DIPOLE AND QUADRUPOLE TESTS OF THE ISOTROPY OF GAMMA-RAY BURST LOCATIONS

Dipole and quadrupole tests of the isotropy of locations on the sphere are discussed. The statistics [cos theta], the dipole moment to the Galactic center, and [sin2 b - 1/3], the quadrupole moment about the Galactic plane, can be used to search for significant anisotropies in galactic coordinates. Such coordinate-system-based tests are the most powerful tests of dipole and quadrupole anisotropies in the particular coordinate system. Two statistics which have not been previously applied to gamma-ray burst data are described. The Rayleigh-Watson statistic W measures the size of the dipole moment of the locations and the Bingham statistic B measures the deviation of the eigenvalues of a quadrupole-like matrix from the values expected for isotropy. Tests based upon these two statistics search for dipole and quadrupole moments in spherical location data in a coordinate-system-independent, and thus model-independent, manner. They can detect an anisotropy in any direction and yield an analytic statistical significance to any detected anisotropy. The statistical tests are demonstrated herein using a variety of data sets and will be applied to the gamma-rav bursts locations of BATSE in Fishman et al.