Question

How is the distance between stars and the earth calculated?

Answer 1

Using stellar parallax.

Explanation:

Stellar parallax is the apparent angular displacement of a star when observed from two sufficiently-spaced known locations

A convenient choice is the same observatory, with time spacing that is (sufficiently large integer) N days.

If N is the exact number of 24-hour days and `alpha` is the parallax angle, the distance of the star is
`sin(Ntheta/2)/sin(alpha/2)` AU,
where `theta=360/356.256363` deg = 0.985609113 deg.

For conversion to light years (ly), use 1 AU = `1/62900`ly, nearly.

The arc of the Earth's orbit for N days subtends N`theta` deg at the Sun.

Sample Data: N = 7, `alpha` = 0.003" = 8.333 E-07 deg.

The approximation to the distance of the star
= `sin(7X0.9856)/sin(8.333 E-07)`AU

= 8.259 E+06 AU

= 131.3 light years.

(It is assumed that the precision in the Radio Telescope for parallax is as good as 0.001".)