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
Back before we had an idea of Newtonian gravity, we just assumed all of the orbits were on rails and could not change. This would drift over time, but not fast enough to notice for hundreds of years.
If you just treat everything as a two body problem between the planet and Sun (and between the Moon and Earth), you get a pretty accurate model, but things start to drift over time. Even then, that’s not even the best we can do with just Newton. This won’t drift for tens of thousands of years.
Even with an N-body system, you can apply Newton’s laws and make very accurate predictions. The problem is if you are even slightly wrong about the initial conditions, you start to drift over time, and the results can’t end up looking very different. That’s why the 3-body problem is so hard, because we can’t tell what it’s going to do after a small change in initial conditions. This won’t drift for millions of years, but we have no idea what it’s going to do when it drifts without more accurate measurements.
Once we had general relativity, Einstein was able to calculate the orbit of the Moon within a centimeter. A prediction we weren’t able to confirm until we landed on the Moon. You don’t need predictions that advanced to predict an eclipse.
If there is a regular pattern in some phenomenon, you do not need to know why that pattern occurs to be able to predict it in the future. Ancient astronomers did not know about the mechanics of the orbits of the Earth and the Moon, but they could observe that solar and lunar eclipses happened at regular intervals, and that eclipses at specific intervals were very similar, and from those patterns they were able to predict future ones.
Almost all the comments are wrong. It has nothing to do with relative mass ratios. Classical 3 body problems were for a massless 3rd body, you can also predict very well the dynamics of triple star systems no matter what mass ratios are there
The moon is massive, it is the biggest moon in the solar system in relative terms (for a planet)
You can predict eclipses because the moon is so close to Earth you can approximate the gravity from the sun as some constant force plus linear tidal force.
The nice thing is that all the complex effects (relativity, quadrupole moment, tidal forces) just rotate the moon’s orbit. It rotates in one way every 8.85 years and in the other 18.6 years. You can empirically measure those without making complex calculations. This was even used to verify the theory of gravity.
There’s a difference between solving it (providing a precise mathematical formula) and estimating the positions of these three bodies for a certain time frame.
What people used to do in the past was the latter – based on observation, they were able to predict the movement of celestial bodies without knowing the exact rules their movements are based on.
And, to a certain extent, that’s what we’re still doing nowadays to “solve” the n-body problem. We have models that aren’t 100% accurate. The further you go in time, the less precise they are.
Has your dad ever held you by your arms and swung around in a circle? That’s basically how orbits work. Instead of having arms planets have gravity to hold them together.
When your dad swings you around like this he has to lean back quite a bit to not topple over. This is because your weight (~18kg for a 5 year old) is not *that* far away from your dad’s (lets say ~90kg).
Now imagine your dad swinging around a bar of soap. Do you think he’d still have to lean back to not topple over? No, right? That’s because the soap bar is so much lighter than your dad. The bar of soap and your dad are like Earth and the Moon. While Earth/Dad feels a bit of an effect from swinging the moon/soap around it is definitely the moon/soap that feels more.
What if your dad was the sun? What could we use to picture Earth? A grain of sand or a tiny down feather! Your dad is going to be completely unaffected by swinging around a grain of sand.
And so it is with Earth, the Sun and the Moon. The moon might be tugging on Earth ever so slightly but from the perspective of the sun, its almost like we’re not even there. And the same happens in calculations. Using two 2-body problems is basically good enough.
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