3.1 Theoretical Arguments for the Spin Paradigm

There is significant theoretical basis for this paradigm as well. Several models of relativistic jet formation (Blandford & Znajek 1977, Punsly & Coroniti 1990) indicate that the jet power should increase as the square of the black hole angular momentum

(3)

where B_p is the strength of the poloidal (vertical/radial) magnetic field threading the ergospheric and horizon region of the rotating hole. In this model rotational energy is extracted via a Penrose-like process: the frame-dragged accretion disk is coupled to plasma above and outside the ergosphere via the poloidal magnetic field; some plasma is pinched and accelerated upward while some disk material is diverted into negative energy (retrograde) orbits inside the ergosphere, removing some of the hole's rotational energy. The key parameter determining the efficiency of this process is the strength of the poloidal magnetic field. The standard approach (e.g., Moderski & Sikora 1996) to estimating B_p is to set it equal to B, the dominant azimuthal magnetic field component given by the disk structure equations, yielding

(4)

(5)

for Class B (radio galaxy/ADAF) and Class A (quasar/standard disk) objects, respectively. Note that, while the jet is not accretion-powered in this model, the efficiency of extraction is still essentially linear in .

Livio et al. (1999) have pointed out that taking B_p B may greatly overestimate the jet power from this process. Using dynamo arguments they propose that a more realistic estimate for the equilibrium poloidal magnetic field is

(6)

where (H/R) is the ratio of disk half-thickness to radius in the jet acceleration region. For thin disks this yields a jet power of only L_jet = 4 x 10⁴⁴ erg s^-1 m₉^1.1 ( / 0.1)^1.2 j² - less than the observed radio power of the strongest sources and much less than their inferred total jet power (see Bicknell, these proceedings, and Bicknell 1995). However, there are several reasons for believing that even with equation (6) the field still can be quite large in many cases, and the jet power still comparable to equations (4) and (5). Firstly, for advective disks (both the accretion-starved kind [ << 1] and the super-Eddington kind [ 1]) the disk is geometrically thick (H/R ~ 1), yielding B_p B even within the dynamo argument. Thick disks also can occur for an even broader range of accretion rate when the hole and disk spin axes are misaligned: because of the Lens-Thirring effect, the gas follows inclined orbits that do not close, creating shocks and dissipation that bloats the disk into a quasi-spherical, inhomogeneous inflow (Blandford 1994). Furthermore, even when H << R, inside the last stable orbit (or in any other region of the disk where the infall velocity suddenly approaches the free-fall speed) conservation of mass will cause a drop in density and pressure. The toroidal field may then be dynamically important, buckling upward out of the plunging accretion flow, resulting in B_p being comparable to B (Krolik 1999).

Figure 1. Schematic representation of four possible combinations of and j, drawn roughly to scale. The horizon interior to the hole is black, while the boundary of the ergosphere (the ``static limit'') is represented by an ellipse 0.5 x 1.0 Schwarzschild radius in size. Top panels depict non-rotating, Schwarzschild holes (j -> 0), bottom panels Kerr holes (j -> 1). Left panels show low accretion rate (ADAF) tori, right panels high accretion rate standard disk models. However, in the lower right panel the region of the disk experiencing significant frame dragging is bloated (c.f. Blandford 1994). Widths of poloidal magnetic field lines and jet arrows are proportional to the logarithm of their strength.

Figure 1 summarizes the main features of the accretion and spin paradigms and shows the four possible combinations of high and low accretion rate and black hole spin. It is proposed that these states correspond to different radio loud and quiet quasars and galaxies. In the figure poloidal magnetic field strengths are estimated from equation (6), but (H/R) is of order unity for the low cases, and also for the high Kerr case due to Lens-Thirring bloating of the inner disk. Otherwise (H/R) is calculated from the electron scattering/gas pressure disk model of Shakura & Sunyaev (1973), and disk field strengths are computed from that paper or from Narayan et al. (1998), as appropriate. The logarithms of the resulting poloidal field strengths, and corresponding jet powers, are represented as field line and jet arrow widths. In the Kerr cases, the inner disk magnetic field is significantly enhanced over the Schwarzschild cases, due in part to the smaller last stable orbit (flux conservation) and in part to the large (H/R) of the bloated disks. The high accretion rate, Schwarzschild case has the smallest field - and the weakest jet - because the disk is thin, the last stable orbit is relatively large, and the Keplerian rotation rate of the field there is much smaller than it would be in a Kerr hole ergosphere. Enhancement of the poloidal field due to the buoyancy process suggested by Krolik (1999) is ignored here because we find it not to be a factor in the simulations discussed below. If it were important, then the grand scheme proposed here would have to be re-evaluated, as the effect could produce strong jets (up to the accretion luminosity in power) even in the plunging region of Schwarzschild holes. Then even the latter would be expected to be radio loud as well (L_jet ~ 10^43-46 erg s^-1).