Editing Appendix/Ramblings/ForCohlHoward (section)

====Ring-Like Distortions====


<font color="red"><b>Model C</b></font>: &nbsp; ([[Appendix/Ramblings/MacSphCriticalPoints#One-Ring_(Dyson-Wong)_Sequence|additional supporting discussion]])
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{{ ES81 }} claim that the one-ring (Dyson-Wong toroid) sequence bifurcates from the Maclaurin sequence precisely at the point where the spheroid has an eccentricity, <math>e = e_\mathrm{cr} = 0.98523</math> &#8212; in which case, also, <math>\Omega^2 = 0.08726</math> and <math>j^2 = 0.02174</math>.  In support of this conjecture, they point out that, {{ Chandrasekhar67_XXXfull }} &#8212; and {{ Bardeen71 }} have shown that this is <font color="darkgreen">&hellip; a neutral point on the Maclaurin sequence against the perturbation of <math>P_4(\eta)</math> displacement at the surface where <math>\eta</math> is one of the spheroidal coordinates."</font>  This is also the "neutral point" on the Maclaurin sequence labeled "F" in Table I of {{ HE82 }}; and the "bifurcation point" along the Maclaurin sequence that is labeled by the quantum numbers, <math>(n, m) = (4, 0)</math> in Table 1 of {{ HE84 }} as well as in the inset box of the ''left panel'' of our [[#RRSTEMfigure2|RRSTEM Figure 2]] (immediately below).

See also &hellip; {{ AKM2003full }}
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<tr><th align="center" colspan="2">RRSTEM Figure 2</th></tr>
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[[File:ES81OneRingWithInsetBox3.png|600px|One-Ring Sequence]]
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Figure 1 extracted from &sect;2.2, p. 488 of &hellip;<br />{{ CKST95bfigure }}
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  <td align="center" colspan="1">[[File:CKST95bFig1annotated3.png|300px|CKST95b Figure 1]]</td>
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  <td align="center">An analogous illustration appears as Figure 1 (p. 585) of &hellip;<br />{{ HE83figure }}</td>
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''Left panel (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.  Models A and B are not explicitly labeled, but the plot still shows the small solid circular markers (purple and yellow, respectively) that identify the locations of these models on the spheroid sequence.  The point where the Jacobi sequence bifurcates from the Maclaurin spheroid sequence (Model A) is labeled by the quantum numbers, <math>(n, m) = (2, 2)</math>, to indicate what geometric distortion, <math>P_n^m</math>, that is associated with this particular bifurcation.  Drawing from Table 1 of {{ HE84 }}, five other neutral points are marked by red crosses; three carry labels in this ''primary plot'' &#8212; <math>(3, 3), (4, 4), (3, 1)</math> &#8212; and the remaining two are labeled in the ''inset box'' &#8212; <math>(4, 2), (4, 0)</math>.

''Left panel (inset box):''  An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs.  The neutral point that is believed to be associated with the bifurcation point, itself, carries the geometric distortion label, <math>(n, m) = (4, 0)</math>, and is identified as our <font color="red">Model C</font>.  As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details|our separate chapter discussion]], the smooth (pink) curve that connects the spheroid sequence to the one-ring sequence has been defined by the set of 18 equilibrium models presented by {{ ES81 }}; the set of small green square markers identify nine equilibrium models obtained from Table I of the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from Table Ia of {{ Hachisu86a }}.

''Right panel:''  Figure 1 (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 Maclaurin spheroid sequence, where <math>e = 0.985226</math>. Accordingly, we have annotated the reprinted figure to indicate that the axisymmetric equilibrium model associated with point "A" is exactly our <font color="red">Model C</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 deform the spheroid into a ring-like configuration.  
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<font color="red"><b>Model D</b></font>: 
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First ''dynamical'' ring mode instability and bifurcation point to the Maclaurin toroid sequence. Also identified at <math>(e, j^2) = (0.998556, 0.04305)</math> as a <math>P_6</math> bifurcation point in Table 2 (p. 292) of {{ EH85 }}.
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<span id="RRSTEMfigure3">&nbsp;</span>
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[[File:EH85MacToroidWithInsetBox2.png|600px|Maclaurin Toroid Sequence]]
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Figure 6 extracted from &sect;4.2, p. 493 of &hellip;<br />{{ CKST95bfigure }}
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  <td align="center" colspan="1">[[File:CKST95bFig6annotated2.png|400px|CKST95b Figure 1]]</td>
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''Left panel (primary plot):''  The solid, multi-colored curve shows how <math>\tau \equiv T_\mathrm{rot}/|W_\mathrm{grav}|</math> (rather than <math>\Omega^2</math>) varies with <math>0 \le j^2 \le 0.07</math> along the Maclaurin spheroid sequence.  The positions of Models A, B, C and, now also, <font color="red">Model D</font> along this sequence are labeled. The colors of the curve segments and the positions of the (yellow and purple) small circular markers have the same meanings as in [[#RRSTEMfigure1|RRSTEM Figure 1]].  The (pink curve) one-ring sequence from [[#RRSTEMfigure2|RRSTEM Figure 2]] is also displayed.

''Left panel (inset box):''  An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called Maclaurin toroid sequence occurs.  Three neutral points along the Maclaurin spheroid sequence are identified by light-green circular markers, and are labeled according to their respective geometric distortions <math>(P_6, P_8, P_4)</math> &#8212; see Table 2 of {{ EH85 }}.  They have argued that the neutral point that is associated with bifurcation, itself, carries the geometric distortion label, <math>P_6</math>; it has been identified here as our <font color="red">Model D</font>.  As has been detailed in [[Apps/MaclaurinToroid#Maclaurin_Toroid_(EH85)|our separate chapter discussion]], the smooth (violet) curve that connects the spheroid sequence to the Maclaurin toroid sequence has been defined by the set of 15 equilibrium models presented by {{ EH85 }} in their Table 2.  [This data supersedes the modeling of the Maclaurin toroid sequence presented by {{ MPT77 }} in their discovery paper.] 

''Right panel:''  Figure 6 (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 point along the Maclaurin spheroid sequence where <math>e = 0.985226</math>; according to {{ Bardeen71 }}, this neutral point is associated with a <font color="red">P<sub>4</sub></font> geometric distortion and should be associated with the onset of dynamical axisymmetric instability. {{ EH85 }} agree that this is the eccentricity at which the P<sub>4</sub> neutral point resides, but they argue that bifurcation &#8212; and onset of dynamical axisymmetric instability &#8212; should be associated instead with the <font color="red">P<sub>6</sub></font> neutral point, as labeled in our ''inset box'', because "P<sub>6</sub>" is encountered ''earlier'' than "P<sub>4</sub>" along the Maclaurin spheroid sequence.  [As they argue, {{ Bardeen71 }} misidentified the bifurcation point because he only investigated models undergoing a P<sub>4</sub> distortion.]  In accordance with the arguments of {{ EH85 }}, our choice for <font color="red">Model D</font> is the Maclaurin spheroid whose eccentricity is the same as the P<sub>6</sub> neutral point, that is, <math>e = 0.998556</math>.
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