The moon is big…

… you’d think you wouldn’t have much trouble figuring out where it’s at.

Last week saw a huge amount of productivity on the history project, wherein I attempt to simulate ancient planetary models. I’m pretty much done with the Ptolemaic model, except for planetary latitudes. I doubt I’ll bother adding that; longitudes are the really important thing for locating planets, since latitude only ever varies a few degrees from the ecliptic at most, and the interesting bits of the models are figuring out longitude anyway. Plus, I’d like to implement the bulk of the Copernican model too (again, probably just the longitudes except for the Moon), and since I don’t have much more than a week before I ought to hand it in, there’s only so much that can realistically be done.

So, so far the program compares predictions made by Ptolemy’s model (which is geocentric) with predictions generated from NASA’s HORIZONS system which I’ve mentioned before. Things actually match pretty well for the sun and planets, at least to within a couple of degrees of longitude.

The moon, however, is way off. As in, in the neighborhood of 120 degrees off.

I can’t put my finger on the problem. I’ve triple-checked the parameters of Ptolemy’s lunar model, and they all match. The operation of the model itself looks just fine — besides, it doesn’t matter since the mismatch is right at the epoch, where Ptolemy explicitly says where the moon should be. Yet the longitudes predicted by Ptolemy and by the NASA ephemeris tables are utterly unreconcilable.

So what’s the problem? Ptolemy’s initial values for the epoch are, I believe, computed from the model and other observations, so if the model isn’t entirely accurate (and it isn’t), it could be the cause of interpolation error on Ptolemy’s part. But that’s an awfully big interpolation error for an epoch that occured during Ptolemy’s life span. (Running the model for a while, you see his moon overtake the NASA moon periodically, so it doesn’t seem his parameter for the speed of the mean longitude is quite right, but that’s just a hunch.) On the other hand, the NASA ephemeris tables for the moon are based on data from 1962-Jan-20 to 2004-Apr-09; the moon’s motion is complex (or at least appears so, since it’s so close), and extrapolating back to 137-Jul-20 (Ptolemy’s epoch) would well introduce significant error. Yet comparing the rest of Ptolemy’s model against the NASA data shows pretty good matching.

I don’t have access to a time machine, so I can’t just go back and see where the moon really was at that time. I don’t know the details of the NASA model for the moon, so I have no clue what kind of error to expect from extrapolating nearly two millennia back in time. And although Robert R. Newton’s book on Ptolemy certainly argues that Ptolemy fudges some of his observations to fit the model, you wouldn’t think things would be that far off during his own time; 120 degrees’ difference in the moon’s position is pretty tough not to notice. (Also, RRN’s book talks about error he sees in Ptolemy’s model and doesn’t mention anything like what I’m seeing, so it could well be that the model he was using in comparison extrapolates better than my NASA-generated tables.)

My best guess is that it’s extrapolation error in the NASA-generated tables. There’s nothing I can do about that; all I can do is mention the (obvious) problem in the program’s accompanying write-up and restate what I’ve said here about the possible causes.

Now that I’ve thoroughly bored and/or confused you, here’s a fun, surprising fact about Ptolemy’s system. Even though it’s geocentric, at least some component of the predicted longitudes for the planets come from the predicted longitude of the sun. What’s more, the model for Venus has the planet moving around a circle (the epicycle), whose center itself moves around a larger circle (the deferent), which has the Earth near its center. Ptolemy equates the position of the epicycle with the position of the sun on the circle it travels around the Earth. Even better, the radius of the epicycle is 0.719444 times the radius of the deferent. Now, get this: the average distance between Venus and the Sun is 0.72333199 AU (where 1 AU = average distance between the Earth and the Sun). Now, from a modern-day perspective, this looks awfully like a heliocentric model for Venus, and even the planet’s orbital radius is reasonably close!

Ptolemy certainly didn’t frame his theory of Venus in heliocentric terms, even though that’s essentially what’s going on in the model, and it’s certainly how we’d think of the relationship these days. If he had seen that for what it is, we might’ve had a Copernican-style heliocentric model 1400 years earlier.

3 Responses

  1. Why are you trying to reconcile Ptolemy’s prediction with modern-day observed values? Predictions can be wrong, you know…

  2. Of course I don’t expect Ptolemy’s model to give exact answers, but I find it hard to believe that his model for the moon would be so grossly wrong for lunar observations during his own lifetime; 120 degrees is close to the difference between “in the sky” and “not in the sky.” Plus, the book by Robert R. Newton (“The Crime of Claudius Ptolemy”) discusses the errors in the model’s predictions, and he doesn’t mention anything within an order of magnitude of this; he finds that Ptolemy’s lunar model has a maximum error of just over one degree. (He doesn’t go into details about how he produced the “correct” lunar positions to compare to, though.)

    Plus, the NASA ephemeris tables aren’t exactly “modern-day observed values” but are predictions made by some modern-day model, so they could well be the source of the discrepency I’m seeing.

    (And if your questions was just a general “why are you doing this in the first place?”, well, I needed to do a project for my history class, and I was curious just how well Ptolemy’s model worked if it wasn’t seriously challenged until around 1400 years after he created it.)

  3. Paul, I remember when you were at home and trying to explain to me about Ptolemy’s model. I still have it drawn on the back of my guitar music…my guitar dude saw that and was like “what is that? geometry?” and i said it was from when my bro was trying to explain his history project to me.

    Hope you do well in that history project. I’ll be sending you my short story that I was forced to write but ended up later really liking my work. It’s very creative but that’s what you should expect from me :)

    Talk to you later!

    Love Amy

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