Breaking the Inverse Square Law

As we discussed in the last blog (tragically, a full month ago now), it took over a century of dedicated effort on the part of several generations of astronomers before Friedrich Bessel finally succeeded in measuring the distance to a nearby star, 61 Cygni.  Right on his heels, contemporaries of Bessel’s reported similar success with measuring the distance to two additional stars, Alpha Centauri (the closest star of all) and Sirius, the brightest star.  As we discovered, the enormous magnitude of these distances, none less than 25 trillion miles, combined with the finite speed of light, spelled serious trouble for the Genesis account of creation.  

In fact, though, much, much greater trouble for Genesis lay ahead.  I’m going to begin to tell the story of that trouble in this blog.  Before proceeding, though, I want apologize for the long delay since the last installment.  In the course of writing this series, I came to realize that, by providing a little more historical background and detail than first envisioned, I could kill two birds with one stone, and close another gap in most people’s knowledge of astronomy.  We’ve all become used to the stunning photographs provided by the big telescopes of our day, most notably the Hubble Space Telescope.  But have you ever wondered how astronomers actually exploit the crispness of these images to do science; that is, how they use telescopes to answer questions and roll back the boundaries of our ignorance?  The story I’m about to tell provides a very concrete example of one way that astronomers use the power of their telescopes to unlock the secrets of the cosmos.

So let’s get to it.  Encouraged by the initial success of Bessel and his colleagues in the late 1830’s, astronomers spent the better part of the next 60 years trying to measure parallax angles for a vast number of other stars, including some of the brightest in the sky.  Believe it or not, in all that time they managed to obtain reliable angles for only a handful of additional stars, none more than 26 light years away.  The problem was the Earth’s atmosphere, which causes the images of the stars to jump around constantly.  Small thought these jumps are, they prevented astronomers of the time from making precise enough positional determinations to also measure the apparent movement of the stars as the earth travelled from one side of the Sun to the other.   We can only imagine scientists’ frustration with this state of affairs, since their telescopes had long since revealed that the heavens were absolutely teeming with millions upon millions of stars, most too faint to be seen with the naked eye.  Would there never be a way to measure how far away they are?

Clearly, an alternative was needed to the trigonometric parallax method.   When we look up at the night sky, one of the most obvious features of the stars is how much they differ in brightness.  A few, like Sirius, blaze like little beacons.  Others are just barely visible.  On the straightforward assumption that bright stars are closer to us than dim stars, couldn’t astronomers use a star’s brightness to determine its distance?

Mathematically, the answer is a straightforward “yes”!  The relation between the brightness of an object and how far away it is follows a really simple relation known as the inverse square law.  To get a feeling for this law, suppose you took two identical candles, placed one 10 feet away from you and the other 20 feet away from you, and lit them both.  How much brighter would the nearer candle be than the farther candle?

Intuition might suggest that the more distant candle, being twice as far away, would be half as bright.  Not so, however.  The geometry of the situation dictates that exactly four times as much light would reach your eye from the nearer candle than the farther candle, so the farther candle would be only 1/4 as bright.  Note that the denominator of that fraction, 4, is 2 times itself, or 2 squared.  Next, suppose you measure the brightness of the more distant candle after moving it an additional 10 feet away, so it is now 30 feet away from you, or three times as far away as the nearer candle.  Your measurement would reveal that it is only 1/9 as bright (3 squared).  At risk of belaboring the point, if you moved the more distant candle out another 10 feet, making it 40 feet away (4 times the distance of the nearer candle), it would appear only 1/16 as bright (4 squared).  And so on.

Now suppose you had a friend position the second candle instead of yourself, so that you didn’t actually know how far away it was.  Could you use the inverse square law to compute its distance?  Easily!  For example, suppose you measured that candle’s brightness to be only 1/25 the brightness of the nearby candle.  Since 5 squared equals 25, you would know that the more distant candle is five times as far away as the nearer candle, or 50 feet. It’s a classic example of the power of a simple mathematical relation to solve for an important unknown!

The inverse square law applies equally well to stars as to candles.  If two stars shine with the same “intrinsic” (astronomers call it “absolute”) brightness, and one star is four times as far away as the other, the more distant star will be 1/16 as bright.  As long as you know the distance to the nearer star, calculating the distance to the further one is trivial.  

Now, back to the beginning of the 20^th century.  Fired up by the inverse square law, an astronomer of that period might have measured the brightness of a target star that had not yielded a measurable parallax angle (and whose distance was therefore undetermined), and compare that brightness to the brightness of a star of known distance, such as Sirius, which is about 8 light years distant.  Suppose he found that his target star was 1/16 as bright as Sirius.  The inverse square law would tell him that his star was four times as far away, or 32 light years from Earth.

Unfortunately, applying the inverse square law is only that straightforward if the two stars whose brightness is being compared have the same intrinsic brightness, like two identical candles.  In our stellar example, the target star would have to have the same absolute brightness as Sirius.  The problem is, stars do not all shine with equal brightness; they range from thousands of times dimmer than the Sun to a million times brighter.  Is our target star 1/16 as bright as Sirius because it is actually much brighter than Sirius, but also much further away than 32 light years?  Or is it 1/16 as bright because it is the same distance as Sirius (or even closer) but intrinsically much dimmer?  Without additional information about the star, you just can’t tell.

Might there be a way to get around this problem and still utilize the inverse square law?  In the next blog, we’ll find out the answer.