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Appendix A
Here we present the results for the frequency dependent light curve parameters including the entire range of shock parameters (see Table 1). The plots in this Appendix provide a global view of the changes in the slopes of the different stages in the (ν_{m} − S_{m}) plane and the light curve parameters.
A.1. Slopes of the energy loss stages
Figures A.1 and A.2 show the maps of the values obtained for the slopes of the different energy loss stages as a function of d, b, and s, indicating some colour levels (black, solid lines) in the plot to help identifying the values, and providing also a dashed line that separates the case of evolution towards a magnetically dominated from evolution towards particle dominated flows. This line is derived as follows: Taking into account that the magnetic energy density is u_{B} ∝ B^{2} and using B ∝ R^{− b}, we obtain u_{B} ∝ R^{−2b}. For particles, u_{e} ∝ ^{∫}n(γ)γdγ using n(γ) ∝ Kγ^{−}s. Neglecting the evolution of γ with distance, we can assume u_{e} ∝ K, and using K ∝ R^{−k} and k = 2(s + 2)/3 if the jet expands adiabatically, we have u_{B}/u_{e} ∝ R^{−2b + 2(s + 2)/3}. Imposing independence with distance brings the exponent to zero, which requires b = (s + 2)/3. If b< (s + 2)/3, the ratio grows with distance, whereas for b> (s + 2)/3 the ratio decreases with distance. Each panel shows the variation of ϵ_{i} (i = 1 Compton, i = 2 synchrotron, and i = 3 adiabatic) for 2 <s< 3 and 1 <b< 2 and a fixed value of d. The value of d is changing from top to bottom from d = −0.45 to d = 0.45 (see also the figure captions). The left column in both plots shows the maps of values of ϵ_{C} as a function of b and d. The vertical levels indicate that this slope is fairly independent of s for any values of b and d, and that it mainly changes with these two parameters. In the case of ϵ_{S}, the slope of the synchrotron stage, the situation is different, and s and d appear to be the most relevant parameters to determine it, although there is also a smooth gradient of this slope in the direction of b for the extreme values of d. Finally, the third column shows the maps of ϵ_{A}, which is most sensitive to d and b, and only shows a smooth variation with s.
A.2. Slopes of the frequency dependent light curve parameters
Figures A.3–A.6 show maps of the exponents of the frequency dependent light curve parameters as a function of s and b for different values of d. As mentioned earlier, the exponents for the flare amplitude and the flare time scale can be obtained either from the rising edge or the decaying edge of the light curve. In Figs. A.3 and A.4 we present these parameters obtained from the
rising edge and in Figs. A.5 and A.6 for the decaying edge of the light curves. In Figs. A.3 and A.4, each panel shows, from left to right, the variation in the flare amplitude exponent, ϵ_{flare amp.}, the flare time scale exponent ϵ_{flare time scale} and the crossband delay exponent, ϵ_{delay}, for 2 <s< 3 and 1 <b< 2 and a fixed value of d. The value of d is changing from top to bottom from d = 0.45 to d = −0.45 (see also the figure captions). The amplitude of the flare undergoes a stronger variation with frequency for decreasing Doppler factors with distance (Fig. A.3) than for the increasing (Fig. A.4), as indicated by the colourscales. In all cases, the slope grows with increasing s and b. The time lapse between the onset of the flare and the peak at each frequency is more sensitive to changes in frequency for decreasing Doppler factors with distance. This time lapse is more sensitive to b than to s, and the difference among frequencies becomes larger (smaller ϵ_{flare time scale}) for values of b closer to 1. Finally, the time lag between the peaks at different frequencies and a reference one has a similar behaviour with respect to the relevant parameters to the time lapse between onsets and peaks. The main difference is that there is not a large difference in the slopes between positive and negative values of d and that there are clear discontinuities in the values of the ϵ_{time lag} for increasing Doppler factors, at certain values of s.
Fig. A.1
Parameter space plots for the variation of the slopes, ϵ_{i} as function of b and s while keeping the d parameter fixed. The columns show from left to right, the slope of the Compton stage, ϵ_{C}, the slope of the synchrotron stage, ϵ_{S}, and the slope of the adiabatic stage, ϵ_{A}. The exponent for the evolution of the Doppler factor, d, is from top to bottom d = −0.45, d = −0.30, d = −0.15, and d = 0. The black dashed line corresponds to a constant u_{B}/u_{e} ratio with distance (b_{eq} = (s + 2)/3)), i.e. to the left of this line the jet flow tends to be magnetically dominated with distance and to the right the jet tends to be particle energy dominated with distance. 

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Fig. A.2
Same as Fig .A.1 for d = 0.15, d = 0.30, and d = 0.45. 

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In Figs. A.5 and A.6 we show the variation of the exponent for the flare amplitude and the flare time scale obtained from the decaying edge of the light curve. The exponent for the flare amplitude ϵ_{flare amp. decay} decreases with d. For d< 0 the absolute value of the exponent increases with s and b. However, for d> 0 the distribution of ϵ_{flare amp. decay} changes: The exponents still increase with s but larger values are obtained towards b = 1. The exponent for the flare time scale derived from the decaying edge of the light curve, ϵ_{flare time decay} is small, typically <0.05. The value and its distribution depend strongly on d. For d< 0 the distribution is smooth and the values decrease with s and b. Nearly no variation in ϵ_{flare time. decay} is obtained for d> 0 (see second column in Figs. A.5 and A.6).
Figures A.7 and A.8 show the expected time lags (in years) between the peaks at 5 GHz, 15 GHz, and 140 GHz and our reference frequency, 345 GHz (left, central and right columns, respectively), for different values of d (different rows), as a function of s and b. The crossband delays become shorter for increasing Doppler factor with distance as indicated by the colour scales at the top of the panels. The time lags between the reference frequency and low frequencies are typically more sensitive to b increasing as this parameter tends to 1, whereas the time lags between 140 GHz and 345 GHz show significant values only for decreasing Doppler factor with distance and higher sensitivity to the spectral slope s.
Fig. A.3
Parameter space plots for the variation of frequency dependent singledish light curve parameters obtained from the rising edge of the light curves as function of b and s while keeping the d parameter fixed. The columns show from left to right the exponent for the variability amplitude, ϵ_{flare amp.}, the exponent for the variability time scale, ϵ_{flare time scale}, and the exponent for the time lag, ϵ_{delay}. The exponent for the evolution of the Doppler factor, d, is from top to bottom d = −0.45, d = −0.30, d = −0.15, and d = 0. The black dashed line corresponds to a constant u_{B}/u_{e} ratio with distance (b_{eq} = (s + 2)/3)), i.e. to the left of this line the jet flow tends to be magnetically dominated with distance and to the right the jet tends to be particle energy dominated with distance. 

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Fig. A.4
Same as Fig. A.3 for d = 0.15, d = 0.30, and d = 0.45. 

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Fig. A.5
Parameter space plots for the variation of frequency dependent singledish light curve parameters obtained from the decaying edge of the light curves as function of b and s while keeping the d parameter fixed. The columns show from left to right, the exponent for the variability amplitude, ϵ_{var. amp.}, the exponent for the variability time scale, ϵ_{var. time scale}, and the exponent for the crossband delay, ϵ_{delay}. The exponent for the evolution of the Doppler factor, d, is from top to bottom d = −0.45, d = −0.30, d = −0.15, and d = 0. The black dashed line corresponds to a constant u_{B}/u_{e} ratio with distance (b_{eq} = (s + 2)/3)), i.e. to the left of this line the jet flow tends to be magnetically dominated with distance and to the right the jet tends to be particle energy dominated with distance. 

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Fig. A.6
Same as Fig. A.3 for d = 0.15, d = 0.30, and d = 0.45. 

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Fig. A.7
Parameter space plots for the for the time lag between three selected frequencies as function of b and s while keeping the d parameter fixed. The columns show from left to right, the (345−5) GHz time lag, the (345−15) GHz time lag, and the (345−86) GHz time lag. The exponent for the evolution of the Doppler factor, d, is from top to bottom d = −0.45, d = −0.30, d = −0.15, and d = 0. The black dashed line corresponds to a constant u_{B}/u_{e} ratio with distance (b_{eq} = (s + 2)/3)), i.e. to the left of this line the jet flow tends to be magnetically dominated with distance and to the right the jet tends to be particle energy dominated with distance. 

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Fig. A.8
Same as Fig. A.7 for d = 0.15, d = 0.30, and d = 0.45. 

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Fig. A.9
Modified Compton stage model using d = 0. The top row shows the slopes for the different energy loss stages from left to right: compton, synchrotron, and adiabatic stage. The delay between selected frequencies with respect to 345 GHz in years is plotted in the second row from left to right: delay to 5 GHz, delay to 15 GHz and delay to 86 GHz. The third row presents the frequency dependent light curve parameters obtained from the rising edge of the light curve from left to right: flare amplitude, flare time scale and cross frequency delay. The bottom row shows the exponent for the flare amplitude and the flare time scale as derived from the decaying edge of the light curve. The black dashed line corresponds to a constant u_{B}/u_{e} ratio with distance (b_{eq} = (s + 2)/3)), i.e. to the left of this line the jet flow tends to be magnetically dominated with distance and to the right the jet tends to be particle energy dominated with distance. 

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© ESO, 2015