Radiation Transport Around Kerr Black Holes Jeremy David ...

168 CHAPTER 6. ELECTRON SCATTERING
the Compton y parameter for a finite medium of nonrelativistic electrons is
y = 4kT e
m e c 2Max(τ es, τes 2 ). (6.27)
For a low-energy soft photon source with multiple scattering events, the final spectrum
due to inverse-Compton scattering can be calculated using the Kompaneets equation,
which is a form of the Fokker-Plank diffusion equation (Kompaneets, 1957). For
hν kT e , the resulting spectrum takes the power-law form
I ν ∼ ν −α , (6.28)
with
α = 3 2 + √ 9
4 + 4 y . (6.29)
At energies above kT e , the electrons no longer efficiently transfer energy to the photons,
so the spectrum shows a cutoff for hν kT e :
I ν ∼ ν 3 exp(−hν/kT e ). (6.30)
With the assumption of purely elastic scattering, we cannot actually reproduce this
cutoff effect; all photons are scattered equally, and thus the ratio ε f /ε i is independent
of energy. Thus equation (6.26) would predict infinite energy boosts until hν ≫ m e c 2 .
In reality, higher energy photons tend to lose energy in scattering, due to the recoil
of the electron. This effect is relatively easy to calculate from conservation of energy
and momentum in the electron rest frame:
ε f =
ε i
1 + ε i
m ec 2 (1 − cosθ)
(6.31)
To accurately include this effect, we would have to keep track of the real “physical”
energy of each photon, instead of the fiducial redshift method that we currently use
to reconstruct the total spectrum afterwards. Ultimately, this is just a matter of
computational intensity and no real conceptual difficulty. To first-order, we can treat
the thermal photon source as a monochromatic emitter at E 0 = 3kT em , which should
give a reasonable approximation to the true solution.
Before we can actually produce such a spectrum, we must first define the electron
temperature and density profile through which the photons will scatter. Like
relativistic jets, there is still no real consensus in the literature as to what exactly
produces the electron corona surrounding the black hole and accretion disk. By measuring
the power-law part of the continuum spectrum [see e.g. Sunyaev & Truemper