SSC/FreeEnergy/PolytropesEmbedded/Pt2

Free Energy of Embedded Polytropes

Part I:   Synopsis	Part II:  Truncated Polytropes	Part III:  Free-Energy of Bipolytropes
		IIIA:  Focus on (5, 1) Bipolytropes	IIIB:  Focus on (0, 0) Bipolytropes	IIIC:  Overview

Free-Energy of Truncated Polytropes

In this case, the Gibbs-like free energy is given by the sum of three separate energies,

$\mathfrak{𝔊}$	$=$	$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+{\mathfrak{𝔖}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}+P_{e}V$
	$=$	$-3{𝒜}\left[\frac{G{M}^2}{R}\right]+n{ℬ}\left[\frac{K{M}^{(n+1)/n}}{R^{3/n}}\right]+\frac{4\pi }{3}\cdot P_e{R}^3,$

where the constants,

${𝒜}\equiv \frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}$

and

${ℬ}\equiv (\frac{4\pi }{3}{)}^{-1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}},$

and, as derived elsewhere,

Structural Form Factors for Pressure-Truncated Polytropes $(n\ne 5)$
${\widetilde{\mathfrak{𝔣}}}_M$ $=$ $(-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }})$ ${\widetilde{\mathfrak{𝔣}}}_W$ $=$ $\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}[{\tilde{\theta }}^{n+1}+3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]$ ${\widetilde{\mathfrak{𝔣}}}_A$ $=$ $\frac{1}{(5-n)}\{6{\tilde{\theta }}^{n+1}+(n+1)[3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]\}$
As we have shown separately, for the singular case of $n=5$, ${\mathfrak{𝔣}}_M$ $=$ $(1+{\ell }^2{)}^{-3/2}$ ${\mathfrak{𝔣}}_W$ $=$ $\frac{5}{2^4}\cdot {\ell }^{-5}\left[\ell \right({\ell }^4-\frac{8}{3}{\ell }^2-1)(1+{\ell }^2{)}^{-3}+{\tan }^{-1}(\ell )]$ ${\mathfrak{𝔣}}_A$ $=$ $\frac{3}{2^3}{\ell }^{-3}[{\tan }^{-1}(\ell )+\ell ({\ell }^2-1)(1+{\ell }^2{)}^{-2}]$ where, $\ell \equiv \tilde{\xi }/\sqrt{3}$

In general, then, the warped free-energy surface drapes across a four-dimensional parameter "plane" such that,

$\mathfrak{𝔊}$

$=$

$\mathfrak{𝔊}(R,K,M,P_e).$

In order to effectively visualize the structure of this free-energy surface, we will reduce the parameter space from four to two, in two separate ways: First, we will hold constant the parameter pair, $(K,M)$; giving a nod to Kimura's (1981b) nomenclature, we will refer to the resulting function, ${\mathfrak{𝔊}}_{K,M}(R,P_e)$, as a "Case M" free-energy surface because the mass is being held constant. Second, we will hold constant the parameter pair, $(K,P_e)$, and examine the resulting "Case P" free-energy surface, ${\mathfrak{𝔊}}_{K,P_e}(R,M)$.

Virial Equilibrium and Dynamical Stability

The first (partial) derivative of $\mathfrak{𝔊}$ with respect to $R$ is,

$\frac{\partial \mathfrak{𝔊}}{\partial R}$

$=$

$\frac{1}{R}[3{𝒜}G{M}^2{R}^{-1}-3{ℬ}K{M}^{(n+1)/n}{R}^{-3/n}+4\pi P_e{R}^3];$

and the second (partial) derivative is,

$\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}$

$=$

$\frac{1}{R^2}[-6{𝒜}G{M}^2{R}^{-1}+(\frac{n+3}{n})3{ℬ}K{M}^{(n+1)/n}{R}^{-3/n}+8\pi P_e{R}^3].$

The virial equilibrium radius is identified by setting the first derivative to zero. This means that,

$3{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$

$=$

$3{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3.$

This expression can be usefully rewritten in the following forms:

Virial Equilibrium Condition
Case 1:	$3(n+3){ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$ $=$ $3(n+3){𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+4\pi (n+3)P_e{R}_{\mathrm{e}\mathrm{q}}^3$
Case 2:	$-6n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}$ $=$ $8\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3-6n{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$
Case 3:	$8\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3$ $=$ $6n{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}-6n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}$

Dynamical stability is determined by the sign of the second derivative expression evaluated at the equilibrium radius; setting the second derivative to zero identifies the transition from stable to unstable configurations. The criterion is,

$0$

$=$

$[-6n{𝒜}G{M}^2{R}^{-1}+3(n+3){ℬ}K{M}^{(n+1)/n}{R}^{-3/n}+8\pi n{P}_e{R}^3{]}_{R_{\mathrm{e}\mathrm{q}}}$

Case 1 Stability Criterion

Using the "Case 1" virial expression to define the equilibrium radius means that the stability criterion is,

$0$	$=$	$-6n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+3(n+3){𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+4\pi (n+3)P_e{R}_{\mathrm{e}\mathrm{q}}^3+8\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3$
	$=$	${𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}[3(n+3)-6n]+4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3[(n+3)+2n]$
$\Rightarrow 4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3[3(n+1)]$	$=$	${𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}[3(n-3)]$
$\Rightarrow 4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^4(n+1)$	$=$	${𝒜}G{M}^2(n-3)$

Case 2 Stability Criterion

Using the "Case 2" virial expression to define the equilibrium radius means that the stability criterion is,

$0$	$=$	$8\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3-6n{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}+3(n+3){ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}+8\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3$
	$=$	$16\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3-[3(n-3)]{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$
$\Rightarrow 16\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^3$	$=$	$[3(n-3)]{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$
$\Rightarrow 16\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^{3(n+1)/n}$	$=$	$[3(n-3)]{ℬ}K{M}^{(n+1)/n}$

Case 3 Stability Criterion

Using the "Case 3" virial expression to define the equilibrium radius means that the stability criterion is,

$0$	$=$	$-6n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+3(n+3){ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}+6n{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}-6n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}$
	$=$	$-12n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}+[6n+3(n+3)]{ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$
$\Rightarrow 9(n+1){ℬ}K{M}^{(n+1)/n}{R}_{\mathrm{e}\mathrm{q}}^{-3/n}$	$=$	$12n{𝒜}G{M}^2{R}_{\mathrm{e}\mathrm{q}}^{-1}$
$\Rightarrow R_{\mathrm{e}\mathrm{q}}^{n-3}$	$=$	$\left[\frac{4n{𝒜}}{3(n+1){ℬ}}{]}^n\right(\frac{G}{K}{)}^n{M}^{n-1}$

Case M

Now, in our discussion of "Case M" sequence analyses, the configuration's radius is normalized to,

$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

$\equiv$

$[G^n{K}^{-n}{M}^{n-1}{]}^{1/(n-3)}.$

Our "Case 3" stability criterion directly relates. We conclude that the transition from stability to dynamical instability occurs when,

$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$[\frac{4n{𝒜}}{3(n+1){ℬ}}{]}^n$
$\Rightarrow [\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{(n-3)/n}$	$=$	$\frac{4n}{15(n+1)}(\frac{4\pi }{3}{)}^{1/n}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_A{\widetilde{\mathfrak{𝔣}}}_M^{(n-1)/n}}$

Also in the "Case M" discussions, the external pressure is normalized to,

$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

$\equiv$

$[G^{-3(n+1)}{K}^{4n}{M}^{-2(n+1)}{]}^{1/(n-3)}.$

If we raise the "Case 1" stability criterion expression to the $(n-3)$ power, then divide it by the "Case 3" stability criterion expression raised to the fourth power, we find,

$\Rightarrow [P_e{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$\left[\frac{{𝒜}G{M}^2(n-3)}{4\pi (n+1)}{]}^{n-3}\right\{\left[\frac{4n{𝒜}}{3(n+1){ℬ}}{]}^n\right(\frac{G}{K}{)}^n{M}^{n-1}{\}}^{-4}$
	$=$	$\left[\frac{{𝒜}(n-3)}{4\pi (n+1)}{]}^{n-3}{G}^{n-3}{M}^{2(n-3)}\right[\frac{3(n+1){ℬ}}{4n{𝒜}}{]}^{4n}(\frac{K}{G}{)}^{4n}{M}^{4(1-n)}$
	$=$	$\left[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\right[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n}{K}^{4n}{M}^{-2(n+1)}{G}^{-3(n+1)}$
$\Rightarrow [\frac{P_e}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$\left[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\right[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n}$
	$=$	$\left[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\right[\frac{3(n+1)}{4n}{]}^{4n}\left[\frac{5{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W}{]}^{3(n+1)}\right(\frac{3}{4\pi }{)}^4[\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}}{]}^{4n}$
	$=$	$\left(\frac{3}{4\pi }{)}^4\right[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\left[\frac{3(n+1)}{4n}{]}^{4n}\right[\frac{5}{{\widetilde{\mathfrak{𝔣}}}_W}{]}^{3(n+1)}{\widetilde{\mathfrak{𝔣}}}_M^{2(n+1)}{\widetilde{\mathfrak{𝔣}}}_A^{4n}$

Case P

Flipping around this expression for $[P_e{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, we also can write,

$[M{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$

$=$

$\left[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\right[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n}{K}^{4n}{G}^{-3(n+1)}{P}_e^{3-n}.$

Now, in our "Case P" discussions we normalized the mass to

$M_{\mathrm{S}\mathrm{W}\mathrm{S}}$

$\equiv$

$(\frac{n+1}{n}{)}^{3/2}{G}^{-3/2}{K}^{2n/(n+1)}{P}_e^{(3-n)/[2(n+1)]}.$

Hence, we have,

$[\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}\right[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n}(\frac{n+1}{n}{)}^{-3(n+1)}$
	$=$	$\left[\frac{(n-3)}{4\pi n}{]}^{n-3}\right(\frac{3}{4}{)}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n},$

where the constants,

${𝒜}\equiv \frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}$

and

${ℬ}\equiv (\frac{4\pi }{3}{)}^{-1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}}.$

So we can furthermore conclude that,

$[\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{(n-3)}{4\pi n}{]}^{n-3}\right(\frac{3}{4}{)}^{4n}\{\frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}{\}}^{-3(n+1)}\{(\frac{4\pi }{3}{)}^{-1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}}{\}}^{4n}$
	$=$	$\left(\frac{3}{4\pi }{)}^4\right[\frac{(n-3)}{4\pi n}{]}^{n-3}\left(\frac{3{\widetilde{\mathfrak{𝔣}}}_A}{4}{)}^{4n}\right[\frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}{]}^{(n+1)}.$

Our expression for $[M{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$ can also be combined with the "Case 2 stability criterion" to eliminate the mass entirely, giving,

$\{16\pi n{P}_e{R}_{\mathrm{e}\mathrm{q}}^{3(n+1)/n}{\}}^{2n}$	$=$	$\{[3(n-3)]{ℬ}K{\}}^{2n}[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{4n}{K}^{4n}{G}^{-3(n+1)}{P}_e^{3-n}$
$\Rightarrow R_{\mathrm{e}\mathrm{q}}^{6(n+1)}$	$=$	$\left[\frac{3(n-3)}{16\pi n}{]}^{2n}\right[\frac{(n-3)}{4\pi (n+1)}{]}^{n-3}[\frac{3(n+1)}{4n}{]}^{4n}{{𝒜}}^{-3(n+1)}{{ℬ}}^{6n}{K}^{6n}{G}^{-3(n+1)}{P}_e^{3(1-n)}$
$\Rightarrow R_{\mathrm{e}\mathrm{q}}^{2(n+1)}$	$=$	$\left\{\right[\frac{(n-3)}{4\pi n}{]}^{2n}\left[\frac{(n-3)}{4\pi n}{]}^{n-3}\right[\frac{(n+1)}{n}{]}^{4n+(3-n)}(\frac{3}{4}{)}^{6n}{\}}^{1/3}{{𝒜}}^{-(n+1)}{{ℬ}}^{2n}{K}^{2n}{G}^{-(n+1)}{P}_e^{(1-n)}$
	$=$	$\left[\frac{(n-3)}{4\pi n}{]}^{(n-1)}\right[\frac{(n+1)}{n}{]}^{(n+1)}(\frac{3}{4}{)}^{2n}{{𝒜}}^{-(n+1)}{{ℬ}}^{2n}{K}^{2n}{G}^{-(n+1)}{P}_e^{(1-n)}.$

Finally, recognizing that in our "Case P" discussions we normalized the radius to

$R_{\mathrm{S}\mathrm{W}\mathrm{S}}$

$\equiv$

$(\frac{n+1}{n}{)}^{1/2}{G}^{-1/2}{K}^{n/(n+1)}{P}_e^{(1-n)/[2(n+1)]},$

we have,

$[R_{\mathrm{e}\mathrm{q}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{(n-3)}{4\pi n}{]}^{(n-1)}\right(\frac{n+1}{n}{)}^{(n+1)}\left(\frac{3}{4}{)}^{2n}{{𝒜}}^{-(n+1)}{{ℬ}}^{2n}\right\{R_{\mathrm{S}\mathrm{W}\mathrm{S}}(\frac{n+1}{n}{)}^{-1/2}{\}}^{2(n+1)}$
$\Rightarrow [\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{(n-3)}{4\pi n}{]}^{(n-1)}\right(\frac{3}{4}{)}^{2n}{{𝒜}}^{-(n+1)}{{ℬ}}^{2n}$
	$=$	$[\frac{n}{n-3}{]}^{(1-n)}(4\pi {)}^{1-n}\left(\frac{3}{4}{)}^{2n}\right[\frac{1}{5}\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2}{]}^{-(n+1)}\left[\right(\frac{4\pi }{3}{)}^{-1/n}\frac{{\widetilde{\mathfrak{𝔣}}}_A}{{\widetilde{\mathfrak{𝔣}}}_M^{(n+1)/n}}{]}^{2n}$
	$=$	$[\frac{n}{n-3}{]}^{(1-n)}(4\pi {)}^{1-n-2}{3}^{2n+2}{4}^{-2n}\left[\frac{5{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W}{]}^{(n+1)}\right[\frac{{\widetilde{\mathfrak{𝔣}}}_A^{2n}}{{\widetilde{\mathfrak{𝔣}}}_M^{2(n+1)}}]$
	$=$	$\left[\frac{n}{n-3}{]}^{(1-n)}\right[\frac{3^2\cdot 5}{4\pi {\widetilde{\mathfrak{𝔣}}}_W}{]}^{(n+1)}[\frac{{\widetilde{\mathfrak{𝔣}}}_A}{4}{]}^{2n}.$

Case M Free-Energy Surface

It is useful to rewrite the free-energy function in terms of dimensionless parameters. Here we need to pick normalizations for energy, radius, and pressure that are expressed in terms of the gravitational constant, $G$, and the two fixed parameters, $K$ and $M$. We have chosen to use,

$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$	$\equiv$	$\left[\right(\frac{G}{K}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}{]}^{1/(n-3)},$
$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$	$\equiv$	$[\frac{K^{4n}}{G^{3(n+1)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(n+1)}}{]}^{1/(n-3)},$

which, as is detailed in an accompanying discussion, are similar but not identical to the normalizations used by Horedt (1970) and by Whitworth (1981). The self-consistent energy normalization is,

$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

$\equiv$

$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}{R}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^3.$

As we have demonstrated elsewhere, after implementing these normalizations, the expression that describes the "Case M" free-energy surface is,

${\mathfrak{𝔊}}_{K,M}^{*}\equiv \frac{{\mathfrak{𝔊}}_{K,M}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}=-3{𝒜}(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{-1}+n{ℬ}(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{-3/n}+\left(\frac{4\pi }{3}\right)\frac{P_e}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^3,$

Given the polytropic index, $n$, we expect to obtain a different "Case M" free-energy surface for each choice of the dimensionless truncation radius, $\tilde{\xi }$; this choice will imply corresponding values for $\tilde{\theta }$ and ${\tilde{\theta }}^{\text{'}}$ and, hence also, corresponding (constant) values of the coefficients, ${𝒜}$ and ${ℬ}$.

Case P Free-Energy Surface

Again, it is useful to rewrite the free-energy function in terms of dimensionless parameters. But here we need to pick normalizations for energy, radius, and mass that are expressed in terms of the gravitational constant, $G$, and the two fixed parameters, $K$ and $P_e$. As is detailed in an accompanying discussion, we have chosen to use the normalizations defined by Stahler (1983), namely,

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

The self-consistent energy normalization is,

$E_{\mathrm{S}\mathrm{W}\mathrm{S}}\equiv \left(\frac{n}{n+1}\right)\frac{G{M}_{\mathrm{S}\mathrm{W}\mathrm{S}}^2}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$

$=$

$(\frac{n+1}{n}{)}^{3/2}{G}^{-3/2}{K}^{3n/(n+1)}{P}_{\mathrm{e}}^{(5-n)/[2(n+1)]}.$

After implementing these normalizations — see our accompanying analysis for details — the expression that describes the "Case P" free-energy surface is,

${\mathfrak{𝔊}}_{K,P_e}^{*}\equiv \frac{{\mathfrak{𝔊}}_{K,P_e}}{E_{\mathrm{S}\mathrm{W}\mathrm{S}}}$

$=$

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

Given the polytropic index, $n$, we expect to obtain a different "Case P" free-energy surface for each choice of the dimensionless truncation radius, $\tilde{\xi }$; this choice will imply corresponding values for $\tilde{\theta }$ and ${\tilde{\theta }}^{\text{'}}$ and, hence also, corresponding (constant) values of the coefficients, ${𝒜}$ and ${ℬ}$.

Summary

	DFB Equilibrium	Onset of Dynamical Instability
Case M:	$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{n-3}$ $=$ $\left[\frac{4\pi }{(n+1{)}^n}\right]{\tilde{\xi }}^{(n-3)}(-{\tilde{\xi }}^2\widetilde{{\theta }^{\text{'}}}{)}^{(1-n)}$	$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$ $=$ $\left[\frac{4n}{15(n+1)}{]}^n\right(\frac{4\pi }{3})\frac{{\widetilde{\mathfrak{𝔣}}}_W^n}{{\widetilde{\mathfrak{𝔣}}}_A^n{\widetilde{\mathfrak{𝔣}}}_M^{(n-1)}}$
	$[\frac{P_{\mathrm{e}}}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{n-3}$ $=$ $[\frac{(n+1{)}^3}{4\pi }{]}^{(n+1)}{\tilde{\theta }}^{(n+1)(n-3)}(-{\tilde{\xi }}^2\widetilde{{\theta }^{\text{'}}}{)}^{2(n+1)}$	$[\frac{P_{\mathrm{e}}}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$ $=$ $\left(\frac{3}{4\pi }{)}^4\right[\frac{(n-3)}{4\pi n}{]}^{n-3}\left(\frac{n+1}{n}{)}^{3(n+1)}\right[\frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}{]}^{n+1}(\frac{3{\widetilde{\mathfrak{𝔣}}}_A}{4}{)}^{4n}$
Case P:	$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}^2$ $=$ $\left(\frac{n}{4\pi }\right){\tilde{\xi }}^2{\tilde{\theta }}^{n-1}$	$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2$ $=$ $\left[\frac{3^2\cdot 5}{4\pi {\widetilde{\mathfrak{𝔣}}}_W}\right]\left[\frac{n-3}{n}{]}^{(n-1)/(n+1)}\right[\frac{{\widetilde{\mathfrak{𝔣}}}_A}{4}{]}^{2n/(n+1)}$
	$[\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}^2$ $=$ $\left(\frac{n^3}{4\pi }\right){\tilde{\theta }}^{n-3}(-{\tilde{\xi }}^2\widetilde{{\theta }^{\text{'}}}{)}^2$	$[\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2$ $=$ $\left[\frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}\right]\left(\frac{3}{4\pi }{)}^{4/(n+1)}\right[\frac{(n-3)}{4\pi n}{]}^{(n-3)/(n+1)}(\frac{3{\widetilde{\mathfrak{𝔣}}}_A}{4}{)}^{4n/(n+1)}$
In all four cases, the expression on right intersects (is equal to) the expression on the left when the following condition applies: For $n\ne 5$:       $2(9-2n){\tilde{\theta }}^{n+1}$ $=$ $3(n-3)[(-{\tilde{\theta }}^{\text{'}}{)}^2-(-\frac{\tilde{\theta }{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }})];$ For $n=5$:       $\left[\frac{2^4\cdot 5}{3}\right]{\ell }^3$ $=$ $(1+{\ell }^2{)}^3{\tan }^{-1}\ell +\ell ({\ell }^4-1).$

If (for $n\ne 5$) we adopt the shorthand notation,

$\mathrm{{\rm Y}}$	$\equiv$	$[3(-{\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]=3[(-{\tilde{\theta }}^{\text{'}}{)}^2-(-\frac{\tilde{\theta }{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }}\left)\right],$
and
$\tau$	$\equiv$	${\tilde{\theta }}^{n+1},$

then the critical condition becomes,

$(n-3)\mathrm{{\rm Y}}$

$=$

$2(9-2n)\tau ,$

and at the critical state, the expressions for the structural form-factors become,

${\widetilde{\mathfrak{𝔣}}}_A$	$=$	$\frac{1}{(5-n)}[6\tau +(n+1)\mathrm{{\rm Y}}]$
	$=$	$\frac{1}{(5-n)}\{6+(n+1)[\frac{2(9-2n)}{n-3}\left]\right\}\tau$
	$=$	$\frac{1}{(5-n)}\left[\frac{6(n-3)+2(9-2n)(n+1)}{n-3}\right]\tau$
	$=$	$\frac{1}{(5-n)}\left[\frac{4n(5-n)}{n-3}\right]\tau$
	$=$	$\frac{4n\tau }{(n-3)};$

${\widetilde{\mathfrak{𝔣}}}_W$	$=$	$\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}[\tau +\mathrm{{\rm Y}}]$
	$=$	$\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}\{1+[\frac{2(9-2n)}{n-3}\left]\right\}\tau$
	$=$	$\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}\left[\frac{3(5-n)}{n-3}\right]\tau$
	$=$	$\frac{3^2\cdot 5\tau }{(n-3){\tilde{\xi }}^2}$
$\Rightarrow \frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}$	$=$	$\left[\frac{(n-3){\tilde{\xi }}^2}{3^2\tau }{]}^3\right(-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }}{)}^2=3^2\left[\frac{(n-3){\tilde{\xi }}^2}{3^2\tau }{]}^3\right(-\frac{{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}}{{\tilde{\xi }}^3}{)}^2$
	$=$	$\left[\frac{(n-3{)}^3}{3^4{\tau }^3}\right](-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^2.$

Hence (1),

$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$\left[\frac{4\pi }{(n+1{)}^n}\right]\left[\frac{4n}{15}{]}^n\right(\frac{1}{3}\left)\right[\frac{3^2\cdot 5}{4n{\tilde{\xi }}^2}{]}^n{\widetilde{\mathfrak{𝔣}}}_M^{1-n}$
	$=$	$\left[\frac{4\pi }{(n+1{)}^n}\right]\left[\frac{1}{{\tilde{\xi }}^{2n}}\right](\frac{-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}}{{\tilde{\xi }}^3}{)}^{1-n}$
	$=$	$\left[\frac{4\pi }{(n+1{)}^n}\right]{\tilde{\xi }}^{n-3}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{1-n}$

Q.E.D.

And (2),

$[\frac{P_{\mathrm{e}}}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$\left(\frac{3}{4\pi }{)}^4\right[\frac{(n-3)}{4\pi n}{]}^{n-3}\left(\frac{n+1}{n}{)}^{3(n+1)}\right[\frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}{]}^{n+1}(\frac{3{\widetilde{\mathfrak{𝔣}}}_A}{4}{)}^{4n}$
	$=$	$3^4(4\pi {)}^{-(n+1)}(\frac{n-3}{n}{)}^{n-3}\left(\frac{n+1}{n}{)}^{3(n+1)}\right[\frac{(n-3{)}^3}{3^4{\tau }^3}{]}^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)}[\frac{3n\tau }{n-3}{]}^{4n}$
	$=$	$3^4(4\pi {)}^{-(n+1)}(\frac{n-3}{n}{)}^{n-3}\left(\frac{n+1}{n}{)}^{3(n+1)}{n}^{3(n+1)}\right[\frac{n-3}{n}{]}^{3(n+1)}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)}[\frac{n}{n-3}{]}^{4n}{\tau }^{4n-3(n+1)}{3}^{4n-4(n+1)}$
	$=$	$[\frac{(n+1{)}^3}{4\pi }{]}^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)}{\tau }^{n-3}$
$\Rightarrow [\frac{P_{\mathrm{e}}}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{n-3}$	$=$	$[\frac{(n+1{)}^3}{4\pi }{]}^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)}{\tilde{\theta }}^{(n+1)(n-3)}.$

Q.E.D.

And (3),

$[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{3^2\cdot 5}{4\pi {\widetilde{\mathfrak{𝔣}}}_W}{]}^{n+1}\right[\frac{n-3}{n}{]}^{(n-1)}[\frac{{\widetilde{\mathfrak{𝔣}}}_A}{4}{]}^{2n}$
	$=$	$\left[\frac{3^2\cdot 5}{4\pi }{]}^{n+1}\right[\frac{n-3}{n}{]}^{(n-1)}\left[\frac{n\tau }{n-3}{]}^{2n}\right[\frac{(n-3){\tilde{\xi }}^2}{3^2\cdot 5\tau }{]}^{n+1}$
	$=$	$\left[\frac{1}{4\pi }{]}^{n+1}\right[\frac{n}{n-3}{]}^{n+1}[(n-3){\tilde{\xi }}^2{]}^{n+1}{\tau }^{n-1}$
$\Rightarrow [\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2$	$=$	$\left(\frac{n}{4\pi }\right){\tilde{\xi }}^2{\tilde{\theta }}^{n-1}$

Q.E.D.

And (4),

$[\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2(n+1)}$	$=$	$\left[\frac{5^3{\widetilde{\mathfrak{𝔣}}}_M^2}{{\widetilde{\mathfrak{𝔣}}}_W^3}{]}^{n+1}\right(\frac{3}{4\pi }{)}^4\left[\frac{(n-3)}{4\pi n}{]}^{(n-3)}\right(\frac{3{\widetilde{\mathfrak{𝔣}}}_A}{4}{)}^{4n}$
	$=$	$\left\{\right[\frac{(n-3{)}^3}{3^4{\tau }^3}](-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^2{\}}^{n+1}{3}^4(4\pi {)}^{-(n+1)}(\frac{n-3}{n}{)}^{(n-3)}[\frac{3n\tau }{n-3}{]}^{4n}$
	$=$	$[\frac{n^3}{4\pi }{]}^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)}{\tau }^{n-3}$
$\Rightarrow [\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}{]}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2$	$=$	$\left[\frac{n^3}{4\pi }\right](-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^2{\tilde{\theta }}^{n-3}$

Q.E.D.

See Also