We all know that Newton’s laws of motion (among other laws) allows one to predict planetary positions. In contrast, it is said the three body problem (i.e., a system of three orbiting bodies) drove Newton crazy for positions cannot be predicted and hence (I believe) the basis of chaoic physics. If the three body problem is such a problem, then how do we manage to predict planetary positions with extreme accuracy time and again?
There’s no analytic solution to the three body problem, but it can be solved numerically with some accuracy.
Here’s a good web page on the subject: http://www.umich.edu/~lowbrows/reflections/2006/dsnyder.17.html
In short, I think we still have not solved the three-body problem - we can’t create a single equation that shows all of the motions of three bodies for all cases. There are a few cases we can work out (shown in that web page), and a lot more cases that we can kind of “brute force” (i.e. we use a two-body equation for the main motion we see, and then do some additional two-body equations for variances from that primary course.) The brute force approach is not half bad for many purposes, especially since we’re generally dealing with objects of hugely different sizes.
As our computers get bigger and better, we can continue brute forcing ever-more difficult situations by doing two-body calculations for each thing with every other thing and adding up all the vectors for each unit of time.
I’m not a physicist or a DE guy, but I’ll weigh in anyway. We can predict the positions of the planets (relative to the sun) because they are periodic. Kepler managed to determine the positions of the planets to a high degree of accuracy by studying the data collected by Tycho Brahe. This was all done before Newton came up with his models.
According to wikipedia, the sun contains 99.86% of the solar system’s mass, so even if you just considered the interactions of each planet with the sun as a collection of two-body problems, you’d get a pretty good approximation.
One thing about the three-body problem confuses people: You can absolutely, positively state the equations for an n-body problem. No problem whatsoever. Solving the equations is the really difficult part. As mentioned, there is no closed form solution like there is for the 2-body problem, e.g., involving old familiar functions like sine and cosine.
The equations can be used to numerically calculate solutions. But a big issue is that the solutions are incredibly sensitive to initial and calculated errors. Be off in the 3rd digit somewhere and after a few hundred orbits you might have a planet being ejected from a double star system that isn’t.
Who claims that the three-body problem drove Newton crazy? It appears to me that Newton only thought a little about it. It appears to me that it wasn’t realized that it was a hard problem until after Newton’s time.
Turning lead into gold was the problem that drove Newton crazy.
I thought it was attempting definitive Biblical prophesy that drove him crazy.
Roughly speaking, here’s how it is done. Say you want to predict the earth’s orbit. So you solve the two body problem for the earth and sun (or maybe for the earth-moon system and the sun). You do the same for each of the other planets. You now have what might be called a first order solution to the solar system. Now you take the orbit, say of Venus and work out how that interacts with the earth’s orbit (the effects are linear, so the results can be added). The technical is how Venus perturbs the orbit. Then Mars, then Saturn and Jupiter. I think you can forget the other planets but maybe not. Uranus was found by working out how it was perturbing Jupiter and simiarly for Neptune. Then you go through each of the other planets and see how their orbits are being perturbed. This gives you second order results. Repeat until further iterations give corrections too small to measure. Needless to say, modern computing has rendered a very painful process painless.
I wish I could produce the cite, but I once read about someone who built a dedicated computer for the purpose of carrying out these computations and he ran the solar system for 10 billion years in the future. Then he moved ever planet one meter (or maybe one km) and did it again. For the eight inner planets, the results hardy changed at all. Pluto, however, was in a completely different place the second time. Another reason, perhaps, for not calling it a planet.
I recall calculations done by “real” astronomers, where a few? billion years into the future, you can’t tell whether things are still pretty much still normal in our solar system, or mars has crashed into the earth, or a planet has been actually ejected from our solar system.
In other words, long term, our solar system is chaotic and you can’t be sure how its going to turn out. It isnt the beautiful “clockwork” system folks used to think it was.
Given that the sun will become a red giant in five billion years, it’s hard to care what will happen to Pluto in ten billion years.
Technically true…but it does show things arent as stable as appear at first glance.
Now, if things “went to shit” orbit wise on the order of trillions of years, then yeah it would much more of a :rolleyes:
Just to add to the fun of long range solar system projections – your equations must change as the sun ejects mass.
For the purposes of predicting approximately where in the sky to look for a planet, or when it will rise, the periodic model is fine. However, we do much more precise things than that. For example, lots of folks have studied the stability of the solar system, or the way the planets tend to herd asteroids into the asteroid belt, or tried to figure out how Pluto got to be on such an elliptical orbit. The moon and the planets pass in front of stars, and by timing these occultations we keep refining orbital estimates, and there are thousands of amateurs who participate in occultation timing (Sky and Telescope magazine has features just for them). I think this whole discipline of precise planetary orbit calculation goes beyond the Keplerian model.
Though I think (not sure) that the orbits of GPS satellites that are calculated in real time, and the ephemeris information that all GPS receivers regularly download (which contain Keplerian coefficients), do take this approach. I’d think the Moon would be a big influence, and while the satellites don’t effect each other I would expect each one is influenced by the Earth, the Moon, and the Sun, all sufficiently to matter. But AFAIK they are making it much simpler than that. This seems wrong somehow, but it’s what I gather from bits and pieces here and there - anybody here know more???