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

Abstract The initial Lorentz factor (Γ 0 ) and jet half-opening angle ( θ 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 Γ 0 and θ jet values. Using this sample, we reexamine the correlations between Γ 0 and the prompt emission parameters (isotropic energy E iso , peak luminosity L iso , and peak energy E p ). Our results confirm the previously reported Γ 0 − E iso ( L iso ), Γ 0 − E p,z , and E iso ( L iso )− E p −Γ 0 relations for both homogeneous interstellar medium (ISM) and Wind density profiles (Wind). Notably, we find that the short GRB 090510 complies with the Γ 0 − E iso relation, but significantly deviates from the Γ 0 − L iso and Γ 0 − E p,z relations. We systematically investigate the influence of θ jet on Γ 0 and find a weak dependence, which reads Γ 0 (ISM) ∝ θ jet − 0.18 ± 0.15 (ISM) and Γ 0 (Wind) ∝ θ jet − 0.72 ± 0.14 (Wind). Additionally, we report, for the first time, three new three-parameter correlations, i.e., E iso ∝ θ jet − 0.52 ± 0.26 (ISM) Γ 0 2.21 ± 0.28 (ISM) and E iso ∝ θ jet − 0.65 ± 0.21 (Wind) Γ 0 2.08 ± 0.19 (Wind); L iso ∝ θ jet − 0.35 ± 0.27 (ISM) Γ 0 2.46 ± 0.29 (ISM) and L iso ∝ θ jet − 0.29 ± 0.28 (Wind) Γ 0 2.37 ± 0.25 (Wind); and E p , z ∝ θ jet − 0.15 ± 0.19 (ISM) Γ 0 1.11 ± 0.21 (ISM) and E p , z ∝ θ jet − 0.17 ± 0.23 (Wind) Γ 0 1.09 ± 0.21 (Wind). These tight E iso (or L iso , E p,z )−Γ 0 − θ jet correlations likely arise from the combined effects of radiation mechanisms, jet structure, and outflow dynamics in GRBs. In addition, we further explore the relations between the initial Lorentz factor and the jet-corrected energy, and find Γ 0 (ISM) ∝ E γ , 52 0.22 ± 0.04 (ISM) and Γ 0 (Wind) ∝ E γ , 52 0.40 ± 0.06 (Wind); and Γ 0 (ISM) ∝ L γ , 52 0.20 ± 0.04 (ISM) and Γ 0 (Wind) ∝ L γ , 52 0.30 ± 0.06 (Wind). 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.