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Dynamics of Astrophysical Discs
During Lent term 2013, I will be teaching the Mathematical Tripos Part III course on the dynamics of astrophysical discs. Topics covered include: basic orbital dynamics, particle and fluid discs, accretion disc evolution, vertical structure of discs, local disc models, instabilities in discs, and satellite-disc interactions.

Lecture 1 (17/01/2013): Introduction (pdf)

Lecture 2 (22/01/2013): Orbital Dynamics

• Equations of motion, circular orbits
• Keplerian orbits, small eccentricity approximation
• Perturbed orbits: epicyclic oscillations
• Precession

Lecture 3 (24/01/2013): Eccentricity evolution in particle discs I

• Equations in fixed frame
• Perturbations due to binary companion
• Eccentricity evolution

Lecture 4 (29/01/2013): Eccentricity evolution in particle discs II

• Eccentricity evolution
• Simulation (mpg)

Lecture 5 (31/01/2013): Viscous Accretion Discs I

• Mechanics of accretion
• Viscosity as proxy for turbulent flow
• Derivation surface density diffusion equation

Lecture 6 (05/02/2013): Viscous Accretion Discs II

• Boundary conditions
• Green's function solution (mpg, pdf)
• Background on the use of Green's functions (pdf)

Lecture 7 (07/02/2013): Vertical structure I

• Hydrostatic equilibrium
• Scaling and importance of H/r
• Energy equation

Lecture 8 (12/02/2013): Vertical structure II

• Radiative disc models
• Opacity laws
• Algebraic approximate solution

Lecture 9 (14/02/2013): Local Disc Models

• Local approximation, expansion potential
• Orbital motion in shearing sheet
• Boundary conditions (shear-periodic)(pdf)

Lecture 10 (19/02/2013): Shearing waves

• Shearing wave example 1 (mpg), example 2 (mpg)
• Amplitude evolution equation
• Stability (axisymmetric, non-axisymmetric and Couette)

Lecture 11 (21/02/2013): Vortices in discs I

• Introduction (pdf)
• Kida solution and stability
• Vortex rotation (mpg), stretching (mpg), oscillation anticyclonic (mpg), oscillation cyclonic (mpg), unstable vortex (mpg)

Lecture 12 (26/02/2013): Vortices in discs II

• Particle dynamics in vortex core: linear system with drag
• Limits of weak and strong drag, general solution
• Condition for decaying solutions (pdf)

Lecture 13 (28/02/2013): Gravitational instability

• Basic equations, perturbing potential
• Linear perturbations, shearing waves
• Condition for axisymmetric instability, Toomre Q
• Non-axisymmetric instability (pdf, mpg, mpg)

Lecture 14 (05/03/2013): Satellite-disc interaction I

• Test particle orbits with perturbing potential
• Change in "orbital elements" (mpg)
• Impulse approximation

Lecture 15 (07/03/2013): Satellite-disc interaction II

• Angular momentum flow (pdf)
• Gap opening (mpg)
• Satellite migration, horseshoe orbits (mpg)

Lecture 16 (12/03/2013): Magneto-rotational instability

• Equations of MHD
• Shearing wave perturbations, amplitude equations
• MRI dispersion relation, instability criterion
• Global MRI simulations (disc, satellite)

Example Sheet 1: pdf (Example class 20/02/2013, 2-3pm, MR5)
Example Sheet 2: pdf (Example class 12/03/2013, 2-3pm, MR4)
Example Sheet 3: pdf (Example class 01/05/2013, 2-4pm, MR14)