Template:LSU CT99CommonTheme3

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Now, beginning with Version 2 of our expression for the Gravitational Potential of Axisymmetric Mass Distributions, let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's anchor ring in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, what we will refer to as the,

Gravitational Potential of an Axisymmetric Mass Distribution (Version 3)



<math>~ - 2G a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta} \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>

<math>\mathrm{where:}~~~\mu^2 = \frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .</math>