- Paul H., Bangor
Your excellent question is otherwise known as the "Parallax Paradox." How did astronomers know which stars to choose when trying to first measure parallax angles? Before we proceed with the answer, it might be worthwhile to quickly discuss the parallax method. For a more thorough discussion, click on the web-page that covers the method in greater detail: https://usm.maine.edu/planet/could-you-please-explain-parallax-again-how-does-it-work-when-youre-measuring-distances-stars
Stars that are closest to the solar system will, when observed at different times of year, exhibit parallax shifts relative to the more distant stars. Imagine as you're walking along the street, you see a stop sign at an intersection ahead of you. From your perspective, it appears precisely aligned with a telephone pole on the street's other side. If you move off to one side, let's just say about three feet, however, the stop sign will appear to then be either to the left or the right of the pole, depending on the direction in which you shifted your position. The closer you are to the stop sign, the greater the position shift will appear. Your perspective on the stop sign changed and consequently, the stop sign's apparent position relative to the background also changed. This example illustrates the concept of "parallax shift."
Astronomers measure the distances to the local stars in this manner, for they can ascertain a star's distance directly from its parallax angle. To return to our previous example, when you're walking on the street, it is rather easy to see that the stop sign is in the foreground and the telephone pole is behind it. How did astronomers know which stars were closest to us when none of the stellar distances had been determined?
Simply, they knew that the stars exhibiting the largest proper motions were likely the closest stars to us. Every star in our sky moves through the galaxy at rather rapid speeds. Our Sun, for instance, is moving around the galactic nucleus at about 486,000 miles per hour! Other star systems are traveling at comparable velocities. However, the distances separating our solar system and other stars are so vast that their proper motions relative to us are quite slight. Astronomers can measure these motions, of course, and know that the stars which exhibit the greatest proper motions, such as Barnard's Star, are the closest to us. (Barnard's Star is about six light years away.)
Let's regard the star 61 Cygni as an example, as it was the first star whose parallax angles was accurately measured. English astronomer James Bradley (1693-1762) first observed 61 Cygni in September 1753 and through these observations determined that it was a binary (or double) star. In 1792, Italian astronomer Giuseppe Piazzi (1746-1826) also observed 61 Cygni and noticed that over the course of slightly less than forty years, the star system shifted remarkably relative to proximate stars due to its inherent motion. Piazzi suggested that other astronomers might want to attempt to measure 61 Cygni's parallax angle, as its high proper motion seemed indicative of close proximity.
In 1838, after many years of failed attempts to measure 61 Cygni's parallax, German astronomer Friedrich Wilhelm Bessel (1784-1846) used the newly designed heliometer to measure 61 Cygni's parallax angle. His was the first accurate parallax measurement of a star. He determined that 61 Cygni's parallax angle was about 0.313 arc-seconds, corresponding to a distance of 3.2 parsecs, or 10.4 light years.