Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

====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>