Editing ParabolicDensity/GravPot

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=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I: &nbsp; Gravitational Potential]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II: &nbsp; Spherical Structures]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III: &nbsp; Axisymmetric Equilibrium Structures]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV: &nbsp; Triaxial Equilibrium Structures (Exploration)]]
&nbsp;
  </td>
</tr>
</table>

==Gravitational Potential==

In an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids|accompanying chapter]] titled, ''Properties of Homogeneous Ellipsoids (1),'' we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids.  In that discussion, we largely followed the derivations of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  In the latter part of the nineteenth-century, {{ Ferrers1877full }} showed that very similar analytic expressions can be derived for ellipsoids that have certain inhomogeneous mass distributions.  Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>\rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, ,</math>
  </td>
</tr>
</table>
that is, configurations with ''parabolic density distributions''.  Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as [[ThreeDimensionalConfigurations/FerrersPotential|Ferrers potential]].

===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]].

===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]].

=See Also=

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