Dynamics of Astrophysical Disks

During Lent term 2025, 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.)

References on specific disk types

• 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.)

Schedule:

Lecture 1: Introduction

• Survey of astrophysical disk systems
• Basic physical and observational properties

Lecture 2: Orbital Dynamics I

• 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.

Lecture 3: Equations of AFD

• 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).

Lecture 4: Viscous Accretion Disks I

• 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.

Lecture 5: Viscous Accretion Disks II

• 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.

Lecture 6: Viscous Accretion Disks III

• Spectrum of steady disks
• Complications and observed SEDs
• Time-dependent solutions
• Greens functions

Lecture 7: Viscous Accretion Disks IV + Vertical Disk Structure

• Algebraic similarity solutions
• Vertical hydrostatic equilibrium
• Important length and time scales

Lecture 8: Radiative Disk Models

• Isothermal and polytropic disk models
• Radiative disk models and opacity laws
• Approximate algebraic solution for an alpha disk

Lecture 9: Dwarf Novae + Local Disk Models

• Thermal instability and outbursts in dwarf novae
• The shearing sheet
• Orbital motion in the shearing sheet

Lecture 10: Incompressible Dynamics, Shearing Inertial Waves, and Centrifugal Instability

• Symmetries and boundary conditions of the shearing sheet
• Incompressible disk dynamics and equations
• Inertial shearing waves
• Centrifugal instability and Rayleigh's criterion

Lecture 11: Vortices in Disks

• Introduction to vortices
• Enstrophy conservation
• Kida vortex solution and its stability

Lecture 12: Density Waves and Gravitational Instability

• Compressible disk dynamics and equations
• Density waves
• Axisymmetric gravitational instability
• Non-axisymmetric instability and `gravitoturbulence'

Lecture 13: Satellite-Disk Interactions I

• Test particle orbits in the presence of an embedded satellite
• Excitation of epicyclic oscillations

Lecture 14: Satellite-Disk Interactions II

• Angular momentum transfer between embedded satellites and their disks
• Gap opening
• Planet migration

Lecture 15: Magnetohydrodynamics in disks

• Equations of MHD
• Derivation of the axisymmetric MRI dispersion relation

Lecture 16: Magnetorotational Instability

• 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