3 Energy from a black hole and debris torus?

Two large reservoirs of energy are in principle available: the binding energy of the orbiting debris, and the spin energy of the black hole. The first can provide up to 42% of the rest mass energy of the torus, for a maximally rotating black hole: the second can provide up to 29% (for a maximal spin rate) of the mass of the black hole itself. How can the energy be transformed into outflowing relativistic plasma after such a coalescence event? There seem to be two options. The first is that some of the energy released as thermal neutrinos is reconverted, via collisions outside the dense core, into electron-positron pairs or photons. The rate of this process depends on the square of the neutrino luminosity. The second option is that strong magnetic fields anchored in the dense matter convert the rotational energy of the system into a Poynting-dominated outflow, rather as in pulsars. Let us consider these two options in turn.

(i) Neutrinos could give rise to a relativistic pair-dominate wind if they converted into pairs in a region of low baryon density (e.g. along the rotation axis, away from the equatorial plane of the torus). The $\nu\bar\nu \to {\rm e}^+ {\rm e}^-$ process can tap the thermal energy of the torus produced by viscous dissipation. For this mechanism to be efficient, the neutrinos must escape before being advected into the hole; on the other hand, the efficiency of conversion into pairs (which scales with the square of the neutrino density) is low if the neutrino production is too gradual. Typical estimates suggest a limit of $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}10^{51}$ erg [42, (Ruffert 1997]; [, Ruffert et al. 1997]; [44, Ruffert & Janka 1998]; [36, Popham et al. 1998)], except perhaps in the "collapsar" or failed SN Ib case where [36, Popham et al. (1998)] estimate 10^52.3 ergs for optimum parameters. If the pair-dominated plasma were collimated into a solid angle $\Omega_j$ then of course the apparent "isotropized" energy would be larger by a factor $(4\pi/\Omega_j)$, but unless $\Omega_j$ is $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}10^{-2} -10^{-3}$ this may fail to satisfy the apparent isotropized energy of 10^53.5 ergs implied by a redshift z=3.4 for GRB 971214.

(ii) An alternative way to tap the torus energy is via magnetic fields threading the torus [31, (Paczynski 1991]; [28, Narayan et al. 1992]; [22, Mészáros & Rees 1997b]; [16, Katz & Piran 1997)]. Even before the BH forms, a NS-NS merging system might lead to winding up of the fields and dissipation in the last stages before the merger [19, (Mészáros & Rees 1992]; [50, Vietri 1997a)].

The above mechanisms tap the rotational energy available in the debris torus. However, a hole formed from a coalescing compact binary is guaranteed to be rapidly spinning, and, being more massive, could contain a larger reservoir of energy than the torus; this energy, extractable in principle through MHD coupling to the rotation of the hole by the [2, Blandford & Znajek (1977)] (B-Z) effect, could be even larger than that contained in the orbiting debris [22, (Mészáros & Rees 1997b]; [32, Paczynski 1998)]. Collectively, any such MHD outflows have been referred to as Poynting jets.

Simple scaling from the familiar results of pulsar theory tells us that fields of order 10¹⁵ G, are needed to carry away the rotational or gravitational energy in the time scales of tens of seconds [49, (Usov 1994]; [47, Thompson 1994)]. If the magnetic fields do not thread the BH, then a Poynting outflow can at most carry the gravitational binding energy of the torus. For a maximally rotating and for a non-rotating BH this is 0.42 and 0.06 of the torus rest mass, respectively. The torus mass in a NS-NS merger is $M_{\rm t}\sim 0.1\ M_\odot$ [44, (Ruffert & Janka 1998)], and for an NS-BH or WD-BH merger it may be $M_{\rm t} \sim 1\ M_\odot$ [32, (Paczynski 1998]; [7, Fryer & Woosley 1998)]. The extractable energy could amount to several times $10^{53}\ \epsilon (M_{\rm t}/M_\odot)$ ergs, where $\epsilon$ is the efficiency in converting gravitational into MHD jet energy. Tori masses even higher than $\sim 1\ M_\odot$ may occur in scenarios involving massive supernovae. Conditions for the efficient escape of a high-$\Gamma$ jet may, however, be less propitious if the "engine" is surrounded by an extensive envelope.

If magnetic fields of comparable strength thread the BH, its rotational energy offers an extra (and even larger) source of energy that can in principle be extracted via the B-Z mechanism [22, (Mészáros & Rees 1997b)]. For a maximally rotating BH, this is $0.29 M_{\rm bh} c^2$ ergs, multiplied, of course, by some efficiency factor. A near-maximally rotating black hole is guaranteed in a NS-NS merger. The central BH will have a mass of about $2.5\ M_\odot$; the NS-BH merger and hypernova models may not produce quite such rapidly-spinning holes, but the hole masses are larger, so the expected rotational energy should be comparable. Spinning holes can thus power a jet of up to $\sim 1.5 \ 10^{54}$ ergs. Even allowing for low total efficiency (say 30%), a system powered by the torus binding energy would only require a modest beaming of the $\gamma$-rays by a factor $(4\pi/\Omega_j)\sim 20$, or no beaming if the jet is powered by the B-Z mechanism, to produce the equivalent of an isotropic energy of 10^53.5 ergs.