During Lent term 2024, I will be giving 16 lectures
on the dynamics of astrophysical discs, as part of Part III of the
Cambridge Mathematical Tripos.
Lectures will be at 11am on Tuesdays and Thursday in MR12
There will be three examples classes and a revision class
in Easter term.
On this webpage I will post the course schedule, pictures, movies, and
other material that appear in
the lectures, as well as suggestions for additional reading, original
references, example sheets, etc.
Introductory references and general review articles
- Ogilvie, lecture notes and slides on accretion disk dynamics (here.)
- Latter, Ogilvie & Rein (2018), review chapter covering rings and disks (pdf.)
- Frank, King & Raine (2002). Accretion Power in Astrophysics, 3rd edn, CUP. (Textbook on classical disk theory.)
- Pringle (1981), ARA&A, 19, 137. (ADS link.) (Succinct review article on viscous disks.)
- Balbus (2003), ARA&A, 41, 555. (ADS link.) (Clear and concise account of instabilities and waves in disks.)
- Esposito (2010), AREPS, 38, 383. (ADS link.) (Gentle recent review of Saturn's rings.)
- Goldreich & Tremaine (1982), ARA&A, 20, 249. (ADS link.) (Detailed account of the physics of planetary rings.)
- Hellier (2001), Cataclysmic Variable Stars: how and why they vary, Springer-Verlag. (Very readable book on CVs.)
- Armitage (2011), ARA&A, 49, 195. (ADS link.) (Good reference on the dynamics of protoplanetary disks.)
- Ferrarese and Ford (2005), SSRv, 116, 523. (link.) (Well written and thorough account of AGN. The first 20 pages are worth reading for an overview on the subject.)
Example sheets & classes
- Example Class 1: 1:30-3pm, Tuesday 27th Feb, room MR3. First problem sheet (pdf)
- Example Class 2: 1:30pm-3pm, Tuesday 12th Mar, room MR3. Second problem sheet (pdf)
- Example Class 3: 1:30-3pm, Tuesday 23rd Apr, room MR5. Third problem sheet (pdf)
- Revision Class: TBC (pdf)
Schedule:
Lecture 1: Introduction
- Survey of astrophysical disk systems
- Basic physical and observational properties
- Equations of motion, circular orbits
- Characteristic frequencies
- Perturbed orbits: epicyclic oscillations
- Precession
A very basic but engaging account of Keplerian orbits can be found in Chapter 4 of David Tong's lecture notes on dynamics (link).
Those wanting more detail can consult Mark Wyatt's notes from his Part III courses, which are posted near the bottom of his homepage (link).
Finally, the classic text on orbital dynamics is `Solar System Dynamics' by Murray and Dermott (link). However, it is fairly hardcore.
- Elementary mechanics of accretion
- Equations of astrophysical fluid dynamics
The mechanical description of accretion using two orbitting particles is lifted from Section 1.2 in Lynden-Bell and Pringle (1974) (ADS link).
Further details on the equations of astrophysical fluid dynamics can be found in this document, written by Gordon Ogilvie. This should serve as a useful reference. For a derivation of the equations you could read chapters 2-4 of Cathie Clarke's book `Principles of astrophysical fluid dynamics' (link).
- Viscosity as proxy for turbulent flow
- Derivation of the diffusion equation
Those interested in learning more about turbulence could read `A first course in turbulence' by Tennekes and Lumley (first two chapters are relevant to today's lecture) and 'Turbulence' by Peter Davidson (chapter 1 and maybe 5).
For another derivation of the diffusion equation, see the notes by Ogilvie (here, lecture 3, and here, lecture 4). Finally, to calculate the components of the stress tensor in cylindrical polar coordinates you might find it useful to consult the resources on tensor calculus in curvilinear coordinates located here and here.
- Boundary conditions
- Steady accretion disks
For another take, and further details, on boundary conditions and steady accreting disks please consult lectures 3 and 4 in this old incarnation of the course, and in lecture 5 in this more recent version.
- Spectrum of steady disks
- Complications and observed SEDs
- Time-dependent solutions
- Greens functions
Slides showing temperature profiles in certain DNe and the SEDs of various disk classes can be found here
Sijme-Jan Paardekooper has some background notes on Greens functions which might be useful. (link.) More involved notes on Greens functions can be found in the IB Methods course. Section 10.1 in Josza's notes may be helpful. (link.)
A movie of a diffusing Greens function is located here.
The derivation of the Greens function in the case that the mean viscosity is a constant can be found in the classic paper by Lynden-Bell and Pringle (1974) in Section 2.2. (ADS link.) However, I would consult the far more accessible derivation in Ogilvie's notes. ( link.)
- Algebraic similarity solutions
- Vertical hydrostatic equilibrium
- Important length and time scales
This image taken by Cassini emphasises how thin Saturn's rings are, which in principle helps us constrain the kinetic temperature of the ring matter. In fact, the actual thickness of 1 km that is observed is due to large-scale corrugations and bending waves, rather than the vertical pressure gradient.
- Isothermal and polytropic disk models
- Radiative disk models and opacity laws
- Approximate algebraic solution for an alpha disk
Most of this material is in the book by Frank et al. (2002), `Accretion power in astrophysics'. If you can't get your hands on the book you could have a look at the Section III.B in the review article by Balbus and Hawley (1998) (ADS link), which discusses the alpha-disk model in detail and works through an approximate solution for an alpha disk with Kramers opacity.
- Thermal instability and outbursts in dwarf novae
- The shearing sheet
- Orbital motion in the shearing sheet
Again, `Accretion power in astrophysics' is a decent reference for thermal instability in CVs. A more involved (and opinionated) review article is by Lasota (ADS link). There is no single `discovery paper' that first outlined the CV limit cycle, the closest might be Faulkner et al. (1983) (ADS link). Rather, the main ideas were developed collectively by researchers in the late 70s and early 80s.
The shearing sheet model was first developed by Goldreich and Lynden-Bell in the 60s (ADS link), in order to study galactic disks. A good discussion of how the shearing sheet/box is implemented numerically is in Hawley et al. (1995) (ADS link). It includes a discussion of shearing periodic boundary conditions.
- Symmetries and boundary conditions of the shearing sheet
- Incompressible disk dynamics and equations
- Inertial shearing waves
- Centrifugal instability and Rayleigh's criterion
The formal derivation of the equations of incompressible fluid dynamics in the shearing sheet can be found in this paper.
A gif of inertial waves in a rotating terrestrial experiment can be found here. Note how the wave packet slowly spreads diagonally to the bottom left, while the wavecrests propagate quickly in the perpendicular direction. Two movies of shearing inertial waves in the shearing sheet can be viewed here and here. The left panel in these movies shows the amplitude of the waves as they shear through kx=0. The right panel shows the wavecrests shearing out to a trailing configuration. Note the transient growth in the second movie.-
I showed that accretion disks are linearly hydrodynamically stable, because their squared epicyclic frequency is positive. Researchers have argued, however, that there may be a `subcritical' transition to turbulence if the disk is perturbed sufficiently strongly. The argument is that the transient growth in a shearing wave might yield large enough amplitudes to set off this process. Two papers that discuss this can be found here and here. There are a number of others. However, numerical simulations and careful laboratory experiments have yet to see this bypass mechanism. See also the discussion in Balbus and Hawley (2006). (link.)
Lecture 11: Vortices in Disks
- Introduction to vortices
- Enstrophy conservation
- Kida vortex solution and its stability
ALMA observations of PP disk asymmetries can be found in this Nature paper by Casassus et al. (2013). (link.) They may or may not correspond to embedded vortices.
The elliptical vortices we looked at were first calculated by Kida (1981) (link). He took the rotating elliptical vortex solution discovered by Kirchoff in the late 19th century and then added a background shear flow. In the second example sheet we will go through the derivation. A short document outlining the derivation of the Kirchhoff solution can be downloaded here.
A movie showing the evolution of an unstable cyclonic vortex can be viewed here. This paper explores 3D instabilities that can attack a Kida vortex.
A youtube movie of the subcritical baroclinic instability simulated by Wlad Lyra can be viewed here.
- Compressible disk dynamics and equations
- Density waves
- Axisymmetric gravitational instability
- Non-axisymmetric instability and `gravitoturbulence'
- Test particle orbits in the presence of an embedded satellite
- Excitation of epicyclic oscillations
- Angular momentum transfer between embedded satellites and their disks
- Gap opening
- Planet migration
- Equations of MHD
- Derivation of the axisymmetric MRI dispersion relation
- Analysis of the dispersion relation
- Importance of disk thickness, magnetic diffusion, and magnetic field strength on the stability criterion
- Physical interpretation of dispersion relation
- Survey of numerical simulations