Editing Appendix/Ramblings/MacSphCriticalPoints (section)

====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>
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  <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>
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  </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>
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