Editing Appendix/Ramblings/ForCohlHoward (section)

===(Axisymmetric) Maclaurin Spheroids===
The following table lists four "critical points" drawn from Table 4 (Appendix A, p. 611) of {{ HTE87 }}; go [[Appendix/Ramblings/MacSphCriticalPoints#HTE87|here]] for additional details.

<span id="RRSTEMtable1">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="85%">
<tr>
  <td align="center" colspan="9">
<b>RRSTEM Table 1</b><br />
Established Critical-Point Models Along the (Axisymmetric) Maclaurin Spheroid Sequence
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">
Model
  </td>
  <td align="center" rowspan="2">
<math>e</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>
  <td align="center" rowspan="6">
[[File:CrossSectionsAnnotated.png|300px|Meridional Cross Sections]]
  </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>A</b></font>
  </td>
  <td align="center"><math>0.812670</math></td>
  <td align="center"><math>0.093557</math></td>
  <td align="center"><math>0.1375</math></td>
  <td align="center"><math>4.555\times 10^{-3}</math></td>
  <td align="left"><math>P_2^2(\eta)\cos(2\phi)</math></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>B</b></font>
  </td>
  <td align="center"><math>0.95289</math></td>
  <td align="center"><math>0.11006</math></td>
  <td align="center"><math>0.2738</math></td>
  <td align="center"><math>1.280\times 10^{-2}</math></td>
  <td align="left"><math>P_2^2(\eta)\cos(2\phi)</math></td>
  <td align="center"><math>n' = 0</math></td>
  <td align="center">Dynamical</td>
</tr>

<tr>
  <td align="center" rowspan="1">
<font color="red"><b>C</b></font>
  </td>
  <td align="center"><math>0.98522</math></td>
  <td align="center"><math>0.087271</math></td>
  <td align="center"><math>0.3589</math></td>
  <td align="center"><math>2.174\times 10^{-2}</math></td>
  <td align="left"><math>P_4(\eta)</math></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>D</b></font>
  </td>
  <td align="center"><math>0.998556</math></td>
  <td align="center"><math>0.036820</math></td>
  <td align="center"><math>0.4511</math></td>
  <td align="center"><math>4.304\times 10^{-2}</math></td>
  <td align="left"><math>P_6(\eta)</math></td>
  <td align="center"><math>n' = 0</math></td>
  <td align="center">Dynamical</td>
</tr>

<tr>
  <td align="left" colspan="9">
<sup>&dagger;</sup>Following {{ EH85full }} &#8212; see especially their Appendix and their Table 2 &#8212; <math>P_{2n}</math> is the Legendre polynomial.  See also, p. 429 of {{ Bardeen71full }}.
  </td>
</tr>

<tr>
  <td align="left" colspan="9">
Given the value of the eccentricity, <math>e</math>, we immediately know from the [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|accompanying detailed discussion]] that &hellip;

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

<tr>
  <td align="right">
<math>
\frac{c}{a }
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1 - e^2)^{1 / 2}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Omega^2 \equiv \frac{\omega_0^2}{4\pi G \rho }
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2e^2} \biggl[
(3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e} 
- 3(1-e^2)
\biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tau \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
  </td>
  <td align="center">
<math>= </math>
  </td>
  <td align="left">
<math>~ \frac{3}{2e^2}\biggl[ 1 - \frac{e(1-e^2)^{1 / 2}}{\sin^{-1} e}\biggr] - 1 \, ;</math>
  </td>
</tr>

<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{6}{5^2} \biggl[ (3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e^3} - \frac{3(1-e^2)}{e^2}\biggr](1 - e^2)^{-2 / 3}  \, .</math>
  </td>
</tr>
</table>
 </td>
</tr>
</table>

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

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"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>"
</font>
</td></tr>
<tr><td align="right">
— Drawn from pp. 475 - 476 of {{ Lebovitz67_XXXIV }} 
</td></tr></table>

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

<span id="RRSTEMfigure1">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="70%">
<tr><th align="center" colspan="2">RRSTEM Figure 1</th></tr>
<tr>
  <td align="center">
<b>EFE Diagram</b><br />

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

[[File:OurHE84Fig1annotated2.png|425px|OurHE84Fig1]]
  </td>
</tr>

<tr>
  <td align="left" colspan="2">
''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.
  </td>
</tr>
</table>

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

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"&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]].
</td></tr>
<tr><td align="right">
— Drawn from pp. 475 - 476 of {{ Lebovitz67_XXXIV }} 
</td></tr></table>

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


<font color="red"><b>Model C</b></font>: &nbsp; ([[Appendix/Ramblings/MacSphCriticalPoints#One-Ring_(Dyson-Wong)_Sequence|additional supporting discussion]])
<table border="0" align="center" width="95%"><tr><td align="left">
{{ 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 }}
</td></tr></table>
<span id="RRSTEMfigure2">&nbsp;</span>
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">RRSTEM Figure 2</th></tr>
<tr>
  <td align="center" colspan="1" rowspan="3">
[[File:ES81OneRingWithInsetBox3.png|600px|One-Ring Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="1">
Figure 1 extracted from &sect;2.2, p. 488 of &hellip;<br />{{ CKST95bfigure }}
  </td>
</tr>
<tr>
  <td align="center" colspan="1">[[File:CKST95bFig1annotated3.png|300px|CKST95b Figure 1]]</td>
</tr>
<tr>
  <td align="center">An analogous illustration appears as Figure 1 (p. 585) of &hellip;<br />{{ HE83figure }}</td>
</tr>

<tr>
  <td align="left" colspan="2">
''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.  
  </td>
</tr>
</table>


<font color="red"><b>Model D</b></font>: 
<table border="0" align="center" width="95%"><tr><td align="left">
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 }}.
</td></tr></table>
<span id="RRSTEMfigure3">&nbsp;</span>
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">RRSTEM Figure 3</th></tr>
<tr>
  <td align="center" colspan="1" rowspan="2">
[[File:EH85MacToroidWithInsetBox2.png|600px|Maclaurin Toroid Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="1">
Figure 6 extracted from &sect;4.2, p. 493 of &hellip;<br />{{ CKST95bfigure }}
  </td>
</tr>
<tr>
  <td align="center" colspan="1">[[File:CKST95bFig6annotated2.png|400px|CKST95b Figure 1]]</td>
</tr>

<tr>
  <td align="left" colspan="2">
''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>.
  </td>
</tr>
</table>

====Finite-Amplitude Oscillations of the Maclaurin Spheroid====

In &sect;53 (pp. 172 - 184) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] we find a discussion titled, "''A class of finite-amplitude oscillations of the Maclaurin spheroid''".  The primary focus is on simulations presented by {{ Rossner67_XXXVIIIfull }} &#8212; which may provide excellent points of comparison for our own investigations into the oscillatory motions of initially stable &#8212; but perturbed &#8212; Maclaurin spheroids,  and the nonlinear evolution of initially unstable Maclaurin spheroids.