The dust detector system onboard Galileo records dust impacts in circumjovian space since the spacecraft has been injected into a bound orbit about Jupiter in December 1995. This is the sixth in a series of papers dedicated to presenting Galileo and Ulysses dust data. We present data from the Galileo dust instrument for the period January to December 1996 when the spacecraft completed four orbits about Jupiter (G1, G2, C3 and E4). Data were obtained as high resolution realtime science data or recorded data during a time period of 100 days, or via memory read-outs during the remaining times. Because the data transmission rate of the spacecraft is very low, the complete data set (i. e. all parameters measured by the instrument during impact of a dust particle) for only 2% (5353) of all particles detected could be transmitted to Earth; the other particles were only counted. Together with the data for 2883 particles detected during Galileos interplanetary cruise and published earlier, complete data of 8236 particles detected by the Galileo dust instrument from 1989 to 1996 are now available. The majority of particles detected are tiny grains (about 10 nm in radius) originating from Jupiters innermost Galilean moon Io. These grains have been detected throughout the Jovian system and the highest impact rates exceeded $rm 100 min^{-1}$. A small number of grains has been detected in the close vicinity of the Galilean moons Europa, Ganymede and Callisto which belong to impact-generated dust clouds formed by (mostly submicrometer sized) ejecta from the surfaces of the moons (Kruger et al., Nature, 399, 558, 1999). Impacts of submicrometer to micrometer sized grains have been detected thoughout the Jovian system and especially in the region between the Galilean moons.
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