Editing SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3 (section)

====Strategically Incorporate Step Function====
As we have discussed separately, a useful expression for the specific entropy of any individual Lagrangian fluid element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{\mathfrak{R}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\mu (\gamma-1)} \cdot \ln \biggl(\frac{P}{\rho^\gamma}\biggr)
\, .
</math>
  </td>
</tr>
</table>
How does <math>s/\mathfrak{R}</math> vary as a function of the Lagrangian mass shell (or Lagrangian radial coordinate)?  In the case of a spherical bipolytropic configuration: <math>s = s_c</math> (a constant) throughout the core; <math>s = s_e</math> (another constant) throughout the envelope; and a [[Appendix/Mathematics/StepFunction|unit step function]],

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">
<math>
H(\zeta) =  
\begin{cases}
0; & ~~ \zeta < 0 \\
1; & ~~ \zeta > 0
\end{cases}
</math>
<p><br />
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, &sect;2.1 (p. 123), Eq. (2.1.6)
  </td>
</tr>
</table>

can be introduced to accomplish the instantaneous jump from <math>s_c</math> to <math>s_e</math> at the core/envelope interface.  Specifically, after defining,

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  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{r_i} - 1  \, ,</math>
  </td>
</tr>
</table>

we obtain the correct physical description of the variation of specific entropy with mass shell, <math>m</math>, via the expression,

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

<tr>
  <td align="right">
<math>s(\zeta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>s_c + H(\zeta)\cdot (s_e - s_c) \, .</math>
  </td>
</tr>
</table>
Adopting the [[Appendix/Mathematics/StepFunction|''half-maximum convention'']] &#8212; which states that <math>H(\zeta = 0) = \tfrac{1}{2}</math> &#8212; we acknowledge that the functional value of the specific entropy at the interface is, <math>s_i = \tfrac{1}{2}(s_c + s_e)</math>.  Also, from our [[Appendix/Mathematics/StepFunction|accompanying brief discussion of the behavior of the unit step function]], we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dH(\zeta)}{d\zeta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\delta(\zeta)    \, ,</math>
  </td>
</tr>
</table>
where, <math>\delta(\zeta)</math> is the Dirac delta function.  We conclude, therefore, that precisely at the interface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ds(\zeta)}{d\zeta}\biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[(s_e - s_c) \cdot \delta(\zeta) \biggr]_i = (s_e - s_c)   \, .</math>
  </td>
</tr>
</table>
Generally speaking, the two parameters, <math>\mu, \gamma</math>, and the mass density, <math>\rho</math>, also will exhibit a step-function behavior at the interface of each equilibrium bipolytrope.  The following table summarizes how we model the radial variation of these quantities.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" rowspan="2">Quantity</td>
  <td align="center" rowspan="2">Functional Behavior</td>
  <td align="center" colspan="3">At Interface</td>
</tr>

<tr>
  <td align="center">Value</td>
  <td align="center">Derivative wrt <math>\zeta</math></td>
  <td align="center">Derivative wrt <math>r</math></td>
</tr>

<tr>
  <td align="center">Specific Entropy</td>
  <td align="center"><math>s(\zeta) = s_c + H(\zeta)\cdot (s_e - s_c)</math></td>
  <td align="center"><math>s_i \equiv s(0) = \tfrac{1}{2}(s_c + s_e)</math></td>
  <td align="center"><math>\frac{ds(\zeta)}{d\zeta}\biggr|_i = (s_e - s_c)</math></td>
  <td align="center"><math>\frac{ds}{dr}\biggr|_i = (s_e - s_c)/r_i</math></td>
</tr>

<tr>
  <td align="center">Mean Molecular Weight</td>
  <td align="center"><math>\mu(\zeta) = \mu_c + H(\zeta)\cdot (\mu_e - \mu_c)</math></td>
  <td align="center"><math>\mu_i \equiv \mu(0) = \tfrac{1}{2}(\mu_c + \mu_e)</math></td>
  <td align="center"><math>\frac{d\mu(\zeta)}{d\zeta}\biggr|_i = (\mu_e - \mu_c)</math></td>
  <td align="center"><math>\frac{d\mu}{dr}\biggr|_i = (\mu_e - \mu_c)/r_i</math></td>
</tr>

<tr>
  <td align="center">Ratio of Specific Heats</td>
  <td align="center"><math>\gamma(\zeta) = \gamma_c + H(\zeta)\cdot (\gamma_e - \gamma_c)</math></td>
  <td align="center"><math>\gamma_i \equiv \gamma(0) = \tfrac{1}{2}(\gamma_c + \gamma_e)</math></td>
  <td align="center"><math>\frac{d\gamma(\zeta)}{d\zeta}\biggr|_i = (\gamma_e - \gamma_c)</math></td>
  <td align="center"><math>\frac{d\gamma}{dr}\biggr|_i = (\gamma_e - \gamma_c)/r_i</math></td>
</tr>
</table>

The step-function that arises in a proper description of the density distribution must be handled with a bit more care.  Throughout the core,
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<tr>
  <td align="right">
<math>
\rho^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(1 + \frac{\xi^2}{3} \biggr)^{-5/2}  \, ,</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{3}{2\pi}\biggr)^{1 / 2} \xi \, ;</math>
  </td>
</tr>
</table>
and throughout the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\rho^*_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \biggl[ \frac{\eta_i}{\sin(\eta_i - B)}\biggr]
\biggl[ \frac{\sin(\eta - B)}{\eta}\biggr] \, ,</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_i^{-2} (2\pi)^{-1 / 2}\eta \, .</math>
  </td>
</tr>
</table>
The complete functional expression for the normalized mass density can therefore be written as,

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

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  <td align="right">
<math>\rho^*(\zeta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho^*_\mathrm{core}(\zeta) + H(\zeta)\cdot \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \frac{dH(\zeta)}{d\zeta}
+ H(\zeta)\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
\, .</math>
  </td>
</tr>
</table>

Sanity check:
<ul>
  <li><math>\zeta < 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \cancelto{0}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{0}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
=
\frac{d\rho^*_\mathrm{core}}{dr^*}
\, .</math>
  </td>
</tr>
</table>
  </li>

  <li><math>\zeta > 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \cancelto{0}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{1}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
=
\frac{d\rho^*_\mathrm{env}}{dr^*}
\, .</math>
  </td>
</tr>
</table>
  </li>
  <li><math>\zeta = 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{d\rho^*_\mathrm{core}}{dr^*} \biggr]_i
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i \frac{1}{r_i} \cancelto{1}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{1/2}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} + \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
+ \frac{1}{r_i} \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i 
\, .</math>
  </td>
</tr>
</table>
  </li>
</ul>