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===Animation=== The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having nineteen different total angular momentum values, <math>~L</math>, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] points out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy. <div align="center" id="Figure3"> <table border="0" cellpadding="5" align="center"> <tr> <th align="center" colspan="1"><font size="+1">Figure 3:</font> Animation </th> </tr> <tr><td align="center"> <table border="1" cellpadding="0" align="center"> <tr> <td align="center" colspan="1" bgcolor="#CCFFFF"> [[File:JacobiMaclaurin2.gif|640px|Animation related to Fig. 3 from Christodoulou1995]] </td> </tr> </table> </td></tr> </table> </div> <!-- "OVERLAID" IMAGES SHOWING RELATIONSHIP BETWEEN B/W CONTOUR PLOT AND (1) OUR JACOBI SEQUENCE PLOT; (2) THE RIEMANN SEQUENCE DIAGRAM FROM EFE; AND (3) OUR COLORED-CONTOUR PLOT <div align="center" id="Figure3"> <table border="0" cellpadding="5" align="center"> <tr> <th align="center" colspan="1"><font size="+1">Figure 3:</font> Overlaying Free-Energy Contour Diagrams and Equilibrium Sequence Diagrams </th> </tr> <tr><td align="center"> <table border="1" cellpadding="0" align="center"> <tr> <td align="center" colspan="2"> [[File:OverlapAttempt3.png|250px|Jacobi Sequence]] </td> <td align="center" colspan="2"> [[File:OverlapAttempt1.png|250px|Chandrasekhar Figure2]] </td> <td align="center" colspan="2"> [[File:VisTrailsFig3e.png|250px|Christodoulou1995Fig3 Flipped]] </td> </tr> </table> </td></tr> </table> </div> --> <div align="center" id="Table1"> <table border="1" cellpadding="5" align="center" width="500px"> <tr> <td align="center" colspan="8"><b><font size="+1">Table 1:</font> Parameter Values Associated with Each Frame of the Figure 3 Animation</b><br /> (parameter values associated with Figures 1 & 2 are highlighted in pink) </td> </tr> <tr> <td align="center" colspan="4">Maclaurin Spheroid</td> <td align="center" rowspan="2"><math>~L^\dagger</math></td> <td align="center" colspan="3">Jacobi Ellipsoids</td> </tr> <tr> <td align="center"><math>~e</math></td> <td align="center"><math>~\frac{c}{a}</math></td> <td align="center"><math>~E_L</math></td> <td align="center"><math>~E_\mathrm{plot}^\ddagger</math></td> <td align="center"><math>~\frac{b}{a}</math></td> <td align="center"><math>~\frac{c}{a}</math></td> <td align="center"><math>~E_\mathrm{Jac}</math></td> </tr> <tr> <td align="center">0.650</td> <td align="center">0.7599342</td> <td align="center">-8.9018255</td> <td align="center">0.0</td> <td align="center">2.8270256</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.675</td> <td align="center">0.7378177</td> <td align="center">-8.8165100</td> <td align="center">0.0</td> <td align="center">2.9985043</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.700</td> <td align="center">0.7141428</td> <td align="center">-8.7216343</td> <td align="center">0.0</td> <td align="center">3.1820090</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.725</td> <td align="center">0.6887489</td> <td align="center">-8.6155943</td> <td align="center">0.0</td> <td align="center">3.3796768</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.750</td> <td align="center">0.66143783</td> <td align="center">-8.4963506</td> <td align="center">0.0</td> <td align="center">3.5942337</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.775</td> <td align="center">0.6319612</td> <td align="center">-8.3612566</td> <td align="center">0.0</td> <td align="center">3.8292360</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.790</td> <td align="center">0.6131068</td> <td align="center">-8.2711758</td> <td align="center">0.0</td> <td align="center">3.9819677</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.795</td> <td align="center">0.6066094</td> <td align="center">-8.2394436</td> <td align="center">0.0</td> <td align="center">4.0351072</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.800</td> <td align="center">0.6000000</td> <td align="center">-8.2067933</td> <td align="center">0.0</td> <td align="center">4.0894508</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.805</td> <td align="center">0.5932748</td> <td align="center">-8.1731817</td> <td align="center">0.0</td> <td align="center">4.1450581</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.810</td> <td align="center">0.5864299</td> <td align="center">-8.1385621</td> <td align="center">0.0</td> <td align="center">4.2019932</td> <td align="center">---</td> <td align="center">---</td> <td align="center">---</td> </tr> <tr> <td align="center">0.815</td> <td align="center">0.5794610</td> <td align="center">-8.1028846</td> <td align="center">0.0064</td> <td align="center">4.2603252</td> <td align="center">0.880967</td> <td align="center">0.545588</td> <td align="center">-8.102934</td> </tr> <tr> <td align="center">0.820</td> <td align="center">0.5723635</td> <td align="center">-8.0660955</td> <td align="center">0.0524</td> <td align="center">4.3201286</td> <td align="center">0.797543</td> <td align="center">0.516311</td> <td align="center">-8.066596</td> </tr> <tr> <td align="center">0.825</td> <td align="center">0.5651327</td> <td align="center">-8.0281369</td> <td align="center">0.1116</td> <td align="center">4.3814839</td> <td align="center">0.744298</td> <td align="center">0.496028</td> <td align="center">-8.029578</td> </tr> <tr> <td align="center">0.830</td> <td align="center">0.5577634</td> <td align="center">-7.9889461</td> <td align="center">0.1665</td> <td align="center">4.4444785</td> <td align="center">0.702967</td> <td align="center">0.479341</td> <td align="center">-7.991848</td> </tr> <tr> <td align="center">0.835</td> <td align="center">0.5502499</td> <td align="center">-7.9484555</td> <td align="center">0.2140</td> <td align="center">4.5092074</td> <td align="center">0.668439</td> <td align="center">0.464724</td> <td align="center">-7.953367</td> </tr> <tr> <td align="center">0.840</td> <td align="center">0.5425864</td> <td align="center">-7.9065917</td> <td align="center">0.2551</td> <td align="center">4.5757737</td> <td align="center">0.638420</td> <td align="center">0.451485</td> <td align="center">-7.914095</td> </tr> <tr> <td align="center">0.845</td> <td align="center">0.5347663</td> <td align="center">-7.8632747</td> <td align="center">0.2912</td> <td align="center">4.6442903</td> <td align="center">0.611646</td> <td align="center">0.439241</td> <td align="center">-7.873990</td> </tr> <tr> <td align="center" bgcolor="pink">0.850</td> <td align="center" bgcolor="pink">0.5267827</td> <td align="center" bgcolor="pink">-7.8184175</td> <td align="center">0.3232</td> <td align="center" bgcolor="pink">4.7148806</td> <td align="center" bgcolor="pink">0.587337</td> <td align="center" bgcolor="pink">0.427750</td> <td align="center" bgcolor="pink">-7.833003</td> </tr> <tr> <td align="left" colspan="8"><sup>†</sup>Here, the units of angular momentum are as used in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. In order to convert to units of <math>~L</math> as used in EFE (see, for example, Table I, in Chapter 5, §32), multiply by <math>~[2^2/(3\cdot 5^{10})]^{1/6} = 0.0717585</math>. </tr> <tr> <td align="left" colspan="8"><sup>‡</sup><math>~E_\mathrm{plot}</math> is a normalized value of <math>~E_L</math> that has been used for plotting purposes. It's definition is: <math>E_\mathrm{plot} = 0.25*\biggl\{ \log_{10}\biggl[0.0001 + \frac{(E_L + |E_\mathrm{Jac}|)}{|E_\mathrm{Jac}|} \biggr] + 4\biggr\}</math> </tr> </table> </div> <pre> PROPERTIES OF VARIOUS MACLAURIN SPHEROIDS eccenL covera omega2 ellChandra L 0.815 5.7946096D-01 3.7625539D-01 3.0571438D-01 4.2603252D+00 0.820 5.7236352D-01 3.8058727D-01 3.1000578D-01 4.3201286D+00 0.825 5.6513273D-01 3.8489420D-01 3.1440854D-01 4.3814839D+00 0.830 5.5776339D-01 3.8917054D-01 3.1892894D-01 4.4444785D+00 0.835 5.5024994D-01 3.9341001D-01 3.2357378D-01 4.5092074D+00 0.840 5.4258640D-01 3.9760569D-01 3.2835048D-01 4.5757737D+00 0.845 5.3476630D-01 4.0174986D-01 3.3326712D-01 4.6442903D+00 0.850 5.2678269D-01 4.0583395D-01 3.3833257D-01 4.7148806D+00 0.855 5.1862800D-01 4.0984835D-01 3.4355656D-01 4.7876802D+00 0.860 5.1029403D-01 4.1378236D-01 3.4894980D-01 4.8628384D+00 0.865 5.0177186D-01 4.1762394D-01 3.5452411D-01 4.9405200D+00 0.870 4.9305172D-01 4.2135955D-01 3.6029264D-01 5.0209081D+00 0.875 4.8412292D-01 4.2497391D-01 3.6626999D-01 5.1042063D+00 0.880 4.7497368D-01 4.2844972D-01 3.7247248D-01 5.1906420D+00 0.885 4.6559102D-01 4.3176729D-01 3.7891846D-01 5.2804708D+00 0.890 4.5596052D-01 4.3490417D-01 3.8562861D-01 5.3739810D+00 0.895 4.4606614D-01 4.3783459D-01 3.9262639D-01 5.4714996D+00 0.900 4.3588989D-01 4.4052888D-01 3.9993856D-01 5.5733994D+00 0.905 4.2541157D-01 4.4295266D-01 4.0759585D-01 5.6801086D+00 0.910 4.1460825D-01 4.4506586D-01 4.1563375D-01 5.7921218D+00 0.915 4.0345384D-01 4.4682147D-01 4.2409362D-01 5.9100155D+00 0.920 3.9191836D-01 4.4816395D-01 4.3302405D-01 6.0344667D+00 0.925 3.7996710D-01 4.4902713D-01 4.4248265D-01 6.1662784D+00 0.930 3.6755952D-01 4.4933139D-01 4.5253852D-01 6.3064134D+00 0.935 3.5464771D-01 4.4897998D-01 4.6327550D-01 6.4560401D+00 0.940 3.4117444D-01 4.4785386D-01 4.7479681D-01 6.6165969D+00 0.945 3.2707033D-01 4.4580450D-01 4.8723156D-01 6.7898831D+00 0.950 3.1224990D-01 4.4264348D-01 5.0074442D-01 6.9781934D+00 </pre> <pre> PROPERTIES OF JACOBI ELLIPSOIDS THAT HAVE THE SAME ANGULAR MOMENTA (L) AS THE ABOVE MACLAURIN SPHEROIDS e b/a c/a A1 A2 A3 omega2 a L_C L energy 0.815 0.880967 0.545588 0.474189 0.557354 0.968456 0.371826 1.983364 0.305714 4.260325 -8.102934 0.820 0.797543 0.516311 0.441622 0.589410 0.968968 0.366634 2.088279 0.310006 4.320129 -8.066596 0.825 0.744298 0.496028 0.419233 0.611283 0.969484 0.361394 2.165672 0.314409 4.381484 -8.029578 0.830 0.702967 0.479341 0.400927 0.629069 0.970004 0.356104 2.232633 0.318929 4.444479 -7.991848 0.835 0.668439 0.464724 0.384983 0.644489 0.970527 0.350761 2.293990 0.323574 4.509207 -7.953367 0.840 0.638420 0.451485 0.370620 0.658325 0.971055 0.345362 2.351945 0.328350 4.575774 -7.914095 0.845 0.611646 0.439241 0.357403 0.671009 0.971588 0.339905 2.407740 0.333267 4.644290 -7.873990 0.850 0.587337 0.427750 0.345063 0.682812 0.972126 0.334386 2.462170 0.338333 4.714881 -7.833003 0.855 0.564969 0.416851 0.333417 0.693915 0.972668 0.328802 2.515795 0.343557 4.787680 -7.791082 0.860 0.544173 0.406427 0.322334 0.704450 0.973216 0.323150 2.569038 0.348950 4.862838 -7.748172 0.865 0.524676 0.396390 0.311717 0.714514 0.973770 0.317425 2.622239 0.354524 4.940520 -7.704210 0.870 0.506269 0.386673 0.301490 0.724181 0.974329 0.311624 2.675686 0.360293 5.020908 -7.659127 0.875 0.488788 0.377221 0.291594 0.733511 0.974895 0.305741 2.729638 0.366270 5.104206 -7.612848 0.880 0.472100 0.367988 0.281979 0.742554 0.975467 0.299772 2.784332 0.372472 5.190642 -7.565289 0.885 0.456097 0.358937 0.272604 0.751348 0.976047 0.293710 2.840003 0.378918 5.280471 -7.516357 0.890 0.440687 0.350033 0.263435 0.759930 0.976635 0.287549 2.896880 0.385629 5.373981 -7.465947 0.895 0.425792 0.341249 0.254440 0.768329 0.977230 0.281283 2.955205 0.392626 5.471500 -7.413941 0.900 0.411344 0.332556 0.245593 0.776572 0.977835 0.274902 3.015230 0.399939 5.573399 -7.360207 0.905 0.397284 0.323929 0.236868 0.784683 0.978449 0.268398 3.077231 0.407596 5.680109 -7.304592 0.910 0.383556 0.315345 0.228242 0.792685 0.979073 0.261762 3.141512 0.415634 5.792122 -7.246923 0.915 0.370112 0.306780 0.219694 0.800598 0.979708 0.254980 3.208416 0.424094 5.910016 -7.187000 0.920 0.356903 0.298209 0.211202 0.808442 0.980356 0.248041 3.278337 0.433024 6.034467 -7.124589 0.925 0.343885 0.289608 0.202744 0.816240 0.981017 0.240927 3.351732 0.442483 6.166278 -7.059417 0.930 0.331013 0.280950 0.194298 0.824010 0.981692 0.233621 3.429144 0.452539 6.306413 -6.991160 0.935 0.318242 0.272206 0.185842 0.831774 0.982384 0.226102 3.511226 0.463276 6.456040 -6.919428 0.940 0.305523 0.263344 0.177348 0.839558 0.983094 0.218342 3.598775 0.474797 6.616597 -6.843746 0.945 0.292805 0.254325 0.168790 0.847385 0.983825 0.210310 3.692789 0.487232 6.789883 -6.763530 0.950 0.280029 0.245105 0.160133 0.855288 0.984579 0.201966 3.794537 0.500744 6.978193 -6.678040 </pre>
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