Editing SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt2 (section)

====P-V Diagram====

For an arbitrary value of the adiabatic exponent, <math>~\gamma_g</math>, it isn't possible to invert this virial relation to obtain an analytic expression for <math>~\chi_\mathrm{ad}</math> as a function of <math>~\Pi_\mathrm{ad}</math>.  But, as written, the virial relation dictates the behavior of <math>~\Pi_\mathrm{ad}</math> as a function of <math>~\Chi_\mathrm{ad}</math>.  Figure 4 displays this <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> behavior for a number of different values of <math>~\gamma_g</math>.

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'''Figure 4:''' <font color="darkblue">Equilibrium Adiabatic P-V Diagram </font> 
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The curves shown here, on the right, trace out the function,
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<math>
~\Pi_\mathrm{ad} = (\Chi_\mathrm{ad}^{4-3\gamma_g} - 1)/\Chi_\mathrm{ad}^4 \, ,
</math>
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for six different values of <math>~\gamma_g</math> &#8212; specifically, for <math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled &#8212;  and show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, pressure-truncated adiabatic sphere with an equilibrium radius <math>~\Chi_\mathrm{ad}</math>.  The solid red curve identifies the behavior of an isothermal <math>~(\gamma_g=1)</math> system.  The mathematical solution becomes unphysical wherever the pressure becomes negative.
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[[File:AdabaticBoundedSpheres_Virial.jpg|450px|center|Equilibrium Adiabatic P-R Diagram]]
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For physically relevant solutions, both <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> must be nonnegative.  Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
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<math>
1 \le \Chi_\mathrm{ad} \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;
</math>

<math>
0 < \Chi_\mathrm{ad} \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .
</math>
</div>

Each of the <math>~\Pi_\mathrm{ad}(\Chi_\mathrm{ad})</math> curves drawn in Figure 4 exhibits an extremum.  In each case this extremum occurs at a configuration radius, <math>~\Chi_\mathrm{extreme}</math>, given by,
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<math>
\frac{\partial\Pi_\mathrm{ad}}{\partial\Chi_\mathrm{ad}} = 0 \, ,
</math>
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that is, where,
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<math>
4 - 3\gamma_g \Chi_\mathrm{ad}^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \Chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .
</math>
</div>
For each value of <math>~\gamma_g</math>, the corresponding dimensionless pressure is,
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<math>
~\Pi_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .
</math>
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In terms of the polytropic index, the equivalent limiting expressions are,
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<math>~\Chi_\mathrm{extreme} = \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~\Pi_\mathrm{extreme}^{(n-3)} = (4n)^{-4n} (n-3)^{n-3} [3(n+1)]^{3(n+1)} \, .</math>
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(In a [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Stability|separate, related discussion of the free-energy function]], we demonstrate that this "extremum" also serves as a dividing line between dynamically stable and unstable models along a given curve.)


In examining the group of plotted curves, notice that, for <math>~\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic &#8212; that is, for all positive &#8212; values of <math>~\Pi_\mathrm{ad}</math>.  Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant &#8212; ''i.e.,'' positive &#8212; pressures only for values of <math>~\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.


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