Editing SSC/Stability/BiPolytropes/HeadScratching (section)

==Differentiate M<sup>*</sup> With Respect to &ell;<sub>i</sub>==

In an accompanying discussion titled, [[SSC/StabilityConjecture/Bipolytrope51#New_Derivation|''New Derivation'']], we examined how the core mass-fraction <math>(\nu)</math> varies with <math>\ell_i \equiv \xi_i/\sqrt{3}</math>.  Here, we want to examine how the total mass <math>(M^*_\mathrm{tot})</math> varies with <math>\ell_i</math>.  In what follows, we borrow heavily from various analytic expressions that have been obtained via this separate ''New Derivation''; and, as in this earlier analysis, numerical evaluations (in parentheses) come from '''Example #1''' for which, <math>\mu_e/\mu_c = 0.25</math> and <math>\xi_i = 0.5</math>, which implies that <math>m_3 = 0.75</math> and <math>\ell_i = (12)^{-1 / 2}</math>.

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

<tr>
  <td align="right">
<math>M^*_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[- \frac{\eta_s^2}{\theta_i} \cdot \biggl(\frac{d\phi}{d\eta}\biggr)_s \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[\frac{A\eta_s}{\theta_i} \biggr]
= 40.09338625</math>,
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(1 + \ell_i^2)^{-1 / 2} = 0.960768923</math>,
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{m_3 \ell_i}{(1+\ell_i^2)} = 0.199852016</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}\biggl[1 + (1 - m_3)\ell_i^2 \biggr] 
= \frac{49}{6\sqrt{3}}
= 4.715027199\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\pi}{2} +  \tan^{-1}\Lambda_i\biggr) +  \frac{m_3\ell_i }{(1 + \ell_i^2)} 
= 3.132453649\, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{1 / 2}
=
\frac{m_3\ell_i}{(1+\ell_i^2)}\biggl\{
1 + \frac{1}{m_3^2 \ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr]^2
\biggr\}^{1 / 2}
= 0.963267676 \, .
</math>
  </td>
</tr>
</table>
Now, the differentiation:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\frac{d\theta_i}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\ell_i (1 + \ell_i^2)^{-3 / 2} 
= - 12 (13)^{-3/2}
= - 0.256015475 (7)
\, ;</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{2(1-m_3)\ell_i}{m_3\ell_i} - \frac{1}{m_3\ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr] 
=
\frac{1}{m_3\ell_i^2}\biggl\{2(1-m_3)\ell_i^2 - [1 + (1 - m_3)\ell_i^2 ] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{48}{3}\biggl\{\frac{1}{24} - \frac{49}{48} \biggr\} = -\frac{47}{3}
= -15.66666666
\, ;
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{9}{13}
-
\frac{2\cdot 3^2}{13^2}
+
\biggl[\frac{2509}{2^2\cdot 3^3}\biggr]^{-1} \biggl( - \frac{47}{3}\biggr)
=
\frac{1}{13^2}\biggl\{9\cdot 13 - 2\cdot 3^2 - \biggl[\frac{2^2\cdot 3^2 \cdot 13 \cdot 47}{193}\biggr] 
\biggr\}
=
\frac{3^2}{13^2 \cdot 193}\biggl\{11 \cdot 193 - 2^2 \cdot 13 \cdot 47 
\biggr\}
=
-0.088573443
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1+\Lambda_i^2)^{1 / 2} \biggl[ \frac{m_3 }{(1+\ell_i^2)} - \frac{2m_3 \ell_i^2}{(1+\ell_i^2)^2} \biggr]
+
\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr]\frac{\Lambda_i }{(1+\Lambda_i^2)^{1 / 2}} \frac{d\Lambda_i}{d\ell_i}
= 4.819904715(4)\biggl[ 0.585798817(5) \biggr] + 0.195503386(6)\cdot \biggl(-\frac{47}{3}\biggr)
= -0.239391902
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{13\cdot 193}{2^2 \cdot 3^3} \biggr)^{1 / 2} \biggl[ 3^2 \cdot 13  - 2 \cdot 3^2  \biggr]\frac{1}{13^2}
+
\frac{49}{2\cdot 3^{3/2}}\biggl[ \frac{3^2 }{13} \biggr]\cdot \frac{1}{2\cdot 3^{1 / 2}}
\biggl(\frac{2^2 \cdot 3^3}{13\cdot 193} \biggr)^{1 / 2} \biggl( - \frac{47}{3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3\cdot 11^2\cdot 193}{2^2 \cdot 13^3} \biggr)^{1 / 2} 
-
\biggl(\frac{3 \cdot 7^4 \cdot 47^2}{2^2 \cdot 13^{3}\cdot 193} \biggr)^{1 / 2}  
=
\frac{3^{1 / 2}}{2\cdot 13^{3/2} \cdot 193^{1 / 2}}\biggl[11\cdot 193  - 7^2 \cdot 47  \biggr]
=
- \biggl[ \frac{2^2 \cdot 3^{5} \cdot 5^2}{13^{3} \cdot 193} \biggr]^{1 / 2}
= - 0.239391901
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl(\frac{\pi}{2}\biggr)^{1 / 2}  \frac{dM^*_\mathrm{tot}}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\ell_i}\biggl[\frac{A\eta_s}{\theta_i}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\eta_s}{\theta_i}\frac{dA}{d\ell_i}
+
\frac{A}{\theta_i}\frac{d\eta_s}{d\ell_i}
-
\frac{A\eta_s}{\theta_i^2}\frac{d\theta_i}{d\ell_i}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
-\biggl[\frac{(3.132453649)}{(0.960768923)} \biggr]0.239391901
-
\biggl[ \frac{( 0.963267676 )}{(0.960768923)} \biggr]0.088573443
+
\biggl[ \frac{(0.963267676 )(3.132453649)}{(0.960768923)^2} \biggr]0.256015475
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
- 0.78050(8)
- 0.088803
+ 0.836874
=
-0.032426 \, .
</math>
  </td>
</tr>
</table>