Radiation Transport Around Kerr Black Holes Jeremy David ...

6.1. PHYSICS OF SCATTERING 165
colored according to their energy-at-infinity E ∞ = −p t , either blue- or redshifted with
respect to their energy in the emitter frame E 0 . For the black hole with a/M = 0.99,
the ISCO is located inside the ergosphere, so all photons are forced to move forward
in φ, and some are even created with negative energies (p t > 0). As mentioned above,
the relativistic beaming is done automatically by the Lorentz boost from the emitter
to the coordinate (or ZAMO) frame, so the blue photons are clearly bunched more
tightly together, as required by the invariance of I ν /ν 3 .
As in Chapter 2, the photon’s position and momentum are tabulated at each step
along its path. However, now we have to check at each interval to see if the photon
scatters off an electron. Conveniently, the Runge-Kutta algorithm takes shorter steps
as smaller r, where the electron density tends to be highest, so we can reliably use
the differential formula for the optical depth to electron scattering:
dτ es = κ es ρds. (6.19)
The density ρ is defined in the ZAMO frame and the opacity κ es is given by the
classical cross section derived above in equation (6.7)
κ es = σ T
m p
= 8π 3
r 2 0
m p
= 0.4 cm 2 /g. (6.20)
The proper distance ds in equation (6.19) is calculated from the path segment dxˆµ i as
in equation (2.67). For relatively small steps, the probability of scattering after each
step is given by dτ es 0.1.
If the photon does in fact experience a scattering event, we first transform into
the ZAMO basis, where the electron temperature is defined. Given the electron
temperature, we assume an isotropic distribution of velocities as defined in equation
(6.11), and pick an electron 4-velocity with random direction in that basis. Next we
must transform to the electron rest frame, in which the photon scatters according to
the Thomson cross section from equation (6.7), reducing a difficult problem in curved
spacetime to a simple classical problem with a single variable—the scattering angle
θ. This set of transformations from coordinate basis to ZAMO basis to electron rest
frame is shown schematically in Figure 6-4 (again the ZAMO basis is denoted by ˆµ
subscripts, and the electron frame by ˜µ, not to be confused with the emitter frame
defined earlier).
The transformation from the ZAMO basis to the electron frame is defined by a
Lorentz boost in the direction of the electron 4-velocity uˆµ → e˜x . Since the scattering
probability is symmetric around this axis, the rotational degree of freedom that fixes
the other spatial axes eỹ and e˜z is completely arbitrary. One convenient form of the