# How is the distance between stars and the earth calculated?

Mar 18, 2016

#### 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
$\sin \frac{N \frac{\theta}{2}}{\sin} \left(\frac{\alpha}{2}\right)$ AU,
where $\theta = \frac{360}{356.256363}$ deg = 0.985609113 deg.

For conversion to light years (ly), use 1 AU = $\frac{1}{62900}$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
= $\sin \frac{7 X 0.9856}{\sin} \left(8.333 E - 07\right)$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".)