Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

=====General Determination of Vertical Coordinate (&zeta;)=====
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Given values of the three parameters, <math>e</math>, <math>\varpi</math>, and <math>\phi_\mathrm{choice}</math>, this last expression can be viewed as a quadratic equation for <math>\zeta</math>.  Specifically,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\alpha \zeta^2 + \beta\zeta + \gamma \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\frac{1}{2}  A_{ss} a_\ell^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{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\}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\beta</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - A_s 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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} \frac{\sin^{-1}e}{e}  \biggr\} 
\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) 
- 
\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2}  
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\gamma</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{2} A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{8e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1 / 2} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) 
\, . 
</math>
  </td>
</tr>
</table>
The solution of this quadratic equation gives,

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

<tr>
  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta \pm \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
  </td>
</tr>
</table>

Should we adopt the ''superior'' (positive) sign, or is it more physically reasonable to adopt the ''inferior'' (negative) sign?  As it turns out, <math>\beta</math> is intrinsically negative, so the quantity, <math>-\beta</math>, is positive.  Furthermore, when <math>\gamma</math> goes to zero, we need <math>\zeta</math> to go to zero as well.  This will only happen if we adopt the ''inferior'' (negative) sign.  Hence, the physically sensible root of this quadratic relation is given by the expression,

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

<tr>
  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta - \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
  </td>
</tr>
</table>

<!--
Given that in this physical system, <math>\zeta = z^2/a_\ell^2</math> must be positive, we must choose the superior root.  We conclude therefore that,

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

<tr>
  <td align="right">
<math>\frac{z^2}{a_\ell^2}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2} - \beta \biggr\}
\, .
</math>
  </td>
</tr>
</table>
<font color="red">But check this statement because it appears that <math>\beta</math> will sometimes be negative.</font>
-->

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

<span id="QuantitativeExample">Here we present a quantitatively accurate depiction</span> of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution.  We closely follow the discussion of [[Apps/MaclaurinSpheroids#Example_Equi-gravitational-potential_Contours|equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids]].  In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with &hellip;
<table border="0" align="center" width="80%">
<tr>
  <td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td>
  <td align="center"><math>e = 0.81267 \, ,</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td>
  <td align="center"><math>A_s = 0.96821916 \, ,</math></td>
  <td align="center"><math>I_\mathrm{BT} = 1.360556 \, ,</math></td>
</tr>
<tr>
  <td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td>
  <td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td>
  <td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td>
</tr>
</table>

[<font color="red">NOTE:</font> &nbsp; Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence &#8212; see the first model listed in Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and/or see Tables 1 and 2 of [[ThreeDimensionalConfigurations/JacobiEllipsoids|our discussion of the Jacobi ellipsoid sequence]].  It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.]

The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at <math>(\varpi, z) = (1, 0)</math>.  That is, when,
<!--
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\frac{1}{2}  A_{ss} a_\ell^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\beta</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
A_{\ell s}a_\ell^2  - A_s 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\gamma</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{2} A_{\ell \ell} a_\ell^2  
-
A_\ell   
</math>
  </td>
</tr>
</table>
-->

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

<tr>
  <td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{max} </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
A_\ell   
- \frac{1}{2} A_{\ell \ell} a_\ell^2 = 0.3515026 \,  . 
</math>
  </td>
</tr>
</table>
So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < \phi_\mathrm{choice}|_\mathrm{max} </math>, recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by,

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

<tr>
  <td align="right">
<math>\phi_\mathrm{choice} </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{2}  A_{ss} a_\ell^2  \cancelto{0}{\zeta^2}   
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\cancelto{0}{\zeta}
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)  
- \frac{1}{2} A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2} A_{\ell \ell} a_\ell^2  \chi^2
- A_\ell \chi
+ \phi_\mathrm{choice} \, ,  
</math>
  </td>
</tr>
</table>
where,
<div align="center"><math>\chi \equiv \frac{\varpi^2}{a_\ell^2} \, .</math></div>
The solution to this quadratic equation gives,

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

<tr>
  <td align="right">
<math>\chi_\mathrm{eqplane} </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{A_{\ell \ell} a_\ell^2}\biggl\{
A_\ell \pm \biggl[A_\ell^2 - 2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}\biggr]^{1 / 2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{A_\ell}{A_{\ell \ell} a_\ell^2}\biggl\{
1 - \biggl[1 - \frac{2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}}{A_\ell^2}\biggr]^{1 / 2}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>
Note that, again, the physically relevant root is obtained by adopting the ''inferior'' (negative) sign, as has been done in this last expression.