Spatial Distribution

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 $L$; let $S$ be the peak flux, $V$ the volume of the sphere, whose radius $r$ is the distance to the GRB source; then, in connection with the minimum detectable flux $S_{min}$ for a fixed detector, there is the maximum volume: $V_{max}$, whose radius $r_{max}$ can be obtained from the following: $S_{min} = L / (4\pi r^2_{max})$. Under these assumptions, the following relationship between the relative volume and the relative peak flux follows:

$$\frac{V}{V_{max}} \ = \ \Big ( \frac{S_{min}}{S} \Big )^{3/2}$$ (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 $V/V_{max}$ distribution is uniform in the allowed range $0\leq V/V_{max} \leq 1$, hence with a mean value $<V/V_{max}> = 0.5$. In the case of 557 BATSE bursts, the corresponding mean value turned out to be $<V/V_{max}> = 0.33 \pm 0.01$ ([Kouveliotou, 1992]). Another analogous test is represented by the LogN-LogS curve: when considering the cumulative distribution $N(>S)$, representing the number of bursts, whose flux intensity is greater than $S$, the case of a homogeneous distribution is characterized by the following proportionality relationship:

$$N(>S) \ \propto \ S^{-3/2}$$ (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.

Figure: LogN-LogS Distribution for 799 BATSE bursts (from [Kouveliotou, 1992]).