Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3C (section)

====Behavior of Equilibrium Sequence====

Here we reprint Figure 1 from an [[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|accompanying chapter wherein the structure of five-one bipolytropes has been derived]]. It displays detailed force-balance sequences in the <math>~q - \nu</math> plane for a variety of choices of the ratio of mean-molecular-weights, <math>~\mu_e/\mu_c</math>, as labeled.

[[File:PlotSequencesBest02.png|450px|center|Five-One Bipolytropic Equilibrium Sequences for Various ratios of the mean molecular weight]]

=====Limiting Values=====
Each sequence begins <math>~(\ell_i = 0)</math> at the origin, that is, at <math>~(q,\nu) = (0,0)</math>.  As <math>~\ell_i \rightarrow \infty</math>, however, the sequences terminate at different coordinate locations, depending on the value of <math>~m_3 \equiv 3(\mu_e/\mu_c)</math>.  In deriving the various limits, it will be useful to note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1+\ell_i^2)}{m_3\ell_i}-\ell_i</math>
  </td>
</tr>

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{m_3^2\ell_i^2}\biggl\{ 1 + (2 -2m_3 + m_3^2) \ell_i^2 + (1 -m_3)^2 \ell_i^4 \biggr\}</math>
  </td>
</tr>
</table>
</div>

Examining the three relevant parameter regimes, we see that:

* For <math>~\mu_e/\mu_c < \tfrac{1}{3}</math>, that is, <math>~m_3 < 1</math> &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan^{-1} \Lambda_i \biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx </math>
  </td>
  <td align="left">
<math>~\tan^{-1} \biggl[\frac{(1 -m_3)}{m_3} \biggr]\ell_i</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx </math>
  </td>
  <td align="left">
<math>~\frac{\pi}{2} -  \biggl[\frac{m_3}{(1 -m_3)\ell_i} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{q}\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
1 + \frac{(1 + \ell_i^2)}{m_3 \ell_i}\biggl[\pi - \frac{m_3}{(1 -m_3)\ell_i} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
\frac{m_3 + \pi \ell_i}{m_3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ q\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{1 + (\pi \ell_i/m_3)} \rightarrow 0 \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
and
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl(\frac{\nu}{q}\biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_i^2}{1 + \Lambda_i^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~m_3^2\ell_i^4 \biggl\{ 1 + (2 -2m_3 + m_3^2) \ell_i^2 + (1 -m_3)^2 \ell_i^4 \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~m_3^2\biggl\{ \ell_i^{-4} + (2 -2m_3 + m_3^2) \ell_i^{-2} + (1 -m_3)^2 \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\nu}{q}\biggr|_{\ell_i\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{m_3}{1-m_3} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \nu \biggr|_{\ell_i\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{m_3}{1-m_3}\biggr]\frac{1}{1 + (\pi \ell_i/m_3)} \rightarrow 0 \, . </math>
  </td>
</tr>
</table>
</div>

*  For <math>~\mu_e/\mu_c = \tfrac{1}{3}</math>, that is, <math>~m_3 = 1</math> &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan^{-1} \Lambda_i </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\tan^{-1} \biggl(\frac{1}{\ell_i}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \tan^{-1} \Lambda_i \biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx </math>
  </td>
  <td align="left">
<math>~\frac{1}{\ell_i}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{q}\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
1 + \frac{(1 + \ell_i^2)}{\ell_i}\biggl[\frac{\pi}{2} + \frac{1}{\ell_i }\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\pi}{2}\biggr)\ell_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ q\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\pi \ell_i} \rightarrow 0</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
and
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl(\frac{\nu}{q}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_i}{(1 + 1/\ell_i^2)^{1/2}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \nu \biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\ell_i \biggl(\frac{2}{\pi \ell_i} \biggr) = \frac{2}{\pi} \approx 0.63662</math>
  </td>
</tr>

</table>
</div>


*  For <math>~\mu_e/\mu_c > \tfrac{1}{3}</math>, that is, <math>~m_3 > 1</math> &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan^{-1} \Lambda_i \biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx </math>
  </td>
  <td align="left">
<math>~\tan^{-1} \biggl[-\biggl(\frac{m_3-1}{m_3} \biggr)\ell_i\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx </math>
  </td>
  <td align="left">
<math>~-\frac{\pi}{2} +  \biggl[\frac{m_3}{(m_3-1)\ell_i} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{q}\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
1 + \frac{(1 + \ell_i^2)}{m_3 \ell_i}\biggl[ \frac{m_3}{(m_3-1)\ell_i} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
1 + \frac{(1 + 1/\ell_i^2)}{(m_3-1) } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
1 + \frac{1}{(m_3-1) } = \frac{m_3}{(m_3-1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ q\biggr|_{\ell_i \rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{(m_3-1)}{m_3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
and
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl(\frac{\nu}{q}\biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_i^2}{1 + \Lambda_i^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~m_3^2\ell_i^4 \biggl\{ 1 + (2 -2m_3 + m_3^2) \ell_i^2 + (m_3-1)^2 \ell_i^4 \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~m_3^2\biggl\{ \ell_i^{-4} + (2 -2m_3 + m_3^2) \ell_i^{-2} + (m_3-1)^2 \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\nu}{q}\biggr|_{\ell_i\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{m_3}{m_3-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \nu \biggr|_{\ell_i\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{m_3}{m_3-1} \biggl[\frac{m_3 - 1}{m_3}\biggr] \rightarrow 1 \, . </math>
  </td>
</tr>

</table>
</div>

Summarizing:

* For <math>~\mu_e/\mu_c < \tfrac{1}{3}</math>, that is, <math>~m_3 < 1</math> &nbsp;  &nbsp;  &nbsp; &hellip; &nbsp;  &nbsp;  &nbsp;  <math>~(q,\nu)_{\ell_i \rightarrow \infty} = (0, 0) \, .</math>


*  For <math>~\mu_e/\mu_c = \tfrac{1}{3}</math>, that is, <math>~m_3 = 1</math> &nbsp;  &nbsp;  &nbsp; &hellip; &nbsp;  &nbsp;  &nbsp;  <math>~(q,\nu)_{\ell_i \rightarrow \infty} = (0, \tfrac{2}{\pi}) \, .</math>


*  For <math>~\mu_e/\mu_c > \tfrac{1}{3}</math>, that is, <math>~m_3 > 1</math> &nbsp;  &nbsp;  &nbsp; &hellip; &nbsp;  &nbsp;  &nbsp;  <math>~(q,\nu)_{\ell_i \rightarrow \infty} = [(m_3-1)/m_3, 1] \, .</math>

=====Turning Points=====

Let's identify the location of two turning points along the <math>~\nu(q)</math> sequence &#8212; one defines <math>~q_\mathrm{max}</math> and the other identifies <math>~\nu_\mathrm{max}</math>.  They occur, respectively, where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln q}{d\ln \ell_i} = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{d\ln \nu}{d\ln \ell_i} = 0 \, .</math>
  </td>
</tr>
</table>
</div>

In deriving these expressions, we will use the relations,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{d\Lambda_i}{d\ell_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{m_3\ell_i^2} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where, 
<div align="center">
<math>~m_3 \equiv 3\biggl(\frac{\mu_e}{\mu_c}\biggr) \, .</math>
</div>


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

<tr>
  <td align="right">
<math>~q </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr] \biggr\}^{-1} \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{d\ln q}{d\ln \ell_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_i}{q} \cdot ( -q^2) \frac{d}{d\ell_i} \biggl\{
\frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-q\ell_i \biggl\{-\frac{1}{\eta_i^2}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]\frac{d\eta_i}{d\ell_i}
+ \frac{1}{\eta_i(1+\Lambda_i^2)} \frac{d\Lambda_i}{d\ell_i}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q\ell_i \biggl\{\frac{(1-\ell_i^2)}{m_3 \ell_i^2}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{(1+\ell_i^2)}{m_3^2 \ell_i^3(1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{q}{m_3^2}{\ell_i^2}\biggl\{m_3 \ell_i (1-\ell_i^2) \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{(1+\ell_i^2)}{(1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nu </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\ell_i q}{(1+\Lambda_i^2)^{1/2}} \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{d\ln \nu}{d\ln \ell_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_i}{\nu} \biggl\{
\frac{q}{(1+\Lambda_i^2)^{1/2}} + \frac{q}{(1+\Lambda_i^2)^{1/2}} \frac{d\ln q}{d\ln \ell_i}
- \frac{\ell_i q \Lambda_i }{(1+\Lambda_i^2)^{3/2}} \frac{d\Lambda_i}{d\ell_i}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q \ell_i}{ \nu(1+\Lambda_i^2)^{1/2}}\biggl\{
1 + \frac{d\ln q}{d\ln \ell_i}
+ \frac{\Lambda_i }{m_3 \ell_i (1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

In summary, then, the <math>~q_\mathrm{max}</math> turning point occurs where,

<div align="center">
<table border="0" cellpadding="5" align="center">
<!--
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ m_3 \ell_i (1-\ell_i^2) \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{(1+\ell_i^2)}{(1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
\, ;
</math>
  </td>
</tr>
-->
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (1+\Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{(1+\ell_i^2)}{m_3 \ell_i (1-\ell_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
\, ;
</math>
  </td>
</tr>
</table>
</div>
 
and the <math>~\nu_\mathrm{max}</math> turning point occurs where,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{\Lambda_i }{m_3 \ell_i (1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr]  
+ \frac{q \ell_i^3  (1-\ell_i^2)}{m_3} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{q\ell_i^2}{m_3^2}\cdot \frac{(1+\ell_i^2)}{(1+\Lambda_i^2)} \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
+ \frac{q \ell_i^3  (1-\ell_i^2)}{m_3} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \biggl[ \frac{\Lambda_i }{m_3 \ell_i (1+\Lambda_i^2)}  
+ \frac{q\ell_i^2}{m_3^2}\cdot \frac{(1+\ell_i^2)}{(1+\Lambda_i^2)} \biggr] \cdot \biggl[ 1 - \ell_i^2(1-m_3) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
+ \frac{q \ell_i^3  (1-\ell_i^2)}{m_3} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
+ \frac{1}{m_3 \ell_i} \biggl[ \frac{\Lambda_i }{(1+\Lambda_i^2)}  
+ \frac{q\ell_i^3}{m_3}\cdot \frac{(1+\ell_i^2)}{(1+\Lambda_i^2)} \biggr] \cdot \biggl[ 1 - \ell_i^2(1-m_3) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="0" width="80%" cellpadding="5" align="center"><tr><td align="left">
<font color="red"><b>NOTE:</b></font>&nbsp; As we show [[#Limiting_Values|above]], for the special case of <math>~m_3 = 1</math> &#8212; that is, when <math>~\mu_e/\mu_c = \tfrac{1}{3}</math>, precisely &#8212; the equilibrium sequence (as <math>~\ell_i \rightarrow \infty</math>) intersects the <math>~q = 0</math> axis at precisely the value, <math>~\nu = 2/\pi</math>.  As is illustrated graphically in [[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|Figure 1 of an accompanying chapter]], no <math>~\nu_\mathrm{max}</math> turning point exists for values of  <math>~m_3 > 1</math>.  
</td></tr>
</table>

For the record, we repeat, as well, that the transition from stable to dynamically unstable configurations occurs along the sequence when,

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

<tr>
  <td align="right">
<math>~\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~20 \biggl\{ \frac{\Lambda_i}{\eta_i}  + \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{20(1+\Lambda_i^2)(1+\ell_i^2)}{m_3\ell_i} \biggl\{ \frac{\Lambda_i}{(1+\Lambda_i^2)}  + \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ m_3 \ell_i (\ell_i^4-1) + m_3(1+\ell_i^2)^3\cdot \tan^{-1}\ell_i </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~20\ell_i^2 (1+\Lambda_i^2)(1+\ell_i^2) \biggl\{ \frac{\Lambda_i}{(1+\Lambda_i^2)}  + \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{m_3 \ell_i (\ell_i^4-1) + m_3(1+\ell_i^2)^3\cdot \tan^{-1}\ell_i }{ 20\ell_i^2 (1+\ell_i^2)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Lambda_i  + (1+\Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr]  \, .</math>
  </td>
</tr>

</table>
</div>


In order to clarify what equilibrium sequences do not have any turning points, let's examine how the <math>~q_\mathrm{max}</math> turning-point expression behaves as <math>~\ell_i \rightarrow \infty</math>.

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

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

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \frac{(1+\ell_i^2)}{ m_3 \ell_i (\ell_i^2-1)   } \biggl[ 1 + \ell_i^2(m_3-1) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{ 1 + \frac{1}{m_3^2\ell_i^2}\biggl[ (m_3-1) \ell_i^2-1 \biggr]^2 \biggr\} \biggl\{ \frac{\pi}{2} 
+ \biggl[ -\frac{\pi}{2} - \frac{1}{\Lambda_i} + \frac{1 }{3\Lambda_i^3} + \mathcal{O}(\Lambda_i^{-5} )\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \frac{(1+\ell_i^2) \ell_i^2(m_3-1)}{ m_3 \ell_i (\ell_i^2-1)   } \biggl[ 1 + \frac{1}{\ell_i^2(m_3-1)} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{ 1 + \frac{(m_3-1)^2 \ell_i^2}{m_3^2}\biggl[ 1 - \frac{1}{ (m_3-1) \ell_i^2 } \biggr]^2 \biggr\} \cdot  
\frac{1}{(-\Lambda_i)} \biggl[  1 - \frac{1 }{3\Lambda_i^2} + \cancelto{0}{\mathcal{O}(\Lambda_i^{-4} )}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \frac{(1+\ell_i^2)}{ \ell_i (\ell_i^2-1)   } \cdot \frac{m_3}{(m_3-1)} \biggl[ 1 + \frac{1}{\ell_i^2(m_3-1)} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ 1 - \frac{2}{ (m_3-1) \ell_i^2 } + \frac{m_3^2}{(m_3-1)^2 \ell_i^2}  +  \frac{1}{ (m_3-1)^2 \ell_i^4 } \biggr] \cdot  \frac{m_3}{(m_3-1)\ell_i} \biggl[1 - \frac{1}{(m_3-1)\ell_i^2}\biggr]^{-1}
\biggl\{  1 - \frac{m_3^2}{3(m_3-1)^2\ell_i^2} \biggl[1 - \frac{1}{(m_3-1)\ell_i^2}\biggr]^{-2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~\biggl(1 + \frac{1}{\ell_i^2}  \biggr)   \biggl[ 1 + \frac{1}{\ell_i^2(m_3-1)} \biggr]\biggl[1 - \frac{1}{(m_3-1)\ell_i^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1 - \frac{1}{\ell_i^2}  \biggr) \biggl[ 1 - \frac{2}{ (m_3-1) \ell_i^2 } + \frac{m_3^2}{(m_3-1)^2 \ell_i^2}  +  \frac{1}{ (m_3-1)^2 \ell_i^4 } \biggr] 
\biggl\{  1 - \frac{m_3^2}{3(m_3-1)^2\ell_i^2} \biggl[1 - \frac{1}{(m_3-1)\ell_i^2}\biggr]^{-2} \biggr\} 
</math>
  </td>
</tr>

</table>
</div>

The leading-order term is unity on both sides of this expression, so they cancel; let's see what results from keeping terms <math>~\propto \ell_i^{-2}</math>.

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

<tr>
  <td align="right">
<math>~ \frac{1}{\ell_i^2} \biggl[ 1 + \frac{1}{(m_3-1)}  - \frac{1}{(m_3-1)}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\ell_i^2} \biggl[- 1 - \frac{2}{ (m_3-1) } + \frac{m_3^2}{(m_3-1)^2 }   - \frac{m_3^2}{3(m_3-1)^2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{2}{ (m_3-1) } + \frac{2m_3^2}{3(m_3-1)^2 }   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 6(m_3-1)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 6(m_3-1) + 2m_3^2   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 6m_3^2-12m_3 + 6 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 6m_3+6 + 2m_3^2   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ m_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2} \, .  
</math>
  </td>
</tr>
</table>
</div>

We therefore conclude that the <math>~q_\mathrm{max}</math> turning point does not appear along any sequence for which,

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

<tr>
  <td align="right">
<math>~m_3</math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~\frac{3}{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\, .</math>
  </td>
</tr>
</table>
</div>


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

<tr>
  <td align="right">
<math>~ \frac{1}{\ell_i^2} \biggl[ 2 + \frac{1}{\alpha m_3} \biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\ell_i^2}\biggl[
\frac{1}{\alpha^2 }  - \frac{2}{ m_3 \alpha  } -\frac{1}{3\alpha^2 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ 2\alpha^2 m_3 + \alpha \biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ \biggl[
\frac{2m_3}{3}  - 2\alpha  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2\alpha^2 m_3 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ 
\frac{2m_3}{3}  - 3\alpha  
</math>
  </td>
</tr>
</table>
</div>
-->

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Five-One Bipolytrope Equilibrium Sequences in <math>~q - \nu</math> Plane</td>
</tr>
<tr>
  <td align="center" width="50%">
Full Sequences for Various <math>~\frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center" width="50%">
Magnified View with Turning Points and Stability Transition-Points Identified
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
[[File:Qvsnu51combined.png|750 px|Five-One Sequences]]
  </td>
</tr>
</table>
</div>