Editing SSC/Structure/BiPolytropes/Analytic15 (section)

====Obtained from Chandrasekhar's U-constraint====
We shall now demonstrate that the same expression for the scaling coefficient, <math>~B_0</math>, can alternatively be obtained from Chandrasekhar's U-constraint, without assuming that <math>~\phi_i = 1</math>, after taking into account the result that already has been obtained from the V-constraint.  As [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|described above]], the U-constraint is an alternative interface condition that may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\xi_i \theta_i}{(-d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\eta_i \phi_i^{5}}{(-d\phi/d\eta)_i} \, ,</math>
  </td>
</tr>
</table>
</div>
which, in the particular case being examined here, becomes &#8212; again, see [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on the "U<sub>5F</sub>" and "U<sub>1E</sub>" function notations used by {{ Murphy83a }} &#8212; 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( U_\mathrm{5F} \biggr)_i </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)} \, .
</math>
  </td>
</tr>
</table>
</div>

Now, from our [[SSC/Structure/BiPolytropes/Analytic15#First_Constraint|discussion, above, of the first constraint]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-2\sin^2\Delta_i)\kappa_i \, .</math>
  </td>
</tr>
</table>
</div>

Hence, Chandrasekhar's U-constraint becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)^2 \kappa_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ B_0^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\sin^4\Delta_i}{\xi_i^2 (3-2\sin^2\Delta_i)^2} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ B_0^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1}  \, ,</math>
  </td>
</tr>
</table>
</div>
which, as predicted, is identical to what we learned from the third interface condition, alone.