Built on Facts

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:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

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

i-88516ca505d84f969b7ab15a965c3fb3-2.png

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

i-d18063683dcc0d42b9be45451a84d1e3-3.png

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.

Comments

  1. #1 joemac53
    December 30, 2009

    Once again showing my age, but I noticed that one right away in the theater 30 something years ago. Young nerd now an old nerd.

  2. #2 GP
    December 30, 2009

    Isn’t Alpha Centauri’s parallax angle a little *less* than one arcsecond?

    [Yes! Fixed. -Matt]

  3. #3 Uncle Al
    December 30, 2009

    Betelgeus is a prime candidate for distance measeurement for being bright, a timely supernova candidate, and not too far away (~600 ly). Getting a tight number is remarkably difficult. The “correct” value is still being debated.

    Uncle Al votes for Betelgeus going supernova on 21 Decemer 2012 because the univserse is rich with irony.

  4. #4 rob
    December 30, 2009

    let’s start the Nova Pool. Uncle Al has Dec 21, 2012. I will pick Apr 1, 2012.

  5. #5 Joshua Zelinsky
    December 30, 2009

    I’ve always found it fascinating how long it took to get accurate parallax measurements. There’s a fascinating book on the history of parallax. Alan Hirschfield’s “Parallax:The Race to Measure the Cosmos” which is highly readable and quite fun with lots of other historical astronomy thrown in.

  6. #6 oscar zoalaster
    December 30, 2009

    If Betelgeuse goes supernova on 21 December 2012 we will not know about it until approximately 21 December 2652. But calculating when it happened will provide some amusement for historians….

  7. #7 Carl Brannen
    December 31, 2009

    Clearly Mr. Solo was using distance to indicate time by using the speed of light as the conversion factor. That would indicate a trip of less than, uh, hmm, let’s see, carry the 1, ah, yes, 39.2 years.

  8. #8 Prof. Bleen
    December 31, 2009

    In practice, I believe there is another factor that has to be corrected for in calculating parallax: the apparent tiny side-to-side motion caused by the Earth moving tangentially to the star under observation (the “aberration of light”). When the Earth is moving at right angles to the incoming light rays (or has a tangential component of motion), the astronomer has to lean her telescope ever so slightly into the star’s light, just as you have to lean your umbrella forward when you’re walking into the rain. The result is I forget the magnitude of the effect, but since the Earth’s orbital speed is about 30 km/s, it isn’t very large. (Actually, according to Wikipedia, it ranges up to about 20 arcseconds, so it is actually large compared to stellar parallaxes.)

  9. #9 Alex
    January 1, 2010

    that thing about han solo’s line is fascinating

  10. #10 Gue
    November 15, 2010

    Hi,
    Have a video on the hipparcos satelitte,
    one shows the launch

    Hope its interesting