Editing ParabolicDensity/GravPot (section)

===Uniform-Density Reminders===
We begin by reminding the reader that, [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_0|for a uniform-density configuration]], the "interior" potential will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_\mathrm{grav}(\mathbf{x})</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>

[[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Triaxial_Configurations_(a1_%3E_a2_%3E_a3)|As we have shown in a separate presentation]], if the three principal axes of the configuration are unequal in length and related to one another such that <math>a_1 > a_2 > a_3 </math>, the appropriate expressions for the four leading coefficients are,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}\biggr]  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
I_\mathrm{BT}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] \, .
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eqs. (33), (34) &amp; (35)</font> 
</div>

As can readily be demonstrated, this scalar potential satisfies the differential form of the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
</div>

As we have [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Oblate_Spheroids_(a1_=_a2_%3E_a3)|also demonstrated]], if the longest axis, <math>a_1</math>, and the intermediate axis, <math>a_2</math>, of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius <math>a_1</math> and the object is referred to as an '''oblate spheroid'''.  For homogeneous oblate spheroids, evaluation of the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> gives, 

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</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>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
A_1  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>A_3</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>
</tr>

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ,
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">&sect;4.5, Eqs. (48) &amp; (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1 / 2} \, .
</math>
</div>

Note the following, [[Apps/MaclaurinSpheroids#Gravitational_Potential|separately derived]] limits:

<table align="center" border=1 cellpadding="8">
<tr>
  <td colspan="3" align="center">
'''Table 1:''' &nbsp;[[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols|Limiting Values]]
  </td>
</tr>
<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<b><math>e \rightarrow 0</math></b>
  </td>
  <td align="center">
<b><math>\frac{a_3}{a_1} \rightarrow 0</math></b>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>\frac{\sin^{-1}e}{e}</math></b>
  </td>
  <td align="center">
<math>1 + \frac{e^2}{6} + \mathcal{O}\biggl(e^4\biggr)</math>
  </td>
  <td align="center">
<math>\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2
- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>A_1 = A_2</math></b>
  </td>
  <td align="center">
<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
  <td align="center">
<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>A_3</math></b>
  </td>
  <td align="center">
<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
  <td align="center">
<math>2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2
- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>I_\mathrm{BT}</math></b>
  </td>
  <td align="center">
<math>2</math>
  </td>
  <td align="center">
<math>0</math>
  </td>
</tr>
</table>

Hence, for a uniform-density sphere <math>(e = 0)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_\mathrm{grav}(\mathbf{x})</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-\pi G \rho_c \biggl[ 2a_1^2 - \frac{2}{3}\biggl(x^2 + y^2 +z^2 \biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-2\pi G \rho_c a_1^2\biggl[ 1 - \frac{1}{3}\biggl(\frac{r}{a_1} \biggr)^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-\frac{3GM}{2a_1}\biggl[ 1 - \frac{1}{3}\biggl(\frac{r}{a_1} \biggr)^2 \biggr] \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), &sect;5.8.9, p. 36, Eq. (5.8.23)
  </td>
</tr>
</table>
This matches the expression for the gravitational potential inside (and on the surface) of a uniform-density sphere, as we have derived in an [[SSC/Structure/UniformDensity#UniformSpherePotential|accompanying chapter]].