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Astron. Astrophys. 345, 769-777 (1999)

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4. Results and discussion

For each source we have compiled flux densities at different frequencies from the literature, mostly from Kühr et al. (1979) and from the CATS database (Verkhodanov et al. 1997). Our new measures at 230 GHz have been added to the compilation. All flux densities have been brought to the BGPW scale (Baars et al. 1977). Examples of source spectra are shown in Fig. 1. For the fit algorithm described in Sect. 4.2 we have assumed flux densities less than 3[FORMULA] to be upper limits only.

[FIGURE] Fig. 1. Selected source spectra (left: type a, right: type b; see Sect. 4.3). The [FORMULA]-values refer to the pure synchrotron loss models. The self-absorbed part of the spectrum has been modelled in a subsequent fit procedure.

Most of the sources show significant departure from the classical power law which describes a zero age transparent synchrotron spectrum from a relativistic electron population with power law energy distribution. The deviations from the power law are of the following type: a) a low-frequency turnover (the most conspicuous deviation); b) a steepening at high frequencies. High-frequency flattening, if any, is quite rare. The above deviations are interpreted as due to synchrotron self-absorption and to particle energy losses, respectively. In order to describe them, we have fitted the compiled flux densities with a synchrotron aged spectrum [FORMULA] (described in the next section), modified by low-frequency absorption, as follows (Pacholczyk 1970):


where [FORMULA] is the frequency at which the optical depth is equal to 1. In case of an homogeneous synchrotron self-absorbed source [FORMULA], while [FORMULA] is the not aged spectral index in the transparent frequency range.

4.1. The synchrotron aged spectrum model

We assume that the radio source evolution is described by a continuous injection model , where the sources are continuously replenished by a constant flow of fresh relativistic particles with a power law energy distribution, with exponent [FORMULA]. It is well known that, under these assumptions, the radio spectrum has a standard shape (Kardashev 1962), with spectral index [FORMULA] below a critical frequency [FORMULA] and [FORMULA] above [FORMULA]. If there is no expansion and the magnetic field is constant, the frequency [FORMULA] (in GHz) depends on the elapsed time since the source formation, [FORMULA] (in Myrs), the intensity of the magnetic field B (in µG) and the magnetic field equivalent to the microwave background [FORMULA] (in µG) as:


The whole spectral shape cannot be described by an analytic equation, the two behaviours described being only the asymptotical ones, and has to be computed numerically. This model is referred as continuous injection (CI). Fitting the spectral data to the numerically computed CI spectrum, one obtains the break frequency [FORMULA], from which the source age is obtained if the magnetic field is known.

This simple model does not consider expansion effects, which may be important if the source is young. So, the simplest variant of the original model is the one in which the radiating particles loose energy through expansion and the magnetic field changes according to flux conservation (Kardashev 1962, CIE). An alternative possibility to be considered, in an expanding source, is that the magnetic field changes less rapidly than in the flux conserving assumption because of a continuous magnetic flux input associated to the fresh particles injection. We set [FORMULA], where for m = 2 we get the flux conserving expansion. This is referred as CIm model. In our special case we assume m = 1, consistent with the models applied by Baldwin (1982) or Begelman (1996). Although the theoretical background to these models is in the Kardashev paper, we have decided to present in the Appendix the detailed development. The spectral shapes of the CIE and CIm models have been computed again numerically. The break frequency for the CIE model is sixteen times higher than in the CI case, for equal elapsed time and final magnetic field intensity (Kardashev 1962). In the CIm model, instead, one finds that the break frequency is [FORMULA] larger than in the CI model. The asymptotic behaviours at frequencies lower and larger than the break frequency are the same as in the CI model. However the steepening occurs over a broader frequency interval. In order to better emphasize the differences between these three models it is useful to consider, together with the usual flux-frequency representation, the point-to-point spectral index defined as


Both representations are shown in Fig. 2. While in the flux-frequency plane the differences are hardly visible, they can be much better traced in the point-to-point spectral index behaviour. Note that the displayed figures only show the theoretical differences. In practice the observed spectra permit only fits in the flux-frequency plane.

[FIGURE] Fig. 2. The three continuous injection models described in the text. The solid line corresponds to the CI, the dashed line to the CIE and the dot-dashed line to the CIm. On the top the flux density (arbitrary units) is plotted as a function of the ratio [FORMULA] (flux-frequency plane), on the bottom the point-to-point spectral index is shown as a function of [FORMULA]. The injection spectral index is the same for all three models [FORMULA].

4.2. The spectral fits

Spectral fits to the spectra have been made with the CI, CIE, and CIm models. They allow us to determine the non-aged spectral index [FORMULA] and the break frequency [FORMULA], that, together with the normalization, are the three free parameters characterizing all the models. The best fits are, surprisingly, obtained with the CI model. The other models have steepenings which are too gradual for the majority of the spectra. An example is given in Fig. 3. The reduced [FORMULA] of the models including adiabatic expansion and a variable magnetic field is always greater (typically twice) than that of the CI model (Fig. 4). The CI fits appear quite good, even in cases of high [FORMULA] values, which, at visual inspection, appear more due to an under-estimation of the flux density errors than to a poor fit of the spectral model on the data.

[FIGURE] Fig. 3. Three different fit results for the source 2252+12. The solid line corresponds to a pure CI model, the dashed line represents the CIE model, the dot-dashed line stands for the CIm model. Note the different qualities of the fits, expressed by the various [FORMULA] values!

[FIGURE] Fig. 4. [FORMULA] histograms for the CI, CIE, and CIm models. On the bottom right the histogram of the ratio between the CIm and the CI [FORMULA] is also shown.

The majority of the spectra show a clear break frequency, with a change of slope [FORMULA]. We stress that there is no evidence for spectral steepening with [FORMULA] significantly larger than 0.5. Only a few sources are fitted by simple power laws. In these cases [FORMULA] could be either very high ([FORMULA] GHz) or very low ([FORMULA]). In these sources we have preferred the low frequency choice, since for the high frequency one would have implied abnormally high values for [FORMULA] as compared to the other sources.

We also note that only a very small fraction of the CSS sources, if any, shows some evidence of flux density excess at high frequency, as it would be caused by a flat spectrum core or by thermal dust emission. Perhaps the only case is 3C138, where the core, known from VLBI observations, shows up in the integrated spectrum at 230 GHz only.

It may appear surprising that the fits with the assumed spectral model are so good. In fact the sources of the sample consist of several components, as lobes and jets and hot spots, where physical conditions can differ from one another and therefore also break frequencies may be different. It is likely that the spectrum is dominated by the brighter component(s). In addition, one could imagine that some confusion might have occurred between genuine spectral steepening due to energy losses and the low-frequency turnover due to absorption processes. We feel that this is a minor problem, but it is difficult to quantify it (see, however, next section).

The break frequencies range from a few hundred MHz to tens of GHz. At low frequencies, as said above, the limit is set by confusion with the effects of absorption processes. The injection spectral index [FORMULA] ranges from 0.35 to 0.8, with [FORMULA]. The typical errors of the break frequencies and the injection spectral indices as given by the fit algorithm are up to 40% and 0.05, respectively. The results of the CI fits are compiled in Table 2.


Table 2. Physical parameters

4.3. Radiative ages and the nature of CSS sources

In order to determine radiative ages, from Eq. (2) and variants of the other models, the magnetic field B has to be known. We stress that the age depends rather strongly on B, which is somewhat uncertain. We take the equipartition value [FORMULA]. Our assumption is motivated by the fact that [FORMULA] accounts rather well for the low-frequency turnover in terms of synchrotron self-absorption. We are aware that this is not a proof for equipartition. Other authors (e.g. Bicknell et al. 1997) prefer instead thermal absorption. In any case, [FORMULA] represents a poor statistical upper limit to B, in the sense that, were it larger by a factor of four, the low frequency turnovers would be systematically higher than observed by [FORMULA] 30%.

Using the value of [FORMULA], we obtain, from Eq. (2), radiative ages [FORMULA] ranging from [FORMULA] to [FORMULA] years. Since the intrinsic magnetic fields of the CSS sources in our sample strongly overweight the magnetic field equivalent to the cosmic microwave background ([FORMULA]), the latter can be neglected in Eq. (2). Therefore, if the source magnetic field deviates by a factor f from the field determined for equipartition [FORMULA], the radiative ages will change by [FORMULA]. These radiative ages do not necessarily represent the source ages, but rather the radiative ages of the dominant source component(s). Only when the lobes, which have accumulated the electrons produced over the source lifetime, dominate the source spectrum, the radiative age [FORMULA] is likely to represent the age of the source. If, instead, the spectrum is dominated by a jet or by hot spots, the radiative age likely represents the permanence time of the electrons in that component and is expected to be less (perhaps much less) than the source age. In addition, dominant jets or hot spots might have their break frequency up-shifted by Doppler effects.

The existing structure information on our sample, mostly from MERLIN and VLBI observations, allows us to split the sources in two groups: those in which the overall spectrum is dominated by lobes (classified "type a" in Table 2); those in which the spectrum is dominated by a bright jet or hot spot (classified "type b" in Table 2). For the B3 VLA sample the available information does not yet allow such a morphological sub-division.

The sources of class b have radiative ages systematically lower than those of class a, as we expected. Furthermore, while the synchrotron age is well correlated with the source size for class a, it seems that there is no correlation at all between the linear size and the radiative age for class b sources (see Fig. 5). We have further computed for each source the expansion velocity [FORMULA], where LS is the source largest dimension. The two classes show very different distributions:


Provided the assumed magnetic field is reasonably correct, the above values for the radiative ages indicate that the CSS sources are young . We further note that the ages, and corresponding expansion velocities, are not far (somewhat larger) from the recent results on expansion of CSOs by Owsianik et al. (1998) and Owsianik & Conway (1998). As the radiative ages are strongly dependent on the assumed magnetic field, a field only a factor two lower than assumed would be required for a better agreement.

[FIGURE] Fig. 5. Linear size as a function of the synchrotron age for type a (filled dots) and type b (open dots) sources. The B3 VLA sources have been excluded. The horizontal dashed lines represent the selection limits of the linear size distribution of the sources in our samples while the diagonal lines reflect constant values of [FORMULA].

In order to maintain the frustration scenario , in which the sources' lifetimes are [FORMULA] years, their equipartition magnetic field should be decreased by a factor [FORMULA].

One could think that the correlation between linear sizes and the radiative ages shown in Fig. 5 could be a partial consequence of the equipartition assumption. In fact, [FORMULA] implies that [FORMULA]. This is not the case for the following reasons:

1) no correlation between linear sizes and radiative ages is obvious for type b sources.

2) In particular, the break frequencies seem to be correlated with the linear sizes for type a sources (Fig. 6).

[FIGURE] Fig. 6. Linear size as a function of the break frequency (in the source rest frame) for type a (filled dots) and type b (open dots) sources. The B3 VLA sources have been excluded. The horizontal dashed lines represent the selection limits of the linear size distribution of the sources in our samples.

These means, at least for class a sources, that the break frequency is an effective clock indicating the source age. The correlation seen in Fig. 5 is not an artifact completely introduced by the assumption of equipartition in Eq. (2).

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999