## 4. The three-stage approachIn 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 The remarks of Sect. 3.1 concerning the number of outbursts and the quoted values of the reduced apply equally here. ## 4.1. ParameterizationThe self-absorbed synchrotron spectrum emitted by electrons with a power law energy distribution of the form can be expressed - by generalizing the homogeneous case (e.g. Pacholczyk 1970; Stevens et al. 1995) - as where is equal to the optical depth at frequency . and are respectively the flux density and the frequency corresponding to an optical depth of . At high frequency () the medium is optically thin () and the spectrum follows a power law of index , whereas at low frequency () it is optically thick () and the spectral index is . In the case of a homogeneous source, . The maximum of the spectrum is reached at the turnover frequency corresponding to an optical depth of . is obtained by differentiating Eq. (3): By developing the exponential of Eq. (4) to the third order, we obtain a good approximate: . We can now rewrite Eq. (3) according to the turnover values , and 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 ; 2) the peaking phase
for and 3) the declining phase for
. The subscripts
"" and
"" refer to the end of the rising
phase and the end of the peaking phase, respectively. We assume that
during each stage We thus need ten parameters: , , , , , , , , and , to describe the evolution of the spectral turnover in the three dimensional -space. The model of MG85 predicts that both the optically thin and thick 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 and to have different values during these two stages. The transition during the intermediate peaking phase from the values in the rising phase ( and ) to the values in the declining phase ( and ) is assumed to be linear with the logarithm of time . This adds the four parameters , , and to the model, having thus a total of fourteen parameters to fully define the evolution of a typical flare in the -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 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 () instead of 72. ## 4.2. ResultsTo 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 of the overall fit is now of . The higher frequency light curves are relatively better described here than with the first approach (compare Figs. 2 and 5). The start times 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 = 1984.1 instead of 1983.4. This later seems to be in better agreement with the observations, but the behaviour of 3C 273 during 1984-1985 is still poorly described.
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 increases during the first 50 days ( year) with decreasing turnover frequency as . 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 and of . The optically thin spectral index is found to be clearly steeper in the rising phase than in the declining phase. It is flattening by during the peaking phase from to . The optically thick spectral index is found to be more constant with a slight tendency to steepen with time. It has a mean value of and is steepening by during the peaking phase.
For each outburst we obtain logarithmic shifts in amplitude , frequency and time , which are similar to those obtained by the first approach (Sect. 3). The dispersions of , and are 0.20, 0.34 and 0.27, respectively. A possible correlation of with either or is again not significant: the Spearman's test probability that stronger correlations could occur by chance is 40 %. On the contrary, the strong correlation observed between and is most probably real (Spearman's test probability ). © European Southern Observatory (ESO) 1999 Online publication: August 25, 1999 |