User:Tohline/SSC/BipolytropeGeneralization

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

We also want to ensure that envelope pressure matches the core pressure at the interface. This means,

<math>~K_e \rho_{ie}^{\gamma_e}</math>	<math>~=</math>	<math>~K_c \rho_{ic}^{\gamma_c}</math>
<math>\Rightarrow ~~~~\frac{K_e}{K_c} </math>	<math>~=</math>	<math>~\rho_{ic}^{\gamma_c} \rho_{ie}^{-\gamma_e} </math>
	<math>~=</math>	<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \rho_\mathrm{norm}^{\gamma_c - \gamma_e}</math>
	<math>~=</math>	<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \biggl\{ \frac{3}{4\pi} \biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{1/(3\gamma_c -4)} \biggr\}^{\gamma_c - \gamma_e}</math>
<math>\Rightarrow ~~~~\frac{K_e}{K_c} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(\gamma_c - \gamma_e)/(3\gamma_c -4)} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1}</math>	<math>~=</math>	<math>~\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1} </math>
	<math>~=</math>	<math> \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} \biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{\rho_\mathrm{norm}}{ \bar\rho } \biggr)^{\gamma_e - \gamma_c} </math>
	<math>~=</math>	<math> \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} \biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{ R}{R_\mathrm{norm}} \biggr)^{3(\gamma_e - \gamma_c)} </math>

Hence, we can write the normalized (and dimensionless) free energy as,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<math> - \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} + \biggl\{ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} + \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggl( \frac{ \rho_{ic} }{ \rho_{ie} } \biggr)\biggr\} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} \biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_c-1)} \, . </math>

Keep in mind that, if the envelope and core both have uniform (but different) densities, then <math>~\rho_{ic} = \rho_c</math>, <math>~\rho_{ie} = \rho_e</math>, and

<math> \frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ; ~~~~~ \frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ; ~~~~~ \frac{\rho_e}{\rho_c} = \frac{q^3(1-\nu)}{\nu (1-q^3)} \, . </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>~=~</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>

<math>~>~</math>

<math>~ 0 \, .</math>

Examples

(0, 0) Bipolytropes

Review

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>

Renormalize

Let's renormalize these energy terms in order to more readily relate them to the generalized expressions derived above.

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

Also,

<math>~\biggl[ \frac{GM_\mathrm{tot}^2}{R} \biggr]^{3\gamma_c -4}</math>	<math>~=</math>	<math> ~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} G^{3\gamma_c -4} M_\mathrm{tot}^{6\gamma_c -8} \biggl( \frac{G}{K_c} \biggr) M_\mathrm{tot}^{2-\gamma_c} </math>
	<math>~=</math>	<math> ~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} \biggl[ \frac{ G^{3\gamma_c -3} M_\mathrm{tot}^{5\gamma_c -6} }{K_c} \biggr] </math>
	<math>~=</math>	<math> ~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} E_\mathrm{norm}^{3\gamma_c-4} \, . </math>
<math> \Rightarrow ~~~~\frac{GM_\mathrm{tot}^2}{R^4 P_i} </math>	<math>~=</math>	<math> ~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} \biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3\gamma_c - 3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} </math>
	<math>~=</math>	<math> \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} \, . </math>

Hence,

<math>~\Lambda</math>

<math>~\equiv~</math>

<math> \frac{3}{2^2 \cdot 5} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} \, .</math>

Given that <math>~\rho_{ic}/\bar\rho = \nu/q^3</math> for the <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, we can finally write,

<math>~\frac{R^3 P_i}{E_\mathrm{norm}}</math>

<math>~=</math>

<math>~\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} \, ,</math>

and,

<math>~\Lambda</math>

<math>~\equiv~</math>

<math> \frac{3}{2^2 \cdot 5} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{-\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} \, .</math>

Also, note that,

<math>~\Lambda \biggl[ \frac{R^3 P_i}{E_\mathrm{norm}} \biggr]</math>	<math>~=</math>	<math> \frac{3}{2^2 \cdot 5} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{-\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} </math>
	<math>~=</math>	<math> \frac{3}{2^2 \cdot 5} \frac{\nu^2}{q^4} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} \, .</math>

Hence the renormalized gravitational potential energy becomes,

<math> \frac{W_\mathrm{grav}}{E_\mathrm{norm}} </math>

<math>~=</math>

<math> - \biggl( \frac{3\pi}{5} \biggr) \frac{\nu^2}{q} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} \cdot f \, ;</math>

and the two, renormalized contributions to the thermal energy become,

<math>~\frac{U_\mathrm{core}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_c-1)} \biggl[ \frac{S_\mathrm{core}}{ E_\mathrm{norm} } \biggr]</math>	<math>~=~</math>	<math> \frac{4\pi q^3 (1 + \Lambda) }{3(\gamma_c-1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} \, ,</math>
<math>~\frac{U_\mathrm{env}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_e-1)} \biggl[ \frac{S_\mathrm{env}}{ E_\mathrm{norm} } \biggr]</math>	<math>~=~</math>	<math> \frac{4\pi}{3(\gamma_e-1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} \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>

Finally, then, we can state that,

<math>~\mathfrak{f}_{WM}</math>	<math>~\equiv</math>	<math>~\frac{\pi \nu^2}{q} \cdot f \, ,</math>
<math>~s_\mathrm{core}</math>	<math>~\equiv</math>	<math> 1 - \Lambda \, , </math>
<math>~(1-q^3) s_\mathrm{env}</math>	<math>~\equiv</math>	<math> (1-q^3) + \Lambda\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>

Going back to the renormalized expression for <math>~\mathfrak{G}^*</math>, we can write,

<math>~\mathfrak{G}^*</math>	<math>~=</math>	<math> - \frac{3}{5} \biggl( \frac{\pi \nu^2}{q} \biggr) \cdot f \chi^{-1} + \biggl\{ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} + \frac{(1-q^3) s_\mathrm{env} }{(\gamma_e - 1)} \biggl[ \frac{ \nu }{ q^3} \biggr]\biggr\} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \chi^{3-3\gamma_c} </math>
	<math>~=</math>	<math> - \frac{3}{5} \biggl( \frac{\pi \nu^2}{q} \biggr) \cdot f \chi^{-1} + \biggl\{ \frac{\nu }{(\gamma_c - 1)} + \frac{(1-q^3) }{(\gamma_e - 1)} \biggl[ \frac{ \nu }{ q^3} \biggr]\biggr\} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \chi^{3-3\gamma_c} </math>
		<math> + \biggl\{ - \frac{\nu }{(\gamma_c - 1)} + \frac{B_\Lambda }{(\gamma_e - 1)} \biggl[ \frac{ \nu }{ q^3} \biggr]\biggr\} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \Lambda \chi^{3-3\gamma_c} </math>
	<math>~=</math>	<math> - \frac{3}{5} \biggl( \frac{\pi \nu^2}{q} \biggr) \cdot f \chi^{-1} + \biggl( \frac{4\pi}{3} \biggr) \biggl[ \frac{q^3 }{(\gamma_c - 1)} + \frac{(1-q^3) }{(\gamma_e - 1)} \biggr] \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c} \chi^{3-3\gamma_c} </math>
		<math> + \biggl( \frac{\pi}{5} \cdot \frac{\nu^2}{q} \biggr) \biggl[ - \frac{1}{(\gamma_c - 1)} + \frac{B_\Lambda }{q^3(\gamma_e - 1)} \biggr] \chi^{-1} </math>
	<math>~=</math>	<math> \biggl( \frac{\pi}{5} \cdot \frac{\nu^2}{q} \biggr) \biggl[\frac{B_\Lambda }{q^3(\gamma_e - 1)} - \frac{1}{(\gamma_c - 1)} - 3f \biggr] \chi^{-1} + \biggl( \frac{4\pi}{3} \biggr) \biggl[ \frac{q^3 }{(\gamma_c - 1)} + \frac{(1-q^3) }{(\gamma_e - 1)} \biggr] \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c} \chi^{3-3\gamma_c} </math>

Virial Equilibrium and Stability Evaluation

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>~=~</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>

Or, given the above renormalization, this expression can be written as,

<math> \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{4-3\gamma_c } \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} </math>	<math>~=~</math>	<math>\biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) </math>
<math> \Rightarrow ~~~~ \frac{R}{R_\mathrm{norm}} </math>	<math>~=~</math>	<math> \biggl\{ \biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} \biggr\}^{1/(4-3\gamma_c)} \, . </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!

Best of the Best

One Derivation of Free Energy

<math>~\mathfrak{G}^*</math>	<math>~=</math>	<math> - \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} + \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_c-1)} </math>
		<math> ~+ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggl( \frac{K_e}{K_c} \biggr) \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(\gamma_c-\gamma_e)/(3\gamma_c -4)} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_e-1)} \, . </math>

Another Derivation of Free Energy

Hence the renormalized gravitational potential energy becomes,

<math> \frac{W_\mathrm{grav}}{E_\mathrm{norm}} </math>

<math>~=</math>

<math> - \biggl( \frac{3\pi}{5} \biggr) \frac{\nu^2}{q} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} \cdot f \, ;</math>

and the two, renormalized contributions to the thermal energy become,

<math>~\frac{U_\mathrm{core}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_c-1)} \biggl[ \frac{S_\mathrm{core}}{ E_\mathrm{norm} } \biggr]</math>	<math>~=~</math>	<math> \frac{4\pi q^3 (1 + \Lambda) }{3(\gamma_c-1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} </math>
	<math>~=~</math>	<math> \frac{\nu (1 + \Lambda) }{(\gamma_c-1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c-1} \chi^{3-3\gamma_c} \, ,</math>
<math>~\frac{U_\mathrm{env}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_e-1)} \biggl[ \frac{S_\mathrm{env}}{ E_\mathrm{norm} } \biggr]</math>	<math>~=~</math>	<math> \frac{2 (2\pi) }{3(\gamma_e-1)} \biggl[ \frac{R^3 P_{ie}}{ E_\mathrm{norm} } \biggr] \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>
	<math>~=~</math>	<math> \frac{2 (2\pi) }{3(\gamma_e-1)} \biggl[ \frac{\mathrm{BigTerm}}{E_\mathrm{norm}} \biggr] R^3 K_e \rho_{ie}^{\gamma_e} </math>
	<math>~=~</math>	<math> \frac{2 (2\pi) }{3(\gamma_e-1)} \biggl[ \frac{\mathrm{BigTerm}}{E_\mathrm{norm}} \biggr] R^3 K_e \rho_\mathrm{norm}^{\gamma_e} \biggl( \frac{\rho_{ie}}{\bar\rho} \biggr)^{\gamma_e} \biggl( \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr)^{\gamma_e} </math>
	<math>~=~</math>	<math> \frac{2 (2\pi) }{3(\gamma_e-1)} \biggl[ \frac{\mathrm{BigTerm}}{E_\mathrm{norm}} \biggr] ( \rho_\mathrm{norm} R_\mathrm{norm}^3) K_e \rho_\mathrm{norm}^{\gamma_e-1} \biggl[ \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e} \chi^{3-3\gamma_e} </math>
	<math>~=~</math>	<math> \frac{2 (2\pi) }{3(\gamma_e-1)} \biggl[ \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e} \chi^{3-3\gamma_e} \biggl[ \mathrm{BigTerm}\biggr] \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \frac{K_e}{E_\mathrm{norm}} \biggl( \frac{3}{4\pi} \biggr)^{\gamma_e-1} \biggl[ \frac{G^3 M_\mathrm{tot}^2 }{K_c^3}\biggr]^{(\gamma_e-1)/(3\gamma_c-4)} </math>
	<math>~=~</math>	<math> \frac{(1-\nu)}{ (1-q^3) (\gamma_e-1)} \biggl[ \frac{3}{4\pi} \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl( \frac{K_e}{K_c} \biggr) \chi^{3-3\gamma_e} \biggl[ \mathrm{BigTerm}\biggr] \frac{K_c M_\mathrm{tot}}{E_\mathrm{norm}} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2}\biggr]^{(1-\gamma_e)/(3\gamma_c-4)} </math>
	<math>~=~</math>	<math> \frac{(1-\nu)}{ (1-q^3) (\gamma_e-1)} \biggl[ \frac{3}{4\pi} \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl( \frac{K_e}{K_c} \biggr) \chi^{3-3\gamma_e} \biggl[ \mathrm{BigTerm}\biggr] \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2}\biggr]^{(1-\gamma_e)/(3\gamma_c-4)} \biggl[ \frac{K_c^{3\gamma_c-4} M_\mathrm{tot}^{3\gamma_c-4}}{ G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6} K_c^{-1}} \biggr]^{1/(3\gamma_c-4)} </math>
	<math>~=~</math>	<math> \frac{(1-\nu)}{ (1-q^3) (\gamma_e-1)} \biggl[ \frac{3}{4\pi} \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl( \frac{K_e}{K_c} \biggr) \chi^{3-3\gamma_e} \biggl[ \mathrm{BigTerm}\biggr] \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2}\biggr]^{(1-\gamma_e)/(3\gamma_c-4)} \biggl[ \frac{K_c^{3}}{G^{3} M_\mathrm{tot}^{2}} \biggr]^{(\gamma_c-1)/(3\gamma_c-4)} </math>
	<math>~=~</math>	<math> \frac{(1-\nu)}{ (1-q^3) (\gamma_e-1)} \biggl[ \frac{3}{4\pi} \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl( \frac{K_e}{K_c} \biggr) \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2}\biggr]^{(\gamma_c-\gamma_e)/(3\gamma_c-4)} \biggl[ \mathrm{BigTerm}\biggr] \chi^{3-3\gamma_e} </math>

Finally, then, we can state that,

<math>~\mathfrak{f}_{WM}</math>	<math>~\equiv</math>	<math>~\frac{\pi \nu^2}{q} \cdot f \, ,</math>
<math>~s_\mathrm{core}</math>	<math>~\equiv</math>	<math> 1 + \Lambda \, , </math>
<math>~(1-q^3) s_\mathrm{env}</math>	<math>~\equiv</math>	<math> (1-q^3) + \Lambda\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>

Note,

<math>~\Lambda</math>

<math>~\equiv~</math>

<math> \frac{3}{2^2 \cdot 5} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{-\gamma_c} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} = \frac{\pi}{5} \biggl( \frac{\nu}{q} \biggr) \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{1-\gamma_c} \chi_\mathrm{eq}^{3\gamma_c - 4} \, .</math>

We also want to ensure that envelope pressure matches the core pressure at the interface. This means,

<math>~K_e \rho_{ie}^{\gamma_e}</math>	<math>~=</math>	<math>~K_c \rho_{ic}^{\gamma_c}</math>
<math>\Rightarrow ~~~~\frac{K_e}{K_c} </math>	<math>~=</math>	<math>~\rho_{ic}^{\gamma_c} \rho_{ie}^{-\gamma_e} </math>
	<math>~=</math>	<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \rho_\mathrm{norm}^{\gamma_c - \gamma_e}</math>
	<math>~=</math>	<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \biggl\{ \frac{3}{4\pi} \biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{1/(3\gamma_c -4)} \biggr\}^{\gamma_c - \gamma_e}</math>
<math>\Rightarrow ~~~~\frac{K_e}{K_c} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(\gamma_c - \gamma_e)/(3\gamma_c -4)} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1}</math>	<math>~=</math>	<math>~\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1} </math>
	<math>~=</math>	<math> \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} \biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{\rho_\mathrm{norm}}{ \bar\rho } \biggr)^{\gamma_e - \gamma_c} </math>
	<math>~=</math>	<math> \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} \biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{ R}{R_\mathrm{norm}} \biggr)^{3(\gamma_e - \gamma_c)} </math>

Keep in mind that, if the envelope and core both have uniform (but different) densities, then <math>~\rho_{ic} = \rho_c</math>, <math>~\rho_{ie} = \rho_e</math>, and

<math> \frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ; ~~~~~ \frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ; ~~~~~ \frac{\rho_e}{\rho_c} = \frac{q^3(1-\nu)}{\nu (1-q^3)} \, . </math>

Summary

Understanding Free-Energy Behavior

Step 1: Pick values for the separate coefficients, <math>\mathcal{A}, \mathcal{B},</math> and <math>\mathcal{C},</math> of the three terms in the normalized free-energy expression,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<math>~ -~ 3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}}{(1-\gamma_c)} ~\chi^{3-3\gamma_c} - \frac{\mathcal{C}}{(1-\gamma_e)} ~\chi^{3-3\gamma_e} </math>

then plot the function, <math>\mathfrak{G}^*(\chi)</math>, and identify the value(s) of <math>~\chi_\mathrm{eq}</math> at which the function has an extremum (or multiple extrema).

Step 2: Note that,

<math>~\mathcal{A}</math>	<math>~\equiv</math>	<math>~\frac{\pi \nu^2}{5q} \biggl\{ 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] \biggr\}</math>
<math>~\mathcal{B}</math>	<math>~\equiv</math>	<math>~\nu \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \biggl[ 1+\Lambda_\mathrm{eq} \biggr]</math>
<math>~\mathcal{C}</math>	<math>~\equiv</math>	<math>~(1-\nu)\biggl( \frac{K_e}{K_c} \biggr)^* \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e-1} \biggl\{ 1 + \frac{\Lambda_\mathrm{eq}}{(1-q^3)}\biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_c}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \biggr\}</math>
	<math>~\equiv</math>	<math>~ \nu \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \biggl\{ \frac{(1-q^3)}{q^3} + \frac{\Lambda_\mathrm{eq}}{q^3} \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_c}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \biggr\} \chi_\mathrm{eq}^{3(\gamma_e - \gamma_c)}</math>

where (see, for example, in the context of its original definition),

<math>~\Lambda_\mathrm{eq} \equiv \frac{3}{2^2\pi \cdot 5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4 P_i} \biggr) \frac{\nu^2}{q^4} </math>

<math>~=~</math>

<math> \frac{\pi}{5} \biggl( \frac{\nu}{q} \biggr) \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{1-\gamma_c} \chi_\mathrm{eq}^{3\gamma_c - 4} </math>

and, where,

<math>\biggl( \frac{K_e}{K_c} \biggr)^* \equiv \frac{K_e}{K_c} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(\gamma_c - \gamma_e)/(3\gamma_c -4)} </math>

<math>~=</math>

<math> \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{1-\nu}{1-q^3} \biggr]^{-\gamma_e} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c} \chi_\mathrm{eq}^{3(\gamma_e - \gamma_c)} \, . </math>

Also, keep in mind that, if the envelope and core both have uniform (but different) densities, then <math>~\rho_{ic} = \rho_c</math>, <math>~\rho_{ie} = \rho_e</math>, and

<math> \frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ; ~~~~~ \frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ; ~~~~~ \frac{\rho_e}{\rho_c} = \frac{q^3(1-\nu)}{\nu (1-q^3)} \, . </math>

Step 3: An analytic evaluation tells us that the following should happen. Using the numerically derived value for <math>~\chi_\mathrm{eq}</math>, define,

<math>~\mathcal{C}^' \equiv \mathcal{C} \chi_\mathrm{eq}^{3(\gamma_c - \gamma_e)} \, .</math>

We should then discover that,

<math>\frac{\mathcal{A}}{\mathcal{B} + \mathcal{C}^'} = \chi_\mathrm{eq}^{4-3\gamma_c} = \frac{1}{\Lambda_\mathrm{eq}} \cdot \frac{\pi}{5} \biggl( \frac{\nu}{q} \biggr) \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{1-\gamma_c} \, .</math>

Check It

<math>~\mathcal{B} + \mathcal{C}^'</math>

<math>~=</math>

<math>~ \nu \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\nu}{q^3} \biggr]^{\gamma_c-1} \biggl\{ \biggl[ 1+\Lambda_\mathrm{eq} \biggr] + \frac{(1-q^3)}{q^3} + \frac{\Lambda_\mathrm{eq}}{q^3} \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_c}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \biggr\} </math>

<math>\Rightarrow~~~~\mathcal{A} \biggl[ \Lambda_\mathrm{eq} \cdot \frac{5}{\pi} \biggl( \frac{q}{\nu^2} \biggr) \biggr]</math>

<math>~=</math>

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

<math> \Rightarrow~~~~\Lambda_\mathrm{eq} \biggl\{ 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] \biggr\} </math>

<math>~=</math>

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

<math> \Rightarrow~~~~\Lambda_\mathrm{eq} \biggl\{ \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] \biggr\} </math>

<math>~=</math>

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

<math>\Rightarrow~~~~ \frac{1}{\Lambda_\mathrm{eq}} </math>	<math>~=</math>	<math> q^3 \biggl\{ \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] \biggr\} - \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_c}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] </math>
<math>\Rightarrow~~~~ \frac{2}{\Lambda_\mathrm{eq}} </math>	<math>~=</math>	<math> 5\biggl( \frac{\rho_e}{\rho_c} \biggr) ( q - q^3 ) + \frac{2}{q^2}\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (1 - q^5 ) - \frac{5}{2}( q^3 - q^5 ) \biggr] - 5 \biggl( \frac{\rho_e}{\rho_c}\biggr) (-2 + 3q - q^3) - \frac{3}{q^2} \biggl( \frac{\rho_e}{\rho_c}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) </math>
	<math>~=</math>	<math> 5\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ ( q - q^3 ) - (-2 + 3q - q^3) \biggr] + \frac{1}{q^2}\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ 2(1 - q^5 ) - 5( q^3 - q^5 ) - 3(-1 +5q^2 - 5q^3 + q^5) \biggr] </math>
	<math>~=</math>	<math> 10\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ 1-q \biggr] + \frac{5}{q^2}\biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ 1 - 3q^2 + 2q^3 \biggr] </math>
<math>\Rightarrow~~~~ \frac{1}{\Lambda_\mathrm{eq}} \biggl[ \frac{2q^2}{5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^{-1} \biggr] </math>	<math>~=</math>	<math> 2q^2 (1-q )+ \biggl( \frac{\rho_e}{\rho_c} \biggr) ( 1 - 3q^2 + 2q^3 ) </math>

Fortunately, this precisely matches our earlier derivation, which states that,

<math>~\frac{1}{\Lambda}</math>

<math>~=</math>

<math>\frac{5}{2}(g^2-1) = \frac{5}{2}\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>

Playing With One Example

By setting,

<math>~\gamma_c = 6/5; ~~~ \gamma_e = 2</math>
<math>~\mathcal{A}</math>	<math>~\mathcal{B}</math>	<math>~\mathcal{C}</math>
2.5	1.0	2.0

a plot of <math>~\mathfrak{G}^*</math> versus <math>~\chi</math> exhibits the following, two extrema:

extremum	<math>~\chi_\mathrm{eq}</math>	<math>~\mathfrak{G}^*</math>		<math>~\chi_\mathrm{eq}^{3(\gamma_c - \gamma_e)}</math>	<math>~\mathcal{C}^'</math>	<math>~\chi_\mathrm{eq}^{4-3\gamma_c}</math>	<math>~\frac{\mathcal{A}}{\mathcal{B} +\mathcal{C}^'}</math>
MIN	<math>1.1824</math>	<math>-0.611367</math>	<math>~~~~\Rightarrow~~~~</math>	<math>0.66891</math>	<math>1.3378</math>	<math>1.0693</math>	<math>1.0694</math>
MAX	<math>9.6722</math>	<math>+0.508104</math>	<math>~\Rightarrow</math>	<math>0.004313</math>	<math>0.008625</math>	<math>2.4786</math>	<math>2.4786</math>

The last two columns of this table confirm the internal consistency of the above-derived relationships. But what does this mean in terms of the values of <math>~\nu</math>, <math>~q</math>, and the related ratio of densities at the interface, <math>~\rho_e/\rho_c</math>?

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>.

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation