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

====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.