Hot-spot model for accretion disc variability as random process

II. Mathematics of the power-spectrum break frequency

¹ Astronomical Institute, Academy of Sciences, Boční II 1401, 14131 Prague, Czech Republic
e-mail: pechacek@ig.cas.cz
² Observatoire Astronomique de Strasbourg, 67000 Strasbourg, France
³ Copernicus Astronomical Center, Bartycka 18, 00716 Warsaw, Poland

Received: 5 September 2012
Accepted: 10 June 2013

Abstract

Context. 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.

Aims. 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.

Methods. 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.

Results. We suggest that our general methodology can be useful 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.