Constraining the initial Lorentz factor of gamma-ray bursts under different circumburst mediums

Abstract

The initial Lorentz factor ($Γ_{\text{0}}$) plays a crucial role in uncovering the physical characteristics of gamma-ray bursts (GRBs). Previous studies have indicated that the ambient medium density index $k$ for GRBs falls in the range of 0 - 2, rather than exactly equal to 0 (homogeneous interstellar ambient) or 2 (typical stellar wind). In this work, we aim to constrain the $Γ_0$ of GRBs considering their distinct circumburst medium. We select a total of 33 GRBs for our analysis, comprising 7 X-ray GRBs and 26 optical GRBs. Subsequently, by utilizing the deceleration time of fireball $t_{\rm p}$, we derive the $Γ_0$ for the 33 GRBs assuming the radiation efficiency of $η=$ 0.2. The inferred initial Lorentz factor was found to be from 50 to 500, consistent with previous studies. We then investigate the correlation between the $Γ_0$ and the isotropic energy $E_{\rm γ,iso}$ (as well as the mean isotropic luminosity $L_{\rm γ,iso}$), finding very tight correlations between them, i.e., $Γ_0$ $\propto$ $E^{0.24}_{\rm γ,iso,52}$ ($Γ_0$ $\propto$ $L^{0.20}_{\rm γ,iso.49}$) with $η$=0.2. Additionally, we verify the correlation among $Γ_0$, the isotropic energy $E_{\rm γ,iso}$ (or $L_{\rm γ,iso}$) and the peak energy $E_{\rm{p,z}}$, i.e., $E_{\rm γ,iso,52}$ $\propto$ $Γ^{1.36}_0$$E^{0.82}_{\rm{p,z}}$ ($L_{\rm γ,iso,49}$ $\propto$ $Γ^{1.05}_0$$E^{0.66}_{\rm{p,z}}$) under the same radiation efficiency ($η$=0.2).