5 Intrinsic time scales

A question which has remained largely unanswered so far is what determines the characteristic duration of bursts, which can extend to tens, or even hundreds, of seconds. This is of course very long in comparison with the dynamical or orbital time scale for the "triggers'', which is measured in milliseconds. While bursts lasting hundreds of seconds can easily be derived from a very short, impulsive energy input, this is generally unable to account for a large fraction of bursts which show complicated light curves. This hints at the desirability for a "central engine'' lasting much longer than a typical dynamical time scale.

Observationally [18, (Kouveliotou et al. 1993)] the short ($\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}2$ s) and long ($\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$\gt$}}}2$ s) bursts appear to represent two distinct subclasses, and one early proposal to explain this was that accretion induced collapse (AIC) of a white dwarf (WD) into a NS plus debris might be a candidate for the long bursts, while NS-NS mergers could provide the short ones [14, (Katz & Canel 1996)]. As indicated by [42, Ruffert et al. (1997)], $\nu\bar\nu$ annihilation will generally tend to produce short bursts $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}1$ s in NS-NS systems, requiring collimation by 10^-1-10^-2, while [36, Popham et al. (1998)] argued that in collapsars and WD/He-BH systems longer $\nu\bar\nu$ bursts may be possible.

An acceptable model requires that the surrounding torus should not completely drain into the hole, or be otherwise dispersed, on too short a time scale. There have been some discussions in the literature of possible "runaway instabilities'' in relativistic tori [29, (Nishida et al. 1996]; [1, Abramowicz et al. 1998]; [5, Daigne & Mochkovitch 1997)]: these are analogous to the runaway Roche lobe overflow predicted, under some conditions, in binary systems. These instabilities can be virulent in a torus where the specific angular momentum is uniform throughout, but are inhibited by a spread in angular momentum. In a torus that was massive and/or thin enough to be self-gravitating, bar-mode gravitational instabilities could lead to further redistribution of angular momentum and/or to energy loss by gravitational radiation within only a few orbits. Whether a torus of given mass is dynamically unstable depends on its thickness and stratification, which in turn depends on internal viscous dissipation and neutrino cooling.

The disruption of a neutron star (or any analogous process) is almost certain to lead to a situation where violent instabilities redistribute mass and angular momentum within a few dynamical time scales (i.e. in much less than a second). A key issue for gamma ray burst models is the nature of the surviving debris after these violent processes are over: what is the maximum mass of a remnant disc/torus which is immune to very violent instabilities, and which can therefore in principle survive for long enough to power the bursts? It is the mass of this residual torus - i.e. what is left after violent instabilities on a dynamical timescale have done their work - that is the relevant $M_{\rm t}$ in the above expressions (in Sect. 3) for the extractable energy of the torus.

If the trigger is to liberate its energy over a period 10-100 s via Poynting flux - either through a relativistic wind "spun off'' the torus or via the B-Z mechanism - the required field is a few times 10¹⁵ G. A weaker field would extract inadequate power; on the other hand, if the large-scale field were even stronger, then the energy would be dumped too fast to account for the longer complex bursts. It is not obvious why the fields cannot become even higher. Note that the virial limit is $B_{\rm v}\sim 10^{17}$ G.

[17, Kluzniak & Ruderman (1998)] note that, starting with 10¹² G, it only takes of order a second for simple winding to amplify the field to 10¹⁵ G; amplification in a newly-formed torus could well occur more rapidly, for instance via convective instabilities, as in a newly formed neutron star (cf. [48, Thompson & Duncan 1993]; [47, Thompson 1994)]. Kluzniak and Ruderman suggest, however, that the amplification may be self-limiting because magnetic stresses would then be strong enough for flares to break out. A magnetic field configuration capable of powering the bursts is likely to have a large scale structure. Flares and instabilities occurring on the characteristic (millisecond) dynamical time scale would cause substantial irregularity or intermittency in the overall outflow that would manifest itself in internal shocks. There is thus no problem in principle in accounting for sporadic large-amplitude variability, on all time scales down to a millisecond, even in the most long-lived bursts. Note also that it only takes a residual torus (or even a cold disk) of $10^{-3}\ M_\odot$to confine a field of 10¹⁵ G, which can extract energy from the black hole via the B-Z mechanism. Even if the evolution time scale for the bulk of the debris torus were no more than a second, enough may remain to catalyse the extraction of energy from the hole at rate adequate to power a long-lived burst.