Power Spectra of Black Holes and Neutron Stars as a Probe of Hydrodynamic Structure of the Source: Diffusion Theory and Its Application to Cygnus X‐1 and Cygnus X‐2 X‐Ray Observations

Abstract

We present a model of Fourier power density spectrum (PDS) formation in accretion-powered X-ray binary systems derived from diffusion theory. Timing properties of X-ray emission are considered a result of diffusive propagation of the driving perturbations in a bounded medium. We prove that the integrated power Px of the resulting PDS is only a small fraction of the integrated power Pdr of the driving oscillations, which is distributed over the disk. Furthermore, we demonstrate that Px is inversely proportional to the characteristic frequency of the driving oscillations ν_dr, which likely scales with the frequency of the local gravity waves in the disk (Keplerian frequency). Because νdr increases toward soft states, we conclude that Px declines toward soft states. This dependence Px~ν^{-1}_dr explains the well-known observational phenomenon that the X-ray variability power decreases when the source evolves to softer states. The resulting PDS continuum is a sum of 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 or corona). The LF PDS component has a power-law shape with an 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 bounded extended medium, the diffusion timescale, and the dependence law of viscosity versus radius. We apply our model of the PDS to RXTE and EXOSAT timing data from Cygnus X-1 and Cygnus X-2, which describes adequately the spectral transitions in these sources. The presented PDSs are shown in frequency range from 10^{-8} to 10^{2} Hz, 10 orders of magnitude.