Editing SSC/Stability/BiPolytropes/HeadScratching (section)

===Central Boundary Condition===

The central boundary condition is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dx_\mathrm{core}}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>

In order to kick-start the integration outward from the center of the configuration, we will following the procedure that has been detailed in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|accompanying discussion]].  At the center of the configuration <math>(\xi_1 = 0)</math>, we label the fractional displacement function as <math>x_1</math> &#8212; value to be set later, perhaps in an effort to help secure the proper matching conditions at the interface &#8212; then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center.  Specifically, given that <math>n = 5, \gamma_g = 6/5</math>, and <math>\alpha_g = -1/3</math> in the core, we will set,

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

<tr>
  <td align="right">
<math>
x_2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  \Delta_\xi^2}{60} \biggr] 
=
x_1 \biggl[ 1 - \frac{\mathfrak{F}  \Delta_\xi^2}{10} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\mathfrak{F}
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\alpha_g \biggr]
=
\frac{1}{6}\biggl[ 5 \sigma_c^2  + 4 \biggr]
\, .</math>
  </td>
</tr>
</table>