Launching proton-dominated jets from accreting Kerr black holes: the case of M87

A general relativistic model for the formation and acceleration of low mass-loaded jets from systems containing accreting black holes is presented. The model is based on previous numerical results and theoretical studies in the Newtonian regime, but modified to include the effects of space-time curvature in the vicinity of the event horizon of a spinning black hole. It is argued that the boundary layer between the Keplerian accretion disk and the event horizon is best suited for the formation and acceleration of the accretion-powered jets in active galactic nuclei and micro-quasars. The model is based on matching the solutions of three different regions: i- a weakly magnetized Keplerian accretion disk in the outer part, where the transport of angular momentum is mediated through the magentorotational instability, ii- a strongly magnetized, advection-dominated and turbulent-free boundary layer (BL) between the outer cold accretion disk and the event horizon, where the plasma rotates sub-Keplerian and iii- a transition zone (TZ) between the BL and the overlying corona, where the electrons and protons are thermally uncoupled, highly dissipative and rotate super-Keplerian. Our model predicts the known correlation between the Lorentz-factor and the spin parameter of the BH. It also shows that the effective surface of the BL, through which the baryons flow into the TZ, shrinks with increasing the spin parameter, implying therefore that low mass-loaded jets most likely originate from around Kerr black holes. When applying our model to the jet in the elliptical galaxy M87, we find a spin parameter in the range 0.99 - 0.998, a transition radius of about 30 gravitational radii and a fraction of 0.05 - 0.1 of the mass accretion rate goes into the TZ, where the plasma speeds up its outward-oriented motion to reach a Lorentz factor of 2.5 - 5.0 at the transition radius.

💡 Research Summary

The paper presents a general‑relativistic framework for the formation and acceleration of low‑mass‑loaded, proton‑dominated jets launched from accreting Kerr black holes, with a specific application to the jet in the giant elliptical galaxy M87. The authors build on earlier Newtonian‑regime studies and numerical simulations, but explicitly incorporate spacetime curvature near the event horizon. Central to the model is the identification of a three‑zone structure that bridges the outer Keplerian accretion disk and the black‑hole horizon:

1. Outer Keplerian Disk (Region i). This is a weakly magnetized, cold disk where angular momentum transport is mediated by the magnetorotational instability (MRI). The flow follows near‑Keplerian rotation and supplies mass to the inner regions.

2. Boundary Layer (BL, Region ii). Situated between the disk and the horizon, the BL is strongly magnetized, advection‑dominated, and essentially turbulence‑free. The plasma rotates sub‑Keplerian, and a thin “effective surface” allows a fraction of the inflowing baryons to leak upward into the overlying corona. The authors show analytically that the area of this surface shrinks dramatically as the dimensionless spin parameter (a) approaches unity, implying that high‑spin black holes naturally produce jets with low mass loading.

3. Transition Zone (TZ, Region iii). Above the BL lies a hot corona where electrons and protons become thermally decoupled. The TZ is highly dissipative; strong electric resistivity and ion–electron decoupling generate a large electromagnetic pressure that accelerates the proton‑rich plasma to super‑Keplerian rotation. The model assumes that a fraction (\eta) (5–10 %) of the accretion rate (\dot M) is transferred from the BL to the TZ.

By matching the solutions of the relativistic magnetohydrodynamic (MHD) equations in each region—conserving mass, energy, angular momentum, and magnetic flux—the authors derive a scaling relation between the jet Lorentz factor (\gamma) and the black‑hole spin: \