Editing SSC/Structure/BiPolytropes/AnalyzeStepFunction (section)

=More Careful Examination of Step Function Behavior=

The ideas that are captured in this chapter have arisen as an extension of our [[SSC/Structure/BiPolytropes/Analytic51Renormalize|accompanying "renormalization" of the Analytic51 bipolytrope]].

==Discontinuous Density Distribution==

===Expectations===

From among the set of [[SSC/Perturbations#Governing_Equations|governing relations]] that apply to spherically symmetric configurations, we focus, first, on the combined,

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} </math><br />
</div>

At the interface between the core and envelope of an equilibrium bipolytrope, both the core and the envelope must satisfy the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{\rho}\frac{dP}{dr}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{GM_r}{r^2} \, .</math>
  </td>
</tr>
</table>
Now, the quantity on the right-hand side of this expression must be the same at the interface, when viewed either from the perspective of the core or from the perspective of the envelope.  Therefore, at the interface, the equilibrium configuration must obey the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{1}{\rho}\frac{dP}{dr}\biggr]_{\mathrm{env}, i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{1}{\rho}\frac{dP}{dr}\biggr]_{\mathrm{core} , i }
\, .
</math>
  </td>
</tr>
</table>

<!--
But ''structurally'' for polytropic configurations, we know that, <math>d\ln P = [(n+1)/n]d\ln\rho</math>.  Hence, at the interface we must find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{(n_e+1)}{n_e\rho}\frac{d\ln \rho}{dr}\biggr]_{\mathrm{env}, i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{(n_c + 1)}{n_c\rho}\frac{d\ln \rho}{dr}\biggr]_{\mathrm{core} , i }
</math>
  </td>
</tr>
</table>
-->

Now, if we set <math>\rho_e = (\mu_e/\mu_c) \rho_c</math> at the interface, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{dP}{dr}\biggr]_{\mathrm{env}, i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mu_e}{\mu_c}\biggl[\frac{dP}{dr}\biggr]_{\mathrm{core} , i }
\, .
</math>
  </td>
</tr>
</table>

===Check Behavior===
In [[SSC/Structure/BiPolytropes/Analytic51#Step_4:_Throughout_the_core_(0_≤_ξ_≤_ξi)|step 4 of our accompanying analysis]], we find that from the perspective of the core,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \, .
</math>
  </td>
</tr>
</table>
Hence, at the interface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*}\biggr|_\mathrm{core} = \biggl\{ \frac{d\xi}{dr^*} \cdot \frac{dP^*}{d\xi} \biggr\}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2 \biggl(\frac{2\pi}{3}\biggr)^{1/2} \biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-4} \xi\biggr]_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl(\frac{4\pi}{3}\biggr) \theta_i^8 ~ r^*_i
\, .
</math>
  </td>
</tr>
</table>
While, in [[SSC/Structure/BiPolytropes/Analytic51#Step_8:_Throughout_the_envelope_(ηi_≤_η_≤_ηs)|step 8 of that analysis]], we find from the perspective of the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\theta_i^6 \phi^2 \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \, .
</math>
  </td>
</tr>
</table>
Hence, at the interface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*}\biggr|_\mathrm{env} = \biggl\{ \frac{d\eta}{dr^*} \cdot \frac{dP^*}{d\eta} \biggr\}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i (2\pi)^{1/2} \biggr\}
2\theta_i^6 \biggl[ \phi \cdot \frac{d\phi}{d\eta}\biggr]_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(2\pi)^{1/2}\theta^{8}_i  \biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)  \biggr\}
\biggl[ 3^{1 / 2} \theta_i^{-3} \biggl(\frac{d\theta}{d\xi}\biggr)_i\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\tfrac{2}{3}(6\pi)^{1/2}\theta^{8}_i  \biggl( \frac{\mu_e}{\mu_c} \biggr) \xi_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~2\biggl( \frac{2\pi}{3}\biggr)^{1/2}\theta^{8}_i  \biggl( \frac{\mu_e}{\mu_c} \biggr) 
\biggl[ \biggl( \frac{2\pi}{3}\biggr)^{1 / 2} r_i^*\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\biggl(\frac{4\pi}{3}\biggr) \theta^{8}_i \biggl( \frac{\mu_e}{\mu_c} \biggr)r_i^*  \, .
</math>
  </td>
</tr>
</table>
Gratifyingly, we find as expected that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{dP^*}{dr^*}\biggr]_{\mathrm{env}, i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mu_e}{\mu_c}\biggl[\frac{dP^*}{dr^*}\biggr]_{\mathrm{core} , i }
\, .
</math>
  </td>
</tr>
</table>