Using a sample of 14 BeppoSAX and 74 Swift GRBs with measured redshift we tested the correlation between the intrinsic peak energy of the time-integrated spectrum , E _ { p,i } , the isotropic-equivalent peak luminosity , L _ { p,iso } , and the duration of the most intense parts of the GRB computed as T _ { 0.45 } ( “ Firmani correlation ” ) .
For 41 out of 88 GRBs we could estimate all of the three required properties .
Apart from 980425 , which appears to be a definite outlier and notoriously peculiar in many respects , we used 40 GRBs to fit the correlation with the maximum likelihood method discussed by D ’ Agostini , suitable to account for the extrinsic scatter in addition to the intrinsic uncertainties affecting every single GRB .
We confirm the correlation .
However , unlike the results by Firmani et al. , we found that the correlation does have a logarithmic scatter comparable with that of the E _ { p,i } - E _ { iso } ( “ Amati ” ) correlation .
We also find that the slope of the product L _ { p,iso } T _ { 0.45 } is equal to \sim 0.5 , which is consistent with the hypothesis that the E _ { p,i } - L _ { p,iso } - T _ { 0.45 } correlation is equivalent to the E _ { p,i } - E _ { iso } correlation ( slope \sim 0.5 ) .
We conclude that , based on presently available data , there is no clear evidence that the E _ { p,i } - L _ { p,iso } - T _ { 0.45 } correlation is different ( both in terms of slope and dispersion ) from the E _ { p,i } - E _ { iso } correlation .