5. Electrodynamics immersed in accretion
A general requirement for realizing the vacuum potential drop is that the density of the ambient plasma is not exceedingly larger than the characteristics Goldreich-Julian density (Goldreich & Julian 1969) :
where is the cyclic frequency of the accelerator in Hz. This is the density of space charge that arises in the magnetosphere of an unipolar inductor type device in the absence of any externally injected plasma. It is because of the unavoidable presence of this space charge that the actual available potential drop in the magnetosphere becomes considerably smaller than the vacuum value . And one of the fundamental and pending problem of pulsar type mechanism is to self-consistently evaluate the value of the modified lower voltage . If it were not not so, many of the X-ray binaries containing either rapidly spinning NSs or modestly spinning white dwarfs would be strong soures of ultra-high energy cosmic rays. Even for isolated radio pulsars like Crab and Vela, the absence of emission of either ultra-high energy cosmic rays and gamma rays (the observed gamma rays lie in the TeV-range) would suggest that, even in the ideal cases, it is difficult to realize the vacuum potential drop . Further, when external plasma is present, the problem becomes even more poorly defined, and, all estimates based on the vacuum value of become highly suspect. In particular, if the external plasma density , at a certain stage, the pulsar type mechanism ceases to operate. This is corroborated by noting the fact that althogh most of the bright X-ray binaries contain NSs with sufficiently high magnetic field, in general, they do not operate in the pulsar mode , i.e., they undergo spin up rather than spin down process. Also, intermittently, during stages of low accretion rate, usually considerably lower than the Eddington rate, some of the pulsars in the X-ray binaries, undergo brief spells of spin-down.
For the the innermost circular orbit, we have
implying a value of
On the other hand, for accretion at the Eddington rate, g s-1, in the approximation of spherical accretion, by using Eq. (18), we find that, the density of the accretion plasma near the innermost region would be
Further, we define a ratio to quantify the degree of contamination of the extraneous plasma:
For a supermassive BH of having a value of G, or for an accreting millisecond binary pulsar with a low G, and whose accretion disk may be almost touching it, we find a value of . Note that, even if we consider the disk to be ideal and thin, there is alwaya a corona above and below the disk. Further, in all realistic cases, in particular, in case of minor deviation from exact axisymmertic accretion geometry, there may be a small component of quasi-spherical flow with density . This is more true for a thick accretion disks or tori. Such extremely weak quasi-spherical flows or corona need not inhibit the formation of effectively baryon free funnels along the symmetry axis, however they can certainly quench the vacuum potential drop estimated in Eq. (16).
Although, most of the accretion power of AGNs is primarily manifest in the form of ultraviolet or X-rays, we note that relativistic bipolar flows with luminositities, sometimes, comparable to seems to be a common features in active galactic nuclei. We also note that radiation driven or hydrodynamic jet models have difficulty in achieving a value of , (Begelman 1994) whereas most of the superluminal flows as well as gamma-ray observations require a value of . Here we may add that for the recently observed galactic `micro quasars', in some cases, the inferred value of while in one case, the value of (Mirabel & Rodriguez 1997, Hjellming & Rupen 1995). And though for radiation driven jets, it is generally found difficult to obtain values of , recently, by using the socalled "Compton Rocket Effect" Reanaud & Henri (1997) have attempted to show that radiation driven disk-jet mechanisms can overcome such limitations, and, in fact, attain a value of in the AGN context. However, the value of the terminal Lorentz factor, in this model, is given by
where is the mean energy of the disk photons in units of For the AGN problem, the value of is in the ultraviolet range, but for the GRB problem, , and therefore, again, we have .
Thus in the absence of alternative more successful theories about such relativistic flows (Begelman 1994), in a rather generous manner, let us assume here that it is the magnetically dominated disk-BH-jets which are working in the AGNs. By considering ergs s-1, we estimate, for so that typical and .
In contrast, note that, except for the recently discovered galactic superluminal sources mentioned above, most of the known bright X-ray binaries (with accretion disks) do primarily emit the accretion power in X-rays and not by relativistic beams. The radio-jets observed in many X-ray binaries like Cyg X-3, SS433, Sco X-1, are non-ralativistic with luminosities insignificant compared to Eddington luminosities. And in the light of the recent optimistic work on radiation driven jets (Reanaud and Henri 1997), it is entirely possible that, purely electromagnetic jet mechanism is neither necessary nor functional in the X-ray binaries. Even, if one assumes that, for the stellar mass cases too, ergs -1, one would obtain a maximum effective value of G with a corresponding value of . In the framework of this probable relativistic jet-mediated liberation of accretion power, we may explain the general absence of such activity in the X-ray binaries by restricting . Recall again that there will be extraneous plasma also be in the form an accretion disk corona and whose density could be proportional to the accretion rate.
The GRB disk case is similar to a stellar mass X-ray binary case with the difference that in the former case, we assume need a much higher value of G. But, on the other hand, note that for the GRB case, assuming that is accreted in s, we will have an accretion rate of g s, so that
Thus, even for an assumed high value of G for a GRB-disk, we will have , and with a value of , the density of the quasi-spherical flow , probably . Also note that, while we observe the AGN or X-ray binaries in a quasi-steady state, probably, atlest thousands of years after their formation, in the GRB disk case, what is really formed after the catastrophic collison is a torus or simply a cloud (Ruffert et al. 1997) which may be settled into a more-steady torus. And for such cases, even if the accretion were limited by the Eddington rate, the chance of a minor quasi-spherical accretion would be a genuine difficulty in achieving any idealized electromagnetic accretion machine. Thus it is very difficult to see how electromagnetic modes of energy extraction based on the idea of accelerators immersed in vacuum can work for the GRB case. In fact, the work of Ruffert et al. (1997) shows that, it is indeed likely that most of the accretion power associated with the GRB-disk is used in producing neutrinos.
© European Southern Observatory (ESO) 1998
Online publication: November 9, 1998