SSC/Structure/BiPolytropes/Analytic51

BiPolytrope with n_c = 5 and n_e = 1

Part I:  (nc,ne) = (5,1) BiPolytrope

Part II:  Example Models

Part III:  Limiting Mass

Part IV:  Free Energy

Eggleton, Faulkner & Cannon (1998) Analytic (nc, ne) = (5, 1)

Comment by J. E. Tohline on 30 March 2013: As far as I have been able to determine, this analytic structural model has not previously been published in a refereed, archival journal. Subsequent comment by J. E. Tohline on 23 June 2013: Last night I stumbled upon an article by Eagleton, Faulkner, and Cannon (1998) in which this identical analytically definable bipolytrope has been presented. Insight drawn from this article is presented in an additional subsection, below.

Here we construct a bipolytrope in which the core has an $n_c=5$ polytropic index and the envelope has an $n_e=1$ polytropic index. This system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. In deriving the properties of this model, we will follow the general solution steps for constructing a bipolytrope that we have outlined elsewhere.  
 
 
 

Steps 2 & 3

Based on the discussion presented elsewhere of the structure of an isolated $n=5$ polytrope, the core of this bipolytrope will have the following properties:

$\theta (\xi )=[1+\frac{1}{3}{\xi }^2{]}^{-1/2}\Rightarrow {\theta }_i=[1+\frac{1}{3}{\xi }_i^2{]}^{-1/2};$

$\frac{d\theta }{d\xi }=-\frac{\xi }{3}[1+\frac{1}{3}{\xi }^2{]}^{-3/2}\Rightarrow (\frac{d\theta }{d\xi }{)}_i=-\frac{{\xi }_i}{3}[1+\frac{1}{3}{\xi }_i^2{]}^{-3/2}.$

The first zero of the function $\theta (\xi )$ and, hence, the surface of the corresponding isolated $n=5$ polytrope is located at ${\xi }_s=\mathrm{\infty }$. Hence, the interface between the core and the envelope can be positioned anywhere within the range, $0<{\xi }_i<\mathrm{\infty }$.

Step 4: Throughout the core (0 ≤ ξ ≤ ξ_i)

Specify: $K_c$ and ${\rho }_0\Rightarrow$
$\rho$	$=$	${\rho }_0{\theta }^{n_c}$	$=$	${\rho }_0(1+\frac{1}{3}{\xi }^2{)}^{-5/2}$
$P$	$=$	$K_c{\rho }_0^{1+1/n_c}{\theta }^{n_c+1}$	$=$	$K_c{\rho }_0^{6/5}(1+\frac{1}{3}{\xi }^2{)}^{-3}$
$r$	$=$	$[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{3}{2\pi }{)}^{1/2}\xi$
$M_r$	$=$	$4\pi \left[\frac{(n_c+1)K_c}{4\pi G}{]}^{3/2}{\rho }_0^{(3-n_c)/(2n_c)}\right(-{\xi }^2\frac{d\theta }{d\xi })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$

Equations (A2) from 📚 Eggleton, Faulkner, & Cannon (1998) present the same relations but adopt the following notations:* $P$ $=$ $p_0{\theta }^6$         $\Rightarrow$ $K_c$ $=$ $p_0{\rho }_0^{-6/5};$ $\rho$ $=$ ${\rho }_0{\theta }^5$       where: $\theta$ $=$ $a_{\mathrm{e}\mathrm{f}\mathrm{c}}[a_{\mathrm{e}\mathrm{f}\mathrm{c}}^2+r^2{]}^{-1/2}=[1+(\frac{r}{a_{\mathrm{e}\mathrm{f}\mathrm{c}}}{)}^2{]}^{-1/2}$ $a_{\mathrm{e}\mathrm{f}\mathrm{c}}^2$ $\equiv$ $\frac{18p_0}{4\pi G{\rho }_0^2}$         $\Rightarrow$ $a_{\mathrm{e}\mathrm{f}\mathrm{c}}^2$ $=$ $\frac{18K_c{\rho }_0^{6/5}}{4\pi G{\rho }_0^2}=3[\frac{K_c}{G{\rho }_0^{4/5}}\cdot (\frac{3}{2\pi }\left)\right]$           $\Rightarrow$ $a_{\mathrm{e}\mathrm{f}\mathrm{c}}$ $=$ $\sqrt{3}\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{3}{2\pi }{)}^{1/2}=\sqrt{3}\left(\frac{r}{\xi }\right)$           $\Rightarrow$ $\frac{r}{a_{\mathrm{e}\mathrm{f}\mathrm{c}}}$ $=$ $\frac{\xi }{\sqrt{3}};$ Hence, $\theta$ $=$ $[1+\frac{{\xi }^2}{3}{]}^{-1/2},$ which matches our expression for the core's polytrope function, $\theta$. Now, look at the EFC98 expression for the core's integrated mass. $M_r$ $=$ $\frac{4\pi {\rho }_0{a}_{\mathrm{e}\mathrm{f}\mathrm{c}}^3}{3}\cdot \frac{r^3}{(a_{\mathrm{e}\mathrm{f}\mathrm{c}}^2+r^2{)}^{3/2}}$   $=$ $\frac{4\pi {\rho }_0}{3}\left[a_{\mathrm{e}\mathrm{f}\mathrm{c}}^3\right](r/a_{\mathrm{e}\mathrm{f}\mathrm{c}}{)}^3[1+(r/a_{\mathrm{e}\mathrm{f}\mathrm{c}}{)}^2{]}^{-3/2}$   $=$ $\frac{4\pi {\rho }_0}{3}[\frac{K_c}{G{\rho }_0^{4/5}}\cdot (\frac{3^2}{2\pi }\left){]}^{3/2}\frac{{\xi }^3}{3^{3/2}}\right[1+\frac{{\xi }^2}{3}{]}^{-3/2}$   $=$ $\frac{4\pi }{3}\left(\frac{3}{2\pi }{)}^{3/2}\right[\frac{K_c^3}{G^3}\cdot \frac{{\rho }_0^2}{{\rho }_0^{12/5}}{]}^{1/2}{\xi }^3[1+\frac{{\xi }^2}{3}{]}^{-3/2}$   $=$ $\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}\right]\left(\frac{2\cdot 3}{\pi }{)}^{1/2}{\xi }^3\right(1+\frac{{\xi }^2}{3}{)}^{-3/2}$ This expression matches ours.

Step 5: Interface Conditions

			Setting $n_c=5$, $n_e=1$, and ${\varphi }_i=1\Rightarrow$
$\frac{{\rho }_e}{{\rho }_0}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{n_c}{\varphi }_i^{-n_e}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5$
$\left(\frac{K_e}{K_c}\right)$	$=$	${\rho }_0^{1/n_c-1/n_e}(\frac{{\mu }_e}{{\mu }_c}{)}^{-(1+1/n_e)}{\theta }_i^{1-n_c/n_e}$	$=$	${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}$
$\frac{{\eta }_i}{{\xi }_i}$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^{(n_c-1)/2}{\varphi }_i^{(1-n_e)/2}$	$=$	$3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2$
$(\frac{d\varphi }{d\eta }{)}_i$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}{\theta }_i^{-(n_c+1)/2}{\varphi }_i^{(n_e+1)/2}\right(\frac{d\theta }{d\xi }{)}_i$	$=$	$3^{1/2}{\theta }_i^{-3}(\frac{d\theta }{d\xi }{)}_i$

Step 6: Envelope Solution

Adopting equation (8) of 📚 M. Beech (1988, Astrophysics and Space Science, Vol. 147, issue 2, pp. 219 - 227), the most general solution to the $n=1$ Lane-Emden equation can be written in the form,

$\varphi =A\left[\frac{\sin (\eta -B)}{\eta }\right],$

where $A$ and $B$ are constants. The first derivative of this function is,

$\frac{d\varphi }{d\eta }=\frac{A}{{\eta }^2}[\eta \cos (\eta -B)-\sin (\eta -B)].$

Equations (A2) from 📚 Eggleton, Faulkner, & Cannon (1998) present the same relations but adopt the following notations:* $P$ $=$ $p_0{\theta }^6$         $\Rightarrow$ $K_c$ $=$ $p_0{\rho }_0^{-6/5};$

From Step 5, above, we know the value of the function, $\varphi$ and its first derivative at the interface; specifically,

${\varphi }_i=1\mathrm{a}\mathrm{n}\mathrm{d}(\frac{d\varphi }{d\eta }{)}_i=3^{1/2}{\theta }_i^{-3}(\frac{d\theta }{d\xi }{)}_i\mathrm{a}\mathrm{t}{\eta }_i=3^{1/2}{\xi }_i\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2.$

From this information we can determine the constants $A$ and $B$; specifically,

${\eta }_i-B={\tan }^{-1}({\mathrm{\Lambda }}_i^{-1})=\frac{\pi }{2}-{\tan }^{-1}({\mathrm{\Lambda }}_i),$

$A=\frac{{\varphi }_i{\eta }_i}{\sin ({\eta }_i-B)}={\varphi }_i{\eta }_i(1+{\mathrm{\Lambda }}_i^2{)}^{1/2},$

where,

${\mathrm{\Lambda }}_i=\frac{1}{{\eta }_i}+\frac{1}{{\varphi }_i}(\frac{d\varphi }{d\eta }{)}_i.$

Step 7

The surface will be defined by the location, ${\eta }_s$, at which the function $\varphi (\eta )$ first goes to zero, that is,

${\eta }_s=\pi +B=\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i).$

Equations (A2) from 📚 Eggleton, Faulkner, & Cannon (1998) present the same relations but adopt the following notations:* $P$ $=$ $p_0{\theta }^6$         $\Rightarrow$ $K_c$ $=$ $p_0{\rho }_0^{-6/5};$

Step 8: Throughout the envelope (η_i ≤ η ≤ η_s)

					Knowing: $K_e/K_c$ and ${\rho }_e/{\rho }_0$ from Step 5   $\Rightarrow$
$\rho$	$=$	${\rho }_e{\varphi }^{n_e}$	$=$	${\rho }_0\left(\frac{{\rho }_e}{{\rho }_0}\right)\varphi$	$=$	${\rho }_0\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi$
$P$	$=$	$K_e{\rho }_e^{1+1/n_e}{\varphi }^{n_e+1}$	$=$	$K_c{\rho }_0^{6/5}\left(\frac{K_e{\rho }_0^{4/5}}{K_c}\right)(\frac{{\rho }_e}{{\rho }_0}{)}^2{\varphi }^2$	$=$	$K_c{\rho }_0^{6/5}{\theta }_i^6{\varphi }^2$
$r$	$=$	$[\frac{(n_e+1)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(1-n_e)/(2n_e)}\eta$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{K_e{\rho }_0^{4/5}}{K_c}{)}^{1/2}(2\pi {)}^{-1/2}\eta$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
$M_r$	$=$	$4\pi \left[\frac{(n_e+1)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{(3-n_e)/(2n_e)}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{K_e{\rho }_0^{4/5}}{K_c}{)}^{3/2}\left(\frac{{\rho }_e}{{\rho }_0}\right)\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$

An examination of their equations (A3) reveals that EFC98 continue to use $\theta$ to represent the polytropic function throughout the envelope — for clarity, we will attach the subscript ${\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}$ — whereas we use $\varphi$. Henceforth we will assume that these functions are interchangeable, that is, ${\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}\leftrightarrow \varphi$, and examine whether or not their various physical parameter expressions match ours. Comment by J. E. Tohline: As detailed in the text, there appears to be a type-setting error in both of these expressions; as published by EFC98, the exponent on the coefficient of theta_i should be 6 and 5, respectively, whereas it appears as 4. In their Eqs. (A3), EFC98 state that, throughout the envelope, $P$ $=$ $p_0{\theta }_c^4{\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}^2=K_c{\rho }_0^{6/5}{\theta }_i^4{\varphi }^2$         and, $\rho$ $=$ $\frac{{\rho }_0}{\alpha }\cdot {\theta }_c^4{\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}={\rho }_0\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^4\varphi ,$ where, in both expressions, we have replaced their label for the value of the polytropic function at the core-envelope interface, ${\theta }_c$, with our label, ${\theta }_i$. Both of their expressions match ours EXCEPT … NOTE:   in both of their expressions, ${\theta }_i$ is raised to the 4th power whereas, according to our derivation, this interface value should be raised to the 6th power in the expression for pressure and it should be raised to the 5th power in the expression for the density. We are confident that our expressions are the correct ones and therefore presume that type-setting errors are present in both of the EFC98 expressions. We state that the envelope's polytropic function has the form, $\varphi$ $=$ $-\frac{A}{\eta }\cdot \sin (B-\eta ),$ where, $\eta$ $=$ $\left[\frac{G{\rho }_0^{4/5}}{K_c}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^2(2\pi {)}^{1/2}r.$ EFC98 state that, ${\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}$ $=$ $\frac{B_{\mathrm{e}\mathrm{f}\mathrm{c}}}{r}\cdot \sin [\beta (r_s-r)],$ where, $\beta r$ $=$ $\frac{3{\theta }_c^2}{\alpha }[a_{\mathrm{e}\mathrm{f}\mathrm{c}}{]}^{-1}r$   $=$ $3{\theta }_c^2\left(\frac{{\mu }_e}{{\mu }_c}\right)\left\{\sqrt{3}\right[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}(\frac{3}{2\pi }{)}^{1/2}{\}}^{-1}r$   $=$ $\left[\frac{G{\rho }_0^{4/5}}{K_c}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^2(2\pi {)}^{1/2}r.$ We therefore conclude that $\beta r=\eta$, and $\beta r_s=B$. If, as we assume to be the case, ${\theta }_{\mathrm{e}\mathrm{f}\mathrm{c}}=\varphi$, it must also be the case that, $\frac{B_{\mathrm{e}\mathrm{f}\mathrm{c}}}{r}$ $=$ $-\frac{A}{\eta }$ $\Rightarrow B_{\mathrm{e}\mathrm{f}\mathrm{c}}$ $=$ $-\frac{A}{\beta }.$ Our expression for the integrated mass throughout the envelope is, $M_r$ $=$ $\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right\{-A[\eta \cos (\eta -B)-\sin (\eta -B)]\}$   $=$ $-A\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right[\sin (B-\eta )+\eta \cos (B-\eta )].$ According to EFC98, $M_r$ $=$ $\frac{4\pi {\rho }_0\alpha }{9}\left[B_{\mathrm{e}\mathrm{f}\mathrm{c}}{a}_{\mathrm{e}\mathrm{f}\mathrm{c}}^2\right]\{\sin [\beta (r_s-r)]+\beta r\cos [\beta (r_s-r)]\}$   $=$ $\frac{4\pi {\rho }_0}{9}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right[-\frac{A{a}_{\mathrm{e}\mathrm{f}\mathrm{c}}^2}{\beta }\left]\right[\sin (B-\eta )+\eta \cos (B-\eta )]$   $=$ $-A\cdot \frac{4\pi {\rho }_0}{9}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right[\sin (B-\eta )+\eta \cos (B-\eta )\left]\right\{\left[\frac{G{\rho }_0^{4/5}}{K_c}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^2(2\pi {)}^{1/2}{\}}^{-1}\left\{3\right[\frac{K_c}{G{\rho }_0^{4/5}}\cdot \left(\frac{3}{2\pi }\right)\left]\right\}$   $=$ $-A\cdot \left(\frac{2}{\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-2}[\sin (B-\eta )+\eta \cos (B-\eta )][\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}$

Examples

Normalization

The dimensionless variables used in Tables 1 & 2 are defined as follows:

${\rho }^{*}$	$\equiv$	$\frac{\rho }{{\rho }_0}$	;	$r^{*}$	$\equiv$	$\frac{r}{[K_c^{1/2}/(G^{1/2}{\rho }_0^{2/5})]}$
$P^{*}$	$\equiv$	$\frac{P}{K_c{\rho }_0^{6/5}}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_0^{1/5})]}$
$H^{*}$	$\equiv$	$\frac{H}{K_c{\rho }_0^{1/5}}$	.

Parameter Values

The $2^{\mathrm{n}\mathrm{d}}$ column of Table 1 catalogues the analytic expressions that define various parameters and physical properties (as identified, respectively, in column 1) of the $(n_c,n_e)=(5,1)$ bipolytrope. We have evaluated these expressions for various choices of the dimensionless interface radius, ${\xi }_i$, and have tabulated the results in the last few columns of the table. The tabulated values have been derived assuming ${\mu }_e/{\mu }_c=1$, that is, assuming that the core and the envelope have the same mean molecular weights.

Table 1: Properties of $(n_c,n_3)=(5,1)$ BiPolytrope Having Various Interface Locations, ${\xi }_i$
Accompanying spreadsheet with parameter values

Parameter	${\xi }_i$	0.5	1.0	3.0	For bipolytropic models having ${\mu }_e/{\mu }_c=1.0$, this figure shows how the interface location, ${\eta }_i$ (solid purple curve), the surface radius, ${\eta }_s$ (green circular markers), and the parameter, ${\tan }^{-1}{\mathrm{\Lambda }}_i$ (orange circular markers), vary with ${\xi }_i$ (ordinate) over the range, $0\le {\xi }_i\le 12$. The three horizontal, red-dashed line segments identify the values of ${\xi }_i$ for which numerical values of these (and other) parameters have been listed in the table shown here on the left.
${\theta }_i$	$(1+\frac{1}{3}{\xi }_i^2{)}^{-1/2}$	0.96077	0.86603	0.50000
$-(\frac{d{\theta }_i}{d\xi }{)}_i$	$\frac{1}{3}{\xi }_i(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}$	0.14781	0.21651	0.12500
$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}\equiv r_i^{*}$	$(\frac{3}{2\pi }{)}^{1/2}{\xi }_i$	0.34549	0.69099	2.07297
${\rho }_i^{*}{|}_c=(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\rho }_i^{*}{|}_e$	$(1+\frac{1}{3}{\xi }_i^2{)}^{-5/2}$	0.81864	0.48714	0.03125
$P_i^{*}$	$(1+\frac{1}{3}{\xi }_i^2{)}^{-3}$	0.78653	0.42188	0.01563
$H_i^{*}{|}_c=\frac{n_c+1}{n_e+1}\left(\frac{{\mu }_e}{{\mu }_c}\right)H_i^{*}{|}_e$	$6(1+\frac{1}{3}{\xi }_i^2{)}^{-1/2}$	5.76461	5.19615	3.00000
$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$(\frac{6}{\pi }{)}^{1/2}({\xi }_i{\theta }_i{)}^3$	0.15320	0.89762	4.66417
$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\eta }_i$	$\sqrt{3}{\theta }_i^2{\xi }_i$	0.79941	1.29904	1.29904
$-(\frac{d\varphi }{d\eta }{)}_i$	$\sqrt{3}{\theta }_i^{-3}(-\frac{d\theta }{d\xi }{)}_i=\frac{{\xi }_i}{\sqrt{3}}$	0.28868	0.57735	1.73205
${\mathrm{\Lambda }}_i$	$\frac{1}{{\eta }_i}+(\frac{d\varphi }{d\eta }{)}_i$	0.96225	0.19245	-0.96225
$A$	${\eta }_i(1+{\mathrm{\Lambda }}_i^2{)}^{1/2}$	1.10940	1.32288	1.80278
$B$	${\eta }_i-\frac{\pi }{2}+{\tan }^{-1}({\mathrm{\Lambda }}_i)$	- 0.00523	-0.08163	-1.03792
${\eta }_s$	$\pi +B$	3.13637	3.05996	2.10367
$-(\frac{d\varphi }{d\eta }{)}_s$	$\frac{A}{{\eta }_s}$	0.35372	0.43232	0.85697
$\left(\frac{{\mu }_e}{{\mu }_c}\right)\cdot [R^{*}\equiv r_s^{*}]$	$\frac{{\eta }_s}{\sqrt{2\pi }{\theta }_i^2}$	1.35550	1.62766	3.35697
$(\frac{{\mu }_e}{{\mu }_c}{)}^2{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}$	$\left(\frac{2}{\pi }{)}^{1/2}{\theta }_i^{-1}\right(-{\eta }^2\frac{d\varphi }{d\eta }{)}_s=(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}$	2.88959	3.72945	6.05187
$\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{{\rho }_c}{\overline{\rho }}$	$\frac{{\eta }_s^2}{3A{\theta }_i^5}$	3.61035	4.84326	26.1844
$(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\cdot [\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}]$	$\sqrt{3}\left(\frac{{\xi }_i^3{\theta }_i^4}{A{\eta }_s}\right)$	0.05302	0.24068	0.77070
$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\cdot [q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}]$	$\sqrt{3}\left[\frac{{\xi }_i{\theta }_i^2}{{\eta }_s}\right]$	0.25488	0.42453	0.61751

Alternatively, if given ${\mu }_e/{\mu }_c$ and the value of the parameter, ${\eta }_i$, then we have,

$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\eta }_i$	$=$	$\frac{3^{3/2}{\xi }_i}{3+{\xi }_i^2}$
$\Rightarrow 0$	$=$	${\xi }_i^2-\left[\right(\frac{{\mu }_e}{{\mu }_c}\left)\frac{3^{3/2}}{{\eta }_i}\right]{\xi }_i+3$
$\Rightarrow {\xi }_i$	$=$	$\sqrt{3}\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{3}{2{\eta }_i}\{1\pm \sqrt{1-\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{2{\eta }_i}{3}{]}^2}\}.$

It must be understood, therefore, that the interface location is restricted to the range,

$0$

$\le {\eta }_i\le$

$\frac{3}{2}\left(\frac{{\mu }_e}{{\mu }_c}\right),$

and that this upper limit on ${\eta }_i$ is associated with a model whose core radius is, ${\xi }_i=\sqrt{3}$. Also,

${\mathrm{\Lambda }}_i$

$=$

$\frac{1}{{\eta }_i}-\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{3}{2{\eta }_i}\{1\pm \sqrt{1-\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{2{\eta }_i}{3}{]}^2}\}.$

Profile

Once the values of the key set of parameters have been determined as illustrated in Table 1, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in step #4 and step #8, above. Table 2 summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, ${\rho }^{*}(r^{*})$, the normalized gas pressure, $P^{*}(r^{*})$, and the normalized mass interior to $r^{*}$, $M_r^{*}(r^{*})$. For all profiles, the relevant normalized radial coordinate is $r^{*}$, as defined in the $2^{\mathrm{n}\mathrm{d}}$ row of Table 2. Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of Table 2.

Table 2: Radial Profile of Various Physical Variables

Variable	Throughout the Core $0\le \xi \le {\xi }_i$	Throughout the Envelope† ${\eta }_i\le \eta \le {\eta }_s$	Plotted Profiles
			${\xi }_i=0.5$	${\xi }_i=1.0$	${\xi }_i=3.0$
$r^{*}$	$(\frac{3}{2\pi }{)}^{1/2}\xi$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
${\rho }^{*}$	$(1+\frac{1}{3}{\xi }^2{)}^{-5/2}$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi (\eta )$
$P^{*}$	$(1+\frac{1}{3}{\xi }^2{)}^{-3}$	${\theta }_i^6[\varphi (\eta ){]}^2$
$M_r^{*}$	$\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })$
†In order to obtain the various envelope profiles, it is necessary to evaluate $\varphi (\eta )$ and its first derivative using the information presented in Step 6, above.

[As of 28 April 2013] For the interface locations ${\xi }_i=0.5,1.0,\mathrm{a}\mathrm{n}\mathrm{d}3.0$, Table 2 provides profiles for three values of the molecular weight ratio: ${\mu }_e/{\mu }_c=1.0,1/2,\mathrm{a}\mathrm{n}\mathrm{d}1/4$. In all nine graphs, blue diamonds trace the structure of the $n_c=5$ core; the core extends to a radius, $r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$, that is independent of molecular weight ratio but varies in direct proportion to the choice of ${\xi }_i$. Specifically, as tabulated in the fourth row of Table 1, $r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}=0.34549,0.69099,\mathrm{a}\mathrm{n}\mathrm{d}2.07297$ for, respectively, ${\xi }_i=0.5,1,\mathrm{a}\mathrm{n}\mathrm{d}3$. Notice that, while the pressure profile and mass profile are continuous at the interface for all choices of the molecular weight ratio, the density profile exhibits a discontinuous jump that is in direct proportion to the chosen value of ${\mu }_e/{\mu }_c$.

Throughout the $n_e=1$ envelope, the profile of all physical variables varies with the choice of the molecular weight ratio. In the Table 2 graphs, red squares trace the envelope profile for ${\mu }_e/{\mu }_c=1.0$; green triangles trace the envelope profile for ${\mu }_e/{\mu }_c=1/2$; and purple crosses trace the envelope profile for ${\mu }_e/{\mu }_c=1/4$. The surface of the bipolytropic configuration is defined by the (normalized) radius, $R^{*}$, at which the envelope density and pressure drop to zero; the values tabulated in row 16 of Table 1 — $1.35550,1.62766,\mathrm{a}\mathrm{n}\mathrm{d}3.35697$ for, respectively, ${\xi }_i=0.5,1,\mathrm{a}\mathrm{n}\mathrm{d}3$ — correspond to a molecular weight ratio of unity and, hence also, to the envelope profiles traced by red squares in the Table 2 graphs. As the molecular weight ratio is decreased from unity to $1/2$ and, then, $1/4$ for a given choice of ${\xi }_i$, the (normalized) radius of the bipolytrope increases roughly in inverse proportion to ${\mu }_e/{\mu }_c$ as suggested by the formula for $R^{*}$ shown in Table 1. This proportional relation is not exact, however, because the parameter ${\eta }_s$, which also appears in the formula for $R^{*}$, contains an implicit dependence on the chosen value of the molecular weight ratio through the parameter ${\eta }_i$.

For a given choice of the interface parameter, ${\xi }_i$, the (normalized) mass that is contained in the core is independent of the choice of the molecular weight ratio. However, the (normalized) total mass, $M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}$, varies significantly with the choice of ${\mu }_e/{\mu }_c$; as suggested by the expression provided in row 17 of Table 1, the variation is in rough proportion to $({\mu }_e/{\mu }_c{)}^{-2}$ but, as with $R^{*}$, this proportional relation is not exact because the parameters ${\eta }_s$ and $A$ which also appear in the formula for $M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}$ harbor an implicit dependence on the molecular weight ratio.

Model Sequences

For a given choice of ${\mu }_e/{\mu }_c$ a physically relevant sequence of models can be constructed by steadily increasing the value of ${\xi }_i$ from zero to infinity — or at least to some value, ${\xi }_i\gg 1$. Figure 1 shows how the fractional core mass, $\nu \equiv M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$, varies with the fractional core radius, $q\equiv r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/R$, along sequences having six different values of ${\mu }_e/{\mu }_c$, as detailed in the figure caption. The natural expectation is that an increase in ${\xi }_i$ along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the sequences marked by blue diamonds (${\mu }_e/{\mu }_c=1$) and by red squares (${\mu }_e/{\mu }_c=$½). But the behavior is different along the other four illustrated sequences. For sufficiently large ${\xi }_i$, the relative radius of the core begins to decrease; then, as ${\xi }_i$ is pushed to even larger values, eventually the relative core mass begins to decrease. Additional properties of these equilibrium sequences are discussed in an accompanying chapter.

Figure 1: Analytically determined plot of fractional core mass ($\nu$) versus fractional core radius ($q$) for $(n_c,n_e)=(5,1)$ bipolytrope model sequences having six different values of ${\mu }_e/{\mu }_c$: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, ${\xi }_i$, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.

The variation of $\nu$ with $q$ for a seventh analytically determined model sequence — one for which ${\mu }_e/{\mu }_c=1/5$ — is mapped out by a string of blue diamond symbols in the left-hand side of Figure 2. It behaves in an analogous fashion to the ${\mu }_e/{\mu }_c=$¼ (purple asterisks) sequence displayed in Figure 1. It also quantitatively, as well as qualitatively, resembles the sequence that was numerically constructed by 📚 M. Schönberg & S. Chandrasekhar (1942, ApJ, Vol. 96, pp. 161 - 172) for models with an isothermal core ($n_c=\mathrm{\infty }$) and an $n_e=3/2$ envelope; Fig. 1 from their paper has been reproduced here on the right-hand side of Figure 2.

Figure 2: Relationship to Schönberg-Chandrasekhar Mass Limit
Analytic BiPolytrope with $n_c=5$, $n_e=1$, and ${\mu }_e/{\mu }_c=1/5$	Edited excerpt from Schönberg & Chandrasekhar (1942)	Figure from Henrich & Chandraskhar (1941)
(Above) Plot of fractional core mass ($\nu$) versus fractional core radius ($q$) for the analytic bipolytrope having ${\mu }_e/{\mu }_c=1/5$. The behavior of this analytically defined model sequence resembles the behavior of the numerically constructed isothermal core models presented by (center) 📚 Schönberg & Chandrasekhar (1942) and by (far right) 📚 L. R. Henrich & S. Chandrasekhar (1941, ApJ, Vol. 94, pp. 525 - 536).

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