Editing SSC/Structure/BiPolytropes/51RenormaizePart3 (section)

==Yabushita75 Plot==

===Specify Desired Abscissa and Ordinate===
Here our desire is to generate a plot that is analogous to the one that appears as Fig. 1 (p. 445) of {{ Yabushita75 }}.  We need to plot the core mass versus the central density, and the total mass versus central density where,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{norm} 
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i \biggr]
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2} \biggl[ \xi_i^3 \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3/2} \biggr]
=
M_\mathrm{norm} 
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \xi_i^3 \theta_i^4 \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{norm} 
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s
=
M_\mathrm{norm} 
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} \biggl( \frac{2}{\pi} \biggr)^{1/2} \eta_s A
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_\mathrm{norm}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]
\, .
</math>
  </td>
</tr>
</table>
As a check against earlier derivations, note as well that,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>M_\mathrm{norm} 
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i \biggr]
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2} \biggl[ \xi_i^3 \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3/2} \biggr]
\biggl\{
M_\mathrm{norm} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s
\biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
3^{1 / 2}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2} \theta_i
\biggl[ \xi_i^3 \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3/2} \biggr]
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s^{-1}
\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center">

<tr>
  <td align="center">[[File:Yabushita75MuRatio100MassesLabeled.png|400px|Yabushita75 Fig.1]]</td>
</tr>
</table>

<table border="0" align="center" width="80%">

<tr>
  <td align="left">
Figure Caption: &nbsp; Analogous to Figure 1 in {{ Yabushita75full }}, the burnt-orange colored curve shows how the core mass varies with <math>\xi_i</math> and the blue curve shows how the configuration's total mass varies with <math>\xi_i</math>.  More specifically, given that <math>\mu_e/\mu_c = 1</math>, the blue curve is a plot of the function, <math>[(2/\pi)^{1 / 2}\eta_s A]</math>, and the burnt-orange curve is a plot of the function, <math>[(6/\pi)^{1 / 2}\xi_i^3 \theta_i^4 ]</math>.
  </td>
</tr>
</table>

===Compare with Earlier Derivation===
From our [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|earlier derivation]], we know that,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sqrt{3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl[
\frac{\xi_i^3 \theta_i^4}{A\eta_s}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sqrt{3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl[
\frac{\xi_i^3 \theta_i^4}{\eta_s}
\biggr]\biggl[
-\eta_s \biggl(\frac{d\phi}{d\eta}\biggr)_s
\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sqrt{3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \theta_i\biggl[
\xi_i^3 \biggl(1 + \frac{1}{3}\xi_i^2\biggr)^{-3 / 2}
\biggr]\biggl[
-\eta_s^2 \biggl(\frac{d\phi}{d\eta}\biggr)_s
\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

Also, our [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|earlier derivation]] gave,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl[\frac{\eta_s^2}{3A\theta_i^5} \biggr]
</math>
  </td>
</tr>

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

\frac{1}{3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-5} \biggl[- \frac{1}{\eta_s}\cdot \biggl(\frac{d\phi}{d\eta}\biggr)_s\biggr]^{-1}
\, .
</math>
  </td>
</tr>
</table>

Hooray! These both match our "new normalization" derivation.

===Locations of Extrema===

====Maximum Core Mass====

Since the core mass is given by an analytic expression, we should be able to determine analytically at what location <math>(\xi_i)</math> its maximum occurs. Specifically, given that,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(1 + \frac{\xi_i^2}{3}\biggr)^{-1 / 2} \, ,
</math>
  </td>
</tr>
</table>

<table border="1" align="right" cellpadding="5"><tr><td align="center">[[File:DFBsequenceB.png|300px|center|Pressure-truncated polytropic sequences]]</td></tr><td align="left">[[SSC/Structure/PolytropesEmbedded#DFBsequences|Pressure-truncated equilibrium polytropic sequences]].</td></tr></table>
the maximum occurs when the first derivative of the function goes to zero, that is,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\frac{1}{M_\mathrm{norm}} \cdot \frac{dM_\mathrm{core}}{d\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2}
\frac{d}{d\xi_i}\biggl[\xi_i^3 \theta_i^4  \biggr]
~~\rightarrow ~~ 0
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d}{d\xi_i}\biggl[\xi_i^3 \biggl(1 + \frac{\xi_i^2}{3}\biggr)^{-2}  \biggr]
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl(1 + \frac{\xi_i^2}{3}\biggr)  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\xi_i^2}{9}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \xi_i^2  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
9 \, .
</math>
  </td>
</tr>
</table>
The burnt-orange colored, vertical dashed line in the above figure has been placed at <math>\xi_i = 3</math>; it intersects the point along the core-mass curve where the core mass is a maximum.  In a [[SSC/Structure/PolytropesEmbedded#Some_Tabulated_Values|separate discussion]] of pressure-truncated polytropic spheres, this has also been identified as the location of the maximum mass along <math>n=5</math> equilibrium sequence.  It is comforting to see that the same turning point arises whether or not an "envelope" has been added to the <math>n=5</math> polytropic core.

====Maximum Total Mass====
Similarly we should be able to derive an analytic expression for the location along the bipolytropic sequence where the configuration's total mass acquires its maximum value.  Drawing from [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|our detailed discussion of the properties of various model parameters]], we can write,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{3 / 2} \biggl( \frac{2}{\pi} \biggr)^{- 1/2}\frac{1}{M_\mathrm{norm}} \cdot M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_s A
</math>
  </td>
</tr>

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

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \theta_i^2
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} - 3^{-1 / 2}\xi_i \, .
</math>
  </td>
</tr>
</table>
Rewriting these terms gives,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \biggl(\frac{3}{3 + \xi_i^2} \biggr)
=
3^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i (3 + \xi_i^2)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\eta_i}{d\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \frac{d}{d\xi_i}\biggl[ \xi_i (3 + \xi_i^2)^{-1} \biggr]
=
3^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl\{
(3 + \xi_i^2)^{-1}
-
2 \xi_i^2 (3 + \xi_i^2)^{-2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl\{
(3 + \xi_i^2)
-
2 \xi_i^2 
\biggr\}(3 + \xi_i^2)^{-2}
=
3^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl\{
3 - \xi_i^2 
\biggr\}(3 + \xi_i^2)^{-2} \, ;
</math>
  </td>
</tr>
</table>
and,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{- 3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \xi_i^{-1} (3 + \xi_i^2) - 3^{-1 / 2}\xi_i 
=
3^{- 3 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \xi^{-1}\biggl\{ 3 + \biggl[ 1 - 3\biggl(\frac{\mu_e}{\mu_c}\biggr)\biggr]\xi_i^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d\Lambda_i}{d\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{- 3 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{d}{d\xi_i}\biggl\{ 3\xi^{-1} + \biggl[ 1 - 3\biggl(\frac{\mu_e}{\mu_c}\biggr)\biggr]\xi_i \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{- 3 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl\{\biggl[ 1 - 3\biggl(\frac{\mu_e}{\mu_c}\biggr)\biggr] -\frac{3}{\xi_i^2} \biggr\}
\, .
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{3 / 2} \biggl( \frac{2}{\pi} \biggr)^{- 1/2}\frac{1}{M_\mathrm{norm}}\cdot \frac{dM_\mathrm{tot}}{d\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\eta_i + \frac{\pi}{2} + \tan^{-1}(\Lambda_i)\biggr]
\frac{d}{d\xi_i}\biggl[\eta_i (1 + \Lambda_i^2 )^{1 / 2} \biggr] 
+
\biggl[\eta_i (1 + \Lambda_i^2 )^{1 / 2} \biggr] 
\frac{d}{d\xi_i}\biggl[\eta_i + \frac{\pi}{2} + \tan^{-1}(\Lambda_i)\biggr]
</math>
  </td>
</tr>

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\overbrace{\biggl[1 + \eta_i\Lambda_i + \frac{\pi}{2}\Lambda_i + \Lambda_i\tan^{-1}(\Lambda_i)\biggr]
\biggl\{
\eta_i (1 + \Lambda_i^2 )^{-1 / 2} \frac{d\Lambda_i}{d\xi_i}
\biggr\}}^\mathrm{TERM1}
+
\underbrace{\biggl[2\eta_i + \frac{\pi}{2} + \tan^{-1}(\Lambda_i)\biggr]
\biggl\{
(1 + \Lambda_i^2 )^{1 / 2} \frac{d\eta_i}{d\xi_i} 
\biggr\}}_\mathrm{TERM2}
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="14">Maximum Total Mass ''a la'' {{ Yabushita75 }}</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>\eta_i</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>\frac{d\eta_i}{d\xi_i}</math></td>
  <td align="center"><math>\frac{d\Lambda_i}{d\xi_i}</math></td>
  <td align="center">TERM1</td>
  <td align="center">TERM2</td>
  <td align="center">Error</td>
  <td align="center"><math>M_\mathrm{tot}/M_\mathrm{norm}</math></td>
  <td align="center">[[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|LAWE]]</td>
  <td align="center">[[SSC/Stability/BiPolytropes/51Models#Stability|Implicit<br />Scheme]]</td>
</tr>

<tr>
  <td align="center">1.000</td>
  <td align="center" bgcolor="lightgreen">1.66846298</td>
  <td align="center">1.4989514</td>
  <td align="center">-0.2961544</td>
  <td align="center">0.0335876</td>
  <td align="center">-0.592299</td>
  <td align="center">-0.1499536</td>
  <td align="center">+0.1499536</td>
  <td align="center"><math>1.58\times 10^{-9}</math></td>
  <td align="center">3.4698691</td>
  <td align="center" bgcolor="yellow">1.6686460157</td>
  <td align="center" bgcolor="yellow">1.6639103365</td>
</tr>
</table>