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    The Physics of Galaxy Clusters

    Clusters of galaxies are the largest and most recently gravitationally-collapsed objects in the Universe. Hence they provide us the opportunity to study an "ecosystem" - a volume that is a high-density microcosm of the rest of the Universe. Clusters are excellent laboratories for studying the rich astrophysics of baryons and dark matter. At the same time, they are extremely rare events, forming at sites of constructive interference of long waves in the primordial density fluctuations. Hence, they are very sensitive tracers of the growth of structure in the universe and the cosmological parameters governing it, which puts them into focus of constraining the properties of Dark Energy or to test whether our understanding of gravity is complete.

    These lectures will explain how clusters form and grow. We will encounter the rich and interesting astrophysics that governs the physics of dark matter and baryons in clusters. We will see how we can take advantage of these physical processes to observe clusters and deepen our understanding of the underlying fundamental physics. To this end we will frequently use the powerful technique of order of magnitude estimates, a very useful tool for contemporary research in astrophysics. The lectures aim at students who

    • wish to extend and deepen their understanding of theoretical physics;
    • are interested in astronomy and astrophysics; or
    • (intend to) carry out a masters thesis or Ph.D. dissertation on an astronomical or astrophysical subject.

    Lecture course

    • In this class, we will follow the instructional strategy of a "flipped classroom" (also called "inverted classroom"), which is a type of blended learning focused on student engagement and active learning.

    • The lecture notes are available as a PDF file. If you find any typos or mistakes, please drop me a note.

    • There will be weekly reading assignments, questions on the content (see below) and exercises that help you to prepare the class for the upcoming week. Please upload your answers to this moodle page (reading assignment questions weekly, exercises bi-weekly).

    • The class meets in person every Wednesday, 14:15 - 15:45 in room (starting October 18, 2023) and the exercises take place every second Thursday, 16:15 - 17:45 in room (starting October 19, 2023). During the class, we will discuss the topic of the week, do some order of magnitude problems, and I will show some more lengthy derivations. Ideally, I would appreciate if you brought a lot of input so that we can have an active discussion with many questions on our topic of galaxy clusters and theoretical astrophysics.

    Weekly Assignments

    1. Assignment for lecture 2 - due Oct 24, 202.
    2. Assignment for lecture 3 - due Oct 31, 2023.
    3. Assignment for lecture 4 - due Nov 7, 2023.
    4. Assignment for lecture 5 - due Nov 14, 2023.
    5. Assignment for lecture 6 - due Nov 21, 2023.
    6. Assignment for lecture 7 - due Nov 28, 2023.
    7. Assignment for lecture 8 - due Dec 5, 2023.
    8. Assignment for lecture 9 - due Dec 12, 2023.
    9. Assignment for lecture 10 - due Dec 19, 2023.
    10. Assignment for lecture 11 - due Jan 9, 2024.
    11. Assignment for lecture 12 - due Jan 16, 2024
    12. Assignment for lecture 13 - due Jan 23, 2024.
    13. Assignment for lecture 14 - due Jan 30, 2024.
    14. Assignment for lecture 15 - due Feb 6, 2024.


    1. Exercise 1 - due Oct 31, 2023.
    2. Exercise 2 - due Nov 14, 2023.
    3. Exercise 3 - due Nov 28, 2023.
    4. Exercise 4 - due Dec 12, 2023.
    5. Exercise 5 - due Jan 9, 2024.
    6. Exercise 6 - due Jan 23, 2024.
    7. Exercise 7 - due Feb 6, 2024.


    • Overview and background:
      • What is a galaxy cluster? Insights from observations at various wavelengths
      • Why are clusters interesting?
        Tools for cosmology and laboratories for high-energy and plasma astrophysics

    • Evolution of the dark component:
      • When do clusters form? ⇒ statistics and power spectra
      • Where do cluster form? ⇒ non-linear evolution
      • How do clusters form? ⇒ spherical collapse model
      • How many clusters are there? ⇒ Press-Schechter mass function
      • What is the structure of a cluster? ⇒ halo density profiles, virial masses

    • Evolution of the baryonic component:
      • Non-radiative physics
        • Adiabatic Processes and Entropy
        • Basic Conservation Equations
        • Buoyancy Instabilities
        • Vorticity and Turbulence
        • Shocks and jump conditions
        • Entropy generation by accretion and hierarchical merging
        • Scaling relations (ideal and real)

      • Radiative physics
        • Radiative cooling and star formation
        • Energy feedback (supernovae, active galactic nuclei)
        • Transport processes of gas:
          conduction, thermal stability (without and with magnetic fields)

      • Non-thermal processes
        • Non-thermal radio emission
        • Origin and transport of magnetic fields, magneto-hydrodynamic turbulence
        • Acceleration of cosmic rays (to first and second order), transport equation

    • Cluster astrophysics and cosmology:
      • Optical: galaxy properties and virial theorem
        • Transforming galaxy populations: tidal effects, dynamical friction, ram pressure
        • Weighting clusters (1): virial theorem

      • Gravitational lensing with clusters
        • Deflection angle, lens equation
        • Einstein radius, lensing potential
        • Weighting clusters (2): strong and weak cluster lensing

      • X-ray cluster astrophysics at high-resolution
        • Weighting clusters (3): hydrostatic equilibrium masses (and biases)
        • Cluster population and evolution
        • Intracluster medium turbulence
        • Merger shocks and electron equilibration
        • Magnetic draping and cold fronts

      • Sunyaev-Zel'dovich (SZ) effect: the thermal energy content
        • Thermal, kinetic and relativistic SZ effect
        • SZ scaling relation and power spectrum
        • SZ effect of AGN bubbles and shocks

      • Radio halos and relics: watching shocks and plasma physics at work
        • Cluster shocks
        • Radio halos and relics
        • Radio galaxies and jellyfish tails

      • Cluster cosmology
        • Cosmological parameter estimates
        • How clusters probe cosmology
        • Cluster probe the nature of dark matter

    Lecture Plan:

    Credit Points:

    Students who wish to obtain credit points are invited to prepare the lectures by reading and working through the weekly assigments posted on this web site and doing the exercises. Those include comprehension questions, order of magnitude problems and quantitative homework problems. You need to achieve 50% of the points to be admitted to an oral exam of 20 to 30 min at the end of the lecture course. A successfull participation of the lectures is rewarded with two credit points.


    Unfortunately, there does not exist a perfect book on this topic. Hence I decided to provide lecture notes in LaTeX form that I will finalize throughout the course. Here is a selection of books that I found quite useful if you want to extend your knowledge about processs that we encounter during the lectures:

    • Overview and Review Article:
    • Background on Cosmology:
      • Bartelmann, M.: Lectures on Cosmology
      • Peacock, J.: Cosmological physics, Cambridge University Press.
      • Peebles, P.J.E.: Principles of physical cosmology, Princeton University Press.
      • Padmanabhan, T.: Structure formation in the universe, Cambridge University Press.

    • Theoretical Physics and Astrophysics:
      • Thorne, K.S. & Blandford R.D.: Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, Caltech lecture notes for download, textbook available from Princeton University Press.
      • Landau L.D. & Lifshitz E.M.: Course of Theoretical Physics, Volumes 1, 2, 5, 6, 8, Butterworth-Heinemann.
      • Shu, F.H.: The Physics of Astrophysics: Gas dynamics, University Science Books.
      • Bartelmann, M.: Theoretical Astrophysics: An Introduction, Wiley-VCH.