Editing Appendix/Ramblings/ForCohlHoward (section)

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

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

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