How is the error in proper motion measurement related to the total observation span?

Answer

The error is inversely proportional to the total observation span

The determination of proper motion, which describes the apparent sideways movement of a star across the sky, fundamentally requires observing the target over a considerable length of time to accumulate enough positional change data to distinguish it from noise. The mathematical relationship dictates that the uncertainty associated with the calculated proper motion decreases as the duration of the observation baseline increases. Consequently, highly precise proper motion measurements for slow-moving objects cannot be achieved quickly; they necessitate decades of dedicated data collection or the full operational lifetime of a long-duration space mission.

How is the error in proper motion measurement related to the total observation span?

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