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

===Part II:  Curve Intersections===
====Early Thoughts====
Notice that, in each frame of the [[#OverlapPlots|above comparison figure]], the "Virial" curve intersects and crosses the "Stahler" curve at two locations.  In each plot these two crossing points are identified by filled black circles and, in each plot, the crossing point that lies farthest along the curve &#8212; again, starting from near the origin and moving around the curve in a counter-clockwise direction &#8212; is associated with the equilibrium model on the Stahler curve that is defined by the same value of <math>~\tilde\xi</math> that was used to define the coefficients of the virial theorem mass-radius relation.  This is not surprising, as the virial theorem ''should'' be precisely satisfied by every one of the equilibrium models along the Stahler sequence, as long as the value of <math>~\tilde\xi</math> that is used to define the coefficients of the free-energy function and, in turn, the virial theorem mass-radius relation is identical to the value of <math>~\tilde\xi</math> that defines the truncation radius of the detailed force-balance model.  For example, the "Virial" curve that appears in the top-right panel of the comparison figure &#8212; a panel whose title includes the notation, <math>~\xi = 3</math> &#8212; intersects the "Stahler" curve at <math>~\mathcal{Y}_\mathrm{max}</math>, that is, at the location of the detailed force-balance model that, as previously explained, has a truncation radius, <math>~\tilde\xi = 3</math>.  

It is not (yet) clear to us what physical significance should be ascribed to the model along the Stahler sequence that is identified by the ''second'' crossing of the "Virial" curve, given that the value of <math>~\tilde\xi</math> associated with the truncation radius of this ''second'' detailed force-balance model is not the same as the value of <math>~\tilde\xi</math> that was used to define the coefficients of the "Virial" curve.  We note that, at least for the range of values of <math>~\tilde\xi</math> sampled in the above figure, this second crossing point seems to hover around the same limited segment of the Stahler sequence.  

By direct analogy with [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|discussions of Bonnor-Ebert spheres]], the "maximum mass" model associated with <math>~\mathcal{Y}_\mathrm{max}</math> along the Stahler mass-radius relation has important physical significance in astrophysics.  For a given applied external pressure, however, no models exist above some limiting mass &#8212; identified, here, by <math>~\mathcal{Y}_\mathrm{max}</math>.

====Analysis Philosophy====
The mass-radius relationship that derives from detailed force-balanced models is a physically meaningful and reliable statement of how a configuration's equilibrium radius will vary if its mass is changed.  (It must be accepted that the configuration's structural form factors will change as it settles into each new equilibrium state, so such an "evolution" must occur on a ''secular'' time scale.)  From the outset, however, the mass-radius relationship derived via the virial theorem &#8212; which, itself, derives from an analysis of the free energy function &#8212; should not be relied upon for the same physical insight.  Consider, for example, that the scalar virial theorem is obtained from an analysis of the free-energy function by varying a system's size ''while holding constant all coefficients in the free-energy expression''; this means that the system's mass as well as its structural form factors is held fixed while searching for an extremum in the free energy.  The temptation, then, is to use the virial theorem to predict what the configuration's new equilibrium size will be if the system's mass is changed while holding the coefficients in the virial theorem constant.  This means holding the structural form factors constant but not simultaneously holding the mass constant, and this differs from the constraints put on the free-energy function analysis that led to the virial theorem expression in the first place!

But we can combine the two analyses &#8212; the detailed force-balance analysis and the free-energy analysis &#8212; in the following meaningful way.  Use the detailed force-balance analysis to identify the properties of an equilibrium state, specifically, for a given mass, determine the system's equilibrium radius and its accompanying structural form factors.  (The virial theorem will be satisfied by this same set of determined parameter values.)  Then, holding both the mass and the structural form factors constant, see how the free energy of the system varies as the configuration's size changed.  In this manner the system's ''dynamical'' stability can be ascertained.

In summary:  The mass-radius relationship determined from an analysis of detailed force-balanced models defines the physically correct ''secular'' evolutionary track for the system; while, an analysis of the free energy variations about an equilibrium state will answer the question of ''dynamical'' stability.

====Quantitative Study====
The preceding philosophical statements not withstanding, it is still worth understanding the relationship &#8212; if any &#8212; between the pair of models that are identified by the "second crossing" of the Stahler sequence by the "Virial" curve.

<div align="center" id="SecondaryOverlap">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="6">
More Information on Secondary Overlap Points
  </td>
</tr>
<tr>
  <td align="center" colspan="2">From [[#OverlapPlots|above table]]</td>
  <td align="center" colspan="4">Determined here</td>
</tr>
<tr>
  <td align="center">
<math>~\mathcal{X} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~\mathcal{Y} \equiv \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~\tilde\xi_-</math>
  </td>
  <td align="center">
<math>~\mathcal{Y}_-</math>
  </td>
  <td align="center">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
</tr>
<tr>
<td align="center">
0.388938
</td>
<td align="center">
0.289568
</td>
  <td align="center">
0.72447
  </td>
<td align="center">
0.289568
</td>
  <td align="center">
0.20491
  </td>
  <td align="center">
0.75191
  </td>
</tr>

<tr>
<td align="center">
0.468918
</td>
<td align="center">
0.570916
</td>
  <td align="center">
0.98268
  </td>
<td align="center">
0.570916
</td>
  <td align="center">
0.20889
  </td>
  <td align="center">
0.75395
  </td>
</tr>

<tr>
<td align="center">
0.507387
</td>
<td align="center">
0.798441
</td>
<td align="center">
1.17380
</td>
<td align="center">
0.798441
</td>
<td align="center">
0.21252
</td>
<td align="center">
0.75652
</td>
</tr>

<tr>
<td align="center">
0.515168
</td>
<td align="center">
0.859518
</td>
<td align="center">
1.22572
</td>
<td align="center">
0.859520
</td>
<td align="center">
0.21360
</td>
<td align="center">
0.75740
</td>
</tr>

<tr>
<td align="center">
0.518588
</td>
<td align="center">
0.888969
</td>
<td align="center">
1.25104
</td>
<td align="center">
0.888968
</td>
<td align="center">
0.21414
</td>
<td align="center">
0.75785
</td>
</tr>

<tr>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904143
</td>
<td align="center">
1.26419
</td>
<td align="center">
0.904142
</td>
<td align="center">
0.21443
</td>
<td align="center">
0.75810
</td>
</tr>

</table>
</div>


====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>