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Astron. Astrophys. 340, 447-456 (1998)

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6. Summary and discussion

In the past, purely electromagnetic modes of GRB origin, mostly involving (isolated) pulsar magnetospheres, have been considered from time to time in the context of galactic models or at best for the so-called galactic-halo models. Energy constraints discouraged extension of such models for the cosmological perspective because of the following simple reason. The radio pulsars are probably the best studied high energy astrophysics objects and there are strong observational reasons to believe that the pulsar magnetic fields do not exceed [FORMULA]G and pulsars are not born with a period shorter that [FORMULA]ms (Bhattacharya & van den Heuvel 1991). While, the constraint on the initial spin period persists, recent observations have found that, there are a class of pulsars with superstrong magnetic fields of few [FORMULA] G (Kouvelotou et al. 1998). However, given their age of few thousand years, they are very slow rotators with [FORMULA] few s compared to the X-ray pulsars or normal radio pulsars. The quiesent electromagnetic activity of such pulsars are completely insignificant for any kind of GRB activity, however, the intermittent release of the stored magnetic energy, plausibly fuels the SGR activity. As far has classical GRBs are concerned, these, magnetars may not be relevant.

In fact, recently, Spruit & Phinney (1998) have suggested that pulsars might be born as slow rotators because the magnetic locking between the core and the envelope of the progenitor prevents the core from spinning rapidly. This suggestion may be generally true although there must be exceptions like Crab for which the initial spin has been estimated to be [FORMULA] ms (Glendenning 1996). In particular, Spruit & Phinney envisage that magnetars are born as slow rotators with [FORMULA] s.

On the other hand, the elaborate realistic numerical computations (though primarily Newtonian) to study the physics of NS-NS collisions (Ruffert et al. 1997) showed that the value of [FORMULA] could be as low as [FORMULA] erg, which falls short of the previously assumed requisite value [FORMULA] ergs by several orders of magnitude. In view of such discouraging results (from the GRB view point), it became, probably, important and necessary to reconsider the electromagnetic models for the cosmological GRBs too. However, in this paper, we probed some of the consequences for extending the normal electromagnetic models involving stellar mass compact objects with [FORMULA] erg/s to scenarios with intended [FORMULA] erg/s or even higher.

A hot and fluid like NS with a assumed value of [FORMULA], as was believed to be required for the GRB problem in the pre-GRB971214 era , is likely to lose practically the entire initial RKE by gravitational radiation because of the equatorial ellipticity associated with its Jacobi Ellipsoid configuration. It is then expected to settle to a much lower value of [FORMULA] within [FORMULA]s of its birth. Even then, it will be subject to r-mode instability. If the intial [FORMULA]-cooling is rapid enough, [FORMULA]s, neutrino flux driven conduction may set up a super strong magnetic field; but, by this time, the bulk viscosity and gravitational radiation may bring down the spin to [FORMULA] ms (we have already mentioned that such strong field magnetars might actually be born with [FORMULA] s). Thus, these models will fail to explain GRBs. And with an actual value of [FORMULA] erg, such models become even much more fragile.

We also recall here that the approximate formation time of a NS out of the proto-NS is determined by the fairly long neutrino diffusion time scale of [FORMULA]s (Shapiro & Teukolsky 1983) and this has been confirmed by SN1987A. Since this time scale is comparable to or, in fact larger than, most of the GRB time scales, it is unjustified to consider the sudden birth of a NS (of high magnetic field) in isolation and then seperately study its probable spin-down or cooling on a time scale of few seconds. Such supposed prompt spin down has to be studied as an integral part of the final stages of the preceding collapse process. Following the discussion of Sect. 1, as soon as, the proto-NS would be envisaged to acquire high spin, the associated gravitational damping would constrain the value of [FORMULA]ms or to a much higher value. In fact, the formation stages of the torus would also certainly involve wobbling and spinning of the would be torus material. And this would again mean huge loss of angular momentum and kinetic energy from the system, though, it would practically be impossible to estimate such losses. The formation of NS always involve release of its BE [FORMULA] ergs primarily in the form of neutrinos and antineutrinos. And this may indeed entail strong neutrino driven convection, and, yet it is certain that most of the neutron stars, except the few magnetars, are not born with super strong magnetic field. So our understanding of genesis of super strong NS is far from complete and we need to invoke the idea of such strong fields with caution.

Recently, Kluzniak & Ruderman (1997) have considered a scenario in which the new born pulsar has a (poloidal) magnetic field [FORMULA], but the differential rotation winds it up to a toroidal field of [FORMULA]G. In this case, [FORMULA] is moderate and the the main relativistic beam arises because of the annihilation of such strong toroidal magnetic fields. However, following Duncan & Thompson (1992) and Usov (1992), even if we assume the creation of a super strong magnetic field [FORMULA]G in new born pulsars on the plea that, unlike macroscopically turbulent accretion disks (actually torus), the core of a pulsar is non-turbulent, even though hot, it is far more difficult to conceive of a field [FORMULA]G in the absence of real macroscopic quantum coherence. And in any case, because of ultrarapid rapid gravitational energy loss, such models are unlikely to work.

In the NS-NS or NS-BH merger scenarios, for the resultant hot disk whose temperature is higher than the superconductivity transition temperature and which is macroscopically turbulent, it is extremely difficult to conceive that it possesses a magnetic field higher than [FORMULA]G. And note that, in the post GRB971214 era , one needs to artificially further over stretch the value of [FORMULA] to probably [FORMULA]G.

Could there be yet more novel variety of electromagnetic models of GRBs involving compact objects? Recently, Usov (1998) purported to show that the bare surface of a new born strange star would spontaneously emit pairs with luminosity [FORMULA] erg/s for about [FORMULA]s. However, we have already shown that this idea can not work simply because the time scale to transport thermal energy from the core of the new born strange star could be as large as [FORMULA]s, rather than [FORMULA]s, as assumed by Usov. The only way to transport the thermal energy on the desired short time scale of [FORMULA]s or shorter would be to consider the emission of [FORMULA] pairs (Mitra 1998a).

As to the disk-jet case, the resultant value of [FORMULA] would be much lower than the one estimated using vacuum electrodynamics because Ruffert et al. (1997) show that the disk would be as hot, [FORMULA]MeV, and therefore, even for an idealized thin disk geometry and no quasi-spherical accretion, there would be fairly dense plasma around the disk (corona) and which would tend to quench any direct electromagnetic mode. Most probably, with accretion rates exceeding the Eddington rate by factors as large as [FORMULA], all such electromagnetic modes of energy extraction, appropriate for radio pulsars, may not be applicable to the GRB-disk at all.

6.1. Mode of energy release

In general accretion energy will be channelized into X-rays and to neutrinos along with the relativistic beam, if any. Given this fact, it is not really possible to ascertain how much of [FORMULA] would go into [FORMULA], where [FORMULA] is the duration of the beam. Similarly, the life time of the disk ([FORMULA]) also can not be predicted, and let us just assume it to be [FORMULA]s. Thus, the value of [FORMULA] is to be set to the desired value by keeping [FORMULA] a free parameter and conveniently ignoring other competetive and probably much more likely process of neutrino production.

Yet, we can say that, as is the case for highly super-Eddington accretion, accretion power will primarily produce [FORMULA]. In fact this is the most natural process of cooling of hot and dense astrophysical plasma. We may look at it in the following way. One of the primary task assigned by Nature to the electro-weak interaction is the drainage of energy from physical and astrophysical systems. For macroscopic bodies at relatively lower densities and temperatures, it is the electromagnetic part which undertakes the responsibility of relieving systems of excess energy and help arrive at (new) equilibrium situations. On the other hand, at high densities and temperatures, it is the weak processes which take up this responsibility, and hot astrophysical bodies cool by emission of neutrinos through pure electroweak processes (like at the late stages of evolution of massive stars) or through various URCA type processes (Chiu & Morrison 1960, Chiu & Stabler 1961).

One may try to further appreciate this problem from this simple argument: Even assuming that NS-NS or NS-BH collision process results in a disk of necessary mass, why must it get accreted within [FORMULA] when the disk of Saturn can stay put indefinitely? Well, the answer would be that there is viscous dissipation and associated loss of angular momentum of the disk material. The nature and quantitative value of disk viscosity is most uncertain and even if the source of viscosity is considered to be disk magnetic field, the dissipation of energy in the disk necessarily means that accretion energy is primarily lost in the form of heat and neutrinos . Thus the BH-disk GRB model of Woosley (1993) appropriately considers neutrino production as the main source of accretion energy liberation. It is a different matter that, this model, has difficulty in accounting for a value of [FORMULA] erg. Finally, note that, even in the AGN context, the physics of magnetically dominated disk-jets is poorly understood and "hydromagnetic propulsion as a mechanism for accelerating jets has become attractive largely through a process of elimination" (Begelman 1994) of other competetive ideas.

It hardly requires a reminder that most of the other original cosmological GRB models too logically envisaged that whether it is a NS-NS collision or a NS-BH collision, the energy is liberated in the form of [FORMULA] (and in gravitational radiation) rather than in the form of any (direct) relativistic [FORMULA] beam (Goodman, Dar, & Nussinov 1987, Paczynski 1990, Haensel, Paczynski, & Amsterdamski 1991, Rees and Meszaros 1992, Meszaros & Rees 1992, Piran 1992, Mochkovitch et al. 1993). Now falling back on these works, we feel that, irrespective of the details, it is much more likely that the powerful [FORMULA] FB is indeed due to the annihilation of [FORMULA]. There may be another reason why we may be compelled to invoke neutrinos in this problem.

The sub-ms time structures found in many GRBs (Bhat et al. 1992) have enhanced the general view that the dimension of the central engine of the GRBs is [FORMULA]cm. Now we repeat the already oft-repeated argument that the fact that for GRB971214, the luminosity is [FORMULA] erg/s, with a probable larger value of [FORMULA], or for GRB970508, the value of [FORMULA] erg/s would suggest a temperature of the emission zone, [FORMULA] K. At such high temperatures, even if one has no neutrino to start with, photoneutrino processes (Chiu & Morrison 1960) ensure that neutrinos become much more numerous than [FORMULA] pairs:


The situation for such high temperature pair plasma becomes somewhat like the early universe at corresponding temperature. In most of the realistic astrophysical situations, neutrinos, being chargeless and weak particles, have much better chance of escaping than [FORMULA] pairs (the pair plasma is already dominated by pairs at such temperatures). Thus, unless we conceive of highly fine tuned and optimistic models, the very existence of such extremely high values of [FORMULA] would imply that the primary energy release mechanism from the central engine is in the form of neutrinos.

In the context of the accretion induced collapse (of white dwarfs) model, it has been estimated that, the efficiency of the pair production via the annihilation of [FORMULA] is


Here the geometrical factor [FORMULA], [FORMULA] is the luminosity of the neutrino beam, [FORMULA] is the crosssection of the process averaged over three flavours, and [FORMULA] is the mean energy of the neutrinos (Goodman et al. 1987, Piran 1992). For the supernova case where [FORMULA] erg/s and [FORMULA]MeV, one obtains a very low value of [FORMULA].

If we directly and naively use this value of [FORMULA] to understand the origin of [FORMULA] erg, we would require a value of [FORMULA] erg! But note here that, since [FORMULA] we find


As [FORMULA] increases beyond, say, [FORMULA] erg/s, with a corresponding decrease in [FORMULA] (a deeper gravitational well is necessary to have

incresed [FORMULA]), the value of [FORMULA] would also increase considerably. Therefore, eventually, we may have


By pursuing such probable higher values of [FORMULA], one might partially approach a situation, where the process [FORMULA] is in thermal equilibrium with [FORMULA]. It does not mean that such a limit is exactly attained, but depending on the unknown details of the evolution of the central engine, a value of [FORMULA] may be achieved. For instance, a value of [FORMULA] erg/s and [FORMULA] erg might self-consistently explain the origin of [FORMULA] erg.

Here we note that some of the GRBs are stronger than even GRB971214. For instance, the fluence of GRB980329 in the (50-300)KeV band is [FORMULA] erg cm-2, which is approximately five times larger than that of GRB971214. In fact the radio afterglow has been detected for this burst too (Taylor et al. 1998). If this burst too lie at [FORMULA], it may be possible to to postulate that the GRBs actually constitute a "standard candle" probably within a factor of 10, with respect to the value of [FORMULA]. Small variations in microphysics from one case to another may induce a spread in the duration, [FORMULA] (of the neutrino burst, not necessarily same as the duration of the observed GRB) with considerable spread in the value of [FORMULA]. Then, because of the non-linear nature of the conversion efficiency (Eq. 30), there is additional spread, probably spanning two orders of magnitude, in the eventual value of [FORMULA]. If the baryon contamination is above a critical value (Shemi & Piran 1990), the resultant GRB could be very feeble and most of the FB energy would go into accelerating the baryons residing inside the FB and those lying ahead (like previuosly ejected mass shells and presupernova wind). Such events would be detectable in the electromagnetic band only if it occurs in a nearby galaxy. It may be tempting to explain the association between the weak event GRB 980425 ([FORMULA] ergs) with the mildly relativistic extraordinary supernova event SN 1998bw having a kinetic energy of [FORMULA] erg occuring at [FORMULA] (Galama et al. 1998, Soffita et al. 1998). We predict that the ejecta of SN 1998bw is not highly beamed though it could be quasi spherical and quasi symmetric contrary to several suggestions to this effect. Future observations should confirm (or reject) this simple prediction.

Such high value of [FORMULA] erg, with an associated value of [FORMULA] erg/s would demand a value of [FORMULA]GeV. This difficult conclusions should not be avoided with the plea that the existing models/theories are unable to explain the origin of such prodigious neutrino luminosities. In fact the existing paradigm, that gravitational collapse can not directly yield a value of [FORMULA] larger than the BE of canonical NS, [FORMULA] erg, and any attempt to harness more energy by studying collapse of relatively more massive proto-NS would give birth to a black hole with hardly any appreciable energy output completely fails to explain the genesis of SN 1998bw. Of course, one can try to avoid this difficulty by proposing "collapsar" models (Wooseley 1993, Wooseley, Eastman, and Schmidt 1998). Although, the intention of the present paper is not to outline any (new) GRB model, we would only like to point out that, recently, we have shown that, for continued general relativistic spherical collapse, the entire origina mass energy can be radiated (Mitra 1998b,c). But more importantly, we have shown that general relativity actually inhibits the formation of "trapped surfaces", the regions wherefrom even radiation can not move out. This opens up the possibilty that gravitational collapse of sufficiently massive stellar cores ([FORMULA]) does not end up as a quiet passage to a black hole. Thus, the collapse of a [FORMULA] stellar core may account for this required [FORMULA]. In fact, we predict that, once the new generation giant neutrino detectors become fully operational, it may be possible to detect neutrino bursts of value [FORMULA] erg with neutrino energies reaching [FORMULA] GeV, in coincidence with the GRB events.

We are also aware here of the fact that, luminosity apart, for explaining GRBs, one needs to conceive of situations where the baryon load of the beam is modest and the value of [FORMULA] is indeed high; but we would address all such questions in a latter work.

To conclude, we did not attempt to predict the value of [FORMULA] by presuming some canonical value of [FORMULA] and [FORMULA]. It is plausible that these two parameters actually vary substantially from case to case even for the afterglow regime. And, probably only when these parameters have an appreciable value [FORMULA] and [FORMULA], radio counterparts are found. Conversely, the absence of detectable radio afterglow for most of the GRBs may be ascribed to actual occurrence of relatively lower vales of these parameters.

Whether the blast wave produes detectable radio afterglow or not, as ensiaged previously (Mitra 1998d), there should be some remnants (GRBR), like defunct supernova remnants, of these events. These remnants may be found in nearby galaxies or in the Milky way.

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Online publication: November 9, 1998