Radiation Transport Around Kerr Black Holes Jeremy David ...

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Over a sample time T f ≫ T l , the number of segments with a lifetime between T and
T + dT is given by
dN(T) = T f
e −T/T l
dT.
Tl
2
(B.9)
Assuming for the time being that each coherent section of the signal is given by
the sinusoidal function f(t) used above, we can sum all the individual segments to
give the total light curve I(t) with corresponding power spectrum
Ĩ 2 (ν) =
=
∫ ∞
0
G 2 (ν, T)dN(T)
( A
2πT f
) 2 ∫ ∞
0
sin 2 [π(ν − ν 0 )T]
π 2 (ν − ν 0 ) 2
= 2A 2 T l 1
T f 1 + 4π 2 Tl 2(ν
− ν 0) 2.
T f
Tl
2
e −T/T l
dT
(B.10)
Hence we find the shape of the resulting power spectrum is a Lorentzian peaked
around ν 0 with characteristic width
∆ν = 1
2πT l
.
(B.11)
Since the boxcar window represents an instantaneous formation and subsequent
destruction mechanism, the resulting power spectrum contains significant power at
high frequencies, a general property of discontinuous functions. A smoother, Gaussian
window function 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 )
(B.12)
where again the characteristic width is given by ∆ν = 1/(2πT). After integrating
over the same distribution of lifetimes dN(T) as above, we get the power spectrum
where we have defined
Ĩ 2 (ν) = 4πN spot A 2 T [
l √π(1 ]
z 3 + 2z 2 )erfc(z)e z2 − 2z , (B.13)
T f
z ≡
1
4πT l (ν − ν 0 ) .
(B.14)
For large z (near the peak at ν = ν 0 ), equation (B.13) can be approximated by the