Comprehensive Statistical Analysis of Initial Lorentz Factor and Jet Opening Angle of Gamma-Ray Bursts

The initial Lorentz factor ($\Gamma_{0}$) and jet half-opening angle ($\theta_{\rm jet}$) of gamma-ray bursts (GRBs) are critical physical parameters for understanding the dynamical evolution of relativistic jets and the true energy release of GRBs. We compile a sample of 89 GRBs that exhibit an onset bump feature in their early optical or GeV light curves, 42 of which also display a jet break feature, and derive their $\Gamma_{0}$ and $\theta_{\rm jet}$ values. Using this sample, we re-eaxmine the correlations between $\Gamma_{0}$ and the prompt emission parameters (isotropic energy $E_{\rm iso}$, peak luminosity $L_{\rm iso}$, and peak energy $E_{\rm p}$). Our results confirm the previously reported $\Gamma_{0}$$-$$E_{\rm iso}$ ($L_{\rm iso}$), $\Gamma_{0}$$-$$E_{\rm p,z}$, and $E_{\rm iso}$ ($L_{\rm iso}$)$-$$E_{\rm p}$$-$$\Gamma_{0}$ relations for both homogeneous interstellar medium (ISM) and wind density profiles (Wind). Notably, we find that the short GRB 090510 complies with the $\Gamma_{0}$$-$$E_{\rm iso}$ relation, but significantly deviates from the $\Gamma_{0}$$-$$L_{\rm iso}$ and $\Gamma_{0}$$-$$E_{\rm p,z}$ relations. We systematically investigate the influence of $\theta_{\rm jet}$ on $\Gamma_{0}$ and find a weak dependence. Additionally, we report, for the first time, three new three-parameter correlations, i.e., $E_{\rm iso}$ (or $L_{\rm iso}$, $E_{\rm p,z}$)$-$$\Gamma_{0}$$-$$\theta_{\rm jet}$ correlations. In addition, we further explore the relations between the initial Lorentz factor and the jet-corrected energy. We also find that the jet-corrected correlations remain significant, suggesting that these relations are intrinsic to the physical nature of GRBs. However, the increased dispersion after correction implies that underlying differences persist among individual GRBs.