BL Lacertae

6 Time correlations

In the Introduction we mentioned a number of studies devoted to the search for periodicities in the BL Lacertae light curves. The derived periods range from 0.31 to $14\rm ~yr$. The light curves obtained by the WEBT collaboration extend to $\sim$$240\rm ~d$, so that possible periodicities longer than, say, a couple of months cannot be looked for. On the other hand, the dense sampling allows one to test the existence of characteristic times of variability down to very short periods.

We applied the discrete correlation function (DCF) method (Edelson & Krolik 1988; Hufnagel & Bregman 1992) to the BL Lacertae fluxes, paying attention to the edge effects, as warned by Peterson (2001). Figure 11 shows the autocorrelation of the uncorrected, galaxy-subtracted R-band fluxes, with a $100\rm ~d$ maximum lag. The central, wide maximum tells us that the R light curve continues to correlate with itself for time shifts shorter than a month; for larger time lags the pre-outburst phase starts to overlap significantly with the outburst one, and the DCF drops to negative values, implying anticorrelation. No significant feature appears, which means that no reliable characteristic variability time scale is found.

Figure 11: Autocorrelation function for the uncorrected, galaxy-subtracted R-band fluxes.

Correction of fluxes for variation of the Doppler factor as described in the previous section removes the signature of the long-term trend. As one can see in Fig. 12, where the autocorrelation for the corrected R-band fluxes is presented (zoomed on smaller time lags), also in this case no significant time scale is recognizable.

Figure 12: Autocorrelation function for the R-band fluxes after subtracting the host galaxy contribution and correcting for Doppler factor variation.

However, if we restrict the DCF analysis to the data from the core WEBT campaign (see Fig. 2), we see that a not negligible signal comes out at a $\sim$$7\rm ~h$ time scale, as shown by Fig. 13. This feature is particularly evident in the second and fourth weeks of the core campaign. Nevertheless, one has to notice that its significance might be affected by the lack of information during observing gaps.

Figure 13: The same as Fig. 12, but restricted to the core WEBT campaign and zoomed on short time scales.

In the above analysis we have neglected that not only fluxes, but also time scales are affected by Doppler boosting. A reliable temporal analysis should take this into account.

It is known that time intervals are changed by a factor $\delta^{-1}$ by relativistic Doppler effect. Under the assumption that fluxes F are modified proportionally to $\delta^3$, we have that $\delta(t) / \delta_{\rm min} = \left[ F(t)/F_{\rm min} \right] ^{1/3} = \left[ s(t) / s_{\rm min} \right] ^{1/3}$, where s(t) is the value of the spline representing the flux base-level variations at time t, and $s_{\rm min}$ is the spline minimum. Consequently, the corrected time t'_n for the nth data point can be calculated as

$$\begin{eqnarray*}t'_n = t'_{n-1}+ \Delta t_n \left[{\int_{t_{n-1}}^{t_n} {s(t)~{\rm d}t}} \over {\Delta t_n s_{\rm min}} \right] ^{1/3}, \end{eqnarray*}$$

where $\Delta t_n=t_n - t_{n-1}$ is the observed time interval between the (n-1)th and nth data points.

By applying the above time correction to the R-band light curve, the total duration of the observing period dilates from 241.71 to $285.57\rm ~d$.

As for the R-band flux autocorrelation, time correction does not introduce any significant change in the results: the $\sim$$7\rm ~h$characteristic time scale of variability is confirmed, and no other signal comes out.

Finally, DCF analysis has been applied to search for the possible existence of time delays between the flux variations in the B and R bands: no significant measurable (greater than a few minutes) delay has been found.

BL Lacertae