# How far is a parsec?

Feb 13, 2016

$3.26 . L i g h t$ $y e a r s$

Feb 13, 2016

It is the distance that a star presents a second of parallax.

#### Explanation:

Parallax in the aparent movement that a close star seems to make in relation to the background stars, due to the movement of the Earth around the Sun.

Imagine a triangle in which the base is the Earth-Sun distance and the opposite angle is one arc second.

There for $\frac{U A}{p a r \sec} = \tan \left(1 \sec o n d\right)$

We must turn 1 second into radians:

$\frac{1 s}{3600 \frac{s}{\mathrm{de} g r e e s}} \cdot \left(\pi \frac{r a \mathrm{di} a n s}{180 \mathrm{de} g r e e s}\right) = 4.84 \times {10}^{-} 6$

For so close angle $\tan \left(\alpha\right) \approx \alpha$

$\frac{149.5 \times {10}^{6} k m}{p a r \sec} = 4.84 \times {10}^{-} 6$

$p a r \sec = \left(\frac{149.5 \times {10}^{6}}{4.84 \times {10}^{-} 6}\right) = 30.9 \times {10}^{12} k m$

How many light years is this?

One light year is $365 \cdot 24 \cdot 3600 \cdot 299.8 \times {10}^{3} = 9.45 \times {10}^{12} k m$

So one parsec is $\frac{30.9 \times {10}^{12} k m}{9.45 \times {10}^{12} k m} = 3.26 l y$