Editing ParabolicDensity/Axisymmetric/Structure (section)

====Now Play With Radial Pressure Gradient====
After multiplying through by <math>\rho/\rho_c</math>, the last term on the RHS of the <math>\hat{e}_\varpi</math> component is given by the expression,
<table border="0" cellpadding="5" align="center">

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

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
\, .
 </math>
  </td>
</tr>
</table>
If we replace the normalized pressure by <math>P^*_\mathrm{deduced}</math>, the first term on the RHS of the <math>\hat{e}_\varpi</math> component becomes,

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

<tr>
  <td align="right"><math>\frac{\partial P^*_\mathrm{deduced}}{\partial\chi} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\partial}{\partial \chi}\biggl\{
\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 + P_c^*
\biggr\}
</math>
  </td>
</tr>

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

Hence,

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

<tr>
  <td align="right">
<math>
\frac{1}{\chi^3} \cdot \frac{j^2}{(\pi G\rho_c a_\ell^4)} \cdot \frac{\rho}{\rho_c}
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial P_\mathrm{deduced}^*}{\partial \chi} \biggr] 
- \frac{\rho}{\rho_c} \cdot \frac{\partial }{\partial \chi}\biggl[ \frac{\Phi_\mathrm{grav}}{(-~\pi G\rho_c a_\ell^2)} \biggr]
</math> 
  </td>
</tr>
</table>