Question

How do astronomers measure the distance to other stars? How accurate are their measurements?

Answer 1

Parallax `angle = alpha`, for the star G, between exactly N days. During N days, O turns through `angle theta` about Sun (S). The distance of the star OG = `sin(theta/2)/sin(alpha/2)` AU.

Explanation:

Let G denote the star.. Determine the chord distance between the two positions `O_1 and O_2` of O, for the period of N days, in the orbit of O, about S.

Use the `triangle`s `O_1SO_2 and O_1GO_2` and equate the values

The star's distance is approximated by `O_1G` that is nearly `O_2G`

Observer's distance from the Sun is nearly 1 AU = 149597870 km.
For N days O turns around S through `theta`
= N X (360/365.256363) deg

The star's distance
= `OS sin(theta/2)/sin(alpha/2)=sin(theta/2)/sin(alpha/2)` AU-

If the precision for angular measurement is 1/1000 deg. N = 30 days will be sufficient for distances of single-digit light years. For larger distances, N has to be increased.

For Alpha Centauri A, at 4.2 ly = 4.2 X 62900 AU from us and with N = 30 days, `alpha` from this formula is 0..006 deg

For N = 30, `theta` = 29.568 deg.