How was it determined that a parsec is 3.26 light years?

Parallax is an effective way to measure distance to nearby stars because it relies on geometry. When astronomers use parallax, they are measuring how a star appears to move against its background. The unit parsec refers to the distance that an object would have to be from the Earth to have a parallax angle of 1 arc-second.

In the figure above, the angle, #alpha# is the angle that is measured by the Earth on opposite sides of the sun. The parallax angle, #p# is half of this angle.

#p=1/2 alpha#

If we define #p# to be #1 " arc-second"#, then our object will be 1 parsec away. Since light from the sun takes 8 minutes and 20 seconds to reach the Earth, we know that;

#1 " AU" = 8.33 " light minutes"#

We can use this information to convert our parsec into light years with the tangent formula.

#tan(p) = (8.33 " light minutes")/d#

Or;

#d = (8.33 " light minutes")/tan(p)#

If we convert #p# to radians, than we can use the small angle approximation, #tan(theta) ~~ theta#.

#1" arc-second" = 4.85 xx 10^-6 " radians"#

Plugging this in for #p# and using the small angle approximation;

#d = (8.33 " light minutes")/(4.85 xx 10^-6)#

# = 1.72 xx 10^6 " light-minutes"#

# = 3.26 " light years"#