Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

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

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<math>~L_\mathrm{tot} </math>
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<math>~=</math>
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<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>
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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.
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<math>~J_\mathrm{Kep}</math>
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<math>~=</math>
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<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>
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&nbsp;
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<math>~=</math>
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<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>
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&nbsp;
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<math>~=</math>
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<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>
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It makes sense, then, to write the total angular momentum as,

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  <td align="right">
<math>~L_\mathrm{tot} </math>
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<math>~=</math>
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<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>
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where, <math>~\mathfrak{J} =1 </math> when one assumes that the primary star is spherical, but when tidal distortions are taken into account,
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<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>

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<font size="+1"><b>Figure 1:</b></font> "Roche" Binary Sequences with Point-Mass Secondary and <math>~M/M^' = 1</math>
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Our Constructed Diagram
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Extracted from Fig. 10 of  [http://adsabs.harvard.edu/abs/1993ApJS...88..205L LRS93Supplement] 
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[[File:DarwinP1compare.png|600px|Compare to LRS93S Fig10]]
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[[File:LRS93SFig10.png|400px|LRS93S Fig10]]
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<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'').
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