2018MNRAS.473.5142C

Assuming that early optical emission is dominated by external reverse shock (RS) in the standard model of gamma-ray bursts (GRBs), we intend to constrain RS models with an initial Lorentz factor Γ₀ of the outflows based on the ROTSE-III observations. We consider two cases of RS behaviour: relativistic shock and non-relativistic shock. For a homogeneous interstellar medium (ISM) and the wind circum-burst environment, constraints can be achieved by the fact that the peak flux F_ν at the RS crossing time should be lower than the observed upper limit F_ν, limit_. We consider the different spectral regimes in which the observed optical frequency ν_opt may locate, which are divided by the orders for the minimum synchrotron frequency ν_m and the cooling frequency ν_c. Considering the homogeneous and wind environments around GRBs, we find that the relativistic RS case can be constrained by the (upper and lower) limits of Γ₀ in a large range from about hundreds to thousands for 36 GRBs reported by ROTSE-III. Constraints on the non-relativistic RS case are achieved with limits of Γ₀ ranging from ∼30 to ∼350 for 26 bursts. The lower limits of Γ₀ achieved for the relativistic RS model are disfavored based on the previously discovered correlation between the initial Lorentz factor Γ₀ and the isotropic gamma-ray energy E_γ, iso_ released in the prompt phase.