Closing the assumption gap in Blandford–Znajek constraints on gamma-ray burst inner engines

The 2019 Blandford–Znajek (BZ) framework analysis of Lloyd-Ronning et al. inferred that observed gamma-ray burst (GRB) luminosities require horizon magnetic fields of roughly $5\times10^{14}$–$10^{17}$ G on rapidly spinning ($a\approx0.9$) stellar-mass black holes, implying progenitor fields of $\sim10^6$ G for collapsars and $\sim10^{15}$ G for neutron-star mergers under magnetic flux conservation — but every input to that inference was assumed: a radiative efficiency $\eta=0.1$, a jet opening angle of $10^\circ$, crude flux conservation from progenitor to horizon, unmodeled MRI saturation and magnetically-arrested-disk microphysics, neglected quantum-electrodynamic (QED) pair production and photon splitting at fields far above the quantum critical field, a simple disk scaling with no neutrino-annihilation power, and an untested high-spin assumption. This problem asks for these inputs to be determined physically — through end-to-end collapse/merger simulations, strong-field QED magnetosphere theory, and observational inference — and for the engine-parameter constraints (horizon field/flux, spin, mass, accretion rate) to be re-derived self-consistently, thereby establishing whether a BZ jet can power observed GRBs and what the true distribution of GRB inner-engine parameters is.

Background

Gamma-ray bursts (GRBs) are the most luminous electromagnetic explosions known. Long GRBs (duration of seconds to minutes) accompany the gravitational collapse of massive stripped-envelope stars ("collapsars"), while short GRBs (lasting less than about two seconds) are associated with binary neutron-star mergers. In the leading model for their power source, the Blandford–Znajek (BZ) process, rotational energy is extracted from a spinning (Kerr) black hole by magnetic field lines that thread the event horizon and are held in place by an accretion disk; the jet luminosity scales approximately as $L_{\rm BZ}\approx10^{53}\,{\rm erg\,s^{-1}}\,(a/0.9)^2(\phi_{\rm BH}/10^{29}\,{\rm G\,cm^2})^2(5M_\odot/M_{\rm BH})^2$, where $a$ is the dimensionless black-hole spin, $\phi_{\rm BH}$ the magnetic flux through the horizon, and $M_{\rm BH}$ the black-hole mass. Efficient extraction requires a force-free magnetosphere, meaning enough ambient plasma to screen the electric field component along the magnetic field.

Lloyd-Ronning, Fryer, Miller, Prasad, Torres and Martin (Mon. Not. R. Astron. Soc. 485, 203, 2019) inverted this relation against observed GRB luminosities: assuming the prompt gamma-ray luminosity is a fraction $\eta=0.1$ of the BZ power, a jet half-opening angle $\theta=10^\circ$, and spin $a\approx0.9$, they found that long GRBs with black-hole masses of $2$–$10\,M_\odot$ require horizon fields of roughly $5\times10^{14}$–$10^{17}$ G, and short GRBs with masses of $0.5$–$4\,M_\odot$ require $\sim10^{15}$–$10^{17}$ G. Under magnetic flux conservation from progenitor to horizon, $B_pR_p^2\approx B_{\rm BH}R_{\rm BH}^2$, these imply progenitor fields of $\sim10^6$ G for single massive stars and $\sim10^{15}$ G for neutron-star mergers — the latter a demanding requirement on merger progenitors.

Each input to this inference is an assumption rather than a derived quantity, and the paper itself flags them all. (i) The radiative efficiency $\eta$ (the fraction of jet power radiated in gamma rays) and the opening angle $\theta$ are not independently observable, so the inferred horizon fields and accretion rates inherit unknown distributions. (ii) Flux conservation from progenitor to horizon is acknowledged as crude: how magnetic flux is generated, transported, redistributed, or lost during collapse or merger and disk formation is unknown. (iii) The field near the horizon depends on the nonlinear saturation of the magnetorotational instability (MRI), the instability of differentially rotating magnetized disks that amplifies seed fields through turbulence, and on whether the system reaches a magnetically arrested disk (MAD) state — a configuration in which accumulated horizon-threading flux becomes strong enough to halt accretion and maximize BZ efficiency; whether and how a MAD is achieved and maintained in GRB engines is unsettled. (iv) At fields far above the quantum critical field $B_Q=m_e^2c^3/e\hbar\approx4.4\times10^{13}$ G, QED processes become important: magnetic pair production ($\gamma\to e^+e^-$, which supplies the charges that maintain force-free conditions) and photon splitting ($\gamma\to\gamma\gamma$, a third-order process that suppresses pair production and has been invoked to explain radio-quiet pulsars and magnetars). Their quantitative impact on the force-free condition and effective BZ power in the GRB black-hole environment was neglected. (v) The scaling of disk density and accretion rate with black-hole mass, and any supplementary jet power from neutrino–antineutrino annihilation ($\nu\bar\nu\to e^+e^-$) near the disk, were left unmodeled. (vi) Whether progenitor angular-momentum transport — including magnetic coupling in stellar interiors, such as the Tayler–Spruit dynamo, a mechanism in which magnetic instabilities in radiative stellar zones move angular momentum outward — allows enough collapsing systems to form rapidly spinning black holes with substantial disks to validate the $a\approx0.9$ assumption is undetermined.

Problem Statement

Determine whether the Blandford–Znajek framework's constraints on gamma-ray burst inner engines survive when its assumed inputs are replaced by physically determined ones, and re-derive the engine-parameter constraints self-consistently. The common objective is a physically grounded version of the inference performed by Lloyd-Ronning et al. (2019); the following determinations are its mutually dependent inputs, each feeding the same BZ luminosity relation $L_{\rm GRB}=\eta L_{\rm BZ}$ and the progenitor requirements derived from it:

1. Magnetic flux supply to the horizon: how flux is generated, transported, redistributed, or lost during core collapse or neutron-star merger and disk formation — replacing the flux-conservation estimate $B_pR_p^2\approx B_{\rm BH}R_{\rm BH}^2$ — and the resulting distribution of horizon fluxes (equivalently, field strengths) across the GRB-producing population.
2. Magnetospheric plasma conditions in the inferred horizon-field regime $B\sim10^{15}$–$10^{17}$ G: the quantitative effect of QED pair production and photon splitting on the force-free condition and the effective BZ power.
3. Disk microphysics: MRI nonlinear saturation, the resulting field-strength distribution in the disk and near the horizon, and whether and how a magnetically arrested disk state is achieved and maintained.
4. The distribution of remnant black-hole spins implied by progenitor angular-momentum transport, testing the $a\approx0.9$ assumption for the GRB-producing subset.
5. The true distributions of the radiative efficiency $\eta$ and the jet opening angle $\theta$ (including jet structure) that convert observed isotropic-equivalent luminosities into engine power.
6. The scaling of disk density and accretion rate with black-hole mass, and the contribution of neutrino-annihilation power relative to the BZ luminosity.

The problem is complete when each input is determined from physics or observation rather than assumed, and the revised black-hole-mass versus horizon-field (and spin, accretion-rate, progenitor-field) constraints for long and short GRBs — including their predictions for progenitor magnetic fields and remnant properties — are derived from those determinations, so that the BZ hypothesis for GRBs is quantitatively confirmed, refined, or refuted.

References

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