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 equlibrium:	<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

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>~\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>

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