Editing Appendix/Ramblings/MacSphCriticalPoints (section)

==Introduction==

As has been explained in, for example, our [[ThreeDimensionalConfigurations/RiemannStype#Fig2|accompanying discussion of Riemann S-type ellipsoids]], the "EFE Diagram"  refers to a two-dimensional parameter space defined by the pair of axis ratios (b/a, c/a), ''usually'' covering the ranges, 0 &le; b/a &le; 1 and 0 &le; c/a &le; 1.  The classic/original version of this diagram appears as Figure 2 on p. 902 of {{ Chandrasekhar65_XXV }}; a somewhat less cluttered version appears on p. 147 of Chandrasekhar's [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  

The version of the EFE Diagram shown in the left-hand panel of the following figure, highlights &hellip;
<ul>
  <li>The Maclaurin spheroid sequence &#8212; the vertical line that runs from <math>(b/a, c/a) = (1, 1)</math> (a nonrotating, sphere) to <math>(b/a, c/a) = (1, 0)</math> (an infinitesimally thin disk), aligning with the right-hand boundary of the Diagram;</li>
  <li>The Jacobi ellipsoid sequence &#8212; the curve running through the purple circular markers that connects the origin of the diagram <math>(b/a, c/a) = (0, 0)</math> to the point <math>(b/a, c/a) = (1, 0.582724)</math> where it intersects (and bifurcates from) the Maclaurin spheroid sequence.</li>
  <li>The pair of self-adjoint sequences (USA and LSA) as [[#Self-Adjoint_Sequences|discussed in detail above]] &#8212; the black dot-dashed curves that run from the origin of the diagram to points identified by yellow circular markers where they intersect the Maclaurin spheroid sequence: at, respectively, <math>(b/a, c/a) = (1, 1)</math> for the USA and <math>(b/a, c/a) = (1, 0.30333)</math> for the LSA.</li>
</ul>

To aid subsequent discussion, in this EFE diagram we have broken the Maclaurin spheroid sequence into three differently colored segments: &nbsp; <b>Blue</b> extends from the nonrotating sphere (point of USA intersection) to the point where the Jacobi ellipsoid sequence intersects; <b>Orange</b> extends from the Jacobi intersection point to the point of LSA intersection; and <b>Black</b> extends from the LSA intersection to the end of the Maclaurin sequence (the infinitesimally thin disk).

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center">
<b>EFE Diagram</b><br />

[[File:OurEFEannotated.png|400px|OurEFE]]
  </td>
  <td align="center">
<b>&Omega;<sup>2</sup> vs. j<sup>2</sup> Diagram</b><br />

[[File:OurHE84Fig1annotated.png|525px|OurHE84Fig1]]
  </td>
</tr>
</table>

Now, as we consider examining the stability of individual models &#8212; or the behavior of equilibrium model sequences &#8212; whose configurations are not constrained to have purely ellipsoidal shapes, the two-dimensional parameter space <math>(b/a, c/a)</math> associated with the EFE diagram proves to be of little use.  In Figures 1 through 6 of an [[Apps/MaclaurinSpheroidSequence|accompanying overview]], we show how the Maclaurin spheroid sequence behaves when displayed in six alternate 2D-parameter-space diagrams that have been proposed/used &#8212; to varying degrees of success &#8212; by various research groups over the years.  Following the [[Apps/EriguchiHachisu/Models|extensive set of related work published by Eriguchi, Hachisu and their collaborators over the decade of the 1980s]], here we adopt the &Omega;<sup>2</sup> versus j<sup>2</sup> diagram as displayed in the right-hand panel of the above figure, where, <math>\Omega^2 \equiv \omega_0^2/(4\pi G \rho)</math>.

The blue, orange, and black segments of the (vertical straight line) Maclaurin spheroid sequence that have been highlighted in our version of the EFE diagram map respectively to the blue, orange, and black segments of the (curved, double-valued) Maclaurin spheroid sequence that appears in our version of the &Omega;<sup>2</sup> versus j<sup>2</sup> diagram.  Also shown (purple circular markers) is the Jacobi ellipsoid sequence, along with its intersection with the Maclaurin spheroid sequence; and (see the yellow circular markers) the points where the USA and LSA sequences intersect the Maclaurin sequence.  As an additional reference point, in the diagram on the right, the small green square marker identifies the [[Apps/MaclaurinSpheroidSequence#OmegaMax|point along the Maclaurin spheroid sequence where &Omega;<sup>2</sup> attains its maximum value]] &#8212; <math>(\Omega^2, j^2)= (0.112333, 0.010105)</math>; in our EFE diagram, a green square marker identifies this same Maclaurin spheroid &#8212; <math>(b/a, c/a) = (1, 0.36769)</math>.