So how far away are the stars, anyway? When I look up at the sky and see stars like Rigel or Procyon, how far am I really seeing? And how do we know, anyway? We've never travelled to any star. How do we know how far away they are?
Let's start small. A centimeter is about the width of your thumbnail. A meter, 1 x 10^2 centimeters, is about the same length as a yardstick. The diameter of the Earth is on the order of 10^8 centimeters. That's a hundred thousand times the length of a meter stick, or a hundred thousand meter sticks placed end to end! That's pretty hard to imagine! And it gets harder. The average distance between the Earth and the Sun is about 1.5 x 10^13 centimeters - this distance is known as the astronomical unit (AU). AUs are used for measuring distances in the solar system, but you can imagine, when dealing with the distances between stars, that the exponents mount up pretty quickly!
The nearest star to the Earth, besides our own sun, is Proxima Centauri. P.Centauri is about 4 x 10^18 cm, or 270,000 AU away. The "average" star, so to speak, is about ten times as far away - 3 x 10^19 cm, or over 2 million AU! Instead of writing out all these exponents all the time, astronomers have invented something called the parsec to express stellar distances. One parsec is equal to 206,265 AU. P.Centauri is about 1.3 pc away, and the "average" star is about 10 pc.
At this point, you're probably thinking, how do we know all this stuff? How do we know that Proxima Centauri is so far away, or that the other stars are further? Have we ever measured it? Of course not! Nobody has gone out with a ruler and measured the distances between stars, you're thinking. But we do have a way of figuring out how far away the stars are. All you need is some basic geometry, and a really good way to measure angles on the sky (a camera or CCD helps too, of course!).
We can see from the sketch above that we have, essentially, an isoceles (two sides equal) triangle that can be divided into two identical right triangles. By applying geometry, we see that we have:
q here is known as the parallax angle and is a measurable quantity, if one is careful and has good equipment. For the sake of the math, because we're dealing with such small angles here (on the order of 0.01 arc seconds, where 60 arc seconds make up an arc minute, and 60 arc minutes make up a degree!), we make an approximation to make the calculation easier. At such small angles, the approximation:
Now we have, in general form:
Incidentally, this geometry tells us where the concept of the parsec came from - 1 parsec is the distance that results in a parallax angle of 1 arc second as seen from Earth.
