Stellar Dynamics and Structure of Galaxies 

Michaelmas Term, 24 Lectures – Professor C. Clarke 

1. Introduction to gravitating systems in the Universe. [1] 

2. 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] 

3. 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 Vc(R); escape velocities Vesc(R). [2] 

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

5. Axisymmetric density distribution. General axisymmetric solution of ∇2Φ = 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] 

6. 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] 

7. 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] 

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

*The topics starred in the schedules will be lectured, but questions will not be set on them in examinations. 

Recommended Reading 

• Goldstein Classical Mechanics, Addison-Wesley (2nd edition 1980). 
• † Binney, J. & Tremaine, S.D. Galactic Dynamics, Princeton University Press (2008). 
• 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)