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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:

\begin{displaymath}
\frac{V}{V_{max}} \ = \ \Big ( \frac{S_{min}}{S} \Big )^{3/2}
\end{displaymath} (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:
\begin{displaymath}
N(>S) \ \propto \ S^{-3/2}
\end{displaymath} (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]).
\begin{figure}\begin{center}
\epsfig{file=logN_logS.ps, width=9.0cm}\end{center}\end{figure}


next up previous contents
Next: Spectral Properties Up: The Prompt Emission Previous: Temporal Properties   Contents
Cristiano Guidorzi 2003-07-31