Editing SSC/Structure/BiPolytropes/51RenormaizePart2 (section)

===Model C Finite-Difference Representation===
Here we examine a discrete representation of a model along the <math>\mu_e/\mu_c = 0.31</math> sequence whose core/envelope interface is located at <math>\xi_i = 2.69697</math>; whose core mass-fraction is <math>\nu = 0.19161</math>; and for which, <math>m_\mathrm{surf} = 2.145465292</math>.

====Treatment of the Core====

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center"><b>Table C1</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Table1Again.png|900px|Equilibrium Structure of Model C]]</td>
</tr>
</table>

<font color="red"><b>STEP1:</b></font>&nbsp; &nbsp; Divide the core into <math>(N_c+1)</math> grid lines &#8212; that is, into <math>N_c</math> radial zones &#8212; associating the first "grid line" with the center of the core and the last grid line with the radial location of the core/envelope interface; in <b>Table C1</b>, we have set <math>N_c = 20</math>.   Choosing <math>0 \le \tilde{M}_r \le \nu</math> as the principal Lagrangian coordinate, and using the available analytic expressions, assign values to the following physical quantities at each grid line:
<ul>
  <li>
Mass (see column titled <font color="darkgreen">tilde M_r</font> in <b>Table C1</b>): &nbsp; &nbsp; Set <math>(\Delta m)_c = \nu/(N_c)</math>; then, for <math>n = 1 ~\mathrm{thru}~ (N_c + 1)</math>, set <math>\tilde{M}_r = (n - 1)(\Delta m)_c \, .</math>
  </li>
  <li>
Polytropic radial coordinate (see column titled <font color="darkgreen">xi from M_r</font> in <b>Table C1</b>): &nbsp; &nbsp; Given that, <math>c_m = m_\mathrm{surf}^{-1} ( \mu_e/\mu_c)^2 (6/\pi)^{1 / 2} = 0.0619017</math>, determine the value of <math>\xi</math> associated with each gridline's value of <math>\tilde{M}_r</math> from the expression,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3^{1 / 2}\biggl[ 3\biggl(\frac{c_m}{\tilde{M}_r}\biggr)^{2/3} - 1\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>
For example, at the 21<sup>st</sup> gridline (associated with the core/envelope interface), this expression gives the expected, <math>\xi_i = 2.69697</math>.
  </li>
  <li>
Given the value of <math>\xi</math> at each gridline, determine the associated values of <math>\tilde{r}, \tilde{\rho}, \tilde{P}</math> &#8212; see the columns in <b>Table C1</b> titled <font color="darkgreen">tilde r, tilde rho, tilde P</font> &#8212; using the appropriate analytic expressions for the ''Core'' [[#VariableProfiles|as provided above]].  For example, at the 21<sup>st</sup> gridline (associated with the core/envelope interface), we find, <math>\tilde{r} = 0.003739</math>, <math>\tilde\rho = 2.5555 \times 10^{5}</math>, and <math>\tilde{P} = 3.0830 \times 10^{6}</math>.

  </li>
</ul>

<font color="red"><b>STEP2:</b></font>&nbsp; &nbsp; Building upon the results of <font color="red"><b>STEP1</b></font>, determine the value of <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math> at each gridline in the (initially) equilibrium model; see the column of <b>Table C1</b> titled <font color="darkgreen">M/(r pi r^4)</font>. 
<ul>
  <li>
As stated in the [[#Takeaway|above ''Takeaway Expression'']], this will simultaneously provide a precise evaluation of the pressure gradient, <math>d\tilde{P}/d\tilde{M}_r</math>, at each gridline ''when the configuration is in equilibrium.''
  </li>
  <li>
After a perturbation is introduced into the (initially equilibrium) configuration, both the pressure gradient and the quantity, <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math>, will deviate from their equilibrium values and, quite generally, from each other.  (Actually, <math>\tilde{M}_r</math> will not vary because, by definition, it is our time-invariant Lagrangian fluid marker; but the pressure gradient and the denominator of the second term, <math>4 \pi \tilde{r}^4</math>, will vary.)  Then, as expressed by the [[#NormalizedEuler|above ''Normalized Euler Equation'']], the sum of these two perturbed quantities will dictate the strength and direction of the unbalanced acceleration that will be felt by the Lagrangian fluid element at each gridline.
  </li>
</ul>

<font color="red"><b>STEP3:</b></font>&nbsp; &nbsp; Our ''discrete representation'' of <b>Model C</b> will be constructed in such a way as to preserve, at each gridline location, the analytically determined values of the Lagrangian marker, <math>\tilde{M}_r</math>, and the corresponding value of (the initial) <math>\tilde{r}</math>.  In doing so, we must expect that our ''discrete'' evaluation of <math>\tilde\rho</math> and <math>\tilde{P}</math> will differ from values determined in the continuum model.  We choose to adopt the following paths toward evaluation of these two scalar quantities:
<ul>
  <li>
Given that, in <font color="red"><b>STEP1</b></font>, we established a grid on which the <math>\tilde{M}_r</math> spacing between gridlines is uniform, we choose here to evaluate <math>\tilde{P}</math> midway between gridlines and to evaluate the pressure gradient via the (2<sup>nd</sup>-order accurate) expression,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\biggl[\frac{d\tilde{P}}{d\tilde{M}_r}\biggr]_n</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{\tilde{P}_{n + 1/2} - \tilde{P}_{n - 1/2}}{(\Delta m)_c} \, .
</math>
  </td>
</tr>
</table>
Note that the ''difference'' between the pair of discrete mid-zone values of the pressure that appears in the numerator of the term on the right-hand-side of this expression straddles the discrete grid in such a way that the left-hand-side pressure ''gradient'' is centered on the n<sup>th</sup> gridline.  This is as desired because the pressure ''gradient'' should be compared with <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math>, which is also evaluated on each gridline.
  </li>
  <li>
We will also evaluate <math>\tilde{\rho}</math> midway between gridlines.  Then, at the center of each ''core'' grid zone, we can use the exact relationship between the normalized pressure and normalized density &#8212; namely,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\tilde{P}_{n+1/2}</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\biggl[\tilde\rho_{n+1/2}\biggr]^{6/5} \, ,
</math>
  </td>
</tr>
</table>
to determine <math>\tilde{P}_{n+1/2}</math> from <math>\tilde{\rho}_{n+1/2}</math> or, after inversion, to determine <math>\tilde{\rho}_{n+1/2}</math> from <math>\tilde{P}_{n+1/2}</math> for all <math>1 \le n \le N_c</math>.
  </li>
</ul>
<span id="STEP4"><font color="red"><b>STEP4:</b></font>&nbsp; &nbsp; By design,</span> the mass contained within every spherical shell of our discrete model is <math>(\Delta m)_c</math> and &#8212; even after a perturbation is introduced &#8212; for all <math>1 \le n \le N_c</math>, the differential volume of the various shells is, 
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>(\Delta ~ \mathrm{Vol})_{n+1/2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{4\pi}{3}\biggl[ \tilde{r}^3_{n+1} - \tilde{r}^3_{n}\biggr] \, .
</math>
  </td>
</tr>
</table>
In an effort to satisfy the continuity equation, throughout our discrete model we will relate the gas density of each spherical shell to the bounding radii of that shell via the expression,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\rho_{n+1/2} = \frac{(\Delta m)_c}{(\Delta ~ \mathrm{Vol})_{n+1/2}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3(\Delta m)_c}{4\pi}\biggl[ \tilde{r}^3_{n+1} - \tilde{r}^3_{n}\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">

<div align="center"><font color="red"><b>ASIDE</b></font></div>

Unperturbed &hellip;

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{3(\Delta m)_c}{4\pi \rho_{n+1/2}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \tilde{r}^3_{n+1} - \tilde{r}^3_{n}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \tilde{r}_{n+1/2} + \frac{\Delta \tilde{r}}{2} \biggr]^3 
- 
\biggl[ \tilde{r}_{n+1/2} - \frac{\Delta \tilde{r}}{2} \biggr]^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tilde{r}_{n+1/2}^3 \biggl\{
\biggl[ 1 + \frac{\Delta \tilde{r}}{2\tilde{r}_{n+1/2}} \biggr]^3 
- 
\biggl[ 1 - \frac{\Delta \tilde{r}}{2\tilde{r}_{n+1/2}} \biggr]^3
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>\tilde{r}_{n+1/2}^3 \biggl\{
\biggl[ 1 + \frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}} \biggr] 
- 
\biggl[ 1 - \frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}} \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>3\Delta \tilde{r} \cdot \tilde{r}_{n+1/2}^2 \, . 
</math>
  </td>
</tr>
</table>


----


Perturbed &hellip;

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{1}{\tilde\rho_0}\biggl[1 + \frac{\delta \tilde\rho}{\tilde\rho_0}\biggr]^{-1} \frac{3(\Delta m)_c}{4\pi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\tilde{r}^3_{n+1}\biggl[1 + \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1}\biggr]^3 
-
\tilde{r}^3_{n}\biggl[1 + \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]^3 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
\frac{1}{\tilde\rho_0}\biggl[1 - \frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] \frac{3(\Delta m)_c}{4\pi}</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\tilde{r}^3_{n+1}\biggl[1 + 3\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1}\biggr] 
-
\tilde{r}^3_{n}\biggl[1 + 3\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
\frac{1}{\tilde\rho_0}\frac{3(\Delta m)_c}{4\pi}
-
\frac{1}{\tilde\rho_0}\biggl[\frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] \frac{3(\Delta m)_c}{4\pi}
</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\tilde{r}^3_{n+1}
+ 
\tilde{r}^3_{n+1}\biggl[3\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1}\biggr] 
-
\tilde{r}^3_{n} 
-
\tilde{r}^3_{n}\biggl[3\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
-
\frac{1}{\tilde\rho_0}\biggl[\frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] \frac{(\Delta m)_c}{4\pi}
</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\tilde{r}^3_{n+1}\biggl[\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1}\biggr] 
-
\tilde{r}^3_{n}\biggl[\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
-
\biggl[\frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] 
</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl\{
\tilde{r}^3_{n+1}\biggl[\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1}\biggr] 
-
\tilde{r}^3_{n}\biggl[\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl\{\tilde{r}_{n+1/2}^3\biggl[1 + \frac{\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\tilde{r}_{n+1/2}^3\biggl[1 - \frac{\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl\{\tilde{r}_{n+1/2}^3\biggl[1 + \frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\tilde{r}_{n+1/2}^3\biggl[1 - \frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl\{
\tilde{r}_{n+1/2}^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
+
\tilde{r}_{n+1/2}^3\biggl[\frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\tilde{r}_{n+1/2}^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}
+
\tilde{r}_{n+1/2}^3\biggl[\frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl\{
\tilde{r}_{n+1/2}^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\tilde{r}_{n+1/2}^3
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}
+
\tilde{r}_{n+1/2}^3\biggl[\frac{3\Delta \tilde{r}}{2\tilde{r}_{n+1/2}}\biggr]
\biggl[ \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
+
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 \tilde{r}_{n+1/2}^3}{(\Delta m)_c}
\biggl\{
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n} \biggr\}
+
\frac{4\pi \tilde\rho_0 }{(\Delta m)_c}
\biggl[3\Delta \tilde{r} \cdot \tilde{r}^2_{n+1/2}\biggr]
\biggl[ \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
+
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]\frac{1}{2} \, .
</math>
  </td>
</tr>
</table>

----

Combined &hellip;

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{3(\Delta m)_c}{4\pi \rho_{n+1/2}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>3\Delta \tilde{r} \cdot \tilde{r}_{n+1/2}^2  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
-
\biggl[\frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] 
</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{4\pi \tilde\rho_0 \tilde{r}_{n+1/2}^3}{(\Delta m)_c}
\biggl\{
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n} \biggr\}
+
3\biggl[ \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
+
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]\frac{1}{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
\frac{4\pi \tilde\rho_0 \tilde{r}_{n+1/2}^3}{(\Delta m)_c}
\biggl\{
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
-
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n} \biggr\}
</math>
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
-
3\biggl[ \biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n+1} 
+
\biggl( \frac{\delta \tilde{r}}{\tilde{r}} \biggr)_{n}\biggr]\frac{1}{2}
-
\biggl[\frac{\delta \tilde{\rho}}{\tilde\rho_0}\biggr] 
\, .
</math>
  </td>
</tr>

</table>
This should be compared with the [[SSC/Perturbations#Continuity_Equation|more traditional derivation of the linearized continuity equation]], which gives,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>4\pi \rho_0 r_0^3 \frac{dx}{dm}  </math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>- 3 x - d \, .</math>
  </td>
</tr>
</table>


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

<ul>
  <li>
Values of the normalized density computed in this manner have been recorded in the column titled <font color="darkgreen">rho_FD</font> of <b>Table C1</b>; the subscript "FD" stands for "Finite Difference".  For example, in the shell just before the core/envelope interface <math>(n = 20)</math>, we find <math>\tilde\rho_{n+1/2} = 2.8168 \times 10^{5}</math>.
  </li>
  <li>
We have determined the value of the normalized pressure that corresponds to each of these "Finite Difference" values of the density using the algebraic relation presented above in <font color="red"><b>STEP3</b></font>; their values have been recorded in the column titled <font color="darkgreen">P_FD</font> of <b>Table C1</b>.  For example, in the shell just before the core/envelope interface <math>(n = 20)</math>, we find <math>\tilde{P}_{n+1/2} = [\tilde\rho_{n+1/2}]^{6/5} = 3.4651 \times 10^{6}</math>.
  </li>
  <li>
From our determination of <math>\tilde{P}_{n+1/2}</math> throughout the core, values of the normalized pressure ''gradient'' have been computed in the manner described above in <font color="red"><b>STEP3:</b></font>, and have been recorded in the column titled <font color="darkgreen">(dP/dM)_FD</font> of <b>Table C1</b>.  For example, at the <math>n = 20</math> gridline, we find <math>[d\tilde{P}/d\tilde{M}_r]_{n} = -9.281 \times 10^{7}</math>.
  </li>
</ul>


<font color="red"><b>STEP5:</b></font>&nbsp; &nbsp; Throughout the core, compare evaluation of the finite-difference representation of the (absolute value of the) pressure gradient to an evaluation of the unperturbed and analytically prescribed profile of the quantity, <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math>.  The left-hand segment of <b>Figure C1</b> provides such a comparison; actually, for reasons that will become clear later, we have multiplied both quantities by <math>4\pi \tilde{r}^2</math> before plotting.

<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center"><b>Figure C1</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Fig1.png|750px|Finite-Difference Structure of Model C]]</td>
</tr>

<tr>
  <td align="left">
The smooth, solid curves (blue for the core and green for the envelope) show the analytically prescribed behavior of the quantity, <math>\tilde{M}_r/\tilde{r}^2</math> as a function of <math>\tilde{M}_r</math> throughout the unperturbed <b>Model C</b>.  The solid, circular markers (colored dark orange throughout the core and light orange across the envelope) identify how our finite-difference representation of the pressure gradient &#8212; more specifically, the quantity, <math>(4\pi \tilde{r}^2)|d\tilde{P}/d\tilde{M}_r|_n</math> &#8212; varies with <math>\tilde{M}_r</math> throughout the equilibrium configuration.  We use the difference between these two quantities as a measure of the ''error'' introduced by our specified finite-difference representation of the equilibrium model. For example, the small solid dots and accompanying (interpolated) dashed curve that appear in <b>Figure C1</b> (blue for the core and green for the envelope) show how, 
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\mathrm{error}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\mathrm{amp}~ \times \biggl\{ \frac{\tilde{M}_r}{\tilde{r}^2} - (4\pi \tilde{r}^2)\biggl|\frac{d\tilde{P}}{d\tilde{M}_r}\biggr|_n \biggr\} \, ,
</math>
  </td>
</tr>
</table>
varies with mass shell throughout the equilibrium model, after setting <math>\mathrm{amp}~= -50</math>; this data has been recorded in the column titled "<font color="red">Alternate_FD</font> error" in <b>Table C1</b>.
  </td>
</tr>
</table>

====Behavior at the Interface====

It is worth pointing out that the second derivative of the pressure (with respect to <math>\tilde{M}_r</math>) exhibits a discontinuous jump at the interface.  Specifically,

The smooth, solid curves in <b>Figure C1</b> (blue for the core and green for the envelope) show the analytically prescribed behavior of the quantity, <math>\tilde{M}_r/\tilde{r}^2</math> as a function of <math>\tilde{M}_r</math> throughout the unperturbed <b>Model C</b>.  These curves intersect at the core/envelope interface (marked by the vertical, black dashed line), which means that the quantity, <math>\tilde{M}_r/\tilde{r}^2</math> has the same value whether viewed from the perspective of the core or from the perspective of the envelope.  But, as the figure illustrates, the curves exhibit different slopes at the interface.

Quite generally we can write,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3}\frac{d\tilde{r}}{d\tilde{M}_r} 
=
\frac{1}{\tilde{r}^2}\biggl\{ 1 - 2 \cdot \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>

This means that, for the core,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ \frac{d\xi}{d\tilde{M}_r} \cdot \frac{d\tilde{r}}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} 
\biggl[ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\biggl\{ 
\frac{1}{3\xi^2 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
\biggr\} 
 \biggr] 
\cdot 
\biggl[ \mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2}\biggl\{1 
-
2\tilde{M}_r 
\biggl[ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\frac{1}{3\xi^3 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2}\biggl[ 1 
- 
\frac{2}{3 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr) 
 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}^2}\biggl( 1 - \frac{2}{3}\xi^2 \biggr) \, .
</math>
  </td>
</tr>
</table>
Specifically at the interface (from the perspective of the core),
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}_i^2}\biggl( 1 - \frac{2}{3}\xi_i^2 \biggr) \, .
</math>
  </td>
</tr>
</table>


<!--
----


<math>
\mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\biggl\{ 
\frac{1}{3\xi^2 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
\biggr\} 
</math><br />
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math><br />
<math>
\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
</math>

----

<math>A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\biggl\{ 
\eta\sin(\eta-B)
\biggr\} 
</math><br />
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta </math><br />
<math>\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
A\biggl[ \sin(\eta-B) - \eta\cos(\eta-B) \biggr] \, ,</math>
----

-->

And for the envelope,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ \frac{d\eta}{d\tilde{M}_r} \cdot \frac{d\tilde{r}}{d\eta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ 
A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\biggl\{ 
\eta\sin(\eta-B)
\biggr\} 
\biggr]^{-1} \cdot 
\biggl[ \mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} \biggl\{1
-
\frac{2\tilde{M}_r}{\eta} \biggl[ 
A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\eta\sin(\eta-B)
\biggr]^{-1} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} \biggl\{1
-
\frac{2}{\eta^2} 
\biggl[ 1 - \eta\cot(\eta-B) \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Now, pulling from [[SSC/Structure/BiPolytropes/Analytic51#Step_6:_Envelope_Solution|our original derivation]], we appreciate that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\eta_i \cot(\eta_i - B) 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\eta_i \Lambda_i
</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)~ \sqrt{3} \theta_i^2 \xi_i  \biggr] 
\frac{\xi_i}{\sqrt{3}} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 - \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)\biggr]
</math>
  </td>
</tr>
</table>

Hence, at the interface (from the perspective of the envelope) we find,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\biggl\{
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
\biggr\}_{i, \mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{\eta_i^2} 
\biggl[ 1 - \eta_i \cot(\eta_i-B) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{\eta_i^2} 
\biggl[ \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
2 \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \biggr]^{-2} 
\biggl[ \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{3} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

<!--
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\theta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\frac{\xi_i}{\sqrt{3}} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] 
\, ,</math></td>
</tr>
</table>
-->

<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center"><b>Figure C2</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Fig2.png|750px|Model C Slopes at Interface]]</td>
</tr>

<tr>
  <td align="left">
A pair of line-segments with arrowheads has been added to Figure C1:  
<ul>
  <li>The red arrow is tangent to the solid blue curve at the core/envelope interface; its slope is,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}_i^2}\biggl( 1 - \frac{2}{3}\xi_i^2 \biggr) = -9.177751 \times 10^{4}\, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>The green arrow is tangent to the solid green curve at the core/envelope interface; its slope is, 
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{3} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
\biggr\} =-4.552725\times 10^5\, .
</math>
  </td>
</tr>
</table>
  </li> 
</ul>
They illustrate that the ''slope'' of the function, <math>\tilde{M}_r/\tilde{r}^2</math>, has a discontinuous jump at the interface.  Given that, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{d\tilde{P}}{d\tilde{M}_r} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{\tilde{M}_r}{\tilde{r}^2} \, ,</math></td>
</tr>
</table>
in the <b>Model C</b> equilibrium configuration, this also illustrates that this bipolytropic model has a discontinuous jump in the ''second-derivative'' of the pressure (with respect to <math>\tilde{M}_r</math>) at the core/envelope interface.
  </td>
</tr>
</table>


Note that this discontinuity will disappear in a model for which the two slopes have the same value, that is, when,

<table border="0" align="center" cellpadding="8">

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

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

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

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

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl(\frac{\mu_e}{\mu_c} - 1\biggr)\xi_i^2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 \, .
</math>
  </td>
</tr>
</table>
This will never occur in this bipolytropic model.

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
<div align="center"><b>Evaluation of the Logarithmic Derivative</b><br /><math>d\ln \tilde{r}/d\ln \tilde{M}_r</math></div>
At the [[#Behavior_at_the_Interface|beginning of this subsection]], we demonstrated that, quite generally,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2}\biggl\{ 1 - 2 \cdot \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>
It is therefore the case that, at the interface and from the perspective of the core,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r} \biggr\}_{i, \mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3}\biggr) \, ;
</math>
  </td>
</tr>
</table>
while, at the interface but from the perspective of the envelope,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r} \biggr\}_{i, \mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3}\biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} 
\, .
</math>
  </td>
</tr>
</table>

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

====Treatment of the Envelope====

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center"><b>Table C2</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Table2.png|1000px|Equilibrium Structure of Model C Envelope]]</td>
</tr>
</table>

<font color="darkgreen"><b>STEP11:</b></font>&nbsp; &nbsp; Divide the envelope into <math>(N_e+1)</math> grid lines &#8212; that is, into <math>N_e</math> radial zones &#8212; associating the first "grid line" with the location of the core/envelope interface and the last grid line with the radial location of the surface of the bipolytropic configuration; in <b>Table C2</b>, we have set <math>N_e = 20</math>.   Choosing <math>\nu \le \tilde{M}_r \le 1</math> as the principal Lagrangian coordinate, and using the available analytic expressions, assign values to the following physical quantities at each grid line:
<ul>
  <li>
Mass (see 1<sup>st</sup> column titled <font color="darkgreen">tilde M_r</font> in <b>Table C2</b>): &nbsp; &nbsp; Set <math>(\Delta m)_e = (1-\nu)/(N_e)</math>; then, for <math>n = 1 ~\mathrm{thru}~ (N_e + 1)</math>, set <math>\tilde{M}_r = \nu + (n - 1)(\Delta m)_e \, .</math>
  </li>
  <li>
First and last polytropic radial coordinates:  In the [[#VariableProfiles|above summary of various physical variable profiles]], we have provided expressions for the envelope's polytropic radial coordinate at the core/envelope interface <math>(\eta_i)</math> and at the surface <math>(\eta_s)</math>.  Evaluating these expressions for <b>Model C</b>, we find,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\eta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3}  \xi_i \biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1} = 0.42286 \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{\pi}{2} + \eta_i + \tan^{-1}\biggl\{
\frac{\xi_i}{\sqrt{3}} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\xi_i^2}\biggl(1+\frac{1}{3}\xi_i^2\biggr) - 1\biggr] 
\biggr\} = 2.67311  
\, ,</math>
</td>
</tr>
</table>
These two values appear, respectively, as the first and last numbers in the (2<sup>nd</sup>) column of <b>Table C2</b>, titled <font color="darkgreen">eta_guess</font>.
  </li>
  <li>
All other polytropic radial coordinates (see the "<math>N_e - 1</math>" numerical values that are highlighted in yellow in the column titled <font color="darkgreen">eta_guess</font> in <b>Table C2</b>): &nbsp; &nbsp; Next, we need to determine what value of <math>\eta</math> is associated with each Lagrangian fluid marker (''i.e.,'' each gridline) that lies between the interface and the surface.  Referring again to the  [[#VariableProfiles|above summary of various physical variable profiles]], we can determine analytically the value of <math>\tilde{M}_r</math> that is associated with any selected value of <math>\eta</math>, via the relation,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\tilde{M}_r(\eta)
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
A\biggl[ \sin(\eta-B) - \eta\cos(\eta-B) \biggr] \, ,
</math>
  </td>
</tr>
</table>
but what we really need is to invert this relation to obtain <math>\eta(\tilde{M}_r)</math>.  Unfortunately, unlike our experience with the core, for which we were able to invert the <math>\tilde{M}_r(\xi)</math> relation to obtain an analytic prescription for <math>\xi(\tilde{M}_r)</math> &#8212; see <font color="red"><b>STEP1</b></font> above &#8212; here, we are unable to analytically invert our <math>\tilde{M}_r(\eta)</math> expression.

Instead, for each gridline we ''guessed'' the value of <math>\eta</math> which, when plugged into the <math>\tilde{M}_r(\eta)</math> expression, would give the value of <math>\tilde{M}_r</math> that was already assigned to that gridline; by trial-and-error, we revised our guess for <math>\eta</math> until the desired value of the normalized mass (see the column titled <font color="darkgreen">tilde M_r</font>) was obtained (usually, to 6 or 7 digit accuracy).  <b>Table C2</b> provides the following record:  At each gridline, our final/best iterative "guess" for <math>\eta</math> is highlighted in yellow in the (2<sup>nd</sup>) column titled <font color="darkgreen">eta_guess</font>; the value of <math>\tilde{M}_r</math> that is obtained by plugging this "best guess" value of <math>\eta</math> into the <math>\tilde{M}_r(\eta)</math> relation is recorded in the (3<sup>rd</sup>) column titled <font color="darkgreen">M_r from eta</font>; and the (4<sup>th</sup>) column titled <font color="darkgreen">error</font> shows the fractional difference between this value and the desired value &#8212; as was our goal, all measured errors are zero, to at least six significant digits.
  </li>
  <li>
Given the "best guess" value of <math>\eta</math> at each gridline, determine the associated values of <math>\tilde{r}(\eta), \tilde{\rho}(\eta), \tilde{P}(\eta)</math> &#8212; see the (5<sup>th</sup>, 6<sup>th</sup>, and 7<sup>th</sup>) columns in <b>Table C2</b> titled <font color="darkgreen">tilde r, tilde rho, tilde P</font> &#8212; using the appropriate analytic expressions for the ''Envelope'' [[#VariableProfiles|as provided above]].  For example, at the 20<sup>th</sup> gridline (associated with the first gridline just inside the surface) where our "best guess" is <math>\eta = 2.3765850</math>, we find, <math>\tilde{r} = 0.021014</math>, <math>\tilde\rho = 5.29458 \times 10^{3}</math>, and <math>\tilde{P} = 1.37706 \times 10^{4}</math>.
  </li>
</ul>


<font color="darkgreen"><b>STEP12:</b></font>&nbsp; &nbsp; As was done above, in <font color="red"><b>STEP2</b></font>, for the core, determine here for the envelope the value of <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math> at each gridline in the (initially) equilibrium model; see the (8<sup>th</sup>) column of <b>Table C2</b> titled <font color="darkgreen">M/(r pi r^4)</font>.


<font color="darkgreen"><b>STEP13:</b></font>&nbsp; &nbsp; As was done in <font color="red"><b>STEP3</b></font> for the core &#8212; but, here, for the envelope &#8212; we choose to evaluate <math>\tilde{P}</math> midway between gridlines and to evaluate the pressure gradient via the (2<sup>nd</sup>-order accurate) expression,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\biggl[\frac{d\tilde{P}}{d\tilde{M}_r}\biggr]_n</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{\tilde{P}_{n + 1/2} - \tilde{P}_{n - 1/2}}{(\Delta m)_e} \, .
</math>
  </td>
</tr>
</table>
Notice that the denominator on the right-hand-side of this expression is <math>(\Delta m)_e</math> (for the envelope) rather than <math>(\Delta m)_c</math> (for the core).  And, as with our treatment of the core, the ''difference'' between the pair of discrete mid-zone values of the pressure that appears in the numerator of this right-hand-side term straddles the discrete grid in such a way that the left-hand-side pressure ''gradient'' is centered on the n<sup>th</sup> gridline.  

At the center of each ''envelope'' grid zone, we can use the exact relationship between the normalized pressure and normalized density &#8212; namely,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{\tilde{P}_{n+1/2}}{ [\tilde\rho_{n+1/2}]^{2} }</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2}
\biggl\{
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi
\biggr\}^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6} (\mathcal{m}_\mathrm{surf}\theta_i)^{-4} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \tilde{P}_{n+1/2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6} (\mathcal{m}_\mathrm{surf}\theta_i)^{-4} \biggl[\tilde\rho_{n+1/2} \biggr]^{2}
</math>
  </td>
</tr>
</table>
&#8212; to determine <math>\tilde{P}_{n+1/2}</math> from <math>\tilde{\rho}_{n+1/2}</math> or, after inversion, to determine <math>\tilde{\rho}_{n+1/2}</math> from <math>\tilde{P}_{n+1/2}</math> for all <math>1 \le n \le N_e</math>.

<font color="darkgreen"><b>STEP14:</b></font>&nbsp; &nbsp; Akin to our treatment of the core (<font color="red"><b>STEP4</b></font>), the mass contained within every spherical shell of our discrete model of the ''envelope'' is <math>(\Delta m)_e</math> and,
in an effort to satisfy the continuity equation, throughout our discrete model we will relate the gas density of each spherical shell to the bounding radii of that shell via the expression,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\rho_{n+1/2} = \frac{(\Delta m)_e}{(\Delta ~ \mathrm{Vol})_{n+1/2}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3(\Delta m)_e}{4\pi}\biggl[ \tilde{r}^3_{n+1} - \tilde{r}^3_{n}\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
<ul>
  <li>
Values of the normalized density computed in this manner have been recorded in the (9<sup>th</sup>) column titled <font color="darkgreen">rho_FD</font> of <b>Table C2</b>.  For example, in the first shell just inside the surface <math>(n = 20)</math>, we find <math>\tilde\rho_{n+1/2} = 2.458565 \times 10^{3}</math>.
  </li>
  <li>
We have determined the value of the normalized pressure that corresponds to each of these "Finite Difference" values of the density using the algebraic relation presented just above; their values have been recorded in the (10<sup>th</sup>) column titled <font color="darkgreen">P_FD</font> of <b>Table C2</b>.  For example, in the first shell just inside the surface <math>(n = 20)</math>, we find <math>\tilde{P}_{n+1/2} = 4.91237\times 10^{-4} [\tilde\rho_{n+1/2}]^{2} = 2.96930 \times 10^{3}</math>.
  </li>
  <li>
From our determination of <math>\tilde{P}_{n+1/2}</math> throughout the envelope, values of the normalized pressure ''gradient'' have been computed in the manner described above in <font color="darkgreen"><b>STEP13</b></font>, and have been recorded in the (11<sup>th</sup>) column titled <font color="darkgreen">(dP/dM)_FD</font> of <b>Table C2</b>.  For example, at the <math>n = 20</math> gridline, we find <math>[d\tilde{P}/d\tilde{M}_r]_{n} = -4.549 \times 10^{5}</math>.
  </li>
</ul>


<font color="darkgreen"><b>STEP15:</b></font>&nbsp; &nbsp; Throughout the envelope, compare evaluation of the finite-difference representation of the (absolute value of the) pressure gradient to an evaluation of the unperturbed and analytically prescribed profile of the quantity, <math>\tilde{M}_r/(4\pi \tilde{r}^4)</math>.  The right-hand segment of <b>Figure C1</b> provides such a comparison; actually, for reasons that will become clear later, we have multiplied both quantities by <math>4\pi \tilde{r}^2</math> before plotting.



<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center"><b>Figure C3</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Fig3.png|750px|Finite-Difference Structure of Model C with 41 gridlines]]</td>
</tr>

<tr>
  <td align="left">
Same as <b>Figure C2</b>, except higher resolution with <math>N_e = N_c = 40</math>.
  </td>
</tr>
</table>
varies with mass shell throughout the equilibrium model, after setting <math>\mathrm{amp}~= -50</math>; this data has been recorded in the column titled "<font color="red">Alternate_FD</font> error" in <b>Table C1</b>.
  </td>
</tr>
</table>