Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

===10<sup>th</sup> Try===

====Repeating Key Relations====

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

<tr>
  <td align="left"><font color="orange"><b>Density:</b></font></td>
  <td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- A_\ell \chi^2  - A_s \zeta^2  
 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2)  \chi^4 \biggr]
\, .
</math>
  </td>
</tr>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot  \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>
</table>

From the [[#Starting_Key_Relations|above (9<sup>th</sup> Try) examination]] of the vertical pressure gradient, we determined that a reasonably good approximation for the normalized pressure throughout the configuration is given by the expression,

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 \biggr]\chi^0
+ \biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2
- \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )
\biggr]\chi^2  
+  \biggl[- A_{\ell s}a_\ell^2 \zeta^2 \biggr]\chi^4 + ~\mathrm{const.}
</math>
  </td>
</tr>
</table>
If we set <math>\chi = 0</math> &#8212; that is, if we look along the vertical axis &#8212; this approximation should be particularly good, resulting in the expression,

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

<tr>
  <td align="right"><math>P_z \equiv \biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta \biggr\}_{\chi=0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>P_c^* - A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Note that in the limit that <math>z \rightarrow a_s</math> &#8212; that is, at the pole along the vertical (symmetry) axis where the <math>P_z</math> should drop to zero &#8212; we should set <math>\zeta \rightarrow (1 - e^2)^{1 / 2}</math>.  This allows us to determine the central pressure.

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

<tr>
  <td align="right"><math>P_c^* </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>A_s (1-e^2) - \frac{1}{2}A_{ss}a_\ell^2 (1-e^2)^2 - \frac{1}{2}(1-e^2)^{-1}A_s(1-e^2)^2 + \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  (1-e^2)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>A_s (1-e^2)  - \frac{1}{2}A_s(1-e^2) + \frac{1}{3}A_{ss} a_\ell^2  (1-e^2)^2 - \frac{1}{2}A_{ss}a_\ell^2 (1-e^2)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{2}A_s(1-e^2) - \frac{1}{6}A_{ss} a_\ell^2  (1-e^2)^2 \, .
</math>
  </td>
</tr>
</table>

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

This means that, along the vertical axis, the pressure gradient is,

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

<tr>
  <td align="right"><math>P_z \equiv \biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta \biggr\}_{\chi=0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>P_c^* - A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right"><math>\frac{\partial P_z}{\partial\zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>- 2A_s \zeta + 2A_{ss}a_\ell^2 \zeta^3 + 2(1-e^2)^{-1}A_s\zeta^3 - 2(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^5 \, .
</math>
  </td>
</tr>
</table>
This should match the more general "<font color="orange">vertical pressure gradient</font>" expression when we set, <math>\chi=0</math>, that is,

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

<tr>
  <td align="right"><math>\biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta} \biggr\}_{\chi=0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 - \cancelto{0}{\chi^2} - \zeta^2(1-e^2)^{-1}\biggr]\cdot \biggl[
2A_{\ell s}a_\ell^2 \zeta \cancelto{0}{\chi^2} - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[- 2A_s \zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
+ \zeta^2(1-e^2)^{-1} \biggl[2A_s \zeta  -  2A_{ss} a_\ell^2  \zeta^3   \biggr]
</math>
  </td>
</tr>
</table>

<b><font color="red">Yes! The expressions match!</font></b>

====Shift to &xi;<sub>1</sub> Coordinate====

In an [[ParabolicDensity/Axisymmetric/Structure/Try1thru7#Setup|accompanying chapter]], we defined the coordinate,

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

<tr>
  <td align="right"><math>\biggl(\frac{\xi_1}{a_s}\biggr)^2</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 
=
\chi^2 + \zeta^2(1-e^2)^{-1} \, .  
</math>
  </td>
</tr>
</table>
Given that we want the pressure to be constant on <math>\xi_1</math> surfaces, it seems plausible that <math>\zeta^2</math> should be replaced by <math>(1-e^2)(\xi_1/a_s)^2 = [(1-e^2)\chi^2 + \zeta^2]</math> in the expression for <math>P_z</math>.  That is, we might expect the expression for the pressure at any point in the meridional plane to be,

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

<tr>
  <td align="right"><math>P_\mathrm{test01}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^1 
+ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^2 
- \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^1 
+ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr] 
- \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr] 
+ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \frac{1}{3} A_{ss} a_\ell^2  
\biggl[ (1-e^2)^2\chi^6 + 2(1-e^2)\chi^4\zeta^2 + \chi^2\zeta^4 \biggr]
- \frac{1}{3}A_{ss} a_\ell^2  
\biggl[ (1-e^2)\chi^4\zeta^2 + 2\chi^2\zeta^4 + (1-e^2)^{-1}\zeta^6 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\chi^0 \biggl\{
P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6
\biggr\}
+ \chi^2 \biggl\{
-A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2
-\frac{1}{3}A_{ss}a_\ell^2\zeta^4
- \frac{2}{3}A_{ss} a_\ell^2\zeta^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \chi^4 \biggl\{
\frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2
- \frac{2}{3}A_{ss} a_\ell^2(1-e^2)\zeta^2
- \frac{1}{3}A_{ss} a_\ell^2(1-e^2)\zeta^2
\biggr\}
+ \chi^6 \biggl\{
- \frac{1}{3} A_{ss} a_\ell^2  (1-e^2)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\chi^0 \biggl\{
P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6
\biggr\}
+ \chi^2 \biggl\{
-A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \chi^4 \biggl\{
\frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2
\biggr\}
+ \chi^6 \biggl\{
- \frac{1}{3} A_{ss} a_\ell^2  (1-e^2)^2
\biggr\}
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" width="6%">&nbsp;</td>
  <td align="center" width="47%">Integration over <math>\zeta</math></td>
  <td align="center">Pressure Guess</td>
</tr>

<tr>
  <td align="center"><math>\chi^0</math></td>
  <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 </math></td>
  <td align="left">
<math>
P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^2</math></td>
  <td align="right">
<math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math>
  </td>
  <td align="left">
<math>
-A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^4</math></td>
  <td align="right">
<math>- A_{\ell s}a_\ell^2 \zeta^2 </math>
  </td>
  <td align="left">
<math>
\frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^6</math></td>
  <td align="right">
none
  </td>
  <td align="left">
<math>
- \frac{1}{3} A_{ss} a_\ell^2  (1-e^2)^2
</math>
  </td>
</tr>
</table>

====Compare Vertical Pressure Gradient Expressions====
From our [[#Starting_Key_Relations|above (9<sup>th</sup> try) derivation]] we know that the vertical pressure gradient is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s ) - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\biggr]\zeta  
+  \biggl[ 2A_{ss} a_\ell^2  -  2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 
+ \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 2A_s (\chi^2-1) + 2A_{\ell s}a_\ell^2 (1 - \chi^2)\chi^2 \biggr]\zeta  
+  \biggl[ 2A_{ss} a_\ell^2(1  -  \chi^2 )
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi^2 + 2(1-e^2)^{-1}A_s \biggr]\zeta^3 
+ \biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1}\biggr] \zeta^5 
\, .
</math>
  </td>
</tr>
</table>

By comparison, the vertical derivative of our "test01" pressure expression gives,

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

<tr>
  <td align="right"><math>P_\mathrm{test01}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\chi^0 \biggl\{
P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6
\biggr\}
+ \chi^2 \biggl\{
-A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \chi^4 \biggl\{
\frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2
\biggr\}
+ \chi^6 \biggl\{
- \frac{1}{3} A_{ss} a_\ell^2  (1-e^2)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial P_\mathrm{test01}}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\chi^0 \biggl\{
-2A_s\zeta + 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^3 - 2A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^5
\biggr\}
+ \chi^2 \biggl\{
2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)\zeta - 4A_{ss}a_\ell^2\zeta^3
\biggr\}
+ \chi^4 \biggl\{
- 2A_{ss} a_\ell^2(1-e^2)\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\zeta^1\biggl\{
- 2A_s
+ 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)\chi^2 
- 2A_{ss} a_\ell^2(1-e^2)\chi^4
\biggr\}
+ 
\zeta^3\biggl\{
2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]
- 4A_{ss}a_\ell^2\chi^2
\biggr\}
+
\zeta^5\biggl\{
- 2A_{ss} a_\ell^2(1-e^2)^{-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\zeta^1\biggl\{
2A_s (\chi^2- 1) 
+ 2A_{ss}a_\ell^2(1-e^2)\chi^2 (1-\chi^2) 
\biggr\}
+ 
\zeta^3\biggl\{
2A_{ss}a_\ell^2(1-2\chi^2) + 2(1-e^2)^{-1}A_s
\biggr\}
+
\zeta^5\biggl\{
- 2A_{ss} a_\ell^2(1-e^2)^{-1}
\biggr\}
</math>
  </td>
</tr>
</table>

Instead, try &hellip;

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2p_2\biggl(\frac{\rho}{\rho_c}\biggr)\frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr]
+ 
3p_3\biggl(\frac{\rho}{\rho_c}\biggr)^2  \frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{2p_2 
+ 
3p_3\biggl(\frac{\rho}{\rho_c}\biggr) \biggr\} \frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{2p_2 
+ 
3p_3\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggr\} \frac{\partial}{\partial\zeta}\biggl[ 1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{(2p_2 + 3p_3) 
- 3p_3\chi^2 - 3p_3\zeta^2(1-e^2)^{-1}  \biggr\} 
\biggl[ - 2\zeta(1-e^2)^{-1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{
6p_3\chi^2\zeta(1-e^2) - 2(2p_2 + 3p_3)(1-e^2)\zeta  + 6p_3\zeta^3  
\biggr\} 
</math>
  </td>
</tr>
</table>

Compare the term inside the curly braces with the term, from the beginning of this subsection, inside the square brackets, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
2A_{\ell s}a_\ell^2 \chi^2\zeta 
- 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2}{e^4} \biggl[(3-e^2) - \Upsilon \biggr]\chi^2\zeta - \biggl[\frac{4}{e^2}\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr] \zeta  
+ \frac{4}{3e^4}\biggl[\frac{4e^2-3}{(1-e^2)} + \Upsilon \biggr] \zeta^3   
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3e^4(1-e^2)}\biggl\{
6 \biggl[(3-e^2) - \Upsilon \biggr](1-e^2)\chi^2\zeta - \biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr](1-e^2) \zeta  
+ 4\biggl[(4e^2-3) + \Upsilon \biggr] \zeta^3
\biggr\}   \, .
</math>
  </td>
</tr>
</table>

<font color="red"><b>Pretty Close!!</b></font>

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
Alternatively: &nbsp; according to the third term, we need to set,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
6p_3
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4\biggl[(4e^2-3) + \Upsilon \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ \Upsilon 
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3}{2}p_3 + (3 - 4e^2)
</math>
  </td>
</tr>
</table>
in which case, the first coefficient must be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
\biggl[(3-e^2) - \Upsilon \biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(3-e^2) - \frac{3}{2}p_3 + (4e^2 - 3 ) \biggr]
=
\biggl[ 3e^2 - \frac{3}{2}p_3 \biggr] \, .
</math>
  </td>
</tr>
</table>
And, from the second coefficient, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
2(2p_2 + 3p_3)
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ 2p_2
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^2\biggl(3-\Upsilon\biggr) - 3p_3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 3p_3 + 6e^2 - 2e^2\biggl[ \frac{3}{2}p_3 + (3 - 4e^2) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 3p_3 + 6e^2 - \biggl[ 3e^2 p_3 + 6e^2 - 8e^4 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
8e^4 - 3p_3(1+e^2) \, ;</math>
  </td>
</tr>
</table>
or,

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

<tr>
  <td align="right">
<math>p_2</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4e^4 - (1+e^2)\biggl[(4e^2-3) + \Upsilon \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4e^4 - (1+e^2)(4e^2-3) - (1+e^2)\Upsilon </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4e^4 - [4e^2-3 + 4e^4-3e^2 ] - (1+e^2)\Upsilon </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - e^2 - (1+e^2)\Upsilon </math>
  </td>
</tr>
</table>

----

SUMMARY:

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>p_2</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - e^2 - (1+e^2)\Upsilon = e^4(A_{\ell s}a_\ell^2) - e^2\Upsilon \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
p_3
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2}{3}\biggl[(4e^2-3) + \Upsilon \biggr]
=
e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \, .
</math>
  </td>
</tr>
</table>

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


<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
Note: &nbsp; according to the first term, we need to set,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
p_3
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(3-e^2) - \Upsilon \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ \Upsilon
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(3-e^2) - p_3 \biggr] \, ,
</math>
  </td>
</tr>
</table>
in which case, the third coefficient must be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
4\biggl[(4e^2-3) + \Upsilon \biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4\biggl[(4e^2-3) + (3-e^2) - p_3 \biggr]
=
4\biggl[3e^2- p_3 \biggr] \, .
</math>
  </td>
</tr>
</table>
And, from the second coefficient, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>
2(2p_2 + 3p_3)
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ 2p_2
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^2\biggl(3-\Upsilon\biggr) - 3p_3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^2\biggl[3-[(3-e^2) - p_3]\biggr] - 3p_3</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^2\biggl[e^2 + p_3\biggr] - 3p_3</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^4 + (2e^2 - 3)p_3 \, ;
</math>
  </td>
</tr>
</table>
or,

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

<tr>
  <td align="right">
<math>
2p_2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^4 + (2e^2 - 3)\biggl[(3-e^2) - \Upsilon  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^4 + (2e^2 - 3)(3-e^2) - (2e^2 - 3)\Upsilon 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2e^4 + (6e^2 - 2e^4 -9 +3e^2) - (2e^2 - 3)\Upsilon 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
9(e^2  -1 ) - (2e^2 - 3)\Upsilon 
</math>
  </td>
</tr>
</table>

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

Better yet, try &hellip;

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test03}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[ 1 - \beta\biggl(1 -  \frac{\rho}{\rho_c} \biggr)\biggr]  
=
p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[ (1 - \beta) + \beta\biggl(\frac{\rho}{\rho_c} \biggr)\biggr]  

</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test03}}{P_c}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\cdots</math>
  </td>
</tr>
</table>
where, in the case of a [[SSC/Structure/OtherAnalyticModels#Pressure|spherically symmetric parabolic-density configuration]], <math>\beta = 1 / 2</math>.  Well &hellip; this wasn't a bad idea, but as it turns out, this "test03" expression is no different from the "test02" guess.  Specifically, the "test03" expression can be rewritten as,

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test03}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_2 (1 - \beta)\biggl(\frac{\rho}{\rho_c}\biggr)^2   
+ p_2\beta \biggl(\frac{\rho}{\rho_c}\biggr)^3 \, ,
</math>
  </td>
</tr>
</table>
which has the same form as the "test02" expression.

====Test04====

From above, we understand that, analytically,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s ) - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\biggr]\zeta  
+  \biggl[ 2A_{ss} a_\ell^2  -  2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 
+ \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 2A_s (\chi^2-1) + 2A_{\ell s}a_\ell^2 (1 - \chi^2)\chi^2 \biggr]\zeta  
+  \biggl[ 2A_{ss} a_\ell^2(1  -  \chi^2 )
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi^2 + 2(1-e^2)^{-1}A_s \biggr]\zeta^3 
+ \biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1}\biggr] \zeta^5 
\, .
</math>
  </td>
</tr>
</table>

Also from above, we have shown that if,

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3  
</math>
  </td>
</tr>
</table>

<table border="1" width="60%" align="center" cellpadding="5"><tr><td align="left">

SUMMARY from test02:

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

<tr>
  <td align="right">
<math>p_2</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - e^2 - (1+e^2)\Upsilon = e^4(A_{\ell s}a_\ell^2) - e^2\Upsilon \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
p_3
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2}{3}\biggl[(4e^2-3) + \Upsilon \biggr]
=
e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

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

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{
6p_3\chi^2\zeta(1-e^2) - 2(2p_2 + 3p_3)(1-e^2)\zeta  + 6p_3\zeta^3  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{
6\biggl[ e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \biggr]\chi^2\zeta(1-e^2) 
- 2\biggl[2e^4(A_{\ell s}a_\ell^2) + 3e^4(A_{ss}a_\ell^2) \biggr](1-e^2)\zeta  
+ 6\biggl[ e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \biggr]\zeta^3  
\biggr\} 
</math>
  </td>
</tr>
</table>


----


Here (test04), we add a term that is linear in the normalized density, which means,

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

<tr>
  <td align="right"><math>\frac{P_\mathrm{test04}}{P_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{P_\mathrm{test02}}{P_c} 
+
p_1 \biggl(\frac{\rho}{\rho_c}\biggr)   
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test04}}{P_c}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]
+
\frac{\partial}{\partial \zeta}\biggl[p_1 \biggl(\frac{\rho}{\rho_c}\biggr)\biggr]
=
\frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]
+
p_1 \frac{\partial}{\partial \zeta}\biggl[ 1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
</math>
  </td>
</tr>
</table>