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What is the proper terminology for when a satellite or spaceship passes into a planet's shadow or night side? Is it crossing the terminator? Passing into eclipse? If there is a different name for coming back into the light, what would that be as well?

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SUMMARY: The word is 'eclipse'

If you visit https://www.skyandtelescope.com/observing/a-jupiter-almanac/ and click on "Phenomena of Jupiter's Moons in 2018" you get to https://s22380.pcdn.co/wp-content/uploads/WebJphenTab2018.pdf . The latter document refers to such events as "an eclipse by Jupiter's shadow".

Since Sky and Telescope is a fairly reliable source, I'd say this is a good answer, though an answer from NASA or IAU would be better.

Note that it is also a lunar eclipse as viewed from Jupiter and a solar eclipse as viewed from the moon being shadowed.

I realize you're talking about Earth's artificial satellites, and this answer involves Jupiter's natural satellites, but I think the same terminology applies.

I happen to be working on Jovian lunar eclipse calculations and ran across this resource incidentally.

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