Editing ThreeDimensionalConfigurations/HomogeneousEllipsoids (section)

==Example Evaluations==

Here we adopt the notation mapping, <math>~(a_1, a_2, a_3) ~\leftrightarrow~ (a,b,c)</math>.  In general, for a given pair of axis ratios, <math>~(\tfrac{b}{a}, \tfrac{c}{a})</math>, a determination of the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, requires evaluation of elliptic integrals.  For practical applications, we have decided to evaluate these special functions using the fortran functions provided in association with the book, ''Numerical Recipes in Fortran'';  in order to obtain the results presented in our Table 2, below, we modified those default (single-precision) routines to generate results with double-precision accuracy.  Along the way (see results posted in our Table 1), we pulled cruder evaluations of both elliptic integrals, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math>, from the printed special-functions table found in a CRC handbook.

As we developed/debugged the numerical tool that would allow us to determine the values of these three coefficients for arbitrary choices of the pair of axis ratios, it was important that we compare the results of our calculations to those that have appeared in the published literature.  As a primary point of comparison, we chose to use ''The properties of the Jacobi ellipsoids'' as tabulated in &sect;39 (Chapter 6) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  In particular, for twenty-six separate axis-ratio pairs, Chandrasekhar's Table IV lists the values of the square of the angular velocity, <math>~\Omega^2</math>, and the total angular momentum, <math>~L</math>, of an equilibrium Jacobi ellipsoid that is associated with each axis-ratio pair.  We should be able to duplicate &#8212; or, via double-precision arithmetic, improve &#8212; Chandrasekhar's tabulated results using the expressions for "omega2",
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega^2}{\pi G\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B_{12}</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (5)</font> </td></tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[\frac{A_1 - (b/a)^2A_2}{1-(b/a)^2} \biggr] \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">using, [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;21, Eqs. (105) &amp; (107)</font></td></tr>
</table>
</div>
and, for "angmom",
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{L}{(GM^3)^{1/2}(abc)^{1/6}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{3}}{10}\biggl[ \frac{a^2 + b^2}{(abc)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (16)</font></td></tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{3}}{10}\biggl[ \frac{1 + (b/a)^2}{(b/a)^{2/3}(c/a)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Or, in connection with the free-energy discussion found in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, &amp; J. E. Tohline (1995, ApJ, 446, 472)],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{5L}{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 \biggr]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}\biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 \biggr]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}</math>
  </td>
</tr>
</table>
</div>

<span id="Table1">&nbsp;</span>
<table align="center" cellpadding="5" border="1">
<tr>
  <th align="center" colspan="12"><font size="+1">Table 1:&nbsp; Example Evaluations</font></th>
</tr>
<tr>
  <th align="center" colspan="2">Given</th>
  <th align="center" colspan="10">Determined using calculator and (crude) CRC tables of elliptic integrals</th>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>~\frac{a_3}{a_1}</math></td>
  <td align="center" colspan="2"><math>~\theta</math></td>
  <td align="center" rowspan="2"><math>~k</math></td>
  <td align="center" colspan="2"><math>~\sin^{-1}k</math></td>
  <td align="center" rowspan="2"><math>~F(\theta,k)</math></td>
  <td align="center" rowspan="2"><math>~E(\theta,k)</math></td>
  <td align="center" rowspan="2"><math>~A_1</math></td>
  <td align="center" rowspan="2"><math>~A_2</math></td>
  <td align="center" rowspan="2"><math>~A_3</math></td>
</tr>
<tr>
  <td align="center">radians</td>
  <td align="center">degrees</td>
  <td align="center">radians</td>
  <td align="center">degrees</td>
</tr>
<tr>
  <td align="right">1.00</td>
  <td align="right">0.582724</td>
  <td align="right">0.94871973</td>
  <td align="right">54.3576</td>
  <td align="right">0.00000000</td>
  <td align="right">0.00000000</td>
  <td align="right">0.000000</td>
  <td align="right">0.94871973</td>
  <td align="right">0.94871973</td>
  <td align="right">0.51589042</td>
  <td align="right">0.51589042</td>
  <td align="right">0.96821916</td>
</tr>
<tr>
  <td align="right">0.96</td>
  <td align="right">0.570801</td>
  <td align="right">0.96331527</td>
  <td align="right">55.1939</td>
  <td align="right">0.34101077</td>
  <td align="right">0.34799191</td>
  <td align="right">19.9385</td>
  <td align="right">0.975</td>
  <td align="right">0.946</td>
  <td align="right">+0.4937</td>
  <td align="right">+0.5319</td>
  <td align="right">+0.9744</td>
</tr>
<tr>
  <td align="right">0.60</td>
  <td align="right">0.433781</td>
  <td align="right">1.12211141</td>
  <td align="right">64.292</td>
  <td align="right">0.88788426</td>
  <td align="right">1.09272580</td>
  <td align="right">62.609</td>
  <td align="right">1.3375</td>
  <td align="right">0.9547</td>
  <td align="right">0.3455</td>
  <td align="right">0.6741</td>
  <td align="right">0.9803</td>
</tr>
</table>


<b>With regard to our Table 1 (immediately above):</b> To begin with, we picked three axis-ratio pairs from Table IV of EFE, and considered them to be "given."  For each pair, we used a hand-held calculator to calculate the corresponding values of the two arguments of the elliptic integrals, namely, <math>~\theta</math> and <math>~k</math>, as [[#Evaluation_of_Coefficients|defined above]].  By default, each determined value of <math>~\theta</math> is in radians.  Because the published CRC special-functions tables quantify both arguments of the special functions in angular ''degrees'', we converted <math>~\theta</math> from radians to degrees (see column 4 of Table 1) and, similarly, we converted <math>~\sin^{-1}k</math> to degrees (see column 7 of Table 1).  For the axisymmetric configuration &#8212; the first row of numbers in Table 1, for which <math>~a_2/a_1 = 1</math> &#8212; the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, were determined to eight digits of precision using the [[#Oblate_Spheroids|appropriate expressions for oblate spheroids]].  Note that, in this axisymmetric case, <math>~F(\theta,0) = E(\theta,0) = \theta</math>, but these function values are irrelevant with respect to the determination of the <math>~A_\ell</math> coefficients.

<div align="center" id="Table2">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="1">
<font size="+1">Table 2:&nbsp; Double-Precision Evaluations</font><p></p>
Related to Table IV in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>
</th>
</tr>
<tr><td align="left">
<pre>
                                                                                                                                 precision
         b/a      c/a              F                   E                  A1                  A2                  A3          [2-(A1+A2+A3)]/2

        1.00   0.582724          -----               -----          5.158904180D-01     5.158904180D-01     9.682191640D-01        0.0D+00
        0.96   0.570801     9.782631357D-01     9.487496699D-01     5.024584655D-01     5.292952683D-01     9.682462661D-01        4.4D-16
        0.92   0.558330     1.009516282D+00     9.489290273D-01     4.884500698D-01     5.432292722D-01     9.683206580D-01        0.0D+00
        0.88   0.545263     1.042655826D+00     9.492826127D-01     4.738278227D-01     5.577100115D-01     9.684621658D-01        2.2D-16
        0.84   0.531574     1.077849658D+00     9.498068890D-01     4.585648648D-01     5.727687434D-01     9.686663918D-01        2.2D-16

        0.80   0.517216     1.115314984D+00     9.505192815D-01     4.426242197D-01     5.884274351D-01     9.689483451D-01       -4.4D-16
        0.76   0.502147     1.155290552D+00     9.514282210D-01     4.259717080D-01     6.047127268D-01     9.693155652D-01        2.2D-16
        0.72   0.486322     1.198053140D+00     9.525420558D-01     4.085724682D-01     6.216515450D-01     9.697759868D-01       -4.4D-16
        0.68   0.469689     1.243931393D+00     9.538724717D-01     3.903895871D-01     6.392680107D-01     9.703424022D-01        2.2D-16
        0.64   0.452194     1.293310292D+00     9.554288569D-01     3.713872890D-01     6.575860416D-01     9.710266694D-01        4.4D-16

        0.60   0.433781     1.346645618D+00     9.572180643D-01     3.515319835D-01     6.766289416D-01     9.718390749D-01       -3.3D-16
        0.56   0.414386     1.404492405D+00     9.592491501D-01     3.307908374D-01     6.964136019D-01     9.727955606D-01       -6.7D-16
        0.52   0.393944     1.467522473D+00     9.615263122D-01     3.091371405D-01     7.169543256D-01     9.739085339D-01        4.4D-16
        0.48   0.372384     1.536570313D+00     9.640523748D-01     2.865506903D-01     7.382563770D-01     9.751929327D-01       -2.2D-16
        0.44   0.349632     1.612684395D+00     9.668252052D-01     2.630231082D-01     7.603153245D-01     9.766615673D-01        8.9D-16

        0.40   0.325609     1.697213059D+00     9.698379297D-01     2.385623719D-01     7.831101146D-01     9.783275135D-01        0.0D+00
        0.36   0.300232     1.791930117D+00     9.730763540D-01     2.132011181D-01     8.065964525D-01     9.802024294D-01        2.2D-15
        0.32   0.273419     1.899227853D+00     9.765135895D-01     1.870102340D-01     8.307027033D-01     9.822870627D-01       -1.3D-15
        0.28   0.245083     2.022466812D+00     9.801112910D-01     1.601127311D-01     8.553054155D-01     9.845818534D-01       -2.4D-15
        0.24   0.215143     2.166555572D+00     9.838093161D-01     1.327137129D-01     8.802197538D-01     9.870665333D-01        1.4D-14

        0.20   0.183524     2.339102805D+00     9.875217566D-01     1.051389104D-01     9.051602520D-01     9.897008376D-01       -1.6D-14
        0.16   0.150166     2.552849055D+00     9.911267582D-01     7.790060179D-02     9.296886827D-01     9.924107155D-01       -3.4D-14
        0.12   0.115038     2.831664019D+00     9.944537935D-01     5.180880535D-02     9.531203882D-01     9.950708065D-01        1.4D-13
        0.08   0.078166     3.229072310D+00     9.972669475D-01     2.817821170D-02     9.743504218D-01     9.974713665D-01        3.9D-13
        0.04   0.039688     3.915557866D+00     9.992484565D-01     9.281550546D-03     9.914470033D-01     9.992714461D-01        9.8D-13
</pre>
</td></tr>
</table>
</div>

<b>With regard to our Table 2 (immediately above):</b> Next, given each pair of axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> &#8212; copied from Table IV of EFE (see columns 1 and 2 of our Table 2)  &#8212; we used some fortran routines from [http://numerical.recipes/ Numerical Recipes] to calculate <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> (see columns 3 and 4 of our Table 2); we converted the routines to accommodate double-precision arithmetic.  We subsequently evaluated the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, (columns 5, 6, &amp; 7 of Table 2) using the expressions given above, then demonstrated that, in each case, the three coefficients sum to 2.0 to better than twelve digits accuracy.