Editing Appendix/Ramblings/ForCohlHoward (section)

====Second-Harmonic Distortions====

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"The disturbances of the Maclaurin spheroids that belong to the second harmonics lead to instability for smaller values of the angular momentum than do those that belong to any higher harmonic &hellip; [Of] the five oscillation frequencies associated with these disturbances &hellip; one is able to distinguish two critical points &hellip;  The first, when <math>e</math> is 0.8127</font> &#8212; our <font color="red">Model A</font> &#8212; <font color="darkgreen">is the point of bifurcation where the Maclaurin and Jacobi sequences intersect.  Although &sigma;<sup>2</sup> vanishes at this critical point, it is positive on either side, so that instability does not set in here, at least in the absence of further effects.  The second, when <math>e</math> is 0.9529 </font>&#8212; our <font color="red">Model B</font> &#8212; <font color="darkgreen"> represents the onset of </font> [dynamical]<font color="darkgreen"> instability, and also the most flattened Maclaurin spheroid that can lie on a Riemann sequence of type S &hellip;</font>"
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— Drawn from pp. 475 - 476 of {{ Lebovitz67_XXXIV }} 
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<font color="red"><b>Model A</b></font>:
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This is the point along the Maclaurin spheroid sequence where the Jacobi sequence bifurcates.  Some of the quantitative characteristics of this critical axisymmetric configuration are identified in Table IV (Chapter 6, &sect;39, p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  See also &hellip; the first line in Table B1 (p. 446) of {{ Bardeen71 }}; and Appendices D.3 & D.4 (pp. 485-486) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>].
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<span id="RRSTEMfigure1">&nbsp;</span>
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<b>EFE Diagram</b><br />

[[File:OurEFEannotated2.png|300px|OurEFE]]
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<b>&Omega;<sup>2</sup> vs. j<sup>2</sup> Diagram</b><br />

[[File:OurHE84Fig1annotated2.png|425px|OurHE84Fig1]]
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''Left panel:''  This version of the [[ThreeDimensionalConfigurations/RiemannStype#Summary|familiar EFE Diagram]] displays the horn-shaped region &#8212; bounded by the lower (LSA) and upper (USA) self-adjoint sequences &#8212; where all equilibrium, S-type Riemann ellipsoids reside.  Axisymmetric <math>(b/a = 1)</math>, uniformly rotating equilibrium configurations belong to the Maclaurin spheroid sequence which, as indicated, functions as the right-hand vertical boundary of the EFE diagram. Maclaurin spheroids that have an eccentricity less than that of <font color="red">Model A</font> <math>(e = 0.812670)</math> lie along the ''blue'' segment of the sequence; the ''orange'' segment is populated by Maclaurin spheroids with eccentricities larger than that of Model A but less than that of <font color="red">Model B</font> <math>(e = 0.95289)</math>; all other Maclaurin spheroids <math>(0.95289 < e \le 1)</math> lie along the ''black'' segment.

''Right panel:''  The solid, multi-colored curve shows how the (square of the) dimensionless rotation frequency, <math>\Omega^2</math>, varies with the (square of the) dimensionless total angular momentum, <math>j^2</math>, along the Maclaurin spheroid sequence.  As in the accompanying  (''left panel'') EFE diagram, <font color="red">Model A</font> and <font color="red">Model B</font> mark the ends of the differently colored curve segments.  

''Both panels:''  A pair of small yellow circular markers identify the points where, respectively, the USA and LSA sequences intersect the Maclaurin spheroid sequence; and the small green square marker identifies where <math>\Omega^2</math> has its maximum value along the Maclaurin spheroid sequence. The sequence of uniformly rotating Jacobi ellipsoids is identified by the series of small solid purple markers; <font color="red">Model A</font> is the axisymmetric equilibrium configuration at the point where the Jacobi ellipsoid sequence intersect (bifurcates from) the Maclaurin spheroid sequence.  Similarly, <font color="red">Model B</font> lies at the intersection of the LSA with the Maclaurin spheroid sequence.
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<font color="red"><b>Model B</b></font>:
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From p. 141 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we find that this model, <font color="darkgreen">"&hellip; the Maclaurin spheroid on the verge of dynamical instability, is the first member of the</font> [Riemann S-type ellipsoid] <font color="darkgreen"> self-adjoint sequence <math>x = + 1</math>."</font>  Some of the quantitative characteristics of this critical (axisymmetric) Maclaurin spheroid are identified in Table VI (Chapter 7, &sect;48, p. 142) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; for example, <math>c/a = a_3/a_2 = 0.30333 \Rightarrow e = 0.95289</math>.  In Table B1 (p. 446) of {{ Bardeen71 }}, this is referred to as the "First nonaxisymmetric dynamical instability."
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"&hellip; to what kind of configuration does [this] instability lead?  {{ Rossner67_XXXVIII }} has answered this question with the aid of Riemann's formulation of the nonlinear, ordinary differential equations describing the motion of a liquid ellipsoid.  Integrating the equations numerically, Rossner found that the configuration neither gets disrupted nor finds its way to a new steady state, but performs a complicated unsteady motion."</font>  &#8212; See [[#Finite-Amplitude_Oscillations_of_the_Maclaurin_Spheroid|further reference to Rossner's work, below]].
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— Drawn from pp. 475 - 476 of {{ Lebovitz67_XXXIV }} 
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