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

====Plotting the Virial Theorem Relation====

[[#ConciseFreeEnergyExpression|Drawing from our above derivations, the concise free-energy expression]] that reflects the properties of pressure-truncated <math>n = 5</math> polytropic configurations is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{18}{5}\cdot \mathcal{A}_{M_\ell}  \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} 
+~ 5\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, , 
</math>
  </td>
</tr>
</table>
</div>
where,
<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>
  <td align="left">
<math>~
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  
=\frac{1}{3^2\cdot 5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^2 \tilde\mathfrak{f}_W \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3}{4\pi}\biggr)^{1/5} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{6/5}} 
= \frac{1}{3}( 4\pi)^{-1/5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6/5}
\biggl[ \tilde\theta^6 + \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

The virial theorem which is derived from this free-energy expression provides a mass-radius relationship to be compared with the detailed force-balance relationship presented by Stahler.
Because our intent is to make this comparison, we begin with the virial theorem as written in terms of the variables, <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math>, and specialized for the case of <math>~n = 5</math> polytropic configurations.  Written in terms of the (constant) coefficients in the free-energy expression, we have
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathcal{X}^4  
- \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot ( \mathcal{X} \mathcal{Y}^3 )^{2/5}  
+~ \frac{9 \mathcal{A}_{M_\ell}}{10\pi}\cdot \mathcal{Y}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ;</math>
  </td>
</tr>
</table>
</div>
or, [[#ConciseVirialXY|from above]], the
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>'''</font><br />

<math>
\mathcal{X}^4   ~- ~
(\mathcal{X} \mathcal{Y}^3)^{2/5} 
\biggl[\frac{\tilde\xi}{4\pi (-\tilde\theta^')}\biggr]^{6/5} \biggl[\tilde\theta^{6} +  \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr]
~+ ~ \mathcal{Y}^2 \biggl[\frac{\tilde\xi}{(- \tilde\theta^')} \biggr]^2 \frac{\tilde\mathfrak{f}_W}{2\cdot 5^2 \pi} 
= 0 \, ,
</math>
</td></tr>
</table>
</div>
where, specifically for <math>~n = 5</math> polytropic configurations &#8212; see our [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|summary of the radial profiles of physical variables]] and our [[SSCpt1/Virial/FormFactors#Summary_.28n.3D5.29|determination of expressions for the structural form-factors]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
( 1+ \ell^2 )^{-1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tilde\theta^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
- 3^{-1/2} \ell ( 1+ \ell^2 )^{-3/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\frac{5}{2^4} \cdot \ell^{-5}  
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\ell^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~  
\frac{\tilde\xi^2}{3} \, .
</math>
  </td>
</tr>
</table>
</div>

Once numerical values have been assigned to the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, the mass-radius relationship given by the scalar virial theorem can be compared quantitatively with Stahler's (detailed force-balance) mass-radius relationship.  The simplest, physically reasonable approximation would be to assume uniform-density structures, in which case, <math>~\tilde\mathfrak{f}_M = \tilde\mathfrak{f}_W = \tilde\mathfrak{f}_A = 1</math>, and accordingly, <math>~\mathcal{A}_{M_\ell} = 5^{-1}</math> and <math>~\mathcal{B}_{M_\ell} = (4\pi/3)^{-1/5}</math>.  But a better approximation would be to assign values to the structural form-factors that properly represent the properties of at least one detailed force-balanced model.  By way of illustration, the following table details what the proper values are for the two free-energy coefficients, and other relevant parameters, specifically for the model along Stahler's sequence that sits at <math>~\mathcal{Y}_\mathrm{max}</math> &#8212; that is, the model whose truncation radius is <math>~\tilde\xi = 3</math>.  As is recorded in the table, in this case the precise values of the free-energy coefficients are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[  1 +
\biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Notice that, by choosing <math>~\tilde\xi = 3</math>, the evaluation of <math>~\tilde\mathfrak{f}_W</math> is particularly simple, in part, because <math>~\tan^{-1}(\ell) = \tan^{-1} \sqrt{3} = \pi/3</math>, but also because the term <math>~(\ell^4 - 8\ell^2/3 - 1)</math> equals zero.


<div align="center" id="ExampleXi3">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="3">
Determination of Coefficient Values in the Specific Case of <math>~\tilde\xi = 3</math>
  </th>
</tr>
<tr>
  <td align="center">
Quantity
  </td>
  <td align="center">
Analytic Evaluation
  </td>
  <td align="center">
Numerical
  </td>
</tr>
<tr>
  <td align="center">
<math>~\ell</math>
  </td>
  <td align="center">
<math>~3^{1/2}</math>
  </td>
  <td align="left">
<math>~1.732051</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\tilde\theta</math>
  </td>
  <td align="center">
<math>~2^{-1}</math>
  </td>
  <td align="left">
<math>~0.5</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\tilde\theta^'</math>
  </td>
  <td align="center">
<math>~2^{-3}</math>
  </td>
  <td align="left">
<math>~0.125</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{3^2\cdot 5}{2^6\pi} \biggr)^{1/2}</math>
  </td>
  <td align="left">
<math>~0.473087</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2}</math>
  </td>
  <td align="left">
<math>~1.774078</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{5^2 \pi^2}{2^8\cdot 3^7} \biggr)^{1/2}</math>
  </td>
  <td align="left">
<math>~0.020993</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2}</math>
  </td>
  <td align="left">
<math>~0.268711</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[  1 +
\biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>
  </td>
  <td align="left">
<math>~0.830395</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~G^*</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{3\cdot 5^3}{2^{12}\pi} \biggr)^{1/2} [  2^3\pi + 3^{5/2} ]</math>
  </td>
  <td align="left">
<math>~6.951544</math>
  </td>
</tr>

<tr>
  <td align="left">
Virial: 

<math>~\mathcal{X}^4 - \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot (\mathcal{X}\mathcal{Y}^3 )^{2/5}</math>

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~~~~~~~~~~+\frac{9 \mathcal{A}_{M_\ell}}{10\pi} \cdot \mathcal{Y}^2</math>
  </td>
  <td align="left">
<math>~ \frac{3^4\cdot 5^2}{2^{12}\pi} - 
\frac{3^4\cdot 5^2}{2^{12}\pi} \biggl[  1 + \biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~~~~~~~~~~+ \biggl( \frac{3^5\cdot 5^4}{2^{18} \pi^2} \biggr)^{1/2}
</math>
  </td>
  <td align="left">
Sums to zero, 

exactly!
  </td>
</tr>
</table>
</div>


The curve traced out by the light-blue diamonds in each panel of the following comparison figure displays Stahler's analytically prescribed mass-radius relation; this curve is identical in all six panels and is the same as the [[#Plotting_Stahler.27s_Relation|curve displayed above]] in connection with our description of Stahler's mass-radius relation.  Each point along this "Stahler" curve identifies a model having a different truncation radius, <math>~\tilde\xi</math>; as plotted here, starting near the origin and moving counter-clockwise around the curve, <math>~\tilde\xi</math> is varied from 0.05 to 42.5.  As foreshadowed by the above discussion, the model having the greatest mass <math>~(\mathcal{Y}_\mathrm{max})</math> along the Stahler sequence &#8212; highlighted by the red filled circle in most of the figure panels &#8212;  is defined by <math>~\tilde\xi = 3</math>.  

In each figure panel, the curve traced out by the orange triangles &#8212; or, in one case, the orange triangles &amp; light purple diamonds &#8212; displays the mass-radius relation defined by the virial theorem.  These "Virial" curves are all defined by the same virial theorem polynomial expression, as just presented, but the coefficient of the <math>~\mathcal{Y}^2</math> term and the coefficient of the <math>~(\mathcal{X}\mathcal{Y}^3)^{2/5}</math> cross term &#8212; essentially, the value of <math>~\mathcal{A}_{M_\ell}</math> and the value of <math>~\mathcal{B}_{M_\ell}</math>, respectively &#8212; have different values in the six separate figure panels.  In each case, a value has been specified for the parameter, <math>~\tilde\xi</math> (as identified in the title of each figure panel), and this, in turn, has determined the values of the two (constant) free-energy coefficients.  For example, in the top-right figure panel whose title indicates <math>~\tilde\xi = 3</math>, the "Virial" curve traces the mass-radius relation prescribed by the virial theorem after the values of the free-energy coefficients have been set to values that correspond to a detailed force-balanced model with this specified truncation radius, that is (see the above table),  <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>.  Columns 2 and 3, respectively, of the table affixed to the bottom of the following comparison figure list the values of <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> that have been used to define the "Virial" curve in each of the six figure panels, in accordance with the value of <math>~\tilde\xi</math> listed in column 1 of the table.


<div align="center" id="OverlapPlots">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 5'' Polytropes
  </th>
</tr>
<tr>
<td align="center" colspan="3">
[[File:Xi1.26419.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi2.0.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi3.0.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
</tr>
<tr>
<td align="center" colspan="3">
[[File:Xi3.5.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi3.850652.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi9.8461.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
</tr>

<tr>
  <td align="center" rowspan="2">
<math>~\tilde\xi</math>
  </td>
  <td align="center" colspan="2">
Free-Energy Coefficients
  </td>
  <td align="center" colspan="2">
Primary Overlap
  </td>
  <td align="center" colspan="2">
Secondary Overlap

(see [[#SecondaryOverlap|further elaboration below]])
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
  <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>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~\mathcal{Y}</math>
  </td>
</tr>

<tr>
<td align="center">
1.26419
</td>
<td align="center">
0.214429
</td>
<td align="center">
0.758099
</td>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904142
</td>
<td align="center">
0.388938
</td>
<td align="center">
0.289568
</td>
</tr>

<tr>
<td align="center">
2
</td>
<td align="center">
0.233842
</td>
<td align="center">
0.779836
</td>
<td align="center">
0.540671
</td>
<td align="center">
1.544775
</td>
<td align="center">
0.468918
</td>
<td align="center">
0.570916
</td>
</tr>

<tr>
<td align="center">
[[#ExampleXi3|3]]
</td>
<td align="center">
0.268711
</td>
<td align="center">
0.830395
</td>
<td align="center">
0.473087
</td>
<td align="center">
1.774078
</td>
<td align="center">
0.507387
</td>
<td align="center">
0.798441
</td>
</tr>

<tr>
<td align="center">
3.5
</td>
<td align="center">
0.288708
</td>
<td align="center">
0.861503
</td>
<td align="center">
0.434310
</td>
<td align="center">
1.744359
</td>
<td align="center">
0.515168
</td>
<td align="center">
0.859518
</td>
</tr>

<tr>
<td align="center">
3.850652
</td>
<td align="center">
0.303490
</td>
<td align="center">
0.884746
</td>
<td align="center">
0.408738
</td>
<td align="center">
1.699778
</td>
<td align="center">
0.518588
</td>
<td align="center">
0.888969
</td>
</tr>

<tr>
<td align="center">
9.8461
</td>
<td align="center">
0.601012
</td>
<td align="center">
1.313904
</td>
<td align="center">
0.186424
</td>
<td align="center">
0.904141
</td>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904143
</td>
</tr>
</table>
</div>


As is [[#Quantitative_Study|discussed more fully, below]], in each of the six panels of the above comparison figure, the "Virial" curve intersects the "Stahler" curve at two locations.  These points of intersection are identified by the black filled circles in each figure panel.  In each case, the intersection point that is farthest along on the Stahler sequence &#8212; as determined by starting at the origin and moving counter-clockwise along the sequence &#8212; identifies the "Primary Overlap" between the two curves.  That is to say, the <math>~(\mathcal{X}, \mathcal{Y})</math> coordinates of this point (see columns 4 and 5 of the table affixed to the bottom of the figure) are the coordinate values that are obtained by plugging the specified value of <math>\tilde\xi</math> (see the title of the figure panel or column 1 of the affixed table) into [[#Parametric_Relations|Stahler's pair of parametric relations]].  The second point of intersection in each panel &#8212; which we will refer to as the "Secondary Overlap" points and whose coordinates are provided in columns 6 and 7 of the affixed table &#8212; appears to be fortuitous and of no particularly significant astrophysical interest.


The mass-radius diagram displayed in the top-right panel of the [[#OverlapPlots|above comparison figure]] has been reproduced in the upper-left panel of the [[#GraphicalDepictionXi3|following figure]] &#8212; in this case, with a coordinate aspect ratio that is closer to 1:1  &#8212; along with color images of the corresponding free-energy surface, viewed from two different perspectives, and a three-column table listing the 3D coordinates, <math>~(X,Y,Z) = (R_\mathrm{eq}/R_\mathrm{SWS}, M_\mathrm{limit}/M_\mathrm{SWS}, \mathfrak{G}^*)</math>, of the seventeen points that have been used to define the displayed "Virial" curve.  To be more explicit, the rainbow-colored free-energy surface, <math>~\mathfrak{G}^*(R,M_\mathrm{limit})</math>, has been defined by the free-energy function appropriate to pressure-truncated <math>~n=5</math> polytropic configurations [[#Plotting_the_Virial_Theorem_Relation|as defined above]], that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{18}{5}\cdot \mathcal{A}_{M_\ell}  \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} 
+~ 5\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, , 
</math>
  </td>
</tr>
</table>
</div>

with the values of the two free-energy coefficients set to the values that correspond to a <math>~\tilde\xi = 3</math> virial curve as discussed above, namely, <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>.


<div align="center" id="GraphicalDepictionXi3">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="2">
Free-Energy Surface for ''n = 5'' Polytropic Configurations Truncated at <math>~\tilde\xi = 3</math>
  </th>
  <th align="center" colspan="1">
Radius
  </th>
  <th align="center" colspan="1">
Mass
  </th>
  <th align="center" colspan="1">
Free

Energy
  </th>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:VirialEquilibria.png|150px|Virial Mass-Radius Relation]]
</td>
<td align="center" rowspan="8" bgcolor="CCFFFF">
[[File:FreeEnergyFaceOn.png|325px|Virial Mass-Radius Relation on top of Free-Energy Surface]]
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.3152
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.1345
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.8231
</td>
</tr>

<tr>
<td align="right" rowspan="1">
0.4159
</td>
<td align="right" rowspan="1">
0.3470
</td>
<td align="right" rowspan="1">
1.9947
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.4731
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.5735
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
3.1095
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="white">
0.5035
</td>
<td align="right" rowspan="1" bgcolor="white">
0.7661
</td>
<td align="right" rowspan="1" bgcolor="white">
3.9591
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="white">
0.5114
</td>
<td align="right" rowspan="1" bgcolor="white">
0.8347
</td>
<td align="right" rowspan="1" bgcolor="white">
4.2409
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.5297
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.0759
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
5.1453
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.5310
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.4036
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.1560
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="white">
0.5250
</td>
<td align="right" rowspan="1" bgcolor="white">
1.5010
</td>
<td align="right" rowspan="1" bgcolor="white">
6.4046
</td>
</tr>

<tr>
<td align="center" colspan="2" rowspan="9" bgcolor="#CCFFFF">
[[File:FreeEnergySurface.png|450px|Free-Energy Surface]]
</td>
<td align="right" rowspan="1" bgcolor="white">
0.5185
</td>
<td align="right" rowspan="1" bgcolor="white">
1.5680
</td>
<td align="right" rowspan="1" bgcolor="white">
6.5606
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.5114
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.6206
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.6741
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
0.4731</font>
</td>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
1.7741</font>
</td>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
6.9515</font>
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.4343
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.8287
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
7.0247
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="red"><font color="white">
0.3984</font>
</td>
<td align="right" rowspan="1" bgcolor="red"><font color="white">
1.8333</font>
</td>
<td align="right" rowspan="1" bgcolor="red"><font color="white">
7.0301</font>
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.3379
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.7764
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.9923
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.2911
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.6909
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.9541
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.2260
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.5229
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.9091
</td>
</tr>

<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.1545
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.2749
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.8786
</td>
</tr>
</table>
</div>

In the bottom panel of this figure, the undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy.  Twelve small colored dots identify extrema &#8212; either the bottom of a valley or the top of a ridge &#8212; in the free-energy function and therefore trace the mass-radius relation defined by the scalar virial theorem.  The 3D coordinates of these twelve points are provided in the three-column table that is affixed to the righthand edge of the figure:  The coordinates of the (only) red dot are provided in row 13 of the table (red has also been assigned as the "bgcolor" of this table row); the equilibrium configuration having the greatest mass along the intersecting Stahler sequence is identified by the (only) black dot (coordinates are provided in row 11 of the table and, correspondingly, bgcolor="black" for that row); bgcolor="lightblue" has been assigned to the other rows of the affixed table that provide coordinates of the other 10, blue dots.

The upper-right panel of this figure presents the two-dimensional projection that results from viewing the identical 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-mass <math>~(X-Y)</math> plane. From this vantage point, the twelve small colored dots cleanly trace out the <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> equilibrium sequence that is defined by the scalar virial theorem, exactly reproducing the "Virial" curve that is depicted in the mass-radius diagram shown in the upper-left panel of the figure.