Joel2: Created page with "Suppose we rewrite (Version 1 of) the above-highlighted Key integral expression such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates $~(\varpi^', z^')$ to a toroidal-coordinate system $~(\eta^',\theta^')$ whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, $~(\varpi_a,z_a)$ . This desired mapping is handled via the pair of relations,..."

2024-06-21T23:45:36Z

Created page with "Suppose we rewrite (Version 1 of) the above-highlighted Key integral expression such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,..."

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Suppose we rewrite (Version 1 of) the [[#Part_I|above-highlighted Key integral expression]] such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,
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<math>~\varpi^' = \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>
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      and      
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<math>~(z^' - z_a) = \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>
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and the corresponding expression for each differential mass element is,
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<math>~\delta M(\eta^',\theta^') = \biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^'</math>.
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This gives, what we will refer to as the,
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<font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 2)'''</font>
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<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
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<math>~=</math>
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<math>~
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{\varpi^{1 / 2}} \biggr] \biggl[ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^{- 1 / 2}K(\mu)
\biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^' </math>
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<math>~=</math>
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<math>~
- 2G \biggl( \frac{\varpi_a^5}{\varpi} \biggr)^{1 / 2} \iint\limits_\mathrm{config}
\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,</math>
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where the square of the argument of the elliptic integral is,
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<math>~\mu^2</math>
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<math>~=</math>
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<math>~
\frac{ 4\varpi \varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')}\biggl\{ \biggl[ \varpi+ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 +
\biggl[z- z_a - \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 \biggr\}^{-1} \, .
</math>
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