Editing Appendix/Ramblings/ForCohlHoward (section)

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

<font color="red"><b>Model F</b></font>:
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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]].)
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<span id="RRSTEMfigure4">&nbsp;</span>
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<tr><th align="center" colspan="2">RRSTEM Figure 4</th></tr>
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[[File:PearAndDumbbellModelF2.png|350px|Pear-Shaped Sequence]]
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''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!
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