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

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3. The light-curve approach

We describe here an approach in which we minimize the number of model-dependent constraints. The light curve of each outburst at a given frequency is described by a simple analytical function. The choice of this function is purely empirical and does not rely on any physical model. The evolution with frequency of the outburst's light curve is left as free as possible. This model has therefore many free parameters, which can adapt to a wide range of different situations.

3.1. Number of outbursts

One crucial parameter of the decomposition is the number of outbursts. Pushed by the wish to reproduce the small features seen in the light curves, one is tempted to add always more outbursts to the fit. In Türler et al. (1999b), we published the results of a decomposition into nineteen outbursts using an approach which is similar to that described below. Here, we try to minimize as much as possible the number of outbursts to better constrain their spectral and temporal evolution. We end with twelve flares, which are absolutely necessary to describe the main features of the light curves.

The aim of the decomposition is not to reproduce the detailed structure of the light curves, but to derive the main characteristics of the outbursts. As a consequence, the [FORMULA] that we shall obtain will be statistically completely unacceptable and will have no meaning in terms of the probability that the model corresponds to what is observed. We will however refer to the obtained values of the reduced [FORMULA] (cf. Sect. 3.3), because it is the usual way to express the quality of a fit.

3.2. Parameterization

At a given frequency [FORMULA], we model the light curve [FORMULA] of a single outburst of amplitude [FORMULA], starting at time [FORMULA] and peaking at [FORMULA] by


if [FORMULA] and by


if [FORMULA]. The exponents [FORMULA] and [FORMULA] define the shape of the light curve at frequency [FORMULA] and [FORMULA] is the [FORMULA]-folding decay time of the flare at frequency [FORMULA]. Different time profiles of an outburst defined by Eqs. (1) and (2) are shown in Fig. 1.

[FIGURE] Fig. 1. Model light curve of an outburst defined by Eqs. (1) and (2), which starts at time [FORMULA] and peaks at an amplitude of [FORMULA] Jy. The three different line types show the effect of varying [FORMULA] and [FORMULA]

Rather than constraining the outburst parameters ([FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]) at each of the twelve light curve's frequencies, we describe their logarithm by a cubic spline which we parameterize at only four frequencies spaced by 0.75 dex and covering the 3 - 600 GHz range (see Fig. 3). This reduces the number of free parameters by a factor three, while keeping the parameterization completely model-independent. We thus need a total of [FORMULA] parameters to fully characterize the spectral and temporal evolution of an outburst, i.e. a surface in the three dimensional [FORMULA]-space (cf. Fig. 4).

We impose that all individual outbursts are self-similar, in the sense that they all have the same evolution pattern, i.e. the same shape of the surface in the [FORMULA]-space. What we allow to change from one outburst to the other is the normalization in flux S, frequency [FORMULA] and time t, which changes, respectively, the amplitude of the outburst (strong or weak), the frequency at which the emission peaks (high- or low-frequency peaking) and the time scale of the evolution (long-lived or short-lived). A change in normalization corresponds to a shift of the position of the outburst's characteristic surface in the [FORMULA]-space. To define this position, we take the point of maximum flux as an arbitrary reference point on the surface. On average among all individual outbursts, this point is located at [FORMULA] and this average normalization defines what we call the typical outburst of 3C 273. We denote by [FORMULA], [FORMULA] and [FORMULA] the logarithmic shifts of this point with respect to the average position, i.e. [FORMULA]. These [FORMULA] logarithmic shifts plus the 12 different start times [FORMULA] of the flares give a total of 48 parameters used to define the specificity of all outbursts.

The superimposed decays of the outbursts that started before 1979 are simply modelled by an hypothetical event of amplitude [FORMULA] at time [FORMULA] and decaying with the [FORMULA]-folding time [FORMULA] of the typical outburst at frequency [FORMULA]. The variation of the amplitude [FORMULA] with frequency is modelled by a cubic spline as for the five other variables, but parameterized at four slightly lower frequencies ([FORMULA] 0.5, 1.0, 1.5 and 2.0), due to the fact that [FORMULA] is only well constrained for the radio light curves. Finally, we assume a constant contribution to the light curves due to the quiescent emission of the jet's hot spot 3C 273A. This emission is modelled with a power law spectrum as given in Türler et al. (1999a).

To summarize, this first parameterization uses a total of 72 ([FORMULA]) parameters to adjust the 4352 observational points in the twelve light curves. The great number of free parameters still leaves more than four thousand degrees of freedom (d.o.f.) to the fit. The simultaneous fitting of the twelve light curves is performed by many iterative fits of small subsets of the 72 parameters.

3.3. Results

Fig. 2 shows three representative light curves among the twelve fitted simultaneously with the outbursts parameterized as described in Sect. 3.2. The major features of the light curves are reproduced by the model with only about one outburst every 1.5 year starting simultaneously at all frequencies. The overall fit has a reduced [FORMULA] value of [FORMULA]. The main discrepancy between the model and the observations arises during 1984-1985, when the very different light curve features in the millimetre and radio domains cannot be correctly described by the 1983.4 flare alone.

[FIGURE] Fig. 2a-c. Three of twelve light curves fitted by the cumulative sum (solid line) of twelve outbursts (dotted lines) parameterized with the light-curve approach described in Sect. 3. 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]). Each outburst starts simultaneously at all frequencies at the epoch [FORMULA] shown in panel a . The dashed line is the contribution of the jet's hot spot 3C 273A

The obtained evolution of the parameters with frequency for the typical outburst is shown in Fig. 3. The amplitude [FORMULA] of the light curve has a maximum at [FORMULA] GHz. Both the rise time [FORMULA] and the [FORMULA]-folding decay time [FORMULA] increase monotonically with wavelength. If we extrapolate the cubic spline to low frequency, it is striking to see that both [FORMULA] and [FORMULA] tend to very high values of the order of 10 years at 1 GHz, while the amplitude of the outburst would still be significant ([FORMULA] Jy). Due to the lack of submillimetre observations before 1981, the amplitude [FORMULA] is not constrained at frequencies above [FORMULA] GHz. The increase of [FORMULA] at these frequencies - due to the spline - is probably not real, but does not affect the fit because the corresponding decay time is short ([FORMULA] year). The two exponents [FORMULA] and [FORMULA] which describe the shape of the outburst's light curve are both higher at radio frequencies than in the mm/submm domain. As a consequence, the light curves at higher frequencies have a steeper rise just after the start of the outburst and a steeper decay just after the peak (see Figs. 1 and 7a).

[FIGURE] Fig. 3. Evolution with frequency of the six parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA], defined in Sect. 3.2. All these functions are a cubic spline passing through the four points with frequencies fixed at [FORMULA] 0.5, 1.0, 1.5 and 2.0 for [FORMULA] and at [FORMULA] 0.5, 1.25, 2.0 and 2.75 for the five other parameters

The five parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] define the typical outburst that can be represented in three dimensions in the [FORMULA]-space as shown in Fig. 4d. The three other panels of Fig. 4 show the three Cartesian projections of this surface. The frequency and time axes cover the same logarithmical range of 4 dex, so that the dotted diagonal in Fig. 4c corresponds to [FORMULA]. At least at low frequencies, both the maximum of the spectra and of the light curves follow quite well this diagonal. The outburst's evolution is thus amazingly symmetric in Figs. 4a and 4b. The maximum amplitude of the typical outburst is of [FORMULA] Jy and is reached after [FORMULA] months at a frequency of [FORMULA] GHz. The frequency [FORMULA] of the spectrum's maximum is steadily decreasing with time (Fig. 4c). The corresponding flux density [FORMULA] is first increasing with decreasing frequency [FORMULA] according to [FORMULA], whereas it decreases as [FORMULA] during the final decline of the outburst (Fig. 4b). This behaviour corresponds qualitatively to what is expected by shock models (e.g. MG85).

[FIGURE] Fig. 4a-d.  Logarithmic spectral and temporal evolution of the typical outburst obtained by the light-curve approach described in Sect. 3. Panel d shows the three-dimensional representation in the [FORMULA]-space. The other panels show the three Cartesian projections: a light curves at different frequencies spaced by 0.2 dex; b spectra at different times spaced by 0.2 dex; c contour plot in the frequency versus time plan. The thick solid and dashed lines show the evolution of the maximum of the spectra and of the light curves, respectively. They follow quite well the dotted diagonal in panel c corresponding to [FORMULA]. The star dot shows the point of maximum development of the typical outburst. The open circles show for each outburst the position that would have this maximum according to the shifts [FORMULA], [FORMULA] and [FORMULA]. The arrows in panel b show the frequency distribution of the twelve light curves

At frequencies above the spectral turnover ([FORMULA]), the spectral index [FORMULA] is first of [FORMULA] and steepens very slightly to [FORMULA] at the maximum development of the outburst. The somewhat chaotic behaviour during the final declining phase - due to the abrupt change in the parameter [FORMULA] - does not enable us to define a reasonable spectral index during this last stage. At frequencies below the spectral turnover ([FORMULA]), the spectral index [FORMULA] is smoothly steepening with time from [FORMULA] to [FORMULA]. This is what is expected from a synchrotron source that starts inhomogeneous and progressively becomes homogeneous (e.g. Marscher 1977).

The twelve individual outbursts have different amplitudes ranging from 5 Jy up to 32 Jy for the 1983.0 flare studied by MG85. The corresponding dispersion [FORMULA] of the amplitude shifts [FORMULA] is [FORMULA], which is slightly smaller than the dispersion of the frequency shifts [FORMULA] ([FORMULA]) and the time shifts [FORMULA] ([FORMULA]). The amplitude shifts [FORMULA] are obviously not correlated with either [FORMULA] or [FORMULA] (Fig. 4a and b). This is confirmed by a Spearman rank-order test (Bevington 1969), which yields that the observed correlations could occur by chance with a probability of more than 60 %. On the contrary, the shifts [FORMULA] and [FORMULA] align well along the [FORMULA] line (Fig. 4c) and the Spearman's test probability of [FORMULA] 0.01 % confirms that this anti-correlation is very significant.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999