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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.

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-

For single-digit light-year distances, N = 30 days will work well if the angular measurement precision is 1/1000 deg. For longer distances, N must 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.

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Answer 2

Luca de

A variety of methods, such as parallax, stellar brightness, and standard candles like Cepheid variables or Type Ia supernovae, are used by astronomers to measure distances to other stars. The precision of these measurements varies depending on the method and the particulars of the observation, but contemporary methods can achieve accuracies within a few percent for nearby stars and galaxies, while more distant objects may have uncertainties of several percent or more.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.