The Amati relation in the “fireshell” model

Abstract

Context. The cosmological origin of Gamma-Ray Bursts (GRBs) has been firmly established, with redshifts up to z = 6.29. GRBs are possible candidates to be used as distance indicators to test cosmological models in a redshift range hardly achievable by other cosmological probes. Asserting the validity of the empirical relations among GRB observables is now crucial for their calibration. Aims. Motivated by the relation proposed by Amati and collaborators, we look within the model for a relation betwee n the peak energy Ep of theFtotal time-integrated spectrum of the afterglow and the tot al energy of the afterglow Ea f t, which in our model encompasses and extends the prompt emission. Methods. The fit within the fireshell model, as for the canonical GRB0 50315, uses the complete arrival time coverage given by the Swift satellite. It is performed simultaneously, self- consistently and recursively in the four BAT energy bands (15-25 keV, 25-50 keV, 50-100 keV and 100-150 keV) as well as in the XRT one (0.2-10 keV). It uniquely determines the two free parameters charac- terizing the GRB source, the total energy E e ± tot of the e ± plasma and its baryon loading B, as well as the effective CircumBurst medium (CBM) distribution. We can then build two sets of gedanken GRBs varying the total energy of the electron-positron plasma E e ± tot and keeping the same baryon loading B of GRB050315. The first set assumes for the effective CBM density the one obtained in the fit of GRB050315. The second set assumes instead a constant CBM density equal to the average value of the GRB050315 prompt phase. Results. For the first set of gedanken GRBs we find a relation Ep∝ (Ea f t) a , with a = 0.45± 0.01, whose slope strictly agrees with the Amati one. Such a relation, in the limit B→ 10 −2 , coincides with the Amati one. Instead, in the second set of gedanken GRBs no correlation is found. Conclusions. Our analysis excludes the Proper-GRB (P-GRB) from the prompt emission, extends all the way to the latest afterglow phases and is independent on the assumed cosmological model, since all gedanken GRBs are at the same redshift. The Amati rela- tion, on the other hand, includes also the P-GRB, focuses on the prompt emission only, and is therefore influenced by the in strumental threshold which fixes the end of the prompt emission, and depe nds on the assumed cosmology. This may well explain the intrinsic scatter observed in the Amati relation.