Editing ParabolicDensity/Axisymmetric/Structure (section)

====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;

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<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,

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<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>,

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<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,

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<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>


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<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,

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<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>