Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1 (section)

===Model Sequences===
<!-- [[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]-->
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted.  Half-a-dozen such model sequences are shown in Figure 1, above.  Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>.  
<table border="1" cellpadding="10" width="90%" align="center">
<tr><td align="left">
<font color="maroon">'''ASIDE'''</font> [18 March 2015]:  Many months after I penned the above description of "evolution" along an equilibrium model sequence, I started analyzing in detail the paper by  {{ Kimura81bfull }}.  The following excerpt from &sect;3 of his paper shows that Kimura presented essentially the same description of "evolution along a sequence" several decades ago:
<!-- [[File:Kimura1981bExcerpt.png|450px|center|border|Excerpt from section 3 of Kimura (1981b)]] -->
<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"It can be seen that if a certain quantity, say <math>Q_1</math>, is fixed, then a sequence of bounded polytropes is constructed by varying a truncation parameter <math>\zeta_1</math> in the range <math>\zeta_0 < \zeta_1 < \zeta_f</math>, where <math>\zeta_0</math> is the dimensionless radius of the inner boundary, and <math>\zeta_f</math> that of the free surface.  Such a sequence will be termed a '<math>Q_1</math>-sequence'."
</font>
</td></tr>
<tr><td align="right">
— Drawn from {{ Kimura81bfull }} 
</td></tr></table>

Kimura uses the subscript "1" to denote the equilibrium value of any physical quantity "<math>Q</math>"; in Figure 1 above, we are holding the equilibrium mass fixed while allowing the external pressure and the configuration volume to vary, so Kimura would say that the figure displays various "M_1 sequences."  And, as is explained more fully in [[SSC/Structure/PolytropesEmbedded#Kimura.27s_Presentation|an accompanying discussion]], his "truncation parameter" is essentially the same as our truncation radius &#8212; specifically, <math>\zeta_1 = (n+1)^{1/2}\tilde\xi</math>.  When projected onto our discussion, the physically relevant range of truncation parameter values is, <math>0 \le \tilde\xi \le \xi_1</math>, where <math>\xi_1</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden radius of an ''isolated'' (unbounded) polytropic sphere]].
</td></tr>
</table>

To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5 ~\tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \tilde\mathfrak{f}_W} 
= \frac{\tilde\theta^{n+1}}{\tilde\theta^{n+1} + 3(\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>
Note that the last expression for <math>\eta_\mathrm{ad}</math> has been obtained after inserting the analytic expression for the structural form-factor, <math>\tilde\mathfrak{f}_W</math> that &#8212; as has been explained in an [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|accompanying discussion]] &#8212; we derived with the help of {{ VH74full }}.