Editing SSC/Structure/BiPolytropes/51RenormaizePart3

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=BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1=

After studying {{ Yabushita75full }} in depth, here we renormalize [[SSC/Structure/BiPolytropes/Analytic51|our original construction]] of bipolytropic models with <math>(n_c, n_e) = (5, 1)</math> such that both entropy values, <math>(K_c, K_e)</math>, are held fixed along each model sequence.

==Original Derivation==
===Throughout the Core===

Drawing from our original derivation, [[SSC/Structure/BiPolytropes/Analytic51#Step_4:_Throughout_the_core_(0_≤_ξ_≤_ξi)|throughout the core]] &hellip;
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>K_c</math> and <math>\rho_0 ~\Rightarrow</math>
  </td>
  <td colspan="2">
&nbsp;
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta^{n_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
</tr>
</table>

</div>

===Throughout the Envelope===
And [[SSC/Structure/BiPolytropes/Analytic51#Step_8:_Throughout_the_envelope_(ηi_≤_η_≤_ηs)|throughout the envelope]],

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="5">
&nbsp;
  </td>
  <td align="left" colspan="2">
Knowing:  <math>K_e/K_c</math> and <math>\rho_e/\rho_0</math> from Step 5 &nbsp; <math>\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_e \phi^{n_e}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl(\frac{\rho_e}{\rho_0}\biggr) \phi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \biggl(\frac{K_e \rho_0^{4/5}}{K_c}\biggr) \biggl(\frac{\rho_e}{\rho_0}\biggr)^{2} \phi^{2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5}  \theta^{6}_i \phi^{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G \rho_0^{4/5}} \biggr]^{1/2} \biggl( \frac{K_e \rho_0^{4/5}}{K_c} \biggr)^{1/2} (2\pi)^{-1/2}\eta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G \rho_0^{4/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{K_e \rho_0^{4/5}}{K_c} \biggr)^{3/2} \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>
</table>

</div>

===Interface Conditions===

And [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|at the interface]] &hellip;

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td colspan="3">
&nbsp;
  </td>
  <td align="left" colspan="2">
Setting <math>n_c=5</math>, <math>n_e=1</math>, and <math>\phi_i = 1 ~~~~\Rightarrow</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho_e}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{1/2} \theta_i^{- 3} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
  </td>
</tr>
</table>
</div>

==New Normalization==

From one of the interface conditions, we see that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~  \rho_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{- 2} \theta^{-4}_i \biggr]^{5 / 4}
=
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]
=
\rho_\mathrm{norm}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]
\, .
</math>
  </td>
</tr>
</table>

<!--  BEGIN Alternate01 -->
<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
<div align="left">Alternate01</div>
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m-2} \theta^{-4}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m}\biggr]^{5 / 4} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{-1}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{5(m-2)/4} \theta^{-5}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \rho_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{\biggl[ \biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m}\biggr]^{- 5 / 4}}_{\rho_\mathrm{alt}}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{5(m-2)/4} \theta^{-5}_i
</math>
  </td>
</tr>
</table>
<font color="red"><b>ABANDONED</b></font>
</td></tr></table>
<!-- END alternate01 -->

Hence, throughout the core, we have,

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

<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</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]
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]
^{6/5} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
K_c \biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} 
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 } \theta^{-6}_i \biggr]\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}
=
P_\mathrm{norm}\biggl[  
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 } \theta^{-6}_i \biggr]\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G} \biggr]^{1/2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]^{- 2 / 5}
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G} \biggr]^{1/2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggr]
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi
=
r_\mathrm{norm} 
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggr]
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1 / 2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]^{-1 / 5}
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3} \biggr]^{1 / 2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i \biggr]
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
</tr>

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

And, throughout the envelope &hellip;

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

<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_\mathrm{norm} \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} \biggr]\phi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5}  \theta^{6}_i \phi^{2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]^{6/5}  
\theta^{6}_i \phi^{2}</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
K_c \biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggr]  \phi^{2}
=
P_\mathrm{norm}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggr]  \phi^{2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G \rho_0^{4/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G } \biggr]^{1/2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]^{- 2 / 5}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G } \biggr]^{1/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2} (2\pi)^{-1/2}\eta
=
r_\mathrm{norm}(2\pi)^{-1/2}\eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} 
\biggl[ \biggl( \frac{K_e}{K_c} \biggr)^{-5 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \biggr]^{-1 / 5}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} 
\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} 
\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)</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </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)</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="10">

<tr>
  <td align="center" colspan="1">Adopted Normalizations</td>
</tr>
<tr>
  <td align="center">
<table border="0" align="center" width="70%" cellpadding="5">
<tr>
  <td align="right"><math>\rho_\mathrm{norm}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\biggl( \frac{K_c}{K_e} \biggr)^{5 / 4} \, ;</math></td>
<td align="center" width="20%">&nbsp; &nbsp;</td>
  <td align="right"><math>P_\mathrm{norm}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>K_c  \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} = \biggl[K_c^{5 } K_e^{-3} \biggr]^{1 / 2}\, ;</math></td>
</tr>

<tr>
  <td align="right"><math>r_\mathrm{norm}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\biggl[ \frac{K_c}{G} \biggr]^{1/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2} = \biggl( \frac{K_e}{G} \biggr)^{1/2} \, ;</math></td>
<td align="center" width="20%">&nbsp; &nbsp;</td>
  <td align="right"><math>M_\mathrm{norm}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3} \biggr]^{1 / 2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} 
=
\biggl[ K_c^{5} K_e G^{-6} \biggr]^{1 / 4} \, .
</math></td>
</tr>
</table>

</td></tr></table>

Note that the configuration's mean density is,

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

<tr>
  <td align="right" colspan="3">
<math>\bar\rho \equiv \frac{3M_\mathrm{tot}}{4\pi R^3} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{4\pi}\biggr) 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
\biggl[r_\mathrm{norm}(2\pi)^{-1/2}\eta_s\biggr]^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{norm}r_\mathrm{norm}^{-3}\biggl( \frac{\mu_e}{\mu_c} 
\biggr)^{-3 / 2} \biggl(\frac{3}{4\pi}\biggr)  
(2\pi)^{3 / 2}\biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\frac{1}{\eta}\cdot \frac{d\phi}{d\eta} \biggr)_s
 </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ K_c^{5} K_e G^{-6} \biggr]^{1 / 4} \biggl( \frac{K_e}{G}\biggr)^{-3 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} 
\biggl[ \biggl(\frac{3^2}{2^4\pi^2}\biggr)  (2\pi)^{3 }\biggl( \frac{2}{\pi} \biggr) \biggr]^{1/2} 
\biggl(-\frac{1}{\eta}\cdot \frac{d\phi}{d\eta} \biggr)_s
 </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3\biggl( \frac{K_c}{K_e}\biggr)^{5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} 
\biggl(-\frac{1}{\eta}\cdot \frac{d\phi}{d\eta} \biggr)_s \, .
 </math>
  </td>
</tr>
</table>
Hence, the central-to-mean density of each equilibrium configuration is,

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

<tr>
  <td align="right" colspan="3">
<math>\frac{\rho_0}{\bar\rho }</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]
\biggl\{ 3\rho_\mathrm{norm}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2} 
\biggl(-\frac{1}{\eta}\cdot \frac{d\phi}{d\eta} \biggr)_s 
\biggr\}^{-1}
\, .
 </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 3
\biggl( \frac{\mu_e}{\mu_c} \biggr) 
\theta^{5}_i \biggl(-\frac{1}{\eta}\cdot \frac{d\phi}{d\eta} \biggr)_s 
\biggr\}^{-1}
\, .
 </math>
  </td>
</tr>
</table>

==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>

=Based on Pressure-Truncated n = 5 Polytrope=

==Chieze87 Normalization==

In a [[SSC/Structure/PolytropesEmbedded#Chieze's_Presentation|subsection]] of our [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|separate discussion of pressure-truncated polytropes]], we highlighted the published work of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&amp;A, 171, 225-232)].  It can [[SSC/Structure/PolytropesEmbedded#Example_Sequences|readily be shown]] that his expressions for <math>P_e</math>, <math>R_\mathrm{eq}</math>, and <math>M_\mathrm{tot}</math> &#8212; in our terminology, <math>P_i</math>, <math>r_i</math>, and <math>M_\mathrm{core}</math> &#8212; are identical to the expressions we [[#Throughout_the_Core|presented above]] in the context of the n = 5 core of our bipolytrope.  Specifically,
[[File:N5Sequence01B.png|400px|right|Chieze87 Figure 3]]
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>P_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i</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{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \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]</math>
  </td>
</tr>
</table>

</div>
If we invert the third expression to determine how the central density depends on the core mass, then use this result to replace <math>\rho_0</math> in the other two expressions, we find that,

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

<tr>
  <td align="right">
<math>P_i \biggl[G^9 K_c^{-10} M_\mathrm{core}^6  \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \xi_i^{18} \biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-12}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{4\pi r_\mathrm{core}^3}{3} \biggr) \biggl[\biggl(\frac{K_c}{G}\biggr)^{15/2} M_\mathrm{core}^{-6}  \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \xi_i^{-15} \biggl(1 + \frac{\xi_i^2}{3} \biggr)^{9}
</math>
  </td>
</tr>
</table>

This leads to panel (a) of [[SSC/Structure/PolytropesEmbedded#Fig3|Figure 3 from that discussion]]; also shown here, on the right.

==Switch from Core Mass to Total Mass==
Now with the bipolytropic model in mind, let's switch from the core mass to the total mass, drawing the following expression from [[#Throughout_the_Envelope|above]] &hellip;
<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{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} 
\theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \rho_0^{1 / 5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot}^{-1}\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} 
\theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s
</math>
  </td>
</tr>
</table>
in which case,
<table border="0" align="center" cellpadding="3">

<tr>
  <td align="right">
<math>P_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
K_c \theta_i^6 \biggl\{
M_\mathrm{tot}^{-1}\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} 
\theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s
\biggr\}^6
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
G^{-9}K_c^{10} M_\mathrm{tot}^{-6} \biggl\{
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-4} 
\biggl( \frac{2}{\pi} \biggr) \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s^2
\biggr\}^3 \, ;
</math>
  </td>
</tr>
</table>

=See Also=

<ul>
  <li>
[https://ui.adsabs.harvard.edu/abs/1974A%26A....32..309G/abstract M. Gabriel &amp; M. L. Roth (1974, A&A, Vol. 32, p. 309)] &hellip; ''On the Secular Stability of Models with an Isothermal Core''
  </li>
  <li>[https://ui.adsabs.harvard.edu/abs/1967AnAp...30..975G/abstract M. Gabriel &amp; P. Ledoux (1967, Annales d'Astrophysique, Vol. 30, p. 975)] &hellip; ''Sur la Stabilit&eacute; S&eacute;culaire des Mode&eacute;les a Noyaux Isothermes''<br />
<p>
In &sect; 1 (p. 442) of {{ Yabushita75 }} we find the following reference: &nbsp; <font color="darkgreen">"A somewhat similar problem has been investigated by [https://ui.adsabs.harvard.edu/abs/1967AnAp...30..975G/abstract Gabriel &amp; Ledoux (1967)]. Gaseous configurations with an isothermal core and polytropic envelope of index 3 were studied by {{ HC41 }} and by {{ SC42 }}.  As is well known there is an upper limit (Sch&ouml;nberg-Chandrasekhar limit) to the mass of the core for the configurations to be in hydrostatic equilibria.  Gabriel &amp; Ledoux have investigated the stability of these configurations and have shown that secular stability is lost at the configuration that corresponds to the Sch&ouml;nberg-Chandrasekhar limit."</font>
  </li>
</ul>


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