Radiation Transport Around Kerr Black Holes Jeremy David ...

4.7. HIGHER ORDER STATISTICS 121
Figure 4-9: The tracks showing how different harmonics of the QPO vary with
respect to one another when the radius of the hot spot orbit varies around a central
value r ≈ r 0 ± 0.2M. The frequencies correspond to a 10M ⊙ black hole with spin
parameter a/M = 0.5.
the distribution of the bicoherence can be written as
b 2 (ν 1 , ν 2 ) ∼ P(δν 1 )P(δν 2 )P(δν 1 + δν 2 ). (4.28)
Expanding equation (4.28) around the center of each peak and defining x ≡ δν 1 /∆ν
and y ≡ δν 2 /∆ν, we see that contours of constant bicoherence have the form
(1 + x 2 + y 2 + x 2 y 2 )(1 + x 2 + 2xy + y 2 ) = const. (4.29)
For small deviations (x, y ≪ 1), these contours can be written
x 2 + y 2 + xy = const, (4.30)
which is the formula for an ellipse with a/b = √ 3, oriented with the semimajor axis
parallel to the line y = −x, as can be seen clearly in Figure 4-8a.
In the second case, where there are many frequencies in the power spectrum due
to hot spots found over a range of radii, there will be phase coherence between the
different harmonics of each individual hot spot, but not with the hot spots at slightly