Inverse Compton Radiation Rybicki & Lightman Chapter Review of the Compton Effect 2 – Review of Thomson Scattering 3 – Klein-Nishima Cross-section 4 – Inverse Compton Scattering 5 – The Compton Catastrophe 6 – Spectrum of IC; Kompaneets Eqn.; y parameter 7 – Synchrotron-Self-Compton 8 -- Sunyaev-Zel’dovich Effect

Inverse Compton E emitted ~ γ 2 E incident Higher-energy photon Low-energy photon Kyle & Skyler 2011

Thomson Scattering hν << mc 2 no electron recoil RESULTS (1) Eγ(before) = Eγ(after) No change in photon energy: “coherent” or “elastic” scattering (2) Thomson cross-section (3) For unpolarized incident photons, cross-section per solid angle:

so... Incident energy flux, erg/s/cm2 Fraction of incident flux radiated into θ,φ per solid angle radiated power per unit solid angle erg/s/ster

The Compton Effect Allow for electron recoil e- photon electron: Kinetic energy after collision = T momentum = p (1) Conservation of momentum (x,y components) Eliminate by squaring and adding (Eqn. A)

(2) Conservation of Energy: or square  (Eqn. B) Subtract (B) from (A): Relativistic momentum for electron after collision 

E is conserved under L.T. so Since So

Compton Scattering After scattering, the photon energy has decreased. Arthur Compton

Note: Can re-write the Compton formula as The scattered photon energy is shifted significantly as the incident photon energy (h ) becomes comparable to the rest energy mc 2 of the electron (=0.511 MeV).

Klein-Nishima Cross-Section Account for quantum mechanical effects: important when Upshot: Cross-section becomes smaller at large photon energies  Compton scattering becomes inefficient at high energies

Inverse Compton Scattering Now consider the case where the electron is moving at relativistic velocity, v We assume that in the rest frame of the electron, Thomson scattering holds, and then consider what happens in the lab frame. photon EfEf Lab Frame Electron Rest Frame

From the formula for Doppler shifts, we can write Now suppose the collision is elastic in the rest frame of the electron, so that For the moment, consider

Thenso the ratio This process converts a low-energy photon to a higher energy photon by a factor of A similar result holds if we instead assume Compton scattering in the rest frame.

NOTES: This energy gain by the photon is the opposite of the energy loss suffered during a Compton scattering event. thus, the process is called INVERSE COMPTON SCATTERING However, this name is slightly misleading because the process is not the inverse of Compton or Thomson scattering, but is rather ordinary Compton or Thomson scattering viewed in a frame in which the electron is highly relativistic.

One can think of the process as an interaction between and electron gas and a “photon gas” - if the electron gas is much “hotter” than the photon gas, energy flows from the electrons to the photons  Inverse Compton - if the photon gas is hotter, energy flows from the photons to the electrons  Compton Heating

Often γ >> 1, so this process can boost, say an IR photon, (with λ=20 microns) to the very hard X-rays or gamma rays  Compton upscattered IR photons may be the physical process which produces X-ray emission in quasars

Even though a particular electron loses a small fraction of its energy in a single encounter, the cumulative effect of many scatterings can be an important process of energy loss – comparable to the energy loss due to synchrotron radiation. Can show (pp in R&L) that magnetic field energy density photon energy density Ratio of power emitted by synch. to power emitted by Compton scattering

The Compton Catastrophe Let τ = the optical depth for Compton scattering In a situation where the electrons are relativistic, each photon increases its energy by γ 2 with each scattering event. Suppose there is a source of synchrotron photons with luminosity = L(sync) The luminosity in Compton scattered photons is L(comp) ~ where K = some constant

However, these Compton photons will themselves be scattered, carrying off another factor of of energy so You’d never see such a source, since the electrons would radiate away their energy instantaneously. This led people to conclude that one must have k<1, i.e. for any viable model for a synchrotron source.

However, two effects cause L(total) to converge: 1 – As the photons attain higher and higher energies, the average scattering will result in a smaller and smaller energy loss to the electron  eventually the photons will “cool” and the electrons will “heat” 2 – The relevant cross-section for scattering becomes the full Klein-Nishima formula which is less than the Thomson cross-section, so Compton scattering becomes less effective as the photon energy increases.

Inverse Compton Spectra So far, we’ve only considered overall energetics. Obviously, we also want to compute the spectrum of the radiation, to compare with observations. The spectrum will depend upon (1) the incident energy spectrum of the photons (2) the energy distribution of the electrons and in general, one has to solve the equation of radiative transfer, to see how many scattering events each photon suffers The approach is to (1) compute what happens when a photon of particular energy Eo scatters off an electron of particular energy (2) average over the photon and electron energy distributions Refn: Longair: High energy Astrophysics

What is the IC spectrum of a single electron? Straight-forward calculation shows that the spectrum of the IC radiation produced by scattering a photon of frequency 0 off an electron with energy  is Very peaked, even more peaked that synchrotron F(x) function

So the spectrum of IC (1) Scattering off a power-law distribution of electron energies, N(E) ~E -p results in a power law for photon energies with spectral index Identical to the spectral index for synchrotron emission. (2) Repeated scatterings off a non-power law distribution can also yield a power law spectrum. (3) Repeated scatterings off non-relativistic electrons can be calculated using the “Kompaneets Equation”

Repeated scatterings off non-relativistic electrons, each with a small energy change, can be calculated using the “Kompaneets Equation” Approach: Consider evolution of phase space density of photons proceeds with time n(x,p) d 3 xd 3 p The result is a differential equation for n which must be solved numerically

Kompaneets Equation f(p) = maxwellian distribution Photon scatters from frequency w’ to w Photon scatters from frequency w to w’ 1+n(w) term: n(w) increased probability due to stimulated emission

(4) Compton y-parameter When y>>1: Spectrum is altered  “saturated” comptonization When y < 1: spectrum is not changed much, “coherent scattering”

Synchrotron-Self-Compton Radiation When a synchrotron source is sufficiently compact that the synchrotron photons can Inverse Compton scatter off the relativistic electrons  emergent spectrum is called “synchrotron self-Compton” of SSC

Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays see Aharonian ApJ 614, 897 Synchrotron Self-Compton

Spectrum of MKN 501 (Seyfert Galaxy) Showing synchrotron peak and SSC peak

Galactic Center Hinton & Arahonian 2006 TeV Gamma rays

Sunyaev-Zel’dovich Effect CMB photons inverse Compton scatter off of hot, X-ray emitting intra-cluster medium in clusters of galaxies  temperature distortion in CMB  Cosmological parameters

Holzapfel et al. 1997, LaRoque et al. 2002

Assume a spherical cluster Need n e and T e

Observations X-rays (false color) –Gives us T e –Temperature profile allows us to calculate Λ ee –Allow us to model n e Radio (contours) –Gives us dθ – ΔT Bonamente et al. 2006

Observations Result:

Carlstrom, Holder, Reese 2002

Constraining Ω M Can constrain Ω M by determining the gas-mass fraction f gas can be measured directly from SZE – Measure d From BBN

Growth of Structure SZE is z independent Can use SZE to find clusters to high z This can be used as a probe of cluster abundance, which is a probe of large scale structure