Revision as of 18:21, 26 May 2014

Bipolytrope Generalization

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Setup

In a more general context, we have discussed a Gibbs-like free-energy function of the generic form,

<math> \mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots </math>

Here we are interested in examining the free energy of isolated, nonrotating, spherically symmetric bipolytropes, so we can drop the term that accounts for the influence of an external pressure and we can drop the kinetic energy term. But we need to consider separately the contributions to the reservoir of thermodynamic energy by the core and envelope. In particular, we will assume that compressions/expansions occur adiabatically, but that the core and the envelope evolve along separate adiabats — <math>~\gamma_c</math> and <math>~\gamma_e</math>, respectively. Guided by our associated discussion of spherically symmetric, polytropic configurations, we have,

<math>~\mathfrak{G}</math>	<math>~=</math>	<math>~W_\mathrm{grav} + \mathfrak{S}_A\biggr\|_\mathrm{core} + \mathfrak{S}_A\biggr\|_\mathrm{env} </math>
	<math>~=</math>	<math> ~W_\mathrm{grav} + \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env} \, . </math>

In addition to the gravitational potential energy, which is naturally written as,

<math>~W_\mathrm{grav}</math>

<math>~- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \cdot \mathfrak{f}_{WM} \, ,</math>

it seems reasonable to write the separate thermal energy contributions as,

<math>~S_\mathrm{core}</math>	<math>~=</math>	<math> ~\frac{3}{2}\biggl[ M_\mathrm{core} \biggl( \frac{P_{i}}{\rho_{ic}} \biggr) \biggr] s_\mathrm{core} = \frac{3}{2}\biggl[ M_\mathrm{core} K_c \rho_{ic}^{\gamma_c-1} \biggr] s_\mathrm{core} \, ,</math>
<math>~S_\mathrm{env}</math>	<math>~=</math>	<math> ~\frac{3}{2}\biggl[ M_\mathrm{env} \biggl( \frac{P_{i}}{\rho_{ie}} \biggr) \biggr] s_\mathrm{env} = \frac{3}{2}\biggl[ M_\mathrm{env} K_e \rho_{ie}^{\gamma_e-1} \biggr] s_\mathrm{env} \, ,</math>

where the subscript "<math>i</math>" means "at the interface," and <math>~\mathfrak{f}_{WM},</math> <math>~s_\mathrm{core},</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (all three functions to be determined) akin to the structural form factors used in our examination of isolated polytropes.

While exploring how the free-energy function varies across parameter space, we choose to hold <math>~M_\mathrm{tot}</math> and <math>~K_c</math> fixed. By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, and densities by, respectively,

<math>~E_\mathrm{norm}</math>	<math>~\equiv</math>	<math>~\biggl[ \frac{G^{3(\gamma_c-1)} M_\mathrm{tot}^{5\gamma_c-6}}{K_c} \biggr]^{1/(3\gamma_c -4)} \, ,</math>
<math>~R_\mathrm{norm}</math>	<math>~\equiv</math>	<math>~\biggl[ \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \biggr]^{1/(3\gamma_c -4)} \, ,</math>
<math>~\rho_\mathrm{norm}</math>	<math>~\equiv</math>	<math>~\biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{1/(3\gamma_c -4)} \, .</math>

Dividing the free-energy expression through by <math>~E_\mathrm{norm}</math> generates,

<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>	<math>~=</math>	<math> - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{E_\mathrm{norm}} \biggr) \biggl( \frac{1}{R} \biggr) \cdot \mathfrak{f}_{WM} + \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_c \rho_{ic}^{\gamma_c-1} }{E_\mathrm{norm}} \biggr] </math>
		<math> ~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_e \rho_{ie}^{\gamma_e-1} }{E_\mathrm{norm}} \biggr] </math>
	<math>~=</math>	<math> - \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl[ \frac{K_c G^{(3\gamma_c -4)}M_\mathrm{tot}^{2(3\gamma_c -4)}}{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} \biggr]^{1/(3\gamma_c -4)} \biggl( \frac{1}{R} \biggr) </math>
		<math> + \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_c^{3\gamma_c -4} }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} \biggr]^{1/(3\gamma_c -4)} \rho_{ic}^{\gamma_c-1} </math>
		<math> ~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_e^{3\gamma_c -4} }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} \biggr]^{1/(3\gamma_c -4)} \rho_{ie}^{\gamma_e-1} </math>

Later it will also be useful to recognize that, in equilibrium <math>~(R = R_\mathrm{eq})</math>, we will demand that <math>~P_{ie} = P_{ic}</math>. As a result, we can choose to write the total thermal energy either entirely in terms of the exponent, <math>~\gamma_c</math> or the exponent, <math>~\gamma_e</math>. Letting the properties of the core take the lead, we can write,

<math>~S_\mathrm{tot}</math>

<math>~R^3 P_{ic} (B_\mathrm{core} + B_\mathrm{env}) = C_\mathrm{core} \biggl( 1 + \frac{B_\mathrm{env}}{B_\mathrm{core}} \biggr) R_\mathrm{eq}^{3-3\gamma_c} \, .</math>

Letting the properties of the envelope take the lead, we obtain,

<math>~S_\mathrm{tot}</math>

<math>~R^3 P_{ie} (B_\mathrm{core} + B_\mathrm{env}) = C_\mathrm{env} \biggl( 1 + \frac{B_\mathrm{core}}{B_\mathrm{env}} \biggr) R_\mathrm{eq}^{3-3\gamma_e} \, .</math>

Free Energy and Its Derivatives

Now, the free energy can be written as,

<math>~\mathfrak{G}</math>	<math>~=~</math>	<math>~U_\mathrm{tot} + W</math>
	<math>~=~</math>	<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env} + W</math>
	<math>~=~</math>	<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] C_\mathrm{core} R^{3-3\gamma_c} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \, .</math>

The first derivative of the free energy with respect to radius is, then,

<math>~\frac{d\mathfrak{G}}{dR}</math>

<math>~ -2 C_\mathrm{core} R^{2-3\gamma_c} -2 C_\mathrm{env} R^{2-3\gamma_e} + A R^{-2} \, .</math>

And the second derivative is,

<math>~\frac{d^2\mathfrak{G}}{dR^2}</math>	<math>~=~</math>	<math>~ -2 (2-3\gamma_c) C_\mathrm{core} R^{1-3\gamma_c} -2 (2-3\gamma_e) C_\mathrm{env} R^{1-3\gamma_e} - 2A R^{-3} \, .</math>
	<math>~=~</math>	<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) C_\mathrm{core} R^{3-3\gamma_c} + (3\gamma_e-2) C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \biggr]</math>
	<math>~=~</math>	<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) S_\mathrm{core} + (3\gamma_e-2) S_\mathrm{env} +W \biggr] \, .</math>

Equilibrium

The radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,

<math>~0 </math>	<math>~=~</math>	<math>~ 2 C_\mathrm{core} R_\mathrm{eq}^{2-3\gamma_c} + 2 C_\mathrm{env} R_\mathrm{eq}^{2-3\gamma_e} - A R_\mathrm{eq}^{-2} </math>
	<math>~=~</math>	<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 C_\mathrm{core} R_\mathrm{eq}^{3-3\gamma_c} + 2 C_\mathrm{env} R_\mathrm{eq}^{3-3\gamma_e} - A R_\mathrm{eq}^{-1} \biggr]</math>
	<math>~=~</math>	<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 S_\mathrm{core} + 2 S_\mathrm{env} +W \biggr]</math>
<math>\Rightarrow ~~~~ 2 S_\mathrm{tot} + W </math>	<math>~=~</math>	<math>~0 \, .</math>

This is the familiar statement of virial equilibrium. From it we should always be able to derive the radius of equilibrium configurations.

Stability

To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius. If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable. Using the above statement of virial equilibrium, that is, setting,

<math>~2 S_\mathrm{tot} + W</math>	<math>~=~</math>	<math>~0 \, ,</math>
<math>\Rightarrow ~~~~ S_\mathrm{env} </math>	<math>~=~</math>	<math>~- S_\mathrm{core} - \frac{W}{2} \, ,</math>

we obtain,

<math>~\frac{d^2\mathfrak{G}}{dR^2}\biggr\|_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ (3\gamma_c-2) S_\mathrm{core} +W - (3\gamma_e-2)\biggl( S_\mathrm{core} + \frac{W}{2}\biggr) \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ 3(\gamma_c-\gamma_e) S_\mathrm{core} + \biggl(2 - \frac{3}{2}\gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ (\gamma_c-\gamma_e) S_\mathrm{core} + \frac{1}{2}\biggl(\frac{4}{3} - \gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ -\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} \biggr]_\mathrm{eq} \, .</math>

So, if when evaluated at the equilibrium state, the expression inside of the square brackets of this last expression is negative, the equilibrium configuration will be dynamically unstable. We have chosen to write the expression in this particular final form because we generally will be interested in bipolytropes for which the adiabatic exponent of the envelope is greater than <math>~4/3</math> and the adiabatic exponent of the core is less than or equal to <math>~4/3</math> — that is, <math>~\gamma_e > 4/3 \ge \gamma_c</math>. Hence, because the gravitational potential energy, <math>~W</math>, is intrinsically negative, the system will be dynamically unstable only if the second term (involving <math>~S_\mathrm{core}</math>) is greater in magnitude than the first term (involving <math>~W</math>).

It is worth noting that, instead of drawing upon <math>~S_\mathrm{core}</math> and <math>~W</math> to define the stability condition, we could have used an appropriate combination of <math>~S_\mathrm{env}</math> and <math>~W</math>, or the <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math> pair. Also, for example, because the virial equilibrium condition is <math>~S_\mathrm{tot} = -W/2</math>, it is easy to see that the following inequality also equivalently defines stability:

<math>~ S_\mathrm{tot}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} </math>

Examples

(0, 0) Bipolytropes

In an accompanying discussion we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:

<math>~ \frac{W}{R_\mathrm{eq}^3 P_i} = - \frac{A}{R_\mathrm{eq}^4 P_i} </math>	<math>~=~</math>	<math>- ~ \frac{3}{5} \biggl[ \frac{GM_\mathrm{tot}^2}{R^4P_i} \biggr] \biggl( \frac{\nu^2}{q} \biggr) f </math>
	<math>~=~</math>	<math>- ~4\pi q^3 \Lambda f \, ,</math>
<math>~\frac{S_\mathrm{core}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{core}</math>	<math>~=~</math>	<math> ~2\pi q^3 (1 + \Lambda) \, ,</math>
<math>~\frac{S_\mathrm{env}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{env}</math>	<math>~=~</math>	<math> 2\pi \biggl[ (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \, ,</math>

where,

<math>~\Lambda</math>	<math>~\equiv~</math>	<math> \frac{3}{2^2 \cdot 5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4 P_i} \biggr) \frac{\nu^2}{q^4} \, ,</math>
<math>~f(q,\rho_e/\rho_c)</math>	<math>~\equiv~</math>	<math>1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl(\frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] </math>
	<math>~=~</math>	<math>1 + \frac{5}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^2) + \frac{1}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) \, ,</math>
<math>~g^2(q,\rho_e/\rho_c)</math>	<math>~\equiv~</math>	<math>1 + \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math>
	<math>~\equiv~</math>	<math>1 + \biggl[ 2\biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q) + \frac{1}{q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (1 - 3q^2 + 2q^3 ) \biggr] \, , </math>

With these expressions in hand, we can deduce the equilibrium radius and relativity stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes using the generalized expressions provided above. For example, from the statement of virial equilibrium <math>~(2S_\mathrm{tot} = - W )</math> we obtain,

<math>~q^3 (1 + \Lambda) + (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) </math>

<math>~q^3 \Lambda \biggl[ 1 + \frac{5}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^2) + \frac{1}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) \biggr] </math>

<math>\Rightarrow ~~~~ \frac{1}{\Lambda}</math>	<math>~=~</math>	<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3) + \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) - \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] </math>
	<math>~=~</math>	<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3 + 2 -3q +q^3) + \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5 +3 - 15q^2+15q^3 -3q^5) </math>
	<math>~=~</math>	<math>\frac{5}{2}\biggl[ 2\biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q) + \frac{1}{q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (1 - 3q^2 + 2q^3 ) \biggr] </math>
	<math>~=~</math>	<math>\frac{5}{2}(g^2-1) </math>
<math>\Rightarrow ~~~~ \biggl[ \frac{P_i}{GM_\mathrm{tot}^2} \biggr] R_\mathrm{eq}^4</math>	<math>~=~</math>	<math>\biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) \, . </math>

And the condition for dynamical stability is,

<math>-\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} </math>	<math>~>~</math>	<math>~0 \, .</math>
<math>\Rightarrow ~~~~ 2\pi q^3 \Lambda \biggl[ \biggl( \gamma_e - \frac{4}{3}\biggr) f - (\gamma_e-\gamma_c) \biggl( 1 + \frac{1}{\Lambda}\biggr) \biggr] </math>	<math>~>~</math>	<math>~0 \, .</math>
<math>~\biggl( \gamma_e - \frac{4}{3} \biggr)f - (\gamma_e - \gamma_c) \biggl[1 + \frac{5}{2}(g^2-1) \biggr]</math>	<math>~>~</math>	<math>~0 \, .</math>

(5, 1) Bipolytropes

In another accompanying discussion we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations? Work in progress!

Related Discussions

Analytic solution with <math>~n_c = 0</math> and <math>~n_e=0</math>.
Analytic solution with <math>~n_c = 5</math> and <math>~n_e=1</math>.

@@ Line 12: / Line 12: @@
 </math>
 </div>
-Here we are interested in examining the free energy of isolated, nonrotating, spherically symmetric bipolytropes, so we can drop the term that accounts for the influence of an external pressure and we can drop the kinetic energy term.  But we need to consider separately the contributions to the reservoir of thermodynamic energy by the core and envelope.  In particularly, we will assume that compressions/expansions occur adiabatically, but that the core and the envelope evolve along separate adiabats &#8212; <math>~\gamma_c</math> and <math>~\gamma_e</math>, respectively.  Guided by our associated discussion of [[User:Tohline/SphericallySymmetricConfigurations/Virial#Dependence_on_Size|spherically symmetric, polytropic configurations]], we have,
+Here we are interested in examining the free energy of isolated, nonrotating, spherically symmetric bipolytropes, so we can drop the term that accounts for the influence of an external pressure and we can drop the kinetic energy term.  But we need to consider separately the contributions to the reservoir of thermodynamic energy by the core and envelope.  In particular, we will assume that compressions/expansions occur adiabatically, but that the core and the envelope evolve along separate adiabats &#8212; <math>~\gamma_c</math> and <math>~\gamma_e</math>, respectively.  Guided by our associated discussion of [[User:Tohline/SphericallySymmetricConfigurations/Virial#Dependence_on_Size|spherically symmetric, polytropic configurations]], we have,
 <div align="center">
 <table border="0" cellpadding="5" align="center">
@@ Line 37: / Line 37: @@
    <td align="left">
 <math>
-~W_\mathrm{grav} + \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env}  \, ,
+~W_\mathrm{grav} + \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env}  \, .
 </math>
    </td>
@@ Line 43: / Line 43: @@
 </table>
 </div>
-where it seems reasonable to write the separate thermal energy contributions as,
+In addition to the gravitational potential energy, which is naturally written as,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~W_\mathrm{grav}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \cdot \mathfrak{f}_{WM} \, ,</math>
+  </td>
+</tr>
+</table>
+</div>
+it seems reasonable to write the separate thermal energy contributions as,
 <div align="center">
 <table border="0" cellpadding="5" align="center">
@@ Line 76: / Line 92: @@
 </table>
 </div>
-where, <math>~s_\mathrm{core}</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (both functions to be determined), and the subscript "<math>i</math>" means "at the interface."
+where the subscript "<math>i</math>" means "at the interface," and <math>~\mathfrak{f}_{WM},</math> <math>~s_\mathrm{core},</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (all three functions to be determined) akin to the [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors|structural form factors]] used in our examination of isolated polytropes.
 While exploring how the free-energy function varies across parameter space, we choose to hold <math>~M_\mathrm{tot}</math> and <math>~K_c</math> fixed.  By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, and densities by, respectively,
@@ Line 119: / Line 135: @@
 </table>
 </div>
+Dividing the free-energy expression through by <math>~E_\mathrm{norm}</math> generates,
 <div align="center">
 <table border="0" cellpadding="5" align="center">
@@ Line 128: / Line 141: @@
 <tr>
    <td align="right">
-<math>~\mathfrak{G}</math>
+<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
    </td>
    <td align="center">
@@ Line 134: / Line 147: @@
    </td>
    <td align="left">
-<math>~W_\mathrm{grav} + \mathfrak{S}_A\biggr|_\mathrm{core} + \mathfrak{S}_A\biggr|_\mathrm{env} </math>
+<math>
+- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{E_\mathrm{norm}} \biggr) \biggl( \frac{1}{R} \biggr) \cdot \mathfrak{f}_{WM}
++ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_c \rho_{ic}^{\gamma_c-1} }{E_\mathrm{norm}} \biggr]
+</math>
    </td>
 </tr>
@@ Line 143: / Line 159: @@
    </td>
    <td align="center">
-<math>~=</math>
+&nbsp;
    </td>
    <td align="left">
-<math>~- A \biggl( \frac{R}{R_0} \biggr)^{-1}
+<math>
-+ C_\mathrm{core} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_c-1)}
+~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_e \rho_{ie}^{\gamma_e-1} }{E_\mathrm{norm}}  \biggr]
-+ C_\mathrm{env} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_e-1)}\, ,
 </math>
    </td>
 </tr>
-</table>
-</div>
-where,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
 <tr>
    <td align="right">
-<math>~A</math>
+&nbsp;
    </td>
    <td align="center">
-<math>~\equiv</math>
+<math>~=</math>
    </td>
    <td align="left">
-<math>\frac{3}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
+<math>
+- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl[ \frac{K_c G^{(3\gamma_c -4)}M_\mathrm{tot}^{2(3\gamma_c -4)}}{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} \biggr]^{1/(3\gamma_c -4)} \biggl( \frac{1}{R} \biggr)
+</math>
    </td>
 </tr>
@@ Line 172: / Line 184: @@
 <tr>
    <td align="right">
-<math>~C_\mathrm{core}</math>
+&nbsp;
    </td>
    <td align="center">
-<math>~=</math>
+&nbsp;
    </td>
    <td align="left">
-<math>\frac{M_\mathrm{core} K}{({\gamma_c}-1)} \biggl[ \frac{3M_\mathrm{core}}{4\pi R_0^3} \biggr]^{\gamma_c-1}
+<math>
-\cdot \biggl[ \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_c}} \biggr]_\mathrm{core}
++ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_c^{3\gamma_c -4}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}}
-\, ,</math>
+\biggr]^{1/(3\gamma_c -4)}  \rho_{ic}^{\gamma_c-1}
+</math>
    </td>
 </tr>
@@ Line 186: / Line 199: @@
 <tr>
    <td align="right">
-<math>~C_\mathrm{env}</math>
+&nbsp;
    </td>
    <td align="center">
-<math>~=</math>
+&nbsp;
    </td>
    <td align="left">
-<math>\frac{M_\mathrm{env} K}{({\gamma_e}-1)} \biggl[ \frac{3M_\mathrm{env}}{4\pi R_0^3} \biggr]^{\gamma_e-1}
+<math>
-\cdot \biggl[ \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_c}} \biggr]_\mathrm{env}
+~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_e^{3\gamma_c -4}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}}
-\, .</math>
+\biggr]^{1/(3\gamma_c -4)}  \rho_{ie}^{\gamma_e-1}
+</math>
    </td>
 </tr>
 </table>
 </div>
-For a given structure, the three coefficients, <math>~A, C_\mathrm{core}</math>, and <math>~C_\mathrm{env}</math>, should remain constant during a radial perturbation of the configuration.
 Later it will also be useful to recognize that, in equilibrium <math>~(R = R_\mathrm{eq})</math>, we will demand that <math>~P_{ie} = P_{ic}</math>.  As a result, we can choose to write the total thermal energy either entirely in terms of the exponent, <math>~\gamma_c</math> or the exponent, <math>~\gamma_e</math>.  Letting the properties of the core take the lead, we can write,

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