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

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3. Accretion disk mode

A NS-NS or a NS-Black Hole (BH) collision may lead to the formation of a transient compact object-disk type configuration (Narayanan, Paczynski, & Piran 1992, Ruffert et al. 1997, Woosley 1993), and following the previous ideas by several authors, (Lovelace 1976, Blandford & Znajek 1977) that such configurations may generate relativistic beams by electromagnetic process, it has been recently proposed that NS-NS or NS-BH collisions may may generate superstrong relativistic winds (Katz 1997, Rees & Meszaros 1997). In very broad terms the power radiated by such scenarios could be understood in terms of the Lovelace (1976) model by assuming the disk to be Keplerian and thin. The magnitude of the induced disk electric field would be (standard polar coordinates are used):

[EQUATION]

The corresponding voltage drop between the inner radius [FORMULA] and the outer radius [FORMULA] would be (Lovelace 1976):

[EQUATION]

where [FORMULA] is the inner radius and [FORMULA] is the outer radius of the disk. In the absence of a any reasonable information about the profile of [FORMULA], one may be forced to consider the most favourable order of magnitude estimate of the foregoing equation by considering [FORMULA], [FORMULA], and [FORMULA]

[EQUATION]

where [FORMULA], [FORMULA], and we have taken [FORMULA] as the radius of the last stable circular orbit and [FORMULA] as the corresponding Keperian value:

[EQUATION]

The current flowing along the disk axis would be [FORMULA] and, using Eq. (14), the maximum beam power output would be

[EQUATION]

For a (stellar mass) NS with weak magnetic fields, the disk may nearly touch the surface and, the above value of [FORMULA] would be appropriate for such cases too. If the above model is extended for such a stellar mass NS, we would obtain a value of [FORMULA] which is quite high for X-ray binaries:

[EQUATION]

where the value of [FORMULA] corresponds to either the disk inner edge touching the NS-surface or the case of a closeby inner edge immersed in the magnetosphere of the NS. This kind of NS-disk model was actually suggested to explain the origin of supposed ultra high energy gamma rays and cosmic rays from the X-ray binary Cygnus X-3 (Chanmungam & Brecher 1985). It was argued, in the eighties, that, Cyg X-3 might contain a rapidly spinning NS, and therefore, either by direct pulsar action or by NS-disk symbiosis, as has been proposed for the present GRB case, should be a strong source of ultra high energy gamma rays and cosmic rays. By overstreching the models, it was even envisaged that a single Cyg X-3 type source could solve the problem of origin of ultra high energy cosmic rays for the entire galaxy. However, it became, clear later, that such theoretical precdictions were completely unfounded. Further, if one has to fit such models into the cosmological GRB scenario, one has to further overstretch such (unsuccessful) models by many orders of magnitude; and one would be compelled to assume an enhanced value of [FORMULA]G to obtain a notional value of [FORMULA] erg s-1.

3.1. Available energy

In case we are considering a BH-disk case, let us estimate first, what could be the order of magnitude of the central BH. From Eq. (15), we find the period associated with the last circular orbit to be

[EQUATION]

Thus in order to explain sub-ms time structure in some of the GRBs (Bhat et al. 1992), we should consider a value of M not exceeding a few solar masses (Woosley 1993). For a spinning canonical NS, the value of RKE is limited by the binding energy (BE), [FORMULA] erg. We note here that for a NS, there is a natural electromagnetic coupling between the disk and the star, and, in principle, it is possible that the supposed beam is partially fed by RKE too. However, it does not at all mean that this is necessarily the case, and, in any case, it does not mean that the entire RKE of the NS is tapped during the finite life time of the disk. If the disk could be considered a permanent structure, a rigid conductor, and rotating synchronously in tandem with the spin of the NS, maintaining a perfect electromagnetic coupling to the NS, it is possible in principle that, given very long time, the entire RKE is harnessed. But the resultant value of [FORMULA] in such a case would be determined by the value of surface magnetic field of the NS, and not by the supposed enhanced value of [FORMULA]G. In other words, in such cases, one reverts back to the pulsar (with intrinsic dipole moment) models. Thus, in the model of Lovelace (1976) and Chamungum & Brecher (1985), the actual source of [FORMULA] is the accretion power and not the RKE of the central compact object. In fact the values of [FORMULA] indicated by either Eq. (16) or Eq. (17), roughly suggest the conversion of an Eddington limited accretion power to electromagnetic power.

This conclusion is even more appropriate for the BH-disk case even though, technically, for a maximally rotating Kerr-Newman BH, the effective value of RKE [FORMULA]. This is so because, unlike a NS-disk case, a Kerr BH has no intrinsic charge, [FORMULA], or magnetic moment, [FORMULA]. Thus there is no natural electromagnetic coupling between the BH and its disk . There are, however, some theoretical estimates, based on purely vacuum electrodynamics, that a stationary axisymmetric BH, placed in an external uniform magneic field,[FORMULA] may acquire charge by the accretion process whose asypmtotic value would be [FORMULA] (Wald 1974). If so, it might be possible to harness the rotational energy of the BH, which, for a maximal case could be [FORMULA] erg (taking a value of [FORMULA]). Similar idea is also expressed by Blandford & Znajek (1977). Given a sufficiently long lived accretion disk (fed by external material), such ideas might be relevant for jets observed in AGNs. Nonetheless, note that, in a strictly axisymmetric and steady case, the accretion process would not deliver any net torque from the BH (Ruffini 1978) though it might be possible that in a transient case some net torque may be derived from the BH. Also note that the BH can acquire a substantial dipole moment only after substantial accretion has taken place. Thus, note that, even in the AGN context, the source of jet power is believed to be primarily the accretion power rather than the stored RKE of the BH (Begelman, Blandford, & Rees 1984).

In the transient GRB case, occurrence of substantial accretion may imply the vanishing of the disk. And unlike the case of a pulsar, the RKE of the BH (with no intrinsic diploe moment), even if it is substantial, can not deliver any power once the disk has vanished. And for the transient GRB event, a very high value of [FORMULA] can be justified by assuming the release of accretion power within a few seconds. Thus, atleast for the GRB problem, the supposed high RKE of the BH can not be meaningfully harnessed and the maximum extractable energy available by all probable processes is the the total BE of the disk of mass [FORMULA] in the gravitational potential of M. In order that the electrodynamic jet mechanism is successful, i.e. the potential drop [FORMULA] across is maintained, the insulating effect of the magnetic field must be effective. This condition is ensured, if for most of the region of the beam, we have [FORMULA] (Lovelace, MacAuslan and Burns 1979), which independently demands that [FORMULA]. For a NS-NS collision case the disk may have a mass of [FORMULA] (Ruffert et al. 1997), and thus the value of

[EQUATION]

[EQUATION]

is insufficient for explaining the energy budget of GRB970508. However, assuming a case of NS-BH collision with [FORMULA] (Meszaros & Rees 1997), it may be possible to satisfy this bare energy requirement ([FORMULA]).

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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