Three-Body Problem: Soluble or Insoluble?

[Isaac Newton]

I won't describe the problem, on the assumption that the people who know the answer to my question will already know what the problem is. It's sometimes called the three orbit problem, and it's a special case of the n-Body Problem.

There's a Wikipedia article about it, though it's not a very good article.

My question:

Is it more correct to say that the problem has been shown forever insoluble?

Or is it more correct to say that it cannot be solved by any known mathematical method?

(I understand there are special cases which are soluble. I understand that it can be "solved" by successive approximations, which grow increasingly inaccurate as the number of iterations increases.)

I'm not a mathematician or astronomer. I'm writing a book, in which I want to use this as an analogy for other things. I want to get it right.

It's my understanding that quantum computers, if they ever exist, will not have infinite computing power. If so, as far as I know, the three body problem will remain insoluble. Correct?

Thanks in advance.

Isaac

 

[Filip Larsen]

Welcome to PF, mr. Newton.

Unlike the two-body problem, it is not possible to find an analytical solution to the three-body problem, that is a solution that would allow you to calculate the position of the bodies as a function of time for some given initial conditions. With various restrictions in the degrees of freedom in a three-body system you can derive some characterization of the possible orbits, but nothing that is considered an analytical solution.

However, it is not especially hard to analyze and solve the three-body (or indeed the n-body) problem if we turn to numerical methods. Such analysis can even include much more complex models (non-uniform planets, non-gravitational forces) that allow for all sorts of special perturbations to be included, so in terms of being able to accurately model and predict, say, the motion of the bodies in the solar system, the computers of today are quite up to the task. I'm not current on the latest research in the area, but to me it seems that the challenge today in getting more accurate celestial science is not the lack of computing power but the uncertainty regarding the precise details of relevant physical laws and measured data.

 

[Isaac Newton]

Welcome to PF, mr. Newton.
However, it is not especially hard to analyze and solve the three-body (or indeed the n-body) problem if we turn to numerical methods. Such analysis can even include much more complex models (non-uniform planets, non-gravitational forces) that allow for all sorts of special perturbations to be included, so in terms of being able to accurately model and predict, say, the motion of the bodies in the solar system, the computers of today are quite up to the task. I'm not current on the latest research in the area, but to me it seems that the challenge today in getting more accurate celestial science is not the lack of computing power but the uncertainty regarding the precise details of relevant physical laws and measured data.

Thank you, Filip.

If I'm not mistaken, you're talking about the method of successive iterations. My understanding is that this is reasonably accurate for millions of years into the future, but beyond a certain number of iterations, adding up to many millions of years, predictions about the future position and velocities of the planets in the solar system become hopelessly inaccurate.

Have I misunderstood?

Cheers,

Isaac

 

[Filip Larsen]

If I'm not mistaken, you're talking about the method of successive iterations. My understanding is that this is reasonably accurate for millions of years into the future, but beyond a certain number of iterations, adding up to many millions of years, predictions about the future position and velocities of the planets in the solar system become hopelessly inaccurate.

Have I misunderstood?

That is correct. Depending on what you want to analyse and how accurate your model and data is, there is a limit to how far into the future you can accurately predict trajectories. In general, both purely gravitational systems and systems which has friction (which in case of n-body simulation could be gravitational energy that ends up as heat in the core of moons and planets) has the possibility for so-called chaotic dynamics.

As you may know from simulation of the weather, presence of chaos effectively limits how far into the future you can predict individual trajectories since chaotic motion will amplify any microscopic inaccuracies in initial data to macroscopic differences later on. The same goes with gravitational systems although the time-frame here is much longer. The possibility of chaos in a system also mean that there is no real hope of finding a simple analytical solution for the motion since such solutions would be incapable of reproducing the structure found in chaotic motion.

 

[Isaac Newton]

That is correct. Depending on what you want to analyse and how accurate your model and data is, there is a limit to how far into the future you can accurately predict trajectories. In general, both purely gravitational systems and systems which has friction (which in case of n-body simulation could be gravitational energy that ends up as heat in the core of moons and planets) has the possibility for so-called chaotic dynamics.

Hello Filip,

I have always had the impression that the three-body problem is inherently chaotic, even in the absence of friction, even if the starting positions, masses and trajectories are known precisely. Is this incorrect?

Isaac

 

[twofish-quant]

I have always had the impression that the three-body problem is inherently chaotic, even in the absence of friction, even if the starting positions, masses and trajectories are known precisely. Is this incorrect?

Depends on the exact type of problem. There are situations with the three body problem which are not chaotic, and there appear to be situations that are. If you have two objects in a circular orbit, and a small satellite that doesn't affect the two other bodies, you don't have chaos. My intuition is that in the general situation where you have two planets and a satellite, you don't get chaos.

Showing that a system is or is not chaotic is sometimes quite difficult, and there is a lot of interesting math involved.

 

[twofish-quant]

Is it more correct to say that the problem has been shown forever insoluble?

It depends what you mean by "solve".

It's my understanding that quantum computers, if they ever exist, will not have infinite computing power. If so, as far as I know, the three body problem will remain insoluble. Correct?

I think that someone has come up with a solution to the three body problem that involves infinite series. The trouble is that once you've come up a three body solution, you add a fourth body, and that causes things to be really complicated.

Also there is a difference between "we don't know how to do this" and "we can prove that it can't be done." A lot of math involves proving that something just cannot be done. For example, Galois proved that you cannot solve a fifth degree polynomial function with a finite number of basic algebra steps.

In the situation with celestial mechanics, things are complex enough so that proving that you *can't* solve the problem for a given definition of solve gets you quite interesting mathematics.

 

[twofish-quant]

One other idea that is important in chaotic systems is something called the Lynapanov exponent. Basically that number gives you an idea of how quickly the system blows up. I think the number for the solar system is five million years.

The other curious thing about the solar system is that the planets appear to be "chaotic but stable." What seems to be the situation is that the solar system is set up so that you can't figure out exactly where the planets are going to be over long periods of time, but it's set up in a way that you can be certain that planets are going to end up flying out of the solar system.

 

[Filip Larsen]

The other curious thing about the solar system is that the planets appear to be "chaotic but stable." What seems to be the situation is that the solar system is set up so that you can't figure out exactly where the planets are going to be over long periods of time, but it's set up in a way that you can be certain that planets are going to end up flying out of the solar system.

I don't think that the presence of chaotic orbits in the region of the planets current state (position and velocity) imply that a planet have to escape the solar system at some point in future time, i.e. that "chaos" implies "blow up", if that is what you mean. As far as I recall, it should be possible to have a solar system where planets in chaotic orbit are bounded in non-overlapping finites regions. Whether that is true or not for our solar system, I don't know. I guess the problem with predicting the precise state of our solar system on a very long term basis, is that it really is not a closed system, but may be affected from (yet unknown) objects from, say, the Oort cloud, nearby stars or similar.

 

[alxm]

Also there is a difference between "we don't know how to do this" and "we can prove that it can't be done." A lot of math involves proving that something just cannot be done. For example, Galois proved that you cannot solve a fifth degree polynomial function with a finite number of basic algebra steps.

And there is also analytical vs numerical. A fifth degree polynomial is unsolvable analytically in that way, but quite easy to solve numerically.

Likewise, the many-body problem may have an analytical solution that we don't know about, but it's entirely possible (perhaps even likely) that this analytical solution is more difficult to calculate than a numerical solution, in which case it's of fairly little practical use.

 

[D H]

It depends what you mean by "solve".

Exactly. Saying that the three body problem is insoluble is rather sloppy. What people really mean when they say that is that the three body problem is insoluble in terms of elementary methods. What constitutes elementary methods is a bit arbitrary and is a bit meaningless. The trig functions are elementary, but that does not mean we can compute an exact values for, say, sin(1) or acos(0).

I think that someone has come up with a solution to the three body problem that involves infinite series.

That was Karl Sundman, and it's pretty much useless. The problem is that the series converges very, very slowly. The three body problem has singularities (there are lots of ways bodies can collide). While those singularities are a space of measure zero, they mess up the series a lot.

 

[ViewsofMars]

I wonder if this might be helpful for this discussion, on March 18, 2010 there was a full press release for the First Clay Mathematics Institute Millennium Prize For Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman.
Full press release: http://www.claymath.org/poincare/millenniumPrizeFull.pdf
http://www.claymath.org/
http://www.eurekalert.org/pub_releases/2010-03/tcmi-tcm031910.php

Congratulations Dr. Perelman.

 

[D H]

The only connection between the Poincaré Conjecture and the n-body problem is that Henri Poincaré worked on both. He worked on a lot of very distinct problems.

 

[ViewsofMars]

D H, I was only addressing *your* mention of "singularities" as was in the pdf I earlier presented.

Perelman's breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. He achieved a complete understanding of singularity formation in Ricci flow, as well as the way parts of the shape collapse onto lower-dimensional spaces. He introduced a new quantity, the entropy, which instead of measuring disorder at the atomic level, as in the classical theory of heat exchange, measures disorder in the global geometry of the space. This new entropy, like the thermodynamic quantity, increases as time passes. Perelman also introduced a related local quantity, the L-functional, and he used the theories originated by Cheeger and Aleksandrov to understand limits of spaces changing under Ricci flow. He showed that the time between formation of singularities could not become smaller and smaller, with singularities becoming spaced so closely – infinitesimally close – that the Ricci flow method would no longer apply. Perelman deployed his new ideas and methods with great technical mastery and described the results he obtained with elegant brevity. Mathematics has been deeply enriched.

 

[D H]

FFS, ViewsofMars. Singularities can appear throughout mathematics. Just because they do appear in two areas does not mean that those two areas have anything to do with one another.

 

[ViewsofMars]

My reponse is on the next page.LOL! ( I wonder if one of those pranksters is up to mischief today!) (tee hee, I love it.)

 

[ViewsofMars]

FFS, ViewsofMars. Singularities can appear throughout mathematics. Just because they do appear in two areas does not mean that those two areas have anything to do with one another.

FFS, D H

"Singularities are to Ricci flow what black holes are to the evolution of the cosmos. Perelman also introduced a kind of geometric entropy, akin to the entropy studied in the exchange of heat, as in a turbine or motor."(Please refer to the link from Ereaka Alert that I earlier presented.) I provided an award given to Perelman who did discuss sigularities. Perhaps you would like to explain sigularities by way of a link (url) in particular the issue of two areas that don't have anything to do with the another.

Also, let's remember the OP Isaac Newton did earlier state, "I'm not a mathematician or astronomer. I'm writing a book, in which I want to use this as an analogy for other things. I want to get it right." His statement may imply an open door policy here on this topic. Math and astronomy, eh? He is writing a book and may wish to include both. Also, Isaac did say, "It's sometimes called the three orbit problem, and it's a special case of the n-Body Problem." The topic is "Three Body Problem."

I need to do my own research on "the three orbit problem", "n-body problem", and "three body problem." Foremost, do we have a problem?

I also have to remember that information changes quite fast in the realm of Science and Math. I do recall in Astromony or Cosmology there is mention of singularity or singularities. I have to hunt for that one.

People do love to learn. Life is a dance. It's fun!

Have a nice day,

Mars

 

[Frame Dragger]

@ViewsofMars: 1.) Why the double post?

@ViewsofMars & D H2.)... what is "FFS" ?

 

[ViewsofMars]

@ViewsofMars: 1.) Why the double post?

@ViewsofMars & D H2.)... what is "FFS" ?

I don't know! I just now saw it. Weird. I was trying to make a few corrections on it and it appeared on this page. I'm going to try to erase the one on the previous page. Sorry.

Ok, I fixed it. I have to run or I'm going to miss my appointment.

 

[ViewsofMars]

@ViewsofMars & D H2.)... what is "FFS" ?

Frame Dragger, FFS means "FOR FURTHER STUDY."

 

[D H]

ViewsofMars, 1/x has a singularity at x=0. That singularity has nothing to do with the Poincaré Conjecture.

FrameDragger, http://en.wikipedia.org/wiki/FFS. You decide.As far as the three body problem is concerned, the oft-cited claim that it is insoluble is essentially wrong. The correct claim is that it is insoluble by means of elementary techniques. In that sense, so what? \int_{-\infty}^x \exp(-t^2) dt is insoluble by elementary techniques, as are the solutions to host of differential equations (e.g., the gamma function, Bessel functions, ...). That these integrals cannot be expressed in terms of elementary functions does not mean we have to throw up our hands in despair. Numerical techniques can do quite a nice job solving these non-elementary integrals.

That said, the n-body problem appears to be chaotic. That means that numerical integration can only take us so far into the future. The numerical integration will yield a definite answer, but that result will essentially be garbage if the integration period is too long. The solar systems Lyapunov time appears to be 5-10 million years. Using numerical techniques on that kind of time scale will yield meaningless nonsense. The distances from the Sun might be close to correct, but where the planets lie on their orbits will be very wrong.

 

[Frame Dragger]

Frame Dragger, FFS means "FOR FURTHER STUDY."

Ahhh... ok... I'll um, take your word on that, even though another definition occurred to me after posting.

A friend is working on the 2-body problem (Numerical GR), and his take on the 3+ is that it's utterly impossible at this time, or any concievable future time. *shrug* I'd say that's a pretty big problem! From what he's said, the 2-body is amazingly complex, and rooted in the need for perfect initial conditions and a LOOOOOT of HPDE's.

EDIT: @D H: OH. So that other definition is applicaple. I am backing away very slowly now!

 

[D H]

The three body problem is not utterly impossible at this time. Think about it this way: How could we have sent vehicles to the Moon or other planets if it were utterly impossible?

 

[Frame Dragger]

The three body problem is not utterly impossible at this time. Think about it this way: How could we have sent vehicles to the Moon or other planets if it were utterly impossible?

I should clarify: impossible to find an exact solution over any given time-scale a la the 2-body problem. I should also add that something may be lost in the translation between my friend, and here, or he may have simplified matters for my sake.

 

[D H]

That the three body problem does not have a solution in the elementary functions is well-known and has been for a long time. That does not mean the three body problem is insoluble. It just means that it doesn't have a solution in the elementary functions.

 

[Frame Dragger]

That the three body problem does not have a solution in the elementary functions is well-known and has been for a long time. That does not mean the three body problem is insoluble. It just means that it doesn't have a solution in the elementary functions.

What is the alternative?

 

[D H]

Numerical techniques.

 

[Frame Dragger]

Numerical techniques.

...Which are unable at present time to achieve that goal. I'm still confused, and I'm not pushing an agenda here, but this flies in the face of what I though I knew.

 

[D H]

To achieve what goal? We do send vehicles to other planets, you know. It can take years to get from here to there. New Horizons, for example, launched in January 2006 and won't arrive at Pluto until July 2015. On route it passed close enough to Jupiter to receive a gravity assist to help it get to Pluto. We have to know where those planets will be (not are) to a very high degree of accuracy to plan and execute missions such as these.

 

[Frame Dragger]

To achieve what goal? We do send vehicles to other planets, you know. It can take years to get from here to there. New Horizons, for example, launched in January 2006 and won't arrive at Pluto until July 2015. On route it passed close enough to Jupiter to receive a gravity assist to help it get to Pluto. We have to know where those planets will be (not are) to a very high degree of accuracy to plan and execute missions such as these.

When I am thinking of the n-body problems I am thinking of degenerate orbits of binary BH's or neutron stars as the simple 2-body model. So the goal is: advancing theory.

 

[twofish-quant]

Numerical techniques.

Or infinite series.

Or use non-elementary functions.

 

[twofish-quant]

When I am thinking of the n-body problems I am thinking of degenerate orbits of binary BH's or neutron stars as the simple 2-body model. So the goal is: advancing theory.

If you want to advance mathematical theory you'd like use topological and geometrical techniques in which the inability to calculate exact results from simple functions is an interesting but not terribly important fact.

 

[Frame Dragger]

If you want to advance mathematical theory you'd like use topological and geometrical techniques in which the inability to calculate exact results from simple functions is an interesting but not terribly important fact.

Is that what you would use in research into gravitational waves? That would explain my confusion, and why I'm just plain wrong about the nature of the problem in general.

 

[Ich]

D H,

Frame Dragger is talking abbout numerical relativity. Full fledged GR.

 

[twofish-quant]

Is that what you would use in research into gravitational waves? That would explain my confusion, and why I'm just plain wrong about the nature of the problem in general.

One way of thinking about it is that you have lots of mathematical tools in the toolbox, and if you can't use tool A, you can use tools B, C, D and E. Anything involving GR you are likely to be using numerical techniques to do any realistic calculations.

 

[D H]

Or infinite series.

Not particularly useful in this problem. The wikipedia article on the n-body problem constains a fairly good discussion of Sundman's series. Scienceworld at wolfram.com also discusses this topic. From the scienceworld article, (emphasis mine):

Since such global regularizations are available for this problem, the restricted problem of three bodies can be considered to be complete "solved." However, this "solution" does not address issues of stability, allowed regions of motion, and so on, and so is of limited practical utility (Szebehely 1967, p. 42). Furthermore, an unreasonably large number of terms (of order 10^8,000,000) of Sundman's series are required into attain anything like the accuracy required for astronomical observations.​

Links:
Wiki: http://en.wikipedia.org/wiki/N-body_problem#Sundman.27s_theorem_for_the_3-body_problem
Scienceworld: http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html

Or use non-elementary functions.

Non-elementary functions were widely used prior to the advent of modern computers. Invariant and mean orbital elements are essentially special-purpose non-elementary functions. These techniques were developed to describe a body orbiting the Earth, for example. Even ignoring perturbing factors from the Sun and Moon, a body orbiting the Earth does not obey Kepler's laws because the Earth is not a point mass. Various techniques were developed to analytically describe orbital behavior about the Earth that account for Earth's oblateness to some extent (usually J2 only). These techniques are still used to some extent; the two-line orbital elements issued by NORAD are one example.

twofish-quant, you missed another formerly widely-used technique in your list of alternatives, perturbation theory. The description of the evolution of the Moon's orbit used to be done using perturbation techniques rather than numerical integration. People are much better at using analytic models than they are are performing lots and lots of rote calculations. Numerical integration didn't really become a viable option until the development of modern computers.

 

[Frame Dragger]

Not particularly useful in this problem. The wikipedia article on the n-body problem constains a fairly good discussion of Sundman's series; see http://en.wikipedia.org/wiki/N-body_problem#Sundman.27s_theorem_for_the_3-body_problem. Scienceworld @wolfram.com also discusses this; see http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html From that latter article, (emphasis mine):

Since such global regularizations are available for this problem, the restricted problem of three bodies can be considered to be complete "solved." However, this "solution" does not address issues of stability, allowed regions of motion, and so on, and so is of limited practical utility (Szebehely 1967, p. 42). Furthermore, an unreasonably large number of terms (of order 10^8,000,000) of Sundman's series are required into attain anything like the accuracy required for astronomical observations.​

Non-elementary functions were widely used prior to the advent of modern computers. Invariant and mean orbital elements are essentially special-purpose non-elementary functions. These techniques were developed to describe a body orbiting the Earth, for example. Even ignoring perturbing factors from the Sun and Moon, a body orbiting the Earth does not obey Kepler's laws because the Earth is not a point mass. Various techniques were developed to analytically describe orbital behavior about the Earth that account for Earth's oblateness to some extent (usually J2 only). These techniques are still used to some extent; the two-line orbital elements issued by NORAD are one example.

twofish-quant, you missed another formerly widely-used technique in your list of alternatives, perturbation theory. The description of the evolution of the Moon's orbit used to be done using perturbation techniques rather than numerical integration. People are much better at using analytic models than they are are performing lots and lots of rote calculations. Numerical integration didn't really become a viable option until the development of modern computers.

Ok, now THAT matches what I thought I knew, and what my friend works at in the API.

EDIT... I take that back... damn... Remember again that as Ich said, I'm talking about pure GR, regularized PDE's using Numerical GR (and therefore computers with a looooot of memory).

 

[D H]

You are still going to have to use numerical techniques, Frame Dragger, if you want to do anything beyond a toy problem. That one has to resort to numerical techniques does not mean the problem is insoluble.

 

[Frame Dragger]

You are still going to have to use numerical techniques, Frame Dragger, if you want to do anything beyond a toy problem. That one has to resort to numerical techniques does not mean the problem is insoluble.

That does make sense. Thanks very much for helping me (start) to understand this. I'm going to hit MTW and some others and recognize that I need to learn a loooooot more.

 

[twofish-quant]

twofish-quant, you missed another formerly widely-used technique in your list of alternatives, perturbation theory. The description of the evolution of the Moon's orbit used to be done using perturbation techniques rather than numerical integration.

It's still pretty widely used in actual calculations. The trouble with numerical integration is that it takes forever to do a calculation so what people do in practice for these sorts of things is that they use numerical calculations to figure out the perturbation elements, and then when people run code that involves the location of the planets they do a quick calculation out of the perturbation elements (VSOP87) rather than a long calculation that reruns the numerical integrations.

People are much better at using analytic models than they are are performing lots and lots of rote calculations.

And analytic or semi-analytic models tend to be very, very fast. The thing about computers is that they make possible analytic calculations that would not otherwise be possible. You can even do symbolic analytic calculations.

The problem is that if you are doing widely different problems, the setup time is painful.

 

[D H]

VSOP87

There is a good reason VSOP hasn't been updated since 1987: It's not a very good model. An analytic model is not going to be as good as a numeric model. There are two leading contenders for high-precision planetary ephemerides, the Development Ephemerides (DE series) from JPL and the Ephemerides of the Planets and the Moon (EPM series) from the Russian Institute of Applied Astronomy. Both are based on numerical integration. To avoid the numerical integration, the released data in the DE series are in the form of sets of Chebyshev polynomial coefficients, with each set applying to a particular planet and a particular span of time. Computing the expansion of the resulting Chebychev polynomials is very fast. In a realistic orbital simulation, models of the Earth's rotation, its atmosphere, and non- gravity model will take much more time than computing ephemerides. (The Moon and Mars are even worse because the gravity models for those objects are really nasty.)

 

[twofish-quant]

There are two leading contenders for high-precision planetary ephemerides, the Development Ephemerides (DE series) from JPL and the Ephemerides of the Planets and the Moon (EPM series) from the Russian Institute of Applied Astronomy. Both are based on numerical integration. To avoid the numerical integration, the released data in the DE series are in the form of sets of Chebyshev polynomial coefficients, with each set applying to a particular planet and a particular span of time.

Cool! Do you know if anyone has put together some open source software that uses these algorithms?

One thing that I'm finding, which seems to be the situation in general is that as computers increase in speed, the types of algorithms which are useful changes quite radically.

 

[Frame Dragger]

Cool! Do you know if anyone has put together some open source software that uses these algorithms?

One thing that I'm finding, which seems to be the situation in general is that as computers increase in speed, the types of algorithms which are useful changes quite radically.

Your comment about speed is very true, but these models are often limited more by memory than raw clock speed. (at least, in the case of work at the the API in Germany). Hell, it's like any other backwards-engineering project... brute force goes a long way, but you have to refine that attack. More power then allows for a more effcient use of memory resources, which ARE severely limiting.

 

[Nabeshin]

EDIT... I take that back... damn... Remember again that as Ich said, I'm talking about pure GR, regularized PDE's using Numerical GR (and therefore computers with a looooot of memory).

So... why use GR for the solar system? It's like demanding we obtain equations of motion for all 10^35 particles (or whatever number) in the apollo rocket before we send it to space...

Or of course a 3-body black hole system or something similar? Although even here I'm not sure you'd need full GR, given that the system is likely unstable and one (or all) of the bodies will become ejected. The scales on which you need to do these full GR calculations is quite small compared to any astronomical scale (order of 10-100s of schwarzschild radii, or <~100km!).

Are you just trying to create an incredibly computationally difficult problem with little to no rewards (i.e. just for fun)?

 

[Frame Dragger]

So... why use GR for the solar system? It's like demanding we obtain equations of motion for all 10^35 particles (or whatever number) in the apollo rocket before we send it to space...

Or of course a 3-body black hole system or something similar? Although even here I'm not sure you'd need full GR, given that the system is likely unstable and one (or all) of the bodies will become ejected. The scales on which you need to do these full GR calculations is quite small compared to any astronomical scale (order of 10-100s of schwarzschild radii, or <~100km!).

Are you just trying to create an incredibly computationally difficult problem with little to no rewards (i.e. just for fun)?

Me?! I'm not modeling anything, and wouldn't know where to start if I wanted to! I'm referring to a friend who works on the 2-body problem re: predicting gravitational waves. I was explaining why I misunderstood that n-body problems are not insoluble. From my understanding, using Newtonian approximations with GR corrections is far more efficient for the solar system, but again, I'm not in a position to know. As I said, this is a friend of mine in Germany, not me, here. Sorry if I was in any way misleading... I thought I was pretty clear.

EDIT: That said, it DOES sound fun to compute down to the Planck Scale... fun, but a terrible waste as you say. I won't deny that it would be amazing to see it however...

 

[D H]

Cool! Do you know if anyone has put together some open source software that uses these algorithms?

http://www.ephemeris.com/software.html
http://www.projectpluto.com/jpl_eph.htm

Fortran source code + data + documentation is at
ftp://ssd.jpl.nasa.gov

 

[Nabeshin]

Me?! I'm not modeling anything, and wouldn't know where to start if I wanted to! I'm referring to a friend who works on the 2-body problem re: predicting gravitational waves. I was explaining why I misunderstood that n-body problems are not insoluble. From my understanding, using Newtonian approximations with GR corrections is far more efficient for the solar system, but again, I'm not in a position to know. As I said, this is a friend of mine in Germany, not me, here. Sorry if I was in any way misleading... I thought I was pretty clear.

EDIT: That said, it DOES sound fun to compute down to the Planck Scale... fun, but a terrible waste as you say. I won't deny that it would be amazing to see it however...

Ahh, ok I don't know why I thought you were doing something. I see your initial post now.

Yeah, we have enough trouble with 2-body problems in numerical GR as it is that to imagine going to three is quite hopeless atm. I mean, simulations already take months running on super computers and generate endless terabytes of data. At least we know how to extract waveforms from these cases by this point!

Again, though, 3-body is an academic exercise at best. I doubt we would expect anywhere within the universe past, present, or future, a 3-body black hole or neutron star system for which these calculations would be necessary.

 

[D H]

So... why use GR for the solar system?

JPL and the Russian Institute of Applied Astronomy do use GR, or rather a post-Newtonian approximation. They model general relativity as a perturbative acceleration on top of that predicted by Newtonian gravity. They are also very careful regarding dynamic time, the time scale in which the equations of motion are propagated. JPL uses its own home-brewed T_eph; they apparently do not like Barycentric Coordinate Time.

 

[Frame Dragger]

Ahh, ok I don't know why I thought you were doing something. I see your initial post now.

Yeah, we have enough trouble with 2-body problems in numerical GR as it is that to imagine going to three is quite hopeless atm. I mean, simulations already take months running on super computers and generate endless terabytes of data. At least we know how to extract waveforms from these cases by this point!

Again, though, 3-body is an academic exercise at best. I doubt we would expect anywhere within the universe past, present, or future, a 3-body black hole or neutron star system for which these calculations would be necessary.

I agree 100%. I was making roughly that point, except that I misunderstood the difference between "theoretically impossible" and "deeply impractical"... which is not a small mistake on my part.

 

[twofish-quant]

Your comment about speed is very true, but these models are often limited more by memory than raw clock speed. (at least, in the case of work at the the API in Germany).

Depends on the calculation. PDE's are very heavily limited by memory. ODE's are not. In any case the cost of memory has gone down as quickly as the cost of CPU.

Hell, it's like any other backwards-engineering project... brute force goes a long way, but you have to refine that attack.

Or you can wait for better computers to come out. Also putting together brute force is sometimes non-trivial.