Editing Apps/DysonWongTori (section)

=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''
* [http://adsabs.harvard.edu/abs/2016ApJ...829...45K W.-T. Kim &amp; S. Moon (2016, ApJ, 829, article id. 45, 22 pp.)] &#8212; ''Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings''
* [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B E. Y. Bannikova, V. G. Vakulik &amp; V. M. Shulga (2011, MNRAS, 411, 557 - 564)] &#8212; ''Gravitational Potential of a Homogeneous Circular Torus: a New Approach''
* [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P D. Petroff &amp; S. Horatschek (2008, MNRAS, 389,156 - 172)] &#8212; ''Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity''
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The following quotes have been taken from [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P Petroff &amp; Horatschek (2008)]:<p></p>
<b>&sect;1:</b> &nbsp; "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincar&eacute; (1885a,b,c) and Dyson (1892, 1893).  Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit.  In particular, Dyson provided a solution to fourth order in the parameter <math>~\sigma = a/b</math>, where <math>~a = r_t</math> provides a measure for the radius of the cross-section of the ring and <math>~b = \varpi_t</math> the distance of the cross-section's centre of mass from the axis of rotation."<p></p>
<b>&sect;7:</b> &nbsp; "In their work on homogeneous rings, Poincar&eacute; and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown.  His result to fourth order is also erroneous as we point out in Appendix B."
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* [http://adsabs.harvard.edu/abs/2006IJMPB..20.3113C P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198)] &#8212; ''Phase Transitions in Self-Gravitating Systems''
* [http://adsabs.harvard.edu/abs/2003MNRAS.339..515A M. Ansorg, A. Kleinw&auml;chter &amp; R. Meinel (2003, MNRAS, 339, 515)] &#8212; ''Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids''
* [http://adsabs.harvard.edu/abs/2001A%26A...375.1091L M. Lombardi &amp; G. Bertin (2001, Astronomy &amp; Astrophysics, 375, 1091 - 1099)] &#8212; ''Boyle's Law and Gravitational Instability''
* [http://adsabs.harvard.edu/abs/1996MNRAS.282..234K W. Kley (1996, MNRAS, 282, 234)] &#8212; ''Maclurin Discs and Bifurcations to Rings''
* [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline, &amp; I. Hachisu (1994, ApJ, 420, 247 - 267)] &#8212; ''The Stability of Thick, Self-Gravitating Disks in Protostellar Systems''
* [http://adsabs.harvard.edu/abs/1991ApJ...374..610B I. Bonnell &amp; P. Bastien (1991, ApJ, 374, 610 - 622)] &#8212; ''The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation''
* [http://adsabs.harvard.edu/abs/1990ApJ...361..394T J. E. Tohline &amp; I. Hachisu (1990, ApJ, 361, 394 - 407)] &#8212; ''The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks''
* [http://adsabs.harvard.edu/abs/1988A%26A...200..127S F. Schmitz (1988, Astronomy &amp; Astrophysics, 200, 127 - 134)] &#8212; ''Equilibrium Structures of Differentially Rotating Self-Gravitating Gases''
* [http://adsabs.harvard.edu/abs/1985Ap%26SS.109...45V P. Veugelen (1985, Astrophysics &amp; Space Science, 109, 45 - 55)] &#8212; ''Equilibrium Models of Differentially Rotating Polytropic Cylinders''
* [http://adsabs.harvard.edu/abs/1984MNRAS.208..279A M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, &amp; R. E. Wilson (1984, MNRAS, 208, 279 - 291)] &#8212; ''Self-Gravity and the Global Structure of Accretion Discs''
* [http://adsabs.harvard.edu/abs/1983A%26A...119..109B P. Bastien (1983, Astronomy &amp; Astrophysics, 119, 109 - 116)] &#8212; ''Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds''
* [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi &amp; D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] &#8212; ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid''
* [http://adsabs.harvard.edu/abs/1980ApJ...236..160T J. E. Tohline (1980, ApJ, 236, 160 - 171)] &#8212; ''Ring Formation in Rotating Protostellar Clouds''
* [http://adsabs.harvard.edu/abs/1980PThPh..63.1957F T. Fukushima, Y. Eriguchi, D. Sugimoto, &amp; G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970)] &#8212; ''Concave Hamburger Equilibrium of Rotating Bodies''
* [http://adsabs.harvard.edu/abs/1978MNRAS.184..709K J. Katz &amp; D. Lynden-Bell (1978, MNRAS, 184, 709 - 712)] &#8212; ''The Gravothermal Instability in Two Dimensions''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, &amp; S. A. Teukolsky (1977, ApJ, 214, 584- 597)] &#8212; ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press &amp; Teukolsky make the following statement: "&hellip; we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincar&eacute;-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) &hellip;"</li></ol>
* [http://adsabs.harvard.edu/abs/1976ApJ...207..736H C. J. Hansen, M. L. Aizenman, &amp; R. L. Ross (1976, ApJ, 207, 736 - 744)] &#8212; ''The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders''
* [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] &#8212; ''Toroidal Figures of Equilibrium''
* [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353)] &#8212; ''Toroidal and Spherical Bubble Nuclei''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. Ostriker (1964, ApJ, 140, 1056)] &#8212; ''The Equilibrium of Polytropic and Isothermal Cylinders''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. Ostriker (1964, ApJ, 140, 1067)] &#8212; ''The Equilibrium of Self-Gravitating Rings''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1529O J. Ostriker (1964, ApJ, 140, 1529)] &#8212; ''On the Oscillations and the Stability of a Homogeneous Compressible Cylinder''
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] &#8212; ''Cylindrical Emden and Associated Functions''
* [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] &#8212; ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.''
* [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 William Duncan MacMillan (1958)], ''The Theory of the Potential'', New York:  Dover Publications
* [https://archive.org/details/foundationsofpot033485mbp Oliver Dimon Kellogg (1929)], ''Foundations of Potential Theory'', Berlin:  Verlag Von Julius Springer
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] &#8212; ''On the Dynamics of Revolving Fluids''
* [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] &#8212; ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] &#8212; ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol>
* [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)]