Introduction

One of the most spectacular phenomena associated with accretion onto black holes is the creation of powerful, highly-relativistic jets. However, despite intense observational and theoretical study, the basic energy source of these relativistic jets remains unknown. Broadly speaking, there are two possibilities. Firstly, jets could be powered by the liberation of gravitational potential energy of accreting matter. If this is the case, the most likely scenario is the formation and subsequent focusing and acceleration of a magnetohydrodynamic (MHD) disk wind (Blandford & Payne 1982). While this mechanism has the appealing feature of potentially being universal to all accreting systems (and therefore allowing a unified model for jets from proto-stellar systems, accreting white dwarfs and accreting neutron stars as well as accreting black holes), it is not clear that such a disk wind can be accelerated to the highly relativistic velocities seen from many black hole systems. The alternative is that jets could be powered by the magnetic extraction of the spin energy of the central black hole using the mechanism described in the seminal paper by Blandford & Znajek (1977; hereafter BZ). The power extracted from a Kerr black hole with dimensionless spin parameter $a_*$ threaded by a magnetic field of strength $B_{\rm H}$ (in the membrane paradigm sense; see Thorne, Price & Macdonald 1986) is

$$L_{\rm BZ}\approx \frac{1}{32}\omega_{\rm F}^2B_{\rm H}^2 r_{\rm H}^2 a_*^2 c$$ (1)

where $r_H$ is the radius of the event horizon and $\omega_F^2=\Omega_F(\Omega_H-\Omega_F)/\Omega_H^2$, with $\Omega_H$ and $\Omega_F$ being the angular velocities of the black hole and magnetic field lines, respectively. It is often argued (e.g., see BZ) that the magnetic field structure adjusts itself such that $\Omega_F=\Omega_H/2$ (Phinney 1983), hence maximizing $\omega_F^2$ to a value of $1/4$. While the initial work of BZ was based on force-free black hole magnetospheres, the basic mechanism is seen to operate in the recent generation of fully relativistic MHD accretion disk simulations (e.g., see Koide et al. 2000; Komissarov 2004, De Villiers et al. 2004, McKinney & Gammie 2004, McKinney 2005a,b,c)

In the past dozen years or so, several studies have cast doubt on whether nature can produce significant hole-threading magnetic fields, leading to the suggestion that the BZ mechanism is insufficient to energize powerful black hole jets. Developing upon work done by van Ballegooijen (1989), Lubow, Papaloizou & Pringle (1994) and Heyvaerts, Priest & Bardou (1996; hereafter HPB) have examined the dragging and concentration of an external field by an MHD turbulent accretion disk. Both sets of authors find that, due to the high effective magnetic diffusivity of such disks, the inward dragging and subsequent concentration of an external field is rather ineffective. In a different approach to this problem, Ghosh & Abramowicz (1997; hereafter GA97) construct force-free black hole magnetosphere models within the radius of marginal stability and show that the field threading the black hole is only of comparable strength to that threading the inner disk. Since the disk-threading field has to be rather weak (with the magnetic pressure at least an order of magnitude less than the total pressure) so as not to quench the magneto-rotational instability (MRI) that drives the accretion itself (Balbus & Hawley 1991, 1998), the inferred black hole threading field would be insufficient to energize the powerful jets in active galactic nuclei (AGN) that we observe. This argument was further developed by Livio, Ogilvie & Pringle (1999) who pointed out that, under these circumstances, the electromagnetic power extracted from the inner regions of the disk would necessarily dominate the black hole spin-energy extraction.

In all of the studies described above, the plunge region of the black hole accretion disk has been neglected. This is the region of the disk within the radius of marginal stability in which the accretion flow is undergoing rapid inwards acceleration (ultimately crossing the event horizon at the velocity of light as seen by a locally non-rotating observer). Unless the magnetic field is extremely strong, this is a region where inertial forces will dominate and the commonly employed force-free approximation will break down. For example, examination of the GA97 steady-state magnetosphere solution shows magnetic field crossing the plunge region with a strength very similar to that in the diffusive region of the disk. This situation seems unlikely -- any dynamically unimportant magnetic field that threads the plunge-region would be swept into the black hole on a dynamical timescale. This means that the actual field threading the plunge region would be very weak. However, it does not imply that the field threading the BH horizon, which is what counts for the BZ effect, is also weak. The field swept in by the plunge region would be ``cleaned'' into some well-ordered configuration threading the black hole (for a full discussion of the ``cleaning'' of a magnetic field by a black hole, see Thorne, Price & McDonald 1986) and can be confined by the inertial forces of the plunging accretion flow even if it achieves strengths appreciably higher than the characteristic field strengths in the inner disk. Since the strength of the BZ mechanism depends on the square of the magnetic field, this enhancement could have major implications for the relative dominance of spin-energy extraction.

In this paper, we extend these previous works by examining the role of the plunge region of a black hole accretion disk in enhancing the horizon-threading flux. In essence, we use the HPB formalism for flux dragging in an accretion disk and impose an inner boundary condition appropriate for the plunge region around a central black hole. This formalism is described in Section 2. Although this analysis is non-relativistic, it should provide guidance about flux enhancement by the plunge region at least in the case of slowly rotating black holes. As reported in Section 3, our analysis confirms the basic intuition discussed in the previous paragraph and uncovers a strong dependence of the equilibrium trapped flux on the disk thickness and the effective magnetic Prandtl number of the disk. Section 4 discusses the sensitivity of our results to the outer boundary condition and then places our results into a wider astrophysical context. Our conclusions are presented in Section 5.