In astronomy, the term proper motion refers to the angular velocity across the sky exhibited by a celestial body. The enormous distances to the stars means that only the closest have proper motions that are large enough to be expressed in arcseconds per year – milliarcseconds per year are more common. Because of these small angular velocities, it is necessary to use a telescope that has a high angular resolution to measure proper motions.

More than two epochs are required to be able to separate the proper motion of a star from its parallax, as both cause the stars to move against the “fixed background” of the night sky. With several epochs of observations it is possible to tell the difference between proper motion and parallax – a star exhibiting proper motion will move uniformly in one direction across the sky, while one displaying parallax will return to its original position after one year of observations tracing out an elliptical path on the sky. However, breaking the degeneracy between proper motion and parallax is not always simple – all stars exhibit both a proper motion, and a parallax at some level, and it can take several epochs to separate the two terms due to experimental uncertainties. It is even harder to uniquely determine these effects for objects in binary and triple systems.

The observational difference between a star that displays proper motion only (left), and one that shows a parallax (right).

The proper motion (μ) has a magnitude and a direction, and is often broken down into the components of right ascension (μ_RA) and declination (μ_Dec) where

μ²=μ_RA² + μ_Dec²

The product of a star’s proper motion μ and distance D yield the transverse velocity V_T= μD (ie the velocity perpendicular to our line of sight). When combined with the radial velocity the 3D space velocity of a star can be obtained.

Prior to the invention of the telescope it was thought that the heavens were unchanging and that the stars did not move relative to each other.