Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 (section)

====Marginal Stability====
As mentioned above, it is widely appreciated that the model having the largest mass &#8212; that is, the model that sits at <math>~\mathcal{Y}_\mathrm{max}</math> &#8212; along the Stahler sequence is of considerable astrophysical significance.  Viewed in terms of a cloud's secular evolution, counter-clockwise along the sequence, something rather catastrophic must happen once the cloud acquires the mass associated with <math>~\mathcal{Y}_\mathrm{max}</math>, because no equilibrium structure is available to the cloud if it gains any additional mass.  It is tempting to associate this point along the Stahler sequence with a dynamical instability, imagining for example that the cloud will begin to dynamically collapse once it reaches this <math>~\mathcal{Y}_\mathrm{max}</math> configuration.  But the "detailed force-balance" technique that is used to define the structure of equilibrium models along the Stahler sequence does not give us any insight regarding a configuration's dynamical stability. 

Our free-energy analysis ''does'' provide this additional insight.  The mass-radius relationship derived from the scalar virial theorem &#8212; which, itself, was derived via a free-energy analysis &#8212; is qualitatively similar to the mass-radius relationship defined (from a detailed force-balance analysis) by the Stahler sequence; in particular, it also exhibits an upper mass limit.  And our free-energy analysis reveals that this "maximum mass" point associated with the virial theorem separates dynamically stable from dynamically unstable models along the sequence.  This realization fuels the temptation just mentioned; that is, it seems to support the idea that the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence is associated with the onset of a dynamical instability along the sequence.  But this is not the case!  Our free-energy analysis has also shown that, when the structural form-factors &#8212; and, most specifically, the coefficients <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> &#8212; are assigned the values appropriate to  the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence, the point of maximum mass associated with the corresponding expression for the virial theorem does not coincide with the configuration at <math>~\mathcal{Y}_\mathrm{max}</math>.  The configuration at <math>~\mathcal{Y} = \mathcal{Y}_\mathrm{max} =  1.774078</math> (also identified as the model having <math>~\tilde\xi = 3.0</math>) is found to be dynamically stable.  Both of these realizations are illustrated graphically in the [[#GraphicalDepictionXi3|above figure]].


Our analysis has shown, instead, that the marginally unstable configuration appears farther along the Stahler sequence when moving in a counter-clockwise direction.  It corresponds to the model having <math>~\tilde\xi = 3.850652</math> instead of <math>~\tilde\xi = 3.0</math>.   While this can be illustrated graphically &#8212; for example, by carefully analyzing and comparing the bottom-center panel with the top-right panel in the [[#OverlapPlots|above figure ensemble]] &#8212; an algebraic demonstration is more definitive.  [[SSC/Virial/PolytropesEmbeddedOutline#Stability_4|Our stability analysis has shown]] that, for any pressure-truncated polytropic configuration, the equilibrium structure associated with the point of marginal instability has,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\biggl( \frac{\mathcal{Y}}{\mathcal{X}^2}\biggr)_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} 
\, .
</math>
  </td>
</tr>

</table>
</div>
For <math>~n=5</math> configurations, this means that the critical model along the equilibrium sequence will have,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{X}_\mathrm{crit}^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\mathcal{A}_{M_\ell}}{10\pi }\biggr] \mathcal{Y}_\mathrm{crit}^2
\, .
</math>
  </td>
</tr>

</table>
</div>
But all configurations along Stahler's equilibrium sequence must also obey the [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|mass-radius relationship]],
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{Y}^2 - 5\mathcal{Y}\mathcal{X} + \frac{20\pi}{3} \mathcal{X}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0
\, .
</math>
  </td>
</tr>

</table>
</div>
Combining these two requirements means,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{Y}_\mathrm{crit}^2 - 5(\mathcal{Y}\mathcal{X})_\mathrm{crit} + \biggl( \frac{2\mathcal{A}_{M_\ell}}{3}\biggr) \mathcal{Y}_\mathrm{crit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\mathcal{Y}_\mathrm{crit}^2 \biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5(\mathcal{Y}\mathcal{X})_\mathrm{crit} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\frac{ \mathcal{X}_\mathrm{crit} }{ \mathcal{Y}_\mathrm{crit} }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{5}\biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>
Now, taking into detailed account the internal structure of pressure-truncated, <math>~n=5</math> polytropic structures as represented in [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|our summary table of Stahler's equilibrium configurations]], we know that, along Stahler's entire sequence,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{ \mathcal{X} }{ \mathcal{Y} }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \biggl( \frac{3\cdot 5}{2^2 \pi} \biggr) \frac{\ell^2}{(1+\ell^2)^{2}}   \cdot
\biggl( \frac{2^2\pi}{3 \cdot 5^3} \biggr) \frac{(1+\ell^2)^{4}}{(\ell^2)^3} 
\biggr\}^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1 + \ell^2}{5\ell^2}
\, ,
</math>
  </td>
</tr>

</table>
</div>
where we have again adopted the shorthand notation,
<div align="center">
<math>~\ell^2 \equiv \frac{\tilde\xi^2}{3} \, .</math>
</div>

We conclude, therefore, that in the marginally unstable model along the Stahler equilibrium sequence,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~1 + \frac{2}{3}\cdot (\mathcal{A}_{M_\ell})_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\frac{1 + \ell_\mathrm{crit}^2}{\ell_\mathrm{crit}^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~(\mathcal{A}_{M_\ell})_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\frac{3}{2} \cdot \ell_\mathrm{crit}^{-2} \, .
</math>
  </td>
</tr>

</table>
</div>
Given that the [[#Plotting_the_Virial_Theorem_Relation|general expression for]] <math>\mathcal{A}_{M_\ell}</math> along the Stahler sequence is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}_{M_\ell} </math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\frac{1}{2^4} 
\biggl[ \ell^{-4} \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr) + \ell^{-5}  (1 + \ell^2)^{3} \tan^{-1}(\ell ) \biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
we deduce that,
<div align="center">
<math>~\ell_\mathrm{crit} = 2.2231751 </math> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
or, equivalently,
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\xi_\mathrm{crit} = 3.850652 \, .</math>
</div>
Hence, also,
<div align="center">
<math>~( \mathcal{X}_\mathrm{crit},  \mathcal{Y}_\mathrm{crit} ) = ( 0.408738, 1.699778 ) \, . </math> 
</div>