Editing Appendix/Ramblings/ForOuShangli (section)

==EFE Diagram (Review)==

<table border="0" cellpadding="12" align="left"><tr><td align="center">'''Figure 1'''</td></tr><tr><td align="center">[[File:EFEdiagram4.png|left|400px|EFE Diagram identifying example models from Ou (2006)]]</td></tr></table>
In the context of our broad discussion of ellipsoidal figures of equilibrium, the label "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_XXVfull }}; 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 here, on the left, highlights four model ''sequences'', all of which also can be found in the original version:
<ul>
<li>''Jacobi'' sequence &#8212; the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from &sect;39, Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in [[ThreeDimensionalConfigurations/JacobiEllipsoids#Table2|Table 2]] of our accompanying discussion of Jacobi ellipsoids.  All of the models along this sequence have <math>f \equiv \zeta/\Omega_f = 0</math> and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, <math>\Omega_f</math>.</li>
<li>''Dedekind'' sequence &#8212; a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair.  All ellipsoidal figures along this sequence have <math>1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>
<li>The X = -1 ''self-adjoint'' sequence &#8212; At every point along this sequence, the value of the key frequency ratio, <math>\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration  <math>(f_-)</math>; specifically, <math>f_+ = f_- = -(a^2+b^2)/(ab)</math>.  The data identifying the location of the small, solid-black markers along this sequence have been drawn from &sect;48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li>
<li>The X = +1 ''self-adjoint'' sequence &#8212; At every point along this sequence, the value of the key frequency ratio, <math>\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration  <math>(f_-)</math>; specifically, <math>f_+ = f_- = +(a^2+b^2)/(ab)</math>.  The data identifying the location of the small, solid-black markers along this sequence have been drawn from &sect;48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li>
</ul>

Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram.  The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by {{ Ou2006 }} and that we have also chosen to use as examples.