Fig. 4: Spectral and temporal evolution in case of adiabatic cooling.

In a we show the α − F relation expected in the case of adiabatic cooling (solid lines). The theoretical curves are computed taking also into account the effect of HLE. The value of λ specifies the evolution of the magnetic field. We adopt a SBPL as spectral shape with α_s = − 1/3 and β_s = 1.5, an initial observed peak frequency of 100 keV and a thickness of the expanding shell that is constant in time. The dot-dashed line is the evolution expected in case of HLE without adiabatic cooling, assuming the same spectral shape and initial observed peak frequency. The error bars represent 1σ uncertainties, calculated via spectral fitting in XSPEC. In b we show the temporal evolution of normalized flux expected in case of adiabatic cooling. δt_obs + 100 s is the time measured from the peak of the decay shifted at 100 s, the typical starting time of the tail emission detected by XRT. We adopt the same parameters as in a, assuming R₀ = 2 × 10¹⁵ cm and Γ = 100. The dot-dashed line is the corresponding HLE model without accounting for adiabatic cooling. τ_ad = R₀/2cΓ² indicates the timescale of adiabatic cooling, which is the same of HLE. The vertical error bars represent 1σ uncertainties and they are calculated via spectral fitting in XSPEC, while horizontal error bars represent the width of the time bin. In the legend we report the name of each GRB.