Editing Appendix/Ramblings/ForCohlHoward (section)

=Recommended Riemann S-Type Ellipsoids for Modeling=
==Introductory Remarks==

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On 4 January 2023, Shawn Walker reintroduced the following strategy:

"<font color="darkgreen">But if I remember correctly, I think the idea was to start with some baby steps.  Perhaps have a fixed domain (the star) and solve the Euler equations in this with perhaps a fixed “fake” gravitational potential.  Just to get something working.  Then add the Poisson equation for the gravity, which would couple with the fluid.</font>"
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Joel's response:  I am completely with you regarding the idea of starting with baby steps.  But if we totally focus on Riemann S-Type ellipsoids, even the first few baby steps can be very realistic.  Immediately below is a table that itemizes my recommendation regarding which ellipsoids should be modeled fist, second, third, etc. and why.  What follows is a short preview.

Every S-Type configuration is a uniform-density ellipsoid (semi-axes: a, b, c) whose gravitational potential -- inside, outside, and on the surface of the ellipsoid -- is known analytically in terms of incomplete elliptic integrals of the 1st and 2nd kind (or simpler, trigonometric functions).  When the chosen ellipsoid is viewed from a frame that is rotating (about its c-axis) with a characteristic frequency, <math>\Omega</math>, the ellipsoidal configuration appears stationary.  As a result, the (analytically specified) gravitational potential does not vary with time.  For our purposes, this can naturally serve as the "fixed fake" potential to which you refer.  

As viewed from the proper (<math>\Omega</math>) rotating frame, we also know the exact solution to the Euler equations throughout every S-Type ellipsoid.  It is a steady-state flow in which all Lagrangian fluid elements move along closed elliptical orbits that &hellip;
<ul>
  <li>lie in planes parallel to the system's (''a-b'') equatorial plane;</li>
  <li>have identical ellipticities (that is, identical axis ratios, <math>b/a</math>);</li>
  <li>have identical orbital frequencies.</li>
</ul>  

So, as we try to build a code that can solve the Euler equations, we can (A) build a coordinate grid that is rotating with the desired (<math>\Omega</math>) frequency; (B) "guess" a velocity flow-field that corresponds to what we know to be the correct, steady-state solution; (C) hold fixed the analytically known gravitational potential; (D) then use the code to integrate the Euler equations forward in time and see how well the configuration maintains a steady-state flow.  We will have successfully debugged the "Euler equations" code if steady-state has been achieved for a variety of different Riemann S-Type ellipsoids.

Once the "Euler equations" code has been properly debugged, we can then add a Poisson solver; this will allow us to examine non-steady-state flows that result from natural instabilities among the set of Riemann S-Type ellipsoids.

----


Other useful attributes of Riemann S-Type ellipsoids &hellip;
<ul>
<li>Each is a (incompressible) uniform-density ellipsoid with semi-axes, <math>(a, b, c)</math>; ''usually'' without loss of generality we are able to set <math>a = 1</math>.</li>
<li>The steady-state velocity flow-field that is associated with each equilibrium configuration can be characterized by the values of two physical parameters: a frequency, <math>\Omega</math>, that is associated with the rate of the ellipsoidal figure's spin about its <math>c</math>-axis; and, a vorticity, <math>\zeta</math>, associated with the elliptical orbital motion of Lagrangian fluid elements ''inside and on the surface of'' the ellipsoid, when viewed from a frame that is rotating with the frequency, <math>\Omega</math>.</li>
</ul>

<!--
Details &hellip;
<ul>
<li>A pair of equilibrium configurations exists if the specified set of axis ratios <math>(b/a, c/a)</math> falls between the "upper-self-adjoint" (USA; <math>x = -1</math>) and "lower-self-adjoint" (LSA; <math>x=+1</math>) sequences; this "horn-shaped" sub-domain is identified in both figure panels (1a) and (1b), immediately below. </li>
<li>At each point in this <math>(b/a, c/a)</math> sub-domain, the pair of equilibrium models are distinguished from one other by the behavior of the fluid flow;</li>
<li>Each is steady-state when viewed from ... </li>
</ul>
-->

<!--
<table border="1" align="center" cellpadding="8">
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  <td align="center" colspan="3">Interesting Domains and/or Equilibrium Sequences in the <math>(b/a, c/a)</math> Diagram</td>
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Figure 2 extracted from &hellip;<br />
{{ Chandrasekhar65_XXVfigure }}
  </td>
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Boundary defined by USA and LSA sequences
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  <td align="center">
Jacobi Sequence:<br />(blue) Points defined by data in Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>; <br />(red) points generated here from [[#Roots_of_the_Governing_Relation|above-defined roots of the governing relation]].
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<b>(1a)</b><br />
[[File:ChandrasekharFig2annotated.png|400px|Chandra Diagram]]
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<b>(1b)</b><br />
[[File:SelfAdjointPlot.png|400px|USA and LSA Plot]]
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<b>(1c)</b><br />
[[File:EFEdiagram02.png|400px|Jacobi/Dedekind Sequence]]
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</table>
-->

==Initial Equilibrium Configurations Recommended for Dynamical Analysis==

===(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.

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<b>RRSTEM Table 1</b><br />
Established Critical-Point Models Along the (Axisymmetric) Maclaurin Spheroid Sequence
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  <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>
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[[File:CrossSectionsAnnotated.png|300px|Meridional Cross Sections]]
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</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>
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<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 }}.
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  <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;

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<math>
\frac{c}{a }
</math>
  </td>
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<math>=</math>
  </td>
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<math>
(1 - e^2)^{1 / 2}
\, ;
</math>
  </td>
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  <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>
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</table>
 </td>
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====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>"
</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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<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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  <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>
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</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>

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<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>
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  <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>
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</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>
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</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.


===Jacobi Ellipsoids===
<span id="RRSTEMtable2">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="85%">
<tr>
  <td align="center" colspan="9">
<b>RRSTEM Table 2</b><br />
Established Critical-Point Models Along the Jacobi Ellipsoid Sequence
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">
Model
  </td>
  <td align="center" rowspan="2">
<math>\frac{b}{a}</math>
  </td>
  <td align="center" rowspan="2">
<math>\frac{c}{a}</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>
</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>E</b></font>
  </td>
  <td align="center"><math>0.2972</math></td>
  <td align="center"><math>0.2575</math></td>
  <td align="center"><math>0.053286</math></td>
  <td align="center"><math>0.1863</math></td>
  <td align="center"><math>1.1507\times 10^{-2}</math></td>
  <td align="center">Dumbbell-shaped</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>F</b></font>
  </td>
  <td align="center"><math>0.432232</math></td>
  <td align="center"><math>0.345069</math></td>
  <td align="center"><math>0.07101</math></td>
  <td align="center"><math>0.1628</math></td>
  <td align="center"><math>7.491\times 10^{-3}</math></td>
  <td align="center">Pear-shaped</td>
  <td align="center">Uniform <math>\omega_0</math></td>
  <td align="center">Secular</td>
</tr>

<tr>
  <td align="left" colspan="9">
Given the value of the square of the angular velocity, <math>\Omega^2 \equiv \omega_0^2/(4\pi G \rho)</math>, we immediately know from the [[ThreeDimensionalConfigurations/JacobiEllipsoids#Angular_Momentum_Constraint|accompanying detailed discussion]] that &hellip;

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

<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{3 \Omega^2}{5^2} \cdot \frac{(a^2 + b^2)^2}{\bar{a}^4} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, p. 103, Eq. (16)</font> <br /><font size="-1">(Note different definition of <math>\Omega^2</math> in EFE)</font></td></tr>
</table>
 </td>
</tr>
</table>

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

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

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]].)
</td></tr></table>

<span id="RRSTEMfigure4">&nbsp;</span>
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">RRSTEM Figure 4</th></tr>
<tr>
  <td align="center" colspan="1" rowspan="1">
[[File:PearAndDumbbellModelF2.png|350px|Pear-Shaped Sequence]]
  </td>
  <td align="left" colspan="1" rowspan="1">
''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!
  </td>
</tr>

</table>

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

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

<ul><li>
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>
</li>
<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>
<li>
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>
<li>
{{ 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>
</ul>


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

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

==Details==

<ul>
  <li>Maclaurin Spheroid Sequence:
  <ol>
    <li>[[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|Standard Presentation]]</li>
    <li>[[Apps/MaclaurinSpheroidSequence#Alternate_Sequence_Diagrams|Alternate Sequence Diagrams]]</li>
    <li>[[Apps/MaclaurinSpheroidSequence#Oblate_Spheroidal_Coordinates|Oblate-Spheroidal Coordinates]]</li>
    <li>Various ''[[AxisymmetricConfigurations/SolutionStrategies#SRPtable|Simple Rotation Profiles]]''
    <li>[[Apps/MaclaurinSpheroidSequence#Bifurcation_Points_Along_Maclaurin-Spheroid_Sequence|Two Particularly Relevant Angular Momentum Distributions]]
      <ul type="-">
        <li>Uniform Rotation &nbsp; &rarr; &nbsp; <math>(n, m) = (4, 0)</math> bifurcation leads to Dyson-Wong "one-ring" sequence</li>
        <li><math>n' = 0</math> &nbsp; &rarr; &nbsp; <math>P_6</math> bifurcation leads to {{ MPT77 }} "Maclaurin Toroid" sequence</li>
      </ul>
    </li>
  </ol>
  </li>
  <li>[[Apps/EriguchiHachisu/Models|Eriguchi, Hachisu, and their various Colleagues]]</li>
  <li>[[Apps/MaclaurinToroid|Maclaurin Toroid Sequence (MPT77 &amp; EH85)]]</li>
  <li>[[Appendix/Ramblings/MacSphCriticalPoints#PhaseTransition|Phase Transition]] to Dyson-Wong (one-ring) sequence</li>
</ul>

<table border="1" cellpadding="8" align="center" width="100%"><tr><td align="center" bgcolor="lightblue">
See also our associated, more detailed discussion of [[Appendix/Ramblings/MacSphCriticalPoints|Critical Points along the Maclaurin Spheroid Sequence]].
</td></tr></table>