# Part II Stellar Dynamics and Structure of Galaxies

## Michaelmas Term, 24 Lectures – Prof M Haehnelt

[The topics starred in the Schedules will be lectured, but questions will not be set on them in examinations.]

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. 

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). 

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. 

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. 

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). 

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. 

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

## Resources

The lecture notes and the example sheets can be found  here.

## Books

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)

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