Editing Appendix/Ramblings/MacSphCriticalPoints (section)

==Axisymmetric Equilibrium Sequences that Display a Topological Change==

In the context of discussions of self-gravitating configurations that are rotating and have a uniform density, there are three well-known ''axisymmetric'' equilibrium sequences:
<ol>
  <li>The Maclaurin spheroid sequence &#8212; first constructed in 1742 by Maclaurin.</li>
  <li>A so-called one-ring (Dyson-Wong toroid) sequence &#8212; the ring-like segment was first mapped out in 1893 by Dyson</li>
  <li>A so-called Maclaurin toroid sequence &#8212; the toroidal segment was first constructed by {{ MPT77 }}.</li>
</ol>
Along the first of these sequences, every equilibrium configuration has a surface that is precisely a spheroid. The other two bifurcate from the Maclaurin spheroid sequence; contain a segment of concave, hamburger-shaped (spheroidal-like) configurations; and, upon further extension, blend into the (separately identified) sequence of toroidal-shaped configurations. In other words, moving along the respective model sequences, we encounter a spheroidal-like to ring-like topological change.  The overriding question that requires nonlinear dynamical modeling to answer is: "Is there a model (or a range of models) along the Maclaurin spheroid sequence that is unstable toward evolution away from that sequence and toward one of the ring-like sequences?"

'''TWO KEY CONCEPTS:'''
<ol type="A">
  <li>The model parameterization that distinguishes the two ring-like sequences from one another is the specified ''distribution of angular momentum.''  Dyson-Wong tori are uniformly rotating, whereas, each Maclaurin toroid has an angular momentum distribution that is the same as is present in all Maclaurin spheroids.  It is this single structural feature that drives the bifurcation points for these sequences to two quite different locations along the Maclaurin spheroid sequence.</li>
  <li>The bifurcation points of both ring-like sequences arise at positions on the Maclaurin spheroid sequence where the models are expected to be violently unstable toward the development of nonaxisymmetric distortions.  Hence, in order to study how unstable spheroids undergo a transition to a ring-like configuration (the topological change), a numerical code must be developed with the capability to suppress all nonaxisymmetric distortions.  This can be accomplished, for example, by adopting a cylindrical coordinate representation of the fluid equations then performing two-dimensional rather than three-dimensional simulations.</li>
</ol>


===One-Ring (Dyson-Wong) Sequence===

It is important to remember &#8212; as emphasized above &#8212; that ''all'' equilibrium models along the one-ring (Dyson-Wong) sequence are uniformly rotating.

====Background Storyline====

<ol>
  <li>Over 125 years ago via a pair of detailed publications &#8212; {{ Dyson1893full }} and {{ Dyson1893Part2full }} &#8212; Dyson demonstrated that a sequence of rapidly rotating, self-gravitating equilibrium models could be constructed that had a uniform density, were uniformly rotating, and had a toroidal (ring) shape. [Configurations that, in every respect except their ''shape'', were like Maclaurin  spheroids.]  See [[Apps/DysonPotential#Dyson_(1893)|our review and discussion]] of this work.</li>
  <li>{{ Wong74full }} tackled this same problem, improving on, and extending Dyson's work.  As a consequence, this sequence of models is often referred to as "Dyson-Wong tori."  See [[Apps/DysonWongTori#Self-Gravitating,_Incompressible_(Dyson-Wong)_Tori|our detailed review and discussion]].</li>
  <li>Motivated by the work of {{ FESB-K80full }}, {{ ES81full }} demonstrated that the Dyson-Wong toroidal sequence can be "smoothly connected" to the Maclaurin spheroid sequence via an intermediate branch of models having a <font color="darkgreen">&hellip; concave hamburger-like shape of equilibrium &hellip;"</font>
<ul>
  <li>Table I of {{ ES81 }} provides quantitative data describing the properties of eighteen models that lie along this combined "one-ring" sequence, such as:  <math>\Omega^2/(4\pi G \rho), j^2, T_\mathrm{rot}/|W_\mathrm{grav}|,</math> and <math>(T_\mathrm{rot} + W_\mathrm{grav})/E_0</math>.  We have copied the values of two of these parameters from their Table I into the first two (pink) columns of our table, immediately below.</li>
  <li>This data has been used to generate the pink-colored one-ring sequence shown in our plot, below; see especially the plot inset.  Over the years, the same set of data has been used to display the behavior of the one-ring sequence in numerous publications; see, for example, the reproduction of Figure 1 (p. 488) from {{ CKST95b }} that we have presented, below.</li>
  <li>Figure 2 of {{ ES81 }} displays meridional-plane cross-sections through five of their eighteen models in an effort to illustrate how the surface geometry smoothly changes along the complete sequence: from spheroid, to "hamburger" shape, to torus.</li>
</ul>
  </li>
  <li>{{ ES81 }} claim that the one-ring 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/(4\pi G \rho) = 0.08726</math> and <math>j^2 = 0.02174</math>.  In support of this conjecture, they point out that, Chandrasekhar (1967; publication XXX) 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 }}.</li>
</ol>

====Model-Sequence Details====

<table border="1" align="center" cellpadding="5">
<tr><td align="center" colspan="13"><b>One-Ring Sequence</b> (see figure inset)<br />as quantitatively described in three separate studies<sup>&dagger;</sup></td></tr>

<tr>
  <td align="center" colspan="3" rowspan="2">{{ ES81 }}<br /><font size="-1">Data extracted from their Table I</font></td>
<td align="center" width="2%" bgcolor="lightgrey" colspan="1" rowspan="21">&nbsp;</td>
  <td align="center" colspan="3">{{ HES82 }}</td></td>
<td align="center" width="2%" bgcolor="lightgrey" colspan="1" rowspan="21">&nbsp;</td>
  <td align="center" colspan="5">{{ Hachisu86a }}</td></td>
</tr>
<tr>
  <td align="center" colspan="2"><font size="-1">Data extracted from their Table I</font></td>
  <td align="center" colspan="1"><font size="-1">Implication</font></td>
  <td align="center" colspan="3"><font size="-1">Data extracted from his Table Ia</font></td>
  <td align="center" colspan="2"><font size="-1">Implication (assuming G = &rho; = 1)</font></td>
</tr>

<tr>
  <td align="center"><math>\frac{\Omega^2}{4\pi G \rho}</math></td>
  <td align="center"><math>j^2</math></td>
  <td align="center"><math>\frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math></td>
  <td align="center"><math>\frac{\Omega^2}{4\pi G \rho}</math></td>
  <td align="center"><math>j</math></td>
  <td align="center"><math>j^2</math></td>
  <td align="center"><math>\Omega^2</math></td>
  <td align="center"><math>M</math></td>
  <td align="center"><math>J</math></td>
  <td align="center"><math>\frac{\Omega^2}{4\pi G \rho}</math></td>
  <td align="center"><math>j^2 = \frac{J^2 \rho^{1 / 3}}{4\pi G M^{10/3}}</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08506</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02243</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3648</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.09635</math></td>
  <td align="center"><math>0.13725</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.018837</math></td>

  <td align="center"><math>0.000</math></td>
  <td align="center"><math>4.17</math></td>
  <td align="center"><math>0.0</math></td>
  <td align="center"><math>0.00</math></td>
  <td align="center"><math>0.00</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08324</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02270</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3654</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.09104</math></td>
  <td align="center"><math>0.14360</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.020621</math></td>

  <td align="center"><math>1.31</math></td>
  <td align="center"><math>2.09</math></td>
  <td align="center"><math>0.960</math></td>
  <td align="center"><math>0.1042</math></td>
  <td align="center"><math>0.00628</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08206</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02269</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3637</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08763</math></td>
  <td align="center"><math>0.14702</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.021619</math></td>

  <td align="center"><math>1.41</math></td>
  <td align="center"><math>1.40</math></td>
  <td align="center"><math>0.666</math></td>
  <td align="center"><math>0.1122</math></td>
  <td align="center"><math>0.0115</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08139</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02251</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3608</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08646</math></td>
  <td align="center"><math>0.14799</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.021901</math></td>

  <td align="center"><math>1.31</math></td>
  <td align="center"><math>1.05</math></td>
  <td align="center"><math>0.483</math></td>
  <td align="center"><math>0.1042</math></td>
  <td align="center"><math>0.0158</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08113</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02224</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3576</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08488</math></td>
  <td align="center"><math>0.14896</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.022189</math></td>

  <td align="center"><math>1.09</math></td>
  <td align="center"><math>0.723</math></td>
  <td align="center"><math>0.307</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0867</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0221</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08119</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02198</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3550</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08379</math></td>
  <td align="center"><math>0.14927</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.022281</math></td>

  <td align="center"><math>1.01</math></td>
  <td align="center"><math>0.811</math></td>
  <td align="center"><math>0.377</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0804</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0227</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08139</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02181</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3534</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08276</math></td>
  <td align="center"><math>0.14904</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.022213</math></td>

  <td align="center"><math>1.01</math></td>
  <td align="center"><math>0.856</math></td>
  <td align="center"><math>0.404</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0804</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0218</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08150</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02177</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3531</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08207</math></td>
  <td align="center"><math>0.14822</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.021970</math></td>

  <td align="center"><math>1.01</math></td>
  <td align="center"><math>0.929</math></td>
  <td align="center"><math>0.447</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0804</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0203</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08183</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02163</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3519</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.08172</math></td>
  <td align="center"><math>0.14714</math></td>
  <td align="center" bgcolor="#7DCEA0"><math>0.021650</math></td>

  <td align="center"><math>0.938</math></td>
  <td align="center"><math>0.953</math></td>
  <td align="center"><math>0.461</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0746</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0199</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08236</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02104</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3461</math></td>
  <td align="center" colspan="3" rowspan="9">&nbsp;</td>

  <td align="center"><math>0.826</math></td>
  <td align="center"><math>0.924</math></td>
  <td align="center"><math>0.445</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0657</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0205</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.08174</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02037</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3378</math></td>

  <td align="center"><math>0.692</math></td>
  <td align="center"><math>0.845</math></td>
  <td align="center"><math>0.398</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0551</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0221</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.07944</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.01992</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3279</math></td>

  <td align="center"><math>0.548</math></td>
  <td align="center"><math>0.721</math></td>
  <td align="center"><math>0.326</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0436</math></td>
  <td align="center" bgcolor="#D7BDE2"><math>0.0252</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.07556</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.01980</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3228</math></td>

  <td align="center"><math>0.408</math></td>
  <td align="center"><math>0.571</math></td>
  <td align="center"><math>0241</math></td>
  <td align="center"><math>0.0325</math></td>
  <td align="center"><math>0.0299</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.07041</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02003</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3176</math></td>

  <td align="center"><math>0.278</math></td>
  <td align="center"><math>0.409</math></td>
  <td align="center"><math>00.154398</math></td>
  <td align="center"><math>0.0221</math></td>
  <td align="center"><math>0.0372</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.06473</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02063</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3140</math></td>

  <td align="center"><math>0.169</math></td>
  <td align="center"><math>0.255</math></td>
  <td align="center"><math>0.0815</math></td>
  <td align="center"><math>0.0134</math></td>
  <td align="center"><math>0.0503</math></td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.05775</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02162</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3119</math></td>

  <td align="center" colspan="5" rowspan="3">&nbsp;</td>
</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.05088</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02304</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3113</math></td>

</tr>

<tr>
  <td align="center" bgcolor="#FDE0DA"><math>0.04399</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.02497</math></td>
  <td align="center" bgcolor="#FDE0DA"><math>0.3122</math></td>

</tr>

<tr>
<td align="left" colspan="13">
<sup>&dagger;</sup>The data drawn from these three separate studies are displayed in the figure inset as follows:
<ul>
  <li>Pink circular markers and accompanying smooth curve: &nbsp; {{ ES81 }}</li>
  <li>Green square markers: &nbsp; {{ HES82 }}</li>
  <li>Light-purple triangular markers: &nbsp; {{ Hachisu86a }}</li>
</ul>
</td>
</tr>
</table>

<span id="PhaseTransition">&nbsp;</span>
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="1" rowspan="2">
[[File:ES81OneRingWithInsetBox.png|800px|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:CKST95bFig1annotated.png|400px|CKST95b Figure 1]]</td>
</tr>
</table>