How was it determined that a parsec is 3.26 light years?
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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, α is the angle that is measured by the Earth on opposite sides of the sun. The parallax angle, p is half of this angle.
If we defines 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;
1AU=8.33 light minutes
We can use this information to convert our parsec into light years with the tangent formula.
If we convert p to radians, than we can use the small angle approximation, tan(θ)≈θ.
Plugging this in for p and using the small angle approximation;