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13 Conclusions

The Cannonball model gives an excellent and extremely simple description of all measured properties of GRB afterglows, including their radio afterglows.

In the CB model, there is an injection bend in the spectrum, at the predicted time-dependent frequency $\rm\nu_b$ of Eq. (14). We have shown that the evidence for the correctness of this prediction is very strong, see Table 2 and Figs. 3 and 4. It is this spectral bend that governs the relative normalization of the radio and the optical AG, again in agreement with observation, as shown in all our figures of AG wide-band spectra.

Since we have always set the electron index to its theoretical value, p=2.2, just three intrinsic parameters are needed to describe an optical or X-ray AG: $\gamma_0$, $x_\infty$ and the normalization; the viewing angle $\theta$, although it must also be fit, is external to the GRB, like the redshift and the absorption in the host and in the Galaxy are. We have shown that, in the CB model, the extension of these results to the radio domain requires the introduction of just one extra parameter: the free-free absorption frequency $\rm\nu_a$ of Eq. (23), and that, in spite of various approximations, this simplest of descriptions is at the moment entirely satisfactory. Notice that what one has to parametrize is a two dimensional surface: the fluence as a function of frequency and time. The shape of this surface is that of a relatively simple ``mountain'', various cuts of which at fixed $\rm t$ or $\nu$are shown if Fig. 1. It would be easy, and it may well be misleading, to overparametrize this rather featureless surface with more than a few parameters.

It is instructive to compare, or so Occam would have thought, the understanding of wide-band AG spectra in the CB model with that in the fireball or firetrumpet models. In the latter, the number of intrinsic parameters varies: seven (e.g. Berger et al. 2001a), eight (e.g. Yost et al. 2001) nine (e.g. Yost et al. 2002) and even thirteen (e.g. Galama et al. 2000). This counting does not include the viewing angle, since the firetrumpets in these works point precisely at the observer[*]. Moreover, even before the ``break'' in the time-evolution - a period during which it is not inconsistent to use the quasi-spherical self-similar approximation of Blandford & McKee (1976) for the expanding material - the ordering of the ``breaks'' in frequency implies a multiple choice of spectral shapes and of their evolution (Granot & Sari 2002).

Countrary to established custom, we are not presenting the $\chi^2$values of our fits, which are generally reasonable and would become quite good if, again following the consuetudinary path, we artificially increased the errors to compensate for scintillations in the radio data and/or uncertainties in attenuation. The reason is that the CB model is a very simplified description of a no doubt very complicated reality (e.g. CBs could be somewhat comet-like, as opposed to spherical, their inner distributions of density, ionization, magnetic field and temperature could be non-trivial, even chaotic, etc.). Even when the physics is much simpler than in the analysis of radio emissions, and the fits are very good - as is the case in our description of optical and X-ray AGs in DDD 2001 - we do not report their quite impressive $\chi^2$values[*]. We view our ``fits'' as rough descriptions, rather than true fits. Under such circumstances, the overintrepretation of a $\chi^2$ test has every chance of being misleading, much more so in models containing many more parameters than the CB model.

For the same reasons, and because of the systematic errors in the data, the values of the parameters we extract from our fits should not be taken entirely at face value, even though the minimization procedure - which attributes to the errors a counterfactual purely statistical origin - results in tiny 1 $\sigma$ spreads for the fitted parameters, and in $\chi^2$ values that are in most cases satisfactory.

In the radio domain, as in every other aspect, the pair SN1998bw/GRB 980425 is particularly fascinating. On the basis of this GRB's observed fluence and distance, and given the (totally trivial but all important) dependence of the fluence on observation angle, we claimed in Dar & De Rújula (2000a) that the only peculiarity of this pair was that it was observed uncharacteristically far from its axis (for a GRB) and uncharacteristically close to it (for a SN). In DDD 2001 we proved that the X-ray AG of GRB 980245 was also what is expected in the CB model, depriving the supernova of its X-ray peculiarity: it did not make the observed X-rays. In this paper, by understanding the magnitude, time- and frequency-dependence of the pair's radio signals - which were not emitted, either, by the SN - we have demonstrated that SN1998bw was also ``radio normal''. Neither this GRB, nor its SN have - in the CB model - anything in particular, except the chance occurrences of the distance and observation angle. Alas, the unique occasion to make a fundamental discovery by actually resolving the SN and the CB, as proposed in Dar & De Rújula (2000a), may now be very difficult, but, as we have explained, not entirely out of the question.

By pure coincidence, the apparent angular velocities of galactic pulsars and cosmological cannonballs are of the same order of magnitude. The analysis of radio scintillations, one of the methods used to measure pulsars velocities, should also be applicable to the GRB ejecta. Thus, it ought to be possible to test the CB-model's prediction of hyperluminal cannonball velocities.


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