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Could there be another planet between Mercury and the Sun? Kepler found a lot of planets orbiting its star even 20 times closer than Mercury. If there were such planets in our system - would it be possible to discover them?

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It is very unlikely. While the hypothetical planet Vulcan between Mercury and the Sun was ruled out many decades ago searches for Vulcanoid asteroids would very likely have detected anything that might qualify as a intra-Mercurial planet.

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No, that is not possible. There are quite a few capable telescopes studying the Sun since some decades. Already Galileo stared at the Sun until he got blind. Don't you think a planet would've been detected if it passed by in images like these? But Sun grazing comets are pretty common visitors in the corona.

It is a bit weird that a couple of percent of exoplanetary systems have hot Jupiters. They probably didn't form there, but migrated. And still stay there somehow.

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