Apps/DysonWongTori - Revision history

Jet53man: /* See Also */

2021-09-13T21:59:40Z

Jet53man: /* Fukushima (2016) */

2021-09-11T16:36:54Z

Fukushima (2016)

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	Figure 1 extracted from [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016)]'''<~~p><~~/p>		Figure 1 extracted from [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016)]'''<br />
	"''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'''"<~~p><~~/p>		"''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'''"<br />
	Astronomical Journal, vol. 152, id. 35, 31 pp. © [https://aas.org/ AAS]		Astronomical Journal, vol. 152, id. 35, 31 pp. © [https://aas.org/ AAS]
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	[[File:FukushimaFig1.png\|325px\|center\|Fukushima (2016~~, AJ, 152, id. 35, 31 pp~~)]]		[[File:FukushimaFig1.png\|325px\|center\|To be inserted: Fig. 1 from Fukushima (2016)]]
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	[[File:ModifiedFukushimaFig1.png\|325px\|center\|Modified Fig. 1 from Fukushima (2016)]]		[[File:ModifiedFukushimaFig1.png\|325px\|center\|To be inserted: Modified Fig. 1 from Fukushima (2016)]]
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	Fukushima also introduces a set of coordinate-dependent functions — <math>~\nu(u), s(\theta), c(\theta),</math> and <math>D(u,\theta) \equiv (u - c)</math> — that, in effect, allow his derivations to be presented in a relatively compact form. [[#Table1\|Table 1, below]], details the relationships between this set of functions and the coordinates/functions used by Wong as well as the coordinates/functions preferred by [http://adsabs.harvard.edu/abs/1953mtp..book.....M Morse & Feshbach (1953)] that we have adopted in an [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Toroidal_Coordinates\|associated discussion]]. Then, for example, the mapping between Fukushima's adopted set of meridional-plane coordinates and ''cylindrical'' coordinates <math>~(\varpi,z)</math> is:		Fukushima also introduces a set of coordinate-dependent functions — <math>~\nu(u), s(\theta), c(\theta),</math> and <math>D(u,\theta) \equiv (u - c)</math> — that, in effect, allow his derivations to be presented in a relatively compact form. [[#Table1\|Table 1, below]], details the relationships between this set of functions and the coordinates/functions used by Wong as well as the coordinates/functions preferred by [http://adsabs.harvard.edu/abs/1953mtp..book.....M Morse & Feshbach (1953)] that we have adopted in an [[2DStructure/ToroidalCoordinates#Toroidal_Coordinates\|associated discussion]]. Then, for example, the mapping between Fukushima's adopted set of meridional-plane coordinates and ''cylindrical'' coordinates <math>~(\varpi,z)</math> is:
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	This last expression not only matches equation (27) in [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] but, as has been pointed out in an [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Chosen_Test_Mass_Distribution\|accompanying discussion]] of the paper by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)], its form matches the familiar algebraic expression for an off-axis circle. The circle — which, here, is associated with a meridional cross-section of the <math>~u</math> = constant torus — has a (cross-sectional) radius,		This last expression not only matches equation (27) in [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] but, as has been pointed out in an [[2DStructure/ToroidalCoordinates#Chosen_Test_Mass_Distribution\|accompanying discussion]] of the paper by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)], its form matches the familiar algebraic expression for an off-axis circle. The circle — which, here, is associated with a meridional cross-section of the <math>~u</math> = constant torus — has a (cross-sectional) radius,
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	<math>~r_t = \frac{a}{\nu} \, ,</math>		<math>~r_t = \frac{a}{\nu} \, ,</math>
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	<td align="center">[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)]</td>		<td align="center">[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)]</td>
	<td align="center">[[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Our Analysis]]</td>		<td align="center">[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Our Analysis]]</td>
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Jet53man: /* Bannikova et al. (2011) */

2021-09-11T16:29:29Z

Bannikova et al. (2011)

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	===Bannikova et al. (2011) ===		===Bannikova et al. (2011) ===

	In §2 of their paper, [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)] develop an integral expression — ultimately, their equation (8) — for the gravitational potential both inside and outside of a uniform-density, axisymmetric, thick torus that has a circular cross-section. They accomplish this by imagining that the (axisymmetric) torus is composed of a set of infinitely thin, axisymmetric rings with their planes being parallel to the torus symmetry plane. [Note that, 28 years earlier, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983)] adopted precisely this same strategy to evaluate the gravitational potential of axisymmetric, inhomogeneous configurations; see our [[~~User:Tohline/~~Apps/ReviewStahler83#Stahler.27s_.281983.29_Rotationally_Flattened_Isothermal_Configurations\|separate discussion of Stahler's work in the context of rotationally flattened, isothermal structures]].] In recording, here, the expression derived by Bannikova et al. (2011), we will substitute parameter names as detailed in the following "mapping" table.		In §2 of their paper, [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)] develop an integral expression — ultimately, their equation (8) — for the gravitational potential both inside and outside of a uniform-density, axisymmetric, thick torus that has a circular cross-section. They accomplish this by imagining that the (axisymmetric) torus is composed of a set of infinitely thin, axisymmetric rings with their planes being parallel to the torus symmetry plane. [Note that, 28 years earlier, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983)] adopted precisely this same strategy to evaluate the gravitational potential of axisymmetric, inhomogeneous configurations; see our [[Apps/ReviewStahler83#Stahler.27s_.281983.29_Rotationally_Flattened_Isothermal_Configurations\|separate discussion of Stahler's work in the context of rotationally flattened, isothermal structures]].] In recording, here, the expression derived by Bannikova et al. (2011), we will substitute parameter names as detailed in the following "mapping" table.

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	<td align="center">[http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)]</td>		<td align="center">[http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)]</td>
	<td align="center">[[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Our Analysis]]</td>		<td align="center">[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Our Analysis]]</td>
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	Recognizing that the density of material in a uniform-density torus is the total mass divided by the [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Total_Mass\|volume of the torus]], ''i.e.,'' it is <math>~M/(2\pi^2\varpi_t r_t^2)</math>, we see that this integral expression is identical to the one [[#General.2C_Two-Dimensional_Integral\|we have professed, above]], is the correct one.		Recognizing that the density of material in a uniform-density torus is the total mass divided by the [[2DStructure/ToroidalCoordinates#Total_Mass\|volume of the torus]], ''i.e.,'' it is <math>~M/(2\pi^2\varpi_t r_t^2)</math>, we see that this integral expression is identical to the one [[#General.2C_Two-Dimensional_Integral\|we have professed, above]], is the correct one.

	===Fukushima (2016)===		===Fukushima (2016)===

Jet53man: /* The Coulomb Potential */

2021-09-11T16:28:19Z

The Coulomb Potential

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	As we have suggested in an [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Special_Case\|accompanying discussion]], this expression should match our separate "special case" evaluation of the potential of a uniform-density torus if this expression is evaluated at <math>~\eta^' = +\infty</math>, and all values of the polar angle, <math>~\theta^'</math>.		As we have suggested in an [[2DStructure/ToroidalCoordinates#Special_Case\|accompanying discussion]], this expression should match our separate "special case" evaluation of the potential of a uniform-density torus if this expression is evaluated at <math>~\eta^' = +\infty</math>, and all values of the polar angle, <math>~\theta^'</math>.
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	<tr><td align="center">Thin Ring Approximation as Presented by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)]</td></tr>		<tr><td align="center">Thin Ring Approximation as Presented by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)]</td></tr>
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	Interestingly, for an example of a derivation of the [[~~User:Tohline/~~Apps/DysonWongTori#Thin_Ring_Approximation\|thin ring approximation, which we have reviewed above]], rather than referencing [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or [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)], Wong (1973) points to an article by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)], which was published in the [http://adsabs.harvard.edu/abs/1965qssg.conf.....R ''Proceedings of the 1<sup>st</sup> Texas Symposium on Relativistic Astrophysics'' (1965), eds., I. Robinson, A. Schild, & E. L. Schucking (Chicago: Chicago University Press)]. Presumably this reflects Wong's personal interactions with [https://en.wikipedia.org/wiki/John_Archibald_Wheeler J. A. Wheeler's] research group at Princeton University, as Thorne was a student of Wheeler.		Interestingly, for an example of a derivation of the [[Apps/DysonWongTori#Thin_Ring_Approximation\|thin ring approximation, which we have reviewed above]], rather than referencing [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or [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)], Wong (1973) points to an article by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)], which was published in the [http://adsabs.harvard.edu/abs/1965qssg.conf.....R ''Proceedings of the 1<sup>st</sup> Texas Symposium on Relativistic Astrophysics'' (1965), eds., I. Robinson, A. Schild, & E. L. Schucking (Chicago: Chicago University Press)]. Presumably this reflects Wong's personal interactions with [https://en.wikipedia.org/wiki/John_Archibald_Wheeler J. A. Wheeler's] research group at Princeton University, as Thorne was a student of Wheeler.

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	[[File:Thorne65Fig1.png\|400px\|~~Figure~~ 1 from Thorne (1965)]]		[[File:Thorne65Fig1.png\|400px\|To be inserted: Fig. 1 from Thorne (1965)]]
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	But Thorne actually presents a solution to a general-relativistic-based initial-value equation that reads,		But Thorne actually presents a solution to a general-relativistic-based initial-value equation that reads,

Jet53man: /* Introducing Toroidal Coordinates */

2021-09-11T16:24:29Z

Introducing Toroidal Coordinates

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	Figure 1 extracted without modification from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)]'''<~~p><~~/p>		Figure 1 extracted without modification from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)]'''<br />
	"''Toroidal and Spherical Bubble Nuclei'''"<~~p><~~/p>		"''Toroidal and Spherical Bubble Nuclei'''"<br />
	Annals of Physics, vol. 77, pp. 279 - 353 © Elsevier Science		Annals of Physics, vol. 77, pp. 279 - 353 © Elsevier Science
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	[[File:Wong1973Figure1.png\|400px\|center\|Wong (1973, Annals of Physics, 77, p. 284)]]		[[File:Wong1973Figure1.png\|400px\|center\|To be inserted: Fig. 1 from Wong (1973, Annals of Physics, 77, p. 284)]]
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Jet53man: /* General, Two-Dimensional Integral */

2021-09-11T16:21:32Z

General, Two-Dimensional Integral

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	====General, Two-Dimensional Integral====		====General, Two-Dimensional Integral====

	In our [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|accompanying statement of this problem]], we have written,		In our [[2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|accompanying statement of this problem]], we have written,
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Jet53man: /* Fukushima (2016) -- Thin Ring */

2021-09-11T16:20:56Z

Fukushima (2016) -- Thin Ring

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	[[File:Fukushima2016Fig4.png\|225px\|~~Our Thin Ring equipotential surface~~]]		[[File:Fukushima2016Fig4.png\|225px\|To be inserted: Fig. 4 from Fukushima (2016)]]
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	[[File:Fukushima2016Fig5.png\|225px\|~~Our Thin Ring equipotential surface~~]]		[[File:Fukushima2016Fig5.png\|225px\|To be inserted: Fig. 5 from Fukushima (2016)]]
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	<td align="center">		<td align="center">
	[[File:Fukushima2016Fig6.png\|225px\|~~Our Thin Ring equipotential surface~~]]		[[File:Fukushima2016Fig6.png\|225px\|To be inserted: Fig. 6 from Fukushima (2016)]]
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Jet53man: /* Thin Ring Approximation */

2021-09-11T16:14:34Z

Thin Ring Approximation

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	{{ ~~User:Tohline/~~Math/EQ TRApproximation }}		{{ Math/EQ TRApproximation }}
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Jet53man: /* Self-Gravitating, Incompressible (Dyson-Wong) Tori */

2021-09-11T15:50:34Z

Self-Gravitating, Incompressible (Dyson-Wong) Tori

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	<!--Much of the introductory material of this chapter has been drawn from the paper by [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline & Hachisu (1990)] titled, ''The Breakup of Self-Gravitating Rings, Tori, and Accretion Disks.''-->		<!--Much of the introductory material of this chapter has been drawn from the paper by [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline & Hachisu (1990)] titled, ''The Breakup of Self-Gravitating Rings, Tori, and Accretion Disks.''-->

	~~{{LSU_HBook_header}}~~


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	<td align="center">Our Separate <br />[[~~User:Tohline/~~Apps/DysonPotential#Dyson_.281893.29\|Chapter Discussion]]		<td align="center">Our Separate <br />[[Apps/DysonPotential#Dyson_.281893.29\|Chapter Discussion]]
	<td align="center">Our Relevant <br />[[#MacMillan_.281930.29\|Discussion, Below]]		<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 />[[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 Separate <br />[[Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Chapter Discussion]]
	<td align="center">Our Relevant <br />[[#Fukushima_.282016.29\|Discussion, below]]		<td align="center">Our Relevant <br />[[#Fukushima_.282016.29\|Discussion, below]]
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Jet53man: Created page with "FORCETOC =Self-Gravitating, Incompressible (Dyson-Wong) Tori=

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 This matches the [[#Our_Integral_Expressions|integral expression that we have derived and repeated, above]].
-=See Also=
+=Additional References=
 * [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.)] &#8212; ''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects''
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 * [http://adsabs.harvard.edu/abs/1885AN....111...37K S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37)] &#8212; ''Zus&auml;tze und Bemerkungen zu Laplace's Untersuchung &uuml;ber die Gestalt der Saturnsringe''
 * Poincar&eacute; (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). &#8212; references copied from paper by [http://adsabs.harvard.edu/abs/1974ApJ...190..675W Wong (1974)]
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