Proper motion Totally Explained

Everything about The Proper Motion totally explained

The proper motion of a star is the measurement of its change in position in the sky over time after improper motions are accounted for. This contrasts with radial velocity which is the time rate of change in distance toward or away from the viewer.

Introduction

Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time. Ursa Major, for example, looks nearly the same now as it did hundreds of years ago. However, precise long-term observations show that the constellations change shape, albeit very slowly, and that each star has an independent motion.
This motion is caused by the true movement of the stars relative to the Sun and solar system through space. It is measured by two quantities: the proper motion angle and the proper motion itself. The first quantity indicates the direction of the proper motion on the celestial sphere (with 0 degrees meaning the motion due north, 90 degrees due east, and so on), and the second quantity gives the motion's magnitude, in seconds of arc per year.
Proper motion may also be given by the angular components in the right ascension (μ_α) and declination (μ_δ). The net proper motion (μ) is given by:

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