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

1 Answer
Mar 17, 2016

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.