What is the time dilatation between the planets Mercury and Earth?
Can a person survive on the dark side of Mercury in the In the L2 position to Mercury and the Sun?
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$\begingroup$ Not specific to your question, but Mercury has craters which never get sunlight and are actually cold like empty space. Some of those craters even have ice in them. It would be a workable location to set up a lander to study Mercury or possible to land some people. The dark side of Mercury would also be explorable for months at a time too. I wouldn't want to live there, but manned exploration of Mercury is possible. Exploration of Venus on the surface would be much more difficult. $\endgroup$– userLTKCommented Mar 17, 2016 at 15:31
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The time dilation on Mercury is mostly negligible. At closest pass to the sun, 47 million KM, the gravitation field strength from the sun is less than 2% the gravitation from Mercury, so that's virtually nothing. Even on the surface of the sun the gravitational time dilation is tiny, but at Mercury distance it's minuscule. What has a bigger effect is the speed of Mercury's orbit, but the effect there would need to be measured in vector combination with the the speed of the Sun around the Milky way and the speed of the Milky way through the universe, so that would rise and fall depending on where Mercury was in it's orbit, and even at plus/minus 180,000 kph, the time dilation effect is very small.
Can a person survive in the L2 position in space? That's an odd question. People can't survive in empty space without a ship anywhere, and within a space ship, survival in space is no longer an issue, so I'm not sure what you're going for there.
Would the ship be hot that close to the sun? Yes, it would need some cooling function, unless it was in shadow, which L2 of Mercury might be. (I'd have to run the numbers).
In any case, All L2 orbits are unstable but the L2 of Mercury is more unstable than most due to Mercury's elliptical orbit and high orbital velocity. Are you asking if L2 would be shaded from the sun by Mercury or did you have something else in mind? L2 is still a good location for an orbit of sorts, as it requires much less adjustment, but it still always requires adjustment cause objects don't stay in L2.
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1$\begingroup$ Just look at the MESSENGER spacecraft which orbited Mercury for ~4 years (it did its first flyby in 2008 and inserted into orbit in 2011). They used a heat shield. $\endgroup$ Commented Mar 17, 2016 at 13:04