Astron. Astrophys. 323, 312-316 (1997)

4. Discussion and conclusions

Up to now, many features of GRBs, such as energy sources, radiation mechanisms et al., still remain mysterious. Among them, the distance scale of GRBs is the most puzzling question. Since the distance is unknown, the theoretical model is difficult to be built to account for the other characteristics of GRBs. Thus, a lot of efforts have been made to estimate the distance scale of GRBs.

Many tests show time dilation of between the BATSE bright bursts and the dimmest bursts (Norris 1994; Norris et al. 1994; Davis et al. 1994), this provides a clue to the distance scale of GRBs. On the other hand, the statistics can also give a distance scale of the dimmest bursts. These two methods are independent, so if the distances determined from these two methods are consistent, it would be a strong evidence in favor of cosmological origin of GRBs. Fenimore & Bloom(1995) calculated the redshift z according to the observed time dilation and the distribution, and found that the results are quite different, thus they conclude that either a large fraction of the observed time dilation is intrinsic to the bursts rather than be the result of expansion of the universe, or strong density evolution and/or luminosity evolution would be required. Here we performed a calculation similar to that of Fenimore & Bloom(1995), but assuming a power-law spectrum of bursts(which in fact adds a free parameter ). Our calculation shows that the redshift determined from these two independent methods are consistent with each other, the redshift of dimmest bursts is about 3.2, rather than a redshift of 1 or 2 (Mao & Paczynski 1992; Wickramasinghe 1993; Norris et al. 1995), and also very different from that of Fenimore & Bloom(1995).

In this paper, the form of Eq. (2) comes from the fact that the observed energy distribution of a source located at cosmological distance will be redshifted. But this form can also represent the evolution of luminosity. Therefore in the present paper, the parameter may indeed contain two effects: one is the redshift contribution, and another is the luminosity evolution. Our results imply that there should be evolution of burst luminosity (since the observed value of mainly lie between 0 and 1, while in our calculation is required to ensure the distance scale to be consistent in two methods), similar to that of AGN.

From Figs. 1-4 we can find that both the value of and the time dilation factor are very sensitive to the parameter . It should be noted that these two quantities depend on in opposite direction, i.e. when increases, the redshift corresponding to the observed value of also increases, while the redshift corresponding to the observed time dilation decreases, and vice versa. Therefore, there should be one best value of which can explain the observed time dilation and statistics simultaneously.

In summary, our results show that, when considering the redshift effects and/or the luminosity evolution, the distance scale of -ray bursts can be determined uniquely, , from the two independent methods, i.e. statistics and the time dilation factor. However, the existence of the latter (time dilation) was suggested by Norris et al. (1994), and questioned by Mitrofanov et al. (1996), its correctness remains to be proved in the future.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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