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

===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,

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<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,

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  <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>

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

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  <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>