Astrophysical implications

Given the caveats discussed above, the results of Section 3 have important implications for the strength of the black hole-threading field and the relevance of the BZ process. Suppose that the magnetic pressure due to the large scale field $B_0$ is a fraction $f$ of the maximum pressure in the accretion disk, $p_{\rm max}$, i.e., $B_0=(8\pi f p_{\rm max})^{1/2}$. Using this together with eqn. 1 and eqn. 17 gives, $L_{\rm BZ}\approx 5\pi\omega_F^2fp_{\rm max} \Upsilon^2r_{\rm H}^2a_*^2c$. Using the expressions for $p_{\rm max}$ for radiation pressure-dominated (RPD) and gas pressure-dominated (GPD) disks from Moderski & Sikora (1996) and GA97, and assuming the usual BZ impedance matching criterion is obeyed, gives

$$L_{\rm BZ} (\hbox{${\rm\thinspace erg}{\rm\thinspace s}^{-1}\,$})... .../10}fM_8^{11/10}\dot{m}^{4/5}_{-4}\Upsilon^2a_*^2 &{\rm GPD} \end{array}\right.$$ (23)

where we have scaled to a black hole mass of $M=10^8M_8\hbox{$\rm\thinspace M_{\odot}$}$ and $\dot{m}=10^{-4}\dot{m}_{-4}$ is the mass accretion rate in Eddington units. This can be directly compared with the expressions for $L_{\rm BZ}$ in GA97 if we set $f\alpha^{-1}\approx 0.1$ (which results from their relation between $\alpha$ and the disk magnetic field). For $\Upsilon=1$ (corresponding to small effective magnetic Prandtl numbers or very thin disks), we find low BZ luminosities that agree very well with those computed by GA97. However, as we have shown, large magnetic Prandtl numbers and/or thick disks can result in large enhancements of the black hole-threading fields, approximately described by $\Upsilon \approx 1+2xP_{\rm m}(h/r)$, where $x\sim {\cal O}(r_{\rm dead}/r_{\rm ms})$. The BZ luminosity is then enhanced by a factor of $\Upsilon^2$.

It is interesting to explore astrophysical consequences of the strong $h/r$ dependence of the equilibrium hole-threading flux $A_*$. There is mounting empirical evidence that black hole systems produce jets only when a geometrically thick accretion disk is present. The best case can be made for the GBHBs, as discussed by Fender, Belloni & Gallo (2004). In their X-ray low-hard (LH) state (a.k.a. the power-law state; McClintock & Remillard 2004) they display steady optically-thick radio cores which, in Cygnus X-1, can be spatially resolved into a jet-like structure by VLBA (Stirling et al. 2001). It is generally believed that the inner regions of the accretion flow in a LH-state GBHB system is radiatively inefficient, hot, and hence geometrically-thick ($h/r\sim 0.5$). However, the radio jet is seen to shut off once the source has made a transition to the high-soft (HS) state (or thermal state; McClintock & Remillard 2004) which is believed to correspond to an inner accretion disk which is radiatively efficient and hence significantly thinner. We postulate that the jet in the LH state is powered by the BZ effect which is enhanced by the flux trapping effect of the plunge region. Some time after a transition to a HS state, the system will possess a disk with a similar accretion rate but significantly reduced thickness. For a fixed accretion rate, the maximum pressure in a disk scales as $p_{\rm max}\propto (h/r)^{-1}$. Using our parameterization for $\Upsilon$, we expect the BZ luminosity scales as $L_{\rm BZ}\propto f(h/r)$, provided $h/r\approxgt 1/xP_{\rm m}$. Hence, due to the inability of a thin disk to trap flux on the black hole, the BZ luminosity of the HS state will be much reduced leading to the suppression of the radio jet.

The actual ${\rm LH}\rightarrow {\rm HS}$ transition itself is particularly interesting. It is during this transition (when the source crosses the ``jet line'' on the X-ray flux/color diagram) that powerful relativistic outflows are produced which, for example, produce the superluminal radio blobs seen from microquasars. It is likely that the transition is driven by the thermal collapse of the LH-state hot disk, producing a structure that eventually evolves into the HS-state cold disk. The nature of the intermediate structure is unclear, however. It has been suggested that the thermal collapse produces a magnetically-dominated region (e.g., Meier 2005) in which MRI-driven turbulence is suppressed and accretion proceeds only through large scale magnetic torques. If the pre-collapse disk is threaded by a large scale magnetic field, this field could readily become dynamically important in the post-collapse disk (since rapid thermal collapse will proceed at constant surface density, producing a disk pressure which scales as $p_{\rm max}\propto h/r$). Subsequent magnetic braking of the disk would lead to rapid inflow, a rapid accretion of poloidal flux onto the black hole, and a rapid increase in the importance of the BZ effect. The powerful ejections seen from GBHBs as they undergo this transition might be the result of such a scenario. The ejections would terminate once the inner disk has ceased to be magnetically dominated (due to the accretion of matter from the outer disk), hence re-establishing a turbulent state with high effective magnetic diffusivity.