-(1-n)/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{[(n-1)-2(n-3)]}(\frac{n}{n+1}{)}^n{\}}^{-1/(n-3)}$
		$+n{{ℬ}}_{M_{\ell }}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/n}\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3(1-n)/[n(n-3)]}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{[(n+1)(n-3)-3(n-1)]}(\frac{n}{n+1}{)}^{-3n}{\}}^{1/[n(n-3)]}$
		$+\frac{4\pi }{3}\cdot \left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{[2(n+1)+3(1-n)]/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{[(3(n-1)-2(n+1)]}(\frac{n}{n+1}{)}^{[3n-3(n+1)]}{\}}^{1/(n-3)}$
	$=$	$-3{{𝒜}}_{M_{\ell }}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-(1-n)/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-5)}(\frac{n+1}{n}{)}^n{\}}^{1/(n-3)}$
		$+n{{ℬ}}_{M_{\ell }}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/n}\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3(1-n)/[n(n-3)]}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-5)}(\frac{n+1}{n}{)}^3{\}}^{1/(n-3)}$
		$+\frac{4\pi }{3}\cdot \left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(5-n)/(n-3)}\left\{\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-5)}(\frac{n+1}{n}{)}^3{\}}^{1/(n-3)}$
	$=$	$\left[\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-5)}\left(\frac{n+1}{n}{)}^3\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(5-n)}{]}^{1/(n-3)}\{-3{{𝒜}}_{M_{\ell }}(\frac{n+1}{n}\left)\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2+n{{ℬ}}_{M_{\ell }}(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/n}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n+1)/n}+\frac{4\pi }{3}\cdot (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3\}.$

Hence, after defining,

${\mathfrak{𝔊}}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{*}$

$\equiv$

$\frac{\mathfrak{𝔊}}{[G^{-3}{K}^n{M}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{n-5}{]}^{1/(n-3)}}(\frac{n}{n+1}{)}^{3/(n-3)}=\frac{\mathfrak{𝔊}}{[K^{6n}{P}_e^{5-n}{]}^{1/[2(n+1)]}}(\frac{nG}{n+1}{)}^{3/2},$

we can write,

${\mathfrak{𝔊}}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{*}$

$=$

$-3{{𝒜}}_{M_{\ell }}\left(\frac{n+1}{n}\right)\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}+n{{ℬ}}_{M_{\ell }}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n+1)/n}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/n}+\frac{4\pi }{3}\cdot (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3.$

Setting the first derivative of this function equal to zero should produce the virial theorem expression. Let's see …

$\frac{\partial {\mathfrak{𝔊}}_{\mathrm{S}\mathrm{W}\mathrm{S}}^{*}}{\partial {𝒳}}$	$=$	$3{{𝒜}}_{M_{\ell }}\left(\frac{n+1}{n}\right)\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-2}-3{{ℬ}}_{M_{\ell }}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n+1)/n}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-(3+n)/n}+4\pi (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2$
	$=$	$3\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-2}\right[{{𝒜}}_{M_{\ell }}\left(\frac{n+1}{n}\right)(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2-{{ℬ}}_{M_{\ell }}\cdot (\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n+1)/n}(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n-3)/n}+\frac{4\pi }{3}\cdot (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4].$

Replacing ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ with ${𝒜}$ and ${ℬ}$, as prescribed by their defined relationships, and setting the expression inside the square brackets equal to zero does, indeed, produce the above, boxed-in viral theorem mass-radius relationship.

Plotting Concise Mass-Radius Relation

Our derived, concise analytic expression for the virial theorem, namely,

$(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^4-(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{(n-3)/n}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{(n+1)/n}+(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^2=0,$

is plotted for seven different values of the polytropic index, $n$, as indicated, in the lefthand diagram of the following composite figure. 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.

Virial Theorem Mass-Radius Relationships
	Digital copy of Figure 17 from … S. W. Stahler (1983) The Equilibria of Rotating Isothermal Clouds. II. Structure and Dynamical Stability The Astrophysical Journal, Vol. 268, pp. 165 - 184
	Stahler (1983) Figure 17
Left-hand Panel:   As detailed below, the three orange-dashed sequences — n = 1, n = 3, and isothermal — are analytically prescribed while the others have been determined via an iterative procedure. Also as detailed below, a solid-yellow circular marker identifies where along each sequence (n > 1) the model with the largest radius resides; for n = 1, the equilibrium sequence asymptotically approaches the maximum radius, $R/R_{\mathrm{S}\mathrm{W}\mathrm{S}}=[15/(8\pi ){]}^{1/2}$, where the mass climbs to infinity. The solid-green circular marker identifies where along each sequence (n ≥ 3) the maximum-mass model resides.

Let's do this again using the mass-radius relation as written explicitly in terms of the normalizations, $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$ and $R_{\mathrm{S}\mathrm{W}\mathrm{S}}$. The relevant, generic nonlinear equation is,

$0$

$=$

$(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4-(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n-3)/n}\left[\right(\frac{3}{4\pi })\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}^{(n+1)/n}+\frac{3}{20\pi }(\frac{n+1}{n}\left)\right(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2.$

Analytically determined roots:

• $n=1$

$\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}=\left(\frac{10\pi }{3}{)}^{1/2}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3[\frac{3\cdot 5}{2^3\pi }-(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2{]}^{-1/2}$         for,     $0\le \frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\le (\frac{3\cdot 5}{2^3\pi }{)}^{1/2}.$

• $n=3$

$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}=\left\{\right[\left(\frac{3}{4\pi }\right)\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}^{4/3}-\left(\frac{1}{5\pi }\right)(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2{\}}^{1/4}$         for,     $0\le \frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}\le (\frac{3^4\cdot 5^3}{2^8\pi }{)}^{1/2}.$

• Isothermal (explained immediately below)

$\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}=\frac{5}{2}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\right)\{1\pm [1-\frac{16\pi }{15}\left(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2{]}^{1/2}\right\}$         for,     $0\le \frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}\le (\frac{3\cdot 5}{2^4\pi }{)}^{1/2}.$

First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.

For Various Values of $n$, Numerically Determined Solutions to the Virial-Equilibrium Relation … $0$ $=$ $(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4-(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{(n-3)/n}\left[\right(\frac{3}{4\pi })\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}^{(n+1)/n}+\frac{3}{20\pi }(\frac{n+1}{n}\left)\right(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2.$
$n=2$	$n=2.8$	$n=3.5$	$n=4$	$n=5$
$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$   $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$       0.3800   0.26562 0.4500   0.477153 0.5000   0.70919 0.5500   1.063602 0.5800   1.39755 0.5950   1.64662 0.6050   1.893915 0.6120   2.22372 0.6131721   2.433375 0.6120   2.64923 0.6050   3.01688 0.5950   3.32037 0.5800   3.658702 0.5500   4.19097 0.5000   4.94599 0.4700   5.38791 0.4500   5.69164	$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$   $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$       0.3800   0.266134 0.4500   0.47971 0.5000   0.71765 0.5250   0.881825 0.5600   1.20977 0.5750   1.427183 0.5850   1.653232 0.5900   1.89304 0.5904492   1.989927 0.5900   2.086584 0.5850   2.32394 0.5750   2.54527 0.5600   2.75612 0.5250   3.07134 0.4500   3.460304 0.3500   3.75881 0.2500   3.97835 0.2000   4.09302 0.1500   4.232786 0.1000   4.430303 0.0700   4.60984 0.0400   4.9057 0.0150   5.47056	$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$   $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$       0.3800   0.26639 0.4500   0.481072 0.5000   0.722406 0.5250   0.89152 0.5600   1.246123 0.5650   1.32113 0.5750   1.52651 0.5800   1.745165 0.5803836   1.823995 0.5800   1.90201 0.5780   2.01647 0.5750   2.11019 0.5600   2.35906 0.5400   2.543602 0.5000   2.7555746 0.4500   2.8890287 0.3800   2.9482952 0.3749583   2.948526 0.3300   2.93161 0.2500   2.829401 0.1500   2.578605 0.1000   2.380925 0.0750   2.2483845 0.0400   1.983015 0.0200   1.726337 0.0100   1.502865	$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$   $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$ 0.1000   0.004224 0.2000   0.034709 0.3000   0.1230901 0.4000   0.31735 0.4500   0.48177 0.5000   0.72493 0.5250   0.89686 0.5400   1.028495 0.5500   1.13574 0.5600   1.26965 0.5730   1.55527 0.5756189   1.750930 0.5730   1.93949 0.5600   2.18983 0.5400   2.376318 0.5250   2.46661 0.5000   2.56895 0.4600   2.657809 0.41184646   2.688999 0.4100   2.68895 0.3800   2.677703 0.3000   2.56612 0.2500   2.44565 0.2000   2.28789 0.1500   2.08747 0.1000   1.82708 0.0750   1.660706 0.0400   1.3470695 0.0200   1.0692 0.0100   0.848625	$\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$   $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$ 0.1000   0.004224 0.2000   0.03471 0.3000   0.123115 0.4000   0.31766 0.4500   0.48278 0.5000   0.72866 0.5250   0.905006 0.5400   1.042907 0.5500   1.15886 0.5600   1.313712 0.5675   1.511304 0.5692185   1.657839 0.5675   1.798532 0.5600   1.97061 0.5400   2.17282 0.5250   2.25888 0.5000   2.34793 0.4600   2.410374 0.4391754   2.417330 0.4000   2.396465 0.3000   2.19848 0.2000   1.84195 0.1000   1.31421 0.0500   0.930314 0.0200   0.58847 0.0100   0.4161145
NOTE:   Along each sequence (fixed value of "n"), the coordinates, $(R/R_{\mathrm{S}\mathrm{W}\mathrm{S}},M/M_{\mathrm{S}\mathrm{W}\mathrm{S}}),$ of the model with the largest radius are typed in a dark green font; the identified coordinate values not only satisfy the virial-balance equation but also the relation, $(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4=\frac{3}{20\pi }(\frac{n-1}{n}\left)\right(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2.$ Similarly, for sequences having n > 3, the coordinates of the model with the maximum mass are highlighted by a yellow background color; the coordinate values not only satisfy the virial-balance equation but also the relation, $(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4=\frac{1}{20\pi }(\frac{n-3}{n}\left)\right(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2.$

From a free-energy analysis of isothermal spheres, we have demonstrated that, when the structural form factors are all set to unity, the statement of virial equilibrium is,

$0$

$=$

$(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4-\frac{3}{4\pi }(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left)\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}})+\frac{3}{20\pi }(\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2,$

where, in order to be consistent with the above polytropic normalizations, we have adopted the isothermal normalizations,

$M_{\mathrm{S}\mathrm{W}\mathrm{S}}{|}_{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}\equiv (\frac{c_s^8}{G^3{P}_e}{)}^{1/2},$

and

$R_{\mathrm{S}\mathrm{W}\mathrm{S}}{|}_{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}\equiv (\frac{c_s^4}{G{P}_e}{)}^{1/2}.$

This is a quadratic equation that can be readily solved to provide an analytic expression for the isothermal mass-radius relation; the relevant expression has already been provided, above.

Confirmation

Rewriting the just-derived virial theorem expression in terms of Stahler's dimensionless radius and mass variables, written in the abbreviated form,

${𝒳}$	$\equiv$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}},$
${𝒴}$	$\equiv$	$\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}},$

we have,

$0$	$=$	$[{𝒳}\cdot \frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{]}^4-[{𝒳}\cdot \frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{]}^{(n-3)/n}[{𝒴}\cdot \frac{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{]}^{(n+1)/n}+[{𝒴}\cdot \frac{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{]}^2$
	$=$	${{𝒳}}^4\left\{\right(\frac{4\pi }{3}{)}^{n/(n+1)}[\frac{3(n+1)}{4\pi n}{]}^{1/2}\frac{{{𝒜}}_{M_{\ell }}^{1/2}}{{{ℬ}}_{M_{\ell }}^{n/(n+1)}}{\}}^4+{{𝒴}}^2\{\left(\frac{4\pi }{3}{)}^{2n/(n+1)}\right[\frac{3(n+1)}{4\pi n}{]}^{3/2}\frac{{{𝒜}}_{M_{\ell }}^{3/2}}{{{ℬ}}_{M_{\ell }}^{2n/(n+1)}}{\}}^2$
		$-{{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}\left\{\right(\frac{4\pi }{3}{)}^{n/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^{1/2}\frac{{{𝒜}}_{M_{\ell }}^{1/2}}{{{ℬ}}_{M_{\ell }}^{n/(n+1)}}{\}}^{(n-3)/n}\right\{\left(\frac{4\pi }{3}{)}^{2n/(n+1)}\right[\frac{3(n+1)}{4\pi n}{]}^{3/2}\frac{{{𝒜}}_{M_{\ell }}^{3/2}}{{{ℬ}}_{M_{\ell }}^{2n/(n+1)}}{\}}^{(n+1)/n}$
	$=$	${{𝒳}}^4\left\{\right(\frac{4\pi }{3}{)}^{4n/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^2\frac{{{𝒜}}_{M_{\ell }}^2}{{{ℬ}}_{M_{\ell }}^{4n/(n+1)}}\right\}+{{𝒴}}^2\left\{\right(\frac{4\pi }{3}{)}^{4n/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^3\frac{{{𝒜}}_{M_{\ell }}^3}{{{ℬ}}_{M_{\ell }}^{4n/(n+1)}}\right\}$
		$-{{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}\left\{\right(\frac{4\pi }{3}{)}^{[(n-3)+2(n+1)]/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^{[(n-3)+3(n+1)]/2n}\frac{{{𝒜}}_{M_{\ell }}^{[(n-3)+3(n+1)]/2n}}{{{ℬ}}_{M_{\ell }}^{[(n-3)+2(n+1)]/(n+1)}}\right\}$
	$=$	$\{{{𝒳}}^4+[\frac{3(n+1)}{4\pi n}\left]{{𝒜}}_{M_{\ell }}{{𝒴}}^2\right\}\left\{\right(\frac{4\pi }{3}{)}^{4n/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^2\frac{{{𝒜}}_{M_{\ell }}^2}{{{ℬ}}_{M_{\ell }}^{4n/(n+1)}}\right\}$
		$-{{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}\left\{\right(\frac{4\pi }{3}{)}^{(3n-1)/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^2\frac{{{𝒜}}_{M_{\ell }}^2}{{{ℬ}}_{M_{\ell }}^{(3n-1)/(n+1)}}\right\}$
	$=$	$\left[\right(\frac{4\pi }{3}){{𝒳}}^4-{{ℬ}}_{M_{\ell }}\cdot {{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}+(\frac{n+1}{n}\left){{𝒜}}_{M_{\ell }}{{𝒴}}^2\right]\left\{\right(\frac{4\pi }{3}{)}^{(3n-1)/(n+1)}\left[\frac{3(n+1)}{4\pi n}{]}^2\frac{{{𝒜}}_{M_{\ell }}^2}{{{ℬ}}_{M_{\ell }}^{4n/(n+1)}}\right\}.$

Replacing ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ with ${𝒜}$ and ${ℬ}$, as prescribed by their defined relationships, the expression inside the square brackets becomes the above, boxed-in mass-radius relationship, namely,

$\frac{4\pi }{3}\cdot {{𝒳}}^4-{ℬ}\cdot {{𝒳}}^{(n-3)/n}\left[{𝒴}\right(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}){]}^{(n+1)/n}+{𝒜}(\frac{n+1}{n}\left)\right[{𝒴}\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}\right){]}^2$

$=$

$0.$

In Terms of Structural Form-Factors

Alternatively, replacing ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ by their expressions in terms of the structural form factors gives,

$\frac{4\pi }{3}\cdot {{𝒳}}^4-{{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}(\frac{4\pi }{3}{)}^{-1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}}+{{𝒴}}^2(\frac{n+1}{5n})\frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}$

$=$

$0.$

Finally, inserting into this relation the expressions presented above for the structural form-factors, ${\widetilde{\mathfrak{𝔣}}}_M$ and ${\widetilde{\mathfrak{𝔣}}}_A$, namely,

${\widetilde{\mathfrak{𝔣}}}_M$	$=$	$[-\frac{3{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{\tilde{\xi }}$
${\widetilde{\mathfrak{𝔣}}}_A$	$=$	${\tilde{\theta }}^{n+1}+\left[\frac{(n+1)}{3\cdot 5}\right]{\tilde{\xi }}^2\cdot {\widetilde{\mathfrak{𝔣}}}_W$

gives us the desired,

Virial Theorem written in terms of ${𝒳}$, ${𝒴}$, and ${\widetilde{\mathfrak{𝔣}}}_W$ $4\pi \cdot {{𝒳}}^4-{{𝒳}}^{(n-3)/n}{{𝒴}}^{(n+1)/n}(4\pi {)}^{-1/n}[\frac{\tilde{\xi }}{(-{\tilde{\theta }}^{\text{'}})}{]}^{(n+1)/n}[{\tilde{\theta }}^{n+1}+\frac{(n+1){\tilde{\xi }}^2}{3\cdot 5}\cdot {\widetilde{\mathfrak{𝔣}}}_W]+{{𝒴}}^2\left(\frac{n+1}{3\cdot 5n}\right)\frac{{\tilde{\xi }}^2}{(-{\tilde{\theta }}^{\text{'}}{)}^2}\cdot {\widetilde{\mathfrak{𝔣}}}_W=0.$

Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes

Now, let's examine the case of pressure-truncated, $n=5$ polytropes. As we have discussed in the context of detailed force-balanced models, Stahler (1983) has deduced that all $n=5$ equilibrium configurations obey the mass-radius relationship,

$(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2-5(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}\left)\right(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}})+\frac{2^2\cdot 5\pi }{3}(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^4$

$=$

$0,$

where, as reviewed above, the mass and radius normalizations, $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$ and $R_{\mathrm{S}\mathrm{W}\mathrm{S}}$, may be treated as constants once the parameters $K$ and $P_e$ are specified. In contrast to this, the mass-radius relationship that we have just derived from the virial theorem for pressure-truncated, $n=5$ polytropes is,

$(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^2-(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{2/5}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^{6/5}+(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{m}\mathrm{o}\mathrm{d}}}{)}^4=0,$

where the mass and radius normalizations,

$M_{\mathrm{m}\mathrm{o}\mathrm{d}}{|}_{n=5}$	$=$	$M_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{3{{ℬ}}_{M_{\ell }}}{4\pi }{)}^{5/3}\right[\frac{2\cdot 5\pi }{3^2{{𝒜}}_{M_{\ell }}}{]}^{3/2},$
$R_{\mathrm{m}\mathrm{o}\mathrm{d}}{|}_{n=5}$	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{3{{ℬ}}_{M_{\ell }}}{4\pi }{)}^{5/6}\right[\frac{2\cdot 5\pi }{3^2{{𝒜}}_{M_{\ell }}}{]}^{1/2},$

depend, not only on $K$ and $P_e$ via the definitions of $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$ and $R_{\mathrm{S}\mathrm{W}\mathrm{S}}$, but also on the structural form factors via the free-energy coefficients, ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$. While these two separate mass-radius relationships are similar, they are not identical. In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions. It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.

This apparent discrepancy is reconciled as follows: The structural form factors — and, hence, the free-energy coefficients — vary from equilibrium configuration to equilibrium configuration. So it does not make sense to discuss evolution along the sequence that is defined by the second of the two polynomial expressions. If you want to know how a given system's equilibrium radius will change as its mass changes, the first of the two polynomials will do the trick. However, the equilibrium radius of a given system can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$, constant; more importantly, the relative stability of a given equilibrium system can be determined by analyzing the behavior of the system's free energy while holding the free-energy coefficients constant. Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression. It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.

Plotting Stahler's Relation

Mass (Y) vs. radius (X) plot

Switching, again, to the shorthand notation,

${𝒳}$	$\equiv$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}},$
${𝒴}$	$\equiv$	$\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}},$

the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways.

Quadratic Equation

One way is to recognize that the polynomial is a quadratic equation whose solution is,

${{𝒴}}_{\pm }$

$=$

$\frac{5}{2}{𝒳}\{1\pm [1-\left(\frac{2^4\cdot \pi }{3\cdot 5}\right){{𝒳}}^2{]}^{1/2}\}.$

In the figure shown here on the right — see also the bottom panel of Figure 2 in our accompanying discussion of detailed force-balance models — Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the positive root while the segment derived from the negative root is shown in orange. The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at

${{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\equiv [\frac{3\cdot 5}{2^4\pi }{]}^{1/2}\approx 0.54627.$

We note that, when ${𝒳}={{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, ${𝒴}=(5{{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/2)\approx 1.36569$. Along the entire sequence, the maximum value of ${𝒴}$ occurs at the location where $d{𝒴}/d{𝒳}=0$ along the segment of the curve corresponding to the positive root. This occurs along the upper segment of the curve where ${𝒳}/{{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{3}/2$, at the location,

${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\equiv [\frac{3^3\cdot 5^2}{2^6}{]}^{1/2}{{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=[\frac{3^4\cdot 5^3}{2^{10}\pi }{]}^{1/2}\approx 1.77408.$

Parametric Relations

The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations. Drawing partly from our above discussion and partly from a separate discussion where we provide a tabular summary of the properties of pressure-truncated $n=5$ polytropes, these are,

${𝒳}{|}_{n=5}$	$=$	$(\frac{5}{4\pi }{)}^{1/2}\tilde{\xi }{\tilde{\theta }}^2=\{\frac{3\cdot 5}{2^2\pi }\left[\frac{{\tilde{\xi }}^2/3}{(1+{\tilde{\xi }}^2/3{)}^2}\right]{\}}^{1/2},$
${𝒴}{|}_{n=5}$	$=$	$(\frac{5^3}{4\pi }{)}^{1/2}\tilde{\theta }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})=[\left(\frac{3\cdot 5^3}{2^2\pi }\right)\frac{({\tilde{\xi }}^2/3{)}^3}{(1+{\tilde{\xi }}^2/3{)}^4}{]}^{1/2}.$

The entire sequence will be traversed by varying the Lane-Emden parameter, $\tilde{\xi }$, from zero to infinity. Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, ${{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, is associated with an embedded $n=5$ polytrope whose truncated, dimensionless Lane-Emden radius is,

$\tilde{\xi }{|}_{{{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=3^{1/2}.$

Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, is associated with an embedded $n=5$ polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,

$\tilde{\xi }{|}_{{{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=3.$

Referring back to our review of turning points along equilibrium sequences and, especially, the work of 📚 H. Kimura (1981, PASJapan, Vol. 33, pp. 299 - 312), we appreciate that the point that corresponds to the maximum mass, ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, is the turning point that Kimura refers to as the "extremum in M1" along a p1 sequence. As we have highlighted, according to Kimura, this point should occur along the sequence where $h_G=0$, that is, where the following condition applies: $\frac{{\tilde{\theta }}^{n+1}}{({\tilde{\theta }}^{\text{'}}{)}^2}=\frac{(n-3)}{2}.$ For the specific case being studied here, namely, $n=5$ polytropic configurations, we therefore expect from Kimura's work that $[{\tilde{\theta }}^6/({\tilde{\theta }}^{\text{'}}{)}^2]=1$ at the "maximum mass" turning point. Given that, $\tilde{\xi }{|}_{{{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=3$         $\Rightarrow$         ${\tilde{\theta }}_{n=5}=\frac{1}{2}$     and      ${\tilde{\theta }}_{n=5}^{\text{'}}=-\frac{1}{8},$ we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum mass matches Kimura's identification of the location of that turning point. We appreciate, as well, that the point corresponding to the maximum normalized equilibrium radius, ${{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, is the turning point that Kimura would reference as the "extremum in r1" along a p1 sequence. Following Kimura's analysis we have shown that this point occurs along the sequence where the following condition applies: $\frac{\xi (-{\theta }^{\text{'}})}{\tilde{\theta }}=\frac{2}{(n-1)},$ that is, for the specific case being studied here, we should expect $[\tilde{\xi }(-{\tilde{\theta }}^{\text{'}})/\tilde{\theta }]=1/2$ at the "maximum radius" turning point. Given that, $\tilde{\xi }{|}_{{{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=3^{1/2}$         $\Rightarrow$         ${\tilde{\theta }}_{n=5}=2^{-1/2}$     and      ${\tilde{\theta }}_{n=5}^{\text{'}}=-(2^3\cdot 3{)}^{-1/2},$ we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum radius matches Kimura's identification of the location of that turning point.

Discussion

Notice that if the quadratic equation is used to map out the mass-radius relationship, the parameter, $\tilde{\xi }$, never explicitly enters the discussion. Instead, a radius $0\le {𝒳}\le {{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is specified and the two equilibrium masses associated with ${𝒳}$ — call them, ${{𝒴}}_{+}$ and ${{𝒴}}_{-}$ — are determined. (The values of the two masses are degenerate at both limiting values of ${𝒳}$.) If the pair of parametric relations is used, instead, only one value of the mass is obtained for each specified value of $\tilde{\xi }$. As $\tilde{\xi }$ is increased from $0$ to $\sqrt{3}$, ${𝒳}$ increases monotonically from $0$ to ${{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the corresponding mass is (only) ${{𝒴}}_{-}$; that is, as $\tilde{\xi }$ is increased from $0$ to $\sqrt{3}$, we move away from the origin in a counter-clockwise direction along the lower segment (colored orange in the above figure) of the plotted equlibrium sequence. Then, as $\tilde{\xi }$ is increased from $\sqrt{3}$ to $\mathrm{\infty }$, we continue to move in a counter-clockwise direction along the equilibrium sequence, but now along the upper segment (colored green in the above figure) of the sequence, back to the origin; that is to say, ${𝒳}$ steadily decreases from ${{𝒳}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ back to $0$ and this time the relevant associated mass is the positive root of the quadratic relation, ${{𝒴}}_{+}$.

Clearly, then, each value of ${𝒳}$ is associated with two different values of the parametric parameter, $\tilde{\xi }$. By inverting the ${𝒳}(\tilde{\xi })$ parametric expression we see that, the two values of $\tilde{\xi }$ associated with a given equilibrium radius are,

${\tilde{\xi }}_{\pm }$

$=$

$\left\{\frac{3}{\alpha }\right[1\pm \sqrt{1-{\alpha }^2}]{\}}^{1/2},$

where,

$\alpha$

$\equiv$

$\frac{({𝒳}/{{𝒳}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}{)}^2}{2-({𝒳}/{{𝒳}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}{)}^2}.$

We note as well that, for a given equilibrium radius, ${𝒳}$, the ratio of the two mass solutions is given by a very simple expression, namely,

$\frac{{{𝒴}}_{-}}{{{𝒴}}_{+}}=\frac{{\tilde{\xi }}_{-}^2}{3}$

or

$\frac{{{𝒴}}_{+}}{{{𝒴}}_{-}}=\frac{{\tilde{\xi }}_{+}^2}{3}.$

This implies, as well, that,

${\tilde{\xi }}_{+}\cdot {\tilde{\xi }}_{-}$

$=$

$3.$

Plotting the Virial Theorem Relation

Drawing from our above derivations, the concise free-energy expression that reflects the properties of pressure-truncated $n=5$ polytropic configurations is,

${\mathfrak{𝔊}}^{*}$

$=$

$-\frac{18}{5}\cdot {{𝒜}}_{M_{\ell }}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}+5{{ℬ}}_{M_{\ell }}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{6/5}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/5}+\frac{4\pi }{3}\cdot (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3,$

where,

${{𝒜}}_{M_{\ell }}$	$=$	$\frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}=\frac{1}{3^2\cdot 5}(\frac{\tilde{\xi }}{-{\tilde{\theta }}^{\text{'}}}{)}^2{\widetilde{\mathfrak{𝔣}}}_W,$
${{ℬ}}_{M_{\ell }}$	$=$	$(\frac{3}{4\pi }{)}^{1/5}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{6/5}}=\frac{1}{3}(4\pi {)}^{-1/5}\left(\frac{\tilde{\xi }}{-{\tilde{\theta }}^{\text{'}}}{)}^{6/5}\right[{\tilde{\theta }}^6+\frac{2}{5}\cdot {\tilde{\xi }}^2{\widetilde{\mathfrak{𝔣}}}_W].$

The virial theorem which is derived from this free-energy expression provides a mass-radius relationship to be compared with the detailed force-balance relationship presented by Stahler. Because our intent is to make this comparison, we begin with the virial theorem as written in terms of the variables, ${𝒳}$ and ${𝒴}$, and specialized for the case of $n=5$ polytropic configurations. Written in terms of the (constant) coefficients in the free-energy expression, we have

${{𝒳}}^4-\frac{3{{ℬ}}_{M_{\ell }}}{4\pi }\cdot ({𝒳}{{𝒴}}^3{)}^{2/5}+\frac{9{{𝒜}}_{M_{\ell }}}{10\pi }\cdot {{𝒴}}^2$

$=$

$0;$

or, from above, the

Virial Theorem written in terms of ${𝒳}$, ${𝒴}$, and ${\widetilde{\mathfrak{𝔣}}}_W$ ${{𝒳}}^4-({𝒳}{{𝒴}}^3{)}^{2/5}[\frac{\tilde{\xi }}{4\pi (-{\tilde{\theta }}^{\text{'}})}{]}^{6/5}[{\tilde{\theta }}^6+\frac{2}{5}\cdot {\tilde{\xi }}^2{\widetilde{\mathfrak{𝔣}}}_W]+{{𝒴}}^2[\frac{\tilde{\xi }}{(-{\tilde{\theta }}^{\text{'}})}{]}^2\frac{{\widetilde{\mathfrak{𝔣}}}_W}{2\cdot 5^2\pi }=0,$

where, specifically for $n=5$ polytropic configurations — see our summary of the radial profiles of physical variables and our determination of expressions for the structural form-factors,

$\tilde{\theta }$	$=$	$(1+{\ell }^2{)}^{-1/2},$
${\tilde{\theta }}^{\text{'}}$	$=$	$-3^{-1/2}\ell (1+{\ell }^2{)}^{-3/2},$
${\mathfrak{𝔣}}_W$	$=$	$\frac{5}{2^4}\cdot {\ell }^{-5}\left[\ell \right({\ell }^4-\frac{8}{3}{\ell }^2-1)(1+{\ell }^2{)}^{-3}+{\tan }^{-1}(\ell )],$
${\ell }^2$	$\equiv$	$\frac{{\tilde{\xi }}^2}{3}.$

Once numerical values have been assigned to the free-energy coefficients, ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$, the mass-radius relationship given by the scalar virial theorem can be compared quantitatively with Stahler's (detailed force-balance) mass-radius relationship. The simplest, physically reasonable approximation would be to assume uniform-density structures, in which case, ${\widetilde{\mathfrak{𝔣}}}_M={\widetilde{\mathfrak{𝔣}}}_W={\widetilde{\mathfrak{𝔣}}}_A=1$, and accordingly, ${{𝒜}}_{M_{\ell }}=5^{-1}$ and ${{ℬ}}_{M_{\ell }}=(4\pi /3{)}^{-1/5}$. But a better approximation would be to assign values to the structural form-factors that properly represent the properties of at least one detailed force-balanced model. By way of illustration, the following table details what the proper values are for the two free-energy coefficients, and other relevant parameters, specifically for the model along Stahler's sequence that sits at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ — that is, the model whose truncation radius is $\tilde{\xi }=3$. As is recorded in the table, in this case the precise values of the free-energy coefficients are,

${{𝒜}}_{M_{\ell }}$	$=$	$(\frac{2^4{\pi }^2}{3^7}{)}^{1/2},$
${{ℬ}}_{M_{\ell }}$	$=$	$\left(\frac{3}{2^{14}\pi }{)}^{1/5}\right[1+\left(\frac{2^6{\pi }^2}{3^3}{)}^{1/2}\right].$

Notice that, by choosing $\tilde{\xi }=3$, the evaluation of ${\widetilde{\mathfrak{𝔣}}}_W$ is particularly simple, in part, because ${\tan }^{-1}(\ell )={\tan }^{-1}\sqrt{3}=\pi /3$, but also because the term $({\ell }^4-8{\ell }^2/3-1)$ equals zero.

Determination of Coefficient Values in the Specific Case of $\tilde{\xi }=3$
Quantity	Analytic Evaluation	Numerical
$\ell$	$3^{1/2}$	$1.732051$
$\tilde{\theta }$	$2^{-1}$	$0.5$
${\tilde{\theta }}^{\text{'}}$	$2^{-3}$	$0.125$
${𝒳}$	$(\frac{3^2\cdot 5}{2^6\pi }{)}^{1/2}$	$0.473087$
${𝒴}$	$(\frac{3^4\cdot 5^3}{2^{10}\pi }{)}^{1/2}$	$1.774078$
${\widetilde{\mathfrak{𝔣}}}_W$	$(\frac{5^2{\pi }^2}{2^8\cdot 3^7}{)}^{1/2}$	$0.020993$
${{𝒜}}_{M_{\ell }}$	$(\frac{2^4{\pi }^2}{3^7}{)}^{1/2}$	$0.268711$
${{ℬ}}_{M_{\ell }}$	$\left(\frac{3}{2^{14}\pi }{)}^{1/5}\right[1+\left(\frac{2^6{\pi }^2}{3^3}{)}^{1/2}\right]$	$0.830395$
$G^{*}$	$(\frac{3\cdot 5^3}{2^{12}\pi }{)}^{1/2}[2^3\pi +3^{5/2}]$	$6.951544$
Virial: ${{𝒳}}^4-\frac{3{{ℬ}}_{M_{\ell }}}{4\pi }\cdot ({𝒳}{{𝒴}}^3{)}^{2/5}$           $+\frac{9{{𝒜}}_{M_{\ell }}}{10\pi }\cdot {{𝒴}}^2$	$\frac{3^4\cdot 5^2}{2^{12}\pi }-\frac{3^4\cdot 5^2}{2^{12}\pi }[1+(\frac{2^6{\pi }^2}{3^3}{)}^{1/2}]$           $+(\frac{3^5\cdot 5^4}{2^{18}{\pi }^2}{)}^{1/2}$	Sums to zero, exactly!

The curve traced out by the light-blue diamonds in each panel of the following comparison figure displays Stahler's analytically prescribed mass-radius relation; this curve is identical in all six panels and is the same as the curve displayed above in connection with our description of Stahler's mass-radius relation. Each point along this "Stahler" curve identifies a model having a different truncation radius, $\tilde{\xi }$; as plotted here, starting near the origin and moving counter-clockwise around the curve, $\tilde{\xi }$ is varied from 0.05 to 42.5. As foreshadowed by the above discussion, the model having the greatest mass $({{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}})$ along the Stahler sequence — highlighted by the red filled circle in most of the figure panels — is defined by $\tilde{\xi }=3$.

In each figure panel, the curve traced out by the orange triangles — or, in one case, the orange triangles & light purple diamonds — displays the mass-radius relation defined by the virial theorem. These "Virial" curves are all defined by the same virial theorem polynomial expression, as just presented, but the coefficient of the ${{𝒴}}^2$ term and the coefficient of the $({𝒳}{{𝒴}}^3{)}^{2/5}$ cross term — essentially, the value of ${{𝒜}}_{M_{\ell }}$ and the value of ${{ℬ}}_{M_{\ell }}$, respectively — have different values in the six separate figure panels. In each case, a value has been specified for the parameter, $\tilde{\xi }$ (as identified in the title of each figure panel), and this, in turn, has determined the values of the two (constant) free-energy coefficients. For example, in the top-right figure panel whose title indicates $\tilde{\xi }=3$, the "Virial" curve traces the mass-radius relation prescribed by the virial theorem after the values of the free-energy coefficients have been set to values that correspond to a detailed force-balanced model with this specified truncation radius, that is (see the above table), ${{𝒜}}_{M_{\ell }}=0.268711$ and ${{ℬ}}_{M_{\ell }}=0.830395$. Columns 2 and 3, respectively, of the table affixed to the bottom of the following comparison figure list the values of ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ that have been used to define the "Virial" curve in each of the six figure panels, in accordance with the value of $\tilde{\xi }$ listed in column 1 of the table.

Comparing Two Separate Mass-Radius Relations for Pressure-Truncated n = 5 Polytropes
$\tilde{\xi }$	Free-Energy Coefficients		Primary Overlap		Secondary Overlap (see further elaboration below)
	${{𝒜}}_{M_{\ell }}$	${{ℬ}}_{M_{\ell }}$	${𝒳}\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	${𝒴}\equiv \frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	${𝒳}$	${𝒴}$
1.26419	0.214429	0.758099	0.520269	0.904142	0.388938	0.289568
2	0.233842	0.779836	0.540671	1.544775	0.468918	0.570916
3	0.268711	0.830395	0.473087	1.774078	0.507387	0.798441
3.5	0.288708	0.861503	0.434310	1.744359	0.515168	0.859518
3.850652	0.303490	0.884746	0.408738	1.699778	0.518588	0.888969
9.8461	0.601012	1.313904	0.186424	0.904141	0.520269	0.904143

As is discussed more fully, below, in each of the six panels of the above comparison figure, the "Virial" curve intersects the "Stahler" curve at two locations. These points of intersection are identified by the black filled circles in each figure panel. In each case, the intersection point that is farthest along on the Stahler sequence — as determined by starting at the origin and moving counter-clockwise along the sequence — identifies the "Primary Overlap" between the two curves. That is to say, the $({𝒳},{𝒴})$ coordinates of this point (see columns 4 and 5 of the table affixed to the bottom of the figure) are the coordinate values that are obtained by plugging the specified value of $\tilde{\xi }$ (see the title of the figure panel or column 1 of the affixed table) into Stahler's pair of parametric relations. The second point of intersection in each panel — which we will refer to as the "Secondary Overlap" points and whose coordinates are provided in columns 6 and 7 of the affixed table — appears to be fortuitous and of no particularly significant astrophysical interest.

The mass-radius diagram displayed in the top-right panel of the above comparison figure has been reproduced in the upper-left panel of the following figure — in this case, with a coordinate aspect ratio that is closer to 1:1 — along with color images of the corresponding free-energy surface, viewed from two different perspectives, and a three-column table listing the 3D coordinates, $(X,Y,Z)=(R_{\mathrm{e}\mathrm{q}}/R_{\mathrm{S}\mathrm{W}\mathrm{S}},M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}/M_{\mathrm{S}\mathrm{W}\mathrm{S}},{\mathfrak{𝔊}}^{*})$, of the seventeen points that have been used to define the displayed "Virial" curve. To be more explicit, the rainbow-colored free-energy surface, ${\mathfrak{𝔊}}^{*}(R,M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}})$, has been defined by the free-energy function appropriate to pressure-truncated $n=5$ polytropic configurations as defined above, that is,

${\mathfrak{𝔊}}^{*}$

$=$

$-\frac{18}{5}\cdot {{𝒜}}_{M_{\ell }}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^2\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-1}+5{{ℬ}}_{M_{\ell }}\cdot \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{6/5}\right(\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^{-3/5}+\frac{4\pi }{3}\cdot (\frac{R}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{)}^3,$

with the values of the two free-energy coefficients set to the values that correspond to a $\tilde{\xi }=3$ virial curve as discussed above, namely, ${{𝒜}}_{M_{\ell }}=0.268711$ and ${{ℬ}}_{M_{\ell }}=0.830395$.

Free-Energy Surface for n = 5 Polytropic Configurations Truncated at $\tilde{\xi }=3$		Radius	Mass	Free Energy
		0.3152	0.1345	0.8231
		0.4159	0.3470	1.9947
		0.4731	0.5735	3.1095
		0.5035	0.7661	3.9591
		0.5114	0.8347	4.2409
		0.5297	1.0759	5.1453
		0.5310	1.4036	6.1560
		0.5250	1.5010	6.4046
		0.5185	1.5680	6.5606
		0.5114	1.6206	6.6741
		0.4731	1.7741	6.9515
		0.4343	1.8287	7.0247
		0.3984	1.8333	7.0301
		0.3379	1.7764	6.9923
		0.2911	1.6909	6.9541
		0.2260	1.5229	6.9091
		0.1545	1.2749	6.8786

In the bottom panel of this figure, the undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. Twelve small colored dots identify extrema — either the bottom of a valley or the top of a ridge — in the free-energy function and therefore trace the mass-radius relation defined by the scalar virial theorem. The 3D coordinates of these twelve points are provided in the three-column table that is affixed to the righthand edge of the figure: The coordinates of the (only) red dot are provided in row 13 of the table (red has also been assigned as the "bgcolor" of this table row); the equilibrium configuration having the greatest mass along the intersecting Stahler sequence is identified by the (only) black dot (coordinates are provided in row 11 of the table and, correspondingly, bgcolor="black" for that row); bgcolor="lightblue" has been assigned to the other rows of the affixed table that provide coordinates of the other 10, blue dots.

The upper-right panel of this figure presents the two-dimensional projection that results from viewing the identical free-energy surface "from above," along a line of sight that is parallel to the free-energy $(Z)$ axis and looking directly down onto the radius-mass $(X-Y)$ plane. From this vantage point, the twelve small colored dots cleanly trace out the $M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}(R_{\mathrm{e}\mathrm{q}})$ equilibrium sequence that is defined by the scalar virial theorem, exactly reproducing the "Virial" curve that is depicted in the mass-radius diagram shown in the upper-left panel of the figure.

See Also