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Astron. Astrophys. 349, 45-54 (1999)

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5. Discussion

The two approaches presented above give comparable results, but differ concerning the existence of a nearly constant peaking phase and the shapes of the spectra (compare Figs. 4b and 6b). The origin of these differences can be understood by comparing the light curve profiles obtained by the two approaches, which are shown in Fig. 7. It is clear that the first approach allowing only a rising phase and a declining phase cannot mimic the three-stage profiles of Fig. 7b resulting from the second approach (Sect. 4.1). On the other hand, only the light-curve approach is able to produce a round peaking phase as seen in Fig. 7a. In a forthcoming paper (Türler et al. in preparation), we will present the results of an hybrid approach, which incorporates the advantages of both approaches in order to better define the properties of the typical outburst.

[FIGURE] Fig. 7a and b. Light curves at different frequencies for the typical outburst obtained with the light-curve approach a and the three-stage approach b . The dotted light curves are spaced by 0.1 dex in frequency and the six solid light curves are at frequencies [FORMULA], in order of increasing time scales

5.1. Do the outbursts correspond to VLBI components?

The decomposition of the light curves into distinct outbursts was motivated by the observation with very long baseline interferometry (VLBI) of distinct components in the jet structure of 3C 273. Since the detection of a new VLBI component (Krichbaum et al. 1990) associated with the strong optical/infrared flare of 1988 in 3C 273 (Courvoisier et al. 1988), there is good evidence that outbursts are related to the ejection of new VLBI knots. To test whether all outbursts are actually associated with superluminal components, we compare in Table 2 the start time [FORMULA] of an outburst - as obtained by the three-stage approach - with the ejection time "[FORMULA] (knot)" of a new VLBI knot as given by Abraham et al. (1996) and Zensus et al. (1990). For each of the eight first outbursts, we can identify one or two possibly associated VLBI components.


Table 2. Relation between VLBI components and the eight first outbursts as obtained with the three-stage approach (Sect. 4). The parameters in this table are defined in Sect. 5.1

To test further this relationship, we compare the flux densities "[FORMULA] (knot)" of the VLBI components observed at epoch [FORMULA] and at a frequency of 10.7 GHz (Abraham et al. 1996) with the flux densities [FORMULA] expected at the same epoch and the same frequency according to the outburst parameters derived here. Table 2 shows that for the five first outbursts there is always one of the possibly associated knots (indicated by an arrow), which has the expected flux. For the three remaining outbursts and especially for the 1988.1 flare, the relation between [FORMULA] (knot) and [FORMULA] is not obvious. At this epoch however, the possibly associated components are still strongly blended by the core emission (component "D") or might even still be part of the unresolved core 2. The total flux [FORMULA] Jy expected by the 1986.3, 1988.1 and 1990.3 outbursts is indeed equal to the observed total flux [FORMULA] Jy of the C9, C10 and D components. These results strongly suggest that there is a close relation between the outbursts and the VLBI knots and hence that our decomposition describes a real physical aspect of the jet.

5.2. How can we understand the peculiarities of individual outbursts?

The relation found between the outbursts and the VLBI knots (Sect. 5.1) has established that our decomposition is not purely mathematical, but does correspond to a physical reality. There should therefore be a physical origin to the clear anti-correlation found between the frequency shifts [FORMULA] and the time shifts [FORMULA] of the individual outbursts. The observed frequency shifts [FORMULA] confirm that 3C 273 emits both low- and high-frequency peaking outbursts (Lainela et al. 1992). The relation between [FORMULA] and [FORMULA] clearly shows that high-frequency peaking flares evolve faster than low-frequency peaking outbursts. The alignment of the shifts along the [FORMULA] line (Figs. 4c and 6c) further suggests the relation [FORMULA].

The origin of this relation could be due to a change [FORMULA] of the Doppler factor [FORMULA], which depends on the flow speed [FORMULA], the Lorentz factor [FORMULA] and the angle to the line of sight [FORMULA]. Observed quantities (unprimed) are related to emitted quantities (primed) as (e.g. Hughes & Miller 1991; Pearson & Zensus 1987):


If we assume that in the jet frame all outbursts are alike (i.e. [FORMULA]), the observed relation [FORMULA] can be interpreted as a change [FORMULA] of the Doppler factor from one outburst to the other. In this case, however, there should also be correlations between [FORMULA] and both [FORMULA] and [FORMULA], which are not observed.

Alternatively, we can consider that the Doppler factor does not change ([FORMULA]) and that the observed relation between [FORMULA] and [FORMULA] is intrinsic and independent of possible flux variations [FORMULA]. Such a correlation might be related to the distance from the core at which the shock forms (Lainela et al. 1992). Indeed, Blandford (1990) shows that for a simple conical jet with constant speed v the frequency of maximum emission [FORMULA] is inversely proportional to the distance down the jet [FORMULA] ([FORMULA]), while the corresponding flux density [FORMULA] is constant. Since the speed v is constant, the turnover frequency [FORMULA] is then also inversely proportional to time ([FORMULA]), as observed. If a shock forms in such an underlying jet at a distance [FORMULA] from the core, both the frequency range of the emission and the time scale of the evolution will depend on the distance [FORMULA], as illustrated in Fig. 8. We therefore propose that short-lived and high-frequency peaking flares are actually inner outbursts, whereas long-lived and low-frequency peaking flares are outer outbursts.

[FIGURE] Fig. 8. Schematic representation of a conical jet with constant speed v in which the frequency of maximum emission [FORMULA] (greyscale) is inversely proportional to the distance down the jet [FORMULA] ([FORMULA]). Let us assume that the peaking phase of an outburst corresponds to an octave of radius ([FORMULA]). This phase has a duration and a frequency range which clearly depend on the distance [FORMULA] at which the shock forms. Inner outbursts peaking from [FORMULA] to [FORMULA] are four times shorter than outer outbursts peaking from [FORMULA] to [FORMULA], while their emission is maximum at four times higher frequencies

This interpretation is supported by the existence of short-lived VLBI components which are only seen close to the core. In our decomposition, the two most short-lived and the most high-frequency peaking outbursts are the two successive flares of 1982.4 and 1983.1. Their start times correspond well to the period from 1981 to 1983 during which only short-lived VLBI components were formed (Abraham et al. 1996). If our interpretation is right, the shifts [FORMULA] and [FORMULA] that we obtain suggest that the 1982.4 flare would have formed about two times closer to the core than the 1983.1 flare and four times closer than the typical outburst.

5.3. What are the constraints for shock models?

According to the shock model of MG85, the optically thin spectral index [FORMULA] should be steeper during the two first stages of the outburst evolution than the usual value of [FORMULA] (Sect. 4.1). A steeper index arises due to the fact that the thickness x of the emitting region behind the shock front is proportional to the cooling time [FORMULA] of the electrons suffering radiative (Compton and/or synchrotron) losses. During the rising and peaking phases, radiative losses are dominant and therefore the thickness x is frequency dependent as [FORMULA], which leads to a steeper optically thin spectral index of [FORMULA]. Until now, the expected flattening of the spectral index by [FORMULA] from the rising and peaking phases to the declining phase was never observed and furthermore the optically thin spectral index [FORMULA] observed at the beginning of the outburst was often found to be already too flat ([FORMULA]) to allow the expected subsequent flattening (Valtaoja et al. 1988; Lainela 1994).

The present result that the optically thin spectral index is flattening with time by [FORMULA] is in good agreement with the change of [FORMULA] expected by the shock model of MG85. The observed flattening of the spectrum is contrary to the steepening with time expected as a result of radiative energy losses by the electrons. The observed behaviour can however also be understood as a change of slope with frequency rather than with time and thus it could conceivably be due to a spectral break that steepens the optically thin spectral index by a factor of 0.5 at higher frequencies. Such a break is expected in the case of continuous injection or reacceleration of electrons suffering radiative losses (Kardashev 1962) and is observed in several hot spots including 3C 273A (Meisenheimer et al. 1989). Whatever the interpretation, the flatter index, [FORMULA], is the relevant index to determine that the electron energy index s ([FORMULA]) is [FORMULA]. This value corresponds to the average value observed in several hot spots (Meisenheimer et al. 1989) and is in agreement with the values expected if the electrons are accelerated by a Fermi mechanism in a relativistic shock (e.g. Longair 1994).

The long flat peaking phase observed in 3C 273 contrasts with the complete absence of this stage in 3C 345 (Stevens et al. 1996). This difference is surprising, because the outburst's evolution is otherwise very similar in these two objects with nearly the same indices for the rising and the declining phases: [FORMULA] in 3C 273 and -0.86 in 3C 345 and [FORMULA] in 3C 273 and [FORMULA] in 3C 345 ([FORMULA]). A value of [FORMULA] was also found in several other sources by Valtaoja et al. (1988). This decrease of the turnover flux with decreasing frequency is steeper than expected by the simplest model of MG85; i.e. with a conical adiabatic jet having a constant Doppler factor [FORMULA]. With [FORMULA] and a magnetic field B oriented perpendicular to the jet axis, their model predicts [FORMULA]. This discrepancy between the observations and the shock model of MG85 was already pointed out by Stevens et al. (1996). We refer the reader to their discussion of two more general cases of the MG85 model: 1) a straight non-adiabatic jet and 2) a curved adiabatic jet. With the observed values of the indices [FORMULA] and [FORMULA], these authors could determine the two free parameters of the model. In our case, with the constraints of all six indices [FORMULA] and [FORMULA] ([FORMULA]), we could not find a good agreement with either of the two models mentioned above. In a forthcoming paper (Türler et al. in preparation), we will further discuss this point and explore whether a non-conical non-adiabatic curved jet can well describe the observations.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999