Editing ParabolicDensity/Axisymmetric/Structure

__FORCETOC__ 
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=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I: &nbsp; Gravitational Potential]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II: &nbsp; Spherical Structures]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III: &nbsp; Axisymmetric Equilibrium Structures]]
&nbsp;[[ParabolicDensity/Axisymmetric/Structure/Try1thru7|Old: 1<sup>st</sup> thru 7<sup>th</sup> tries]]<br />
&nbsp;[[ParabolicDensity/Axisymmetric/Structure/Try8thru10|Old: 8<sup>th</sup> thru 10<sup>th</sup> tries]]
  </td>
  <td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV: &nbsp; Triaxial Equilibrium Structures (Exploration)]]
&nbsp;
  </td>
</tr>
</table>

==Axisymmetric (Oblate) Equilibrium Structures==

===Tentative Summary===

====Known 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>
  <td align="left">&nbsp;</td>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\partial}{\partial\zeta} \biggl[\frac{ \Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s}a_\ell^2 )\chi^2 \zeta - 2A_s \zeta  + 2(A_{s s} a_\ell^2) \zeta^3
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left">&nbsp;</td>
  <td align="right">
and, &nbsp; &nbsp; <math>\frac{\partial}{\partial\chi} \biggl[\frac{ \Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s}a_\ell^2 )\chi \zeta^2
- 2A_\ell \chi  
+ 2(A_{\ell \ell} a_\ell^2)  \chi^3
\, .
</math>
  </td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
  </td>
  <td align="right">[1.7160030]</td>
</tr>

<tr>
  <td align="right">
<math>
A_\ell
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
  </td>
  <td align="right">[0.6055597]</td>
</tr>

<tr>
  <td align="right"><math>A_s</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
  </td>
  <td align="right">[0.7888807]</td>
</tr>

<tr>
  <td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
=
\biggl[\frac{1}{2}-\frac{(A_s - A_\ell)}{4e^2}\biggr]
\, ;
</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">[0.3726937]</td>
</tr>

<tr>
  <td align="right">
<math>a_\ell^2 A_{ss} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{3}\biggl\{
\frac{( 4e^2 - 3 )}{e^4(1-e^2)} 
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
=
\frac{2}{3}\biggl[ (1-e^2)^{-1} - \frac{(A_s-A_\ell)}{e^2} \biggr]
\, ;
</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">[0.7021833]</td>
</tr>

<tr>
  <td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2) 
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] 
\biggr\} 
=
\frac{(A_s - A_\ell)}{e^2}
\, ,
</math>
  </td>
  <td align="right">[0.5092250]</td>
</tr>
</table>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2  \biggr]^{1 / 2} \, .
</math>
</div>

<font color="red">NOTE: &nbsp; The posted numerical evaluations (inside square brackets) assume that the configuration's eccentricity is</font> <math>e = 0.6 \Rightarrow a_s/a_\ell = 0.8</math>.

Drawing from our separate "[[ParabolicDensity/Axisymmetric/Structure/Try8thru10#6th_Try|6<sup>th</sup> Try]]" discussion &#8212; and as has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example &#8212; for the axisymmetric configurations under consideration, the <math>\hat{e}_z</math> and <math>\hat{e}_\varpi</math> components of the Euler equation become, respectively,</span>
<table border="1" align="center" cellpadding="10"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]  
</math>
  </td>
</tr>

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

Multiplying the <math>\hat{e}_z</math> component through by length <math>(a_\ell)</math> and dividing through by the square of the velocity <math>(\pi G \rho_c a_\ell^2)</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]\frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\frac{\rho_c}{\rho}\cdot \frac{\partial }{\partial \zeta}\biggl[ \frac{P}{(\pi G\rho_c^2 a_\ell^2)} \biggr] 
- \frac{\partial }{\partial \zeta}\biggl[ \frac{\Phi}{(-~\pi G\rho_c a_\ell^2)} \biggr] 
</math>
  </td>
</tr>

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

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

Multiplying the <math>\hat{e}_\varpi</math> component through by length <math>(a_\ell)</math> and dividing through by the square of the velocity <math>(\pi G \rho_c a_\ell^2)</math>, we have,

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

<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{j^2}{\varpi^3} \cdot \frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi_\mathrm{grav}}{\partial\varpi}\biggr] \frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
</tr>

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

====Play With Vertical Pressure Gradient====

<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 )\zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
- \chi^2 \biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
- \zeta^2(1-e^2)^{-1}\biggl[ (2A_{\ell s}a_\ell^2 \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>
(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3 
- (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\zeta  -  2A_{ss} a_\ell^2 \chi^2 \zeta^3 
- (1-e^2)^{-1}\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta^3  +  2A_{ss} a_\ell^2  \zeta^5   \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>
</table>
Integrate over <math>\zeta</math> gives &hellip;

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

<tr>
  <td align="right"><math>P^*_\mathrm{deduced} \equiv \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>
\overbrace{\biggl[ (A_{\ell s}a_\ell^2 \chi^2 - A_s ) - (A_{\ell s}a_\ell^2 \chi^4 - A_s \chi^2)\biggr]}^\mathrm{coef1}\zeta^2  
+  \underbrace{\frac{1}{2}\biggl[ A_{ss} a_\ell^2  -  A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(A_{\ell s}a_\ell^2 \chi^2 - A_s )\biggr]}_\mathrm{coef2}\zeta^4 
+  \overbrace{\frac{1}{3}\biggl[ - (1-e^2)^{-1}A_{ss} a_\ell^2 \biggr]}^\mathrm{coef3} \zeta^6 + ~\mathrm{const}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</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>
<!-- NOTE:  &nbsp; The integration constant must be the dimensionless central pressure, <math>P_c^*</math>. -->

If I am interpreting this correctly, <math>P_\mathrm{deduced}^*</math> should tell how the normalized pressure varies with <math>\zeta</math>, for a fixed choice of <math>0 \le \chi \le 1</math>.  Again, for a fixed choice of <math>\chi</math>, we want to specify the value of the "const." &#8212; hereafter, <math>C_\chi</math> &#8212; such that <math>P_\mathrm{deduced}^* = 0</math> at the surface of the configuration; but at the surface where <math>\rho/\rho_c = 0</math>, it must also be true that,

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

<tr>
  <td align="right">at the surface &nbsp; &hellip; &nbsp;</td>
  <td align="right"><math>\zeta^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(1-e^2)\biggl[ 1 - \chi^2 - \cancelto{0}{\frac{\rho}{\rho_c}} \biggr] 
= (1-e^2)(1-\chi^2)
\, .
</math>
  </td>
</tr>
</table>
Hence <font color="red">(numerical evaluations assume &chi; = 0.6 as well as e = 0.6)</font>,

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

<tr>
  <td align="right"><math>-~C_\chi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\overbrace{\biggl[ (A_{\ell s}a_\ell^2 \chi^2 - A_s ) - (A_{\ell s}a_\ell^2 \chi^4 - A_s \chi^2)\biggr]}^{\mathrm{coef1} ~=~ -0.38756}\biggl[ (1-e^2)( 1 - \chi^2 )  \biggr]  
+  \underbrace{\frac{1}{2}\biggl[ A_{ss} a_\ell^2  -  A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(A_{\ell s}a_\ell^2 \chi^2 - A_s )\biggr]}_{\mathrm{coef2} ~=~ 0.69779}\biggl[ (1-e^2)( 1 - \chi^2 )  \biggr]^2 
+ \overbrace{\frac{1}{3}\biggl[ - (1-e^2)^{-1}A_{ss} a_\ell^2 \biggr]}^{\mathrm{coef3} ~=~ -0.36572} \biggl[ (1-e^2)( 1 - \chi^2 )  \biggr]^3 
= -~0.66807 \, .
</math>
  </td>
</tr>
</table>
<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''Central Pressure'''</div>

At the center of the configuration &#8212; where <math>\zeta = \chi = 0</math> &#8212; we see that,

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{2}\biggl[ A_s (1-e^2) \biggr]
+  \frac{1}{6}\biggl[ A_{ss} a_\ell^2(1-e^2)^2  \biggr] 
</math>
  </td>
</tr>
</table>
Hence, the central pressure is,

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

<tr>
  <td align="right"><math>P^*_c \equiv \biggl[P^*_\mathrm{deduced}\biggr]_\mathrm{central} = C_\chi\biggr|_{\chi=0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}\biggl[ A_s (1-e^2) \biggr]
-  \frac{1}{6}\biggl[ A_{ss} a_\ell^2(1-e^2)^2  \biggr] \, .
</math>&nbsp; &nbsp; &nbsp; [0.2045061]
  </td>
</tr>
</table>

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


<table border="0" align="center" cellpadding="8" width="80%">
<tr>
  <td align="left">
For an oblate-spheroidal configuration having eccentricity, <math>e=0.6 ~\Rightarrow~ a_s/a_\ell = 0.8</math>, the figure displayed here, on the right, shows how the normalized gas pressure <math>(P^*_\mathrm{deduced}/P^*_c)</math> varies with height above the mid-plane <math>(\zeta)</math> at three different distances from the symmetry axis:  (blue) <math>\chi = 0.0</math>, (orange) <math>\chi = 0.6</math>, and (gray) <math>\chi = 0.75</math>.
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" rowspan="2">circular<br />marker<br />color</td>
  <td align="center" rowspan="2">chosen<br /><math>\chi</math></td>
  <td align="center" colspan="2">resulting &hellip;</td>
</tr>
<tr>
  <td align="center">surface <math>\zeta</math></td>
  <td align="center">mid-plane<br />pressure</td>
</tr>
<tr>
  <td align="center"><font color="blue">blue</font></td>
  <td align="center"><math>0.00</math></td>
  <td align="center"><math>0.8000</math></td>
  <td align="center"><math>1.00000</math></td>
</tr>
<tr>
  <td align="center"><font color="orange">orange</font></td>
  <td align="center"><math>0.60</math></td>
  <td align="center"><math>0.6400</math></td>
  <td align="center"><math>0.32667</math></td>
</tr>
<tr>
  <td align="center"><font color="gray">gray</font></td>
  <td align="center"><math>0.75</math></td>
  <td align="center"><math>0.52915</math></td>
  <td align="center"><math>0.13085</math></td>
</tr>
</table>
  </td>
  <td align="center">
[[File:FerrersVerticalPressureD.png|center|500px|Ferrers Vertical Pressure ]]
  </td>
</tr>
</table>

Inserting the expression for <math>C_\lambda</math> into our derived expression for <math>P^*_\mathrm{deduced}</math> gives,


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

<tr>
  <td align="right"><math>P^*_\mathrm{deduced} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(\mathrm{coef1}) \cdot \biggl[ \zeta^2 - (1-e^2)( 1 - \chi^2) \biggr] 
+  (\mathrm{coef2} )\cdot \biggl[ \zeta^4 - (1-e^2)^2( 1 - \chi^2)^2 \biggr]
+  ( \mathrm{coef3}) \cdot \biggl[ \zeta^6 - (1-e^2)^3( 1 - \chi^2)^3\biggr]
\, .
</math>
  </td>
</tr>
</table>


----


Note for later use that,

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

<tr>
  <td align="right"><math> \frac{\partial C_\chi}{\partial\chi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
&hellip;
  </td>
</tr>
</table>

====Isobaric Surfaces====

By design, the mass within our oblate-spheroidal configuration is distributed in such a way that iso-density surfaces are concentric spheroids.  As stated earlier, the relevant mathematically prescribed density distribution is,

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

<tr>
  <td align="right">
<math>\frac{\rho(\chi, \zeta)}{\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>
</table>

In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)  In an effort to make this determination for our <math>e = 0.6</math> spheroid, we first examine the iso-density surface for which <math>\rho/\rho_c = 0.3</math>.  Via the expression,

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

<tr>
  <td align="right">
<math>\zeta^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1-e^2)\biggl[1 - \chi^2 - \frac{\rho}{\rho_c} \biggr] 
=
0.64 \biggl[1 - \chi^2 - 0.3 \biggr]
\, ,</math>
  </td>
</tr>
</table>

we can immediately determine that our three chosen radial cuts <math>(\chi = 0.0, 0.6, 0.75)</math> intersect this iso-density surface at the vertical locations, respectively, <math>\zeta = 0.66933, 0.46648, 0.29665</math>; these numerical values have been recorded in the following table.  The table also contains coordinates for the points where our three cuts intersect the <math>(e = 0.6)</math> iso-density surface for which <math>\rho/\rho_c = 0.6</math>.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" rowspan="2">diamond<br />marker<br />color</td>
  <td align="center" rowspan="2">chosen<br /><math>\rho/\rho_c</math></td>
  <td align="center" rowspan="2">chosen<br /><math>\chi</math></td>
  <td align="center" colspan="2">resulting &hellip;</td>
</tr>
<tr>
  <td align="center">&nbsp; &nbsp; <math>\zeta</math> &nbsp; &nbsp;</td>
  <td align="center">normalized<br />pressure</td>
</tr>
<tr>
  <td align="center" rowspan="3"><font color="darkgreen">green</font></td>
  <td align="center" rowspan="3"><math>0.3</math></td>
  <td align="center" rowspan="1"><math>0.00</math></td>
  <td align="center" rowspan="1"><math>0.66933</math></td>
  <td align="center" rowspan="1"><math>0.060466</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.60</math></td>
  <td align="center" rowspan="1"><math>0.46648</math></td>
  <td align="center" rowspan="1"><math>0.057433</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.75</math></td>
  <td align="center" rowspan="1"><math>0.29665</math></td>
  <td align="center" rowspan="1"><math>0.055727</math></td>
</tr>
<tr>
  <td align="center" rowspan="3"><font color="purple">purple</font></td>
  <td align="center" rowspan="3"><math>0.6</math></td>
  <td align="center" rowspan="1"><math>0.00</math></td>
  <td align="center" rowspan="1"><math>0.50596</math></td>
  <td align="center" rowspan="1"><math>0.292493</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.60</math></td>
  <td align="center" rowspan="1"><math>0.16000</math></td>
  <td align="center" rowspan="1"><math>0.280361</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.75</math></td>
  <td align="center" rowspan="1">n/a</td>
  <td align="center" rowspan="1">n/a</td>
</tr>
</table>
For each of these five <math>(\chi,\zeta)</math> coordinate pairs, we have used our above derived expression for <math>P^*_\mathrm{deduced}/P^*_c</math> to calculate the "normalized pressure" at the relevant point inside the configuration.  These results appear in the last column of the table; they also have been marked in the accompanying figure: dark green diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.3</math> and purple diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.6</math>. Notice that the normalized density is everywhere lower than <math>0.6</math> along the <math>\chi = 0.75</math> cut, so the final row in the table has been marked "n/a" (not applicable).

The dark green diamond-shaped markers in the figure  &#8212; along with the associated tabular data &#8212; show that at three separate points along the <math>\rho/\rho_c = 0.3</math> iso-density surface, the normalized pressure is ''nearly'' &#8212; but not exactly &#8212; the same; its value is approximately <math>0.057</math>.  Similarly, the purple diamond-shaped markers show that at two separate points along the <math>\rho/\rho_c = 0.6</math> iso-density surface, the normalized pressure is nearly the same; in this case its value is approximately <math>0.28</math>.  This seems to indicate that, throughout our configuration, the isobaric surfaces are almost &#8212; but not exactly &#8212; aligned with iso-density surfaces.

====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>

===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>

=See Also=

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