Editing ParabolicDensity/GravPot (section)

===Parabolic Density Distribution===


<div align="center">SUMMARY &#8212; copied from [[ThreeDimensionalConfigurations/Challenges#Trial_.232|accompanying, ''Trial #2'' Discussion]]</div>

After studying {{ Ferrers1877full }} and the relevant sections of both [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription.  As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials| separate chapter]], the potential that it generates is sometimes referred to as a [[ThreeDimensionalConfigurations/FerrersPotential|''Ferrers'' potential]], for the exponent, n = 1.

In our [[ThreeDimensionalConfigurations/FerrersPotential|accompanying discussion]]  we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{6}  \biggl(3A_{11}x^4 +  3A_{22}y^4 + 3A_{33}z^4  \biggr) 
\, ,
</math>
  </td>
</tr>
</table>

where,
<table border="0" align="center" cellpadding="10" width="80%">
<tr>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i \ne j</math></td></tr>
<tr>
  <td align="right">
<math>A_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (107)</font> ]</td></tr>
</table>

  </td>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i = j</math></td></tr>
<tr>
  <td align="right">
<math>2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{a_i^2} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (109)</font> ]</td></tr>
</table>

  </td>
</tr>
</table>
More specifically, in the three cases where the indices, <math>i=j</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3A_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_1^2} - (A_{12} + A_{13}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>3A_{22}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_2^2} - (A_{21} + A_{23}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>3A_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_3^2} - (A_{31} + A_{32}) \, .
</math>
  </td>
</tr>
</table>

<!--
In the case of a spherical configuration, we have:
<div align="center">
<math>A_1 = A_2 = A_3 = \tfrac{2}{3} \, ,</math>
</div>
in which case,
<div align="center">
<math>A_{12} = A_{13} = A_{23} = 0 ,</math> &nbsp; &nbsp; &nbsp; <math>A_{11} = A_{22} = A_{33} = \tfrac{2}{3a_1^2} \, ,</math>
</div>
and the expression for the spherically symmetric potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a_1^2 
- \frac{2}{3}\biggl(r^2 \biggr) 
+ \frac{1}{3a_1^2}  \biggl(x^4 +  y^4 + z^4  \biggr) 
\, ,
</math>
  </td>
</tr>
</table>
-->

In the case of an axisymmetric <math>(a_m = a_\ell)</math>, but nearly spherical configuration,

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

<tr>
  <td align="right">
<math>A_1=A_2</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}(e^4)\biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_3</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}(e^4)\biggr] \, .</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>A_{13} = A_{23}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{A_1 - A_3}{a_1^2e^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{2}{3a_1^2e^2}\biggl[\biggl( 1 - \frac{e^2}{5} \biggr) - \biggl( 1 + \frac{2e^2}{5} \biggr) + \mathcal{O}(e^4)\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3a_1^2}\biggl[\frac{3}{5} + \mathcal{O}(e^2)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{2}{5a_1^2} \, .
</math>
  </td>
</tr>
</table>
These results match our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#For_Spheres_(aℓ_=_am_=_as)|separate derivations in the case of a sphere]].  Specifically,

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

<tr>
  <td align="right">
<math>A_{1} = A_{2} = A_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3} \, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_{11} = A_{12} = A_{13} = A_{22} = A_{23} = A_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5a_1^2} \, .
</math>
  </td>
</tr>
</table>
<span id="ParabolicPotential">Hence, for a sphere with a parabolic density distribution, we find,</span>

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

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a_1^2 - \frac{2}{3} r^2 
+ \frac{2}{5a_1^2}\biggl( x^2y^2 + x^2z^2 + y^2z^2\biggr)
+ \frac{1}{5a_1^2}  \biggl(x^4 +  y^4 + z^4  \biggr) 
\, .
</math>
  </td>
</tr>
</table>

<font color="red">This matches the gravitational potential</font> [[SSC/Structure/OtherAnalyticModels#ParabolicPotential|derived for a parabolic density distribution using spherical coordinates]].