The Fixed Stars Are not Fixed
Fixed stars do have parallax, which is a change in apparent position caused by the orbital motion of the Earth. This effect was small enough to not have been noticed until modern times. It can be used to find the distance to nearby stars. This motion is only apparent.
The fixed stars exhibit real motion as well, however. This motion may be viewed as having components that consist in part of motion of the galaxy to which the star belongs, in part of rotation of that galaxy, and in part of motion peculiar to the star itself within its galaxy.
This real motion of a star is divided into radial motion and proper motion, with "proper motion" being the component across the line of sight. In 1718 Edmund Halley announced his discovery that the fixed stars actually have proper motion. Proper motion was not noticed by ancient cultures because it requires precise measurements over long periods of time to notice. In fact, the night sky today looks very much as it did thousands of years ago, so much so that some modern constellations were first named by the Babylonians.
A typical method to determine proper motion is to measure the position of a star relative to a limited, selected set of very distant objects that exhibit no mutual movement, and that, because of their distance, are assumed to have very small proper motion. Another approach is to compare photographs of a star at different times against a large background of more distant objects. The star with the largest known proper motion is Barnard's Star.
The phrase "fixed star" is technically incorrect, but nonetheless it is used in an historical context, and in classical mechanics.
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