Editing Apps/DysonWongTori (section)

===Fukushima (2016) -- Thin Ring===
As is discussed in considerable depth, [[#Fukushima_.282016.29|below]], [http://adsabs.harvard.edu/abs/2016AJ....152...35F Toshio Fukushima (2016, AJ, 152, id. 35, 31 pp.)] has used ''zonal toroidal harmonics'' to examine the gravitational field external to ring-like objects.  In the first segment of &sect;3 of his paper, Fukushima introduces a potential function, <math>~\Phi_P</math>, that, in his words, "is a special solution of simplified Poisson's equation being valid in the whole space."  From his equations (21), (62), (63), and (64), we see that,
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<math>~\Phi_P</math>
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<math>~=</math>
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<math>~-\frac{GM}{\sqrt{(\varpi + a)^2 + z^2 }} \biggl[ \frac{2}{\pi} \cdot K(m)\biggr] \, ,</math>
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where,
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<math>~K(m)</math>
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<math>~\equiv</math>
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<math>~
\int_0^{\pi/2} \frac{d\varphi}{\sqrt{ 1 - m\sin^2\varphi}}
</math>
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and the parameter,
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<math>~m</math>
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<math>~\equiv</math>
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<math>~
\frac{2\nu}{u + \nu} \, .
</math>
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Given that (see, for example, Fukushima's equations 19 and 7, in conjunction with his Figure 1),
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<math>~2u = \biggl(\frac{p}{q} + \frac{q}{p}\biggr) = \frac{p^2 + q^2}{pq} \, ,</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~2\nu = 2(u^2 - 1)^{1 / 2} = 2 \biggl[ \frac{(p^2 + q^2)^2 - 4p^2q^2}{4p^2q^2} \biggr]^{1 / 2} = \frac{p^2 - q^2}{pq} \, ,</math>
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we recognize that the parameter, <math>~m</math>, can be rewritten as,
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<math>~m</math>
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<math>~=</math>
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<math>~
4\nu \biggl[2u + 2\nu \biggr]^{-1} = \frac{2(p^2 - q^2)}{pq} \biggl[\frac{2p^2}{pq}\biggr]^{-1} = 1 - \frac{q^2}{p^2} \, .
</math>
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Acknowledging furthermore &#8212; see our more extensive [[#Fukushima_.282016.29|discussion, below]] &#8212; the parameter mapping,
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<math>~\biggl[ \rho_2 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ p \biggr]_\mathrm{Fukushima}</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~\biggl[ \rho_1 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ q \biggr]_\mathrm{Fukushima}</math>
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it is clear that Fukushima's parameter, <math>~m</math>, is identical to MacMillan's modulus, <math>~k^2</math>, and that Fukushima's expression for <math>~\Phi_P</math> is identical to the expression for the "[[#RingPotential|Gravitational Potential of a Thin Ring]]" that we have presented above.  Indeed, immediately preceding his equation (62), Fukushima explicitly acknowledges that his expression for <math>~\Phi_P</math> has been drawn from [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)].


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  <th align="center" colspan="4">Graphical Depiction in the Meridional Plane of the Gravitational Potential of a Thin, Axisymmetric Ring</th>
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[[File:TThinRing72cropped.png|380px|Our Thin Ring equipotential surface]]
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Extracted without modification from [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016)]'''<p></p>
"''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'''"<p></p>
Astronomical Journal, vol. 152, id. 35, 31 pp. &copy; [https://aas.org/ AAS]
</td>
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  <td align="center">Fukushima's Figure 4</td>
  <td align="center">Fukushima's Figure 5</td>
  <td align="center">Fukushima's Figure 6</td>
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[[File:Fukushima2016Fig4.png|225px|To be inserted: Fig. 4 from Fukushima (2016)]]
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[[File:Fukushima2016Fig5.png|225px|To be inserted: Fig. 5 from Fukushima (2016)]]
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[[File:Fukushima2016Fig6.png|225px|To be inserted: Fig. 6 from Fukushima (2016)]]
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Our Effort to Reproduce Fukushima's Figures
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[[File:FlatColorContoursCropped.png|225px|Our Thin Ring equipotential surface]]
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[[File:OurFig5Bhalf.png|225px|Our Thin Ring equipotential surface]]
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[[File:OurFig6Chalf.png|225px|Our Thin Ring equipotential surface]]
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Figures 4, 5, &amp; 6 of [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] display, with quantitative accuracy, the behavior of the potential function, <math>~\Phi_P/|\Phi_0|</math>, where <math>~\Phi_0</math> is a normalization factor.  We have extracted digital copies of these three figures and have displayed them, without modification, in the above figure ensemble.  We have evaluated the expression for the potential of an infinitesimally thin ring as derived by MacMillan (1958) and, in the same figure ensemble, have displayed the results in a manner that facilitates comparison with Fukushima's published results; our displayed results incorporate the normalization, <math>~\Phi_0 = GM/(2\pi a)</math>, so, given the high degree of quantitative overlap, we presume that this is the normalization that was adopted by Fukushima.  In the left-most panel of our figure ensemble, we have displayed the absolute value of this same two-dimensional, normalized potential function in the form of a warped surface; this has been done largely for visual effect.  We should point out that our pair of multi-colored contour plots &#8212; both the warped surface and its flat projection onto the meridional plane &#8212; cover an area that extends from the cylindrical-coordinate axes out to the boundaries, <math>~(R/a, Z/a) = (\tfrac{3}{2}, \pm\tfrac{3}{4})</math>, whereas Fukushima's contour plot (his Figure 4) extends twice as far in both directions.