How does a star’s proper motion differ from its annual parallax?

A star’s annual parallax and its proper motion are distinct phenomena that describe different aspects of a star's apparent movement across the sky.

• Annual Parallax: This is the apparent, periodic shift in a star's position caused by the Earth’s revolution around the Sun. Because Earth views stars from different positions in its orbit throughout the year, a nearby star appears to trace a small ellipse against the background of more distant stars. This back-and-forth motion is used by astronomers to calculate the star’s distance from Earth.1
• Proper Motion: This refers to the actual, long-term angular movement of a star across the celestial sphere at right angles to the observer’s line of sight. Unlike the cyclical, annual effect of parallax, proper motion is a cumulative change in position resulting from the star's own physical movement through space relative to the Sun and other stars. It is typically measured in seconds of arc per year and is denoted by the Greek letter $\mu$.2

In summary, while parallax is an apparent, seasonal oscillation reflecting the observer's changing vantage point, proper motion represents the star's genuine, progressive travel through the galaxy.12