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?
orbital-motion – How far ahead can solar and lunar eclipses be predicted?
celestial-mechanicschaos theoryorbital-motionsolar system
Related Solutions
Lasker published a well-known result in 1989 showing that the solar system is chaotic, the inner planets more so than the outer planets. Quoting the Scholarpedia article (written by Lasker himself):
An integration over 200 million years showed that the solar system, and more particularly the system of inner planets (Mercury, Venus, Earth, and Mars), is chaotic, with a Lyapunov time of 5 million years (Laskar, 1989). An error of 15 m in the Earth's initial position gives rise to an error of about 150 m after 10 Ma; but this same error grows to 150 million km after 100 Ma. It is thus possible to construct ephemerides over a 10 million year period, but it becomes essentially impossible to predict the motion of the planets with precision beyond 100 million years.
So one approach to your question, to get at least a qualitative answer, would be to compare a 15-m error in the Earth's initial position to a 70-kg error in its mass. Let's start with the Earth-Sun gravitational potential energy, which depends on the mass of the Earth $\left(m\right)$ and its orbital radius $\left(r\right)$:
$$U\left(r, m\right) = -mr^{-1},$$
in units where $GM_\textrm{Sun} = 1.$ The errors in $U$, one due to the error in $m$ and the other due to the error in $r$ will be
$$\delta U_{m} = r^{-1}\delta m \textrm{ }\textrm{ (magnitude), and}$$
$$\delta U_{r} = mr^{-2}\delta r.$$
The ratio of the errors is
$$\frac{r^{-1}\delta m}{mr^{-2}\delta r} = \frac{\delta m}{m} \frac{r}{\delta r} \approx 6 \times 10^{-14}.$$
You can use SI units to get the numerical result, but you don't have to plug in any values to see what is going on. Because the units cancel, we can just compare the ratio of the errors to the ratio of the values. The ratio of the errors, $\delta m / \delta r$, is approximately 5. But the ratio of the values, $r/m$, is $\approx 10^{-13}.$
So, if the system's sensitivity to the mass error scales in a similar way to its sensitivity to the position error, it seems the mass error will have a much smaller effect than the position error for calculations covering 10 million years. Calculations that cover a longer period are not reliable regardless of the source of error.
We can the path of any moon as an epicycloid: we treat the planet as though it moves in uniform circular motion with radius $r_p$, and the moon as though it moves in uniform circular motion about the planet with radius $r_m$. The net acceleration of the moon will be the vector sum of the planet's acceleration towards the sun (with magnitude $r_p \omega_p^2$) and the moon's acceleration towards the planet (with magnitude $r_m \omega_m^2$). Here, $\omega_p$ is the angular velocity of the planet about the sun, and $\omega_m$ is the angular velocity of the moon about the planet.
Since the planet's acceleration always points towards the sun, the only way for the moon to have a net acceleration vector pointing away from the sun is for its acceleration vector towards the planet to be of greater magnitude: $$ r_m \omega_m^2 > r_p \omega_p^2, $$ or in terms of the respective periods $T_p$ and $T_m$, $$ \frac{r_m}{T_m^2} > \frac{r_p}{T_p^2}. $$ Under these approximations, any moon for which this is satisfied will have some point of "concavity" in its orbit when it is between the Sun and its parent planet.
I whipped up some code using Mathematica and Wolfram's curated data to see which moons satisfy this condition. My initial findings are that only a few of the outermost moons of Jupiter, Saturn, and Neptune do not have points of concavity in their orbits (along with Earth's moon, of course.) This makes a certain amount of sense, since orbital periods of moons aren't significantly longer in the outer solar system than in the inner solar system, while the orbital periods of the planets definitely are.
However, I'm not sure that I've done the coding correctly, since I expected my code to return all moons of a given planet outside a certain radius and this isn't what happened. I will try to fix this and update once I have a better grasp on the real answer.
Best Answer
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
Adams, Fred C., and Gregory Laughlin. "Migration and dynamical relaxation in crowded systems of giant planets." Icarus 163.2 (2003): 290-306.
Espenak and Meeus. "Five Millennium Canon of Solar Eclipses: -1999 to +3000." NASA Technical Publication TP-2006-214141 (2006).
Espenak and Meeus. "Ten Millennium Canon of Long Solar Eclipses." Eclipse Predictions by Fred Espenak and Jean Meeus (NASA's GSFC).
Laskar, Jacques. "A numerical experiment on the chaotic behaviour of the solar system." Nature 338 (1989): 237-238.
Laskar, Jacques. "Large scale chaos and marginal stability in the solar system." Celestial Mechanics and Dynamical Astronomy 64.1-2 (1996): 115-162.
Laskar, Jacques, and Monique Gastineau. "Existence of collisional trajectories of Mercury, Mars and Venus with the Earth." Nature 459.7248 (2009): 817-819.
Lithwick, Yoram, and Yanqin Wu. "Theory of Secular Chaos and Mercury's Orbit." The Astrophysical Journal 739.1 (2011): 31.
Lithwick, Yoram, and Yanqin Wu. "Secular chaos and its application to Mercury, hot Jupiters, and the organization of planetary systems." Proceedings of the National Academy of Sciences (2013): 201308261.
Naoz, Smadar, et al. "Secular dynamics in hierarchical three-body systems." Monthly Notices of the Royal Astronomical Society (2013): stt302.