Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

====Vertical Component====
We will focus, first, on the vertical component.  Specifically, since both <math>\rho</math> and <math>\Phi_\mathrm{grav}</math> are known, the vertical gradient of the (unknown) scalar pressure is
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{\partial P}{\partial z}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \rho ~ \frac{\partial}{\partial z} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
  </td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

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

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell)} \cdot \frac{\partial P}{\partial z}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial z} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell)}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 -  \biggl( \frac{\varpi^2}{a_\ell^2}  + \frac{z^2}{a_s^2}\biggr) \biggr] \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}  
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
  </td>
</tr>

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

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

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

where (unlike above) we are using the dimensionless lengths, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>.  Continuing to streamline this function, we have,
<table border="0" align="center" cellpadding="8">

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

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

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[
- A_s  + (A_{\ell s}a_\ell^2  +  A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4  \biggr]\zeta  
+  
\biggl\{
[A_s(1-e^2)^{-1} + A_{ss} a_\ell^2]  -  [A_{ss} a_\ell^2   + A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\chi^2 
\biggr\}\zeta^3  
-  A_{ss} a_\ell^2(1-e^2)^{-1}  \zeta^5 
\, .  
</math>
  </td>
</tr>
</table>
So, let's see what happens if we assume that the pressure has the form,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>p_0 + p_2 \zeta^2 + p_4\zeta^4 + p_6\zeta^6  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P_\mathrm{vert}}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>2p_2 \zeta + 4p_4\zeta^3 + 6p_6\zeta^5  \, ,</math>
  </td>
</tr>
</table>
in which case,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_0 + \frac{1}{2}\biggl[
- A_s  + (A_{\ell s}a_\ell^2  +  A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4  
\biggr] \zeta^2 
+ 
\frac{1}{4}\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2]  -  [A_{ss} a_\ell^2   + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4 
+ 
\frac{1}{6}\biggl[
-  A_{ss} a_\ell^2 (1-e^2)^{-1} 
\biggr]\zeta^6  
\, .
</math>
  </td>
</tr>
</table>


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

<font color="red">REMINDER:</font>  
From [[#2nd_Try|above]] &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 -  \chi^2  - \zeta^2(1-e^2)^{-1} 
\, .
</math>
  </td>
</tr>
</table>

And, in the case of the spherically symmetric equilibrium configuration, the [[SSC/Structure/OtherAnalyticModels#Pressure|pressure distribution]] derived by {{ Prasad49 }} has the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c}</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[1 + \biggl(\frac{\rho}{\rho_c}\biggr)\biggr]
\, .
</math>
  </td>
</tr>
</table>
In the context of rotationally flattened configurations, therefore, we might expect the (vertical) pressure distribution to be of the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c}</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[1 -  \chi^2  - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[1 -  \chi^2  - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[2 -  \chi^2  - \zeta^2(1-e^2)^{-1}\biggr]
</math>
  </td>
</tr>

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

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

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