Radiation Transport Around Kerr Black Holes Jeremy David ...

4.5. ELECTRON SCATTERING IN THE CORONA 111
Figure 4-6: Simulated power spectrum from a single hot spot light curve where the
emitted photons are scattered exactly once each by a uniform corona of electrons.
The simulated spectra are plotted as dots and asterices, while the analytic model
is a solid line. In (a), the mean free path to scattering is λ = 10M, while (b)
represents a much larger, low density corona with λ = 100M.
has a Gaussian distribution in azimuth with length ∆φ, the original X-ray light curve
will be convolved with a Gaussian window of characteristic time T = ∆φ/(πν φ ). A
Gaussian window in time gives a Gaussian profile in frequency space:
w(t) = exp
( −t
2
2T 2 )
⇔ W(ν) = √ 2π T T f
exp
( −ν
2
2∆ν 2 )
, (4.20)
where again the characteristic width is given by ∆ν = 1/(2πT). The exponential
damping of the Gaussian W(ν) is stronger than the Lorentzian factor [eqn. (4.18)] at
higher frequencies, but both effects (coronal scattering and hot spot stretching) are
probably important in explaining the lack of significant power in the harmonics above
∼ 500 Hz in the RXTE observations. From the central limit theorem, in the limit
of many scattering events, the time delay distribution should also approach that of a
Gaussian, further damping out the higher frequency power. Regardless of the precise
shape of the convolution window in time, this simple analytic model shows how the
scattering time scale can be understood as another expression of the causality limits
on the size of the emission region. For an optically thick corona with length scale
R scat , all frequency modes above ν ∼ c/R scat should be damped out significantly.