Wednesday, August 18, 2010

Ptolemaic astronomy question

Ok, hist of sci people out there, help me out:

In Hilchot Yesodei Hatorah chapter 3, Maimonides lays out the geocentric model of the cosmos, epicycles and all. At 3:8, he notes that Earth is about 40 times larger than the moon, the sun is about 170 times larger than Earth, the sun is the largest of the "stars" (a category that includes planets too), and Mercury is the smallest. Comparing this with the actual data we know now, the sun is of course larger than any of the planets, and Mercury is the second smallest of the heavenly bodies known at that time (the moon is smaller). The sun's radius is 109 times Earth's radius, so the Rambam's number is pretty decent, within a factor of 2. Earth's radius is 3.67 times the radius of the moon, so that figure is considerably further off (by an order of magnitude). The ratio of Earth's volume to the moon's volume is 49, much closer to the Rambam's number of 40 (and he doesn't actually specify which dimension he's talking about), though if we understand the Rambam's ratios to be about volume rather than linear dimension, then the sun-to-Earth ratio is thrown way off.

So my question is this: HOW THE HECK DID HE KNOW? (And by "he", I mean the Rambam himself, or ancient Greek astronomers, or medieval Arab astronomers, or wherever he's getting his data from.) Even if the moon number is considerably further off than the sun number, it's still an impressive feat to know that the moon is smaller than Earth (by any amount) even though the sun is much larger (by an amount that he basically got right) and the sun and moon are the same apparent size in the sky. He knew that the sun was farther away than the moon (which can be reasonably inferred from the sun's (apparent) orbital period being longer), which would mean that the sun is larger than the moon if they're the same apparent size, but it's not clear how he got any sort of quantitative relationship between those sizes (the ratio he gives between the sizes of the sun and the moon doesn't have any obvious mathematical relationship to the ratio of their orbital periods), let alone how he could compare either of them to the size of the Earth. (Did he have some version of Kepler's Third Law?) I know that Eratosthenes measured the circumference of the Earth, but did pre-modern astronomers have any sense of how far away the sun or other celestial bodies were? And as for the sizes of the planets (the ones we would call planets, not the sun and the moon), how could anyone resolve any finite sizes, rather than just seeing them as points of light? I can't blame him for thinking the moon is larger, but how did he know that Mercury is smaller than Venus, Mars, Jupiter, and Saturn?

Pardon me if these questions are ignorant; I would be fascinated to know the answers. Thanks!


  1. Could he have been using brightness? I have no clue how he got the exact size of the sun/moon, but brightness of each object is a pretty good proxy for the relative sizes(with the moon being the only one to break the rule and the one Rambam got wrong).

    Note, this is all pure speculation, I have no clue how Rambam knew the information, but it seems like a logical approach.

  2. Ancient Greek astronomers, I think going back to Hipparchus, and certainly including Ptolemy, knew very well how far away the moon was, and consequently how big it was, from parallax. (In addition, they also knew the size of the moon relative to the Earth, from the curvature of the edge of the Earth's shadow on the moon during a lunar eclipse, which could be used to confirm the parallax measurement.) The Rambam, who learned his astronomy from Ptolemy and from Arab astronomers, even includes a correction to the moon's apparent position in the sky, due to parallax, in the laws of the calendar (Kiddush Hachodesh) in the Mishneh Torah. From this, it is clear that the Rambam was talking about volume when he said that the Earth was 40 times as big as the moon.

    He was presumably also talking about volume when he said that the sun was 170 times as big as the Earth, and in this he was way off. The Greek and Arab astronomers did not have any way to measure the distance of the sun and the planets, since their parallax is too small to measure without a telescope. Aristarchus attempted to measure the distance to the sun (maybe by measuring the parallax of Venus?), and concluded that it was 20 million miles away, but this was really just a lower limit. (The real distance is 93 million miles.) It was enough, however, to correctly conclude that the sun is much bigger than the Earth. I don't know where the Rambam got the idea that the sun was 170 bigger in volume than the Earth, which would require the sun to be only about 5 million miles away. Possibly some other astronomer, after Aristarchus, made another attempt to measure the distance to the sun and planets using parallax, and got results that were even less accurate than Aristarchus.

    The Rambam would have had no good way to tell how big the planets were. But he might reasonably have inferred that Mercury was small from the fact that it was assumed to be relatively near the earth, from its fast orbit, but was still hard to see.

    Actually, the Rambam, and ancient Greek astronomers, could have estimated the order of magnitude of the distance to the sun, if they had realized that meteors are in independent orbit around the sun, and the relative speed of a meteor to the earth, when it enters the atmosphere, is comparable to the speed of the earth in its orbit around the sun (or, as they would have put it before Copernicus, the speed of the sun in its orbit around the Earth). The speed of a meteor entering the Earth's atmosphere can be measured by two people standing a few tens of miles apart, observing the same meteor, and using parallax. (See the Amateur Scientist column in the March 1987 issue of Scientific American, the only time my name appeared in that magazine.) But the ancient Greek astronomers didn't think of this.

  3. Circumference of the Earth was calculated by Eratosthenes in the 3rd century BCE. This would have been known to Rambam

  4. You didn't ask about this, but there's a very nice article by Dr. Jeremy Rosen on a Jewish anti-Copernican religious humanist in the most recent Torah u-Madda Journal

  5. Perhaps 'Rambam' received his info from 'Kabbala?' 'The Zohar' Rabbi Shimon Bar Yochai?'
    Sizes and speeds are probably related to 'The Sefirot.'
    (To me) for instance, the 'male principle,' 'Chomcha' is: 'concept' 'idea' and 'wisdom' is quick, like Mercury. 'Bina' or the feminine principle (also 'Havana' 'Intellectual understanding,'is next) but I can't compare the two...lack of knowledge. 'Daat' is Earth 'Be daat yesod haHaretz (My Hebrew is lacking...lazy to check) "With knowledge HE founded the Earth."