User:Tohline/Apps/DysonWongTori - Revision history

Tohline: /* Self-Gravitating, Incompressible (Dyson-Wong) Tori */

2018-09-21T21:54:06Z

Self-Gravitating, Incompressible (Dyson-Wong) Tori

← Older revision		Revision as of 21:54, 21 September 2018
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	<tr>		<tr>
	<th algin="center" colspan="5">		<th algin="center" colspan="5">
	Schematics of ''Anchor Ring'' and/or ''Thin Ring'' ~~Relevant~~ to ~~Five Research Papers~~		Schematics Illustrating ''Anchor Ring'' and/or ''Thin Ring'' Variable Notation<br />
			as Found in Five Separate Discussions Over a Period of 120+ Years<br />
			<font size="-1">(click on an icon to view full-sized schematic)</font>
	</th>		</th>
	</tr>		</tr>
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	<td align="center">[[File:WongTorusIllustration02.png\|150px\|Wong (1973)]]</td>		<td align="center">[[File:WongTorusIllustration02.png\|150px\|Wong (1973)]]</td>
	<td align="center">[[File:FukushimaTorusIllustration02.png\|150px\|Fukushima (2016)]]</td>		<td align="center">[[File:FukushimaTorusIllustration02.png\|150px\|Fukushima (2016)]]</td>
			</tr>
			<tr>
			<td align="center">Our Separate <br />[[User:Tohline/Apps/DysonPotential#Dyson_.281893.29\|Chapter Discussion]]
			<td align="center">Our Relevant <br />[[#MacMillan_.281930.29\|Discussion, Below]]
			<td align="center">Our Separate <br />[[User:Tohline/Apps/Ostriker64#Polytropic_.26_Isothermal_Tori\|Chapter Discussion]]
			<td align="center">Our Separate <br />[[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Chapter Discussion]]
			<td align="center">Our Relevant <br />[[#Fukushima_.282016.29\|Discussion, below]]
	</tr>		</tr>
	</table>		</table>

Tohline: /* Self-Gravitating, Incompressible (Dyson-Wong) Tori */

2018-09-21T21:21:07Z

Self-Gravitating, Incompressible (Dyson-Wong) Tori

← Older revision		Revision as of 21:21, 21 September 2018
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}


			<table border="1" cellpadding="8" align="center">
			<tr>
			<th algin="center" colspan="5">
			Schematics of ''Anchor Ring'' and/or ''Thin Ring'' Relevant to Five Research Papers
			</th>
			</tr>
			<tr>
			<th align="center">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893)]</th>
			<th align="center">[https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958)]</th>
			<th align="center">[http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Ostriker (1964)]</th>
			<th align="center">[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]</th>
			<th align="center">[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)]</th>
			</tr>
			<tr>
			<td align="center">[[File:DysonTorusIllustration04.png\|150px\|Dyson (1893a,b)]]</td>
			<td align="center">[[File:MacMillanTorusIllustration02.png\|150px\|MacMillan (1958; originally, 1930)]]</td>
			<td align="center">[[File:OstrikerTorusIllustration01.png\|150px\|Ostriker (1964)]]</td>
			<td align="center">[[File:WongTorusIllustration02.png\|150px\|Wong (1973)]]</td>
			<td align="center">[[File:FukushimaTorusIllustration02.png\|150px\|Fukushima (2016)]]</td>
			</tr>
			</table>

	==Overview==		==Overview==

Tohline: /* Overview */

2018-08-24T18:36:24Z

Overview

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	<td>		<td>
	[[File:RoverDinfty.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverDinfty.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = infinity]]
	</td>		</td>
	<td>		<td>
	[[File:RoverD5over1.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverD5over1.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = 5]]
	</td>		</td>
	<td>		<td>
	[[File:RoverD5over2.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverD5over2.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = 2.5]]
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td>		<td>
	[[File:RoverD5over3.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverD5over3.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = 1.667]]
	</td>		</td>
	<td>		<td>
	[[File:RoverD5over4.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverD5over4.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = 1.25]]
	</td>		</td>
	<td>		<td>
	[[File:RoverDunity.png\|300px\|center\|~~Dyson (1893, ''~~The Potential of an Anchor Ring~~'')~~]]		[[File:RoverDunity.png\|300px\|center\|The Potential Exterior to an Anchor Ring; R/d = 1]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Overview */

2018-08-24T16:48:35Z

Overview

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	<table border="0" cellpadding="8" align="center"><tr><td align="center">		<table border="0" cellpadding="8" align="center"><tr><td align="center">
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr><td align="center" colspan="2">		<tr><td align="center" colspan="3">
	'''Figures 1 - 6 extracted without modification from pp. 63-66 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<p></p>		'''Figures 1 - 6 extracted without modification from pp. 63-66 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<p></p>
	''The ~~Theory~~ of ~~the Potential~~'', ~~New York: McGraw-Hill~~		''The Potential of an Anchor Ring'', Phil. Trans. Royal Soc. London. A., Vol. 184
	</td></tr>		</td></tr>
	<tr>		<tr>
	<td>		<td>
	[[File:RoverDinfty.png\|~~460px~~\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]		[[File:RoverDinfty.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
	</td>		</td>
	<td>		<td>
	[[File:RoverD5over1.png\|~~460px~~\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]		[[File:RoverD5over1.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
	</tr></table>		</td>
			<td>
			[[File:RoverD5over2.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</td>
			</tr>
			<tr>
			<td>
			[[File:RoverD5over3.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</td>
			<td>
			[[File:RoverD5over4.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</td>
			<td>
			[[File:RoverDunity.png\|300px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</td>
			</tr>
			</table>
	</td></tr></table>		</td></tr></table>



	==Thin Ring Approximation==		==Thin Ring Approximation==

Tohline: /* Self-Gravitating, Incompressible (Dyson-Wong) Tori */

2018-08-24T16:44:09Z

Self-Gravitating, Incompressible (Dyson-Wong) Tori

← Older revision		Revision as of 16:44, 24 August 2018
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	==Overview==		==Overview==
	In his pioneering work, [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)] and [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D (1893, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106)] used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] extended Dyson's work, using numerical techniques to obtain more accurate — but still approximate — equilibrium structures for incompressible tori having solid body rotation. Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613)] have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.		In his pioneering work, [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)] and [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D (1893, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106)] used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] extended Dyson's work, using numerical techniques to obtain more accurate — but still approximate — equilibrium structures for incompressible tori having solid body rotation. Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613)] have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.

			<table border="0" cellpadding="8" align="center"><tr><td align="center">
			<table border="1" cellpadding="5" align="center">
			<tr><td align="center" colspan="2">
			'''Figures 1 - 6 extracted without modification from pp. 63-66 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<p></p>
			''The Theory of the Potential'', New York: McGraw-Hill
			</td></tr>
			<tr>
			<td>
			[[File:RoverDinfty.png\|460px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</td>
			<td>
			[[File:RoverD5over1.png\|460px\|center\|Dyson (1893, ''The Potential of an Anchor Ring'')]]
			</tr></table>
			</td></tr></table>



	==Thin Ring Approximation==		==Thin Ring Approximation==

Tohline: /* The Coulomb Potential */

2018-07-17T23:20:54Z

The Coulomb Potential

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	====The Coulomb Potential====		====The Coulomb Potential====
	[[File:~~MovieWongN4~~.gif\|right\|~~Modes~~ n = 3]]As [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,		[[File:MovieWongN4b.gif\|300px\|right\|frame\|Contribution to potential by mode n = 3 (magnified by 100)]]As [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* The Coulomb Potential */

2018-07-17T23:08:20Z

The Coulomb Potential

← Older revision		Revision as of 23:08, 17 July 2018
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	====The Coulomb Potential====		====The Coulomb Potential====
	As [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,		[[File:MovieWongN4.gif\|right\|Modes n = 3]]As [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Introducing Toroidal Coordinates */

2018-06-26T03:14:41Z

Introducing Toroidal Coordinates

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	</table>		</table>
	</div>		</div>
	If such a torus has a uniform density, <math>~\rho_0</math>, throughout, and a total charge (mass), <math>~q</math>, then the charge (mass) and density will be related through the toroidal-coordinate expression (see Wong's equation 2.51),		<span id="DensityFormula">
			If such a torus has a uniform density</span>, <math>~\rho_0</math>, throughout, and a total charge (mass), <math>~q</math>, then the charge (mass) and density will be related through the toroidal-coordinate expression (see Wong's equation 2.51),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* The Coulomb Potential */

2018-06-16T01:59:36Z

The Coulomb Potential

← Older revision		Revision as of 01:59, 16 June 2018
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta		\times \cos[m(\psi - \psi^')] \cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
	\end{cases}\, ,		\end{cases}\, ,
	</math>		</math>

Tohline: /* Exterior Solution */

2018-05-26T23:07:16Z

Exterior Solution

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	</div>		</div>

	After making the substitution, <math>~q \rightarrow (-GM)</math>, we ~~acknowledge~~ that this is, indeed, the expression that has been ~~[[#TRApproximation\|~~derived, above,]] for the gravitational potential in the thin ring approximation, <math>~\Phi_\mathrm{TR}</math>.		After making the substitution, <math>~q \rightarrow (-GM)</math>, we see that this is, indeed, the expression that has been derived, above, for the [[#TRApproximation\|gravitational potential in the thin ring approximation]], <math>~\Phi_\mathrm{TR}</math>.