Notes from Under Sky

Size Matters

How can we deduce the size of the moon?

In my previous Notes episode, I talked about astronomy education. The whole thing that prompted this, I think, was a discussion on the silly kinds of questions that we get as amateurs when we host a public star party. "Which end do we look through?" when they have only seen the last 10 people peer into the eyepiece. Stuff like that.

One question that was held up as a sort of canonical example of a silly question was "Which is larger, the moon or the earth?" As if anyone could not know which was larger. Of course the moon is smaller. But wait, how do we really know that? Don't most of us know it from a book, or being taught that? How was it first determined?

The answer to this question, as with many questions, goes back to the Ancient Greeks. When they first pondered this question, their answer was "as big as it looks." This seemed self-evident because the answer is right for a great many things, such as boats, chickens, goats, mother-in-laws, etc. By and large, these things did not look any smaller or larger than they actually were.

But these things also had in common that they could be looked at up close (well, maybe except for the mother-in-law). We have lots of experience with things we can touch and see up close. We can apprehend their size rather easily. We are unlikely to be off by more than a few inches at most when guessing someone's height, for instance.

Such isn't the case with the moon. You can't simply reach up, pull the moon down, measure it with a measuring tape, and tack it back up there. What's more, the moon didn't appear appreciably larger from a mountaintop than it did from the sea, so it was clear that it had to be a long long way away. Gradually it became clear that the usual cues of distance were useless when it came to the moon; it was just too far away for them to be useful. Some other way had to be found to figure out the size of the moon.

It was Aristarchus who first figured out this way. He noticed that during a lunar eclipse, the shadow of the earth was considerably larger than the disc of the moon, so that the moon could enter the earth's shadow rather off center and still be completely in a ruddy darkness. Rather than use the figures Aristarchus obtained, let's give the modern value: the earth's darkest shadow, the umbra, is about 8/3 times as large as the disc of the moon.

Does this mean that the earth is 8/3 times as large (by diameter) as the moon? Not quite. The sun is larger than the moon. We know this because it passes behind the moon during a solar eclipse and still appears to extend the same width. The moon's shadow is the same size as the moon right in front of the moon during an eclipse, but by the time that shadow reaches the surface of the earth, it is this itty bitty little dot, which at best is a couple of hundred kilometers in width. Anyone who has had to chase a solar eclipse and choose from a very narrow range of destinations is aware of this. It is approximately right to say that the shadow of the moon shrinks by its own width over the earth-moon distance.

But if the moon's shadow shrinks that much, so should the earth's, as long as the sun is very far away (and therefore much larger than either the moon and the earth). So if the earth's shadow is 8/3 moon widths at the moon, it should be one more moon width wide at the earth, or 11/3 times as wide as the moon. The earth is therefore about 11/3 times as wide as the moon, too.

Later, Eratosthenes determined the size of the earth to be about 13,000 km in diameter, which makes the moon about 3,500 km in diameter. Since it subtends about 1/2 degree as seen from the earth, the moon must therefore be about 385,000 km away, a fact first determined by the greatest Greek astronomer of all, Hipparchus.

Copyright (c) 2000 Brian Tung