Spectral lags caused by the Doppler effect of fireballs

Abstract

Recently, Shen et al. (2005) studied the contributions of curvature effect of fireballs to the spectral lag and showed that the observed lags could be accounted for by the effect. Here we check their results by performing a more precise calculation with both the formulas presented in Shen et al. (2005) and Qin et al. (2004). Several other aspects which were not considered in Shen et al. (2005) are investigated. We find that in the case of ultra-relativistic motions, both formulas are identical as long as the whole fireball surface is concerned. In our analysis, the previous conclusion that the detected spectral lags could be accounted for by the curvature effect is confirmed, while the conclusion that the lag has no dependence on the radius of fireballs is not true. We find that introducing extreme physical parameters is not the only outlet to explain those observed large lags. Even for the larger lags ($\sim5s$) (see Norris et al. 2005), a wider local pulse ($Δt_{θ,FWHM} = 10^{7}s$) could account for it. Some conclusions not presented in Shen et al. (2005) or those modified in our analysis are listed below: a) lag $\propto Γ^{-ε}$ with $ε>$ 2; b) lag is proportional to the local pulse width and the $FWHM$ of the observed light curves; c) a large lag requires a large $α_0$ and a small $β_0$ as well as a large $E_{0p}$; d) when the rest frame spectrum varies with time, the lag would become larger; e) lag decreases with the increasing of $ R_c$; f) lag $\propto$ E within the certain energy range for a given Lorentz factor; g) lag is proportional to the opening angle of uniform jets when $θ_j < 0.6Γ^{-1}$.