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Could I park a station far out stationary relative to the sun?


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Try it with a ship and see if you stay at 0 orbital velocity. Realistically you shouldn't be able to but Kerbal physics might allow it. I had a ship in a game on an escape trajectory out of the solar system but when that ship isn't focused there is a line indicating it would come back in toward the sun (like a comet's orbit).

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You'd have to be infinitely far out for that since the sun's SoI is infinite. Due to rounding errors in KSP you wouldn't have to go all the way to infinite, but still a long ass way. As in, years of timewarping at the max to get there.

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