The solar system is non-integrable and has chaos.
The sun-earth-moon three-body system might be chaotic.
So, how far into the future can we predict solar eclipses and/or lunar eclipses?
How about 1 million years?
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4$\begingroup$ One is tempted to ask why you need to predict solar and lunar eclipses a million years in the future, and what kind of accuracy you need. $\endgroup$– Williham TotlandCommented Sep 26, 2014 at 12:23
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21$\begingroup$ @WillihamTotland Really? Because we can, of course. you might as well ask why we examine stars 10million light years away. $\endgroup$– Carl WitthoftCommented Sep 26, 2014 at 14:34
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1$\begingroup$ @CarlWitthoft I'm not saying it's a bad or unnecessary question, but a sense of what kind of accuracy is desired is useful to give an answer. The requirements of a sci-fi story aiming for a bit of verisimilitude are different for the requirements associated with a desire to detonate a nuclear visible in the umbra of an eclipse six million years from now, and different from simple curiosity. $\endgroup$– Williham TotlandCommented Sep 26, 2014 at 14:37
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3$\begingroup$ @CarlWitthoft - We can't, of course. Even with the very best numerical integration techniques, the very best models of interactions, and the very best possible measurements, errors tend to exhibit exponential growth. The time scale at which the discrepancy between nearby initial conditions grow by a factor of $e$ is the Lyapunov time. Since $e^{20}$ is a rather large number, trying to go beyond 20 Lyapunov times is a rather pointless endeavor, regardless of the accuracy of the initial conditions. $\endgroup$– David HammenCommented Sep 26, 2014 at 17:21
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3$\begingroup$ @user121330 - That was a generic remark. Regardless of how good the initial estimate is, an estimate of where something is after 20 Lyapunov times is pretty much garbage. I unsuccessfully tried searching for estimates of the Lyapunov time of the Moon's orbit. The paper Kudryavtsev, S. M. "Long-term harmonic development of lunar ephemeris." Astronomy and Astrophysics 471.3 (2007): 1069-1075 suggests to me that the Lyapunov time of the Moon's orbit might be as low as a few thousand years. $\endgroup$– David HammenCommented Sep 26, 2014 at 19:47
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4 Answers
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On predicting planetary orbits
A number of studies have shown that the inner solar system is chaotic, with a Lyapunov time scale of about 5 million years. This 5 million year time scale means that while one can somewhat reasonably create a planetary ephemeris (a time-based catalog of where the planets were / will be) that spans from 10 million years into the past to 10 million years into the future, going beyond that by much is essentially impossible. At a hundred million years, the position of a planet on its orbit becomes complete garbage, meaning that the uncertainties in the planetary positions exceed the orbital radii.
What one can do is forgo the idea of predicting position and instead ask only about parameters that determine the size, shape, and inclination of planetary orbits. This lets one look to secular chaos as opposed to dynamic chaos, which in turn lets attempt to answer the key question, Is the solar system stable?
The answer to this question is "not quite". The key culprit is Mercury, the most chaotic of all of the planets. One factor is its small size, which magnifies perturbations from other planets. Another factor is resonances with Jupiter and Venus. Both of these planets have multiple resonances with Mercury's eccentricity (Jupiter more so than Venus), and Venus also has multiple resonances with Mercury's inclination. These resonances spell doom for Mercury. Mercury is perched on the threshold of secular chaos, and is likely to be ejected from the solar system in a few billion years.
On predicting eclipses
The issue of chaos becomes even more extreme when trying to predict eclipses, particularly solar eclipses. The Sun, Jupiter, and Venus have marked effects on the long-term behavior of the Moon's orbit. Even more importantly, however, the Moon is receding from the Earth due to tidal interactions, and this rate is not constant. The current recession rate is about twice the average rate over the last several hundred million years. Changes in the shape and interconnectivity of the oceans drastically changes the rate at which the Moon recedes from the Earth. The melting of the ice covering Antarctica and Greenland would also significantly change the recession rate, as would the Earth entering another glaciation. Even a small change destroys the ability to make long term predictions of the Moon's orbit.
NASA developed a pair of catalogs of solar eclipses: one covering a 5,000-year period spanning from about 4000 years ago to about 1000 years into the future; the other a 10,000-year catalog of solar eclipses spanning from about 6000 years ago to about 4000 years into the future. The accuracy of these catalog degrades drastically before 3000 years ago and after 1000 years into the figure. Beyond these inner limits, the path of the eclipse over the Earth's surface becomes markedly unreliable, as does the ability to determine whether the eclipse will be partial, total, annular, or hybrid. At the outer time limits of the longer catalog, whether an eclipse did / will occur begins to become a bit dubious.
Because of the Earth's much larger shadow, predictions of lunar eclipses are a bit more reliable, but not much. The problem is that of exponential error growth, which is a characteristic of dynamically chaotic systems. Predictions of lunar eclipses more than a few tens of thousands of years into the future are more or less nonsense. The millions of years asked in the question: No.
The technique of orbital averaging once again can be of aid in determining characteristics of the Moon's orbit (but not position on the orbit). This can be augmented by geological records. Various tidal rhythmites give clues as to the paleological orbit of the Moon. A few rock formations exhibit layering that recorded the number of days in a month and the number of months in a year at the time the rock formation was created.
References
answered Sep 26, 2014 at 16:31
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$\begingroup$ FWIW, the NASA eclipse canons were calculated by Meuss & Espenak using VSOP87 for the Sun and ELP-2000/82 (by Chapront, Chapront-Touzé et al) for the Moon. Note that ELP-2000 was fitted to JPL's Development Ephemeris, initially DE200, but "improved parameters have been published up to DE405". However, the precision of lunar parameters in the JPL DEs has continued to improve in the last decade, with better Lunar Laser Ranging data. $\endgroup$– PM 2RingCommented Oct 31, 2022 at 1:57
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How far ahead can we predict solar and lunar eclipses?
NASA has uncertainty calculations that show how certain we are about when eclipses happen. From a back of the envelope, the eclipses will likely vary by a full day 35 thousand years from now. That said, we have eclipse seasons, so we know eclipses will continue to happen, and at roughly which time of the year.
The Moon is receding from the Earth, so total eclipses will become rarer over time. A more distant orbit is also slower, so the number of eclipses per year will also reduce over time. Eclipses will keep happening until the Sun expands as it starts dying in about 5 or 6 billion years. For a bit, they will be transits, and then we'll be inside the sun's atmosphere, and friction forces could change our orbit significantly. This all supposes that a gravitational body that isn't in the solar system (or one that is!?!) doesn't drastically change the system in some fashion.
The solar system is non-integrable and has chaos. The sun-earth-moon three-body system might be chaotic.
As pointed out in other posts, 4 billion years of orbits suggest a certain system stability.
Chaos means that for slightly different initial conditions, we get wildly varying results. In fact, the stability of the solar system helps demonstrate that gravitational systems are chaotic - if bodies flew off into deep space every time, that would be predictable. As 3 body systems go, Sun/Earth/Moon is stable.
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1$\begingroup$ Re the first sentence, "NASA has uncertainty calculations that show how certain we are about when eclipses happen." Your interpretation of the linked page is incorrect.Those uncertainty calculations are about $\Delta T$, the difference Terrestrial Time and Universal Time. Given a perfect ephemeris, we would be able to exactly predict/postdict the times (Terrestrial Time) solar eclipses occur. An uncertainty in $\Delta T$ results uncertainty in where on Earth a solar eclipse is observable. Ephemerides aren't perfect, but the uncertainties are much smaller than those induced by $\Delta T$. $\endgroup$ Commented Jun 10, 2019 at 9:25
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$\begingroup$ @DavidHammen I think it's safe to say that 'predict eclipses' includes what time they will occur, what the orientation of the Earth will be when they occur, and therefore, where on the Earth they occur. It's completely consistent with the page given. It's not as if I predicted the end of the world or anything here. $\endgroup$ Commented Jun 17, 2019 at 18:21
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Look I know link only answers are terrible, but I don't want to ruin the surprise. Check out this link.
That's on Wikipedia! It's safe to say that if wikipedia knows something, the experts know quite a bit more. We know the dynamics of the Sun-Earth-Moon system really well including the perturbations and the most likely sources of future upsets, so barring a major asteroidal collision or something, we can probably predict the times of lunar and solar eclipses for the next million years. Now, I doubt we'll have as precise of timing as what you've seen for a million-year range, but no doubt we'll give you the day, rough time, and probable location (maybe some geologists could help by predicting plate tectonics for the next million years to get the location). Sorry I can't give you more of a scientific answer. I'll let someone else have that honour; I just wanted to show you how far ahead we can predict already
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3$\begingroup$ "That's on Wikipedia! It's safe to say that if wikipedia knows something, the experts know quite a bit more." Well ... mostly. But I suspect that this is true in this case because it is an uncontroversial and rather geeky topic and that's what Wikipedia does best. $\endgroup$ Commented Sep 26, 2014 at 15:22
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2$\begingroup$ Does it seem plausible that one thousand year predictions with no quoted uncertainties imply million year predictions? $\endgroup$ Commented Sep 26, 2014 at 17:21
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$\begingroup$ @user121330 that really depends on what you consider plausible $\endgroup$– JimCommented Sep 26, 2014 at 17:22
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Given that we exist at all, we can confidently back-track the positions of the planets (and our moon) for a couple billion years. I see no reason, barring rather massive exo-system-sourced objects showing up unexpectedly, that the positions will go chaotic enough to be unpredictable any time in the next couple billion years. I suppose it might depend a bit on the definition of "predict" -- can we predict the onset of an eclipse 10 million years from now to an accuracy of one picosecone? Probably not.
EDIT - even a junior amateur astronomer like me should have done better than that. Not to disappoint everyone, but in the semi-distant future there won't be any solar eclipses at all!
answered Sep 26, 2014 at 11:43
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2$\begingroup$ Whoa!! way too fast there: please say a bit more about the first sentence for those of us with mortal IQ levels :) why does our existence imply an ability to hindtell (or retrodict or whatever the buzzword is) planetary positions. Also, do you know of any estimates of prediction error as a function of time lapse (to quantify your second last sentence)? $\endgroup$ Commented Sep 26, 2014 at 12:15
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2$\begingroup$ @WetSavannaAnimalakaRodVance I'm implying that the fact that the Earth has existed this long, not specifically the human species :-), suggests long-term orbital stability. $\endgroup$ Commented Sep 26, 2014 at 12:48
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$\begingroup$ @CarlWitthoft Thanks: got it now. Brains a bit slow on the uptake. $\endgroup$ Commented Sep 26, 2014 at 12:49
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8$\begingroup$ -1. The first sentence is not true, nor is the second. That we are here is evidence that the Earth's mean motion hasn't changed by much. After several million orbits, even a tiny change in mean motion (or orbital radius) will drastically change the position on that orbit. After hundreds of millions of orbits, the position becomes completely unknowable. The inner solar system is chaotic with a Lyapunov time scale of about five million years. $\endgroup$ Commented Sep 26, 2014 at 13:47
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1$\begingroup$ I read the wikipedia article you linked to... Perhaps you could link to the section in question? Are you referring to 'total solar eclipses'? $\endgroup$ Commented Sep 26, 2014 at 17:57