User:Tohline/2DStructure/AxisymmetricInstabilities
Axisymmetric Instabilities to Avoid
Here we draw heavily from the extensive discussion of instabilities that appears in [T78]
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When constructing rotating equilibrium configurations that obey a barotropic equation of state, keep in mind that certain parameter ranges should be avoided because they will lead to structures that are unstable toward the dynamical development of local, convective-type motions. Here are a few well-known examples.
Rayleigh-Taylor Instability
This bouyancy-driven instability arises when the condition,
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<math>~(- \vec{g} ) \cdot \nabla\rho</math> |
<math>~< </math> |
<math>~0 </math> |
[stable] , |
is violated, where, <math>~\vec{g}</math> is the local gravitational acceleration vector. In the simplest case of spherically symmetric configurations, this means that the mass density must decrease outward.
Høiland Criterion
According to [ T78 ] — see p. 168 — this pair of mathematically expressed conditions has the following meaning:
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"A baroclinic star in permanent rotation is dynamically stable with respect to axisymmetric motions if and only if the two following conditions are satisfied: (i) the entropy per unit mass, <math>~s</math>, never decreases outward, and (ii) on each surface <math>~s</math> = constant, the angular momentum per unit mass, <math>~\Omega \varpi^2</math>, increases as we move from the poles to the equator." |
Schwarzschild Criterion
In the case of nonrotating equilibrium configurations, the Høiland Criterion reduces to the Schwarzschild criterion. That is, thermal convection arises when the condition,
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<math>~(- \vec{g} ) \cdot \nabla s</math> |
<math>~> </math> |
<math>~0 </math> |
[stable] , |
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[ T78 ], p. 167, Eq. (43) |
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is violated, where, <math>~s</math> is the local specific entropy of the fluid. This means that, in order for a spherical system to be stable against dynamical convective motions, the specific entropy must increase outward.
Poincaré-Wavre Theorem
Lord Rayleigh Instability
See Also
- 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
- D. Petroff & S. Horatschek (2008, MNRAS, 389,156 - 172) — Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity
- P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198) — Phase Transitions in Self-Gravitating Systems
- M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091 - 1099) — Boyle's Law and Gravitational Instability
- 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
- 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
- Gunnar Randers (1942, ApJ, 95, 88) — The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.
- F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106) — The Potential of an Anchor Ring. Part II.
- 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.
- 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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© 2014 - 2021 by Joel E. Tohline |