Editing SSC/Structure/BiPolytropes/AnalyzeStepFunction (section)

==Transition==

In transitioning from the core to the envelope, all of the above (green) <font color="green">STEPS</font> will remain the same except for the 1<sup>st</sup> Law's treatment of the pressure.  Following along the lines of our [[Appendix/Ramblings/PatrickMotl#May_5_(following_a_phone_conversation_with_Patrick)|''Ramblings'' idea exchange with Patrick Motl]], the mass-density is generically related to the pressure via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{s \bar\mu (\gamma_g-1)}{\Re}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~P 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\gamma_g - 1) \rho^{\gamma_g}\exp \biggl[\frac{s \bar\mu (\gamma_g-1)}{\Re}\biggr]\, .</math>
  </td>
</tr>
</table>
Given that, in polytropic configurations for which we make the association, <math>\gamma_g = (n+1)/n</math>, the pressure is related to the density via the expression, <math>P = K\rho^{\gamma_g}</math>, we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>K</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\gamma_g - 1) \exp \biggl[\frac{s \bar\mu (\gamma_g-1)}{\Re}\biggr] \, .
</math>
  </td>
</tr>
</table>

===Core===
For the core, we set <math>\gamma_g = 6/5</math>.  Hence,
<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>K_c \rho^{6/5} \, .</math>
  </td>
</tr>
</table>

Normalizing the pressure and the density as we have in [[SSC/Structure/BiPolytropes/Analytic51#Normalization|a closely related discussion of the structure of bipolytropes]], we have throughout the core,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
P^* = \frac{P}{K_c\rho_0^{6/5}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\rho^*)^{6/5} \, .</math>
  </td>
</tr>
</table>

Now, in this normalized expression we see that the polytropic constant for the core is, <math>K^*_c = 1</math>.  This means that the specific entropy of all the fluid in the core is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{5} \exp \biggl[\frac{s^*_c {\bar\mu}_c }{5\Re}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{s^*_c {\bar\mu}_c }{\Re}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
5\ln(5 ) \, .
</math>
  </td>
</tr>
</table>

===Envelope===
For the envelope, we set <math>\gamma_g = 2</math>.  Hence,
<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>K_e \rho^{2} \, .</math>
  </td>
</tr>
</table>

Adopting the same pressure and density normalizations as used in the core, we have throughout the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
P^* = \frac{P}{K_c\rho_0^{6/5}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{K_e}{K_c} \biggr) \rho_0^{4/5} (\rho^*)^2 \, .</math>
  </td>
</tr>
</table>
Now, at the [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface of the equilibrium model]], we know that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{K_e}{K_c}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i
=
\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[1 + \frac{\xi_i^2}{3}\biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Hence, throughout 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>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[1 + \frac{\xi_i^2}{3}\biggr]^2 (\rho^*)^2 
\, ,
</math>
  </td>
</tr>
</table>

in which case we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>K^*_e</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[1 + \frac{\xi_i^2}{3}\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{s^*_e \bar\mu_e}{\Re}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[1 + \frac{\xi_i^2}{3}\biggr]^2 \biggr\} \, .
</math>
  </td>
</tr>
</table>

===<font color="green">STEP 3</font> Clarification===

It is critically important to appreciate that the manner in which the pressure is determined at discrete locations in our finite-difference model must be different in the envelope and the core.  Keeping in mind that, in general,
<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>(\gamma_g - 1) \rho^{\gamma_g}\exp \biggl[\frac{s \bar\mu (\gamma_g-1)}{\Re}\biggr]\, ,</math>
  </td>
</tr>
</table>
for <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have &hellip;

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

<tr>
  <td align="right">
Core <math>(\gamma_g=6/5)</math>: &nbsp; &nbsp; &nbsp; <math>P_{j-1/2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{5}\biggl[ {\bar\rho}_{j-1/2}\biggr]^{6/5} \cdot \exp\biggl[ \frac{\mu_c s^*_c}{5\mathfrak{R}} \biggr]
=
\biggl[ {\bar\rho}_{j-1/2}\biggr]^{6/5} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
Envelope <math>(\gamma_g=2)</math>: &nbsp; &nbsp; &nbsp; <math>P_{j-1/2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ {\bar\rho}_{j-1/2}\biggr]^{2} \cdot \exp\biggl[ \frac{\mu_e s^*_e}{\mathfrak{R}} \biggr]
=
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl[1 + \frac{\xi_i^2}{3}\biggr]^2 \biggl[ {\bar\rho}_{j-1/2}\biggr]^{2} 
\, .
</math>
  </td>
</tr>
</table>

===Interface===
Drawing from our [[SSC/Structure/BiPolytropes/Analytic51|chapter in which the bipolytrope is constructed]], we note the following determination of the density and pressure at the interface, as viewed from the perspective of the core and, separately, from the perspective of the envelope.

<table border="1" align="center">
<tr>
  <td align="center" colspan="2">
Properties at the (Unperturbed) Interface
  </td>
</tr>
<tr>
  <td align="center" width="50%">
<b>Core</b><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{s^*_c \bar\mu_c}{\Re}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
5\ln(5) 
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">
<b>Envelope</b><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{s^*_e \bar\mu_e}{\Re}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta_i^{-4} \biggr\} 
</math>
  </td>
</tr>
</table>
  </td>
</tr>

<tr>
<td align="left">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>(\rho^*)_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta_i^5</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
(P^*)_i 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{5} (\rho^*)_i^{6/5}\exp \biggl[\frac{s_c \bar\mu_c }{5\Re}\biggr]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\rho^*)_i^{6/5}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta_i^{6}</math>
  </td>
</tr>
</table>

</td>
<td align="left">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>(\rho^*)_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
(P^*)_i 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\rho^*)_i^{2}\exp \biggl[\frac{s_e \bar\mu_e }{\Re}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\rho^*)_i^{2}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta_i^{-4} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta_i^6</math>
  </td>
</tr>
</table>

</td>
</tr>
</table>


----

In what follows, we will continue to focus on the interface but, for simplicity, will drop the "i" subscript; instead, we will use a "0" subscript to indicate a property at the ''unperturbed'' interface.


<table border="1" align="center">
<tr>
  <td align="center" colspan="2">
Properties at the ''Perturbed Interface
  </td>
</tr>
<tr>
  <td align="center" width="50%">
<b>Core</b><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{s^*_c \bar\mu_c}{\Re}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
5\ln(5) 
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">
<b>Envelope</b><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{s^*_e \bar\mu_e}{\Re}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta_i^{-4} \biggr\} 
</math>
  </td>
</tr>
</table>
  </td>
</tr>

<tr>
<td align="left">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho^*_c = (\rho^*_c)_0 + (\delta \rho)_c </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta_i^5 + (\delta \rho)_c</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
P^*_c  = (P^*_c)_0 + (\delta P)_c 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \rho^*_c \biggr]^{6/5}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\theta_i^5 + (\delta \rho)_c\biggr]^{6/5}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\theta_i^6\biggl[1 + \frac{6(\delta \rho)_c}{5\theta_i^5}\biggr]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Rightarrow ~~~ (\delta P)_c</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{6\theta_i(\delta \rho)_c}{5}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Rightarrow ~~~ p_c \equiv \frac{(\delta P)_c}{(P^*_c)_0}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{6(\delta \rho)_c}{5\theta_i^5}</math>
  </td>
</tr>
</table>

</td>
<td align="left">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho^*_e = (\rho^*_e)_0 + (\delta\rho)_e</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5+ (\delta\rho)_e</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
P^*_e = (P^*_e)_0 + (\delta P)_e
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \rho^*_e \biggr]^{2}\exp \biggl[\frac{s_e \bar\mu_e }{\Re}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5+ (\delta\rho)_e\biggr]^{2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta_i^{-4} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\theta_i^6\biggl[ 1 + 2(\delta\rho)_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i^{-5}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (\delta P)_e</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>2(\delta\rho)_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Rightarrow ~~~ p_e \equiv \frac{(\delta P)_e}{(P^*_e)_0}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{2(\delta\rho)_e}{\theta_i^5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} </math>
  </td>
</tr>
</table>

</td>
</tr>
</table>

Now, in order for <math>(\delta P)_c = (\delta P)_e</math> at the interface, we must have,

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

<tr>
  <td align="right">
<math>
2(\delta\rho)_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6\theta_i(\delta \rho)_c}{5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~(\delta\rho)_e   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{5}\biggl( \frac{\mu_e}{\mu_c} \biggr) (\delta \rho)_c
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{(\delta\rho)_e }{\theta_i^5}  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{5}\biggr) \frac{(\delta \rho)_c}{\theta_i^5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{(\delta\rho)}{\rho^*_0} \biggr]_e 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{5}\biggr) \biggl[ \frac{(\delta\rho)}{\rho^*_0} \biggr]_c
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~d_e 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\gamma_c}{\gamma_e}\biggr) d_c \, .
</math>
  </td>
</tr>
</table>

We also know that, according to the linearized equation of continuity, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[r_0 \frac{dx}{dr_0} + 3x + d \biggr]_e
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[r_0 \frac{dx}{dr_0} + 3x + d \biggr]_c \, .
</math>
  </td>
</tr>
</table>
Given that, at the interface, <math>x_e = x_c</math>, this means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[r_0 \frac{dx}{dr_0}\biggr]_e   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[r_0 \frac{dx}{dr_0}\biggr]_c - \biggl[\biggl(\frac{\gamma_c}{\gamma_e}\biggr)- 1\biggr] d_c 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[r_0 \frac{dx}{dr_0}\biggr]_c 
+ \biggl[\biggl(\frac{\gamma_c}{\gamma_e}\biggr)- 1\biggr] \biggl\{
\biggl[r_0 \frac{dx}{dr_0}\biggr]_c + 3x 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3x\biggl[\biggl(\frac{\gamma_c}{\gamma_e}\biggr)- 1\biggr] 
+ \biggl(\frac{\gamma_c}{\gamma_e}\biggr) 
\biggl[r_0 \frac{dx}{dr_0}\biggr]_c  \, ;
</math>
  </td>
</tr>
</table>
or, dividing through by <math>x</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[\frac{d\ln x}{d\ln r_0}\biggr]_e   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3\biggl[\biggl(\frac{\gamma_c}{\gamma_e}\biggr)- 1\biggr] 
+ \biggl(\frac{\gamma_c}{\gamma_e}\biggr) 
\biggl[\frac{d\ln x}{d\ln r_0}\biggr]_c  \, .
</math>
  </td>
</tr>
</table>

This exactly matches the interface constraint presented in [[SSC/Stability/BiPolytropes#Interface_Conditions|a related discussion in a separate chapter]].

Now, as has already been stated in [[#Step_5|Step 5, above]], for our analytic guess for the core's displacement function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{d\ln x}{d\ln r_0}\biggr]_c</math>
  </td>
  <td align="center">
<math>=</math>
  <td align="left">
<math>
\biggl(\frac{2}{15}\biggr) \xi^2 \biggl[\frac{\xi^2}{15}-1\biggr]^{-1}
=
2.4526969
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[\frac{d\ln x}{d\ln r_0}\biggr]_e</math>
  </td>
  <td align="center">
<math>=</math>
  <td align="left">
<math>
0.2716182
\, .
</math>
  </td>
</tr>
</table>
where the numerical evaluation has been presented specifically for the core/envelope interface of the critical "Model A" configuration in which, <math>\xi_i = 9.0149598</math>.  Also, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{x}{r_0}   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\alpha_\mathrm{scale}}{\xi} \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggl[1 - \frac{\xi^2}{15}\biggr]
=
(-\alpha_\mathrm{scale}) \cdot 0.7092314 \, ,
</math>
  </td>
</tr>
</table>
we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{dx}{dr_0}\biggr]_c</math>
  </td>
  <td align="center">
<math>=</math>
  <td align="left">
<math>
(-\alpha_\mathrm{scale}) \cdot 1.7395297
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[\frac{d x}{d r_0}\biggr]_e</math>
  </td>
  <td align="center">
<math>=</math>
  <td align="left">
<math>
(-\alpha_\mathrm{scale}) \cdot 0.1926402
\, .
</math>
  </td>
</tr>
</table>