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In the previous post, I showed how a nearby star -- Barnard's Star -- appears to move against the sky over time. Barnard's Star is the second closest to the Sun. The closest star (actually, the closest three stars, all bound into one multiple star system) is the Alpha Centauri system, at 4.3 light years away, about 2/3 the distance to Barnard's Star.
Alpha Centauri consists of three stars, two bright stars (both broadly similar to the Sun, called Alpha Centauri A and Alpha Centauri B) in a close orbit, and a third, much farther out, that is currently slightly closer to the Sun the the other two. This star, a very dim, red star, is sometimes called Proxima Centauri to emphasize its closeness to the Sun; it is the closest star to the Sun we know of.
Unsurprisingly, Alpha Centauri was one of the first stars to have its parallax measured. Since Alpha Centauri is closer to the Sun than Barnard's Star, it shows a larger parallax shift every year. Additionally, it is moving through space more slowly, so its proper motion is quite a bit lower (despite being closer). As a result, the parallax effects are easier to see with Alpha Centauri:
This image shows the main components of Alpha Centauri -- the two brighter stars -- at a high magnification (the same as used last week for Barnard's Star). I have slightly simplified this image by ignoring the orbital motion of the two stars; they actually complete an orbit every 80 years, and it's possible to follow the changes in the orbit on a year-to-year basis.
One thing to note is that Alpha Centauri is not far from the ecliptic pole -- that is, it's well away from the plane of the Earth's orbit. As a result, its parallax over the course of a year looks like a (fairly broad) ellipse. By contrast, Barnard's Star is near the plane of the Earth's orbit, and so its parallax motion is a very thin ellipse (almost side-to-side). The parallax is easier to follow here because of Alpha Centauri's closer position and lower proper motion.
Proxima Centauri shows a larger parallax, but not by enough to matter much. It is in fact one of the least-luminous stars readily visible from Earth. Despite its closeness, you need about a 4-inch telescope and dark skies to see it clearly. It's so dim that even from Alpha Centauri proper, it would be a 5th-magnitude naked eye star -- meaning that if you were in a light-polluted city on a planet orbiting Alpha Centauri A, you wouldn't be able to see it at all.
Are there any closer stars than these? We don't know. If there are any, they must be exceedingly dim, or we would have found them by now. It's possible that the Sun has a sub-stellar companion that's still visible as a faint starlike object; it would have a huge parallax, but since it would be moving in space with the Sun, it wouldn't be moving the way Alpha Centauri and Barnard's Star do. However, the success of the HIPPARCOS mission has led to follow-up missions that will probably be able to answer this question in the next few years.