Parallax, Derivation of the Parsec & Lightyear

"Parallax" is the well-known optical phenomenon that causes a stationary object to appear in two different locations when viewed from two different observers. The deviation in the object's apparent position is large when the observers are near the object, and decreases as the observers move farther away. Nearly all vertebrates exploit this phenomenon to visualize 3D space. These animals compare the differences in an object's location between their two eyes to get a sense for how far away it is.

The same principle can be used to determine the distance from Earth to a given star. Imagine a giant salamander in space named Martin. His nose is the sun, and his left and right eyes represent the position of the Earth in the Spring and Fall, respectively.While we humans on Earth only get a 2D view of the sky at any given time, Martin has 3D depth perception of all the stars in the sky due to parallax.

Image sources:
Space: http://stillsound.files.wordpress.com/2012/04/outer-space-stars.jpeg
Salamandar: http://upload.wikimedia.org/wikipedia/commons/1/15/Fire_salamander_March_2008b.jpg
Sun: http://upload.wikimedia.org/wikipedia/commons/b/b4/The_Sun_by_the_Atmospheric_Imaging_Assembly_of_NASA%27s_Solar_Dynamics_Observatory_-_20100819.jpg
Earth:http://upload.wikimedia.org/wikipedia/commons/2/22/Earth_Western_Hemisphere_transparent_background.png

Since the Earth revolves around the sun, the positions of the stars look slightly different throughout the year based on the Earth's current position on its orbit. This phenomenon has been observed by astronomers since ancient times. It allowed the earliest civilizations to determine the length of a year by recording the number of days until a nearby star returned to its same position in the sky, at the same time of the night.

Parallax can be used to determine the distance from the Earth to nearby stars. By measuring the offset angle of a star's position at opposite times of the year using precise telescopes, it is a simple geometry problem to figure out how far away it is.

As an example, let us assume the angle a = 1 arcsecond in the above diagram. The corresponding distance, d, is defined as the parsec -- a fundamental unit in Astronomy. To determine the value of the parsec given our knowledge of the AU, we make the approximation tan(a) = a. Thus, we have

To get a a sense for the length of a parsec, let us compare it to the Lightyear -- the distance light travels in a year:

Thus, a parsec is slightly over 3 lightyears. As large as this seems, it is only 3/4 of the way to the nearest star, Alpha Centauri. This means that in order to use parallax as a meaningful measuring device, telescopes must be able to resolve angles much smaller than 1 arcsecond.