ESTADO DE LOS PROCEDIMIENTOS (SGC)

In most cases, AGN light curves are not densely sampled enough to pro- duce reliable cross-spectra, but lags and coherence can be crudely estimated with the cross-correlation function (CCF). Time lags between two bands can translate into a shift of the CCF central peak if one light curve is a simple translation in time of the other, or produce an asymmetric peak if the lags are more complex. Fourier frequency dependent time lags, as pro- duced by the model and seen in some AGN, produce skewed CCFs with centroids shifted to positive lags, meaning that the soft band leads the hard one. The CCF peak can remain close to zero lag because the lags associated with the narrow-width (i.e. short time-scale) CCF components are small. However, longer time-scale components are separated by progressively larger lags, producing noticeably better correlations at positive than at negative lags, so the CCF becomes skewed.

As an example, Fig. 4.12 shows a set of auto-correlation Functions (ACF) and CCFs for MCG-6-30-15. In this figure, solid lines represent the ACF for the soft band (0.5-0.7 keV) of three light curves, each approx- imately 100 ks long, taken by XMM-Newton during revolutions 301, 302 and 303. These data were first discussed by Fabian et al. 2002) and we followed the procedure described therein to construct the light curves. The dotted and dashed lines represent the CCF between this soft light curve and the medium (0.7-2.0 keV) and hard (2.0-10.0 keV) light curves. The CCFs are slightly asymmetric and this asymmetry increases with the energy sep- aration of the bands, as would be expected from the lag spectrum results obtained by Vaughan et al. (2003) for the same data set. Notice, however, that in orbit 302 the CCFs are skewed towards negative lags (i.e. soft band lagging hard), contrary to what is seen in the other two orbits and what is obtained from the averaged lag spectra. As we will see, this anomaly can easily result from statistical fluctuations inherent in the variability process (and not associated with any changes to the emitting region), and highlights one of the caveats that must be borne in mind when using the CCF alone to determine time lags.

We used the variability model to produce synthetic light curves that match the MCG-6-30-15 PSD at low and high frequencies (McHardy et al. 2005) and the PSD energy dependence and time lags (Vaughan et al. 2003). The smaller energy dependence and lags in this object, compared to NGC 4051, can be reproduced with closer emissivity indices, where values of γ = 3, 4 and 5 were appropriate to fit the soft, medium and hard bands used above. We used short segments of these simulated light curves to

Figure 4.12: Auto- and cross-correlation functions for three, 100 ks long, observations of MCG-6-30-15 taken by XMM-Newton during orbits 301, 302 and 303. The solid line shows the ACF of the soft (0.5-0.7 keV) light curve, the dotted- and dashed-lines represent the CCF between this and the medium (0.7-2.0 keV) and hard (2.0-10.0 keV) light curves respectively. A CCF centroid displaced to positive lags indicates that the soft band leads the harder band.

compute ACFs and CCFs, to compare with the results of the real data. The segments used were_∼104Rg/clong, which equals 100 ks for a 2×106MBH, similar to the mass estimated for MCG-6-30-15 by Mc_{Hardy et al. (2005)} of 3₋6_×106M. The ACFs for theγ = 3 synthetic light curve segments and the CCF between these and the γ = 4 and 5 segments are displayed in Fig. 4.13, in solid, dashed and dotted lines, respectively. This figure shows that the model can reproduce the asymmetry in the CCF and its energy dependence, but also that quite different CCF characteristics can be expected from the same underlying model parameters.

One thing to note when calculating the ACF of short stretches of red- noise light curves is that the width of the central peak varies. This is a consequence of the stochastic nature of the time series and does not neces- sarily imply a change in the underlying physical conditions. The different ACF central peaks shown in Fig. 4.13 are calculated from segments of a single long (stationary) synthetic light curve. The segments are 104_R

g/c long, covering the frequency range around and above the bend in the PSD. Evidently, the same set of underlying model parameters can produce con- siderably different peak widths.

Secondly, the shift in the CCF centroid can change artificially from pos- itive to negative lags. The time lags produced by the model are strictly positive, as the fluctuations propagate only inwards and harder bands are more centrally concentrated (see Equation 4.3). This produces CCFs skewed towards positive lags. This effect can be seen in the dotted and dashed lines in Fig. 4.13, that represent the CCFs between light curves of indicesγsoft = 3 and γhard = 4 and 5 respectively. In panels a, c and d, the CCF are better correlated at positive than at negative lags, and this positive asymmetry indicates that the ‘hard’ light curve lags the ‘soft’, as expected.

However, it happens occasionally that different fluctuations align by chance, and can make the hard light curve correlate with the soft one bet- ter at a negative lag. If this happens at low Fourier frequencies, where the amplitude of the fluctuations is largest, it can cause the CCF centroid to flip to negative lags, as in panel b of Fig. 4.13. A negative lag centroid occured for approximately one quarter of the segments of this length and for 5% of the cases where segments 10 times longer were used. Fig. 4.14 shows the lag spectra corresponding to panels a and b of Fig. 4.13, in solid lines, and the lag spectrum of the full synthetic light curves in dashed lines, all these calculated for γsoft = 3 and γhard = 5. The large scatter around the underlying lag spectrum is caused only by the small number of points used in the short light curves, as these have no added observational noise. In panel a, the lags follow the underlying spectrum, and so, the correspond-

ing CCF in Fig. 4.13 shows the expected positive-lag asymmetry, while the negative-lags seen in panel b of Fig. 4.13 are caused by the largely outlying lag value at the lowest frequency bin in Fig. 4.14 panel b. In conclusion, the slight asymmetry of the MCG-6-30-15 CCFs and the reversal of the lag sign in the data from orbit 302 can be accounted for by the variability model, without needing to invoke additional effects of, e.g. spectral evolution of the emitting region. A comparison of the fractional lags at, for example, the PSD bend frequency in each case gives a mass-independent measure, so an extrapolation of the lag spectra in NGC 4051 published by Mc_{Hardy et al.} (2004) down to the break time-scale would suggest that the lags in this case are indeed systematically larger than in MCG-6-30-15