Editing Appendix/Ramblings/ForCohlHoward (section)

==Index Symbols for Type I Riemann Ellipsoids==

2 March 2022: &nbsp; Joel requests Howard's assistance in verifying the correct expressions to use for the index symbols, <math>(A_1, A_2, A_3)</math>, when constructing "Type I" Riemann ellipsoids.

As has been [[#Three_Lowest-Order_Expressions|summarized above]], when I have derived expressions for the three lowest-order index symbols, I have used <math>(\ell, m, s)</math> instead of <math>(1, 2, 3)</math> as the subscript notation.  In the context of our discussion of Riemann S-type ellipsoids, it is usually the case that <math>a_1</math> is the longest semi-axis and <math>a_3</math> is the shortest semi-axis, so we can adopt the association, <math>(\ell, m, s) \rightarrow (1, 2, 3)</math>. This is consistent with the set of expressions for <math>A_1</math>,  <math>A_2</math>, and  <math>A_3</math>, that Howard has defined inside of his Mathematica notebook, namely &hellip;

<table border="1" align="center" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 m \sin^3(\phi) } \biggl[
\mathrm{EllipticF}[\phi, m] - \mathrm{EllipticE}[\phi, m] 
\biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 m(1-m) \sin^3(\phi) } \biggl[
\mathrm{EllipticE}[\phi, m] - \cos^2\theta \cdot \mathrm{EllipticF}[\phi, m] - \frac{a_3}{a_2}\cdot\sin^2\theta \sin\phi
\biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 (1-m) \sin^3(\phi) } \biggl[
\frac{a_2}{a_3}\cdot \sin(\phi) - \mathrm{EllipticE}[\phi, m] 
\biggr]
\, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

But I am also interested in developing a better understanding of the structural properties of Type I Riemann Ellipsoids.  For these systems, it is the case that <math>a_2</math> is the longest semi-axis and <math>a_3</math> is the shortest semi-axis, so we should adopt the association, <math>(\ell, m, s) \rightarrow (2, 1, 3)</math>.  In <font color="red">STEP #2</font> of the chapter that I am writing [[3Dconfigurations/DescriptionOfRiemannTypeI#Description_of_Riemann_Type_I_Ellipsoids|about Type I Riemann Ellipsoids]], I claim that the correct expressions for the three lowest-order index symbols are &hellip;
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
2\biggl( \frac{a_1}{a_2}\biggr) \biggl[  \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_1 = 2 - (A_2 + A_3)
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
I would like to know if Howard agrees with this claim.  In particular, I would like to know what values of <math>(A_1, A_2, A_3)</math> he obtains using Mathematica's tools, for the following pairs of model axis ratios.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="5">
Example Models Drawn ''Primarily''<sup>&dagger;</sup> from Table 4 (p. 858) of  {{ Chandrasekhar66_XXVIII }}
  </td>
</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="3">Howard Cohl's 20-Digit Precision Evaluation of Index Symbols</td>
</tr>
<tr>
  <td align="center" bgcolor="red" width="25%"><font color="white">A<sub>1</sub></font></td>
  <td align="center" bgcolor="red" width="25%"><font color="white">A<sub>2</sub></font></td>
  <td align="center" bgcolor="red" width="25%"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="right">1.05263</td>
  <td align="right">0.41667</td>
  <td align="left">0.43008853859109863146</td>
  <td align="left">0.40190463270335853286</td>
  <td align="left">1.1680068287055428357</td>
</tr>
<tr>
  <td align="right">1.25000</td>
  <td align="right">0.50000</td>
  <td align="left">0.50824409128926609544</td>
  <td align="left">0.37944175746381924039</td>
  <td align="left">1.1123141512469146642</td>
</tr>
<tr>
  <td align="right">1.44065</td>
  <td align="right">0.49273</td>
  <td align="left">0.52404662956445493304</td>
  <td align="left">0.32351600902859843592</td>
  <td align="left">1.1524373614069466310</td>
</tr>
<tr>
  <td align="right">1.66666</td>
  <td align="right">0.33333</td>
  <td align="left">0.41804279087942201679</td>
  <td align="left">0.20718018294728778618</td>
  <td align="left">1.3747770261732901970</td>
</tr>
<tr>
  <td align="right">1.36444</td>
  <td align="right">0.09518</td>
  <td align="left">0.14378468450707924448</td>
  <td align="left">0.091526900363135453419</td>
  <td align="left">1.7646884151297853021</td>
</tr>
<tr>
  <td align="right">1.69351</td>
  <td align="right">0.11813</td>
  <td align="left">0.18177227461540501422</td>
  <td align="left">0.084646944102441440238</td>
  <td align="left">1.7335807812821535455</td>
</tr>
<tr>
  <td align="right">1.52303</td>
  <td align="right">0.05315</td>
  <td align="left">0.085943612594485367787</td>
  <td align="left">0.046184257303756474717</td>
  <td align="left">1.8678721301017581575</td>
</tr>
<tr>
  <td align="right">1.78590</td>
  <td align="right">0.06233</td>
  <td align="left">0.10259594310295396216</td>
  <td align="left">0.043583894016884923661</td>
  <td align="left">1.8538201628801611142</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow">1.2500</td>
  <td align="right" bgcolor="yellow">0.4703</td>
  <td align="left">0.48955940032702523984</td>
  <td align="left">0.36486593343389634429</td>
  <td align="left">1.1455746662390784159</td>
</tr>
<tr>
  <td align="left" colspan="5">
<sup>&dagger;</sup>NOTE: &nbsp; The model whose axis-ratios are highlighted with a yellow background has been drawn from Table 6a (p. 871) of  {{ Chandrasekhar66_XXVIII }}
  </td>
</tr>
</table>

If you want to know how I derived the relevant index-symbol expressions, go to my MediaWiki chapter that discusses ''The Gravitational Potential (A<sub>i</sub> coefficients)'' in the context of <b>Ellipsoidal &amp; Ellipsoidal-Like</b> equilibrium structures.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center">[[File:TiledMenuIndexSymbolEvaluations.png|600px|Tiled Menu Chapter titled "Index Symbol Evaluations"]]
</tr>
</table>

More specifically, go to the subsection of this chapter titled, "[[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|Derivation of Expressions for A<sub>i</sub>]]," where I evaluate <math>A_\ell, A_m, A_s</math>.  These expressions will always remain the same.  Then the question is, for your specific problem of interest, which ellipsoidal axis <math>a_1, a_2,</math> or <math>a_3</math> is the "largest, medium-sized, or smallest"?  When considering Riemann Type-I Ellipsoids, <math>a_2</math> is the largest axis, so <math>A_2 \leftrightarrow A_\ell</math>; <math>a_3</math> is the smallest axis, so <math>A_3 \leftrightarrow A_s</math>; and <math>a_1</math> is the medium-sized axis, so <math>A_1 \leftrightarrow A_m</math>.