3 Connections between millimetre flux curves and VLBI components

In order to compare TFD and VLBI events in our data, we decompose the total flux density variations in the Mets $\ddot{\rm a}$hovi 22/37 GHz flux curves into exponential flares of the form

$$\Delta S(t) = \left\{ \begin{array}{ll} \Delta S_{{\rm max}} ... ...max}}-t) / 1.3 \tau}, & t > t_{{\rm max}}. \end{array} \right.$$ (1)

Here $\Delta S_{{\rm max}}$ is the maximum amplitude of the flare, $t_{{\rm max}}$ is the epoch of the flare maximum and $\tau$ is the flare rise timescale. It has been shown earlier that all TFD variations can be modelled to surprising accuracy with a small number of flares consisting of an exponential rise, sharp peak and exponential decay superposed on a constant baseline flux (Valtaoja et al. 1999). This decomposition (described in more detail by Valtaoja et al. 1999) helps us to identify and isolate individual events as well as to estimate amplitudes and timescales of the outbursts.

 

 

Table 2: The percentage of each class of the radio-loud AGN in three different samples. [HPQ = high optical polarization quasar, LPQ = low optical polarization quasar, BLO = BL Lacertae object, GAL = radio galaxy and N/A = No exact classification available].

Class	Our sample	2 Jy sample	EGRET blazars
HPQ	44%	28%	37%
LPQ	26%	32%	30%
BLO	26%	28%	26%
GAL	4%	8%	0%
N/A	0%	4%	7%

Figure 1: An example (quasar 1633+382) of a graph containing TFD measurements, exponential flare model fits, and individual VLBI component flux density vs. time. The top panel presents total flux density observations at 22 GHz from the Mets $\ddot{\rm a}$hovi quasar monitoring program (dots) and our fit (curve), which is a sum of individual exponential model flares superposed on a constant baseline flux. The modelled flares are shown in the middle panel with the epochs of the maximum flux density indicated. The bottom panel displays the flux evolution of the VLBI components at 22 GHz from the VLBA observations by Jorstad et al. (2001a), whose component designations we adopt; the errors in flux density of the VLBI components are approximately 0.05 Jy.

Figure 2: Another sample TFD curve, exponential flare model fits, and individual VLBI component TFD values as a function of time. The 1998.48 data is from VLBA observations by Fredrik Rantakyr $\ddot{\rm o}$ (Rantakyr $\ddot{\rm o}$ et al. 2002). See the caption of Fig. 1 for details.

Next we plotted the TFD decompositions and the flux variations of the VLBI components for each source. Two examples illustrate the results of this comparison: 1633+382 (4C 38.41) and PKS 2230+114 (CTA 102) are shown in Figs. 1 and 2. The component identifications can be found in Jorstad et al. (2001a). Even at first glance, it is evident that there is a clear connection between the millimetre continuum variations and the VLBI component fluxes. Whenever there are enough VLBA observations, the summed flux curve of the VLBI components is similar to the continuum flux curve; only the amplitude of the former is $\sim$90% that of the latter. This is expected with the missing 10% of the flux in the VLBI maps probably just due to the insensitivity of high-frequency VLBI to diffuse emission. There is a slight time shift between the 37 GHz TFD curves and the 43 GHz VLBI component flux curves. This is understandable according to the shock models, since the maximum amplitude of the flare moves from high frequencies to lower frequencies as the shock evolves.

A much more interesting result is that for every superluminal ejection seen in the VLBA data, the TFD decomposition shows a coinciding flare. We examine ejections having zero epochs after the year 1990. For most of our sources, Mets $\ddot{\rm a}$hovi TFD monitoring is rather sparse before this and therefore not suitable for our comparison. We exclude two ejections because of large gaps in the Mets $\ddot{\rm a}$hovi flux curve at their zero epochs (the observation gap in 1994). We require that there be at least three observations of the ejected component and that the observed flux density of the component be greater than 0.1 Jy (the approximate noise level of Mets $\ddot{\rm a}$hovi observations) at some time. In our data, there are 29 ejections of VLBI components fulfilling the above criteria (see Table 3). The TFD flares corresponding to these 29 ejections are identified by comparing the component ejection times with the beginning times of the TFD flares, as well as by comparing the light curves of the VLBI components with those of the decomposed TFD flares.

 

 

Table 3: The zero epochs of the VLBI components ( $t_{{\rm0, VLBI}}$) determined by Jorstad et al. (2001a) and the start times of the corresponding TFD flares ( $t_{{\rm0, TFD}}$). The last column gives the time difference $\Delta {t} = t_{{\rm0, VLBI}}-t_{{\rm0, TFD}}$. The designation of the components follows Jorstad et al. (2001a).

Source	Comp.	$t_{{\rm0, VLBI}}$	$t_{{\rm0, TFD}}$	$\Delta t$ [yr]
0202+149	B	$1994.8 \pm 0.1$	1994.7	0.1
0219+428	B4	$1995.7 \pm 0.4$	1995.7	0.0
	B3	$1995.4 \pm 0.1$	1994.9	0.5
0235+164	B1	$1995.1 \pm 0.2$	1994.8	0.3
0420-014	B	$1995.3 \pm 0.1$	1994.9	0.4
0458-020	B2	$1994.0 \pm 0.1$	1993.6	0.4
0528+134	B4	$1995.5 \pm 0.1$	1995.5	0.0
	B3	$1994.8 \pm 0.1$	1994.7	0.1
	B2	$1994.5 \pm 0.2$	1994.2	0.3
	B1	$1993.4 \pm 0.7$	1993.4	0.0
0827+243	B1	$1994.7 \pm 0.1$	1994.6	0.1
0851+202	B3	$\approx 1996.6$	1996.6	0.0
	B2	$1995.6 \pm 0.1$	1995.6	0.0
1156+295	B3	$1996.3 \pm 0.2$	1996.0	0.3
	B2	$1995.3 \pm 0.1$	1995.1	0.2
1226+023	B5	$1993.4 \pm 1.3$	1993.4	0.0
	B3	$1992.4 \pm 0.3$	1991.9	0.5
	B2	$1991.3 \pm 0.4$	1991.0	0.3
	B1	$1990.4 \pm 0.3$	1990.7	0.3
1253-055	B3	$1995.7 \pm 0.1$	1995.5	0.2
	B2	$1994.2 \pm 0.2$	1993.8	0.4
	E2+B1	$1993.5 \pm 0.2$	1992.4	1.1
1510-089	B1	$1996.1 \pm 0.1$	1996.3	-0.2
	D2	$1994.1 \pm 0.2$	1993.9	0.2
1633+382	B3	$1994.8 \pm 0.1$	1994.7	0.1
2230+114	B3	$1996.1 \pm 0.1$	1996.0	0.1
	B1	$1994.3 \pm 0.1$	1993.9	0.4
2251+158	B3	$1995.6 \pm 0.2$	1995.4	0.2
	B2	$1995.1 \pm 0.1$	1995.0	0.1

We define the beginning of an exponential TFD flare as $t_{{\rm0,TFD}} = t_{{\rm max}}-\tau$, where $\tau$ is the variability timescale (e-folding time). This definition gives the point where $S(t_{{\rm0,TFD}})=\frac{S_{{\rm max}}}{e}$. While there is no mathematical sense in defining the beginning of an exponential function, in reality there must be a starting point to a flare. We could instead estimate the beginning of the flare as the previous local minimum of the flux curve ( $t_{{\rm lm}}$). If we compare $t_{{\rm lm}}$ to $t_{{\rm0, TFD}}$ for an outburst that starts just after a local minimum, we see that the average time difference between the two is 0.0 years with a standard deviation of 0.4 years (see Fig. 3). Therefore, the average values of $t_{{\rm0, TFD}}$ and $t_{{\rm lm}}$ are the same. When two or more closely spaced outbursts blend together, the local minimum is no longer a good indicator of the start of the flare. In such a case the local minimum is near the peak rather than the beginning of the later flare. Hence, $t_{{\rm0, TFD}}$ is a more reliable and practical starting point to a flare.

Figure 3: The time difference between $t_{{\rm0,TFD}} = t_{{\rm max}}-\tau$ and the epochs of the local minima in our data.

We compare the extrapolated ejection epochs of the superluminal knots (from Jorstad et al. 2001a) with the beginning times of the TFD flares. In 28 of the 29 cases we find a TFD flare that occurred within 0.5 yr of the ejection epoch. The only exception is component E2+B1 of 3C 279, for which $t_{{\rm0, VLBI}}$ is not very well determined.

The frequency of large TFD flares ( $\Delta S > 0.3 \cdot S_{{\rm quiescent}}$) estimated from the Mets $\ddot{\rm a}$hovi data is 1 per 1.6 years. On the other hand, the frequency of observed superluminal ejections is approximately 1 per 2.3 years. Using these values we calculate the probability that a superluminal ejection could occur by random chance within a time interval dt before or after the beginning of the TFD flare. The results are given in Table 4. For every applied dt range the expected number of random occurrences is clearly much lower than the observed number of coincidences. The probability that 28 out of 29 ejections would be observed to occur randomly within 0.5 yr of the beginnings of TFD flares is <10^-7. Hence, the correspondence between $t_{{\rm0, VLBI}}$ and $t_{{\rm0, TFD}}$ is real at a very high level of significance.

We therefore find that, at high radio frequencies, the start of a TFD flare precedes the arrival of a new superluminal knot at the position of the brightness centroid of the core of the jet. However, we do not have enough VLBI data to say if the converse is true, i.e., whether there is a new VLBI component for every TFD flare. When we see the new VLBI component for the first time, the flux of the TFD flare is usually already decreasing. This behaviour is analysed in Sect. 4, in which we discuss the so-called core flares.

 

 

Table 4: The probability P of coincidence of superluminal ejections and the start of TFD flares, for both the random and observed case $(P_{{\rm Observed}} = N_{{\rm Associated}} / N_{{\rm Total}})$. The interval dt is the maximum time difference between the ejection of the knot and the beginning of a TFD flare for them to be considered associated. $\rho$ is the probability that the observed number of associations could occur by random chance.

dt [yr]	$P_{{\rm Random}}$	$P_{{\rm Observed}}$	$\rho$
0.5	47%	97%	< 10-7
0.4	40%	90%	< 10-7
0.3	32%	76%	< 10-6
0.2	22%	59%	< 10-4
0.1	12%	41%	< 10-5

The mean time difference between the zero epoch of the VLBI components and the beginning of the TFD flares $\Delta {t} = t_{{\rm0, VLBI}}-t_{{\rm0, TFD}}$ is $+(0.19 \pm 0.04)$ yr (ignoring component E2+B1 in 3C 279). The extrapolated ejection time of a VLBI component is therefore $\sim$0.2 yr after the beginning of the associated TFD flare, on average. This may indicate that the proper motion of a typical VLBI knot accelerates during the early stages in the component's evolution. On the other hand, we note that $t_{{\rm0, VLBI}}$ is the moment when the component is coincident with the brightness centroid of the core. In this case, the TFD flare might begin when the disturbance that creates the shock first hits the inner edge of the core, which would occur before $t_{{\rm0, VLBI}}$.

The fluxes of the VLBI components and the decomposed TFD flares are correlated. In Fig. 4 we plot the VLBI component fluxes vs. the decomposed TFD flare fluxes at the VLBI epochs (from our exponential-flare model fits). The Spearman correlation coefficient of this graph is $r_{{\rm S}}=0.817$; the probability that $r_{{\rm S}}$ would be this high from uncorrelated data $\sim$10^-14. The linear Pearson correlation coefficient $r_{{\rm P}}=0.76$, which corresponds to a probability of $\sim$10^-12 that the correlation is by chance. Furthermore, in 59% of the cases the fluxes differ by less than a factor of two. There is therefore a clear connection between new VLBI components in the jet and mm-wave flares in $\gamma$-ray blazars.

Figure 4: Decomposed TFD flare model fluxes compared with associated VLBI component fluxes at the time of the VLBI observations.