Deriving the Parallax Formula
Surveyors on Earth use triangulation to determine distances. Astronomers
use stellar triangulation, or parallax, to
determine the distances to stars. If you consider the distance to the star
from the Sun equal to the radius of a very large circle with the star at
its midpoint, then:
The apparent (angular) shift is equal to twice the 'parallax angle'
(labeled p). The tangent of p equals the ratio of the length of the
opposite side (labeled a) to the length of the adjacent side (labeled
d). The opposite side is the radius of the Earth's orbit around the Sun
and the length of the adjacent side is the distance to the star. Since
this is a small angle (much less than 10o), we can use the
'small angle approximation', and say that tan(p) is approximately equal
to p (if p is measured in radians).
Rearranging this formula to solve for the distance to the star leaves:
The unit of 'parsec' is defined so that if the parallax
angle is measured in arcseconds from the Earth at six month intervals,
the distance to the star is in parsecs.
- 1 parsec=206,265 AU
- 1 parsec=3.0857X1016 m
- 1 parsec=3.2616 light years