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That is, when did astronomy figure out how to predict when and where a solar eclipse will be visible?

It seems people noticed fairly early on that the Sun, Moon and Earth return to the same approximate relative positions periodically (~18 years and 11 days and 8 hours). This tells them how long it takes for an eclipse to (almost) repeat itself. However, the shadow of the moon on Earth is only ~150km wide during a solar eclipse, and so only a small part of the planet will actually see it happening. Thus:

... since the Saros cycle does not include an integral number of days ... the next eclipse in the cycle will occur one third of the way around the world from the previous eclipse in the cycle. So, the Saros cycle allowed ancient astronomers to predict when a solar eclipse could be expected, but it did not help them figure out where the eclipse would occur.

- http://ds9.ssl.berkeley.edu/solarweek/THURSDAY/history.html

So when did astronomers understood the entire mechanics well enough to be able to predict both the time and place of a solar eclipse?

Inspired by this question on History.SE. Note that I'm not just asking for when one eclipse was correctly predicted (by chance or otherwise), but rather when the physics of it is understood well enough for reliable predictions.

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    $\begingroup$ Would you be looking for post-Copernican-astronomical-theory - that is, when we truly understood the layout of the solar system and thus the mechanisms behind eclipses? $\endgroup$ – HDE 226868 Nov 11 '14 at 16:05
  • $\begingroup$ @HDE226868 I am certainly interested to hear about it being applied to eclipse predictions :) $\endgroup$ – Semaphore Nov 11 '14 at 16:15
  • $\begingroup$ @HDE 226868: As you can see from my answer Copernican astronomy has NOTHING to do with the question. Moon rotates about Earth. This was understood by Hiparchus, Ptolemy and all others. "Copernican astronomy" did not contribute much to the question. As I said there was almost no progress between Ptolemy and Brahe. $\endgroup$ – Alexandre Eremenko Nov 12 '14 at 3:55
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The principle was known long ago, to the Babylonians and Hellenistic Greeks but the accuracy of prediction depends on the detail of the Lunar motion (the motion of the Sun is relatively simple). Without a precise Lunar theory, it was possible to predict that an eclipse is LIKELY to happen on such and such date and time, but not with a 100% certainty, and the place where it will be visible and other features could not be predicted.

Of course there were many cases when an eclipse was predicted and really happened, see, for example http://en.wikipedia.org/wiki/Solar_eclipse_of_August_21,_1560

Another question is how far in advance one could predict. I suppose at the time of Brahe it was possible to predict an eclipse several months in advance.

Let me give an example in 1598 the almanac gave a solar eclipse on March 7 and a lunar eclipse with the error of one hour. This led Kepler to the discovery of the 4-th inequality (see below). An almanac is supposed to make predictions for one year.

Sufficiently precise Lunar theory for accurate predictions for several years in advance, with 100% certainty and where it will be visible, and other features of the eclipse, was developed in the middle of 18 century, as a result of combined efforts of several greatest mathematicians of that time (Clairaut, Euler, T. Mayer).

But there are many records of more or less accurate predictions before that time, beginning from antiquity.

Remark 1. This is very different from the Lunar eclipses which do not require knowledge of fine detail of the Moon motion. Prediction of the Lunar eclipses was possible since antiquity.

Remark 2. The main motivation for these efforts was not the eclipse prediction but more important practical problem: determination of the longitude at sea by the "Method of Lunar distances". But almost at the same time, chronometer was invented, and for about 1/2 of a century the two methods were competing. When chronometers became affordable (in the first half of 19-th century, the method of Lunar distances gradually was displaced by a simpler method, based on a chronometer).

EDIT. Let me elaborate as much as possible without MathJax. First of all, what does it mean to "predict"?

If 10% of the Sun disc area is obscured, is this an eclipse or not? This is what I mean by "features", full or not full, perhaps circular. The same eclipse will be full in one place and partial in another.

Second important question: predict how much in advance? 2 days? 2 months? 2 years? or 1000 years? This makes a great difference.

Now a brief account of what was involved. I assume that the Sun motion is known precisely (it was known to sufficient accuracy to Hipparchus). So we only discuss the Moon. Both Sun and Moon have visible diameter about 1/2 degree. So to predict an eclipse at a given location and time we need to know the Moon motion, say to minutes of angle.

From the time of Hipparchus to this time, the (geocentric) coordinates of the Moon as functions of time are described by a series of the form $$At+E(t)+E'(t)+E''(t)+ \cdots$$ where $t$ is time, $A$ is the "mean motion" and $E$ s are periodic terms depending on the mutual position of Sun and Moon. These periodic terms are called "inequalities". The first inequality is die to Hipparchus and its maximal amplitude is about 6 degrees. Second inequality (a. k. a. evection) was discovered by Ptolemy, its maximal value is 2.5 degrees.

While the first inequality is due to ellipticity of the orbit (Kepler's law, as we know now), the second inequality is due to the influence of the Sun.

There was no much progress between Ptolemy and Tycho, except perhaps that some numerical constants were determined more precisely. Astronomers before Tycho did not measure angles with sufficient precision. Their theories were based on Lunar eclipses, not on the direct angle measurements.

It was Tycho Brahe who discovered the third inequality (a. k. a. variation) which can be as large as 40' (that is greater than the diameter of the sun or of the Moon). Later Kepler and Brahe discovered the forth inequality (of amplitude 11').

At this point the development stopped, because already the second inequality cannot be explained by the Kepler laws (Kepler laws solve the 2 body problem, and here we have 3 bodies: the influence of the Sun is not negligible, as we see already from the second inequality). For example, Jupiter obeys Kepler's laws to very high accuracy, because the influence of other planets is negligible. But Earth-Moon-Sun system is a real 3 body system, and Moon does not obey Kepler's laws with sufficient accuracy.

With Newton's Universal law of gravitation, the first and most important problem was whether it gives anything new besides the Kepler Laws. (After all "explanation of the Kepler Laws was not such a great achievement because the Gravitation Law was derived from these very Kepler laws). The problem was to predict NEW phenomena, or at least (as a first step) to explain the third inequality.

It is known (nowadays) that the three body problem has no closed form solution. This is a system of differential equations based on the law of gravity, but this system cannot be exactly solved. So the major efforts of the best 18 century mathematicians were spent on finding a useful approximate solution. Some of them were stimulated with the practical of this problem which I mentioned above. After the great efforts (the main contributors are Clairaut, Euler and Mayers, but also many others) the theory permitted to compose lunar tables which predicted Moons position for several years in advance with the accuracy to fractions of a minute.

THIS was the decisive test of the law of gravitation. Another test, some times earlier was the correct prediction of the shape of the Earth, confirmed by precise measurements.

Mayers and Euler shared a part of the Longitude Prize given for the solution of the problem of longitude by the British Parliament. (The main part of this prize was awarded to Harrison who invented chronometer at approximately the same time). Nautical Almanac based on Mayers tables was published. It gave predictions for one year. I examined the early issues of the Almanac, and can say that the maximal error in Moon's position was about 0.2'. But every year it had to be corrected. Further mathematical achievements of 19-th century made possible very long term predictions (say, 2000 years) with the accuracy within seconds.

Modern formula contains several THOUSAND periodic terms, "inequalities". So we can easily tell that some eclipse happened on such and such date (say 1545 BC) at such time, when such and such Babylonian king reigned, and the eclipse was 35% at in a given city. This degree of accuracy of "predictions" was only made possible in 20-s century, approximately since 1910.

The previous, very simplified discussion concerns only Moon's motion in longitude. The motion in latitude is even more important for the eclipse prediction, and it is described similarly. The key steps are due to Hipparchus (inclination of the orbit), Brahe (first inequality), Kepler (elliptic orbit), Newton (law of gravity), d'Alembert, Clairaut and Euler (approximate solution of Newton's equations) and Mayer (making of the tables).

I should also mention that at the time of Hipparchus, there existed a different theory, of Babylonian astronomers which had approximately the same accuracy and was based on different mathematics. This theory was practiced as late as 19-th century in India, by Tamil astronomers, and it gave good predictions of Lunar (not Solar!) eclipses.

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    $\begingroup$ Would you mind elaborating on these 18th century developments? Since its only briefly touched upon atm. What did Clairaut/Euler/Mayer achieve that enabled predicting locations? Was this a dramatic departure from what was available before? $\endgroup$ – Semaphore Nov 11 '14 at 14:36
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    $\begingroup$ @Semaphore: I did as much as possible without mathJax $\endgroup$ – Alexandre Eremenko Nov 12 '14 at 3:34
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First of all, I would recommend reading the books of Robert Newton of Johns Hopkins University, especially his 1970 book. Newton understood this subject in far greater detail than practically anyone and his books have extremely clear and detailed discussions of the issues involved. Other books on the same subject are almost childish in comparison with Newton's books.

In ancient times, the Greeks attributed to Hipparchus the first ability to predict eclipses, but it is likely that the Babylonians and Chaldeans had the capability as well and Newton's research suggests it is likely that Hipparchus was simply continuing a tradition that had been established by the Babylonians.

In principle it is easy to predict an eclipse by projecting forwards from previously known eclipses. The farther ahead (or backward) the projection is, however, the more inaccurate it becomes because the motion of the moon is irregular in various ways. Beyond a distance of 150 years or so it becomes increasingly difficult to retroactively predict an eclipse accurately unless contemporary observations are known. And, of course, since we do not know what will happen in the future, predicting an eclipse even using modern methods beyond a few hundred years will have a high probability of error.

For the ancient astronomers the ability to predict eclipses 50 years in advance or less, based on contemporary observations was easily available.

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    $\begingroup$ The work of Robert Newton was very much criticized by astronomers and historians of astronomy. I actually read his book on Ptolemy. On my opinion, this is not real science. In no way I can recommend it as a source of knowledge about history of astronomy. $\endgroup$ – Alexandre Eremenko Oct 17 '17 at 4:44

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