Editing SSC/Virial/PolytropesSummary (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 the figure near the beginning of this discussion.  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>.  To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
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<math>~\eta_\mathrm{ad} </math>
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<math>~\equiv</math>
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<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} = \biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}\, ,</math>
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<math>~\Pi_\mathrm{ad}</math>
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<math>~=</math>
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<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
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<math>~\Chi_\mathrm{ad}</math>
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<math>~=</math>
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<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
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</table>
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====When &eta;<sub>ad</sub> = 0====
For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, <math>\eta_\mathrm{ad}</math>, can be zero for one of two reasons:  Either <math>n=5</math>; or <math>\tilde\theta \rightarrow \theta_{\xi_1} = 0</math>.  In the latter case, all curves converge on the same point, that is, <math>(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0)</math>.  This corresponds to the case of no external medium <math>(P_e = 0)</math> and, hence, the associated equilibrium configuration is the familiar [[SSC/Structure/Polytropes#Polytropic_Spheres|''isolated'' polytropic sphere]].  As can be deduced from our above discussion of the [[SSC/Virial/PolytropesSummary#ConciseVirial|algebraic expression of the virial theorem]], because <math>\Chi_\mathrm{ad} = 1</math>, the equilibrium radius of such a configuration is,
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<math>\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>
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<math>=</math>
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<math>\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>
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</tr>
</table>
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As is demonstrated in an [[SSC/Virial/Polytropes#Strategy2|accompanying discussion]] and also [[SSC/Virial/PolytropesSummary#Physical_Meaning_of_Parameter|mentioned above]], after inserting the relevant expressions for the free-energy coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,
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<math>\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
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<math>=</math>
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<math>\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>
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As we have [[SSC/Virial/Polytropes#Central_and_Mean_Pressure|reviewed elsewhere]] &#8212; see also our [[SSC/Structure/Polytropes#Polytropic_Spheres|detailed discussion of isolated polytropes]] &#8212; this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of [[Appendix/References|Chandrasekhar [C67]]] in association with his discussion of the dimensionless coefficient, <math>W_n</math>, and the central pressure of polytropes.

In the former case &#8212; that is, in the case where <math>\eta_\mathrm{ad} \rightarrow 0</math> because the chosen polytropic index is, <math>n=5</math> &#8212; it must be the case that <math>\Chi_\mathrm{ad} = 1</math> along the entire sequence (see the green curve labeled <math>\gamma = (n+1)/n = 6/5</math> in the accompanying figure).  This means that the expression for the central pressure,
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<tr>
  <td align="right">
<math>\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>=</math>
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<math>\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>
  </td>
</tr>
</table>
</div>
does not explicitly depend on the size of the applied external pressure.  But the central pressure ''does'' depend on the radial location at which the configuration is truncated, via the parameter <math>\tilde\theta^'</math>, which is evaluated at <math>\tilde\xi</math>, rather than at <math>\xi_1</math>.