Maximum spin of black holes driving jets

Unbound outflows in the form of highly collimated jets and broad winds appear to be a ubiquitous feature of accreting black hole systems. The most powerful jets are thought to derive a significant fraction, if not the majority, of their power from the rotational energy of the black hole. Whatever the precise mechanism that causes them, these jets must, therefore, exert a braking torque on the black hole. Consequently, we expect jet production to play a significant role in limiting the maximum spin attainable by accreting black holes. We calculate the spin-up function – the rate of change of black hole spin normalized to the black hole mass and accretion rate – for an accreting black hole, accounting for this braking torque. We assume that the accretion flow on to a Kerr black hole is advection-dominated (ADAF) and construct easy-to-use analytic fits to describe the global structure of such flows based on the numerical solutions of Popham & Gammie. We find that the predicted black hole spin-up function depends only on the black hole spin and dimensionless parameters describing the accretion flow. Using recent relativistic magnetohydrodynamical (MHD) numerical simulation results to calibrate the efficiency of angular momentum transfer in the flow, we find that an ADAF flow will spin a black hole up (or down) to an equilibrium value of about 96 per cent of the maximal spin value in the absence of jets. Combining our ADAF system with a simple model for jet power, we demonstrate that an equilibrium is reached at approximately 93 per cent of the maximal spin value, as found in the numerical simulation studies of the spin-up of accreting black holes, at which point the spin-up of the hole by accreted material is balanced by the braking torque arising from jet production. The existence of equilibrium spin means that optically dim active galactic nuclei (AGNs) that have grown via accretion from an advectiondominated flow will not be maximally rotating. It also offers a possible explanation for the tight correlation observed by Allen et al. between the Bondi accretion rate and jet power in nine, nearby, X-ray luminous giant elliptical galaxies. We suggest that the black holes in these galaxies must all be rotating close to their equilibrium value. Our model also yields a relationship between jet efficiency and black hole spin that is in surprisingly good agreement with that seen in the simulation studies, indicating that our simple model is a useful and convenient description of ADAF inflow – jet outflow about a spinning black hole for incorporation in semi-analytic modelling as well as cosmological numerical simulation studies focusing on the formation and evolution of galaxies, groups and clusters of galaxies.