Power spectra of black holes and neutron stars as a probe of hydrodynamical structure of the source: Diffusion theory and its application to X-ray observations

Abstract

We present a model of Fourier Power Density Spectrum (PDS) formation in accretion powered X-ray binary systems derived from the first principles of the diffusion theory. Timing properties of X-ray emission are considered to be a result of diffusive propagation of the driving perturbations in a bounded medium. We prove that the integrated power of the resulting PDS, Px is only a small fraction of the integrated power of the driving oscillations, Pdr which is distributed over the disk. Furthermore, we demonstrate that the power Px is inversely proportional to the characteristic frequency of the driving oscillations νdr which is likely scaled with the frequency of the local gravity waves in the disk (Keplerian frequency). Keeping in mind that νdr increases towards soft states leads us to conclude that the power Px declines towards soft states. This dependence Px ∝ νdr-1 explains the well-known observational phenomenon that the power of the X-ray variability decreases when the source evolves to softer states. The resulting PDS continuum is a sum of two components, a low frequency (LF) component which presumably originates in an extended accretion disk and a high frequency (HF) component which originates in the innermost part of the source [Compton cloud (CC)]. The LF PDS component has a power-law shape with index of 1.0-1.5 at higher frequencies ("red" noise) and a flat spectrum below a characteristic (break) frequency ("white" noise). This white-red noise (WRN) continuum spectrum holds information about the physical parameters of the bounded extended medium, diffusion time scale and the dependence law of viscosity vs radius. This LF PDS associated with the extended disk dominates in the soft states of the system, while the HF PDS characteristic of innermost CC component is dominant in the low/hard and intermediate states. These PDS LF and HF components directly correspond to the energy spectrum components. Namely: LF WRN is related to thermal emission from an accretion disk, and the HF WRN to the power-law tail, which presents a fraction of the disk emission Comptonized in the Compton cloud. Hence, a change of PDS features correlates with a change of energy spectral features. Analyzing the data for a number of sources we find that the PDS is well represented by a sum of the WRN CC component and the WRN extended disk component. We apply our model of the PDS to a sample of RXTE and EXOSAT timing data from Cyg X-1 and Cyg X-2 which describes adequately the spectral transitions in these sources.