Free-Energy Stability Analysis

Most General Case

Consider a free-energy function of the form,

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

<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>

where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,

<math>~\chi \equiv \frac{R}{R_0} \, ,</math>

is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.

Virial Equilibrium

The first variation (first derivative) of this function with respect to the configuration's radius is,

<math>~\frac{d\mathcal{G}}{d\chi}</math>

<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>

According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,

<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>

Stability

Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,

<math>~\mathcal{G}^'</math>

<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>

The first variation (first derivative) of this function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,

<math>~\frac{d\mathcal{G}^'}{d\chi}</math>

<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>

If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.

Pressure-Truncated Configurations

Expectations

For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,

Free-energy expression:	<math>~\mathcal{G}</math>	<math>~=</math>	<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
Virial equilibrium:	<math>~0</math>	<math>~=</math>	<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math>
Stability indicator:	<math>~\frac{d\mathcal{G}^'}{d\chi}</math>	<math>~=</math>	<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>

Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

<math>~6g \chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
	<math>~=</math>	<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math>
<math>~\Rightarrow ~~~ 6g \chi_\mathrm{eq}^4 +\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~ 2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr] </math>
<math>~\Rightarrow ~~~ \biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~ 2e - \biggl[\frac{af(n+3)}{nb} \biggr] </math>
<math>~\Rightarrow ~~~ \chi_\mathrm{eq}^4\biggr\|_\mathrm{crit} </math>	<math>~=</math>	<math>~ \biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, . </math>

Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

<math>~ \chi_\mathrm{eq}^4\biggr\|_\mathrm{crit} </math>	<math>~=</math>	<math>~ \biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr] </math>
	<math>~=</math>	<math>~ \frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, . </math>

Energies and Structural Form Factors

Old Approach

As has been developed in, for example, our accompanying review, we adopt the following normalizations:

<math>~R_\mathrm{norm}</math>	<math>~=</math>	<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
<math>~P_\mathrm{norm}</math>	<math>~=</math>	<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
<math>~\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R^3_\mathrm{norm}}</math>	<math>~=</math>	<math>~ \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3)} \, ,</math>
<math>~E_\mathrm{norm}</math>	<math>~=</math>	<math>~ \biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>

Then, from separate summaries — both here and here — we can write,

<math>~\frac{M_r(x)}{M_\mathrm{tot}} </math>	<math>~=</math>	<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,</math>
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>	<math>~=</math>	<math> - \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx </math>
	<math>~=</math>	<math> - \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \int_0^{1} 5x \biggl\{\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx\biggr\} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx </math>
	<math>~=</math>	<math> - \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, , </math>
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}} = \frac{U_\mathrm{int}}{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma} \biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma \int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\} </math>
	<math>~=</math>	<math>~\frac{4\pi n}{3} \cdot \chi^{-3/n} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n} \cdot \tilde\mathfrak{f}_A \, ,</math>

where the structural form factors are defined as follows:

<math>~\mathfrak{f}_M </math>	<math>~\equiv</math>	<math>~ \int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
<math>~\mathfrak{f}_W</math>	<math>~\equiv</math>	<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
<math>~\mathfrak{f}_A</math>	<math>~\equiv</math>	<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>

This gives, specifically for specifically for pressure-truncated polytropic configurations,

<math>~\tilde\mathfrak{f}_M</math>	<math>~=</math>	<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
<math>\tilde\mathfrak{f}_W</math>	<math>~=</math>	<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, , </math>
<math>~ \tilde\mathfrak{f}_A </math>	<math>~=</math>	<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1) \biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} \, . </math>

New Approach

In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some earlier derivations.

Mass

Defining <math>~M_\mathrm{tot}</math> as the total mass of the isolated configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily equilibrium) radius; and being careful to define the mean density of the truncated configuration such that,

<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>

we have,

<math>~M_r(r) </math>	<math>~=</math>	<math>~ \int_0^r 4\pi r^2 \rho dr </math>
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>	<math>~=</math>	<math>~ \frac{3}{4\pi} \int_0^r 4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr) \frac{dr}{R_\mathrm{norm}} </math>
	<math>~=</math>	<math>~ \biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3 \int_0^r 3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{dr}{R} </math>
	<math>~=</math>	<math>~ \biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3 \int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi} </math>
	<math>~=</math>	<math>~ \biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3 \int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi} </math>
	<math>~=</math>	<math>~ \biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3} \int_0^\xi 3\xi^2 \theta^n d\xi \, . </math>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

<math>~{\tilde\mathfrak{f}}_M</math>

<math>~\equiv </math>

<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi} 3\xi^2 \theta^n d\xi = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>

Now, from the,

Polytropic Lane-Emden Equation

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math>

we realize that,

<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>

<math>~- \xi^2 \theta^n \, .</math>

So these last two expressions may also be written as,

<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>

<math>~ \biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr] \, ; </math>

and,

<math>~{\tilde\mathfrak{f}}_M</math>

<math>~\equiv </math>

<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi \, .</math>

Modified Internal Energy

Now we want to develop the appropriately scaled integral definition of a "variational" internal energy having the form,

<math>~\frac{U_\Upsilon}{E_\mathrm{norm}}</math>	<math>~\equiv</math>	<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{P}{P_\mathrm{norm}}\biggr) \biggl( \frac{dr}{R_\mathrm{norm}}\biggr) </math>
	<math>~=</math>	<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho}\biggr]^{\gamma_\mathrm{g}} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)^{\gamma_\mathrm{g}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma_\mathrm{g}} \int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{P}{P_c}\biggr) \biggl( \frac{dr}{R}\biggr) </math>
	<math>~=</math>	<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}\chi^{-3/n} {\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \, . </math>

Hence, the coefficient, <math>~f</math>, in the free-energy expression is,

<math>~f = \chi^{3/n}\biggl[ \frac{U_\Upsilon}{E_\mathrm{norm}}\biggr]</math>

<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl\{ {\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \biggr\} \, ;</math>

or, if <math>~\Upsilon_U(\xi) = 1</math>, then,

<math>~f \rightarrow b = \chi^{3/n}\biggl[ \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr]</math>

<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} {\tilde\mathfrak{f}}_A \, ;</math>

where,

<math>~{\tilde\mathfrak{f}}_A</math>

<math>~ \biggl\{ {\tilde\xi}^{-3} \int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi \biggr\} \, . </math>

When <math>~\Upsilon_U(\xi) = 1</math>, then according to Viala & Horedt (1974), this integral over polytropic functions becomes,

<math>~ \int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi </math>	<math>~=</math>	<math>~ \frac{(n+1)}{(5-n)} \biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] </math>
<math>~\Rightarrow~~~{\tilde\mathfrak{f}}_A \equiv {\tilde\xi}^{-3}\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi </math>	<math>~=</math>	<math>~ \frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] \, , </math>

which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> derived earlier.

Modified Gravitational Potential Energy

Similarly, we have,

<math>~\frac{W_\Upsilon}{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~ - \frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\int_0^R \Upsilon_W(r) \biggl(\frac{GM_r}{r}\biggr) 4\pi r^2 \rho dr </math>
	<math>~=</math>	<math>~ - \frac{R_\mathrm{norm}\rho_c R^2}{M_\mathrm{tot}}\int_0^R 4\pi \Upsilon_W(r) \biggl(\frac{M_r}{M_\mathrm{tot}}\biggr) \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2} </math>
	<math>~=</math>	<math>~ - \frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\chi^{-1} \int_0^R 3\Upsilon_W(r) \biggl[\frac{M_r}{M_\mathrm{tot}}\biggr] \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2} </math>
	<math>~=</math>	<math>~ - \biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5} \int_0^\tilde\xi 3\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi </math>
	<math>~=</math>	<math>~ - \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5} \int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \, . </math>

Hence, the coefficient, <math>~e</math>, in the free-energy expression is,

<math>~e = -\chi \biggl[ \frac{W_\Upsilon}{E_\mathrm{norm}}\biggr]</math>

<math>~ \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \biggl\{{\tilde\xi}^{-5} \int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, ; </math>

or, if <math>~\Upsilon_W(\xi) = 1</math>, then,

<math>~e \rightarrow a = -\chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr]</math>

<math>~ \frac{3}{5}\biggl[\frac{1}{{\tilde\mathfrak{f}}_M} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 ~{\tilde\mathfrak{f}}_W \, ; </math>

where,

<math>~{\tilde\mathfrak{f}}_W</math>

<math>~\equiv</math>

<math>~ \biggl\{{\tilde\xi}^{-5} \int_0^\tilde\xi 5\biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, . </math>

Now, according to Viala & Horedt (1974), when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

<math>~W_\mathrm{grav}</math>	<math>~=</math>	<math>~ - \frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5 \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] </math>
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~ - \frac{1}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, . </math>

As we have detailed elsewhere, from this, we have deduced that, for polytropic configurations,

<math>~\tilde\mathfrak{f}_W </math>	<math>~=</math>	<math> {\tilde\xi}^{-5} \int_0^\tilde\xi 5 \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi </math>
	<math>~=</math>	<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, . </math>

Test Virial Equilibrium Condition

If the correct normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is specified, our expectation regarding virial equilibrium is that,

<math>~3nc\chi_\mathrm{eq}^{4 } - 3b\chi_\mathrm{eq}^{(n-3)/n} + an</math>

Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by Horedt (1970), namely,

<math> ~\chi_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} </math>	<math>~=~</math>	<math>~ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)} \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \, , </math>
<math> ~\frac{P_e}{P_\mathrm{norm}} = \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{e}}{P_\mathrm{Horedt}} </math>	<math>~=~</math>	<math>~ \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)} \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, , </math>

where we have taken into account the shift in normalization factors,

Switch from Hoerdt's (1970) Normalization
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>	<math>~=</math>	<math>~ \biggl[\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}\biggr]^{1/(4-3\gamma)} = \biggl[(n+1)^{-1} \biggl( 4\pi \biggr)^{1/n}\biggr]^{n/(n-3)} = \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \, ; </math>
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)} \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} </math>	<math>~=</math>	<math>~ \biggl\{ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3} \biggl( \frac{1}{4\pi} \biggr) \biggr\}^{\gamma/(4-3\gamma)} = \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \, . </math>

We therefore have:

First Term

<math>~3n\biggl[\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr]\chi_\mathrm{eq}^{4 }</math>	<math>~=</math>	<math>~ 4\pi n \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)} </math>
		<math>~ \times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggr\}^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{4(n-1)/(n-3)} </math>
<math>~</math>	<math>~=</math>	<math>~ 4\pi n \biggl[(n+1)^{[3(n+1)-4n]} ( 4\pi )^{[4-(n+1)]} \biggr]^{1/(n-3)} {\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{[2(n+1)+ 4(1-n)]/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{[4(n-1)-2(n+1)]/(n-3)} </math>
<math>~</math>	<math>~=</math>	<math>~ \biggl[ \frac{n}{(n+1) }\biggr] {\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^2 \, . </math>

Second Term

<math>~3b\chi_\mathrm{eq}^{(n-3)/n}</math>	<math>~=</math>	<math>~ 4\pi ~n \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] </math>
		<math>~ \times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggr\}^{(n-3)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n} </math>
	<math>~=</math>	<math>~ \frac{4\pi ~n}{(5-n)} \biggl[ \frac{1}{4\pi} \biggl( - \frac{\tilde\xi}{\tilde\theta^'}\biggr) \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] </math>
		<math>~ \times (n+1)^{-1} ( 4\pi )^{1/n} {\tilde\xi}^{(n-3)/n} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n} </math>
	<math>~=</math>	<math>~ \frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2} \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] </math>
		<math>~ \times {\tilde\xi}^{[(n-3)/n + 3(n+1)/n]} ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)/n - (n+1)/n]} </math>
	<math>~=</math>	<math>~ \frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2} \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} \, . </math>

Third Term

<math>~an</math>	<math>~=</math>	<math>~ \frac{3}{5}\biggl[\biggl( - \frac{\tilde\xi}{3\tilde\theta^'} \biggr) \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \frac{3\cdot 5~n}{(5-n)\tilde\xi^2} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] </math>
	<math>~=</math>	<math>~ \biggl[\frac{M}{M_\mathrm{tot}}\biggr]^2 \frac{n \tilde\xi^4}{(5-n)} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} \, . </math>

Combined

Combining the three terms, the virial expression becomes,

<math>~ (5-n)(n+1)\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^{-2} \biggl[ 3nc\chi_\mathrm{eq}^{4 } + an - 3b\chi_\mathrm{eq}^{(n-3)/n} \biggr]</math>	<math>~=</math>	<math>~ n(5-n) {\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} + n(n+1)\tilde\xi^4 \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} </math>
		<math>~ -n \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} </math>
	<math>~=</math>	<math>~n( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{ (5-n)\tilde\theta_n^{n+1} + (n+1) \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] </math>
		<math>~ - \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] \biggr\} </math>
	<math>~=</math>	<math>~n(n+1) ( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{ 0 \biggr\} \, . </math>

Q. E. D.

The Ledoux Variational Principle

Drawing from a separate presentation of Ledoux's variational principle, let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

<math>~ 2\pi e^{2i\omega t} \biggl\{ - \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0 - \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 + \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0 -\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R} \biggr\} \, . </math>

Our normalization produces,

<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>	<math>~=</math>	<math>~ - \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0 - \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 </math>
		<math>~ - (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0 -\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R} </math>
	<math>~=</math>	<math>~ - \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2\int_0^R x^2 \biggl[\frac{R_\mathrm{norm} R^5}{G\rho_c}\biggr] \biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3}\biggr]^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \omega^2 \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R} - \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 </math>
		<math>~ - (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R} - 3 \gamma_\mathrm{g} x_\mathrm{surface}^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \frac{P_e}{P_\mathrm{norm}}\biggr] </math>
	<math>~=</math>	<math>~ - \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} \int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R} ~- ~\gamma_\mathrm{g}\biggl[\frac{P_c }{P_\mathrm{norm} }\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R} </math>
		<math>~ - (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} \int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R} ~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, . </math>

Given that,

<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>