I go to a planet, and spend 10 years according to that planet's calendar. But I have spent only 5 months in Earth time. Will I have aged 10 years by the time I am back?

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A "Year" on a planet is the time it takes to go around the Sun. None of the planets in the Solar System fit the question, though, so I'll just treat it as a hypothetical planet, around a hypothetical star.

In 5 months, if the planet goes around the star 10 times, it means that a year would be just 12.5 years long. Now, it is not possible to directly calculate the distance of the planet from the star from this, or the mass of the planet. All we know that the planet's centripetal force (from rotation) should balance the gravitational pull for it to be in a stable orbit.

If the star is similar to the Sun in size, the planet will have to be:
1. Much smaller than the Earth, and very close to the star
2. At least as large as the Earth, but farther from the star, moving much faster than the Earth.

When it comes to age, gravity and exposure to radiation are both major factors. So, if you're farther, you will probably age even less than the 5 months. But if you're closer to the Sun, the lower gravity will slow down aging, while the increased radiation will accelerate it. And this is only about physical age, by the way :)


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