Parallax Example
Hold your right forefinger out an arm's length in front of you; keep it in that position. Close your left eye and look at your finger against a distant background.
Now open your left eye but close your right and look at your finger. The finger appears to jump to the right. That's parallax. Distant stars appear to wobble,
instead of jumping, against the starry background.
If we know the distance between your two eyes and the angle between the two lines of sight to your finger, we can calculate the distance to your finger.
Astronomers use this simple technique to measure distances up to almost 500 light years away, except they use the orbit of the Earth about the Sun as a
measurement base.
 First, an astronomer sights on the
star in June; the star appears to be in
position J, in the figure, against a
background of more distant stars.
Then he sights on it again, in
December, when he is the orbit
diameter distance (2 r, in the figure)
away from his first sighting. The star
appears to have jumped to position
D, in the figure. So, the diameter of
the Earth's orbit replaces the distance between your two eyes. The angle between the two lines of sight remains the same for the star example as it did for the
finger example, namely, 2Ø. The astronomer measures this angle by the jump (2Ø) in arc degrees against the star background.
Before we put numbers into this example, we need to define parallax.
The parallax of a star is the angular size (Ø, in the figure) of the elliptical arc that the star appears to trace against the background of stars. This is equal to
about half the jump D to J.
Now, let's plug some numbers into the example. Suppose an astronomer measures the angle 2Ø of apparent star movement from two sky photos taken in
June and December. He determines the angle to be 0.5 arc seconds. Then Ø = 0.5 / 2 = 0.25 arc seconds = 1.212E-6 radians
All star parallaxes (Ø) are small since the stars are so far away. So, we can approximate the tan of Ø by the angle Ø itself:
tan Ø ~ Ø.
This gives us
tan Ø = r / d ~ Ø.
And substituting 149,597,870 kilometers for the radius of Earth, r, and 1.212E-6 radians for the measured parallax angle, Ø, we have d.
Thus d ~ r / Ø = 1.496E8 / 1.212E-6 = 123.43E12 kilometers = 13.05 light years.
Note: one light year equals about 9.4605E12 kilometers.
That's how astronomers determine the distance (d) to a star from its parallax (Ø).
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