Michaelmas Term Courses

1. Introduction to gravitating systems in the Universe. [1]
   Scope and motivation of stellar and galactic dynamics. Galaxies as composite systems of stars, gas, and dark matter; why dynamics is needed to recover the mass distribution and reconstruct formation histories. Collisionless versus collisional components; stellar filling factor and why galaxies can often be treated as collisionless systems whereas gas cannot. Observational overview of simple hot stellar systems, including globular clusters, open clusters, and galaxy clusters. Surface brightness, characteristic radii, velocity dispersion, crossing time, mass-to-light ratio, and the first dynamical evidence for dark matter from galaxy clusters. Introduction to the main tools of galactic dynamics: gravitational potential, density, orbits, observables, and phase-space distribution functions.

2. Orbits in a given potential [4]
   Newtonian gravity and gravitational potential; relation between force, potential, and density. Motion in a central potential; conservation of energy and angular momentum; proof that motion under a radial force law is planar. Equations of motion in plane polar coordinates and the orbit equation. Keplerian motion under the inverse-square law: ellipses, conic sections, bound and unbound trajectories, turning points, and the interpretation of orbital energy. Effective potential, circular orbits, and escape velocity. Kepler's laws and applications to the Galactic Centre and compact objects. Two-body and binary-star dynamics, including centre-of-mass motion, reduced mass, orbital angular momentum, and selected mechanisms of orbital evolution such as mass loss and gravitational radiation. General radial force laws, radial and azimuthal periods, apsidal precession, and worked examples beyond the Kepler problem.

3. Derivation of potential from density distribution [2]
   Derivation and use of Poisson's equation for self-gravitating systems. Integral form of the potential, Green-function approach, and the role of the divergence theorem and Gauss's theorem. From observed structure to dynamical model: deprojection ideas, galaxy morphology as context, and the relation between projected and three-dimensional density profiles. Potentials and forces for spherical mass distributions, including the homogeneous sphere, modified Hubble profile, and power-law density models. Circular-speed and escape-speed curves for spherical systems, and the connection between density slope and orbital behaviour.

4. Circular and nearly circular orbits [2]
   Circular motion in a fixed potential and the relation between angular frequency and the local gradient of the potential. Stability of circular orbits from radial perturbations and the effective potential. Small-amplitude radial oscillations, epicyclic frequency, and the epicyclic approximation. Apsidal precession and the dependence of precession rate on the form of the potential. Extension to more general axisymmetric potentials: circular motion in the mid-plane, vertical perturbations about the plane, vertical oscillation frequency, and the interpretation of near-circular disk orbits in galaxies.

5. Axisymmetric density distributions and disk dynamics [5]
   Gravitational potentials generated by flattened and axisymmetric matter distributions. Why spherical systems are mathematically simpler than disks; Poisson's equation in axisymmetry; Legendre-series solutions and the potential of a ring. Thin-disk potentials and the use of Bessel functions in solving Laplace's and Poisson's equations. Circular velocity in thin disks and explicit examples including Mestel, exponential, and Kuzmin disks. Rotation curves of disk galaxies and of the Milky Way. Oort constants from local stellar kinematics, their relation to shear, vorticity, and epicyclic motion, and modern measurements using Gaia data. Observational rotation curves from H I 21 cm emission, the flat outer rotation curves of spirals, the baryonic Tully-Fisher relation, and the dynamical case for dark matter halos.

6. Collisionless systems, Jeans equations and virial methods [4]
   Meaning and domain of validity of the collisionless approximation. Two-body relaxation, crossing and relaxation times, estimates for globular clusters and galaxies, and the transition between collisional and collisionless behaviour. Gravitational drag, focusing, and dynamical friction. The distribution function in phase space and the collisionless Boltzmann equation, including its form in cylindrical polar coordinates and its relation to continuity of phase-space flow. Jeans equations as velocity moments of the Boltzmann equation and their analogy with fluid equations. Applications to isotropic, spherical, and axisymmetric systems, including asymmetric drift, the local mass density in the Solar neighbourhood (the Oort limit), spherical mass estimators, and the mass-anisotropy-density degeneracy. Tensor and scalar virial theorems and their use in estimating masses of stellar systems.

7. Jeans theorem and distribution functions [3]​​​​​​​
   Integrals of motion and the construction of steady-state stellar systems. Jeans theorem and its implications for equilibrium distribution functions. Building self-consistent models from a chosen potential-density pair. Eddington inversion for isotropic spherical systems and conditions for physical distribution functions. Worked examples of distribution functions in simple potentials, including the harmonic oscillator and the homogeneous sphere. Power-law and double-power-law density models, polytropic distribution functions, the Plummer model, and singular and non-singular isothermal spheres. Interpretation of these models in terms of density structure, velocity dispersion, and the connection between theoretical distribution functions and observable stellar systems.

8. Globular clusters: observations, models, and evolution [3]
   Observed properties of globular clusters: ages from isochrones, age-metallicity trends, Galactic distribution, masses, tidal radii, and their contribution to early star formation and galaxy assembly. Tidal truncation, stripping, and tidal tails as signatures of interaction with the host galaxy. King models as lowered isothermal distribution functions; solution of the associated Poisson equation; core, scale, and tidal radii; projected density and velocity-dispersion profiles. Dynamical evolution of globular clusters through relaxation, evaporation and escape of stars, core collapse, binary heating, and mass segregation. Systems with anisotropic velocity distributions and extensions beyond isotropic King models, including Michie-type models. Use of globular clusters as laboratories for stellar dynamics and as tracers of galaxy formation.

Recommended Reading 

• † Binney, J. & Tremaine, S.D. Galactic Dynamics, Princeton University Press (2008).
• Landau & Lifshitz Mechanics, Pergamon (3rd edition 1976, reprinted 1994).
• Goldstein Classical Mechanics, Addison-Wesley (2nd edition 1980).
• † 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)

1. Introduction: problems with Newtonian gravity, the equivalence principle, gravity as spacetime curvature.

2. Foundations of special relativity: Inertial frames, spacetime geometry, Lorentz transformations, length contraction and time dilation, Minkowski line element, particle worldlines and proper time, Doppler effect, addition of velocities, acceleration and event horizons in special relativity.

3. Manifolds, coordinates and tensors: Concept of a manifold, curves and surfaces, coordinate transformations, Riemannian geometry, intrinsic and extrinsic geometry, the metric tensor, lengths and volumes, local Cartesian coordinates, pseudo- Riemannian geometry, scalar fields, vectors and dual vectors, tensor fields, tensor algebra, covariant differentiation and the metric connection, intrinsic derivative, parallel transport, geodesics.

4. Minkowski space and particle dynamics: Cartesian inertial coordinates, Lorentz transformations, 4-tensors and inertial bases, 4-vectors and the lightcone, 4-velocity, 4-acceleration, 4-momentum of massive and massless particles, relativistic mechanics, arbitrary coordinate systems.

5. Electromagnetism: Lorentz force, the current 4-vector, the electromagnetic field tensor and field equations, the electromagnetic 4-potential.

6. Spacetime curvature: Locally-inertial coordinates, weak gravitational fields, intrinsic curvature, the curvature tensor, the Ricci tensor, parallel transport, geodesic deviation and tidal effects, physical laws in curved spacetime.

7. Gravitational field equations: the energy-momentum tensor, perfect fluids, relativistic fluid dynamics, the Einstein equations, the weak field limit, the cosmological constant.

8. Schwarzschild spacetime: static isotropic metrics, solution of empty-space field equations, Birkhoff’s theorem, gravitational redshift, trajectories of massive particles and photons. Singularities, radially-infalling particles, event horizons, Eddington- Finkelstein coordinates, gravitational collapse, tidal forces.

9. Experimental tests of general relativity: precession of planetary orbits, the bending of light.

10. Friedmann–Robertson–Walker spacetime: the cosmological principle, comoving coordinates, the maximally-symmetric 3-space, the FRW metric, geodesics, cosmological redshift, the cosmological field equations.

11. Linearised gravity and gravitational waves: weak field metric, linearised field equations, Lorenz gauge, wave solutions of linearised field equations.*

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

Recommended Reading

• M. P. Hobson, G. P. Efstathiou and A. N. Lasenby General Relativity: An Introduction for Physicists, CUP 2006