Cross-Eyed Stars

Still working my way through Neal Stephenson's Quicksilver, and I'm now about half way through it (and thus about 1/6 of the way through the whole Baroque Cycle). The book is about the intrigues and adventures of Daniel Waterhouse and Jack Shaftoe, but it's actually about the birth of the modern world through the twin developments of science and finance.

There's no shortage of science. One nearly throwaway vignette involves Robert Hooke attempting to empirically resolve a debate concerning the nature of stars, roughly, are they all sort of pasted on a crystal dome not so far beyond the edge of the soar system or are they independent sources of light scattered throughout deep space very far beyond the planets? Hooke's adversary argues against the latter possibility (though not we take it for granted) in what's pretty much the following way:

Hold out a finger at arm's length and close an eye. Notice the position of the finger relative to more distant background objects. Now close that eye and open the other. See how the finger seems to have shifted. The stars don't shift as you move around the earth, therefore the stars aren't scattered throughout the universe. Hooke is not moved by this argument, since the shifts might just be too small to see. What's needed is a sensitive measuring instrument and the largest possible side-to-side motion in an effort to make the apparant shift of the stars as large as possible. As it happens, the motion of the earth as it orbits the sun is a pretty darn enormous side-to-side motion, and so the largest shifts of nearby stars with respect to farther stars will happen over that cycle. A picture from Wikipedia will make the situation more clear:

Call the distance from the earth to the near star d, and the orbital radius of the earth r. It's clear from trigonometry that

You can take my word for it that for small angles x, arcsin(x) is approximately equal to x. These angles are really darn small, so the approximation is really darn good. (The next order correction is ~x^3.) As such, we can safely say that

The nearest naked-eye visible star is about 4.37 light years away and the earth's orbital radius is around 75,000,000 km. Convert and plug in, and the parallax angle is a little under 1 arcsecond. The arcseond is a unit of angle equal to 1/3600 of a degree. A degree is roughly the angle subtended by the width of your little finger at arm's length, so it's a tiny and difficult angle to measure even with a good telescope. Atmospheric distortion makes this doubly difficult even with telescopes of high resolving power. Space telescopes can do much better, and in fact missions such as Hipparcos have measured many stellar distances with great precision using this method.

Incidentally the famous unit of measurement called the parsec is defined as the distance at which an object shows a parallax of 1 arcsecond. It's equal to around 3.26 light years. That makes Han Solo's famous "...the ship that made the Kessel Run in less than twelve parsecs" remark a little suspect, but that's a story for another day.