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How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or fe

by ko27 5y ago

How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?

amelius 5y ago
The system is chaotic so there is a strong dependence on the initial conditions. I suppose that if you don't know these precisely, then at some point even the best computer simulation can't help you much.
db48x 5y ago
We can do it accurately, to any precision you care to pay for. Since n-body gravitation is a chaotic system, getting more precise predictions requires more precise measurements of the current state of the solar system. When it’s not possible to measure things more precisely, we instead run many simulations with small random perturbations in the current state, then classify the simulations to get probabilities.
Y_Y 5y ago
One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range of possible values, rather than the most probable one (which is normally implicitly done). So if you measure the earth to be (6±1)e24 kg, then you work with something that looks like [5,7]e24 kg i.e. the segment of the number line corresponding to the possible physical realities that led to your measurement. You'll get a range of different outcomes in the end, and their relative probabilities given your priors. You can do this exactly for some systems, but usually you'll do some monte carlo and hope it's valid. This is similar to classical (linear) error propagation where you carry around an "uncertainty", but chaotic systems don't generally allow you to make the assumption of narrow Gaussians used there.
aaaaaaaaaaab 5y ago
It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.
- zelphirkalt 5y ago
  Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?
  - deleted 5y ago
    [deleted]
  - jjgreen 5y ago
    Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos https://en.wikipedia.org/wiki/Control_of_chaos
  - aaaaaaaaaaab 5y ago
    Indeed, there can be islands of stability in the phase space of chaotical systems.
  - wisty 5y ago
    Often yes. How often happens in practice will usually depend on the size of the solution space. A chaotic system is pretty much a random number generator, and random number generators can spit out the same number (or nearby numbers) twice (otherwise they wouldn't be random).
  - db48x 5y ago
    Yes, though frequently the solutions are very similar to each other. For example, if you plot the future of an asteroid in 10,000 different simulations, you’ll probably find that in most of them the asteroid remains in the asteroid belt where it started from, but perhaps in 10% of them it is perturbed enough by Jupiter that its orbit becomes a Trojan, or some other variety. If you look at the details of the 90% where it stays in the asteroid belt, you find that while they are all in different orbits from each other, the differences are not very significant. Just 9,000 rather similar orbits inside the asteroid belt. “Chaotic” usually means that the difference between two similar starting conditions grows without bound the longer you run the simulations forward. But orbits are closed loops; everything about an orbit is periodic. If two orbiting objects start near each other but have different orbital periods, then soon enough they will be far apart from each other. However, if you keep running time forward then they will end up right next to each other again. The distance between them is itself periodic, bounding the total error in a practical sense. Combine that with the overall stability of our solar system, and you find that most objects tend to stay in particular orbital families for quite some time. Most objects are near the bottoms of deep potential wells, and the forces that can push them out of those wells are quite small. It is only once they are pushed near the boundaries of those wells that rapid changes can begin to happen. Of course if it were any other way, then there would be nothing left in the asteroid belt by now. Compare that with Saturn’s rings, which simulations suggest will only last another 100k years, give or take a bit. They must be a relatively recent phenomena.
dekhn 5y ago
we can simulate the solar system to very high accuracy a few hundred years into the future. https://www.nature.com/articles/nphys728 https://www.nature.com/articles/nphys728

by ko27 5y ago

amelius 5y ago

The system is chaotic so there is a strong dependence on the initial conditions. I suppose that if you don't know these precisely, then at some point even the best computer simulation can't help you much.

db48x 5y ago

We can do it accurately, to any precision you care to pay for. Since n-body gravitation is a chaotic system, getting more precise predictions requires more precise measurements of the current state of the solar system. When it’s not possible to measure things more precisely, we instead run many simulations with small random perturbations in the current state, then classify the simulations to get probabilities.

Y_Y 5y ago

One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range of possible values, rather than the most probable one (which is normally implicitly done). So if you measure the earth to be (6±1)e24 kg, then you work with something that looks like [5,7]e24 kg i.e. the segment of the number line corresponding to the possible physical realities that led to your measurement. You'll get a range of different outcomes in the end, and their relative probabilities given your priors. You can do this exactly for some systems, but usually you'll do some monte carlo and hope it's valid. This is similar to classical (linear) error propagation where you carry around an "uncertainty", but chaotic systems don't generally allow you to make the assumption of narrow Gaussians used there.

aaaaaaaaaaab 5y ago

It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.

zelphirkalt 5y ago
Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?
- deleted 5y ago
  [deleted]
- jjgreen 5y ago
  Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos https://en.wikipedia.org/wiki/Control_of_chaos
- aaaaaaaaaaab 5y ago
  Indeed, there can be islands of stability in the phase space of chaotical systems.
- wisty 5y ago
  Often yes. How often happens in practice will usually depend on the size of the solution space. A chaotic system is pretty much a random number generator, and random number generators can spit out the same number (or nearby numbers) twice (otherwise they wouldn't be random).
- db48x 5y ago
  Yes, though frequently the solutions are very similar to each other. For example, if you plot the future of an asteroid in 10,000 different simulations, you’ll probably find that in most of them the asteroid remains in the asteroid belt where it started from, but perhaps in 10% of them it is perturbed enough by Jupiter that its orbit becomes a Trojan, or some other variety. If you look at the details of the 90% where it stays in the asteroid belt, you find that while they are all in different orbits from each other, the differences are not very significant. Just 9,000 rather similar orbits inside the asteroid belt. “Chaotic” usually means that the difference between two similar starting conditions grows without bound the longer you run the simulations forward. But orbits are closed loops; everything about an orbit is periodic. If two orbiting objects start near each other but have different orbital periods, then soon enough they will be far apart from each other. However, if you keep running time forward then they will end up right next to each other again. The distance between them is itself periodic, bounding the total error in a practical sense. Combine that with the overall stability of our solar system, and you find that most objects tend to stay in particular orbital families for quite some time. Most objects are near the bottoms of deep potential wells, and the forces that can push them out of those wells are quite small. It is only once they are pushed near the boundaries of those wells that rapid changes can begin to happen. Of course if it were any other way, then there would be nothing left in the asteroid belt by now. Compare that with Saturn’s rings, which simulations suggest will only last another 100k years, give or take a bit. They must be a relatively recent phenomena.

dekhn 5y ago

we can simulate the solar system to very high accuracy a few hundred years into the future. https://www.nature.com/articles/nphys728 https://www.nature.com/articles/nphys728