Hot-spot model for accretion disc variability as random process - II. Mathematics of the power-spectrum break frequency
(Jun 21, 2013)Abstract
We study some general properties of accretion disc variability in the context
of stationary random processes. In particular, we are interested in
mathematical constraints that can be imposed on the functional form of the
Fourier power-spectrum density (PSD) that exhibits a multiply broken shape and
several local maxima. We develop a methodology for determining the regions of
the model parameter space that can in principle reproduce a PSD shape with a
given number and position of local peaks and breaks of the PSD slope. Given the
vast space of possible parameters, it is an important requirement that the
method is fast in estimating the PSD shape for a given parameter set of the
model. We generated and discuss the theoretical PSD profiles of a
shot-noise-type random process with exponentially decaying flares. Then we
determined conditions under which one, two, or more breaks or local maxima
occur in the PSD. We calculated positions of these features and determined the
changing slope of the model PSD. Furthermore, we considered the influence of
the modulation by the orbital motion for a variability pattern assumed to
result from an orbiting-spot model. We suggest that our general methodology can
be useful in for describing non-monotonic PSD profiles (such as the trend seen,
on different scales, in exemplary cases of the high-mass X-ray binary Cygnus
X-1 and the narrow-line Seyfert galaxy Ark 564). We adopt a model where these
power spectra are reproduced as a superposition of several Lorentzians with
varying amplitudes in the X-ray-band light curve. Our general approach can help
in constraining the model parameters and in determining which parts of the
parameter space are accessible under various circumstances.