Radiation Transport Around Kerr Black Holes Jeremy David ...

200APPENDIX B. SUMMING PERIODIC FUNCTIONS WITH RANDOM PHASES
narrow Lorentzian
Ĩ 2 (ν) ≈ 4πN spot A 2 T l
T f
1
1 + 48π 2 T 2
l (ν − ν 0) 2.
(B.15)
As with the boxcar window, the exponential lifetime distribution has the effect
of narrowing the peak of the net power spectrum compared with that of a single
Gaussian segment of the light curve with length T l . These results are in fact easily
generalized. For any set of localized, self-similar window functions w(t, T) = w(t/T),
the corresponding power spectra W 2 (ν; T) can be approximated near ν = 0 as a
Lorentzian:
W 2 (ν; T) ≈ T 2 1
1 + β 2 T 2 ν2, (B.16)
T 2 f
with β a dimensionless constant over the set of w(t, T). The characteristic width of
W 2 (ν, T) is thus defined as 1/(βT). Integrating over the lifetime distribution dN(T)
from equation (B.9), the net power function is given by
Ĩ 2 1
(ν) ≈ Ĩ2 (ν 0 )
1 + 12β 2 Tl 2(ν
− ν 0) 2.
(B.17)
We see now that the general effect of an exponential distribution of sampling lifetimes
is to decrease the peak width, and thus increase the coherency, by a factor of √ 12.