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Since the Galileo mission will last 4+ years in orbit around Jupiter, Jupiter will be opposite the Sun four times during that period. How will that affect the science gathering and transmission of data to Earth?
The geometrical situation that you describe with occurs when the Sun lies directly between Jupiter and Earth (scientists refer to this as "solar conjunction"). The Sun is a strong source of electromagnetic activity, and it wreaks havoc with the spacecraft's radio signal, essentially reducing the spacecraft's data rate to Earth to 0 for the two and a half weeks centered around conjunction. Mission planners and telemetry engineers define this problem area as occurring when the Sun-Earth-Galileo angle is less than 7 degrees (see figure below); a relatively "quiet" Sun can mean that data can be successfully returned at angles as small as 3-5 degrees.
During Galileo's primary mission, two conjunction periods ran from December 11-28, 1995, and January 11-28, 1997. During the extended mission, solar conjuctions periods ran from February 14, 1998 - March 4, 1998, March 22, 1999 - April 10, 1999 and April 28, 2000 to May 17, 2000. The next solar conjuction period is from July 9 to 28, 2002. Conjunction lowers the amount of data that can be returned to Earth.
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