Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

====Using Our Adopted Notation====
Consider the simple model of two spherical stars in circular orbit about one another, as depicted here on the right.  In addition to the physical parameters explicitly labeled in this diagram, we adopt the following variable notation:  
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* The total system mass is,
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<math>~M_\mathrm{tot} \equiv M + M^' \, ;</math>
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* The ratio of the primary to secondary mass is,
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<math>~\lambda \equiv \frac{M}{M^'} \, ;</math>
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* And the separation between the two centers is,
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<math>~d \equiv r_\mathrm{cm} + r^'_\mathrm{cm} \, .</math>
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[[File:BinarySimpleModel02.png|350px|Simple Binary Model]]
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For a circular orbit, the angular velocity is related to the the system mass and separation via the Kepler relation,
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<math>~\omega^2 d^3 = GM_\mathrm{tot} \, ,</math>
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and the distances, <math>~r_\mathrm{cm}</math> and <math>~r^'_\mathrm{cm}</math>, between the center of each star and the center of mass (cm) of the system must be related to one another via the expression,
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  <td align="right">
<math>~\frac{r^'_\mathrm{cm}}{r_\mathrm{cm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{M}{M^'} = \lambda \, .</math>
  </td>
</tr>
</table>
</div>

Note that the following relations also hold:
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<tr>
  <td align="right">
<math>~M = M_\mathrm{tot} \biggl( \frac{\lambda}{1+\lambda}\biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~M^' = M_\mathrm{tot} \biggl( \frac{1}{1+\lambda}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_\mathrm{cm} = d \biggl( \frac{1}{1+\lambda}\biggr) \, ;</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~r^'_\mathrm{cm} = d \biggl( \frac{\lambda}{1+\lambda}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>

Hence, the orbital angular momentum is,
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  <td align="right">
<math>~L_\mathrm{orb}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [M r^2_\mathrm{cm}  + M^' (r^'_\mathrm{cm})^2]\omega </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ M_\mathrm{tot} d^2 \biggl[\biggl( \frac{\lambda}{1+\lambda}\biggr) \biggl( \frac{1}{1+\lambda}\biggr)^2  + \biggl( \frac{1}{1+\lambda}\biggr) \biggl( \frac{\lambda}{1+\lambda}\biggr)^2 \biggr] \biggl[\frac{GM_\mathrm{tot}}{d^3}\biggr]^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (G M_\mathrm{tot}^3 d)^{1 / 2} \biggl[\frac{\lambda}{(1+\lambda)^2}  \biggr] \, .</math>
  </td>
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</table>
</div>

Assuming that both stars are rotating synchronously with the orbit, their respective spin angular momenta are,
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<tr>
  <td align="right">
<math>~L_M = I_M \omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}MR^2 \omega</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}M_\mathrm{tot} \biggl(\frac{\lambda}{1+\lambda}\biggr) R^2 \biggl[ \frac{G M_\mathrm{tot}}{d^3} \biggr]^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}(GM_\mathrm{tot}^3 d)^{1 / 2} \biggl(\frac{\lambda}{1+\lambda}\biggr) \biggl( \frac{R}{d}\biggr)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~L_{M^'} = I_{M^'} \omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}{M^'}(R^')^2 \omega</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}M_\mathrm{tot} \biggl(\frac{1}{1+\lambda}\biggr) (R^')^2 \biggl[ \frac{G M_\mathrm{tot}}{d^3} \biggr]^{1 / 2}</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5}(GM_\mathrm{tot}^3 d)^{1 / 2} \biggl(\frac{1}{1+\lambda}\biggr) \biggl( \frac{R^'}{d}\biggr)^2 \, .</math>
  </td>
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<span id="SphericalLtot">Hence, the total angular momentum of the system is,</span>
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<tr>
  <td align="right">
<math>~L_\mathrm{tot} = L_\mathrm{orb} + L_M + L_{M^'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(G M_\mathrm{tot}^3 d)^{1 / 2} \biggl[\frac{\lambda}{(1+\lambda)^2}  \biggr] 
+ \frac{2}{5}(GM_\mathrm{tot}^3 d)^{1 / 2} \biggl(\frac{\lambda}{1+\lambda}\biggr) \biggl( \frac{R}{d}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2}{5}(GM_\mathrm{tot}^3 d)^{1 / 2} \biggl(\frac{1}{1+\lambda}\biggr) \biggl( \frac{R^'}{d}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(G M_\mathrm{tot}^3 R)^{1 / 2} \biggl\{ \biggl[\frac{\lambda}{(1+\lambda)^2}  \biggr] \biggl( \frac{d}{R}\biggr)^{1 / 2}
+ \frac{2}{5} \biggl(\frac{1}{1+\lambda}\biggr)\biggl[ \lambda
+  \biggl( \frac{R^'}{R}\biggr)^{2} \biggr] \biggl( \frac{d}{R}\biggr)^{-3/2}
\biggr\}
</math>
  </td>
</tr>
</table>
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If we assume that the two stars have the same (uniform) densities, <math>~\rho</math>, then, following Darwin (1906; see immediately below), the two stellar radii can be related to the mass ratio, <math>~\lambda</math> via the expressions,
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  <td align="right">
<math>~R = a\biggl( \frac{\lambda}{1+\lambda}\biggr) \, ,</math>
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  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
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<math>~R^' = a\biggl( \frac{1}{1+\lambda}\biggr) \, ,</math>
  </td>
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where, the characteristic length scale is,
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<math>a \equiv \biggl(\frac{3M_\mathrm{tot}}{4\pi \rho}\biggr)^{1 / 3} \, .</math>
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Replacing <math>~R</math> and <math>~R^'</math> by these expressions in our equation for <math>~L_\mathrm{tot}</math> results in the simplistic/illustrative expression for <math>~L_1</math> derived by Darwin (1906) and presented in the [[#DarwinL1L2|boxed-in image, below]].  Darwin's expression for <math>~L_2</math> is obtained by using the same expression for <math>~R</math> but treating the secondary as a point mass, that is, setting <math>~R^' = 0</math>.