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

====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>