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Deriving the Parallax formula

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:

parallax diagram

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).

parallax diagram

Rearranging this formula to solve for the distance to the star leaves:

parallax formula

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
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A service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Alan Smale (Director), within the Astrophysics Science Division (ASD) at NASA/GSFC

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