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  • Click here for the 2023-24 Part II MT Lecture Timetable.

  • Click below for detailed information on each course.


Q. Introduction to Astrophysics (non-examinable)

  1. Basics: Scale and content of Universe: Sizes and masses. Magnitudes, HR diagram, Distance determination, The Sun as a typical star, overview of stellar lifecycles, Newtonian mechanics, orbits, tides, blackbodies, continuum radiation mechanisms, spectra and line radiation. [4] 

  2. Telescopes, Instruments, and Observational Techniques: E-M radiation, gamma-rays, X-rays, UV, visible, IR, mm, radio, transparency of the atmosphere, Major space-based and ground-based facilities, S/N calculations. [2] 

  3. White Dwarfs and Neutron Stars: WD origin, structure, neutron star origin, structure, discovery of pulsars, observed properties, evolution, beaming, magnetic fields, magnetic dipoles, pulse timing, the utility of pulsaras, gravitational waves. [3] 

  4. Close Binary Stars: Observations, visual binaries, spectroscopic binaries, eclipsing binaries, masses and radii, consequences of a supernova, equipotentials, mass transfer, accretion discs, magnetic stars, evolution in binary systems, cataclysmic variables, the variety of binary systems, stellar mass black holes. [3] 

  5. Supernovae and Hypernovae: Types, energetics, rates, light curves, spectra, pre-cursors, remnants, radio-active decay, Gamma Ray Bursts (GRBs), discovery, searches, observations, long and short duration GRBs, collapsar-hypernova model, merging of neutron stars and black holes, fireball-shock model, beaming. QSOs as a probe of the intergalactic medium. [3] 

  6. Active Galactic Nuclei: Discovery, observations, classification, energetics, standard model, host galaxies, reverberation mapping, jets, superluminal motion, unified models, QSO population evolution, black holes in non-AGN galaxies. [3] 

  7. Galaxies and Clusters of Galaxies: Structure and content, galaxies within them, hot X-ray gas, magnetic fields, dark matter, virial mass, tidal stripping, S-Z effect, cooling flows. [2] 

  8. Gravitational Lenses: Basic physics, Young diagrams, Einstein rings, critical surface mass density, strong lensing by galaxy clusters, caustics and critical lines, cluster masses, weak lensing, determining the hubble constant, micro-lensing, constraints on halo objects. [2] 

  9. Exoplanets: Discovery methods, statistics of known exoplanets, pulsar planets, hot jupiters, transits, planet formation, dust, proto-planetary discs, Hill radius, future observations, life. [2] 

Recommended Reading 

  • Shu, F.H., The Physical Universe, chaps. 5-10, University Science Books, California, (1982). 
  • Accretion Power in Astrophysics (Cambridge Astrophysics) Hardcover – 17 Jan. 2002 by Juhan Frank, Andrew King, Derek Raine. CUP 
  • † Carroll, B.W. & Ostlie, D.A. An Introduction to Modern Astrophysics (Addison-Wesley) 2017. 

Q. Stellar Dynamics and Structure of Galaxies

  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. . 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. 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)

Q. Structure and Evolution of Stars

  1. Basic Concepts and Observational Properties: Course overview; Mass, Temperature, Luminosity Gravity, composition, Age; Photometry and stellar colours; Spectra and spectral lines;

  2. Distance: Parallax, apparent and absolute magnitudes; Masses from binary stars;

  3. Temperature: Black-body radiation, Wien’s Law; The Hertzsprung-Russell Diagram and spectral classification

  4. Stellar Structure: Timescales; dynamical, thermal nuclear. Energy generation, thermonuclear reactions. Energy transport; opacity, radiative and convective transport. Equations of stellar structure. Hydrostatic equilibrium, Virial Theorem, Pressure. Stellar properties as a function of mass, homology. Degeneracy: Chandrasekar limit.

  5. Stellar Evolution and the Hertzsprung-Russell diagram: Pre-main sequence evolution, Hayashi and Henyey tracks. Post-main sequence evolution: massive stars, supernovae, neutron stars, black holes. Post-main sequence evolution: low-mass stars, planetary nebulae, white dwarfs, Type Ia supernovae. Initial mass function

  6. Observational Tests and Constraints: The mass-luminosity relationship. Stellar abundances. The most massive stars and stellar winds. Supernovae

Recommended Reading 

  • Prialnik, D., An Introduction to the Theory of Stellar Structure and Evolution, Cambridge University Press, 2000, 2nd Edition 2010
  • Lamers, H J G L M and Levesque E M, Understanding Stellar Evolution, IOP Publishing, 2017
  • Guidry, M, Stars and Stellar Processes, Cambridge University Press, 2019
  • LeBlanc, F, An Introduction to Stellar Astrophysics, Wiley, 2010

Q. Relativity (Shared with Physics)

  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

Q. Principles of Quantum Mechanics (Shared with Mathematics)

  1. Dirac formalism: Bra and ket notation, operators and observables, probability amplitudes, expectation values, complete commuting sets of operators, unitary operators. Schrodinger equation, wave functions in position and momentum space. [3]

  2. Time evolution operator: Schrodinger & Heisenberg pictures, Heisenberg equations of motion. [2]

  3. Harmonic oscillator: Analysis using annihilation, creation and number operators. Significance for normal modes in physical examples. [2]

  4. Multiparticle systems: Composite systems and tensor products, wave functions for multiparticle systems. Symmetry or antisymmetry of states for identical particles, Bose and Fermi statistics, Pauli exclusion principle. [3]

  5. Perturbation theory: Time-independent theory; second order without degeneracy, first order with degeneracy. [2]

  6. Angular momentum: Analysis of states ljm> from commutation relations. Addition of angular momenta, calculation of Clebsch-Gordan coefficients. Spin, Pauli matrices, singlet and triplet combinations for two spin half states. [4]

  7. Translations and rotations: Unitary operators corresponding to spatial translations, momenta as generators, conservation of momentum and translational invariance. Corresponding discussion for rotations. Reactions, parity, intrinsic parity. [3]

  8. Time-dependent perturbation theory: Interaction picture. First-order transition probability, the golden rule for transition rates. Application to atomic transitions, selection rules based on angular momentum and parity, absorption, stimulated and spontaneous emission of photons. [3]

  9. Quantum basics: Quantum data, qubits, no cloning theorem. Entanglement, pure and mixed states, density matrix. Classical determinism versus quantum probability, Bell inequality for singlet two-electron state, GHZ state. [2]

Recommended Reading

  • E. Merzbacher Quantum Mechanics, 3rd edition. Wiley 1998
  • B.H. Bransden and C.J. Joachain Quantum Mechanics, 2nd edition. Pearson
  • J. Binney and D. Skinner The Physics of Quantum Mechanics. Cappella Archive, 3rd edition
  • P.A.M. Dirac The Principles of Quantum Mechanics. Oxford University Press 1967, reprinted 2003
  • S. Weinberg Lectures on Quantum Mechanics. CUP, 2nd ed., 2015
  • J.J. Sakurai and J.J. Napolitano Modern Quantum Mechanics. CUP 2017

Further information about this course is available on the Department of Mathematics course pages.

Examples papers are available on the DAMTP Examples page.