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From the angular distribution of the BATSE catalog
(fig. ), the observed dipole and
quadrupole relative to the galaxy are perfectly consistent with
null values, i.e. with an isotropic distribution; this
property clearly supports an extragalactic origin or, at least,
from an extended dark halo surrounding our Galaxy;
to distinguish between these two possible models, two kind
of analysis have been performed: either to test the angular
isotropy with high precision ([Briggs et al., 1996], [Hakkila et al., 1994],
[Tegmark et al., 1996b]), or to search for a possible population
of GRB sources around M31 ([Klose, 1995], [Lamb, 1995]);
while the latter did not find any evidence, the former over
the years set severe constraints, thus making unlikely a
galactic origin of GRBs; these constraints regarded burst
repetition as well ([Meegan et al., 1995], [Tegmark et al., 1996a]).
The distribution of sources in space shows a significant
deviation from the homogeneous flat space distribution
([Pendleton et al., 1995]), but consistent with a cosmological
distribution. Furthermore, the short duration burst distribution
looks consistent with a homogeneous Euclidean distribution
([Katz & Canel, 1996]).
On this subject, according to a standard candle model within
an Euclidean geometry space,
let us suppose that all bursts have the same luminosity ;
let be the peak flux, the volume of the sphere, whose radius
is the distance to the GRB source; then, in connection with
the minimum detectable flux for a fixed detector,
there is the maximum volume: , whose radius
can be obtained from the following:
.
Under these assumptions, the following relationship between
the relative volume and the relative peak flux follows:

(1) 
The ratio expressed by eq. is suitable for testing
the spatial distribution of a GRB set: for a homogeneous distribution
within an Euclidean space, the distribution is uniform
in the allowed range
, hence with a mean value
. In the case of 557 BATSE bursts, the corresponding
mean value turned out to be
([Kouveliotou, 1992]). Another analogous test is represented by the
LogNLogS curve: when considering the cumulative distribution ,
representing the number of bursts, whose flux intensity is greater than
, the case of a homogeneous distribution is characterized by the
following proportionality relationship:

(2) 
According to the interpretation of eq. , from fig. ,
obtained from a sample of BATSE bursts, it comes out that the spatial
distribution cannot be consistent with the homogeneous case, since there are
fewer faint bursts than predicted by this case.
Next: Spectral Properties
Up: The Prompt Emission
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Cristiano Guidorzi
20030731