Editing Appendix/Ramblings/TurningPoints (section)

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

<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">
Extracted from p. 229 of<br />
{{ Darwin06figure }}
  </td>
</tr>
<tr>
  <td align="left">
<!-- [[File:DarwinText01.png|700px|Roche limit]] -->
<font color="darkgreen">"Finally the solution for Roche's limit and for the ratio of the axes of the ellipsoid in limiting stability may be taken to be as follows &hellip;
<table border="0" align="center" width="80%" cellpadding="5">
<tr>
  <td align="center" width="20%"><math>\gamma</math></td>
  <td align="center" width="20%"><math>\sin^{-1}\kappa</math></td>
  <td align="center" width="20%"><math>\cos\gamma</math></td>
  <td align="center" width="20%"><math>\cos\beta</math></td>
  <td align="center"><math>r/a</math></td>
</tr>
<tr>
  <td align="center">61<sup>&deg;</sup> 8&half;'</td>
  <td align="center">78<sup>&deg;</sup> 52'</td>
  <td align="center">0.4827</td>
  <td align="center">0.5114</td>
  <td align="center">2.4553</td>
</tr>
</table>
with uncertainty of unity in the las place of decimals in <math>r</math> and of half a minute of arc in <math>\sin^{-1}\kappa</math>.</font>
  </td>
</tr>
<tr>
  <td align="center">
Extracted from p. 242 of<br />
{{ Darwin06figure }}
  </td>
</tr>
<tr>
  <td align="left">
<!--[[File:DarwinText02.png|700px|Roche limit]] -->
<font color="darkgreen">
"As stated in the Preface, the radius vector of limiting stability, which has been called 'Roche's limit,' is found to be 2.4553, and the axes of the critical ellipsoid are proportional to the numbers 10000, 5114, 4827.  These may be compared with the 2.44 and 1000, 496, 469 determined by Roche himself.  When we consider the methods which he employed, we must be struck with the closeness to accuracy to which he attained."
</font>
  </td>
</tr>
</table>

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,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{I}{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}a_1^2  \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~j \equiv \frac{J_\mathrm{tot}}{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{r^2}{(1+\cancelto{0}{p})} + \frac{I}{M} \biggr]\Omega_\mathrm{Kep} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
where, in order to ensure that the density of the primary remains constant along an equilibrium sequece, the adopted normalizing length scale is customarily,
<div align="center">
<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>
</div>

in which case, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{J} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R}  \biggr)^{2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>


<table border="1" align="center" cellpadding="5">

<tr>
  <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>
</tr>

<tr>
  <th align="center" colspan="4">
Extracted from Table 1 of [http://adsabs.harvard.edu/abs/1963ApJ…138.1182C Chandrasekhar (1963)]<br />
same as [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] Table XVI</th>
  <th align="center" colspan="5">EFE Check</th>
</tr>
<tr>
  <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>
</tr>
<tr>
<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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <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>
</tr>
<tr>
  <td align="center" colspan="8">
[[File:P0Diagram.png|600px|Roche, Darwin, and Chandrasekhar p=0]]
  </td>
</tr>
</table>