-model
Although measured astronomical data are time domain data, a commonly
applied method works in the frequency domain by analyzing the power
spectrum of the time series. As the observational window function is
convoluted to the true spectrum of the source, artefacts might be produced
in the power spectrum, which make a proper interpretation more difficult
(Papadakis & Lawrence 1995; Priestley 1992). In
most cases, the power spectra are fit by a power law function with an offset
described as 
, with values of 
ranging from 0 to 2
and a mean of about 1.5 (Lawrence & Papadakis 1993). The
value c is often denoted as the "observational noise floor" which
describes the random process comprising the observational errors whereas the
"red noise" component is the signal of interest. In the case of long AGN
observations, however, a flattening at low frequencies occurs which cannot
be modelled by the 
-model (McHardy 1988).
The -model is an ad hoc description of the measured
periodogram, without any direct physical motivation. However, it is
possible to generate time series with a -spectrum using
self-organized criticality models simulating the mass flow within an
accretion disc of the AGN (Mineshige et al. 1994). Such models
produce a stationary time series that exhibits a -power
spectrum by limiting the timescales occurring in the simulated accretion
process. A -model without limited timescales would be
stationary only if the power law slope is smaller than unity (Samorodnitsky
& Taqqu 1994). The observed time series is composed by the
superposition of single luminosity bursts. The slope of the
-spectrum of data simulated in that way is about 1.8,
significantly higher than those measured from real data (Lawrence &
Papadakis 1993). If the inclination of the accretion disk is brought
in as an additional model parameter the slope can be diminished, but not in
a way that leads to convincing results (Abramowicz et al.
1995). Another point that contradicts this assumption is that there
is no correlation between the spectral slope and the type of the Seyfert
galaxy (Green et al. 1993). This correlation should be present
since the Seyfert type is believed to be caused by the inclination of the
line of sight (Netzer 1990).
The periodogram which is used to estimate the true source spectrum is
difficult to interpret in the presence of non-equispaced sampling time
series arising from real astronomical data (Deeter & Boynton
1982 and references therein). The estimation of the
-spectrum is hampered even in the absence of data gaps. This
is due to the finite extent of the observed time series. Therefore, the
transfer function (Fourier transform of the sampling function) is a
sinc-function which will only recover the true spectrum if this is
sufficiently flat (Deeter & Boynton 1982; Deeter
1984). In the case of "red noise" spectra the sidebands of the
transfer function will cause a spectal leakage to higher frequencies which
will cause the spectra to appear less steep (the spectral slope will be
underestimated).
Even periodograms of white noise time series deviate from a perfectly flat
distribution of frequencies as the periodogram is a
-distibuted random variable with a standard deviation equal
to the mean (Leahy et al. 1983). Thus the periodograms
fluctuate and their variances are independent of the number of data points
in the time series. Due to the logarithmic frequency binning, AGN
periodograms will always show this strong fluctuation due to the low number
of periodogram points averaged in the lowest frequency bins (see
Fig. 1 (click here)).
Figure 1: a) EXOSAT ME X-ray lightcurve of the quasar 3C 273
(Jan. 1986), b) corresponding periodogram. Each dot represents the
spectral power at its frequency, stepped with 
. The
periodogram is binned logarithmically (squares indicates a single point
within the frequency bin)
Furthermore, additional modulations can be created in white noise
periodograms if the time series consists of parts which slightly differ in
their means and variances, respectively (Krolik 1992). In the
case of the EXOSAT ME X-ray lightcurves this effect is due to the swapping
of detectors as each detector has its own statistical characteristics which
cannot be totally suppressed (Grandi et al. 1992; Tagliaferri
et al. 1996). Figure 1 (click here)a shows a typical X-ray lightcurve
which mainly consists of uninterrupted 11 ksec observation blocks before
detectors are swapped. If the periodogram frequency corresponds to the
observation block length, the calculated sum of Fourier coefficients equals
its expected white noise value of 
due to the constant mean and
variance within the entire oscillation cycle. At other, mainly lower,
frequencies the Fourier sum yields non-white values due to temporal
correlations caused by different means and variances of observation blocks
located in the test frequency cycle. These deviations from a flat spectrum
will be very strong at frequencies which correspond to twice the observation
block length. The arrows in Fig. 1 (click here)b clearly show this minimum
feature at 
and another shortage of power at 
which
corresponds to the long uninterrupted 72 ksec observation block starting at
the second half of the EXOSAT observation (Fig. 1 (click here)a).
Consequently a model is required which operates in the time domain and avoids any misleading systematical effects occuring in power spectra.