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As you may know, people first figured out distances to the stars by measuring parallax shifts coming from the Earth's orbit. You may be wondering, though: given how far away the stars are, just what does this look like?
The first reliable measurements didn't come until the 1830s, over two centuries after the invention of the telescope and long after the discovery of Uranus, all the Messier objects, many faint planetary satellites, and subtle planetary details. As you can imagine, it's a fairly small effect that's difficult to detect. How small? More below the break.
The following animation shows the motion of Barnard's Star, which is the second-closest star to Earth, over the course of 5 years. This image simulates a very high magnification, with a field of view of about 1.5 arcminutes, or about 1/20th the diameter of the full Moon. Distinguishing this motion is really only possible by careful comparison to other stars over the course of several years.
You'll note that Barnard's star is moving in two ways: it is moving in an (almost) straight line towards the top of the frame, and it is also moving side-to-side very slightly. The straight-line motion is not the parallax; it's the proper motion, or the motion actually due to the star moving in space with respect to the Sun. It's the side-to-side motion that is the parallax, and as you can see, it's quite subtle -- much smaller than the proper motion.
If you have a hard time seeing the parallax in the animation, this still image (containing each frame of the animation, minus the labels) should make it clear:
Subtle as the motion may seem, it's actually now within the realm of amateur equipment. The proper motion of Barnard's Star, in fact, is large enough to be readily noticeable, after just a year or two, by making careful sketches (I've done this myself several times). I have not seen the parallax myself, but about 15 years ago, amateur astronomer Dennis Di Cicco made careful observations of Barnard's Star with off-the-shelf amateur telescope equipment, and drew up a very similar curve to this animation, including the parallax.
So even though it's a very small effect, it's readily measurable by even fairly small telescopes, and in fact, this was the basis for the HIPPARCOS mission, whose data make up the basis of the various star charts on this site, and in many other star chart programs. It measured over 100,000 parallaxes to very high precision, and right now, there are plans for other satellites to measure millions of stars to even higher accuracy.