# What is a parsec?

May 18, 2016

A unit of length.

#### Explanation:

It's definition is a little hard to understand, but it is the distance where 1 Astronomical Unit (AU) subtends an angle of 1 arcsecond (or $\frac{1}{3600}$ of a degree). It is equal to 3.26156 light-years.

See the picture below for a visual.

Let's do the calculations. Let $R$ be the distance of a star 1 parsec away and $r = 1 A U$ be the radius of the Earth's orbit and $\theta$ be the parallax angle which is 1 arc second by definition. As the angle is small we can use the formula $r = R \theta$, where $\theta$ is in radians to relate the values.

Converting $\theta$ to radians gives:
$\theta = \frac{1}{360 \cdot 60 \cdot 60} \cdot \frac{\pi}{180}$
So, we can calculate the value of a parsec $R$ in AU as:
$R = \frac{r}{\theta} = \frac{1}{\theta} = \frac{360 \cdot 60 \cdot 60 \cdot 180}{\pi} = 206 , 264.8 A U$
Now $1 U A = 1.58125 \cdot {10}^{- 5}$ light years.
We ccan now calculate the value of a parsec in light years as:
$R = 206 , 264.8 \cdot 1.58125 \cdot {10}^{- 5} = 3.26156$ light years.