Editing Appendix/Ramblings/TurningPoints (section)

===Incompressible Roche Ellipsoids (&lambda; &#8800; 0)===

Let's see if we can understand the relationship between tabulated data presented by [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, &amp; Shapiro (1993b, ApJS, 88, 205)] &#8212; hereafter, LRS93S &#8212; for the case of incompressible Roche ellipsoids, when <math>~\lambda = p = 1</math>.  After setting <math>~\kappa_{n=0} = 1</math> in their equation (4.8), we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~I</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}a_1^2 M \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>
And, from their equation (7.12),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~J_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{Mr^2}{(1+p)} + I \biggr]\Omega \, .</math>
  </td>
</tr>
</table>
</div>

<span id="Kepler">If we adopt the Keplerian orbital frequency, the expression for the total angular momentum is,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~J_\mathrm{Kep}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{Mr^2}{(1+p)} + \frac{1}{5}a_1^2 M \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggr]\biggl[ \frac{GM_\mathrm{tot}}{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^3)^{1/2} a_1^2\biggl[ \frac{1}{(1+p)}\biggl(\frac{r}{a_1}\biggr)^2 + \frac{1}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggr]\biggl[ \frac{1}{r^3} \biggl( \frac{1+p}{p} \biggr)\biggr]^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(GM^3 R)^{1/2} \biggl( \frac{a_1^4}{r^{3}R} \biggr)^{1/2} \biggl[ \frac{1}{(1+p)}\biggl(\frac{r}{a_1}\biggr)^2 + \frac{1}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggr]
\biggl[ \biggl( \frac{1+p}{p} \biggr)\biggr]^{1/2} \, . </math>
  </td>
</tr>
</table>
</div>

For the specific case of an equal-mass binary sequence &#8212; that is, <math>~\lambda = p = 1</math> &#8212; as considered in the following table, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bar{J}_\mathrm{Kep}\biggr|_{p=1} \equiv (GM^3 R)^{-1/2} J_\mathrm{Kep}\biggr|_{p=1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2}  \biggl( \frac{a_1}{r} \biggr)^{2}\biggl( \frac{r^3}{a_1^3} \cdot \frac{a_1^3}{R^3}\biggr)^{1/6} \biggl[ \biggl(\frac{r}{a_1}\biggr)^2 + \frac{2}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2}  \biggl( \frac{a_1}{r} \biggr)^{3/2} \biggl( \frac{a_2}{a_1}\cdot \frac{a_3}{a_1}\biggr)^{-1/6} \biggl[ \biggl(\frac{r}{a_1}\biggr)^2 + \frac{2}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>

where we also have involved the expression for an equivalent spherical radius given just before their equation (7.21), namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_1 a_2 a_3 = a_1^3\biggl( \frac{a_2}{a_1}\cdot \frac{a_3}{a_1}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>


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

<tr>
  <th align="center" colspan="9"><font size="+1"><b>Table 1:</b></font>&nbsp; Incompressible <math>~(n=0)</math> Roche Ellipsoids with <math>~\lambda = p = 1</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>
  <td align="center"> (9) </td>
</tr>
<tr>
<td align="center"><math>~\cos^{-1}(a_3/a_1)</math>
<td align="center"><math>~a_2/a_1</math>
<td align="center"><math>~a_3/a_1</math>
<td align="center"><math>~\Omega^2</math>
<td align="center"><math>~r/a_1</math>
<td align="center"><math>~\bar{J}_\mathrm{Kep}=L_\mathrm{tot}/(GM^3 R)^{1/2}</math>
<td align="center"><math>~r/R</math>
<td align="center"><math>~\mathfrak{J}</math>
<td align="center"><math>~L_\mathrm{tot}/(GM_\mathrm{tot}^3 R)^{1/2}</math>
</tr>
<tr>
  <td align="center"> 12&deg; </td>
  <td align="center"> 0.98660 </td>
  <td align="center"> 0.97815 </td>
  <td align="center"> 0.009293 </td>
  <td align="center"> 6.5181</td>
  <td align="center"> 1.8498 </td>
  <td align="center"> 6.5959 </td>
  <td align="center"> 1.0104 </td>
  <td align="center"> 0.6540 </td>
</tr>
<tr>
  <td align="center"> 24&deg; </td>
  <td align="center"> 0.94376 </td>
  <td align="center"> 0.91355 </td>
  <td align="center"> 0.036152 </td>
  <td align="center"> 3.9916 </td>
  <td align="center"> 1.5168 </td>
  <td align="center"> 4.1938 </td>
  <td align="center"> 1.0436 </td>
  <td align="center"> 0.5363 </td>
</tr>
<tr>
  <td align="center"> 36&deg; </td>
  <td align="center"> 0.86345 </td>
  <td align="center"> 0.80902 </td>
  <td align="center"> 0.076342 </td>
  <td align="center"> 2.9005 </td>
  <td align="center"> 1.3846 </td>
  <td align="center"> 3.2689 </td>
  <td align="center"> 1.1086 </td>
  <td align="center"> 0.4895 </td>
</tr>
<tr>
  <td align="center"> 48&deg; </td>
  <td align="center"> 0.73454 </td>
  <td align="center"> 0.66913 </td>
  <td align="center"> 0.118726 </td>
  <td align="center"> 2.2266 </td>
  <td align="center"> 1.3353 </td>
  <td align="center"> 2.8215 </td>
  <td align="center"> 1.2360 </td>
  <td align="center"> 0.4721 </td>
</tr>
<tr>
  <td align="center"> 54&deg; </td>
  <td align="center"> 0.64956 </td>
  <td align="center"> 0.58779 </td>
  <td align="center"> 0.134284 </td>
  <td align="center"> 1.9645 </td>
  <td align="center"> 1.3351 </td>
  <td align="center"> 2.7080 </td>
  <td align="center"> 1.3510 </td>
  <td align="center"> 0.4720 </td>
</tr>
<tr>
  <td align="center"> 59&deg; </td>
  <td align="center"> 0.56892 </td>
  <td align="center"> 0.51504 </td>
  <td align="center"> 0.140854 </td>
  <td align="center"> 1.7702 </td>
  <td align="center"> 1.3494 </td>
  <td align="center"> 2.6652 </td>
  <td align="center"> 1.5003 </td>
  <td align="center"> 0.4771 </td>
</tr>
<tr>
  <td align="center"> 60&deg; </td>
  <td align="center"> 0.55186 </td>
  <td align="center"> 0.50000 </td>
  <td align="center"> 0.141250 </td>
  <td align="center"> 1.7335 </td>
  <td align="center"> 1.3542 </td>
  <td align="center"> 2.6627 </td>
  <td align="center"> 1.5390 </td>
  <td align="center"> 0.4788 </td>
</tr>
<tr>
  <td align="center"> 61&deg; </td>
  <td align="center"> 0.53451 </td>
  <td align="center"> 0.48481 </td>
  <td align="center"> 0.141298 </td>
  <td align="center"> 1.6974 </td>
  <td align="center"> 1.3597 </td>
  <td align="center"> 2.6624 </td>
  <td align="center"> 1.5816 </td>
  <td align="center"> 0.4807 </td>
</tr>
<tr>
  <td align="center"> 66&deg; </td>
  <td align="center"> 0.44429 </td>
  <td align="center"> 0.40674 </td>
  <td align="center"> 0.135785 </td>
  <td align="center"> 1.5253 </td>
  <td align="center"> 1.4006 </td>
  <td align="center"> 2.6980 </td>
  <td align="center"> 1.8732 </td>
  <td align="center"> 0.4952 </td>
</tr>
<tr>
  <td align="center"> 69&deg; </td>
  <td align="center"> 0.38813 </td>
  <td align="center"> 0.35837 </td>
  <td align="center"> 0.127424 </td>
  <td align="center"> 1.4388 </td>
  <td align="center"> 1.4278 </td>
  <td align="center"> 2.7557 </td>
  <td align="center"> 2.1105 </td>
  <td align="center"> 0.5073 </td>
</tr>
<tr>
  <td align="center"> 71&deg; </td>
  <td align="center"> 0.35022 </td>
  <td align="center"> 0.32557 </td>
  <td align="center"> 0.119625 </td>
  <td align="center"> 1.3647 </td>
  <td align="center"> 1.4723 </td>
  <td align="center"> 2.8144 </td>
  <td align="center"> 2.3873 </td>
  <td align="center"> 0.5205 </td>
</tr>
<tr>
  <td align="center"> 72&deg; </td>
  <td align="center"> 0.33119 </td>
  <td align="center"> 0.30902 </td>
  <td align="center"> 0.115054 </td>
  <td align="center"> 1.3337 </td>
  <td align="center"> 1.4919 </td>
  <td align="center"> 2.8512 </td>
  <td align="center"> 2.5357 </td>
  <td align="center"> 0.5275 </td>
</tr>
<tr>
  <td align="center"> 73&deg; </td>
  <td align="center"> 0.31213 </td>
  <td align="center"> 0.29237 </td>
  <td align="center"> 0.110044 </td>
  <td align="center"> 1.3028 </td>
  <td align="center"> 1.5140 </td>
  <td align="center"> 2.8938 </td>
  <td align="center"> 2.7072 </td>
  <td align="center"> 0.5353 </td>
</tr>
<tr>
  <td align="center"> 75&deg; </td>
  <td align="center"> 0.27405 </td>
  <td align="center"> 0.25882 </td>
  <td align="center"> 0.098753 </td>
  <td align="center"> 1.2419 </td>
  <td align="center"> 1.5663 </td>
  <td align="center"> 3.0001 </td>
  <td align="center"> 3.1371 </td>
  <td align="center"> 0.5538 </td>
</tr>
<tr>
  <td align="center"> 78&deg; </td>
  <td align="center"> 0.21726 </td>
  <td align="center"> 0.20791 </td>
  <td align="center"> 0.078934 </td>
  <td align="center"> 1.1513 </td>
  <td align="center"> 1.6731 </td>
  <td align="center"> 3.2327 </td>
  <td align="center"> 4.1282 </td>
  <td align="center"> 0.5915 </td>
</tr>
<tr>
  <td align="center"> 81&deg; </td>
  <td align="center"> 0.16126 </td>
  <td align="center"> 0.15643 </td>
  <td align="center"> 0.056499 </td>
  <td align="center"> 1.0599 </td>
  <td align="center"> 1.8353 </td>
  <td align="center"> 3.6139 </td>
  <td align="center"> 5.9641 </td>
  <td align="center"> 0.6489 </td>
</tr>
</table>


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

<tr>
  <th align="center" colspan="10"><font size="+1"><b>Table 2:</b></font>&nbsp; Incompressible <math>~(n=0)</math> Roche Ellipsoids with <math>~\lambda = p = 1</math></th>
</tr>

<tr>
  <th align="center" colspan="7">LRS93 Supplements</th>
  <th align="center" colspan="3">LRS93 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>
  <td align="center">(9)</td>
  <td align="center">(10)</td>
</tr>
<tr>
<td align="center"><math>~r/a_1</math>
<td align="center"><math>~r/R</math>
<td align="center"><math>~a_2/a_1</math>
<td align="center"><math>~a_3/a_1</math>
<td align="center"><math>~\bar\Omega</math>
<td align="center"><math>~\bar{J}</math>
<td align="center"><math>~\bar{E}</math>
<td align="center"><math>~\biggl(\frac{rp^{1/3}}{a_1}\biggr)^3\biggl(\frac{R}{rp^{1/3}}\biggr)^3\biggl( \frac{a_2}{a_1}\cdot \frac{a_3}{a_1}\biggr)^{-1}</math>
<td align="center"><math>~\bar\Omega_\mathrm{Kep}</math>
<td align="center"><math>~\bar{J}_\mathrm{Kep}</math>
</tr>
<tr>
  <td align="center"> 5.0 </td>
  <td align="center"> 5.131 </td>
  <td align="center"> 0.9707 </td>
  <td align="center"> 0.9533 </td>
  <td align="center"> 0.1406 </td>
  <td align="center"> 1.653 </td>
  <td align="center"> -0.6943 </td>
  <td align="center">1.0000</td>
  <td align="center">0.1405</td>
  <td align="center">1.6515</td>
</tr>
<tr>
  <td align="center"> 4.0 </td>
  <td align="center"> 4.202 </td>
  <td align="center"> 0.9441 </td>
  <td align="center"> 0.9139 </td>
  <td align="center"> 0.1901 </td>
  <td align="center"> 1.522 </td>
  <td align="center"> -0.7128 </td>
  <td align="center">0.9998</td>
  <td align="center">0.1896</td>
  <td align="center">1.5180</td>
</tr>
<tr>
  <td align="center"> 3.0 </td>
  <td align="center"> 3.348 </td>
  <td align="center"> 0.8750 </td>
  <td align="center"> 0.8222 </td>
  <td align="center"> 0.2690 </td>
  <td align="center"> 1.408 </td>
  <td align="center"> -0.7349 </td>
  <td align="center">1.0001</td>
  <td align="center">0.2666</td>
  <td align="center">1.3954</td>
</tr>
<tr>
  <td align="center"> 2.7 </td>
  <td align="center"> 3.124 </td>
  <td align="center"> 0.8345 </td>
  <td align="center"> 0.7738 </td>
  <td align="center"> 0.3000 </td>
  <td align="center"> 1.386 </td>
  <td align="center"> -0.7404 </td>
  <td align="center">0.9998</td>
  <td align="center">0.2958</td>
  <td align="center">1.3661</td>
</tr>
<tr>
  <td align="center"> 2.5 </td>
  <td align="center"> 2.989 </td>
  <td align="center"> 0.7981 </td>
  <td align="center"> 0.7330 </td>
  <td align="center"> 0.3222 </td>
  <td align="center"> 1.377 </td>
  <td align="center"> -0.7427 </td>
  <td align="center">1.0002</td>
  <td align="center">0.3160</td>
  <td align="center">1.3506</td>
</tr>
<tr>
  <td align="center"> 2.380 </td>
  <td align="center"> 2.916 </td>
  <td align="center"> 0.7715 </td>
  <td align="center"> 0.7044 </td>
  <td align="center"> 0.3358 </td>
  <td align="center"> 1.375 </td>
  <td align="center"> -0.7432 </td>
  <td align="center">1.0005</td>
  <td align="center">0.3279</td>
  <td align="center">1.3436</td>
</tr>
<tr>
  <td align="center"> 2.2 </td>
  <td align="center"> 2.821 </td>
  <td align="center"> 0.7236 </td>
  <td align="center"> 0.6553 </td>
  <td align="center"> 0.3556 </td>
  <td align="center"> 1.380 </td>
  <td align="center"> -0.7418 </td>
  <td align="center">1.0003</td>
  <td align="center">0.3446</td>
  <td align="center">1.3373</td>
</tr>
<tr>
  <td align="center"> 2.112 </td>
  <td align="center"> 2.783 </td>
  <td align="center"> 0.6960 </td>
  <td align="center"> 0.6281 </td>
  <td align="center"> 0.3648 </td>
  <td align="center"> 1.386 </td>
  <td align="center"> -0.7399 </td>
  <td align="center">0.9998</td>
  <td align="center">0.3518</td>
  <td align="center">1.3366</td>
</tr>
<tr>
  <td align="center"> 2.0 </td>
  <td align="center"> 2.744 </td>
  <td align="center"> 0.6561 </td>
  <td align="center"> 0.5901 </td>
  <td align="center"> 0.3753 </td>
  <td align="center"> 1.399 </td>
  <td align="center"> -0.7358 </td>
  <td align="center">1.0001</td>
  <td align="center">0.3592</td>
  <td align="center">1.3389</td>
</tr>
<tr>
  <td align="center"> 1.801 </td>
  <td align="center"> 2.713 </td>
  <td align="center"> 0.5708 </td>
  <td align="center"> 0.5123 </td>
  <td align="center"> 0.3886 </td>
  <td align="center"> 1.441 </td>
  <td align="center"> -0.7218 </td>
  <td align="center">1.0004</td>
  <td align="center">0.3654</td>
  <td align="center">1.3552</td>
</tr>
<tr>
  <td align="center"> 1.697 </td>
  <td align="center"> 2.724 </td>
  <td align="center"> 0.5184 </td>
  <td align="center"> 0.4664 </td>
  <td align="center"> 0.3908 </td>
  <td align="center"> 1.477 </td>
  <td align="center"> -0.7097 </td>
  <td align="center">1.0000</td>
  <td align="center">0.3632</td>
  <td align="center">1.3727</td>
</tr>
<tr>
  <td align="center"> 1.6 </td>
  <td align="center"> 2.759 </td>
  <td align="center"> 0.4644 </td>
  <td align="center"> 0.4198 </td>
  <td align="center"> 0.3884 </td>
  <td align="center"> 1.524 </td>
  <td align="center"> -0.6939 </td>
  <td align="center">1.0004</td>
  <td align="center">0.3563</td>
  <td align="center">1.3977</td>
</tr>
<tr>
  <td align="center"> 1.5 </td>
  <td align="center"> 2.831 </td>
  <td align="center"> 0.4040 </td>
  <td align="center"> 0.3682 </td>
  <td align="center"> 0.3798 </td>
  <td align="center"> 1.590 </td>
  <td align="center"> -0.6717 </td>
  <td align="center">1.0000</td>
  <td align="center">0.3428</td>
  <td align="center">1.4358</td>
</tr>
<tr>
  <td align="center"> 1.0 </td>
  <td align="center"> 5.312 </td>
  <td align="center"> 0.0823 </td>
  <td align="center"> 0.0811 </td>
  <td align="center"> 0.1685 </td>
  <td align="center"> 2.888 </td>
  <td align="center"> -0.3772 </td>
  <td align="center">0.9996</td>
  <td align="center">0.1334</td>
  <td align="center">2.2859</td>
</tr>
</table>


====Digesting LRS93S Results====

In column (12) of the above table we have combined the data from columns (5), (6), (7), &amp; (8) to demonstrate that LRS93S indeed used the definition of <math>~R</math>, given above for their incompressible, Roche ellipsoid configurations; these terms should zombie to give unity, and they appear to do so, within the accuracy presented by the data from LRS93S.  In column 14 of the table, we have listed the value of <math>~\bar{J}_\mathrm{Kep}</math>, as given by the above expression.  Column 13 lists <math>~\bar{\Omega}_\mathrm{Kep}</math>, as follows:

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

<tr>
  <td align="right">
<math>~\Omega_K</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{GM_\mathrm{tot}}{r^3} \biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{GM}{R^3} \biggl( \frac{1+p}{p}\biggr) \biggl( \frac{R}{r}\biggr)^3\biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{4\pi G \rho}{3}\biggr]^{1/ 2} \biggl( \frac{1+p}{p}\biggr)^{1 / 2} \biggl[\biggl( \frac{a_1}{r}\biggr)^3 \frac{a_2}{a_1} \cdot \frac{a_3}{a_1}\biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \bar\Omega_\mathrm{Kep}\biggr|_{p=1} \equiv \frac{\Omega_\mathrm{Kep}}{(\pi G \rho)^{1/2}}\biggr|_{p=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{8}{3}\biggl( \frac{a_1}{r}\biggr)^3 \frac{a_2}{a_1} \cdot \frac{a_3}{a_1}\biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
The value of the LRS93S correction factor, <math>~(1+\delta)</math>, can be obtained either from the ratio, <math>~\bar{J}/\bar{J}_\mathrm{Kep}</math>, or from the ratio, <math>~\bar{\Omega}/\bar{\Omega}_\mathrm{Kep}</math>.

====Digesting the EFE Results====

The EFE table lists values of <math>~\bar\Omega_\mathrm{Kep}</math>, but not values of <math>~r/a_1</math>.  Inverting the expression just provided gives,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a_1}{r}\biggr)^3 \frac{a_2}{a_1} \cdot \frac{a_3}{a_1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3}{8}\bar\Omega^2_\mathrm{Kep}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{r}{a_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{8}{3\bar\Omega^2_\mathrm{Kep} }\biggl( \frac{a_2}{a_1} \cdot \frac{a_3}{a_1} \biggr) \biggr]^{1 / 3} \, .</math>
  </td>
</tr>
</table>
</div>

In the above "EFE Check" column, we've listed these inferred values of <math>~r/a_1</math>.  Or, we might prefer the ratio, <math>~r/R</math>, which is obtained from the Keplerian frequency via the expression,

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

<tr>
  <td align="right">
<math>~ \Omega_K^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi G \rho}{3} \biggl( \frac{1+p}{p}\biggr) \biggl[\frac{R}{r} \biggr]^{3} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~  \biggl[\frac{R}{r} \biggr]^{3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{3}{4}\biggr] \biggl( \frac{p}{1+p}\biggr) ~\biggl( \frac{\Omega_K^2}{\pi G\rho}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~  \biggl(\frac{r}{R} \biggr)_{p=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{3}{8} \biggl( \frac{\Omega_K^2}{\pi G\rho}\biggr)_{p=1} \biggr]^{-1/3}  \, .</math>
  </td>
</tr>
</table>
</div>


====Comparison====
In the context of our [[#SphericalLtot|simplistic spherical model, above]], we derived the following expression for the total angular momentum:

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

<tr>
  <td align="right">
<math>~L_\mathrm{tot} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(G M_\mathrm{tot}^3 R)^{1 / 2} \biggl\{ \frac{1}{(1+\lambda)}  \biggl( \frac{d}{R}\biggr)^{1 / 2}
+ \frac{2}{5}\biggl[ 1
+  \cancelto{0}{\frac{1}{\lambda}\biggl( \frac{R^'}{R}\biggr)^{2}} \biggr] \biggl( \frac{d}{R}\biggr)^{-3/2}
\biggr\}\biggl(\frac{\lambda}{1+\lambda} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

Rewriting our [[#Kepler|just-derived "Keplerian" expression]] to emphasize the ratio <math>~r/R</math> instead of <math>~r/a_1</math>, and to highlight the system's total mass in the leading ''dimensional'' coefficient, allows us to more readily recognize the overlap with this simpler expression.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~J_\mathrm{Kep}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(GM^3 R)^{1/2} 
\biggl[ \frac{1}{(1+p)}\biggl( \frac{r}{R} \biggr)^{1/2}  + \frac{1}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R} \cdot  \frac{R}{r} \biggr)^{2} \biggl( \frac{r}{R} \biggr)^{1/2} \biggr]
\biggl( \frac{1+p}{p} \biggr)^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(GM_\mathrm{tot}^3 R)^{1/2} \biggl( \frac{p}{1+p} \biggr)^{3/2}
\biggl[ \frac{1}{(1+p)}\biggl( \frac{r}{R} \biggr)^{1/2}  + \frac{1}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R}  \biggr)^{2} \biggl( \frac{r}{R} \biggr)^{-3/2} \biggr]
\biggl( \frac{1+p}{p} \biggr)^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(GM_\mathrm{tot}^3 R)^{1/2} 
\biggl[ \frac{1}{(1+p)}\biggl( \frac{r}{R} \biggr)^{1/2}  + \frac{1}{5}\biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R}  \biggr)^{2} \biggl( \frac{r}{R} \biggr)^{-3/2} \biggr]
\biggl( \frac{p}{1+p} \biggr) </math>
  </td>
</tr>
</table>
</div>

It makes sense, then, to write the total angular momentum as,

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

<tr>
  <td align="right">
<math>~L_\mathrm{tot} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(G M_\mathrm{tot}^3 R)^{1 / 2} \biggl\{ \frac{1}{(1+\lambda)}  \biggl( \frac{d}{R}\biggr)^{1 / 2}
+ \frac{2}{5} \cdot \mathfrak{J}\biggl( \frac{d}{R}\biggr)^{-3/2}
\biggr\}\biggl(\frac{\lambda}{1+\lambda} \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\mathfrak{J} =1 </math> when one assumes that the primary star is spherical, but when tidal distortions are taken into account,
<div align="center">
<math>~\mathfrak{J} =  \frac{1}{2} \biggl( 1 + \frac{a_2^2}{a_1^2}\biggr) \biggl( \frac{a_1}{R}  \biggr)^{2} \, .</math>
</div>

<table border="1" cellpadding="5" align="center" width="80%">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Figure 1:</b></font> "Roche" Binary Sequences with Point-Mass Secondary and <math>~M/M^' = 1</math>
  </td>
</tr>
<tr>
  <td align="center">
Our Constructed Diagram
  </td>
  <td align="center">
Extracted from Fig. 10 of  [http://adsabs.harvard.edu/abs/1993ApJS...88..205L LRS93Supplement] 
  </td>
</tr>
<tr>
  <td align="center">
[[File:DarwinP1compare.png|600px|Compare to LRS93S Fig10]]
  </td>
  <td align="center">
[[File:LRS93SFig10.png|400px|LRS93S Fig10]]
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
<b>Left:</b> &nbsp; Curves showing how the total system angular momentum varies with binary separation when <math>~n=0</math> and the secondary star <math>~(M^')</math> is treated as a point mass.  (Blue dashed curve) Primary star assumed to be a sphere and, hence, <math>~\mathfrak{J} = 1</math>; (Green filled circular markers) Primary star is an (EFE) ellipsoidal configuration with axis ratios specified by columns 2 and 3 of our Table 1, normalized angular momentum specified by column 6 of our Table 1, and binary separation specified by column 7 of our Table 1; (Solid red curve connecting red filled circular markers) Primary star is an (LRS93S) ellipsoidal configuration with axis ratios specified by columns 3 and 4 of our Table 2, normalized angular momentum specified by column 6 of our Table 2, and binary separation specified by column 2 of our Table 2.  The green filled circular markers define the same (EFE) sequence that is presented as a dot-dashed curve in the right-hand panel; the red filled circular markers and associated smoothed curve define the same (LRS93S) sequence that is presented as a solid curve in the right-hand panel.  The purple filled circlular marker identifies the turning point along the (LRS93S) sequence associated with the minimum system angular momentum; the yellow filled circular marker identifies the turning point along the same sequence that is associated with the minimum separation &#8212; the so-called "Roche" limit.

<b>Right:</b> &nbsp; (The following text is largely taken from the Fig. 10 caption of LRS93S)  Equilibrium curves generated by LRS93S showing total angular momentum as a function of binary separation along two incompressible, and three compressible Roche sequences with <math>~M/M^' = 1</math>.  The various curves display results from polytropic configurations having <math>~n=0</math> (''solid line''), <math>~n=1</math> (''dotted line''), <math>~n=1.5</math> (''short-dashed line''), and <math>~n=2.5</math> (''long-dashed line'').  For comparison, the sequence obtained by EFE for <math>~n=0</math> is also drawn (''dotted-dashed line'').
  </td>
</tr>
</table>