Maximal Jet Energy of Gamma-Ray Bursts through the Blandford-Znajek Mechanism

Gamma-ray bursts (GRBs) are among the most energetic events in the universe, driven by relativistic jets launched from black holes (BHs) formed during the collapse of massive stars or after the merger of two neutron stars (NSs). The jet power depends on the BH spin and the magnetic flux accreted onto it. In the standard thin disk model, jet power is limited by insufficient magnetic flux, even when the spin approaches maximum possible value. In contrast, the magnetically arrested disk (MAD) state limits jet energy by extracting significant angular momentum, braking BH rotation. We propose a unified model incorporating both standard thin disk and MAD states, identifying a universal curve for jet power per accretion rate as a function of the magnetic flux ratio, $\Delta_\mathrm{eq} = (\Phi_\mathrm{BH}/\Phi_\mathrm{MAD})_\mathrm{eq}$, at spin equilibrium. For long GRBs (lGRBs), the model predicts a maximum jet energy of $\sim 1.5\%$ of the accretion energy, occurring at $\Delta_\mathrm{eq} \sim 0.4$ where the BH equilibrium spin is $a \sim 0.5$. Both long and short GRBs are unlikely to be produced by a MAD: for short GRBs (sGRBs), this requires an accreted mass orders of magnitude smaller than that available, while for lGRBs, the narrow progenitor mass distribution challenges the ability to produce the observed broad distribution of jet energies. This framework provides a consistent explanation for both standard and luminous GRBs, emphasizing the critical role of magnetic flux. Both long and short GRBs require magnetic flux distributions that peak around $10^{27}\,\mathrm{G\,cm}^2$.