During Lent term 2018, I will be giving 16 lectures
on the dynamics of astrophysical discs, as part of Part III of the
Cambridge Mathematical Tripos.
Lectures take place at 10am on Tuesdays and
Thursdays in room MR5.
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, (old) lecture notes and slides on accretion disk dynamics (here and here.)
- Latter, Ogilvie & Rein., draft 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.)
- Netzer (2006), (pdf.) (Summary of main physical processes in AGN.)
Example sheets & classes
- Example Class 1: 2-4pm 15th February in MR13. First problem sheet (pdf)
- Example Class 2: 2-4pm 1st March in MR13. Second problem sheet (pdf)
- Example Class 3: 2-4pm 15th March in MR2. Third problem sheet (pdf)
- Revision Class: TBC. DAD Exam 2014 (pdf)
Schedule:
Lecture 1 (18/01/2018): Introduction (pdf)
- Survey of astrophysical disk systems
- Basic physical and observational properties
- Equations of motion, circular orbits
- Characteristic frequencies
- Perturbed orbits: epicyclic oscillations
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.
- Precession
- 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). See the book by Frank et al. (2002) for a fuller discussion of the derivation and analysis of the diffusion equation. You can also refer to Section III in Balbus and Hawley (1998) (ADS link). The notes by Ogilvie (here and here) are also a good reference. Finally, to calculate the components of the stress tensor in cylindrical polar coordinates I used some vector calculus results that happen to be listed near the bottom of and this page
- Boundary conditions
- Steady accretion disks
Further details on steady accreting disks are in this document, written by Gordon Ogilvie.
- Spectrum of steady disks
- Complications and observed SEDs
- Time-dependent solutions
- Greens functions
The cartoon I used for the SEDs of PP disks comes from Andrea Isella's thesis, which I can't find online anymore. It does appear on this page which also has some nice introductory material. The grid of observed SEDs I took from Fang et al. (2009), and can be obtained from here. The X-ray binary SED I stole from an old proceedings by Gierlinski and Zdiarski. The paper is here. The schematic representation of AGN spectra I got from an article by Koratkar which you can find at this location.
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.)
The movie of the diffusing Greens function shown in the lecture 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.)
- Vertical hydrostatic equilibrium
- Important length and time scales
- Isothermal and polytropic disk models
- Radiative disk models and opacity laws
This image taken by Cassini emphasises how thin Saturn's rings are. The quoted thickness of 1 km is due to large-scale corrugations and bending waves.
- Approximate algebraic solution for an alpha disk
- Thermal instability in cataclysmic variables
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.
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
- Orbital motion in the shearing sheet
- Symmetries and boundary conditions
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.
- 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 have yet to see this bypass mechanism. See also the discussion in Balbus and Hawley (2006). (link.)
Lecture 11 (22/02/2018): Vortices in Disks
- Introduction to vortices
- Enstrophy conservation
- Kida vortex solution and its stability
Lecture 12 (27/02/2018): Density Waves and Gravitational Instability
- 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
- The impulse approximation
- Angular momentum transfer between embedded satellites and their disks
- Gap opening
- Planet migration
- Equations of MHD
- Derivation of the axisymmetric dispersion relation
- Analysis of dispersion relation
- Importance of disk thickness, magnetic diffusion, and magnetic field strength on stability criterion
- Physical interpretation of dispersion relation
- Survey of numerical simulations