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Do they move the telescope over the year to account for movement? How is that calculated? Does this change with being closer to planets and their gravitational pull?

Asked from a total moron.

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Yes, it's something that's referred to as pointing stability. The telescope will have star trackers to precisely know it's relative position - basically you make sure that you see the correct stars from where it is placed on the spacecraft. It will use reaction wheels to make tiny correction's to its position. Imagine you are in a computer chair and trying to spin yourself without feet or hands touching anything, just by twisting your body. Reaction wheels work on the same principle. As Earth completes a year around the Sun, the gravitational pull from other solar system bodies is very minor on PLATO. That said, keeping a spacecraft in L2 is not easy - there is nothing to "orbit".

And when the reaction wheels get saturated they have to expend propellant to let them spin down. It is a fascinating mechanism.

Here is the Wikipedia about Lagrange Points (L2 is one of these): https://en.m.wikipedia.org/wiki/Lagrange_point

The orbital corrections are minimized at L2, because of the relative distance of the moon and other planets vs size. But that is what is accounted for in the corrections.

James Webb Telescope is at Sun-Earth L2.

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