The Comptonized model (COMP) is a power-law model with a high-energy exponential cutoff: (A.1)where A is the normalization factor at 100 keV in units of ph s-1 cm-2 keV-1, α is the power-law index, and Ep is the peak energy in the νFν space in units of keV.
The Band function (BAND) is a model which a low-energy cutoff power law and a high-energy power law joined together by a smooth transition. It is an empirical function proposed by Band et al. (1993): (A.2)where (A.3)In Eqs. (A.2) and (A.3), A is the normalization factor at 100 keV in units of ph s-1 cm-2 keV-1, α is the low-energy power-law index, β is the high-energy power-law index, Ep is the peak energy in the νFν space in units of keV, and Ec is the characteristic energy where the low-energy power law with an exponential cutoff
ends and the pure high-energy power law starts, in units of keV. We note that when β → −∞ the Band function reduces to the Comptonized model.
The smoothly broken power law (SBPL) is a model of two power laws joined by a smooth transition. It was first parameterized by Ryde (1999) and then re-parameterized by Kaneko et al. (2006): (A.4)where (A.5)In Eqs. (A.4) and (A.5), A is the normalization factor at 100 keV in units of ph s-1 cm-2 keV-1, α and β are the low- and high-energy power-law indices respectively, Eb is the break energy in units of keV, and Δ is the break scale. Unlike the Band function, the break scale is not coupled to the power-law indices, so SBPL is a five-parameter-model if we let Δ free to vary. It is fixed at Δ = 0.3 in all the Fermi GBM GRB catalogs and is therefore adopted in this paper.
The peak energy of SBPL in the νFν space can be found at (A.6)We note that Eq. (A.6) is only valid for α> −2 and β< −2.
Comparison between the average time-resolved and the time-integrated sharpness angles.
© ESO, 2015