Editing Appendix/Ramblings/ForCohlHoward (section)

===Jacobi Ellipsoids===
<span id="RRSTEMtable2">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="85%">
<tr>
  <td align="center" colspan="9">
<b>RRSTEM Table 2</b><br />
Established Critical-Point Models Along the Jacobi Ellipsoid Sequence
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">
Model
  </td>
  <td align="center" rowspan="2">
<math>\frac{b}{a}</math>
  </td>
  <td align="center" rowspan="2">
<math>\frac{c}{a}</math>
  </td>
  <td align="center" rowspan="2">
<math>\Omega^2</math>
  </td>
  <td align="center" rowspan="2">
<math>\tau</math>
  </td>
  <td align="center" rowspan="2">
<math>j^2 = \frac{1}{3}\biggl(\frac{4\pi}{3}\biggr)^{-4 / 3} L_*^2</math>
  </td>
  <td align="center" colspan="3">
Bifurcation Characteristics &hellip;
  </td>
</tr>

<tr>
  <td align="center"><sup>&dagger;</sup>Geometric Distortion</td>
  <td align="center">Angular Mom. Profile</td>
  <td align="center">Instability Type</td>
</tr>

<tr>
  <td align="center" rowspan="1">
<font color="red"><b>E</b></font>
  </td>
  <td align="center"><math>0.2972</math></td>
  <td align="center"><math>0.2575</math></td>
  <td align="center"><math>0.053286</math></td>
  <td align="center"><math>0.1863</math></td>
  <td align="center"><math>1.1507\times 10^{-2}</math></td>
  <td align="center">Dumbbell-shaped</td>
  <td align="center">Uniform <math>\omega_0</math></td>
  <td align="center">Secular</td>
</tr>

<tr>
  <td align="center" rowspan="1">
<font color="red"><b>F</b></font>
  </td>
  <td align="center"><math>0.432232</math></td>
  <td align="center"><math>0.345069</math></td>
  <td align="center"><math>0.07101</math></td>
  <td align="center"><math>0.1628</math></td>
  <td align="center"><math>7.491\times 10^{-3}</math></td>
  <td align="center">Pear-shaped</td>
  <td align="center">Uniform <math>\omega_0</math></td>
  <td align="center">Secular</td>
</tr>

<tr>
  <td align="left" colspan="9">
Given the value of the square of the angular velocity, <math>\Omega^2 \equiv \omega_0^2/(4\pi G \rho)</math>, we immediately know from the [[ThreeDimensionalConfigurations/JacobiEllipsoids#Angular_Momentum_Constraint|accompanying detailed discussion]] that &hellip;

<table align="center" border="0" cellpadding="5">

<tr>
  <td align="right"><math>L_*^2 \equiv \frac{L^2}{GM^3 \bar{a}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{3 \Omega^2}{5^2} \cdot \frac{(a^2 + b^2)^2}{\bar{a}^4} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, p. 103, Eq. (16)</font> <br /><font size="-1">(Note different definition of <math>\Omega^2</math> in EFE)</font></td></tr>
</table>
 </td>
</tr>
</table>

====Pear-Shaped Distortion====

<font color="red"><b>Model F</b></font>:
<table border="0" align="center" width="95%"><tr><td align="left">

According to <font color="#00CC00">Chapter 6, &sect;40</font> of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], a pear-shaped configuration bifurcates from the Jacobi Sequence at <math>(b/a, c/a) = (0.432232, 0.345069)</math>, where <math>\omega_0^2/(\pi G \rho) = 0.284030</math>;  this result has been obtained via the virial relations &#8212; see Eq. (28) on p. 106 &#8212; as well as via a direct perturbation analysis &#8212; see Eq. (57) on p. 110.  From the information provided in the first row of Table I in {{ EHS82 }}, we appreciate as well that <math>\omega_0^2/(4\pi G \rho) = 0.07101; ~j^2 = 0.07821; ~\tau \equiv T/|W| = 0.1628</math>. (This information also has been recorded near the end of our [[ThreeDimensionalConfigurations/JacobiEllipsoids#Bifurcation_from_Maclaurin_to_Jacobi_Sequence|accompanying discussion of the Jacobi ellipsoid sequence]].)
</td></tr></table>

<span id="RRSTEMfigure4">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">RRSTEM Figure 4</th></tr>
<tr>
  <td align="center" colspan="1" rowspan="1">
[[File:PearAndDumbbellModelF2.png|350px|Pear-Shaped Sequence]]
  </td>
  <td align="left" colspan="1" rowspan="1">
''Primary plot:''  As in the ''right panel'' of [[#RRSTEMfigure1|RRSTEM Figure 1, above]], the solid, multi-colored curve shows how <math>\Omega^2</math> varies with <math>0 \le j^2 \le 0.04</math> along the Maclaurin spheroid sequence; for reference, <font color="red">Model A</font> and <font color="red">Model B</font> are labeled. Also as in [[#RRSTEMfigure1|RRSTEM Figure 1]], starting at the Model A bifurcation point, the purple dashed curve identifies the equilibrium sequence of Jacobi ellipsoids. A red cross labeled <font color="red">F</font> identifies the position along the Jacobi ellipsoid sequence that is a "neutral point against the 3<sup>rd</sup>-order harmonic perturbation."  A so-called pear-shaped sequence branches off at this point; here, we have displayed the behavior of this sequence by drawing the properties of equilibrium models from Table I (p. 1071) of {{ EHS82 }}. 

''Inset box:'' Especially Because the pear-shaped sequence is  very short, an ''inset box'' is used to magnify the plotted sequence; this was also done by {{ EHS82 }} &#8212; see their Figure 1 (p. 1072).  A red arrow, that accompanies our <font color="red">Model F</font> label, points to the location along the Jacobi sequence where the relevant neutral point lies.  As is suggested by {{ EHS82 }}, the presumption is that this pear-shaped sequence bifurcates from the Jacobi sequence at the neutral point.  But this is not the case in our plot.  Upon closer inspection, we see that the pear-shaped sequence ''does'' appear to bifurcate from the Jacobi sequence in Figure 1 of {{ EHS82 }} and that this happens because the Jacobi sequence is shifted a bit from ''our'' depiction of the Jacobi sequence location.  Curious!
  </td>
</tr>

</table>

====Dumbbell-Shaped Distortion====

<font color="red"><b>Model E</b></font>:

<ul><li>
According to the last pair of equations on p. 128 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;45</font>, a neutral point belonging to the fourth harmonic (a dumbbell-shaped) distortion arises on the Jacobi Sequence at <math>(b/a) = 0.2972</math> and <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}081~~~~\Rightarrow ~~~~ (c/a) = 0.2575</math>.  Chronologically, this result for <math>(b/a, c/a)</math> appears first in Eq. (93) on p. 635 of {{ Chandrasekhar67_XXXIIfull }}.  Then, in Eq. (66) on p. 302 of {{ Chandrasekhar68_XXXVfull }} &#8212; we find <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}068</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}081</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>
</li>
<li>
According to the first row of properties in Table II of {{ EHS82 }}, we find that <font color="red">Model E</font> is characterized by the properties &hellip;  <math>\Omega^2/(4\pi G \rho) = 0.0532</math>; <math>j^2 = ( 3\cdot 2^{-8} \pi^{-4} )^{1/3}  L^2/(GM^3\bar{a}) = 0.01157</math>; and <math>\tau \equiv T/|W| = 0.1863 </math>.  &nbsp;I have not (yet) found the corresponding value of <math>\Omega^2</math> in any of Chandrasekhar's publications, but if we combine the value of <math>\Omega^2</math> obtained from {{ EHS82hereafter }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we find &hellip; <math>\Omega^2/(\pi G \rho) = 0.2128</math>; from the [[#Angular_Momentum_Constraint|above expression]], <math>L_* = 0.48242</math>; and &nbsp; <math>j^2 = ( 3\cdot 2^{-8} \pi^{-4})^{1/3}  L_*^2 = 0.01149</math>.  &nbsp; This value of <math>j^2</math> is very close to the value obtained by {{ EHS82hereafter }}.</li>
<li>
In the paragraph at the top of the right-hand column of p. 467 of {{Hachisu86bfull }}, we find &hellip; <math>\Omega^2/(4\pi G\rho) = 0.0535</math>; <math>j^2 = 0.01157</math>.</li>
<li>
{{ CKST95bfull }} grab parameter values from a variety of sources.  In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their &sect;3.2 (p. 492), they state &hellip; <math>(b/a, c/a) = (0.29720, 0.25746)</math>; &nbsp;<math>\Omega^2/(4\pi G \rho) = 0.0532790</math>; &nbsp; and, <math>j^2 = 0.0115082</math>.</li>
<li><font color="red">NOTE (23 May 2023):</font> &nbsp; Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]  &#8212; that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> &#8212; into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math>, and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>, and <math>j^2 = 0.011507</math>.</li>
</ul>


<span id="RRSTEMfigure5">&nbsp;</span>
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">RRSTEM Figure 5</th></tr>
<tr>
  <td align="center" colspan="1" rowspan="2">
[[File:PearAndDumbbellModelE4.png|600px|Pear and Dumbbell Sequences]]
  </td>
  <td align="center" colspan="1" rowspan="1">
Figure 4 extracted from &sect;3.2, p. 493 of &hellip;<br />{{ CKST95bfigure }}
  </td>
</tr>
<tr>
  <td align="center" colspan="1">[[File:CKST95bFig4annotatedBetter.png|300px|CKST95b Figure 4]]</td>
</tr>

<tr>
  <td align="left" colspan="2">
''Left panel (primary plot):''  Same as the ''primary plot'' displayed in [[#RRSTEMfigure4|RRSTEM Figure 4, immediately above]].  A red cross labeled <font color="red">E</font> identifies the position along the Jacobi ellipsoid sequence that is a neutral point against a 4<sup>th</sup>-order harmonic perturbation.  A so-called dumbbell-shaped sequence branches off at this point; it, in turn, transitions to a sequence of equal-mass binaries.

''Left panel (inset box):''  An ''inset box'' shows more clearly the sequence of equilibrium models that make up the dumbbell (green markers and curve) and binary (blue markers and curve) sequences.  Here, the dumbbell sequence is defined by a set of equilibrium models drawn from Table II (p. 1073) of {{ EHS82 }} while the binary sequence is defined by a set of equilibrium models drawn from the subsection (p. 243) of Table 1 labeled "N = 0" in {{ HE84b }}.

''Right panel:''  Figure 4 (plus caption) from {{ CKST95b }} has been reprinted here to emphasize its similarity to, and overlap with our ''inset box''.  According to the caption of this reprinted figure, the filled circular marker labeled "A" identifies the bifurcation point on the Jacobi ellipsoid sequence, where <math>(b/a, c/a) = (0.2972, 0.2575)</math>.  Accordingly, we have annotated the reprinted figure to indicate that the Jacobi ellipsoid model associated with point "A" is exactly our <font color="red">Model E</font>.  As is stated in the caption of this reprinted figure, the dotted line XBC denotes the (hypothesized) onset of a secular instability that &#8212; in the nonlinear regime and conserving total angular momentum (vertical dotted line) &#8212; should lead to fission of the ellipsoid into an equal-mass binary system.  
  </td>
</tr>
</table>