Editing SSC/Structure/BiPolytropes/Analytic51/Pt3 (section)

=BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1 (Pt 3)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic51|Part I:&nbsp; (n<sub>c</sub>,n<sub>e</sub>) = (5,1) BiPolytrope]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic51/Pt2|Part II:&nbsp; Example Models]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic51/Pt3|Part III:&nbsp; Limiting Mass]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic51/Pt4|Part IV:&nbsp; Free Energy]]
&nbsp;
  </td>
</tr>
</table>

==Limiting Mass==
===Background===
As early as 1941, Chandraskhar and his collaborators realized that the shape of the model sequence in a <math>\nu</math> versus <math>q</math> diagram, as displayed in Figure 2 above, implies that equilibrium structures can exist only if the fractional core mass lies below some limiting value.  This realization is documented, for example, by the following excerpt from &sect;5 of {{ HC41full }}.

<div align="center">
<table border="1" cellpadding="5" width="60%">
<tr>
  <td align="center" colspan="1">
Text excerpt from &sect;5 (pp. 532 - 533) of<br />{{ HC41figure }}
  </td>
</tr>

<tr>
  <td align="left" colspan="1">
<!-- [[Image:HenrichChandra41a.jpg|600px|center|HenrichChandra1941]] -->
<!-- [[Image:AAAwaiting01.png|600px|center]] -->
<font color="darkgreen">"&hellip; at a fixed central temperature, the fraction of the total mass, <math>\nu</math>, contained in the core increases slowly at first and soon very rapidly as <math>q</math> approaches <math>q_\mathrm{max}</math>.  However, this increase of <math>\nu</math> does not continue indefinitely; <math>\nu</math> soon attains a maximum value <math>\nu_\mathrm{max}</math>.  There exists, therefore, an upper limit to the mass which can be contained in the isothermal core."</font>
  </td>
</tr>

</table>
</div>

Given that our bipolytropic sequence has been defined analytically, it may be possible to analytically determine the limiting core mass of our model.  In order to accomplish this, we need to identify the point along the sequence &#8212; in particular, the value of the dimensionless interface location &#8212; at which <math>d\nu/dq = 0</math> or, equivalently, <math>d\nu/d\xi_i = 0</math>.

Before carrying out the desired differentiations, we will find it useful to rewrite the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>

We obtain,

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>m_3 \biggl( \frac{\ell_i}{1 + \ell_i^2} \biggr) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2]  ~~~ \Rightarrow ~~~ 
</math>
&nbsp;&nbsp;'''<font color="red">Believe it or not &hellip; &nbsp;</font>'''
<math>
(1 + \Lambda^2) = \frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2}{1 + \ell_i^2} \biggr]^{1/2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\pi}{2} \biggr)^{1/2} \frac{M_\mathrm{core}}{9}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\ell_i^3}{(1 + \ell_i^2)^{3/2}} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\pi}{2} \biggr)^{1/2} \frac{M_\mathrm{tot}}{9}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{1/2}
\biggl\{  \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) + m_3 \ell_i (1 + \ell_i^2)^{-1} \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

Hence,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} 
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2}  \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1}
</math>
</div>

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
An interesting limiting case is <math>m_3 = 1</math>, in which case,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\nu
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
(\ell_i^3) (1 + \ell_i^2)^{-1/2}  \biggl[ \ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \biggl(\frac{1}{\ell_i}\biggr) \biggr) \biggr]^{-1} \, ,</math>
  </td>
</tr>
</table>
and the maximum value of <math>\nu</math> along this sequence arises when <math>\ell_i \rightarrow \infty</math>, in which case,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\nu
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\ell_i^2   \biggl[ \ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2}  \biggr) \biggr]^{-1} 
\rightarrow
\frac{2}{\pi}
\, .</math>
  </td>
</tr>
</table>
</td></tr></table>


The condition, <math>d\nu/d\xi_i = 0</math>, also will be satisfied if the condition,
<div align="center">
<math>
\frac{d\ln\nu}{d\ln\ell_i} = 0 \, ,
</math>
</div>
is met.

===Derivation===
My manual derivation gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
(1+\ell_i^2)  \biggl[ \frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr]
\biggl\{ 3  - \frac{(1-m_3)^2 \ell_i^2 (1+\ell_i^2)}{[ 1 + (1-m_3)^2 \ell_i^2 ]} \biggr\} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> (1+\ell_i^2)
\frac{\partial \tan^{-1}\Lambda_i}{\partial \ln \ell_i} 
-m_3\ell_i \biggl\{ \ell_i^2 + 2  - \frac{(1-m_3)^2 \ell_i^2 (1+\ell_i^2)}{[ 1 + (1-m_3)^2 \ell_i^2 ]} \biggr\}
</math> 
  </td>
</tr>
</table>
</div>

where,

<div align="center">
<math>
\frac{\partial \tan^{-1}\Lambda_i}{\partial \ln \ell_i} =  
\frac{[(1-m_3)\ell_i^2 - 1 ] }{m_3\ell_i (1 + \Lambda_i^2)}
= \frac{m_3 \ell_i [(1-m_3)\ell_i^2 - 1 ]}{(1 + \ell_i^2) [ 1 + (1-m_3)^2 \ell_i^2]}  \, .
</math>
</div>

Upon rearrangement, this gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
(1+\ell_i^2)  \biggl[ \frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr]
\biggl\{ 3[ 1 + (1-m_3)^2 \ell_i^2 ]  - (1-m_3)^2 \ell_i^2 (1+\ell_i^2) \biggr\} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
 m_3\ell_i \biggl\{[(1-m_3)\ell_i^2 - 1 ] -(\ell_i^2 + 2)[ 1 + (1-m_3)^2 \ell_i^2 ]  + (1-m_3)^2 \ell_i^2 (1+\ell_i^2) \biggr\} \, ,
</math> 
  </td>
</tr>
</table>
</div>


and further simplification <font color="red">[completed on 19 May 2013]</font> gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math> 
  </td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center">
<math>\xi_i</math>
  </td>
  <td align="center">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
LHS
  </td>
  <td align="center">
RHS
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.0719668  </td>
  <td align="right">
0.0710624  </td>
  <td align="right">
0.2128753  </td>
  <td align="right">
0.0726547  </td>
  <td align="right">
1.8516032  </td>
  <td align="right">
-223.8157  </td>
  <td align="right">
-223.8159  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.1591479  </td>
  <td align="right">
0.1493938  </td>
  <td align="right">
0.4903393  </td>
  <td align="right">
0.1663869  </td>
  <td align="right">
2.1760793  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.1924833  </td>
  <td align="right">
0.1750954  </td>
  <td align="right">
0.6130669  </td>
  <td align="right">
0.2053811  </td>
  <td align="right">
2.2958639  </td>
  <td align="right">
-18.47809  </td>
  <td align="right">
-18.47808  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.3309933  </td>
  <td align="right">
0.2342522  </td>
  <td align="right">
1.4179907  </td>
  <td align="right">
0.4064595  </td>
  <td align="right">
2.761622  </td>
  <td align="right">
-2.601255  </td>
  <td align="right">
-2.601257  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
</tr>

<tr>
  <td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
  </td>
</tr>
</table>

===Limit when m<sub>3</sub> = 0===
It is instructive to examine the root of this equation in the limit where <math>m_3 = 0</math> &#8212; that is, when <math>\mu_e/\mu_c = 0</math>.  First, we note that,
<div align="center">
<math>\Lambda_i\biggr|_{m_3 \rightarrow 0} = \biggl\{ \frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2] \biggr\}_{m_3 \rightarrow 0} = \infty \, .</math> 
</div>
Hence,
<div align="center">
<math>\biggl[\tan^{-1}\Lambda_i\biggr]_{m_3 \rightarrow 0} = \frac{\pi}{2} \, ,</math> 
</div>
and the limiting relation becomes,
<div align="center">
<math>
\pi (1+\ell_i^2) [ 3 + (2-\ell_i^2)\ell_i^2] = 0 \, ,
</math>
</div>
or, more simply,
<div align="center">
<math>
\ell_i^4 - 2\ell_i^2 - 3 = 0 \, .
</math>
</div>
The real root is,
<div align="center">
<math>\ell_i^2 = \frac{1}{2} \biggl[ 2 + \sqrt{4 + 12} \biggr] = 3 ~~~~ \Rightarrow ~~~~ \xi_i = 3 \, .</math>
</div>
For <math>\xi_i = 3</math>, the radius of the core, the mass of the core, and the pressure at the edge of the core are, respectively,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>r^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>r_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} \biggl[ \frac{K_c^{1/2}}{G^{1/2} \rho_0^{2/5}} \biggr] \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>M^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} \rho_0^{1/5}} \biggr] \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>P^*_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>P_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} [ K_c \rho_0^{6/5}] \, .</math>
  </td>

</tr>
</table>
</div>

If we invert the middle expression to obtain <math>\rho_0</math> in terms of <math>M_\mathrm{core}</math>, specifically,
<div align="center">
<math>\rho_0^{1/5} = \biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr] \, ,</math> 
</div>
then we can rewrite <math>r_\mathrm{core}</math> and <math>P_i</math> in terms of, respectively, the ''reference'' radius,
<math>R_\mathrm{rf}</math>, and reference pressure, <math>P_\mathrm{rf}</math>, as defined in 
[[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|our discussion of isolated <math>n=5</math>
polytropes embedded in an external medium]].  Specifically, we obtain,
<div align="center">
<table border="1" cellpadding="5">
<tr>

  <td align="right">
<math>r_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} \biggl[ \frac{K_c^{1/2}}{G^{1/2} } \biggr] \biggl(\frac{3^7}{2^5\pi}\biggr)^{-1}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr]^{-2}</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2^9 \pi}{3^{11}}\biggr)^{1/2} \biggl[ \frac{G^{5/2} M^2_\mathrm{core}}{K_c^{5/2}} \biggr]</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2^9 \pi}{3^{11}}\biggr)^{1/2} \frac{3^3}{2^6} \biggl( \frac{5^5}{\pi} \biggr)^{1/2} R_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{5^5}{2^3 \cdot 3^5} \biggr)^{1/2} R_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

</tr>
<tr>

  <td align="right">
<math>P_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} [ K_c ] \biggl(\frac{3^7}{2^5\pi}\biggr)^{3}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr]^6 </math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^{7}\pi}\biggr)^{3}  \biggl[ \frac{K_c^{10}}{G^{9} M^6_\mathrm{core}} \biggr] </math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^7\pi}\biggr)^{3}  \biggl( \frac{2^{26} \pi^3}{3^{12} 5^9} \biggr) P_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2^{5}\cdot 3^9 }{5^9} \biggr) P_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

</tr>
</table>
</div>

['''<font color="red">26 May 2013</font>''' with further elaboration on '''<font color="red">28 May 2013</font>'''] This is the same result that was obtained when we
[[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|embedded an isolated <math>n=5</math> polytrope in an external medium]].  Apparently, therefore, the physics that leads to the mass limit for a [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert sphere]] is the same physics that sets the {{ SC42 }} mass limit.

==Derivation by Eggleton, Faulkner, and Cannon (1998)==

The analytically prescribable sequence of bipolytropic models having <math>(n_c, n_e) = (5, 1)</math> displays an interesting behavior that extends beyond identification of a Sch&ouml;nberg-Chandrasekhar-like mass limit. After reaching a maximum value of <math>q</math> but before reaching the maximum value of <math>\nu</math>, the sequence bends back on itself.  This means that, even though the fraction of mass enclosed in the core is steadily increasing, the total radius of the configuration is increasing faster than the radius of the core.  Qualitatively, at least, this mimics the behavior exhibited by normal stars as they evolve off the main sequence and up the red giant branch.

As I pondered whether or not to probe this analogy in more depth, I recalled &#8212; even dating back to my years as a graduate student at Lick Observatory &#8212; hearing John Faulkner profess that he finally understood why stars become red giants.  I also recalled the following passage from  [<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>]'s textbook on ''Stellar Interiors'':
<table align="center" width="75%" border="1" cellpadding="10">
<tr>
  <td align="center">
Excerpt from &sect;2.3, p. 55 of [<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>]
  </td>
</tr>
<tr>
  <td align="left">
<font color="darkgreen">"The enormous increase in radius that accompanies hydrogen exhaustion moves the star into the red giant region of the H-R diagram.  While the transition to red giant dimensions is a fundamental result of all evolutionary calculations, a convincing yet intuitively satisfactory explanation of this dramatic transformation has not been formulated.  Our discussion of this phenomenon follows that of [http://adsabs.harvard.edu/abs/1984PhR...105..329I Iben and Renzini (1984)] although we must state that it is not the whole story."</font><sup>&dagger;</sup>
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<sup>&dagger;</sup><font color="darkgreen">"Other attempts include:  {{ EF81 }}; [http://adsabs.harvard.edu/abs/1983A%26A...127..411W Weiss (1983)]; [http://adsabs.harvard.edu/abs/1985ApJ...296..554Y Yahil &amp; Van den Horn (1985)]; [http://adsabs.harvard.edu/abs/1988ApJ...329..803A Applegate (1988)]; [http://adsabs.harvard.edu/abs/1989MNRAS.236..505W Whitworth (1989)]; [http://adsabs.harvard.edu/abs/1992ApJ...400..280R Renzini et al. (1992)]. [http://adsabs.harvard.edu/abs/1991ApJ...372..592B Bhaskar &amp; Nigam (1991)] use an interesting set of dimensional arguments plus notions from polytrope theory.  We suspect the answers may lie in their paper but someone has yet to come along and translate the mathematics into an easily comprehensible physical picture."</font>
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While examining the set of authors who more recently have cited the work by {{ EF81 }}, I discovered a paper by {{ EFC98full }} with the following abstract:

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[[Image:EagletonFaulknerCannon98.jpg|600px|center|Eggleton, Faulkner, &amp; Cannon (1998, MNRAS, 298, 831)]]
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<div align="center">{{ EFC98figure }}</div>
Abstract: &nbsp;<font color="darkgreen">"We present a simple analytic model of a composite polytropic star, which exhibits a limiting Sch&ouml;nberg-Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a &mu;-jump (i.e. a molecular weight jump) at the interface.  Our model consists of an n<sub>c</sub> = 5 core, an n<sub>e</sub> = 1 envelope and a &mu;-jump by a factor &ge; 3; the core mass fraction cannot exceed 2/&pi;.  We use the classic ''U, V'' plane to show that composite models will exhibit a Sch&ouml;nberg-Chandrasekhar limit only if the core is 'soft', i.e. has n<sub>c</sub> &ge; 5, and the envelope is 'hard', i.e. has n<sub>c</sub> < 5; in the critical case (n<sub>c</sub> = 5), the limit only exists if the &mu;-jump is sufficiently large, &ge; 6/(n<sub>e</sub> + 1)."</font>
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This paper uses analytic techniques to derive precisely the same sequence of <math>(n_c, n_e) = (5, 1)</math> bipolytropic models that we have presented above.