Introduction

The X-rays from active galactic nuclei (AGN) are thought to originate from the innermost regions of an accretion disk around a central supermassive black hole. Thus, in principle, the study of these X-rays should allow one to probe the immediate environment of the accreting black hole as well as the exotic physics, including strong-field general relativity, that operates in this environment.

In the past decade X-ray astronomy has begun to fulfill that promise. Both EXOSAT and Ginga discovered iron K-shell features (including the K fluorescent line of cold iron at 6.4keV) in the X-ray spectra of Seyfert galaxies which were interpreted as `reflection' of the primary X-ray continuum by cold, optically-thick material in the immediate vicinity of the black hole (Guilbert & Rees 1988; Lightman & White 1988; Nandra et al. 1989; Nandra, Pounds & Stewart 1990; Matsuoka et al. 1990). It was suggested that this cold reflecting material was the putative accretion disk of AGN models. With the launch of ASCA and the advent of medium resolution spectroscopy, the iron line in several objects was shown to be broad ( FWZI) and skewed (Tanaka et al. 1995; Nandra et al. 1997). The overall line profiles are in good agreement with models for fluorescent line emission from the innermost regions of geometrically-thin black hole accretion disks (Fabian et al. 1989). Such data allow us to address issues such as the location of the radius of marginal stability, the spin of the black hole, and the inclination distribution of various classes of AGN (see Reynolds 1999 and references therein for a review of these studies). In the current, RXTE era, we can now probe the iron line and Compton reflection hump in individual objects in some detail (e.g., MCG-5-23-16, Weaver et al. 1998; MCG-6-30-15, Lee et al. 1998, 1999a; NGC 5548, Chiang et al. 1999).

While these spectral studies have been successful, a complete picture of the AGN phenomenon is not possible without addressing the timing properties. Timing studies are important for two intertwined reasons. Firstly, AGN are inherently variable systems. In general, the variability timescale in a given object is seen to shorten as one considers higher frequency radiation. In the X-ray and -ray bands, dramatic variability has been seen in many Seyfert galaxies with doubling timescales of only a few minutes (e.g. see Reynolds et al. 1995). Although it is poorly understood to date, the nature of this violent variability is a vital component of any final AGN model. Careful characterization of the timing properties, as well as determining the observed spectral evolution during dramatic temporal events, is required if we are to understand this phenomenon.

Secondly, timing studies are needed to break certain degeneracies that exist in models which, to date, have only been constrained by purely spectral data. The spin of the black hole in MCG-6-30-15 provides an excellent example of such a degeneracy -- by fitting the `very-broad' state (Iwasawa et al. 1996) of the iron line in this object with models consisting of a thin, disk-hugging corona, Dabrowski et al. (1997) inferred that the black hole in this AGN must be almost maximally rotating, with a dimensionless spin parameter of a>0.94. However, by including line emission from within the radius of marginal stability, Reynolds & Begelman (1997) showed that a geometry in which the X-ray source is at some height above the disk plane can produce the same line profile even if the black hole is completely non-rotating. While there are subtle spectral differences between the two scenarios (Young, Ross & Fabian 1998) the most obvious way of distinguishing these scenarios is through their timing properties. The Reynolds & Begelman (1997) geometry predicts substantial time delays between fluctuations in the primary power-law continuum and the responding fluctuations in the iron line. More generally, the reverberation characteristics of the iron line contain tremendous information on the mass and spin of the black hole as well as the geometry of the X-ray source (Stella 1990; Reynolds et al. 1999).

The observational situation is more complex. Lee et al. (1999b) and Chiang et al. (1999) have analyzed extensive RXTE datasets for MCG-6-30-15 and NGC 5548, respectively, in order to study the timing properties and spectral variability. In both of these objects, the same pattern of spectral variability is seen. Firstly, the X-ray photon index displays flux-correlated changes in the sense that the source is softer when it is brighter. Secondly, and more surprisingly, the iron line flux was found to be constant over the timescales probed by these direct spectral studies ( ksec). As discussed by both sets of authors, these results are difficult to interpret in the framework of standard X-ray reflection models since the breadth of these lines indicate that they originate from a small region. It appears that some feedback mechanism regulates the amount of iron line emission in order to produce approximately constant iron line flux. Flux-correlated changes in the ionization state of the disk represent one such mechanism (we discuss this in more detail in Section 5 of this paper). Unless this feedback mechanism operates instantaneously, we might still expect variability of the iron line flux on short timescales.

Driven by these motivations, this paper addresses the problem of determining causal relationships between light curves in different X-ray bands, with particular emphasis on timescales shorter than those that can be probed by direct spectroscopy. In particular, we use the long RXTE observation of the bright Seyfert 1 galaxy MCG-6-30-15 reported by Lee et al. (1999a,b) and consider the relationship between the 2-4keV band (hereafter called the continuum band) and the 5-7keV band which contains most of the iron line photons (and hereafter called the line band). An important special case is one in which there is a linear transfer function relating one band to the other:

where a(t) and b(t) are continuum and line band fluxes respectively, and is the transfer function. Such relationships between bands contain much of the important physical information, such as the reverberation characteristics of the iron line.

Mathematically, the linear transfer equation can be easily inverted using Fourier methods to obtain,

where represents the Fourier transform of a(t). However, in real situations, a large number of regularly sampled measurements are required to obtain an accurate deconvolution using this simple method. More often, deconvolution is achieved using maximum entropy techniques or some other regularization method (Horne et al. 1991; Krolik et al. 1991).

Another common approach (and one that is often used with less well sampled data) is to compute cross-correlation functions (CCFs), or some variant thereof which accounts for the finite and irregular sampling often encountered in real data. The discrete correlation function (DCF; Edelson & Krolik 1988) is one example of such a variant. Lee et al. (1999b) apply such methods to the observation of MCG-6-30-15 considered in this paper and detect both phase and time lags between RXTE bands (also see Nowak & Chiang 1999). While these methods are powerful, it can be difficult to separate subtle time leads/lags from the autocorrelation properties of the data.

Here, we take an alternative approach which is heavily based on the method of Press, Rybicki & Hewitt (1992; hereafter PRH92). In essence, we use the correlation properties of the continuum band data to reconstruct an optimal continuum light curve in which the data gaps have been interpolated. Most importantly, we also compute the expected deviation of the continuum flux from the interpolated curve. The reconstructed continuum band light curve is convolved with a trial transfer function and compared with the line band light curve in a sense. We then examine changes in the statistic as a function of the parameters that define the trial transfer function.

Section 2 recaps the PRH92 method. This is then applied to the RXTE data for MCG-6-30-15 in Section 3. The robustness and validity of our approach is demonstrated by applying this method to simulated data (Section 4). Section 5 draws together our results and discusses their implications for the nature of this source. In particular, we argue that the black hole in this AGN has a mass of only . In order to explain the spectral variability, it is suggested that there are flux correlated changes in the ionization state of the surface layers of the accretion disk. Section 6 presents a short summary of the results and relevant astrophysical implications.