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Here on Earth we get to experience lunar and solar eclipses. Is this pure coincidence that everything happens to line up?

Do other planets in the solar system experience this phenomenon or is this something unique on Earth?

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Pretty much every planet with a moon can have eclipses - you'll have seen photographs of Saturn with shadows cast from its moons, like this one from wodumedia.com:

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If you were in a balloon where that shadow is, you would see a solar eclipse. It wouldn't be as exciting as from Earth, as the Sun would appear so much smaller from this distance.

If you want the planet to eclipse the moons, this happens far more often than on Earth, as the moons are so much smaller than Saturn. They often are completely shadowed.

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Eclipses are common, but solar eclipses like the ones our moon causes would be unusual since it takes a combination of factors, the angular size of the moon and sun are nearly the same from the viewpoint of Earth so that the moon (depending on where it is in its slightly ellipitcal path around the Earth) can cover the sun nearly exactly.

NASA has a nice picture of an eclipse as seen from the Mars Curiosity rover showing an eclipse by a moon with an angular size less than that of the Sun.

Phobos eclipsing the Sun

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    $\begingroup$ One also does not need to be on a planetary body to experience an eclipse. NASA also has an absolutely gorgeous picture of Saturn eclipsing the sun as viewed by the orbiting Cassini spacecraft: apod.nasa.gov/apod/ap110904.html $\endgroup$ – aleppke Sep 27 '16 at 21:26
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    $\begingroup$ One of my favourite photographs that, @aleppke. Dr Porco has stated in interviews since that they asked people to wave at Cassini about an hour an a half before the photo was taken, so that when the light hit the probe it would include people waving from Earth. :) $\endgroup$ – Benjamin Nolan Sep 29 '16 at 15:29

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