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  • Click here for the Part II Michaelmas 2024 Lecture Timetable.

  • Click below for detailed information on each course.

 

Q. Introduction to Astrophysics (non-examinable)

  1. Basics: Stellar magnitudes, parallax, use of distance modulus and absolute magnitude, period luminosity relationship for Cepheid variables. Hubble's law, using HR diagrams to estimate distances. Stellar collapse time, Kelvin Helmholtz timescales, the virial theorem, nuclear fusion, stellar evolution, endpoints of stellar evolution. Stars on the main sequence. [2] 

  2. Binary stars, Kepler's law, tides, Planck function, Stefan Boltzmann law, electromagnetic radiation, absorption cross section and optical depth. Continuum radiation types, Compton and inverse Compton scattering. Bremsstrahlung, synchrotron radiation, synchrotron self-absorption, atomic spectra, 21 cm radiation, molecular spectra, the Doppler effect, interstellar absorption, line widths and profiles, luminosity classes and P Cygni profiles. [2] 

  3. Telescopes, instruments and their limitations, adaptive optics, interferometers and signal-to-noise calculations. General review of the current range of instruments astronomers can access both on the ground and in space. Optical and radio telescopes, ultrahigh energy cosmic ray detection. Astronomical instrumentation: detectors including CCD and CMOS detectors. Quantum dot technology and quanta image sensors. Multi-object spectroscopy including fibre fed instruments. Diffraction limit and the need for adaptive optics, earth rotation synthesis and the measurement and importance of signal-to-noise in astronomy. [2] 

  4. White dwarfs, neutron stars and pulsars. Discovery of white dwarfs, degeneracy pressure, mass radius relation for white dwarfs, as they cool, the approach to becoming a black hole: pulsars and their discovery. Pulsar properties and understanding their magnetic field structure. Llife and death of pulsars, pulsars as magnetic dipoles. Using pulsars as precision clocks, gravitational wave detection. Binary stars, supernovae and accretion physics. Measuring properties of binary stars, different kinds of binaries, supernovae in binary systems, gravitational potential in binary systems, Roche lobes, accretion and the Eddington limit. Accretion onto magnetic stars, accretion with significant angular momentum, cataclysmic variables, x-ray boosters and stellar mass black holes. [4] 

  5. Supernovae, gamma ray bursts, SNR evolution. Supernovae types, nuclear burning, core collapse, neutrino detection, importance of P Cygni profiles, light curves, evolution of supernovae remnants. Discovery of gamma ray bursts, GRB characteristics, evidence for their great distance, the data from BATSE. SWIFT data, different kinds of gamma-ray bursts, the fireball shock model, evidence for nearby SNRs in the past. [4] 

  6.  Active Galactic Nuclei. Early discoveries, Quasar discovery, importance of high redshifts, Quasar spectra, AGN energetics, mass accretion rate, host galaxy properties, constraints on Quasar models. Reverberation mapping , beamed radiation in AGN jets, the black holes at the centre of M 87 and possibly the Milky Way. Super luminal expansion, unified model of quasars and their energetics. Models of Quasar evolution, evidence for black holes in relatively normal galaxies, optical and radio rotation curves, dynamics of stars in the centre of the Milky Way, the black hole in our galaxy. [2] 

  7. Galaxies and clusters of galaxies. Galaxy types and general characteristics. Galaxy populations, interactions and collisions. Rich clusters of galaxies and their properties, interaction between galaxies and intergalactic matter and magnetic fields. Hot gas in the centres of clusters, the importance of multi-wavelength imaging , rich clusters of galaxies and their use as cosmological telescopes. Justification for the belief in dark matter, cluster masses from x-ray observations. The Sunyaev-Zeldovich effect. Cooling flows within galaxies and clusters. Importance of weak shocks. Galaxy correlations, including Tully-Fisher relation. [2] 

  8. Gravitational Lensing. General principles of gravitational lensing, Einstein rings, caustics and critical lines for elliptical mass distribution.  Rich clusters of galaxies and their lensing, measuring the Hubble constant, microlensing of exoplanets. [2] 

  9. Exoplanets. Detection of exoplanets, statistics of those found. Information we can extract from a variety of different eclipsing events. Importance of dust in forming exoplanets. Observational support evidence, forces on dust grains. The Hill radius. The difficulty in detecting earth like planets. Plans for future detection programs. [2] 

  10. Magnetic fields in astronomy. Magnetic fields in our Sun, space weather and dangers from it. Measuring magnetic fields, synchrotron radiation, Faraday rotation, Zeeman splitting, equi partition considerations, celebration cosmic rays and the creation of magnetic fields. Extra/intra-galactic magnetic fields. [1] 

  11. Observational Cosmology. Big bang observations, Hubble's law, nuclide abundances, cosmic microwave background, redshifts surveys, CMB experiments, radio surveys, infrared results, dark matter properties, cosmic neutrinos, gravitational waves and background. [2] 

Recommended Reading 

  • 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 

  • † 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)

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.