Editing AxisymmetricConfigurations/PoissonEq (section)

===Norman and Wilson (1978)===

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<td align="center" bgcolor="red">&nbsp; 3D &nbsp;</td>
<td align="center" bgcolor="lightgreen">&nbsp; Cyl &nbsp;</td>
<td align="center" bgcolor="yellow">&nbsp; Sph &nbsp;</td>
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Following the discussion presented in &sect;4.1 of [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], it is sometimes useful to rewrite [[#PotentialA|''Form A'' of the boundary potential]] as,
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<math>~ \Phi_B(r,\theta,\phi)</math>
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<math>~=</math>
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<math>~ 
-4\pi G  
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)}  \, ,
</math>
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<span id="MultipoleMoments">where we have introduced what is commonly referred to as the,
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<font color="#770000">'''Multipole Moments of the Mass Distribution'''</font>
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<math>~q_{\ell m}</math>
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<math>~\equiv</math>
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<math>~
\int (r^')^\ell  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' 
\, .
</math>
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[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.3)
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When written explicitly in terms of cartesian coordinates &#8212; see [[#Ylm|Table 2, below]], for each of the relevant <math>~Y_{\ell m}</math> expressions &#8212; the first few of these moments have the functional representations [[#Multipole_Moments_of_the_Mass_Distribution|derived below]].  For the cases that correspond to positive values of the index, <math>~m</math>, the set of ''multipole moment'' expressions that have been included in our [[#qlm|Table 3 summary]] exactly matches the set of expressions presented as equations (4.4), (4.5), and (4.6) in [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)].  (Jackson's equation 4.7 explains how to map these expressions to the cases corresponding to negative values of the index, <math>~m</math>.)  

<span id="PhiSeriesExpansion">Let's now look at various terms in the summed expression for the boundary potential</span> with each term expressing the contribution for a separate value of the index, <math>~\ell</math>.  Isolating the first three terms from all the rest, for example, we have,

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<math>~ \Phi_B(r,\theta,\phi)</math>
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<math>~=</math>
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<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>
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We recognize, first, that as we consider boundary points that lie farther and farther away from the mass distribution, the magnitude of the first <math>(\ell = 0)</math> term drops off as <math>~r^{-1}</math> &#8212; the expected behavior of the potential outside of a point mass; the second <math>~(\ell=1)</math> term drops off as <math>~r^{-3}</math>; and the third <math>~(\ell=2)</math> term drops off as <math>~r^{-5}</math>.  Below, we have evaluated in more detail the behavior of these first three terms.  This evaluation gives us the,
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<font color="#770000">'''Boundary Potential Written in Terms of Multipole Moments '''</font>
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<math>~ \Phi_B(r,\theta,\phi)</math>
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<math>~=</math>
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<math>~ 
~-~~\frac{GM}{r}
~~- ~~\frac{GM}{r^3} \biggl[ \vec{x} \cdot \vec{x}_\mathrm{com} \biggr]
~~-~~\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr]
~~-~~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>
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where, as defined below, <math>~\vec{x}_\mathrm{com}</math> is the [[#Second_Term|center-of-mass location]], and <math>~Q_{i,j} </math> is the [[#QuadrupoleMomentTensor|traceless quadrupole moment tensor]].

If a modeled mass-density distribution, <math>~\rho( \vec{x}^{~'})</math>, has been configured such that the center-of-mass of the system coincides with the origin of the coordinate system &#8212; that is, if it has been configured such that <math>~\vec{x}_\mathrm{com} = 0</math> &#8212; then the second term in this series can be set to zero.  If, in addition, all terms having <math>~\ell \ge 3</math> are ignored because their values drop off rapidly with distance &#8212; specifically, inverse distance to the <math>~(2\ell + 1)</math> power &#8212; then a reasonably good approximation for the potential on the boundary of the modeled system is given by the expression,
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<math>~ \Phi_B</math>
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<math>~\approx</math>
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<math>~ 
-\frac{GM}{r}
~-\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr] \, .
</math>
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This is precisely the relation that was adopted by [http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman &amp; J. R. Wilson (1978, ApJ, 224, 497 - 511)] when, in the context of star formation, they modeled the ''Fragmentation of Isothermal Rings''  &#8212; see specifically their equation (18).