2025ApJ...991..209Z

The initial Lorentz factor (Γ₀) 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 Γ₀ and θ_jet values. Using this sample, we reexamine the correlations between Γ₀ and the prompt emission parameters (isotropic energy E_iso, peak luminosity L_iso, and peak energy E_p). Our results confirm the previously reported Γ₀-E_iso (L_iso), Γ₀-E_p,z, and E_iso (L_iso)-E_p-Γ₀ relations for both homogeneous interstellar medium (ISM) and Wind density profiles (Wind). Notably, we find that the short GRB 090510 complies with the Γ₀-E_iso relation, but significantly deviates from the Γ₀-L_iso and Γ₀-E_p,z relations. We systematically investigate the influence of θ_jet on Γ₀ and find a weak dependence, which reads Γ₀ (ISM) ∝${\theta }_{{\rm{jet}}}^{-0.18\pm 0.15}$ (ISM) and Γ₀ (Wind) ∝ ${\theta }_{{\rm{jet}}}^{-0.72\pm 0.14}$ (Wind). Additionally, we report, for the first time, three new three-parameter correlations, i.e., ${E}_{{\rm{iso}}}\propto {\theta }_{{\rm{jet}}}^{-0.52\pm 0.26}$ (ISM) ${{\rm{\Gamma }}}_{0}^{2.21\pm 0.28}$ (ISM) and ${E}_{{\rm{iso}}}\propto {\theta }_{{\rm{jet}}}^{-0.65\pm 0.21}$ (Wind) ${{\rm{\Gamma }}}_{0}^{2.08\pm 0.19}$ (Wind); ${L}_{{\rm{iso}}}\propto {\theta }_{{\rm{jet}}}^{-0.35\pm 0.27}$ (ISM) ${{\rm{\Gamma }}}_{0}^{2.46\pm 0.29}$ (ISM) and ${L}_{{\rm{iso}}}\propto {\theta }_{{\rm{jet}}}^{-0.29\pm 0.28}$(Wind) ${{\rm{\Gamma }}}_{0}^{2.37\pm 0.25}$ (Wind); and ${E}_{{\rm{p}},{\rm{z}}}\propto {\theta }_{{\rm{jet}}}^{-0.15\pm 0.19}$ (ISM) ${{\rm{\Gamma }}}_{0}^{1.11\pm 0.21}$ (ISM) and ${E}_{{\rm{p}},{\rm{z}}}\propto {\theta }_{{\rm{jet}}}^{-0.17\pm 0.23}$ (Wind) ${{\rm{\Gamma }}}_{0}^{1.09\pm 0.21}$ (Wind). These tight E_iso (or L_iso, E_p,z)-Γ₀-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 Γ₀ (ISM) ∝${E}_{\gamma ,52}^{0.22\pm 0.04}$ (ISM) and Γ₀ (Wind) ∝${E}_{\gamma ,52}^{0.40\pm 0.06}$ (Wind); and Γ₀ (ISM) ∝${L}_{\gamma ,52}^{0.20\pm 0.04}$ (ISM) and Γ₀ (Wind) ∝${L}_{\gamma ,52}^{0.30\pm 0.06}$ (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.