Editing AxisymmetricConfigurations/PoissonEq (section)

==Multipole Expansions==

===Mass Multipole Moments===

As an extension of the [[#MultipoleMoments|above discussion of]]
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Multipole Moments of the Mass Distribution'''</font>
</td>
</tr>
<tr>
  <td align="right">
<math>~q_{\ell m}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\int (r^')^\ell  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' 
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.3)
  </td>
</tr>
</table>
here we evaluate a set of the leading order mass-multipole moments in terms of their cartesian-coordinate representations.

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

<tr>
  <td align="right">
<math>~q_{00}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ Y_{11}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- ~\int r^'  \biggl[ \sqrt{\frac{3}{8\pi}}\sin\theta ~\biggl(\cos\phi^' - i\sin\phi^' \biggr) \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int r^'  \biggl[ \frac{\sqrt{(x^')^2+(y^')^2}}{r^'} 
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int (x^' - iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
\, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{1,- 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \sqrt{\frac{3}{8\pi}}~\int (x^' + iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{10}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ Y_{10}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ \sqrt{\frac{3}{4\pi}}~\cos\theta^' \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{22}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl\{ \frac{1}{4}\sqrt{\frac{15}{2\pi}} ~\sin^2\theta^' ~\biggl[\cos(2\phi^') - i\sin(2\phi^')\biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (r^')^2  \biggl\{ \frac{(x^')^2+(y^')^2}{(r^')^2}  ~\biggl[\frac{ (x^')^2 - (y^')^2 - 
2i ( x^' y^' )}{(x^')^2+(y^')^2} \biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{2,-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{21}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2  \biggl[ \sin\theta^' \cos\theta^' (\cos\phi^' -i\sin\phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2  
\biggl[ \frac{z^' \sqrt{(x^')^2+(y^')^2}}{(r^')^2} 
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^')  
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{2,-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^')  
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{20}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{20}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ \sqrt{\frac{5}{16\pi}} (3\cos^2\theta^' - 1) \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \, .
</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="qlm">
<table border="1" align="center" cellpadding="8" width="80%">
<tr><th>Table 3: &nbsp; Summary of Cartesian Expressions for Leading Mass-Multipole Moments</th></tr>

<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_{0, 0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{1,\pm 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mp \sqrt{\frac{3}{8\pi}}~\int (x^' \mp iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{1,0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,\pm 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' \mp iy^')^2~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,\pm 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mp \sqrt{\frac{15}{8\pi}} \int z^'(x^' \mp iy^')  
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

</table>

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

===Contributions to the Gravitational Potential in Terms of Multipole Moments===

Expanding on the [[#PhiSeriesExpansion|above discussion of the gravitational potential, written as a series expansion in terms of mass-multipole moments and spherical harmonics]],
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \Phi_B(r,\theta,\phi) \biggr|_{\ell=0} +~ \Phi_B(r,\theta,\phi)\biggr|_{\ell=1} + ~\Phi_B(r,\theta,\phi)\biggr|_{\ell=2} 
-~4\pi G  
\sum_{\ell=3}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)}  \, ,
</math>
  </td>
</tr>
</table>
here we evaluate the first three, leading order terms.

====First Term====
First (term with <math>~\ell = 0</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\biggl[ q_{00}  \biggr] \frac{ Y_{00}(\theta,\phi)}{r} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\biggl[ \frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{\sqrt{4\pi}~r}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{r} \, .  
</math>
  </td>
</tr>
</table>

====Second Term====
Second (terms with <math>~\ell = 1</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3}  
\sum_{m=-1}^{+1} \biggl[ q_{1 m}\biggr] \frac{Y_{1 m}(\theta,\phi)}{r^2}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3r^2}  
\biggl\{ 
\biggl[ q_{1,- 1}\biggr] Y_{1, -1}(\theta,\phi) 
+ \biggl[ q_{1 0}\biggr] Y_{1 0}(\theta,\phi) 
+ \biggl[ q_{1 1}\biggr] Y_{1 1}(\theta,\phi) 
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3r^2}  
\biggl\{ 
\biggl[ + \frac{3}{8\pi}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x - iy) }{r}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \frac{3}{4\pi} \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{z}{r} 
+ \biggl[ \frac{3}{8\pi}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x + iy) }{r}
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{r^3}  
\biggl\{ 
\biggl[ \frac{1}{2}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x - iy) 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
+ \biggl[ \frac{1}{2}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x + iy)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{r^3}  
\biggl\{ 
\biggl[ \int (x^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x ) 
+ \biggl[ \int (y^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (y ) 
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM}{r^3} \biggl[ \vec{x} \cdot \vec{x}_\mathrm{com} \biggr]</math>
  </td>
</tr>
<tr>
  <td align="right">
[https://en.wikipedia.org/wiki/Center_of_mass#A_continuous_volume where]:
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\vec{x}_\mathrm{com} \equiv \frac{1}{M} \int ~\vec{x}^{~'} \rho(\vec{x}^{~'}) d^3x^'\, .
</math>
  </td>
</tr>
</table>

====Third Term====

Third (terms with <math>~\ell = 2</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ q_{2, -2} \biggr] Y_{2,-2}  + \biggl[ q_{2, 2} \biggr] Y_{2,2}  
+ \biggl[ q_{2,-1} \biggr] Y_{2, -1} + \biggl[ q_{2,1} \biggr] Y_{2, 1} 
+ \biggl[ q_{20} \biggr] Y_{20} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ \frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}}  \sin^2\theta e^{-2i\phi}
+ \biggl[\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}}   \sin^2\theta e^{2i\phi}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{-i\phi}
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{i\phi}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr] \sqrt{\frac{5}{16\pi}} (3\cos^2\theta - 1)  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) - i\sin(2\phi) \biggr]
+ \frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) + i\sin(2\phi) \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl[ \cos\phi - i\sin\phi \biggr]
+ \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]   \biggl[ \cos\phi + i\sin\phi \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \frac{5}{2^4\pi}  \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl( \frac{3z^2 - r^2}{r^2} \biggr)  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4G}{r^5} \biggl\{
\frac{3}{2^5 } 
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 - 2i xy\biggr]
+ \frac{3}{2^5 } 
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 + 2i xy \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \frac{3z}{2^3}   \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl[x - i y \biggr]
+ \frac{3z}{2^3} \biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]   \biggl[x + i y \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \biggl( \frac{3z^2 - r^2}{2^4} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 } 
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' + \int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+ \frac{3i xy}{2 } 
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' -  \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  3xz  \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' + \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^'\biggr]  
+ 3i yz\biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' - \int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 } 
\biggl[ \int [2(x^')^2 -  2(y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ \frac{3i xy}{2 } 
\biggl[\int -4i x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  3xz  \biggl[\int 2x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 3i yz\biggl[ \int -2iy^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ 6 xy \biggl[\int x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  6xz  \biggl[\int x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 6yz\biggl[ \int y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  \, .
</math>
  </td>
</tr>
</table>

Rearranging terms, we have,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{3( x^2 - y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{3( x^2  )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
-  \biggl( \frac{x^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
-\frac{3( y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
- \biggl( \frac{y^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( x^2  )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2-3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2 -3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( x^2  )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2-3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [-3(x^')^2 +  3(y^')^2 -3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+\frac{( x^2  )}{2} 
\biggl[ \int [6(x^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [6(y^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ x^2
\biggl[ \int [3(x^')^2  - (r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+ y^2
\biggl[ \int [3(y^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr] \, ,
</math>
  </td>
</tr>
</table>
<span id="QuadrupoleMomentTensor">where,</span> following [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], we have introduced the,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Traceless Quadrupole Moment Tensor'''</font>
</td>
</tr>
<tr>
  <td align="right">
<math>~Q_{i,j}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\int [ 3(x_i^') (x_j^') - (r^')^2 \delta_{ij} ] \rho(\vec{x}^{~'}) d^3x^' \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 138, Eq. (4.9)<br />
[http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman &amp; J. R. Wilson (1978)], p. 501, Eq. (19)<br />
  </td>
</tr>
</table>