Virial Stability of BiPolytropes

BiPolytrope Structural Relations

[Following a discussion that Tohline had with Kundan Kadam on 3 July 2013, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]

We will adopt the following approach:

Properties of the core <math>\cdots</math>
- Uniform density, <math>\rho_c</math>;
- Polytropic constant, <math>K_c</math>, and polytropic index, <math>n_c</math>;
- Surface of the core at <math>r_i</math>;
Properties of the envelope <math>\cdots</math>
- Uniform density, <math>\rho_e</math>;
- Polytropic constant, <math>K_e</math>, and polytropic index, <math>n_e</math>;
- Base of the core at <math>r_i</math> and surface at <math>R</math>.

Use the dimensionless radius,

<math>\xi \equiv \frac{r}{r_i}</math>.

Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.

Expressions for Mass

Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,

<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .

The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,

<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi^3 - 1) \biggr]</math> .

Hence, the mass of the core, the mass of the envelope, and the total mass are, respectively,

<math>M_\mathrm{core} = \frac{4\pi}{3} \rho_c r_i^3 = M_\mathrm{tot} \biggl[ \frac{\rho_c}{\rho_\mathrm{norm}} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]</math> <math>\Rightarrow</math> <math>\frac{\rho_c}{\rho_\mathrm{norm}} = \frac{M_\mathrm{core}}{M_\mathrm{tot}} \biggl( \frac{r_i}{R_0}\biggr)^{-3}</math> ;

<math>M_\mathrm{env} = \frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] = M_\mathrm{tot} (\xi_s^3 - 1) \biggl[ \frac{\rho_e}{\rho_\mathrm{norm}} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]</math> <math>\Rightarrow</math> <math>\frac{\rho_e}{\rho_\mathrm{norm}} = \frac{M_\mathrm{env}}{M_\mathrm{tot}} \biggl( \frac{r_i}{R_0}\biggr)^{-3} (\xi_s^3 - 1)^{-1}</math> ;

<math>M_\mathrm{tot} = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi_s^3 - 1) \biggr] = M_\mathrm{tot} \biggl( \frac{\rho_c}{\rho_\mathrm{norm}} \biggr) \biggl( \frac{r_i}{R_0}\biggr)^3 \biggl[ 1 + \frac{\rho_e}{\rho_c} (\xi_s^3 - 1) \biggr] </math> ;

where, <math>\rho_\mathrm{norm} \equiv 3M_\mathrm{tot} /(4\pi R_0^{3})</math>. Letting <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math> — which also means, <math>M_\mathrm{env}/M_\mathrm{tot} = (1-\nu) </math> — we can write,

<math>\frac{\rho_e}{\rho_c} = \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1} = \frac{(1-\nu)}{\nu (\xi_s^3 - 1)} </math> ,

and,

<math>\nu (\xi_s^3 - 1) \biggl( \frac{\rho_e}{\rho_c} \biggr) = (1-\nu) </math> <math>\Rightarrow</math> <math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^3 - 1) \biggr]^{-1}</math> .

Following the work of Schönberg & Chandrasekhar (1942) — see our accompanying discussion — we are seeking equilibrium configurations in the <math>\nu - q</math> plane where,

<math>\nu</math>	<math>\equiv</math>	<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>, (as also defined here)
<math>q</math>	<math>\equiv</math>	<math>\frac{r_i}{R} = \frac{1}{\xi_s}</math> .

So we can rewrite the above expression as,

<math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^3} - 1\biggr) \biggr]^{-1}</math> ,

or,

<math>\frac{\rho_e}{\rho_c} = \biggl[ \frac{1-\nu}{\nu}\biggr] (\xi_s^3 - 1)^{-1} = \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) \, . </math>

The following figure shows how <math>\nu</math> varies with <math>q</math> for various choices of the mass density ratio, <math>\rho_e/\rho_c</math>. It illustrates that, for a given core-to-total mass ratio, <math>\nu</math>, the relative location of the interface radius, <math>q</math>, can vary between zero and one, but each value of <math>q</math> reflects a different ratio of envelope-to-core mass density.

Energy Expressions

The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,

<math>dW = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .

Hence,

<math>W = W_\mathrm{core} + W_\mathrm{env}</math>	<math> = - G \biggl\{ \int_0^{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_c dr + \int^R_{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_e dr \biggr\} </math>
	<math> = - G \biggl\{ \int_0^1 \biggl( \frac{4\pi }{3} \rho_c r_i^3 \xi^3 \biggr) 4\pi r_i^2 \rho_c \xi d\xi + \int_1^{\xi_s} \frac{4\pi}{3} \rho_c r_i^3 \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] 4\pi r_i^2 \rho_e \xi d\xi \biggr\} </math>
	<math> = - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^1 \xi^4 d\xi + \int_1^{\xi_s} \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] \biggl( \frac{\rho_e}{\rho_c} \biggr) \xi d\xi \biggr\} </math>
	<math> = - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5} + \biggl( \frac{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s} (\xi^3 - 1) \xi d\xi \biggr\} </math>
	<math> = - \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) 3\nu^2 \xi_s \biggl\{ \frac{1}{5} + \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\} </math>

I like the form of this expression. The leading term, which scales as <math>R^{-1}</math>, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of <math>\xi_s</math>, <math>\nu</math>, and density ratio <math>(\rho_e/\rho_c)</math>. Actually, only two internal structural parameters need to be specified — <math>\nu</math> and <math>\xi_s</math> (or, <math>q</math>). From these two, the expression shown above allows the determination of <math>(\rho_e/\rho_c)</math>.

Drawing on expressions developed in our introductory discussion of the virial equation, the internal energy of the bipolytropic configuration is,

<math> U = U_\mathrm{core} + U_\mathrm{env} </math>

<math> \biggl\{ M_\mathrm{core} \biggl[ (1 - \delta_{\infty n_c}) n_c K_c \rho_c^{1/n_c} + \delta_{\infty n_c} c_s^2 \ln\biggl(\frac{\rho_c}{\rho_\mathrm{norm}} \biggr) \biggr] + n_e M_\mathrm{env} K_e \rho_e^{1/n_e} \biggr\} \, , </math>

where we have allowed for either an isothermal (<math>\delta_{\infty n_c} = 1</math>) or an adiabatic (<math>\delta_{\infty n_c} = 0</math>) core and, for normalization purposes, we have introduced,

<math> \rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, , </math>

where, <math>R_0</math> is an, as yet unspecified, radius. This expression for the total internal energy may be rewritten as,

<math> \frac{U}{M_\mathrm{tot}} </math>	<math>=</math>	<math> \nu \biggl[ (1 - \delta_{\infty n_c}) n_c K_c \rho_c^{1/n_c} + \delta_{\infty n_c} c_s^2 \ln\biggl(\frac{\rho_c}{\rho_\mathrm{norm}} \biggr) \biggr] + (1-\nu) n_e K_e \rho_e^{1/n_e} </math>
	<math>=</math>	<math> \nu \biggl\{ (1 - \delta_{\infty n_c}) n_c K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1/n_c} + \delta_{\infty n_c} c_s^2 \ln( \nu R_0^3 r_i^{-3} ) \biggr\} + (1-\nu) n_e K_e \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) (1-\nu)(\xi_s^3-1)^{-1} r_i^{-3} \biggr]^{1/n_e} </math>
	<math>=</math>	<math> \nu \biggl\{ (1 - \delta_{\infty n_c}) n_c K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr) \nu \xi_s^{3} \biggr]^{1/n_c} \biggl(\frac{R}{R_0}\biggr)^{-3/n_c} + ~\delta_{\infty n_c} c_s^2 \biggl[ \ln( \nu \xi_s^3) - 3 \ln \biggl( \frac{R}{R_0} \biggr) \biggr] \biggr\} </math>
		<math>~~~ + (1-\nu) n_e K_e \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr) (1-\nu)(\xi_s^3-1)^{-1} \xi_s^{3} \biggr]^{1/n_e} \biggl(\frac{R}{R_0}\biggr)^{-3/n_e} \, . </math>

Pressure Across the Interface

We will relate <math>K_e</math> to <math>K_c</math> by demanding that initially the pressure is identical in both layers. The relevant algebraic relation will depend on whether the core is isothermal (<math>n_c = \infty</math>), or whether it has a finite polytropic index and therefore adjusts adiabatically to compressions or expansions.

Isothermal Core

In the case of an isothermal core, the condition for pressure balance at the interface is,

Drawing on the above expressions for the mass, we can write,

<math> \frac{\rho_e}{\rho_\mathrm{norm}} = \biggl( \frac{4\pi R_0^3}{3M_\mathrm{tot} } \biggr) \biggl( \frac{3M_\mathrm{env}}{4\pi r_i^3} \biggr) (\xi_s^3 - 1)^{-1} = \frac{(1-\nu) \xi_s^3}{(\xi_s^3-1)} \biggl( \frac{R}{R_0} \biggr)^{-3} = \frac{(1-\nu) }{(1-q^3)} \biggl( \frac{R}{R_0} \biggr)^{-3} \, . </math>

Hence, pressure balance at the interface implies that the constant,

<math> \kappa_I \equiv \frac{c_s^2}{K_e \rho_\mathrm{norm}^{1/n_e}} = \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{\rho_e}{\rho_\mathrm{norm}} \biggr)^{1/n_e} = \biggl[ \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) \biggr] \biggl[ \frac{(1-\nu) }{(1-q^3)} \biggr]^{1/n_e} \biggl( \frac{R}{R_0} \biggr)^{-3/n_e} \, . </math>

<math> \Rightarrow ~~~~ \kappa_I^{n_e} = \biggl( \frac{\nu}{q^3} \biggr) \biggl[ \frac{q^3 (1-\nu) }{\nu (1-q^3)} \biggr]^{n_e+1} \biggl( \frac{R}{R_0} \biggr)^{-3} \, . </math>

This can be rewritten in a variety of, not necessarily more useful, forms, such as,

<math> \biggl(\frac{\nu}{q^3}\biggr) \biggl[ \biggl(\frac{1-\nu}{\nu}\biggr)^{n_e+1} \kappa_I^{-n_e} \biggl( \frac{R}{R_0} \biggr)^{-3} \biggr] = \biggl[ \frac{1-q^3}{q^3} \biggr]^{n_e +1} \, ; </math>

or,

<math> \nu^{-n_e} \xi_s^3 \biggl[ (1-\nu)^{n_e+1} \kappa_I^{-n_e} \biggl( \frac{R}{R_0} \biggr)^{-3} \biggr] = ( \xi_s^3-1 )^{n_e +1} \, ; </math>

or,

<math> \frac{1}{\xi_s^3} (\xi_s^3 -1)^{n_e+1} = q^3 \biggl( \frac{1}{q^3} - 1\biggr)^{n_e+1}= \nu^{-n_e} (1-\nu)^{n_e+1} \kappa_I^{-n_e} \biggl( \frac{R}{R_0} \biggr)^{-3} \, ; </math>

or,

<math> q^{3/(n_e+1)} \biggl( \frac{1}{q^3} - 1\biggr) = q^{-3n_e/(n_e+1)} (1-q^3) = \biggl[\nu^{-n_e}\ (1-\nu)^{n_e+1} \kappa_I^{-n_e} \biggl( \frac{R}{R_0} \biggr)^{-3} \biggr]^{1/(n_e+1)} \, . </math>

Adiabatic Core

In the case of an adiabatic core, the condition for pressure balance at the interface is,

=

<math> \biggl[ \frac{\rho_e^{1+1/n_e}}{\rho_c^{1+1/n_c}} \biggr] = \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{\rho_e^{1/n_e}}{\rho_c^{1/n_c}} \, . </math>

Hence, pressure balance at the interface implies that the constant,

<math> \kappa_A \equiv \biggl( \frac{K_c }{K_e} \biggr) \rho_\mathrm{norm}^{1/n_c - 1/n_e} </math>	=	<math> \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{(\rho_e/\rho_\mathrm{norm})^{1/n_e}}{ (\rho_c/\rho_\mathrm{norm})^{1/n_c}} </math>
	=	<math> \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{(\rho_e/\rho_\mathrm{norm})^{1/n_e}}{ (\rho_c/\rho_\mathrm{norm})^{1/n_c}} \, . </math>
	=	<math> \frac{(1-\nu)}{\nu (\xi_s^3-1)} \biggl[ \frac{\xi_s^3 (1-\nu)}{(\xi_s^3-1)} \biggl( \frac{R}{R_0} \biggr)^{-3}\biggr]^{1/n_e} \biggl[ \nu \xi_s^3 \biggl( \frac{R}{R_0} \biggr)^{-3} \biggr]^{-1/n_c} </math>
	=	<math> (1-\nu)^{1+1/n_e} \nu^{-(1+1/n_c)} (\xi_s^3 - 1)^{-(1+1/n_e)} \xi_s^{(1/n_e - 1/n_c)} \biggl( \frac{R}{R_0} \biggr)^{-3(1/n_e -1/n_c)} </math>

Sanity Check: Uniform Density Configuration

<math> M_\mathrm{core} = \frac{4\pi}{3} r_i^3 \rho_c </math>

Virial Analysis

Free Energy Expression

To within an additive constant, the free energy may now be written as,

<math> \mathfrak{G} = W + U = - A \chi^{-1} + (1-\delta_{\infty n_c}) B_c \chi^{-3/n_c} - \delta_{\infty n_c} B_I \ln\chi + B_e \chi^{-3/n_e} \, , </math>

where, <math>\chi \equiv R/R_0</math> and,

<math> A </math>	<math>=</math>	<math> \biggl( \frac{3GM^2_\mathrm{tot}}{5R_0} \biggr) \nu^2 \xi_s \biggl\{ 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\} \, , </math>
<math>B_c</math>	<math>=</math>	<math> n_c K_c M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_c} \nu^{1+1/n_c} \xi_s^{3/n_c} \, , </math>
<math>B_I</math>	<math>=</math>	<math> 3 M_\mathrm{tot} c_s^2 \nu \, , </math>
<math>B_e</math>	<math>=</math>	<math> n_e K_e M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e} \, . </math>

Summary Expressions (New)

In the above derivations, we have adopted the notation,

<math> \rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, . </math>

Now, guided by the earlier discussion of pressure-bounded isothermal spheres, we choose the following normalization energy and radius:

<math> E_0 = 3M_\mathrm{tot} c_s^2 </math> and <math> R_0 = \frac{GM_\mathrm{tot}}{5c_s^2} \, . </math>

Also, by analogy, it is useful to define the dimensionless parameter,

<math> \Pi_I \equiv \frac{K_e \rho_\mathrm{norm}^{1/n_e}}{c_s^2} = \frac{K_e}{c_s^2} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr]^{1/n_e} = \biggl( \frac{3\cdot 5^3}{4\pi} \biggr)^{1/n_e} \frac{K_e}{c_s^2} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]^{1/n_e} \, . </math>

(It is worth noting that if we set <math>n_e = -1</math>, the dimensionless parameter <math>\Pi_I</math> becomes identical to the parameter <math>\Pi</math> as defined in the context of our discussion of the Bonnor-Ebert sphere. But in order to complete the analogy with the Bonnor-Ebert sphere discussion, we would also need to change the sign on the last term in the above expression for the free energy because in the earlier discussion the external pressure was an external, confining condition whereas here it is included as an internal energy of the system.)

Relevant Expressions for Isothermal Core
<math> \frac{\rho_e}{\rho_c}= \frac{\mu_e}{\mu_c} </math>	<math> \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) </math>
<math> \chi \equiv \frac{R}{R_0} </math>	<math> \Pi_I^{n_e/3} [\nu^{-n_e}\ (1-\nu)^{n_e+1} ]^{1/3} q^{n_e} (1-q^3)^{-(n_e+1)/3} </math>
<math> \chi^3 = \biggl(\frac{R}{R_0} \biggr)^{3} </math>	<math> \Pi_I^{n_e} \biggl( \frac{\rho_e}{\rho_c}\biggr)^{n_e + 1} \biggl[ q^3 + \biggl( \frac{\rho_e}{\rho_c} \biggr)(1-q^3) \biggr]^{-1} </math>
<math> \frac{A}{E_0} </math>	<math> \nu^2 \xi_s \biggl\{ 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\} </math>	<math>= \biggl( \frac{\nu^2}{q} \biggr) \biggl\{ f_A(\nu,q) \biggr\}</math>
<math> \frac{B_e}{E_0} </math>	<math> \Pi_I \biggl( \frac{n_e}{3} \biggr) (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e} </math>
<math> \frac{B_I}{E_0} </math>	<math>\nu </math>
<math> \frac{\mathfrak{G}}{E_0} </math>	<math> - \frac{A}{E_0} \chi^{-1} - \frac{B_I}{E_0} \ln\chi + \frac{B_e}{E_0} \chi^{-3/n_e} </math>	<math> = \nu \biggl[ - \biggl( \frac{\nu}{q} \biggr) \frac{f_A(\nu,q)}{\chi} - \ln\chi + \frac{n_e}{3} \biggl(\frac{1}{q^3}-1\biggr)\biggr] </math>

[On 8 November 2013, J. E. Tohline wrote: I just confirmed that the simpler expression for the normalized total free energy, <math>\mathfrak{G}/E_0</math>, matches the more complicated version. I don't like the result because the third term in the free energy -- the one contributed by the internal energy of the envelope -- is independent of the radius of the configuration, <math>\chi</math>; it works out this way because my expression for <math>\Pi_I</math> has a radial dependence that exactly cancels out the explicit radial dependence that appears in the more complicated expression. But maybe it's okay after all because this expression is intended to show how the free energy varies across the <math>(q,\nu)</math> plane, and the effect of <math>\Pi_I</math> appears implicitly through the specification of <math>\chi</math>, or visa versa.]

Old and Probably Irrelevant Discussion

Summary Expressions (Old)

In the above derivations, we have adopted the notation,

<math> \rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, . </math>

Now, guided by the dimensional aspects of the various coefficients in the free energy expression, we choose the following normalization energy and radius:

<math> E_0 = M_\mathrm{tot} K_e \rho_\mathrm{norm}^{1/n_e} </math> and <math> R_0 = \frac{GM_\mathrm{tot}}{K_e \rho_\mathrm{norm}^{1/n_e}} \, . </math>

When combined with the expression for <math>\rho_\mathrm{norm}</math>, these become,

<math> E_0 = \biggl[ \biggl( \frac{4\pi}{3} \biggr) G^3 M_\mathrm{tot}^{5-n_e} K_e^{-n_e}\biggr]^{1/(3-n_e)} </math> and <math> R_0 = \biggl[ \frac{3M_\mathrm{tot}}{4\pi} \biggl( \frac{K_e}{GM_\mathrm{tot}} \biggr)^{n_e} \biggr]^{1/(3-n_e)} \, . </math>

So, the primary scales are determined after specifying two parameters: <math>M_\mathrm{tot}</math> and <math>K_e</math>. We also obtain,

<math> \kappa_I \equiv \frac{c_s^2}{K_e \rho_\mathrm{norm}^{1/n_e}} = \frac{M_\mathrm{tot} c_s^2}{E_0} = c_s^2 \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{K_e^{n_e}}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(3-n_e)} \, . </math>

Relevant Expressions for Isothermal Core
<math> \frac{\rho_e}{\rho_c} </math>	<math> \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) </math>
<math> \chi \equiv \frac{R}{R_0} </math>	<math> q^{n_e} (1-q^3)^{-(n_e+1)/3} \biggl[\nu^{-n_e}\ (1-\nu)^{n_e+1} \kappa_I^{-n_e} \biggr]^{1/3} </math>
<math> \frac{A}{E_0} </math>	<math> \biggl( \frac{3}{5} \biggr)\nu^2 \xi_s \biggl\{ 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\} </math>
<math> \frac{B_e}{E_0} </math>	<math> n_e(1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e} </math>
<math> \frac{B_I}{E_0} </math>	<math> 3 \kappa_I \nu </math>
<math> \frac{\mathfrak{G}}{E_0} </math>	<math> - \frac{A}{E_0} \chi^{-1} - \frac{B_I}{E_0} \ln\chi + \frac{B_e}{E_0} \chi^{-3/n_e} </math>

Subsequently, we will also find it useful to have expressions for the following coefficient ratios:

<math>\frac{n_e A}{3 B_e}</math>	<math>=</math>	<math> \biggl( \frac{GM^2_\mathrm{tot}}{5R_0} \biggr) \nu^2 \xi_s \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\} \biggl[K_e M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1} </math>
	<math>=</math>	<math> \biggl( \frac{GM_\mathrm{tot}}{5K_e \rho_\mathrm{norm}^{1/n_e} R_0} \biggr) \nu^2 \xi_s \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\} (1 - \nu)^{-(1+1/n_e)} q^{3/n_e} \biggl(\frac{1}{q^3} - 1 \biggr)^{1/n_e} </math>
	<math>=</math>	<math> \biggl( \frac{GM_\mathrm{tot}}{5K_e \rho_\mathrm{norm}^{1/n_e} R_0} \biggr) \nu^2 q^{-1} [ (1 - \nu)^{-(n_e+1)} (1 - q^3 ) ]^{1/n_e} \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\} \, ; </math>

<math>\frac{n_e B_I}{3 B_e}</math>	<math>=</math>	<math> M_\mathrm{tot} c_s^2 \nu \biggl[K_e M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1} </math>
	<math>=</math>	<math> \kappa_I \nu (1 - \nu)^{-(1+1/n_e)} q^{3/n_e} \biggl(\frac{1}{q^3} - 1 \biggr)^{1/n_e} </math>
	<math>=</math>	<math> \kappa_I \nu \biggl[ (1 - \nu)^{-(n_e+1)} (1-q^{3}) \biggr]^{1/n_e} \, ; </math>

<math>\frac{n_e B_c}{n_c B_e}</math>	<math>=</math>	<math> \biggl[K_c M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_c} \nu^{1+1/n_c}\xi_s^{3/n_c} \biggr] \biggl[K_e M_\mathrm{tot} \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1} </math>
	<math>=</math>	<math> \kappa_A \nu^{1+1/n_c}\xi_s^{3/n_c} \biggl[ (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1} </math>
	<math>=</math>	<math> \kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)} \biggr] \biggl[ \xi_s^{3(1/n_c-1/n_e)} (\xi_s^3 - 1)^{1/n_e}\biggr] </math>
	<math>=</math>	<math> \kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)} \biggr] \biggl[ q^{3(1/n_e-1/n_c)} \biggl( \frac{1}{q^3} - 1\biggr)^{1/n_e}\biggr] </math>
	<math>=</math>	<math> \kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)} \biggr] \biggl[ q^{-3n_e/n_c} (1-q^3) \biggr]^{1/n_e} </math>
	<math>=</math>	<math> \kappa_A \biggl[\nu^{1+1/n_c} q^{-3/n_c} \biggr] \biggl[ (1 - \nu)^{-(n_e+1)} (1-q^3) \biggr]^{1/n_e} \, . </math>

Derivatives of Free Energy

<math> \frac{\partial\mathfrak{G}}{\partial \chi} = A \chi^{-2} -(1-\delta_{\infty n_c}) \frac{3}{n_c} B_c \chi^{-(1+3/n_c)} - \delta_{\infty n_c} B_I \chi^{-1} -\frac{3}{n_e} B_e \chi^{-(1+3/n_e)} \, ; </math>

<math> \frac{\partial^2\mathfrak{G}}{\partial \chi^2} = -2 A \chi^{-3} + (1-\delta_{\infty n_c}) \frac{3}{n_c} \biggl(1+\frac{3}{n_c}\biggr) B_c \chi^{-(2+3/n_c)} + \delta_{\infty n_c} B_I \chi^{-2} + \frac{3}{n_e} \biggl(1+\frac{3}{n_e}\biggr) B_e \chi^{-(2+3/n_e)} \, . </math>

Equilibrium Condition

We obtain the equilibrium radius, <math>\chi_E</math>, when <math>\partial\mathfrak{G}/\partial\chi = 0</math>. Hence, the relation governing the equilibrium radius is,

<math> A \chi_E^{-2} </math>	<math>=</math>	<math> (1-\delta_{\infty n_c}) \frac{3}{n_c} B_c \chi_E^{-1- 3/n_c)} +\delta_{\infty n_c} B_I \chi_E^{-1} +\frac{3}{n_e} B_e \chi_E^{-1-3/n_e)} </math>
<math> \Rightarrow ~~~~~ \frac{n_e A}{3B_e} </math>	<math>=</math>	<math> (1-\delta_{\infty n_c}) \frac{n_e B_c}{n_c B_e} \chi_E^{1- 3/n_c} +\delta_{\infty n_c} \frac{n_e B_I}{3B_e} \chi_E + \chi_E^{1-3/n_e} </math>
<math> \Rightarrow ~~~~~ \chi_E^{1-3/n_e} </math>	<math>=</math>	<math> \alpha - (1-\delta_{\infty n_c}) \beta \chi_E^{1- 3/n_c} - \delta_{\infty n_c} \beta_I \chi_E \, , </math>

where,

<math>\alpha \equiv \frac{n_e A}{3B_e} \, ; ~~~ \beta \equiv \frac{n_e B_c}{n_c B_e} \, ; ~~~ \beta_I \equiv \frac{n_e B_I}{3B_e} \, .</math>

Stability

At this equilibrium radius, the second derivative of the free energy has the value,

<math> \chi_E^3 \biggl( \frac{n_e}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr\|_E </math>	<math> = </math>	<math> -2 A\biggl( \frac{n_e}{3B_e} \biggr) + (1-\delta_{\infty n_c}) \biggl( \frac{n_e}{3B_e} \biggr) \frac{3}{n_c} \biggl(1+\frac{3}{n_c}\biggr) B_c \chi_E^{1-3/n_c} + \delta_{\infty n_c} \biggl( \frac{n_e}{3B_e} \biggr) B_I \chi_E + \frac{3}{n_e}\biggl( \frac{n_e}{3B_e} \biggr) \biggl(1+\frac{3}{n_e}\biggr) B_e \chi_E^{1-3/n_e} </math>
	<math> = </math>	<math> -2 \alpha + (1-\delta_{\infty n_c}) \beta \biggl(1+\frac{3}{n_c}\biggr) \chi_E^{1-3/n_c} + \delta_{\infty n_c} \beta_I \chi_E + \biggl(1+\frac{3}{n_e}\biggr) \chi_E^{1-3/n_e} \, , </math>

which, when combined with the condition for equilibrium gives,

<math> \chi_E^3 \biggl( \frac{n_e}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr\|_E </math>	<math> = </math>	<math> -2 \alpha + (1-\delta_{\infty n_c}) \beta \biggl(1+\frac{3}{n_c}\biggr) \chi_E^{1-3/n_c} + \delta_{\infty n_c} \beta_I \chi_E + \biggl(1+\frac{3}{n_e}\biggr) \biggl[ \alpha - (1-\delta_{\infty n_c}) \beta \chi_E^{1- 3/n_c} - \delta_{\infty n_c} \beta_I \chi_E \biggr] </math>
	<math> = </math>	<math> \alpha \biggl(\frac{3}{n_e}-1\biggr) + (1-\delta_{\infty n_c}) \beta \biggl(\frac{3}{n_c}-\frac{3}{n_e}\biggr) \chi_E^{1-3/n_c} - \delta_{\infty n_c} \beta_I \biggl(\frac{3}{n_e}\biggr)\chi_E </math>
<math> \Rightarrow ~~~~ \chi_E^3 \biggl( \frac{n_e^2}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr\|_E </math>	<math> = </math>	<math> \alpha (3-n_e) - (1-\delta_{\infty n_c}) 3\beta \biggl(1 - \frac{n_e}{n_c} \biggr) \chi_E^{1-3/n_c} - \delta_{\infty n_c} 3\beta_I \chi_E \, . </math>

Finally, the equilibrium configuration is stable as long as this second derivative is positive. Hence, for a bipolytrope with an isothermal core (<math>\delta_{\infty n_c} = 1</math>), the configuration is stable as long as,

In the adiabatic case (<math>\delta_{\infty n_c} = 0</math>), the configuration is stable as long as,

Examples

Isothermal Core with <math>n=3/2</math> Envelope

When the core is isothermal and <math>n_e = 3/2</math>, the equilibrium condition is:

<math> \chi_E^{-1} = \alpha - \beta_I \chi_E \, , </math>

<math> \Rightarrow ~~~~ \beta_I \chi_E^2 - \alpha \chi_E + 1 = 0 \, , </math>

<math> \Rightarrow ~~~~ \chi_E = \frac{1}{2\beta_I} \biggl[ \alpha \pm \sqrt{\alpha^2 - 4\beta_I} \biggr] = \frac{\alpha}{2\beta_I} \biggl[ 1 \pm \sqrt{1 - \frac{4\beta_I}{\alpha^2}} \biggr] \, . </math>

At the same time, the condition for stability is,

Isothermal Core with <math>n=1</math> Envelope

When the core is isothermal and <math>n_e = 1</math>, the equilibrium condition is:

<math> \chi_E^{-2} = \alpha - \beta_I \chi_E \, , </math>

<math> \Rightarrow ~~~~ \chi_E^3 - \frac{\alpha}{\beta_I} \chi_E^2 + \frac{1}{\beta_I} = 0 \, . </math>

(We need to solve this cubic equation.)

At the same time, the condition for stability is,

Old (and probably incorrect) cases

Envelope with <math>n=3/2</math>

If we choose an <math>n_e = 3/2</math> envelope, we obtain stability for,

In this case, the equilibrium radius condition is,

<math> \chi_E^2 - \alpha \chi_E + \beta =0 </math>

<math> \Rightarrow ~~~~ \chi_E = \frac{1}{2}\biggl[\alpha \pm \biggl( \alpha^2 -4\beta \biggr)^{1/2} \biggr] = \frac{\alpha}{2}\biggl[1 \pm \biggl( 1 -\frac{4\beta}{\alpha^2} \biggr)^{1/2} \biggr] </math>

Envelope with <math>n=1</math>

If, instead, we choose an <math>n_e = 1</math> envelope, we obtain stability for,