The information about the observed properties of gamma-ray bursts was provided by Dr. Fishman in his introductory paper. I shall summarize what I consider to be the most important for their distance scale. The observed distribution of gamma-ray bursts appears to be isotropic. It is true that a number of researchers found small departures from perfectly uniform distribution by selecting various sub-samples of the bursts (e.g. Lamb & Quashnock 1994), but these small irregularities were not confirmed by the new data, and even the correctness of the original claim is in doubt (Rutledge & Lewin 1993). It is natural that any finite sample must exhibit some statistical fluctuations, giving evidence for things which are not, in fact, there. So far no specific suggestion of anisotropy survived the test of time. To the contrary, the more bursts are known the more isotropic their distribution appears, as the statistical fluctuations average out.
There was a lot of confusion in the 1980s about the correct way of measuring burst intensity. By ``correct'' I mean the way which would be proper to study their radial distribution. After some debate a consensus has been reached: the burst intensity has to be measured in the same units which are used to detect the bursts. Roughly speaking it is the number of photons detected in a time bin of a chosen duration. For example, if a burst is detected when the count of photons in one second intervals exceeds some threshold, then the peak count per second should be used as a measure of the burst intensity.
It is well known that if sources of any kind are uniformly distributed in a 3-dimensional Euclidean space then the number of sources, , that appear to be brighter than some limit is proportional to that limit in power -1.5, that is we have: . The reason is simple: the sources that appear a factor brighter are a factor farther away, and therefore they are detectable in a volume that is a factor larger, and hence they are more numerous by a factor . For this relation to hold, the statistical properties of distant sources should be the same as the statistical properties of nearby sources, and the average number of sources per unit volume should be constant. This relation is observed to be approximately satisfied by the bright stars as well as by the bright galaxies, even though the nature of the two types of objects, as well as their distances, are very different.
The best statistics of bright gamma-ray bursts is provided by the instrument PVO - Pioneer Venus Orbiter, which was collecting information for 13 years, and registered over 200 strong bursts (Fenimore et al. 1992). The distribution of these bursts is well approximated with the relation , indicating that the sources of apparently strong, and presumably relatively nearby bursts, have a uniform number density out to some distance. The most sensitive instrument to date is BATSE, roughly 30 times more sensitive than PVO. The distribution of BATSE bursts may be approximated with the relation (Meegan 1992), indicating that the number of apparently weak, and presumably distant bursts is relatively small, as if there were fewer of them at large distances, or if the space at some large distance was no longer Euclidean. The results from instruments with the sensitivity intermediate between PVO and BATSE is intermediate (Tamblyn & Melia 1993). In fact, when the PVO and BATSE bursts are combined, a gradual transition from a slope close to -1.5 on the bright end to -0.8 at the faint end becomes apparent (Fenimore et al. 1993). It appears that the results from all instruments provide a consistent view: the relatively nearby bursts are approximately uniformly distributed with distance, but beyond some distance the number density of bursters decreases, or the space becomes non-Euclidean.
It is very important to consider the two results together: the isotropic distribution in the sky with the apparent shortage of weak bursts in all directions. The conclusion is: we appear to be at the center, or near the center of a spherically symmetric and bounded distribution of gamma-ray bursters.
Another very important feature of a gamma-ray burst is its spectrum. The spectra are very broad, covering many decades of photon energies, extending in extreme cases to photons as soft as 1 keV, and in some cases to photons as hard as 18 GeV (Hurley et al. 1994). A typical spectrum may be approximated with a broken power law (Schaefer et al. 1992, 1994), and it is so broad that there is no doubt it is non-thermal. I think there is a consensus on this issue.
It is important to note that while it is relatively easy to make a model of a nearly thermal stellar spectrum, regardless of whether the star is radiating mostly in the infrared (like M-type giants and dwarfs), in the optical (as our sun), in the ultraviolet (as O-type stars), or even in the X-rays (as X-ray bursts). A thermal model is uniquely described with its temperature, and a few other well defined parameters. The situation is very different for any class of astronomical objects which have clearly non-thermal spectra, like quasars, radio pulsars, or gamma-ray bursts. There is no unique way to model such a spectrum and there is no consensus among the researchers which of the many models of any of the non-thermal sources is correct. Invariably, the models invoke free parameters and a lot of more or less ad hoc assumptions. While the quantitative models of thermal or nearly thermal sources may even be useful in estimating their distances, nothing of this sort is possible for any of the sources which have non-thermal spectra - the diversity of non-thermal models is limited only by the imagination of theoreticians who study them. I know of no example of a model of a non-thermal source to be successfully used to deduce its distance. Therefore, my conclusion is that currently there is no known way to use the observed spectra to deduce the distance to gamma-ray bursts.
But not everything is lost. Sometimes a spectrum can be used to deduce the distance even if we have no idea how the spectrum was formed: we just use the source as a light bulb, and the fact that it can be seen may be used for the distance estimate. The trick is that the matter between the source and the observer may absorb photons of some energy. For example, interstellar gas is very opaque to the ultraviolet and soft X-rays (cf. Paczynski 1991), while TeV photons cannot travel very far through the intergalactic radiation field (cf. Stecker et al. 1992). If the spectrum of a source located near the plane of our galaxy is found to extend into sub-keV domain than the source must be closer than about 1 kiloparsec from us (well within our galaxy), as otherwise the interstellar gas would prevent the soft photons from reaching us. If the spectrum is seen to extend above 1 TeV then the source cannot be at a distance larger than about 500 Megaparsecs as the intergalactic infrared radiation prevents super TeV photons from reaching us from a larger distance. It is possible that in the future such effects may help us to estimate the distances to gamma-ray bursters.
The bursts show a tremendous variety of intensity variation, the time structure and duration (cf. Fishman et al. 1994). In the past, the rapid time variability was thought to provide some limit on the distance to the sources. This is no longer so. It is well established that if the source expands ultra relativistically then the observed variability is compatible with any distance (Zdziarski et al. 1991). And there are at least two reasons to expect ultra-relativistic expansion of gamma-ray bursters located either in the galactic corona or at cosmological distances. First, at either distance scale, the luminosities are highly super-Eddington with the powerful radiation pressure driving relativistic expansion. Second, the observed spectra show no cut-off at the pair creation limit, implying the gamma-ray photons are streaming nearly radially from the ``photosphere'' out. This is equivalent to an ultra-relativistic flow.
It is my conclusion that at this time we have only one robust distance indicator for gamma-ray bursters: the apparently isotropic distribution of bursts in the sky combined with the evidence that the radial distribution is bounded at the same distance in all directions.
Following Hakkila et al. (1994) I shall use the term `galactic corona' rather than 'galactic halo' for one of the currently popular distributions of gamma-ray bursters. The reason is that the required distribution is unlike any observed or theoretically proposed galactic halo. According to the most popular ideas of the galactic origin of the bursters they are related to the very high velocity neutron stars, which are claimed to have this novel type of distribution.
It is interesting to note that the two most popular distance scales are both extragalactic. The nearest galaxies, the Large and Small Magellanic Clouds, are at the distance of kiloparsecs. A few other galaxies are at about kiloparsec. A typical coronal gamma-ray burst is believed to be at kiloparsecs, i.e. well into the intergalactic space. Of course, in the cosmological hypothesis the bursters are even farther away. The only justification to call the coronal distribution galactic rather than extragalactic is the origin of the bursters: in the coronal scenario they are thought to be ejected from our galaxy.