Editing Appendix/Ramblings/BordeauxSequences (section)

====Fixed Mass Ratio====

Let's define the mass ratio,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{M_t}{M_c} \, ,
</math>
  </td>
</tr>
</table>

and build a sequence along which this ratio is held constant.  For the problem being considered here, the relevant expression for <math>~q</math> is,

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

<tr>
  <td align="right">
<math>~q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2\pi^2 R_\mathrm{eq}^3 \rho \alpha_t \beta_t^2  \biggr]
\biggl[ \frac{4\pi}{3} \rho R_\mathrm{eq}^3 (1 - e^2)^{1 / 2} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3\pi}{2} \biggr)  \alpha_t \beta_t^2 (1 - e^2)^{- 1 / 2}\, .
</math>
  </td>
</tr>
</table>

The sequence is constructed by choosing <math>~q</math> (held fixed), and varying the value of <math>~0 < \beta_t \le \beta_\mathrm{max}</math>; then, for each chosen parameter pair, recognize that,

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

<tr>
  <td align="right">
<math>~
\alpha_t (1 - e^2)^{- 1 / 2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2}{3\pi} \biggr)  \frac{q}{ \beta_t^2  } \, ,</math>
  </td>
</tr>
</table>
that is, from the [[#Surrounding_Torus|above expression for <math>~\alpha_t</math>]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[\frac{1}{3\Omega^2} (1 - e^2)^{1 / 2} \biggr] (1 - e^2)^{- 3 / 2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{2}{3\pi} \biggr)  \frac{q}{ \beta_t^2  } \biggr]^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\Omega^2 (1 - e^2) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{3}\biggl[ \biggl( \frac{3\pi}{2} \biggr)  \frac{ \beta_t^2  }{q} \biggr]^3 \, .</math>
  </td>
</tr>
</table>

But, for central models along the MLS, we also must satisfy the relation [[#Model_Sequences|given above]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Omega^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{2e^2} \, .
</math>
  </td>
</tr>
</table>
Hence, for each <math>~(q, \beta_t)</math> parameter pair, the relevant central-object eccentricity is given by the root of the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{2}(3 - 2e^2)(1 - e^2)^{3 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)^2 }{2e^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{3}\biggl[ \biggl( \frac{3\pi}{2} \biggr)  \frac{ \beta_t^2  }{q} \biggr]^3 \, .</math>
  </td>
</tr>
</table>

Note that, for the limiting value, <math>~\beta_\mathrm{max} = (\alpha_t - 1)</math>, the relevant relation becomes,

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

<tr>
  <td align="right">
<math>~
(\alpha_t - 1)^2\alpha_t (1 - e^2)^{- 1 / 2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2q}{3\pi}    </math>
  </td>
</tr>
</table>
where [[#Surrounding_Torus|again, as above]]
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_t \equiv \frac{a}{R_\mathrm{eq}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{1}{3\Omega^2} (1 - e^2)^{1 / 2} \biggr]^{1 / 3} \, .</math>
  </td>
</tr>
</table>