BL Lacertae

5 Colour indexes

We have seen that the brightness behaviour of BL Lacertae in the period examined here appears as the superposition of rapid flares, typically lasting for about a day or less, on a long-term trend, which is responsible for a transition from a relatively low brightness level to a higher brightness level around $\rm JD=2~451~800$. It is now interesting to analyse the data in order to understand if the mechanism causing rapid flares is of the same nature as the one determining the long-term flux variations. One piece of information is likely to come from colour index analysis, which can reveal whether flux variations imply spectral changes or not.

Since the BL Lacertae host galaxy is relatively bright, we first subtracted its contribution from the observed fluxes in order to avoid contamination in the colour indexes.

According to Scarpa et al. (2000), the R magnitude of the BL Lacertae host galaxy is $R_{\rm host}=15.55 \pm 0.02$, adopting a host galaxy colour V-R=0.61; Mannucci et al. (2001) derived an average effective colour for elliptical galaxies with M_V<-21 of $B-V=0.99 \pm 0.05$. The inferred B-band host galaxy magnitude is thus $B_{\rm host}=17.15$.

Actually, the above magnitudes represent extrapolations to infinity, while an aperture radius of $8 \rm ~ arcsec$ (to be compared with the $4.8 \rm ~ arcsec$ half-light galaxy radius given by Scarpa et al. 2000) was suggested in the data reduction for the source measure, together with radii of 10 and $16 \rm ~ arcsec$ for the edges of the background annulus. By using these parameters and a de Vaucouleurs r^1/4 profile, we estimated that the host galaxy contribution to the observed fluxes is only 59.65% of the whole galaxy flux.

We then transformed both the observed and the host galaxy B and R magnitudes into dereddened fluxes by using the coefficient of Galactic extinction in the B band A_B=1.420given by NED and by deriving extinction in the Cousins' R band by means of Cardelli et al. (1989): A_R=0.9038. Fluxes relative to zero-mag values were taken from the photometric calibration of Bessell (1979). The B and R host galaxy fluxes ( $2.175 ~ \rm mJy$ and $4.266 ~ \rm mJy$, respectively) were reduced by a factor 0.5965 and then subtracted from the observed fluxes, and the point source (reddened) magnitudes derived from the "cleaned'' fluxes.

B-R colour indexes were calculated by coupling B and R data taken by the same instrument within $20\rm ~min$ (in most cases the time separation between the coupled data is in the range 2-$4\rm ~min$). Only B and R data with errors not greater than 0.04 and $0.03\rm ~mag$, respectively, were considered.

The plot of the resulting 620 B-R indexes as a function of time (see Fig. 6) suggests that colours are more sensitive to rapid variations than to the long-term trend.

Figure 6:   Temporal evolution of B-R colour index (upper panel) and Rmagnitude (lower panel) after subtracting the host galaxy contribution from the fluxes.

Moreover, during well sampled flares the B-R index strictly follows the flux behaviour, as shown by Fig. 7, which presents an enlargement of Fig. 6 during the fourth week of the core WEBT campaign. In this sense, we can say that fast flares are due to a chromatic mechanism, which causes a spectral flattening when the source brightens.

Figure 7:   Temporal evolution of B-R colour index (upper panel) and R magnitude (lower panel) during the last week of the core WEBT campaign, after subtracting the host galaxy contribution from the fluxes.

Figure 8 (upper panel) shows the B-R versus R plot. Points are distributed over two separated regions of the figure, according to the brightness level of the source, with a boundary at $R\sim 14.1$. However, inside each region, the colour indexes seem to follow a trend with a similar slope: a bluer-when-brighter behaviour, as already noticed in Fig. 7 for the short-term flares. On the contrary, the long-term variations appear as essentially achromatic.

Figure 8: B-R colour index versus R magnitude (after subtracting the host galaxy contribution from the fluxes) for uncorrected data (upper panel), for data corrected by subtracting the base-level modulations modelled as a cubic spline (middle panel), and for data further corrected for Doppler factor variations (lower panel); different symbols refer to different flux levels of the spline (see Fig. 9): < $10 ~ \rm mJy$ (black squares), between 10 and $12.5\rm~mJy$(blue circles), between 12.5 and $15\rm~mJy$ (red triangles), and > $15 ~ \rm mJy$ (green crosses).

In order to verify the supposed existence of two different mechanisms acting on different time scales, we have tried to model the long-term trend as a modulated base contribution to the source flux density, on which the short-term flares are superposed. We expect that, once fluxes are corrected for this contribution, the B-R versus R plot will contain the signature of one component only, i.e. the chromatic one.

The first step has been to define a flux base level lapping on the flux minima of the R light curve.

Data in the R band were first binned daily for removing effects due to intranight dense sampling, and then binned over $10\rm ~d$; the binned light curve was then fitted with a cubic spline interpolation (Press et al. 1992). The R light curve was divided into two zones, corresponding to the pre-outburst ( $\rm JD<2~451~790$) and outburst phases. The previously derived spline was then proportionally rescaled to pass through the minima of each zone. The result can be seen in Fig. 9, where the upper, grey (green in the electronic version) line traces the original cubic spline interpolation, while the lower, dark (blue) line shows the rescaled spline, representing the pursued base-level modulation due to the achromatic mechanism mentioned above.

Figure 9: R-band fluxes (mJy) after subtraction of the host galaxy contribution as a function of JD; cubic spline interpolation through the binned light curve is represented by the grey (green) line; the rescaled spline (see text for explanation) is shown by the dark (blue) line.

The same spline was used to find the base-level modulation for the B-band fluxes, by proportionally rescaling it to pass through the minimum flux. The resulting flux ratio of the base levels is F_R/F_B=2.332(B-R=1.787).

By seeing Fig. 9, one might object that only a few minimum points are very close to the base level, even when the sampling is good and many local minima could be identified. The point is that local minima are most likely not states where the flaring activity is out: they are presumably due to the superposition of different events started at different times. On the contrary, the detection of "no flaring'' may be a very rare event, if ever happens.

Fluxes were then "corrected'' for their respective base levels by subtracting the shaded (yellow) area shown in Fig. 9 from the R fluxes, and analogously for the B fluxes. "Corrected'' B-R and R values were finally obtained by $B-R = 2.5 \log{(F_R/F_B)} + 0.868$, $R=-2.5 \log{F_R} + 17.125$, where the constants take both the zero-mag fluxes and the Galactic extinction coefficients into account.

The new B-R versus R plot is shown in Fig. 8 (middle panel): apart from a few points mainly coming from the low-brightness peak of the spline (red triangles, see caption to the figure), most of the points are now following a single linear trend, confirming the starting assumption that an achromatic mechanism produces the base-level variations.

However, one can notice that the distribution of data points corresponding to the outburst phase [grey (green) crosses] still extends to higher brightness levels. This is the consequence of the greater amplitude exhibited by the flux variations in the outburst state (see Fig. 9). We have already noticed that, in the logarithmic scale of magnitudes, variation amplitudes are comparable in the pre-outburst and in the outburst phases, which means that flux amplitudes are proportional to the flux level.

We can thus further refine our model for the flux base-level variations, by assuming that the achromatic mechanism is also responsible for the brightness-dependence of the variation amplitudes. A simple explanation for this is obtained by assuming that the base-level oscillations are the result of the variation of the relativistic Doppler factor $\delta=[\Gamma(1-\beta \cos\theta)]^{-1}$, where $\Gamma$ is the Lorentz factor of the bulk motion of the plasma in the jet and $\theta$ is the viewing angle, since fluxes are enhanced proportionally to a certain power of $\delta$ by relativistic boosting.

In order to clean the observed fluxes for this effect, we derived "corrected'' fluxes by rescaling each original flux by the ratio between the minimum value of the spline and the value of the spline at the considered time. In this way, we obtain fluxes normalized to the value of $\delta$ where the spline has its minimum, thus eliminating the effects of the $\delta$variation, in terms of both the base-level variations and the different variation amplitudes. The resulting "cleaned'' light curve is shown in Fig. 10: the variation amplitude is now comparable over all the period, which should mean that we are now seeing the behaviour of the chromatic component alone.

Figure 10: R-band fluxes after subtraction of the host galaxy contribution and correction for the mechanism responsible for the base-level variations (see text for explanation).

As for the colour indexes, we derived B-R and R values from corrected fluxes as already done in the previous case; the bottom panel of Fig. 8 displays the final result: all data fall in a narrower brightness range, as expected, and the linear correlation appears better defined.

BL Lacertae