Editing SSCpt1/Virial/PolytropesEmbeddedOutline (section)

====ASIDE:  Isothermal Configurations====
While our focus in this chapter is on polytropic systems, it is advantageous to review [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] discussion of pressure-truncated isothermal configurations because that discussion includes presentation of a free-energy surface &#8212; see, specifically, Whitworth's Figure 2.   

Setting <math>\delta_{1\eta} = 1</math> in Whitworth's free-energy expression (his equation 10, [[#Coefficient_Definitions|reprinted above]]) gives,  
<div align="center" id="WhitworthFreeEnergyExpression">
<math>
\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = -\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} 
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 
-2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr) - \mathfrak{G}_0 \, ,
</math>
</div>
where, as earlier, we have inserted the additional constant, <math>\mathfrak{G}_0</math>, to accommodate normalization. A segment of the free-energy surface defined by this function is displayed in the right-hand panel of our [[#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|Figure 1, at the top of this page]].  In constructing this figure, <math>\mathfrak{G}_0</math> has been set to a value that ensures that <math>\mathcal{U}</math> is everywhere positive over the displayed domain: <math>0.1 \le P_e/P_\mathrm{rf} \le 1.1</math> and <math>0.3 \le R/R_\mathrm{rf} \le 3.0</math>.     

<span id="ScalarVirialTheorem">
For a given choice of <math>P_e</math>, equilibrium radii are identified by setting <math>d\mathcal{U}/dR = 0</math>, that is, they are defined by the (scalar virial theorem) relation,</span>
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{P_e}{P_\mathrm{rf}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr) - 3  \biggr] 
\, ;
</math>
  </td>
</tr>
</table>

and equilibria are stable  &#8212; that is, <math>[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math> &#8212; if <math>R_\mathrm{eq}/R_\mathrm{rf} > 1</math>.  For physically realistic systems, of course, <math>P_e</math> must be positive &#8212; which means that all equilibria have <math>R_\mathrm{eq}/R_\mathrm{rf} > 3/4</math>.  But, from this algebraic virial theorem expression, it is also clear that physically realistic equilibrium configurations only exist when <math>P_e/P_\mathrm{rf} \le 1</math>.  The sequence of small, colored spherical dots in the right-hand panel of our [[#IntroFigures|Figure 1]] identify parameter-value pairs, <math>(R_\mathrm{eq}, P_e)</math>, associated with fourteen different equilibrium configurations:  Blue dots &#8212; lying along the valley of the free-energy surface &#8212; identify stable configurations; white dots &#8212; balanced along the crest of the surface ridge &#8212; identify dynamically unstable configurations; and the lone red dot identifies the critical neutral equilibrium state, which is also associated with the maximum allowable value of <math>P_e</math> along the equilibrium model sequence.


<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">
<!-- [[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]] -->
Figure 2 extracted without modification from &sect;6 (p. 973) of &hellip;<br />
MNRAS, 195: 967-977, 1981 June<br />
GLOBAL GRAVITATIONAL STABILITY FOR ONE-DIMENSIONAL POLYTROPES<br />
Ant. Whitworth<br />
</td></tr>
<tr>
  <td align="center">[[File:Whitworth81Figure2.png|600px|Figure 2 from Whitworth (1981)]]
</tr>
<tr>
  <td align="left">
Caption (''verbatim''): &nbsp; The continuous lines are potentials controlling radial motions of a uniform-density spherical cloud with isothermal equation of state (i.e., <math>\eta = 1</math>), for several different values of the external pressure <math>P_\mathrm{ex}</math> (as labelled).  The filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state.  The dotted lines are freefall collapse potentials for comparison.
  </td>
</tr>
</table>


Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)] &#8212; reproduced immediately above &#8212; does an excellent job of conveying many of the essential elements of this isothermal free-energy surface within the constraints imposed by a two-dimensional black &amp; white line plot.  In order to construct this compact plot, Whitworth employed a different free-energy normalization parameter for each selected value of the external pressure.  Specifically, he used,
<div align="center">
<math>\mathfrak{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr] 
- \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>
</div>
In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>\mathfrak{G}_0(P_e)</math> normalization function when generating the multicolored, three-dimensional free-energy surface that is displayed (with three different projections) immediately below.  Aside from this pressure-dependent normalization parameter, the multi-colored free-energy surface that has been drawn, here, for comparison with Whitworth's Figure 2 is identical to the one displayed in the right-hand panel of our Figure 1 (see [[#IntroFigures|the top of this page]]).


<div align="center" id="3DIsothermalSurface">
<table border="1" cellpadding="5" align="center" width="80%">
<tr>
<td align="center" colspan="2">
Figure 3: &nbsp; Our Depiction of Whitworth's (1981) 3D Isothermal Free-Energy Surface<br />As Seen From Different Lines of Sight
</td>
<td align="center" rowspan="5">
<table border="0" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="4">
Table 1:Properties of<br />Selected Virial Equilibria
  </th>
</tr>
<tr>
  <td align="center">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>\frac{P_e}{P_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>\mathfrak{G}_0</math>
  </td>
</tr>

<tr>
  <td align="right">
2.395
  </td>
  <td align="right">
0.200
  </td>
  <td align="right">
-0.03205
  </td>
  <td align="right">
-1.8831
  </td>
</tr>

<tr>
  <td align="right">
2.008
  </td>
  <td align="right">
0.30946
  </td>
  <td align="right">
-0.04507
  </td>
  <td align="right">
-1.6786
  </td>
</tr>

<tr>
  <td align="right">
1.800
  </td>
  <td align="right">
0.400
  </td>
  <td align="right">
-0.05548
  </td>
  <td align="right">
-1.5646
  </td>
</tr>

<tr>
  <td align="right">
1.622
  </td>
  <td align="right">
0.50364
  </td>
  <td align="right">
-0.06730
  </td>
  <td align="right">
-1.4666
  </td>
</tr>

<tr>
  <td align="right">
1.452
  </td>
  <td align="right">
0.632
  </td>
  <td align="right">
-0.08213
  </td>
  <td align="right">
-1.3743
  </td>
</tr>

<tr>
  <td align="right">
1.347
  </td>
  <td align="right">
0.72543
  </td>
  <td align="right">
-0.09327
  </td>
  <td align="right">
-1.3205
  </td>
</tr>

<tr>
  <td align="right">
1.270
  </td>
  <td align="right">
0.800
  </td>
  <td align="right">
-0.10252
  </td>
  <td align="right">
-1.2835
  </td>
</tr>

<tr>
  <td align="right">
1.127
  </td>
  <td align="right">
0.93516
  </td>
  <td align="right">
-0.12070
  </td>
  <td align="right">
-1.2263
  </td>
</tr>

<tr>
  <td align="right">
1.071
  </td>
  <td align="right">
0.97564
  </td>
  <td align="right">
-0.12681
  </td>
  <td align="right">
-1.2111
  </td>
</tr>

<tr>
  <td align="right" bgcolor="red">
<font color="white">1.000</font>
  </td>
  <td align="right" bgcolor="red">
<font color="white">1.000</font>
  </td>
  <td align="right" bgcolor="red">
<font color="white">-0.13086</font>
  </td>
  <td align="right" bgcolor="red">
<font color="white">-1.2025</font>
  </td>
</tr>

<tr>
  <td align="right">
0.9612
  </td>
  <td align="right">
0.9897
  </td>
  <td align="right">
-0.12880
  </td>
  <td align="right">
-1.2061
  </td>
</tr>

<tr>
  <td align="right">
0.9061
  </td>
  <td align="right">
0.92636
  </td>
  <td align="right">
-0.11370
  </td>
  <td align="right">
-1.2297
  </td>
</tr>

<tr>
  <td align="right">
0.859
  </td>
  <td align="right">
0.800
  </td>
  <td align="right">
-0.07423
  </td>
  <td align="right">
-1.2835
  </td>
</tr>

<tr>
  <td align="right">
0.822
  </td>
  <td align="right">
0.632
  </td>
  <td align="right">
0.000
  </td>
  <td align="right">
-1.3743
  </td>
</tr>

<tr>
  <td align="right">
0.789
  </td>
  <td align="right">
0.400
  </td>
  <td align="right">
+0.17021
  </td>
  <td align="right">
-1.5646
  </td>
</tr>

<tr>
  <td align="right">
0.767
  </td>
  <td align="right">
0.200
  </td>
  <td align="right">
+0.47300
  </td>
  <td align="right">
-1.8831
  </td>
</tr>
</table>
</td>
</tr>
<tr>
<td align="center" colspan="1" rowspan="1" bgcolor="#C0FFFF">
[[File:EnergyRadiusViewTop5.png|350px|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
<td align="center" colspan="1" rowspan="1" bgcolor="#CCFFFF">
[[File:FEmovie02.gif|350px|Animated Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center" colspan="2" rowspan="1" bgcolor="#C0FFFF">
[[File:EnergyRadiusViewBottom6.png|600px|Whitworth's (1981) Isothermal Free-Energy Surface]]
<!-- [[Image:AAAwaiting01.png|440|Whitworth's (1981) Isothermal Free-Energy Surface]] -->
</td>
</tr>
<tr>
  <td align="left" colspan="2">
Graphical depictions of the free-energy surface, <math>~\mathfrak{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[#WhitworthFreeEnergyExpression|above]].  
*''Upper-right quadrant'':  The undulating free-energy surface is drawn in 3D and viewed from a (time-varying) vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy.  The three coordinate axes are labeled and colored as follows:  Radius (red; hereafter, <math>X</math>), External Pressure (green; hereafter, <math>Y</math>), and Free Energy (blue; hereafter, <math>Z</math>).  The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots:  Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.
*''Upper-left quadrant'':  The two-dimensional projected image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>(Z)</math> axis and looking directly down onto the radius-pressure <math>(X-Y)</math> plane.  From this vantage point, the sixteen small colored dots cleanly trace out the <math>P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]].  
*''Bottom'':  The two-dimensional projected image that results from viewing the free-energy surface "from underneath," along a line of sight that is parallel to the external-pressure <math>(Y)</math> axis and looking directly up at the radius-free-energy <math>(X-Z)</math> plane.  This image can be directly compared with [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] Figure 2.  (Whitworth's black &amp; white plot has been reprinted next to our multi-colored image, and the aspect-ratio of the plot has been modified slightly, in order to facilitate comparison.)  Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image.  For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is peeking from behind the "Radius" axis &#8212; that is, the unstable equilibrium configuration that has a free-energy value of zero &#8212; corresponds to the open circle in Whitworth's plot that is labeled as <math>P_e = 0.632</math> (see also the coordinate values given in our accompanying, Table 1).
  </td>
</tr>
</table>
</div>


Caption to Table 1, which accompanies our Figure 3: &nbsp; Coordinates <math>(X, Y, Z)</math> = <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathfrak{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the free-energy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.]  Seven of the nine equilibrium configurations that are identified in Whitworth's Figure 2 are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red.  For completeness, the value of the corresponding normalization energy, <math>\mathfrak{G}_0(P_e)</math>, is also tabulated.