Editing AxisymmetricConfigurations/PoissonEq (section)

=====Simplification for 2D, Axisymmetric Systems=====
It is easy to show that this last expression for <math>~\Phi_B</math> &#8212; which has been used in our 3D simulations &#8212; is a ''generalization'' of the expression for <math>~\Phi_B</math> that was employed by [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black &amp; Bodenheimer (1975)] for 2D, axisymmetric simulations.  In axisymmetric systems, by definition, physical variables exhibit no variation in the azimuthal coordinate direction.  Hence, in the expression for <math>~\Phi_B</math>: 
* the azimuthal coordinate, <math>~\phi</math>, need not appear explicitly as an independent variable; 
* the index, <math>~m</math>, must be set to zero, so there is no summation over this index; and,
* every surviving spherical harmonic can be written more simply in terms of a Legendre function, namely,
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<math>~Y_{\ell m} \rightarrow Y_{\ell 0}</math>
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<math>~=</math>
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<math>~\sqrt{\frac{(2\ell+1 )}{4\pi}} P_\ell(\chi) \, ,</math>
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  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; where,</td>
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<math>~\chi \equiv \frac{z}{(\varpi^2 + z^2)^{1 / 2}}  \, .</math>
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Note that the argument, <math>~\chi</math>, is still the spherical-coordinate expression, <math>~\cos\theta</math>, but here it has been written in terms of cylindrical coordinates. We have, therefore,

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<math>~ \Phi_B(\varpi, z)\biggr|_{2D}</math>
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<math>~=</math>
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<math>~ 
- G 
\sum_{\ell=0}^{\ell_\mathrm{max}}  P_\ell(\chi) \biggl[ \varpi^2 + z^2 \biggr]^{-(\ell+1)/2} \int  P_\ell(\chi^') \biggl[ (\varpi^')^2 + (z^')^2 \biggr]^{\ell/2} 
~\rho(\varpi^', z^') d^3x^' \, ,
</math>
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where, <math>~d^3x^' = 2\pi \varpi^' d\varpi^' dz^'</math>.  This is precisely the same as equation (5) from [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black &amp; Bodenheimer (1975)]; see also, equations (8) and (9) in [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)].