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

The initial Lorentz factor (${\Gamma }_{\text{0}}$) 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 ${\Gamma }_0$ 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 $t_p$, we derive the ${\Gamma }_0$ 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 ${\Gamma }_0$ and the isotropic energy $E_{\gamma ,\mathrm{iso}}$ (as well as the mean isotropic luminosity $L_{\gamma ,\mathrm{iso}}$), finding very tight correlations between them, i.e., ${\Gamma }_0$ ∝ $E_{\gamma ,\mathrm{iso},52}^{0.24}$ (${\Gamma }_0$ ∝ $L_{\gamma ,\mathrm{iso}.49}^{0.20}$) with η=0.2. Additionally, we verify the correlation among ${\Gamma }_0$, the isotropic energy $E_{\gamma ,\mathrm{iso}}$ (or $L_{\gamma ,\mathrm{iso}}$) and the peak energy $E_{p,z}$, i.e., $E_{\gamma ,\mathrm{iso},52}$ ∝ ${\Gamma }_0^{1.36}$$E_{p,z}^{0.82}$ ($L_{\gamma ,\mathrm{iso},49}$ ∝ ${\Gamma }_0^{1.05}$$E_{p,z}^{0.66}$) under the same radiation efficiency (η=0.2).