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

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 = $\sin \frac{\frac{\theta}{2}}{\sin} \left(\frac{\alpha}{2}\right)$ AU.

#### Explanation:

Let G denote the star.. Determine the chord distance between the two positions ${O}_{1} \mathmr{and} {O}_{2}$ of O, for the period of N days, in the orbit of O, about S.

Use the $\triangle$s ${O}_{1} S {O}_{2} \mathmr{and} {O}_{1} G {O}_{2}$ and equate the values

The star's distance is approximated by ${O}_{1} G$ that is nearly ${O}_{2} G$

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
= $O S \sin \frac{\frac{\theta}{2}}{\sin} \left(\frac{\alpha}{2}\right) = \sin \frac{\frac{\theta}{2}}{\sin} \left(\frac{\alpha}{2}\right)$ 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.