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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.

$p=21 α$

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.

$tan(p)=d8.33lightminutes $

Or;

$d=tan(p)8.33lightminutes $

If we convert $p$ to radians, than we can use the small angle approximation, $tan(θ)≈θ$.

$1arc−second=4.85×10_{−6}$ radians

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

$d=4.85×10_{−6}8.33lightminutes $

=$1.72×10_{6}$ light-minutes

=$3.26$ light years

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