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Warp Propagation in Astrophysical Discs

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Book cover Astrophysical Black Holes

Part of the book series: Lecture Notes in Physics ((LNP,volume 905))

Abstract

Astrophysical discs are often warped, that is, their orbital planes change with radius. This occurs whenever there is a non-axisymmetric force acting on the disc, for example the Lense–Thirring precession induced by a misaligned spinning black hole, or the gravitational pull of a misaligned companion. Such misalignments appear to be generic in astrophysics. The wide range of systems that can harbour warped discs—protostars, X-ray binaries, tidal disruption events, quasars and others—allows for a rich variety in the disc’s response. Here we review the basic physics of warped discs and its implications.

Big whorls have little whorls, which feed on their velocity. And little whorls have lesser whorls, and so on to viscosity.

– Adaptation of The Siphonaptera by Lewis Richardson

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Notes

  1. 1.

    See the definition of the quantity A at the end of page 1189 of [44].

  2. 2.

    See the definition of the quantity A at the end of page 1189 of [44]. In the linear equations it is often convenient to assume the unit tilt vector \(\boldsymbol{l} = \left (l_{x},l_{y},l_{z}\right )\) has l z  ≈ 1 and therefore adopt complex equations for the disc tilt where e.g. \(W = l_{x} + il_{y}\). In this case a diffusion coefficient with non-zero real and imaginary parts has components in the direction of both \(\partial \boldsymbol{l}/\partial R\) and \(\boldsymbol{l} \times \partial \boldsymbol{l}/\partial R\).

  3. 3.

    Strictly speaking the rhs of this relation is missing a factor of order unity dependent on the warp amplitude \(\left \vert \psi \right \vert \), as the forcing in a warp restricts the hydrostatic balance assumed in its derivation. This affects the viscosity coefficients Q 1, Q 2, Q 3 significantly for large \(\left \vert \psi \right \vert \), and so in this case we must use the form of [38] taking this effect into account (Ogilvie, private communication).

  4. 4.

    There is a long history, in both the astronomical and fluids literatures, of modelling turbulent fluids with effective viscosities.

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Acknowledgements

We thank Phil Armitage, Giuseppe Lodato, Steve Lubow, Rebecca Martin, Rebecca Nealon, Daniel Price & Jim Pringle for providing thoughtful comments on the manuscript. We thank Gordon Ogilvie for useful discussions. CN thanks NASA for support through the Einstein Fellowship Programme, grant PF2-130098. Astrophysics research at Leicester is supported by an STFC Consolidated Grant.

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Authors and Affiliations

  1. JILA, University of Colorado & NIST, Boulder, CO, 80309-0440, USA

    Chris Nixon

  2. Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK

    Andrew King

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Correspondence to Chris Nixon .

Editor information

Editors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

    Francesco Haardt

  2. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

    Vittorio Gorini

  3. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

    Ugo Moschella

  4. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

    Aldo Treves

  5. Dipartimento di Fisica, Università di Milano-Bicocca, Milano, Italy

    Monica Colpi

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Nixon, C., King, A. (2016). Warp Propagation in Astrophysical Discs. In: Haardt, F., Gorini, V., Moschella, U., Treves, A., Colpi, M. (eds) Astrophysical Black Holes. Lecture Notes in Physics, vol 905. Springer, Cham. https://doi.org/10.1007/978-3-319-19416-5_2

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