Isn’t the 3 body problem (sun, Earth, Moon) very difficult to solve? How did humans predict future eclipses decades even centuries ago?
In: Planetary Science
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The Sun is much more massive than the Earth, and the Earth is much more massive than the Moon. Gravity is a force proportional to mass, so in this setup, the Sun dominates the equations of motion, while the Moon’s effects are somewhat negligible, allowing for approximate solutions. It is true that the exact solution would be chaotic over very long time periods, but the differences on human timescales are rather small and wouldn’t be easily detected by ordinary observation.
Thus, eclipses could be predicted by assuming the Earth orbits around the Sun in a circle, and the Moon orbits around the Earth in a circle, which produce regular patterns depending on the angles of those orbits. Better approximations could be made by using elliptical orbits instead of circular orbits.
The Sun is so much bigger than the Earth and Moon (and all the other planets) that, to a very good approximation, the Sun may as well be unaffected by the gravity of its planets. The Earth is also quite a bit bigger than the Moon, but the difference isn’t as great, so the Moon’s effect is more noticeable.
To put it in perspective, the Sun contains 99.8% of the mass of the entire Solar System. All the planets may as well be a rounding error. ~~As far as we’re concerned, the Sun is standing still, completely unaffected by the planets (even if *technically* it is a little bit).~~ Some planets do have a noticeable effect on the Sun, but basically there are enough huge size differences that almost any calculation we would want to do simplifies to a two-body problem.
The three-body problem is mainly difficult when all three bodies are close enough in size that they all have a significant effect on each other.
The Sun, the Earth and the Moon is not really a three-body problem because of huge differences in mass between the bodies and in distances between the pairs.
Effectively, you have two separate two-body problem Sun vs (Earth+Moon) and Earth vs Moon; and you can solve them independently and then introduce small perturbations for each from the third body if necessary to achieve required precision.
The solution to a 3 body problem (in which you can given the initial values predict the exact position and momentum of all 3 bodies at any point in time) does not exist. That only means we don’t have a perfect formula for solving it forever and perfectly because it eventually gets chaotic.
We can, however, take a look at the enormous amount of data we have and create reasonable, short term predictions for specific times that seem very long in human years but are very short in the astronomical sense. And in those passing years we have gotten better at gathering said data and analyzing it, and now have computers that can do, as James May one said: Years worth of arithmetic: *snaps* like that.
That and the Earth being MUCH bigger than the Moon and the Sun being MUCH, MUCH, MUCH, MUCH bigger than everything else means the math can be simplified a lot and still work.
The three body problem is so difficult to solve not because it is impossible but because the calculations needed to do it accurately over a long period just pile up rapidly as the motions of each body (it’s not just three bodies, obv, it’s any system of 3 or more bodies) all affect each other. Your initial conditions, presumably at rest or in constant motion, are easy to compute. Once they move, it’s harder and harder to know with precision where things will be simply because the rules of gravitation and motion guiding them initially are all that dictate the motion at the outset, but as time passes, you have to continually factor how their influence on each other is affecting them. In terms of real bodies like the objects of the solar system, there’s also the issue that you’re not actually dealing with three bodies, or 8, or 9, or 100,000. To have extremely precise calculations, you’d need to account for the effects of all the objects in the solar system – the more bodies you can add in, the more precise the projections you can calculate.
Eclipses though are a separate thing from this. Eclipses can be predicted mathematically not through the three body problem (I mean, yes, it’s a three body problem) but through more straightforward math. The positions of the sun, moon and earth do not need to be known with the level of precision that is normally what you’re trying to get when you’re dealing with a three body problem situation. Predicting eclipses is ultimately about knowing when various paths on orbits that are already known will cross one another, and we can know that from observational data. The three body problem applies to predictions of positions of objects that are, like, “purely predictions” – you’re trying to work out mathematically where objects will be and to a precise degree. Once it was figured out formally that the earth was orbiting the sun and that the moon also orbits the earth, all of them in ellipse shapes, it was/is possible to work out eclipse predictions from observations of their real orbits over time and relatively straightforward math. You also don’t have to know exacting positions of the objects – like down to milimeter or whatever arbitrarily small metric you want to use – to predict an eclipse.
The issue with the three body problem is ultimately a calculation problem – it’s not that it’s formally impossible, it’s just that the burden of calculations you have to do spirals out of control quickly to do on paper, or even with a computer. The more objects you have to calculate, the more quickly the calculation burden spirals. In actual reality, it’s also impossible to actually know all the forces acting on the real objects whose positions you’re trying to predict with mathematically certainty into the future, and the less you know of those forces, the more quickly your prediction breaks down.
You know how two similarly sized bodies will both orbit a spot outside of themselves called a barycenter?
The 3 body problem is less about systems with stable barycenters like Sol>Earth>Luna and more about 3 similar bodies orbiting a more chaotic barycenter. There are a few special cases where it works out and is stable if left undisturbed but the vast majority of 3 body orbits self destruct and either devolve to more common trinary systems or destroy or reject one of the bodies.
[PBS Space Time](https://youtu.be/et7XvBenEo8?si=2ePk04L5D1Yp6VyL) Has a fun video with more details.
The 3 body problem can’t be solved but it can be simulated to some accuracy. So you can accurately predict what will happen in the next few hundred years, but maybe not in the next few hundred million years.
This is true of any chaotic system, you can’t accurately predict it forever but you can take the current state and then work forward to where it will go in the near future. Because you can never have a perfect capture of the starting state your predictions will eventually drift out of sync with reality, but you can still make accurate predictions in the near term.
I think it solved 2 Millenia ago according to this video. It gets into the nitty gritty. I’m not much of a geometry math guy so it’s mostly over my head. Interesting video though
Other than the handful of weird specific mathematical solutions to the three body problem, there are also two general solutions:
1. 3rd orbiting the center of mass of the two larger objects at great enough distance that the interplay between the two of them doesn’t throw the orbit off
2. 3rd object Orbiting close around the mid sized object as that one orbits around the larger one at a significantly greater distance
The moon (and most of earths satellites) fall under category 2, and thus their orbits are pretty easy to predict.
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