Editing StabilityVariationalPrinciple (section)

=====Mass=====

Defining <math>~M_\mathrm{tot}</math> as the total mass of the ''isolated'' configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily ''equilibrium'') radius; and being careful to define the mean density of the truncated configuration such that,
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<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>
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we have,
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<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~M_r(r) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \int_0^r  4\pi r^2 \rho dr  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3}{4\pi} \int_0^r  4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr)  \frac{dr}{R_\mathrm{norm}}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^r  3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{dr}{R}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^\xi  3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{d\xi}{\tilde\xi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^\xi  3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{d\xi}{\tilde\xi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}  
\int_0^\xi  3\xi^2 \theta^n  d\xi \, .  
</math>
  </td>
</tr>
</table>
</div>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

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<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
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<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi}  3\xi^2 \theta^n  d\xi  = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>
  </td>
</tr>
</table>
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Now, from the,
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Polytropic Lane-Emden Equation<p></p>
{{ Math/EQ_SSLaneEmden01 }}
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we realize that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi^2 \theta^n \, .</math>
  </td>
</tr>
</table>
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So these last two expressions may also be written as,

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<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr)  {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr]  
\, ; 
</math>
  </td>
</tr>
</table>
</div>
and,

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<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi  \, .</math>
  </td>
</tr>
</table>
</div>