MASS MEASUREMENTS OF AGN FROM MULTI-LORENTZIAN MODELS OF X-RAY VARIABILITY. I. SAMPLING EFFECTS IN THE σ rms-MBH CORRELATION

Recent X-ray variability studies on a large sample of Active Galactic Nuclei (AGN) have suggested that the fractional rms variability amplitude, σrms, seems to correlate with black hole mass, MBH. The fact that the rms amplitude can be easily obtained from the observed X-ray light curves has motivated the theoretical modeling of the σ rms-MBH relationship. A viable approach to addressing this problem is to assume an underlying power spectral density with a suitable mass dependence, derive the functional form of the relationship between fractional rms and mass for a given sampling pattern, and investigate whether the result is consistent with the observations. Inspired by the similarities shared by the timing properties of AGN and X-ray binaries, previous studies have explored model power spectral densities characterized by broken power laws. With the advent of RXTE, however, it has been shown that the band-limited noise spectra of X-ray binaries are not broken power laws but can often be described in terms of a small set of broad Lorentzians, with different amplitudes and centroid frequencies. In this paper, motivated by the latest timing results from X-ray binaries, we consider that AGN broad-band noise spectra consist of a small number of Lorentzian components. This formalism allows, for the first time, to fully account for sampling effects in theoretical models of X-ray variability in an analytic manner. We demonstrate the excellent agreement between our analytical results and Monte Carlo simulations designed to account for the distortions suffered by the underlying power spectral density due to sampling effects. We show that, neglecting sampling effects when deriving the fractional rms from the model power spectrum density can lead to underestimating it by a factor of a few with respect to its true value. We discuss the implications of our results for the derivation of AGN masses from the σ rms-MBH correlation. Subject headings: accretion disks — black hole physics — galaxies:active, nuclei — X-rays:binaries, galaxies