* *

**Orbits in a given potential**. Particle orbit in Newtonian gravity; energy, angular momentum. Radial force law - general orbit is in a plane; equations of motion in cylindrical polars. Inverse square law; bound and unbound orbits, Kepler's laws; escape velocity; binary stars; reduced mass. General orbit under radial force law; radial and azimuthal periods; precession. [4]

**Derivation of potential from density distribution**. Poisson's equation. Description of structure of galaxies. Gravitational potential for spherical systems: homogeneous sphere, modified Hubble profile, power law. Circular orbits; rotation law V_{c}(R); escape velocities V_{esc}(R). [2]

**Nearly circular orbits**. Radial perturbations; epicyclic frequency; stability; apsidal precession. Application to pseudo-black hole potential Φ = -*GM/(r-r*_{s}). Vertical perturbations in axisymmetric potential; vertical oscillation frequency; nodal precession. [2]

**Axisymmetric density distribution**. General axisymmetric solution of ∇^{2}Φ = 0. Potential due to ring of matter; series solution; 18-year eclipse cycle. Potential due to thin disc; rotation curves of Mestel's disc; exponential disc. Rotation curve of the galaxy; Oort's constants. Rotation curves of spiral galaxies; need for dark matter. [5]

**Collisionless systems**. Relaxation time. Estimates for stellar and galaxy clusters. Gravitational drag. The stellar distribution function; collisionless Boltzmann equation. The Jeans equations as moments of the Boltzmann equation. Analogy with fluid equations. Application to mass in the solar neighbourhood (Oort limit). [4]

**Jeans Theorem**. Application to simple systems in which the distribution function depends only on energy. Useful approximate galactic potentials; polytrope, Plummer's model, isothermal sphere. [3]

**Globular cluster evolution**. Models of globular clusters. King models. *Models with anisotropic velocity distributions.* Observational tests. [3]

## Books

Goldstein *Classical Mechanics*, Addison-Wesley (2nd edition 1980).

† Binney, J. & Tremaine, S.D. *Galactic Dynamics*, Princeton University Press (1987).

Landau & Lifshitz *Mechanics*, Pergamon (3rd edition 1976, reprinted 1994).

† Binney, J. & Merrifield, M. *Galactic Astronomy*, Princeton University Press (1998).

Sparke, L.D. & Gallagher, J.S. *Galaxies in the Universe - An Introduction* CUP (2000) (ISBN 0-521-59740-4)