Editing Apps/Ostriker64 (section)

===Potential of a Thin Hoop===
In &sect;IIb of his [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II],  Ostriker (1964) derives an expression for the gravitational potential of a torus in the ''Thin Ring'' approximation, beginning specifically with the [[SR/PoissonOrigin#Step_1|integral form of the Poisson equation]] that is widely referred to in the astrophysics community as an expression for the,
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<font color="#770000">'''Scalar Gravitational Potential'''</font>
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<math>~ \Phi(\vec{x})</math>
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<math>~\equiv</math>
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<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.2, p. 77, Eq. (12)
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(Note: &nbsp; Consistent with the usage favored by his doctoral dissertation advisor in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], throughout his collection of 1964 papers Ostriker adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.)  Employing [[#Coordinate_System|Ostriker's adopted coordinate system]], and recognizing that, <font color="darkgreen">"the distance between the point of integration <math>~(0,0,\theta^')</math> and the point of observation <math>~(r,\phi,0)</math>"</font> is,
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<math>~|\vec{x}^{~'} - \vec{x}|</math>
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<math>~=</math>
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<math>~[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} \, ,</math>
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Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (21)
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this expression for the gravitational potential becomes,
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<math>~ \Phi(r,\phi)</math>
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<math>~=</math>
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<math>~ -G \int \int \rho(r^',\phi^') r^' (R+r^'\cos\phi^')  dr^' d\phi^' \int \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} }  </math>
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&nbsp;
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<math>~=</math>
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<math>~ -G (2\sigma R) \int_0^\pi \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} }  </math>
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  <td align="right" rowspan="4">[[File:WolframAlphaResult.png|300px|WolframAlpha result]]</td>
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&nbsp;
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<math>~=</math>
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<math>~ -\frac{4G \sigma R}{r} \int_0^\pi \frac{\tfrac{1}{2}d\theta^'}{[1 +n^2\sin^2(\tfrac{1}{2}\theta^')]^{1 / 2} }  </math>
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&nbsp;
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<math>~=</math>
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<math>~ -\frac{4G \sigma R}{r} \biggl[ \frac{K(k)}{\sqrt{n^2+1}}  \biggr]  \, ,</math>
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Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (22)
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where,
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<math>~n^2 \equiv \frac{4R(R+r\cos\phi)}{r^2}</math>
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&nbsp; &nbsp; and &nbsp; &nbsp;
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<math>~k \equiv \biggl[ \frac{n^2}{n^2+1} \biggr]^{1 / 2} \, .</math>
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Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (23)
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Mapping back to cylindrical coordinates, for the moment, we recognize that,
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<math>~r^2</math>
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<math>~=</math>
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<math>~(\varpi - R)^2 + z^2</math>
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<math>~\Rightarrow ~~~ n^2</math>
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<math>~=</math>
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<math>~\frac{4R\varpi}{(\varpi - R)^2 + z^2}</math>
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<math>~\Rightarrow ~~~ n^2 + 1</math>
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<math>~=</math>
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<math>~\frac{4R\varpi + (\varpi - R)^2 + z^2}{(\varpi - R)^2 + z^2} = \frac{(\varpi + R)^2 + z^2}{(\varpi - R)^2 + z^2} \, .</math>
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Acknowledging as well that the mass of Ostriker's "thin hoop" is, <math>~M = 2\pi \sigma R</math>, his expression for the potential becomes,
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<math>~\Phi(\varpi,z)</math>
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<math>~=</math>
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<math>~ -\frac{2G M}{\pi} \biggl[ \frac{K(k)}{\sqrt{(\varpi + R)^2 + z^2}}  \biggr] \, ,</math>
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where,

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<math>~k</math>
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<math>~=</math>
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<math>~\biggl[ \frac{4R\varpi}{(\varpi + R)^2 + z^2} \biggr]^{1 / 2} \, .</math>
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After adopting the variable association, <math>~R \leftrightarrow a</math>, it is clear that Ostriker's derived expression is identical to the Key Equation that we have [[Apps/DysonWongTori#Thin_Ring_Approximation|identified elsewhere]] as providing the,
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<td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td>
<td align="center" colspan="1" rowspan="2">[[File:FlatColorContoursCropped.png|220px|Contours for Thin Ring Approximation]]</td>
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{{ Math/EQ TRApproximation }}
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