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I've heard a rumor that Curiosity rover sent a recording about Phobos covering the Sun on the Martian sky. Did it happen? If yes, how frequently could such things be recorded?

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No, because Mars can't have eclipses. Strictly speaking, Mars has only transits. The difference is that Mars's moons are smaller than the Sun as viewed from Mars, thus they don't block out the entire sun. Eclipses are defined as only occurring if the entire sun is blocked, or at least the vast majority. Phobos blocks out only about 60% of the sun at most. This could arguably be called a partial solar eclipse (But definitions are fluid, so...)

And yes, there have been several transits viewed by the rovers Curiosity and Opportunity and photographed. Here's pictures of some of them.

Curiosity enter image description here

Opportunity

enter image description here

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    $\begingroup$ +1 for a good answer, but I thought partial eclipses are also a thing? $\endgroup$
    – March Ho
    Oct 1, 2015 at 6:31
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    $\begingroup$ @MarchHo I suppose it depends on what exactly you mean by an eclipse. For example, is Venus transiting in between the Earth and the sun a solar eclipse? How about Mercury? A spacecraft in Earth orbit? Where to draw the line? $\endgroup$
    – user
    Oct 1, 2015 at 14:18
  • $\begingroup$ @MichaelKjörling I couldn't find a good definition, but en.wikipedia.org/wiki/Solar_eclipses_on_Mars seems to suggest that all transits are also eclipses (of the partial or annular type). $\endgroup$
    – March Ho
    Oct 1, 2015 at 14:27
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    $\begingroup$ Like defining planets, I've learned that the definition of Eclipse, Transit, and Occultation are all similar. Basically, the difference is in the size of the object. Phobos is either eclipse or transit, Deimos would just be transit. Sigh, definitions... $\endgroup$
    – PearsonArtPhoto
    Oct 1, 2015 at 15:08
  • $\begingroup$ I think we can speak of a partial eclipse when the object hiding the sun is big enough to hide it completely, but we're geographically poorly placed for this to be the case (for example, on Earth during an eclipse, when we're not right in the shadow zone). If the object passes completely in front of the sun, but is too small to hide it completely, then it's a transit. $\endgroup$
    – AoooR
    Aug 8, 2023 at 9:12

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