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 $t_{ \rm p}$, we derive the $\Gamma_0$ for the 33 GRBs assuming the radiation efficiency of $\eta =$ 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 $E_{ \rm \gamma,iso}$ (as well as the mean isotropic luminosity $L_{ \rm \gamma,iso}$), finding very tight correlations between them, i.e., $\Gamma_0$ $\propto$ $E^{ 0.24}_{ \rm \gamma,iso,52}$ ($\Gamma_0$ $\propto$ $L^{ 0.20}_{ \rm \gamma,iso.49}$) with $\eta$=0.2. Additionally, we verify the correlation among $\Gamma_0$, the isotropic energy $E_{ \rm \gamma,iso}$ (or $L_{ \rm \gamma,iso}$) and the peak energy $E_{ \rm{ p,z}}$, i.e., $E_{ \rm \gamma,iso,52}$ $\propto$ $\Gamma^{ 1.36}_0$$E^{ 0.82}_{ \rm{ p,z}}$ ($L_{ \rm \gamma,iso,49}$ $\propto$ $\Gamma^{ 1.05}_0$$E^{ 0.66}_{ \rm{ p,z}}$) under the same radiation efficiency ($\eta$=0.2).
Categories: astro-ph.HE