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

Dec 17, 2015

A parsec is defined to be the distance to an object that shows a parallax angle of $1 \text{ arc-second}$. That distance happens to be $3.26 \text{ light years}$.

#### Explanation:

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.

$p = \frac{1}{2} \alpha$

If we define $p$ to be $1 \text{ arc-second}$, 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;

$1 \text{ AU" = 8.33 " light minutes}$

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

$\tan \left(p\right) = \frac{8.33 \text{ light minutes}}{d}$

Or;

$d = \frac{8.33 \text{ light minutes}}{\tan} \left(p\right)$

If we convert $p$ to radians, than we can use the small angle approximation, $\tan \left(\theta\right) \approx \theta$.

$1 \text{ arc-second" = 4.85 xx 10^-6 " radians}$

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

$d = \frac{8.33 \text{ light minutes}}{4.85 \times {10}^{-} 6}$

$= 1.72 \times {10}^{6} \text{ light-minutes}$

$= 3.26 \text{ light years}$