Elementary processes in astrophysics (NAST024)

### Compton scattering

Main study material for this topic is the textbook by Vladimir Karas, in particular, Sections 10.3 through 10.5. You may also check my hand-notes for some derivations.

Please, concentrate on the following issues:

• coordinate system: the classical Compton scattering formula is derived in the frame in which the electron is at rest before the scattering. Only in that particular frame, the photon always looses its energy, while the electron gains it. The inverse Compton scattering is not necessarily a completely reverse process (i.e., the electron does not need to be at rest after the scattering), but in general it refers to an event during which the photon gains energy in the oberver's rest frame.
• please, don't get confused by the notation; quantities ${\gamma }_{\mathsf{x}}$ refer to the electron, while ${e}_{\mathsf{x}}$ are energies of the photon
• you should be able to derive the classical scattering formula, i.e., energy of the photon after the scattering, ${e}_{\mathsf{f}}$, as a function of its initial energy and the scattering angle in the rest frame of the electron before the scattering
• derivation of the energy of photon within the scope of the inverse Compton scattering basically means transformation of the classical formula to arbitrary "observer's" coordinate system. In practise, that system is not completely arbitrary, which comes from the assumptions that are used throughout the derivation: the electron moves at relativistic speed in the observer's frame, ${\gamma }_{\mathsf{i}}^{\mathsf{L}}\gg 1$, while the energy of photon before the scattering is much less than the rest-mass energy of electron, ${e}_{\mathsf{i}}^{\mathsf{L}}\phantom{\rule{0.3em}{0ex}}{\gamma }_{\mathsf{i}}^{\mathsf{L}}\ll 1$
• Formula for energy of the scattered photon in the observer's frame in the form of equation (10.30) in the textbook is obtained with some further assumptions. The most important one perhaps is that while the radiation field is roughly isotropic in the observer's frame, in the rest frame of the electron (either before or after scattering), most of the photons are coming from a narrow angle from the direction opposite to the motion of the electron (as a consequence of aberation)
• you are supposed to understand the idea of Zeldovich-Sunyaev effect, i.e. what are the incoming photons, what is the scattering medium and for which astrophysical measurements this effect can be used for. Note, that the key idea of the inverse Compton scattering holds also for ${\gamma }_{\mathsf{i}}^{\mathsf{L}}<1$, which is the case here (the intergalactic medium is kinematically hot, but the electrons do not move with relativistic velocities)

While key aspect of the topic is to understand the difference between the Compton and inverse Compton scattering, some other sub-topics are considerd less important during the exam, but may be of your professional interest

• I will not ask you to write the Klein-Nishina formula for the Thompson scattering cross section, though it would be good if you are able to sketch roughly its shape
• the formula for net energy transfer due to inverse Compton scattering is usually not a subject of examination
• detailed derivation of the Zeldovich-Sunyaev effect presented in the textbook is actually rather straightforward and you are encouraged to go through it. On the other hand, it relies on the Kompaneets' equation, derivation of which we will skip (interested students may search for it, e.g., in the textbook "Theoretical Astrophysics" by T. Padmanabhan)