Statistical Analysis of BATSE Gamma-Ray Bursts: Self-Similarity and the Amati Relation

The statistical properties of a complete, flux-limited sample of 197 long gamma-ray bursts (GRBs) detected by BATSE are studied. In order to bring forth their main characteristics, care was taken to define a representative set of 10 parameters. A multivariate analysis gives that ~70% of the total variation in parameter values is driven by only three principal components. The variation of the temporal parameters is clearly distinct from that of the spectral ones. A close correlation is found between the half-width of the autocorrelation function (τ) and the emission time (𝒯50); most importantly, this correlation is self-similar in the sense that the mean values and dispersions of both τ and 𝒯50 scale with the duration of the burst (T90). It is shown that the Amati relation can be derived from the sample and that the scatter around this relation is correlated with the value of τ. Hence, τ has a role similar to that of the break in the afterglow light curve (tb) in the Ghirlanda-relation. In the standard GRB-scenario, the close relation between a global parameter (tb) and a local one (τ) indicates that some of the jet-properties do not vary much for different lines of sight. Finally, it is argued that the basic temporal and spectral properties are associated with individual pulses, while the overall properties of a burst is determined mainly by the number of pulses.