Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

===Incompressible Roche Ellipsoids (&lambda; = 0)===

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Extracted from p. 229 of [http://rsta.royalsocietypublishing.org/content/206/402-412/161 G. H. Darwin (1906)]
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[[File:DarwinText01.png|700px|Roche limit]]
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Extracted from p. 242 of [http://rsta.royalsocietypublishing.org/content/206/402-412/161 G. H. Darwin (1906)]
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[[File:DarwinText02.png|700px|Roche limit]]
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Here we examine the results presented by Roche, by Darwin, and by EFE for the case of a point-mass secondary (<math>~(M^')</math> and a primary whose mass <math>~(M)</math> is formally zero.  In this case, we must use a different scheme for normalizing physical quantities.  Because the secondary is not spinning and it has no orbital motion, only the primary contributes to the system's "angular momentum"; but because the primary has no mass, we need to examine its (and, hence, the system's) ''specific'' angular momentum.  Specifically,

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<math>~\frac{I}{M}</math>
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<math>~=</math>
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<math>~\frac{1}{5}a_1^2  \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \, ,</math>
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and,
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<math>~j \equiv \frac{J_\mathrm{tot}}{M}</math>
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<math>~=</math>
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<math>~\biggl[ \frac{r^2}{(1+\cancelto{0}{p})} + \frac{I}{M} \biggr]\Omega_\mathrm{Kep} </math>
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&nbsp;
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<math>~=</math>
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<math>~R^2\biggl[ \biggl(\frac{r}{R}\biggr)^2 + \frac{2}{5} \cdot \mathfrak{J} \biggr]\biggl(\frac{GM^'}{r^3}\biggr)^{1 / 2} </math>
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&nbsp;
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<math>~=</math>
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<math>~(GM^' R)^{1 / 2} \biggl[ \biggl(\frac{r}{R}\biggr)^{1 / 2} + \frac{2}{5} \cdot \mathfrak{J} \biggl(\frac{r}{R}\biggr)^{-3 / 2} \biggr] \, , </math>
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where, in order to ensure that the density of the primary remains constant along an equilibrium sequece, the adopted normalizing length scale is customarily,
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<math>~R^3 \equiv a_1 a_2 a_3 ~~~\Rightarrow ~~~ \frac{R}{a_1} = \biggl( \frac{a_2}{a_1}\cdot \frac{a_3}{a_1} \biggr)^{1 / 3} \, ,</math>
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in which case, 
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<math>~\mathfrak{J} </math>
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<math>~=</math>
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<math>~\frac{1}{2} \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R}  \biggr)^{2} </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{1}{2} \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_2}{a_1}\cdot \frac{a_3}{a_1} \biggr)^{-2 / 3} </math>
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  <th align="center" colspan="8"><font size="+1"><b>Table 3:</b></font>&nbsp; Incompressible <math>~(n=0)</math> Roche Ellipsoids with <math>~\lambda = p = 0</math></th>
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Extracted from Table 1 of [http://adsabs.harvard.edu/abs/1963ApJ…138.1182C Chandrasekhar (1963)]<br />
same as [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] Table XVI</th>
  <th align="center" colspan="5">EFE Check</th>
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  <td align="center"> (1) </td>
  <td align="center"> (2) </td>
  <td align="center"> (3) </td>
  <td align="center"> (4) </td>
  <td align="center"> (5) </td>
  <td align="center"> (6) </td>
  <td align="center"> (7) </td>
  <td align="center"> (8) </td>
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<td align="center"><math>~\cos^{-1}(a_3/a_1)</math></td>
<td align="center"><math>~a_2/a_1</math></td>
<td align="center"><math>~a_3/a_1</math></td>
<td align="center"><math>~\Omega^2</math></td>
<td align="center"><math>~r/R</math></td>
<td align="center"><math>~R/a_1</math></td>
<td align="center"><math>~\mathfrak{J}</math></td>
<td align="center"><math>~j/(GM^' R)^{1/2}</math></td>
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  <td align="center"> 24&deg; </td>
  <td align="center"> 0.93188 </td>
  <td align="center"> 0.91355 </td>
  <td align="center"> 0.022624 </td>
  <td align="center"> 3.8916 </td>
  <td align="center"> 0.9478 </td>
  <td align="center"> 1.0400 </td>
  <td align="center"> 2.0269 </td>
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  <td align="center"> 36&deg; </td>
  <td align="center"> 0.84112 </td>
  <td align="center"> 0.80902 </td>
  <td align="center"> 0.047871 </td>
  <td align="center"> 3.0312 </td>
  <td align="center"> 0.8796 </td>
  <td align="center"> 1.1035 </td>
  <td align="center"> 1.8247 </td>
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  <td align="center"> 48&deg; </td>
  <td align="center"> 0.70687 </td>
  <td align="center"> 0.66913 </td>
  <td align="center"> 0.074799 </td>
  <td align="center"> 2.6122 </td>
  <td align="center"> 0.7791 </td>
  <td align="center"> 1.2352 </td>
  <td align="center"> 1.7333 </td>
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  <td align="center"> 57&deg; </td>
  <td align="center"> 0.57787 </td>
  <td align="center"> 0.54464 </td>
  <td align="center"> 0.088267 </td>
  <td align="center"> 2.4720 </td>
  <td align="center"> 0.68022 </td>
  <td align="center"> 1.4415 </td>
  <td align="center"> 1.7206 </td>
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  <td align="center"> 60&deg; </td>
  <td align="center"> 0.53013 </td>
  <td align="center"> 0.50000 </td>
  <td align="center"> 0.089946 </td>
  <td align="center"> 2.4565 </td>
  <td align="center"> 0.6424 </td>
  <td align="center"> 1.5523 </td>
  <td align="center"> 1.7286 </td>
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  <td align="center"> 61&deg; </td>
  <td align="center"> 0.51373 </td>
  <td align="center"> 0.48481 </td>
  <td align="center"> 0.090068 </td>
  <td align="center"> 2.4554 </td>
  <td align="center"> 0.6292 </td>
  <td align="center"> 1.5964 </td>
  <td align="center"> 1.7329 </td>
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  <td align="center"> 62&deg; </td>
  <td align="center"> 0.49714 </td>
  <td align="center"> 0.46947 </td>
  <td align="center"> 0.089977 </td>
  <td align="center"> 2.4562 </td>
  <td align="center"> 0.6157 </td>
  <td align="center"> 1.6450 </td>
  <td align="center"> 1.7382 </td>
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  <td align="center"> 63&deg; </td>
  <td align="center"> 0.48040 </td>
  <td align="center"> 0.45399 </td>
  <td align="center"> 0.089689 </td>
  <td align="center"> 2.4589 </td>
  <td align="center"> 0.6019 </td>
  <td align="center"> 1.6984 </td>
  <td align="center"> 1.7443 </td>
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  <td align="center"> 66&deg; </td>
  <td align="center"> 0.42898 </td>
  <td align="center"> 0.40674 </td>
  <td align="center"> 0.087201 </td>
  <td align="center"> 2.48202 </td>
  <td align="center"> 0.5588 </td>
  <td align="center"> 1.8959 </td>
  <td align="center"> 1.7694 </td>
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  <td align="center"> 71&deg; </td>
  <td align="center"> 0.34052 </td>
  <td align="center"> 0.32557 </td>
  <td align="center"> 0.077474 </td>
  <td align="center"> 2.5818 </td>
  <td align="center"> 0.4804 </td>
  <td align="center"> 2.4178 </td>
  <td align="center"> 1.8399 </td>
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  <td align="center"> 72&deg; </td>
  <td align="center"> 0.32254 </td>
  <td align="center"> 0.30902 </td>
  <td align="center"> 0.074648 </td>
  <td align="center"> 2.6140 </td>
  <td align="center"> 0.4636 </td>
  <td align="center"> 2.5679 </td>
  <td align="center"> 1.8598 </td>
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  <td align="center"> 75&deg; </td>
  <td align="center"> 0.26827 </td>
  <td align="center"> 0.25882 </td>
  <td align="center"> 0.064426 </td>
  <td align="center"> 2.7455 </td>
  <td align="center"> 0.4110 </td>
  <td align="center"> 3.1728 </td>
  <td align="center"> 1.9359 </td>
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  <td align="center"> 79&deg; </td>
  <td align="center"> 0.19569 </td>
  <td align="center"> 0.19081 </td>
  <td align="center"> 0.047111 </td>
  <td align="center"> 3.0475 </td>
  <td align="center"> 0.3342 </td>
  <td align="center"> 4.6471 </td>
  <td align="center"> 2.0951 </td>
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[[File:P0Diagram.png|600px|Roche, Darwin, and Chandrasekhar p=0]]
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