2 MHD energy extraction

 

As the black hole accretes, it is spun up, so a large fraction of system's energy comes to reside in its rotation. For neutron star mergers the mass accreted (after black hole formation) may be $\Delta M \sim 0.01 - 0.1 \,M\hbox{$\odot$}$ (Ruffert & Janka 1998; Eberl 1998); for neutron star - black hole mergers, the mass is $\sim\!5$ times greater (Eberl et al. 1998). Thus the total rotational energy in the hole is  erg. The energy which may be dissipated in the disk is also large, 0.06 to $0.42\,\Delta M c^2 \sim 10^{53}$ erg. However, it is also possible that the matter in the disk accretes without appreciable emission (advection dominated flow).

A number of mechanisms have been discussed for converting the available energy into directed flow: magnetic interaction extracting rotational energy from the hole (e.g., Blandford & Znajek 1977; Mészáros & Rees 1997); reconnection of magnetic field amplified by differential rotation in the disk (Thompson 1996; Mészáros & Rees 1997); the dynamo action of rotating magnetic field in the disk (Katz 1994, 1997); flare-like activity of the disk field (Galeev 1979); etc. These solutions all have in common the need for magnetic field strength near the equipartition value, $\sim\!10^{15}$ Gauss. Whether this field is actually created is unclear. The MHD solutions are also appropriate chiefly for low viscosity, low entropy disks that last a long time. It might be hard to generate the necessary magnetic field in a disk that made only a few revolutions. In any case, the lifetime of the disk formed, e.g., by neutron star merger is tens of milliseconds for a disk viscosity, $\alpha \sim 0.01$ (Ruffert & Janka 1998; Popham et al. 1998). This is much shorter than most GRBs, so the duration would have to be governed by the external interaction, not by the life time of the engine.

There is an interesting conundrum here in that the MHD models need large fields that would imply large disk viscosity (Balbus & Hawley 1998), but in the merging neutron star case, high viscosity would also cause so much heating that much of the desired energy would be radiated away as neutrinos. On the other hand, the non-MHD models, i.e., those that use neutrinos to transport energy (Ruffert & Janka 1998), implicitly assume a large disk field in order to provide the necessary viscosity for heating. So MHD energy extraction, if it works at all, should also be important in neutrino powered models, especially in the collapsars of the next section.

Generally speaking, MHD models have several advantages. First, MHD energy extraction can potentially be quite efficient and the production of jets can occur naturally, as in active galactic nuclei. The upper bound on the jet energy is about 10⁵³ erg beamed into a small fraction of the sky. If one must confront energetic GRBs at red shift 5 say, this sort of efficiency may be necessary. Second, MHD energy extraction can work for low values of accretion rate (and $\alpha$). Below $0.1 \,M\hbox{$\odot$}\ {\rm s}^{-1}$, even for high $\alpha$, energy extraction from the disk by neutrino emission becomes inefficient (Popham et al. 1998). This is another reason to consider MHD extraction in collapsars.

  

Figure 1: The disk density structure inside a $14\, M\hbox{$\odot$}$ helium star 7.5 s after iron core collapse. The polar density is nearly six orders of magnitude lower than the equatorial density near the inner boundary