Revision as of 19:06, 12 August 2018

Polytropic & Isothermal Tori

Overview

Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — specifically, polytropic and isothermal — fluids as presented in a series of papers by Jeremiah. P. Ostriker:

J. Ostriker (1964, ApJ, 140, 1056) — The Equilibrium of Polytropic and Isothermal Cylinders
J. Ostriker (1964, ApJ, 140, 1067) — The Equilibrium of Self-Gravitating Rings
J. Ostriker (1964, ApJ, 140, 1529) — On the Oscillations and the Stability of a Homogeneous Compressible Cylinder
J. Ostriker (1965, ApJ Supplements, 11, 167) — Cylindrical Emden and Associated Functions

I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of S. Chandrasekhar.

T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.) — Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects
W.-T. Kim & S. Moon (2016, ApJ, 829, article id. 45, 22 pp.) — Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings
E. Y. Bannikova, V. G. Vakulik & V. M. Shulga (2011, MNRAS, 411, 557 - 564) — Gravitational Potential of a Homogeneous Circular Torus: a New Approach
D. Petroff & S. Horatschek (2008, MNRAS, 389,156 - 172) — Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity

The following quotes have been taken from Petroff & Horatschek (2008):

§1: "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincaré (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."

§7: "In their work on homogeneous rings, Poincaré 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."

P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198) — Phase Transitions in Self-Gravitating Systems
M. Ansorg, A. Kleinwächter & R. Meinel (2003, MNRAS, 339, 515) — Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids
M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091 - 1099) — Boyle's Law and Gravitational Instability
W. Kley (1996, MNRAS, 282, 234) — Maclurin Discs and Bifurcations to Rings
J. W. Woodward, J. E. Tohline, & I. Hachisu (1994, ApJ, 420, 247 - 267) — The Stability of Thick, Self-Gravitating Disks in Protostellar Systems
I. Bonnell & P. Bastien (1991, ApJ, 374, 610 - 622) — The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation
J. E. Tohline & I. Hachisu (1990, ApJ, 361, 394 - 407) — The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks
F. Schmitz (1988, Astronomy & Astrophysics, 200, 127 - 134) — Equilibrium Structures of Differentially Rotating Self-Gravitating Gases
P. Veugelen (1985, Astrophysics & Space Science, 109, 45 - 55) — Equilibrium Models of Differentially Rotating Polytropic Cylinders
M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, & R. E. Wilson (1984, MNRAS, 208, 279 - 291) — Self-Gravity and the Global Structure of Accretion Discs
P. Bastien (1983, Astronomy & Astrophysics, 119, 109 - 116) — Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds
Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875) — Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid
J. E. Tohline (1980, ApJ, 236, 160 - 171) — Ring Formation in Rotating Protostellar Clouds
T. Fukushima, Y. Eriguchi, D. Sugimoto, & G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970) — Concave Hamburger Equilibrium of Rotating Bodies
J. Katz & D. Lynden-Bell (1978, MNRAS, 184, 709 - 712) — The Gravothermal Instability in Two Dimensions
P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597) — Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid (includes torus with non-uniform rotation)

Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an equilibrium incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"

C. J. Hansen, M. L. Aizenman, & R. L. Ross (1976, ApJ, 207, 736 - 744) — The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders
C.-Y. Wong (1974, ApJ, 190, 675 - 694) — Toroidal Figures of Equilibrium
C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353) — Toroidal and Spherical Bubble Nuclei
Gunnar Randers (1942, ApJ, 95, 88) — The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.
William Duncan MacMillan (1958), The Theory of the Potential, New York: Dover Publications
Oliver Dimon Kellogg (1929), Foundations of Potential Theory, Berlin: Verlag Von Julius Springer
Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154) — On the Dynamics of Revolving Fluids
F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106) — The Potential of an Anchor Ring. Part II.
1. In this paper, Dyson derives the gravitational potential inside the ring mass distribution
F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95) — The Potential of an Anchor Ring. Part I.
1. In this paper, Dyson derives the gravitational potential exterior to the ring mass distribution
S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37) — Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe
Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by Wong (1974)

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 =Polytropic &amp; Isothermal Tori=
-<!--Much of the introductory material of this chapter has been drawn from the paper by [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline &amp; Hachisu (1990)] titled, ''The Breakup of Self-Gravitating Rings, Tori, and Accretion Disks.''-->
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 ==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 &#8212; but still approximate &#8212; equilibrium structures for incompressible tori having solid body rotation.  Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi &amp; D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline &amp; 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.
+Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible &#8212; specifically, polytropic and isothermal &#8212; fluids as presented in a series of papers by Jeremiah. P. Ostriker:
+* [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''
+I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of [https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar S. Chandrasekhar].
@@ Line 45: / Line 48: @@
 * [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

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Revision as of 19:06, 12 August 2018

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Polytropic & Isothermal Tori

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