# Bipolytrope Generalization

## Setup

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

$\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots$

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 — $~\gamma_c$ and $~\gamma_e$, respectively.

### Review of Isolated Polytrope

If we were configuring isolated polytropes — instead of bipolytropes — the free-energy expression would be, simply,

$\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_A \, ,$

and, following the detailed steps presented in our introductory discussion of the free energy of spherically symmetric, configurations, properly normalized expressions for the two contributing energy terms would be,

 $~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}$ $~=$ $- \int_0^{\chi = R_\mathrm{edge}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*$ $~=$ $- \biggl\{ \biggl( \frac{\rho_c}{\bar\rho} \biggr) \int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \biggr\}_\mathrm{eq} \cdot \chi^{-1} \, ,$ $~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}$ $~=$ $~\frac{1}{({\gamma_g}-1)} \int_0^{\chi=R_\mathrm{edge}^*} 4\pi (r^*)^2 P^* dr^*$ $~=$ $~\frac{1}{({\gamma_g}-1)} \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma} \biggl\{ \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma} \int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\}_\mathrm{eq} \cdot \chi^{3-3\gamma} \, ,$

where,

 $~\frac{M_r(x)}{M_\mathrm{tot} }$ $~=$ $~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, .$

We note that, because $~M_r(x)/M_\mathrm{tot} = 1$ in the limit, $~x \rightarrow 1$, we can write,

 $~\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq}$ $~=$ $~ \biggl[ \int_0^{1} 3x^2 \biggl( \frac{\rho(x)}{\rho_c} \biggr) dx \biggr]^{-1} \, ,$

or, if desired, the central-to-mean density ratio in one or both energy terms could be replaced by a term involving the normalized central pressure and the dimensionless equilibrium radius, $~\chi_\mathrm{eq}$, via the relation,

 $\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}$ $~=$ $~\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .$

### Bipolytrope

When considering an isolated, spherically symmetric bipolytropic configuration, each energy term will be made up of separate contributions coming from the core and envelope, that is,

$\mathfrak{G} = (W_\mathrm{grav} + \mathfrak{S}_A)_\mathrm{core} + (W_\mathrm{grav} + \mathfrak{S}_A)_\mathrm{env} \, .$

#### Partitioning the Mass

The core will be principally defined in terms of two dimensionless parameters — $~q$ and $~\nu$ — which are, respectively, the core's radius relative to the bipolytrope's total radius, and the core's mass relative to the total mass of the bipolytropic configuration, specifically,

 $~q$ $~\equiv$ $~x_i = \frac{r_i}{R_\mathrm{edge}} \, ,$ $~\nu$ $~\equiv$ $~\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .$

Given the separate (equilibrium) density profiles of the core and the envelope while sticking to the notation used in our introductory discussion, we can write,

 $(\mathrm{For}~0 \leq r^* \leq r_i^*)$       $~M_r$ $~=$ $\biggl( \frac{4\pi}{3} \biggr) R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_0^{r^*} 3 (r^*)^2 \rho_\mathrm{core}^* dr^*$ $~=$ $M_\mathrm{tot} \cdot \chi^3 \int_0^{x} 3 \biggl[ \frac{\rho_\mathrm{core}(x)}{{\bar\rho}_\mathrm{core}} \biggr] \biggl[ \frac{M_\mathrm{core}/(x_i R_\mathrm{edge})^3}{M_\mathrm{tot}/R^3_\mathrm{norm}} \biggr] x^2 dx$ $~=$ $M_\mathrm{tot} \biggl( \frac{\nu}{q^3} \biggr) \int_0^{x} 3 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core} x^2 dx \, ;$ $(\mathrm{For}~r_i^* \leq r^* \leq R_\mathrm{edge}^*)$       $~M_r$ $~=$ $M_\mathrm{core} + \biggl( \frac{4\pi}{3} \biggr) R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_{r_i^*}^{r^*} 3 (r^*)^2 \rho_\mathrm{env}^* dr^*$ $~=$ $M_\mathrm{core} + M_\mathrm{tot} \cdot \chi^3 \int_{x_i}^{x} 3 \biggl[ \frac{\rho_\mathrm{env}(x)}{{\bar\rho}_\mathrm{env}} \biggr] \biggl\{ \frac{M_\mathrm{env}/[ (1-x_i^3) R_\mathrm{edge}^3]}{M_\mathrm{tot}/R^3_\mathrm{norm}} \biggr\} x^2 dx$ $~=$ $M_\mathrm{tot} \biggl\{\nu + \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{x_i}^{x} 3 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env} x^2 dx \biggr\} \, .$

EXAMPLES

#### Separate Contributions to Gravitational Potential Energy

Given the separate (equilibrium) density and $~M_r$ profiles of the core and the envelope while sticking to the notation used in our introductory discussion, we can write,

 $~W_\mathrm{grav}\biggr|_\mathrm{core}$ $~=$ $- E_\mathrm{norm} \int_0^{r_i^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*$ $~=$ $- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{\nu}{q^3} \biggr) \int_0^{x_i} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core} dx$ $~W_\mathrm{grav}\biggr|_\mathrm{env}$ $~=$ $- E_\mathrm{norm} \int_{r_i^*}^{\chi = R_\mathrm{edge}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*$ $~=$ $- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{x_i}^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env} dx$

EXAMPLES

#### Separate Thermodynamic Energy Reservoirs

In our introductory discussion of the free energy function for spherically symmetric configurations, we developed expressions that define the separate contributions to the thermodynamic energy reservoir that pertain to the core and envelope of bipolytropic configurations. In that discussion we pointed out that, in general for the core, the pressure drops monotonically from a value of $~P_0$ at the center of the configuration according to an expression of the form,

$~P_\mathrm{core}(x) = P_0 [1 - p_c(x)]$      for      $~0 \leq x \leq q \, ,$

and that, for the envelope, the pressure drops monotonically from a value of $~P_{ie}$ at the interface according to an expression of the form,

$~P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]$      for      $~q \leq x \leq 1 \, ,$

where $~p_c(x)$ and $~p_e(x)$ are both dimensionless functions that will depend on the equations of state that are chosen for the core and envelope, respectively. By prescription, the pressure in the envelope must drop to zero at the surface of the bipolytropic configuration, hence, we should expect that $~p_e(1) = 1$. Furthermore, by prescription, the pressure in the core will drop to a value, $~P_{ic}$, at the interface, so we can write,

$~P_{ic} = P_0 [1 - p_c(q)] \, .$

In equilibrium — that is, when $~R_\mathrm{edge} = R_\mathrm{eq}$ — we will demand that the pressure at the interface be the same, whether it is referenced in the core or in the envelope, that is, we will demand that $~P_{ic} = P_{ie} \, .$ It is therefore strategically advantageous to rewrite the expression for the run of pressure through the core in terms of the pressure at the interface rather than in terms of the central pressure; specifically,

$~P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .$

 KEY QUESTION: How should piecewise pressure be normalized? Here we will operate under two constraints when relating the central pressure, $~P_0$, to the core pressure at the interface, $~P_{ic}$. First, the pressure in the core will drop monotonically from its central value according to the following very general prescription, $~P_\mathrm{core}(x) = P_0 [1 - p_c(x)] \, .$ Second, it is the pressure at the interface, $~P_{ic} = P_\mathrm{core}(q)$, that will be determined by the virial equilibrium condition. There are two possible ways to determine the central pressure from knowledge of $~P_{ic}$, but which is the physically correct method to embrace? Case A: Addition … $~P_0 = P_{ic} + \Delta P \, ,$     where     $\Delta P \equiv P_0 p_c(q) \, ,$ in which case, we should write that, $~P_\mathrm{core}(x) = ( P_{ic} + \Delta P ) - P_0 p_c(x) \, .$ Case B: Multiplication … $~P_{ic} = P_0 [1-p_c(q)] ~~~~\Rightarrow~~~~~ P_0 = \frac{P_{ic}}{[1-p_c(q)]} \, ,$ in which case, we should write that, $~P_\mathrm{core}(x) = P_{ic} \biggl[ \frac{ 1-p_c(x) }{ 1-p_c(q) } \biggr] \, .$ Prior to August 2014, we have been naively implementing "Case A," effectively assuming that the quantity, $~\Delta P$ (as well as $~P_{ic}$), is held fixed as we search for the equilibrium value of $~P_0$. See, for example, the comment dated 12 February 2014 in connection with my discussions with Kundan Kadam, or even the "new derivation" summarized in the table below, where we have set, $~\Delta P = \Pi q^2 \, .$ But we now suspect that "Case B" is the proper approach to embrace because, once the parameter $~q$ has been specified, it allows for the function, $~P_\mathrm{core}(x)$, to scale with the system size in exactly the same way as the interface pressure scales with size.

With these generic expressions for the pressure profile in hand, the separate components of the thermodynamic energy reservoir derived in our introductory discussion are,

 $~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}$ $~=$ $\frac{4\pi }{({\gamma_c}-1)} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} \int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx$ $~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}$ $\frac{4\pi }{({\gamma_e}-1)} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_e} \int_q^1 \biggl[1 - p_e(x) \biggr] x^2 dx \, .$

EXAMPLES

## Generalized Free-Energy Expression

Bringing all of these expressions together, the normalized free-energy function for bipolytropes is,

 $~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}$ $~=$ $~ \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env} + \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}$ $~=$ $~ -3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}_\mathrm{core}}{(1-\gamma_c)} \chi^{3-3\gamma_c} - \frac{\mathcal{B}_\mathrm{env}}{(1-\gamma_e)} \chi^{3-3\gamma_e} \, ,$

where,

 $~\mathcal{A}$ $~\equiv$ $\biggl( \frac{\nu}{q^3} \biggr) \int_0^{q} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core} x dx ~+ \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{1} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env} x dx \, ,$ $~\mathcal{B}_\mathrm{core}$ $~\equiv$ $\frac{4\pi}{3} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q 3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \, ,$ $~\mathcal{B}_\mathrm{env}$ $~\equiv$ $\frac{4\pi}{3} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_q^1 3\biggl[1 - p_e(x) \biggr] x^2 dx \, .$

ASIDE: In some of our older derivations, the function names $~s_\mathrm{core}$ and $~s_\mathrm{env}$ were introduced as a shorthand notation. When referenced to our present, broad treatment of the free-energy function for bipolytropes, we note that,

 $~q^3 s_\mathrm{core}$ $~=$ $~\int_0^q 3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \, ,$ $~(1-q^3) s_\mathrm{env}$ $~=$ $~\int_q^1 3\biggl[1 - p_e(x) \biggr] x^2 dx \, .$

We also previously adopted the coefficient notation $~\mathcal{B}$ (with no subscript) for what is now called $~\mathcal{B}_\mathrm{core}$, and we used $~\mathcal{C}$ for what is now labeled $~\mathcal{B}_\mathrm{env}$.

## Extrema and Virial Equilibrium

Extrema arise in the free-energy function wherever,

 $~\frac{\partial \mathfrak{G}^*}{\partial \chi}$ $~=$ $~0 \, ,$

that is, when,

 $~3\mathcal{A}\chi^{-2} - 3\mathcal{B}_\mathrm{core} \chi^{2-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi^{2-3\gamma_e}$ $~=$ $~0 \, .$

Values of the dimensionless variable, $~\chi$, that provide solutions to this algebraic equation identify the size of equilibrium configurations and will henceforth be labeled with the "eq" subscript, that is,

 $~\chi ~~~ \rightarrow ~~~ \chi_\mathrm{eq}$ $~=$ $~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .$

### Virial Theorem

We can rewrite the equilibrium condition as,

 $~0$ $~=$ $~- \chi_\mathrm{eq}^{-1} \biggl[ -3\mathcal{A}\chi_\mathrm{eq}^{-1} - 3\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{3-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{3-3\gamma_e} \biggr]_\mathrm{eq}$ $~=$ $~- \frac{\chi_\mathrm{eq}^{-1} }{E_\mathrm{norm}} \biggl[ ( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env} + 3 (\gamma_c-1) ( \mathfrak{S}_A )_\mathrm{core} + 3 (\gamma_e-1) ( \mathfrak{S}_A)_\mathrm{env} \biggr]_\mathrm{eq} \, .$

Drawing from our introductory discussion of the reservoir of thermodynamic energy, we note that, for adiabatic systems, $~\mathfrak{G}_A$ is equivalent to the internal energy of the system and therefore its relationship to the thermal energy, $~S_\mathrm{therm}$, is,

$~\mathfrak{G}_A = \frac{2}{3(\gamma-1)} S_\mathrm{therm} \, .$

(This applies separately for the core and the envelope.) We therefore recognize that our derived expression for equilibrium systems is none other than the virial theorem applied to bipolytropic configurations, specifically, in equilibrium,

 $~ ( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env} + 2 [ ( S_\mathrm{therm} )_\mathrm{core} + ( S_\mathrm{therm} )_\mathrm{env} ]$ $~=$ $~0 \, .$

Example Bipolytrope Virial Theorem

Virial theorem for $~(n_c, n_e) = (0, 0)$ bipolytrope:

In equilibrium, we will demand that $~P_{ie} = P_{ic}$ and we will set $~\chi \rightarrow \chi_\mathrm{eq}$. Hence,

 $~\frac{S_\mathrm{therm}}{E_\mathrm{norm}}$ $~=$ $\biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr)_\mathrm{env}$ $~=$ $\frac{3(\gamma_c-1)}{2}\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \frac{3(\gamma_e-1)}{2} \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}$ $~=$ $2\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3-3\gamma_c} \biggl\{ \biggl( \frac{P_0}{P_{ic}} \biggr) \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] \biggr\}$ $~+~ 2\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3-3\gamma_e} \biggl\{ (1-q^3) + b_\xi \biggl(\frac{P_0}{P_{ic} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\}$ $~=$ $2\pi \biggl[ \frac{P_{0}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3} \biggl\{ \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] + \frac{P_{ic} }{P_0} (1-q^3) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr\}$ $~=$ $2\pi \biggl[ \frac{P_{0}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3} \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 + (1- b_\xi q^2) (1-q^3) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr]$ $~=$ $2\pi \biggl[ \frac{P_{0} R_\mathrm{edge}^4}{P_\mathrm{norm} R_\mathrm{norm}^4} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{-1} \biggl[ 1 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 - b_\xi (q^2-q^5) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr]$ $~=$ $2\pi \biggl[ \frac{P_{0} R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{-1} \biggl[ 1 + b_\xi \biggl(\frac{2}{5} q^5 \mathfrak{F} - \frac{3}{5}q^5 + q^5 - q^2 \biggr) \biggr]$ $~=$ $2\pi \biggl[ \frac{1}{b_\xi} \biggl( \frac{3}{2^3 \pi} \biggr) \frac{\nu^2}{q^6} \biggr] \chi_\mathrm{eq}^{-1} \biggl\{ 1 + b_\xi \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr] \biggr\}$

The virial theorem states that, in equilibrium,

 $~\frac{2S_\mathrm{therm}}{E_\mathrm{norm}}$ $~=$ $~- \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \, ,$

which, in turn, implies,

 $\biggl[ \frac{1}{b_\xi} \biggl( \frac{3}{2} \biggr) \frac{\nu^2}{q^6} \biggr] \chi_\mathrm{eq}^{-1} \biggl\{ 1 + b_\xi \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr] \biggr\}$ $~=$ $\frac{3}{5}\biggl(\frac{\nu^2}{q} \biggr) f(\nu,q) \chi_\mathrm{eq}^{-1}$ $\Rightarrow ~~~~ \frac{1}{b_\xi} + \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr]$ $~=$ $\frac{2}{5}\cdot q^5 f$ $\Rightarrow ~~~~ \frac{1}{b_\xi}$ $~=$ $q^2 + \frac{2}{5}q^5 (f - 1 - \mathfrak{F} ) \, .$

Now, a bit of algebra shows that,

 $~\frac{2}{5}q^5 (f - 1 - \mathfrak{F} )$ $~=$ $~\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ 2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \, .$

Hence, we have,

 $~\frac{1}{b_\xi} = \biggl( \frac{2^3 \pi}{3} \biggr) \frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } \biggl( \frac{q^3}{\nu}\biggr)^2$ $~=$ $~q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ 2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr]$ $\Rightarrow ~~~~ \frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 }$ $~=$ $~ \biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl\{ q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ 2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \biggr\} \, .$

This exactly matches the equilibrium relation that was derived from our detailed force-balance analysis of $~(n_c, n_e) = (0, 0)$ bipolytropes.

### More Utilitarian Form

Multiplying the equilibrium condition through by $~(\chi^2/3)$ — and appending the "eq" suffix to $~\chi$, throughout — gives,

 $~\mathcal{A}$ $~=$ $~\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{4-3\gamma_c} + \mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{4-3\gamma_e} \, .$

Inserting the generic definitions of the coefficients $~\mathcal{B}_\mathrm{core}$ and $~\mathcal{B}_\mathrm{env}$ — expressed in shorthand notation as referenced above — and demanding that the interface pressures be identical, gives,

 $~\mathcal{A}$ $~=$ $~\biggl\{ \frac{4\pi}{3} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} q^3 s_\mathrm{core} \biggr\} \chi_\mathrm{eq}^{4-3\gamma_c} + \biggl\{ \frac{4\pi}{3} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} (1-q^3) s_\mathrm{env} \biggr\} \chi_\mathrm{eq}^{4-3\gamma_e}$ $~=$ $~\frac{4\pi}{3} \biggl[ \frac{P_i \chi^4}{P_\mathrm{norm}} \biggr]_\mathrm{eq} [ q^3 s_\mathrm{core} + (1-q^3) s_\mathrm{env} ]$ $~=$ $~\frac{4\pi}{3} \biggl[ \frac{P_i R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} [ q^3 s_\mathrm{core} + (1-q^3) s_\mathrm{env} ] \, .$

EXAMPLES