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

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4. The three-stage approach

In the light-curve approach described above, we model analytically the light curve of an outburst at different frequencies and show that the resulting typical flare is qualitatively in agreement with what is expected by shock models in relativistic jets. It is thus of interest to derive from the data the parameters that are relevant to those models.

The shock model of MG85 and its generalization by Valtaoja et al. (1992) describe the evolution of the shock by three distinct stages: 1) a rising phase, 2) a peaking phase and 3) a declining phase 1. The three-stage approach presented below is similar to that of Valtaoja et al. (1992), in the sense that its aim is simply to qualitatively describe the observations. It contains however more parameters in order to include those which are relevant to test the physical model of MG85.

The remarks of Sect. 3.1 concerning the number of outbursts and the quoted values of the reduced [FORMULA] apply equally here.

4.1. Parameterization

The self-absorbed synchrotron spectrum emitted by electrons with a power law energy distribution of the form [FORMULA] can be expressed - by generalizing the homogeneous case (e.g. Pacholczyk 1970; Stevens et al. 1995) - as


where [FORMULA] is equal to the optical depth [FORMULA] at frequency [FORMULA]. [FORMULA] and [FORMULA] are respectively the flux density and the frequency corresponding to an optical depth of [FORMULA]. At high frequency ([FORMULA]) the medium is optically thin ([FORMULA]) and the spectrum follows a power law of index [FORMULA], whereas at low frequency ([FORMULA]) it is optically thick ([FORMULA]) and the spectral index is [FORMULA]. In the case of a homogeneous source, [FORMULA].

The maximum [FORMULA] of the spectrum [FORMULA] is reached at the turnover frequency [FORMULA] corresponding to an optical depth of [FORMULA]. [FORMULA] is obtained by differentiating Eq. (3):


By developing the exponential of Eq. (4) to the third order, we obtain a good approximate: [FORMULA]. We can now rewrite Eq. (3) according to the turnover values [FORMULA], [FORMULA] and [FORMULA] by


The evolution with time of the self-absorbed synchrotron spectrum of Eq. (5) is assumed to follow three distinct stages: 1) the rising phase for [FORMULA] ; 2) the peaking phase for [FORMULA] and 3) the declining phase for [FORMULA]. The subscripts "[FORMULA]" and "[FORMULA]" refer to the end of the rising phase and the end of the peaking phase, respectively. We assume that during each stage i ([FORMULA]) both the turnover frequency [FORMULA] and the turnover flux [FORMULA] evolve with time as a power law, but with exponents that differ during the three stages:


We thus need ten parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA], to describe the evolution of the spectral turnover in the three dimensional [FORMULA]-space.

The model of MG85 predicts that both the optically thin [FORMULA] and thick [FORMULA] spectral indices should be flatter during the declining phase than during the rising and peaking phases (see Fig. 3 of Marscher et al. 1992). To test whether the spectrum is actually changing from the rising phase to the declining phase, we allow the two spectral indices [FORMULA] and [FORMULA] to have different values during these two stages. The transition during the intermediate peaking phase from the values in the rising phase ([FORMULA] and [FORMULA]) to the values in the declining phase ([FORMULA] and [FORMULA]) is assumed to be linear with the logarithm of time [FORMULA]. This adds the four parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] to the model, having thus a total of fourteen parameters to fully define the evolution of a typical flare in the [FORMULA]-space instead of the twenty parameters used in the first approach (Sect. 3.2).

The specificity of each outburst is modelled with a total of [FORMULA] parameters exactly as described in Sect. 3.2 for the light-curve approach. We do not model again the superimposed decays of the outbursts that started before 1979, but simply use the same exponential decay as obtained by the first approach (Sect. 3.2). The constant contribution of the jet's hot spot 3C 273A is also considered here. The total number of parameters in this second parameterization is a bit less than for the first one: 62 ([FORMULA]) instead of 72.

4.2. Results

To allow a better comparison with the results of the first approach (Sect. 3.2), we show in Fig. 5 the same light curves as in Fig. 2. The reduced [FORMULA] of the overall fit is now of [FORMULA]. The higher frequency light curves are relatively better described here than with the first approach (compare Figs. 2 and 5). The start times [FORMULA] of the outbursts are very similar to those obtained by the first approach, except for the fourth flare which is now starting much later at [FORMULA] = 1984.1 instead of 1983.4. This later [FORMULA] seems to be in better agreement with the observations, but the behaviour of 3C 273 during 1984-1985 is still poorly described.

[FIGURE] Fig. 5a-c. Same as Fig. 2, but with the outbursts parameterized according to the three-stage approach described in Sect. 4. a The 1.1 mm ([FORMULA] GHz) light curve ([FORMULA]); b the 37 GHz light curve ([FORMULA]); c the 8.0 GHz light curve ([FORMULA])

The obtained values of the parameters are given in Table 1. They correspond to the spectral and temporal evolution of the typical outburst shown in Fig. 6. If the tracks followed by the maximum of the spectra and of the light curves are similar to those obtained by the first approach (Fig. 4), the spectral evolution of the outburst derived here is quite different. We obtain that the spectral turnover flux [FORMULA] increases during the first 50 days ([FORMULA] year) with decreasing turnover frequency [FORMULA] as [FORMULA]. The subsequent very flat peaking phase is found to be relatively long, since it lasts 1.5 year and spans nearly one order of magnitude in frequency from 120 GHz to 13.8 GHz. The final declining phase is quite abrupt with a relation between [FORMULA] and [FORMULA] of [FORMULA]. The optically thin spectral index [FORMULA] is found to be clearly steeper in the rising phase than in the declining phase. It is flattening by [FORMULA] during the peaking phase from [FORMULA] to [FORMULA]. The optically thick spectral index [FORMULA] is found to be more constant with a slight tendency to steepen with time. It has a mean value of [FORMULA] and is steepening by [FORMULA] during the peaking phase.

[FIGURE] Fig. 6a-d. Logarithmic spectral and temporal evolution of a typical flare in 3C 273 obtained by the three-stage approach described in Sect. 4. The range covered by the axis is the same as in Fig. 4. All lines and points are defined as described in Fig. 4. The maximum of the spectrum follows the thick solid line. The three different slopes of this line correspond from left to right in panel a to [FORMULA], [FORMULA] and [FORMULA]; in panel b to [FORMULA], [FORMULA] and [FORMULA]; and in panel c to [FORMULA], [FORMULA] and [FORMULA]. The values of these slopes are given in Table 1. Notice in panel b the change of the optically thin spectral index [FORMULA] from -1.1 during the rising phase to -0.5 during the declining phase. This change is responsible for the strange shapes of the high frequency light curves in panel a


Table 1. Values of the parameters defined in Sect. 4.1 corresponding to the evolution of the typical outburst shown in Fig. 6. The two first columns display the fourteen best fit parameters, whereas other related parameters are shown in the last column

For each outburst we obtain logarithmic shifts in amplitude [FORMULA], frequency [FORMULA] and time [FORMULA], which are similar to those obtained by the first approach (Sect. 3). The dispersions [FORMULA] of [FORMULA], [FORMULA] and [FORMULA] are 0.20, 0.34 and 0.27, respectively. A possible correlation of [FORMULA] with either [FORMULA] or [FORMULA] is again not significant: the Spearman's test probability that stronger correlations could occur by chance is [FORMULA] 40 %. On the contrary, the strong correlation observed between [FORMULA] and [FORMULA] is most probably real (Spearman's test probability [FORMULA]).

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999