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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.

    Online course because of the corona pandemics

    • 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, which should hopefully allow differentiated learning styles during online-class time.

    • I will provide you with my lecture notes in LaTeX. The manuscript will be finalized in the upcoming weeks and is available as a PDF file. If you find any typos or mistakes, please drop me a note.

    • There will be weekly reading assignments (see below) and small exercises that help you to prepare the class for the upcoming week.

    • The class will meet on zoom every Tuesday, 12:15 to 13:45 starting April 21, 2020. Please drop me an email to obtain the zoom code and password! 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. I have never done such an online class before, so I hope you can provide me with feedback on what works and what does not so that we can improve together and make this class a success for you!

    Weekly Assignments

    1. Send me an email to get the zoom access code for class on April 21, 2020.
    2. Assignment for lecture 2 - due April 28, 2020.
    3. Assignment for lecture 3 - due May 5, 2020.
    4. Assignment for lecture 4 - due May 12, 2020.
    5. Assignment for lecture 5 - due May 19, 2020.
    6. Assignment for lecture 6 - due May 26, 2020.
    7. Assignment for lecture 7 - due June 2, 2020.
    8. Assignment for lecture 8 - due June 9, 2020.
    9. Assignment for lecture 9 - due June 16, 2020.
    10. Assignment for lecture 10 - due June 23, 2020.
    11. Assignment for lecture 11 - due June 30, 2020.
    12. Assignment for lecture 12 - due July 7, 2020.
    13. Assignment for lecture 13 - due July 21, 2020.


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

    • Cluster physics informed by different observables:
      • Optical: galaxy properties and virial theorem
        • Transforming galaxy populations: ram pressure, tidal effects, dynamical friction
        • Weighting clusters (1): virial theorem

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

      • X-rays: gastrophysics at high-resolution
        • Weighting clusters (3): hydrostatic equilibrium masses (and biases)
        • Kinematics of shocks and cold fronts
        • Probing kinetic equilibrium with collisionless shocks
        • Width of cold fronts - magnetic draping

      • Sunyaev-Zel'dovich (SZ) effect: the thermal energy content
        • Thermal and kinetic SZ effect
        • Properties and SZ scaling relation, SZ power spectrum

      • Radio halos and relics: watching powerful shocks and plasma physics at work

    Lecture Plan:

    • Orders of magnitudes sheet
    • Lecture 1: Overview of clusters across wave bands: optical, X-rays, gravitational lensing
    • Lecture 2: Sunyaev-Zel'dovich effect, the growth of perturbations, power spectra
    • Lecture 3: Hierarchical structure formation, non-linear evolution, spherical collapse
    • Lecture 4: Cluster mass function, halo formation and density profiles
    • Lecture 5: Adiabatic processes and entropy, basic conservation equations
    • Lecture 6: Buoyancy instabilities, vorticity, turbulence
    • Lecture 7: Gravity waves, shocks and jump conditions
    • Lecture 8: Turbulent scaling laws, entropy generation by accretion, cluster scaling relations
    • Lecture 9: Radiative cooling and heating, feedback by supernovae and AGNs
    • Lecture 10: Heat conduction, thermal instability
    • Lecture 11: Non-thermal processes, magnetic fields
    • Lecture 12: Cosmic rays
    • Lecture 13: Optical: galaxy interactions and virial theorem
    • Lecture 14: X-ray cluster astrophysics and Sunyaev-Zel'dovich effect
    • Lecture 15: Radio relics and halos probing shocks and plasma physics, cluster cosmology
    • Tutorial: Historical context, superclusters, overview and clarifying questions

    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. Those include comprehension questions, order of magnitude problems and from time to time quantitative homework problems. We will discuss the solution to these questions and problems in class. In the end, there will be an oral exam of 20 to 30 min. 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.