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Is there a chance for the many geostationary satellites to enter into the moon’s shadow?

If so, how frequently do such eclipses happen and how long do they typically last?

Are such eclipses pre-calculated and the satellite is programmed to handle such events (e.g. sudden drop in temperature associated with eclipse, etc)?

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You can get an approximate answer by looking at how often a solar eclipse crosses the equator. This map for the years 2021 through 2040 implies that satellites around 30 W longitude will experience a deep partial eclipse 5 times during that time period. (2023, 2033, 2028, 2034, 2038). The maximum duration of a total solar eclipse on Earth is just over 7 minutes long, so the duration at geostationary orbit (slightly closer to the Moon) is only slightly longer. Of course, a solar eclipse causes by the Moon only affects the few satellites in the path.

Eclipse paths 2021-2040 From NASA Solar Eclipse Page

A bigger problem are the eclipses that occur each year when all of the geostationary satellites pass through the Earth's shadow. These occur for a few weeks around the March and September equinoxes. At the date of maximum eclipse, the duration is 72 minutes. See Intelsat Eclipse Season.

I would hope that satellite needs no special programming to handle these Earth eclipses. The satellite either gets power directly from the solar cells or battery. It should not need to know that an eclipse will occur and therefore do something different. Therefore, the infrequent lunar eclipses should not require anything special.

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