Constraining External Reverse Shock Physics of GRBs from ROTSE-III Limits

Assuming that the early optical emission is dominated by the external reverse shock (RS) in the standard model of gamma-ray bursts (GRBs), we intend to constrain RS models with the initial Lorentz factor $\Gamma_0$ of the outflows based on the ROTSE-III observations. We consider two cases of the RS behavior: the relativistic shock and the non-relativistic shock. For homogeneous interstellar medium (ISM) and wind circum-burst environment, the constraints can be achieved by the fact that the peak flux $F_{\rm \nu}$ at the RS crossing time should be lower than the observed upper limit $F_{\rm \nu,limit}$. We consider the different spectral regimes that the observed optical frequency $\nu_{\rm opt}$ may locate in, which are divided by the orders for the minimum synchrotron frequency $\nu_{\rm m}$ and the cooling frequency $\nu_{\rm c}$. {\bf Considering the homogeneous and wind environment around GRBs, we find that the relativistic RS case can be constrained by the (upper and lower) limits of $\Gamma_0$ in a large range from about hundreds to thousands for 36 GRBs reported by ROTSE-III. The constraints on the non-relativistic RS case are achieved with limits of $\Gamma_0$ for 26 bursts ranging from $\sim 30$ to $\sim 350$.} The lower limits of $\Gamma_0$ achieved for the relativistic RS model is disfavored based on the previously discovered correlation between the initial Lorentz factor $\Gamma_0$ and the isotropic gamma-ray energy $E_{\rm \gamma, iso}$ released in prompt phase.