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Whether it is Orion or Sagittarius, the speed relative to me is clearly faster than the speed of light. These are not inertial systems, so they are not subject to the constraints of special relativity, but does the fact that their apparent velocity is a circular motion exceeding the speed of light cause any kind of apparent change in time for celestial objects if the argument is based on a celestial coordinate system?

They at least don't seem to be unable to follow, unlike the laser beams I shot at the moon or something.

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does the fact that their apparent velocity is a circular motion exceeding the speed of light cause any kind of apparent change in time for celestial objects if the argument is based on a celestial coordinate system?

No. As you say it is not an inertial frame. The speed of light is not limited to $c$ in non-inertial frames. The speed is not invariant in non-inertial frames, so none of the usual relativistic effects arise.

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