Radiation Transport Around Kerr Black Holes Jeremy David ...

166 CHAPTER 6. ELECTRON SCATTERING
e θ ^
e~
y
v e −
p γ
θ1
v e −
p γ
e φ
e θ
e r
θ
θ0
e^
r
e~
x
p γ
eφ^
ez~
coordinate
basis
ZAMO
basis
electron
rest frame
Figure 6-4: Schematic picture of coordinate transformations from the coordinate
basis e µ in which the geodesic trajectories are integrated, to the ZAMO basis eˆµ in
which the electron density and temperature are defined, to the electron rest frame
basis e˜µ in which the electron scattering angle θ is given simply by the Thomson
cross section for unpolarized radiation.
generalized Lorentz boost is given by Misner, Thorne, & Wheeler (1973):
u µ = [γ, βn j ] (|n| = 1),
Λ t′
t = γ,
Λ t′
j = Λ j′
t = −βγn j ,
Λ j′
k
= Λ k′
j = (γ − 1)n j n k + δ jk . (6.21)
The photon momentum in the electron frame is thus given by p˜µ = Λ˜µˆµ pˆµ .
All that is left to do is calculate the scattering angle θ. Most portable random
number generators produce a random variable X uniformly distributed in the range
[0, 1], and from this we must produce a random variable θ with distribution according
to equation (6.7). The cross section can be re-written in terms of the normalized
probability distribution function
or defining z ≡ cos θ,
f(θ)dθ = 3 2 sin θ(1 + cos2 θ)dθ
f(z)dz = 3 8 (1 + z2 )dz.
(6.22a)
(6.22b)