Editing Apps/MaclaurinSpheroids (section)

====Example Equi-gravitational-potential Contours====
As an example, let's examine the gravitational potential everywhere inside (and  on the surface) of the oblate spheroid whose properties are presented in the first row of model data in  [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table1|Table 1 of our accompanying discussion of the properties of homogeneous ellipsoids]].  That is, let's examine a model with <math>a_1 = 1.0</math> and &hellip;
<table border="0" align="center" width="80%">
<tr>
  <td align="center"><math>\frac{a_3}{a_1} = 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_1 = A_2 = 0.51589042 \, ,</math></td>
  <td align="center"><math>A_3 = 0.96821916 \, ,</math></td>
  <td align="center"><math>I_\mathrm{BT} = 1.360556 \, .</math></td>
</tr>
</table>

In the meridional <math>(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration &#8212; identified by the thick, solid-black curve below, in Figure 1 &#8212; is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\varpi^2}{a_1^2} + \frac{z^2}{a_3^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="2">
<math>1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ z</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\pm a_3 [1 -  \varpi^2]^{1 / 2}  \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; &nbsp; for <math>~0 \le | \varpi | \le 1 \, .</math></td>
</tr>
</table>
Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression,

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

<tr>
  <td align="right">
<math>\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{eff}}{\pi G \rho} +  I_\mathrm{BT}a_1^2 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="1">
<math>\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2 + A_3 z^2   </math>
  </td>
</tr>
</table>

(Notice that, written in this manner, <math>\phi_\mathrm{choice}</math> assumes its minimum value (zero) when <math>(\varpi, z) = (0, 0)</math>, that is, at the center of the configuration.)  This means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>z </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - \biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2\biggr]^{1 / 2} \, .  </math>
  </td>
</tr>
</table>

----

<span id="norotation">'''No Rotation'''</span>

When we do not consider the effects of rotation and plot, instead, just the equi-gravitational-potential surfaces, then 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>z </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - A_1  \varpi^2\biggr]^{1 / 2} \, .  </math>
  </td>
</tr>
</table>

Because we know that the <math>\Phi_\mathrm{grav}</math> = constant surfaces are all less flattened than the configuration itself, we should expect that the largest value of the potential that will arise inside &#8212; actually, on the surface of &#8212; the flattened spheroidal configuration will be found at <math>(\varpi, z) = (1, 0)</math>, that is, when,
<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>\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - A_1  \biggr]^{1 / 2} \, .  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow  ~~~\phi_\mathrm{choice}\biggr|_\mathrm{max}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_1  \, .  </math>
  </td>
</tr>
</table>
So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < A_1 \, ,</math> recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by,
<div align="center">
<math>\varpi = \sqrt{\phi_\mathrm{choice}/A_1} \, .</math>
</div>
Next, we recognize that the largest equipotential surface that fits entirely within the surface of the oblate spheroidal configuration has the value of the potential that is found on the symmetry axis and at the pole of the spheroid, that is, at <math>(\varpi, z) = (0, a_3) \, .</math>  For this case we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{mid}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a_3^2 A_3 \, .</math>
  </td>
</tr>
</table>
Hence, all equipotential surfaces having <math>0 < \phi_\mathrm{choice} \le a_3^2 A_3</math> will lie entirely within the spheroid.  But equipotential surfaces having <math>a_3^2 A_3 < \phi_\mathrm{choice} \le A_1</math> will cut through the surface of the spheroid at the value of <math>\varpi</math> where "the two values of z<sup>2</sup> match," that is, where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_3^2(1-\varpi^2)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{A_3} \biggl[ \phi_\mathrm{choice} - A_1  \varpi^2\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~\varpi </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1   -  a_3^2 A_3 } \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

Therefore, for the example model parameters specified  above, our selection of equipotential surfaces to plot should be guided by the following constraints.
<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center" colspan="2">Equipotential Contour Lies &hellip;</td>
</tr>
<tr>
  <td align="center" colspan="1" width="50%">Entirely Inside Spheroid's Surface</td>
  <td align="center" colspan="1">Partially Outside Spheroid's Surface</td>
</tr>
<tr>
  <td align="center" colspan="1"><math>0 < \phi_\mathrm{choice} < 0.32878</math></td>
  <td align="center" colspan="1"><math>0.32878 < \phi_\mathrm{choice} < 0.51589</math></td>
</tr>
<tr>
  <td align="center" colspan="1"><math>0 \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
  <td align="center" colspan="1"><math>\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
</tr>
<tr><td align="left" colspan="2">
<div align="center">'''Figure 1: &nbsp; Meridional Plane Cross-section'''<br />
[[File:MacAtJacBifurcationJustGravity01.png|550px|center|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]</div>
''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724.  ''Dashed curves'':  Equi-gravitational-potential contours plotted in increments of <math>\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.404 (purple), and 0.479 (orange).
</td>
</tr>
</table>


----

'''With Rotation'''
This expression is only applicable to our physical problem under the following conditions:
<ol>
<li>
The argument of the square root must not be negative, that is, <math>\varpi</math> must be confined to the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0 \le | \varpi |</math>
  </td>
  <td align="center">
<math>\le</math>
  </td>
  <td align="left">
<math>\biggl[ \phi_\mathrm{choice}\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr)^{-1 }\biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Note that, in turn, in order to ensure that the argument of ''this'' square root is not negative, we should only explore rotation rates for which <math>\omega_0^2/(2\pi G \rho) \le A_1 \, .</math>
</li>
<li>
In order that our equipotential surface be relevant only to the ''interior'' of our configuration, for every allowed value of <math>\varpi ,</math> the value of <math>z</math> corresponding to the potential surface must be less than or equal to the value of  <math>z</math> at the surface of the configuration.  That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_3^2(1-\varpi^2) </math>
  </td>
  <td align="center">
<math>\le</math>
  </td>
  <td align="left">
<math>\frac{1}{A_3} \biggl[ \phi_\mathrm{choice} - \biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho} - a_3^2 A_3^2 \biggr) \varpi^2 </math>
  </td>
  <td align="center">
<math>\le</math>
  </td>
  <td align="left">
<math>\phi_\mathrm{choice} - a_3^2 A_3^2 </math>
  </td>
</tr>
</table>
</li>
</ol>