This is covered extensively in every classical mechanics textbook I’ve ever seen, so I’m not going to try to reproduce or explain the math itself (if you want to know that, it’s not hard to find), but I will provide a quick summary.
The “three-body problem” refers to the fact that there is no “analytic” solution to the equations of motion for 3 objects interacting gravitationally under Newtonian dynamics. An “analytic solution” is a single closed-form equation that you can enter initial conditions into and out pops the future dynamical state information at all future times. So if you know exactly the 3-dimensional location coordinates and the 3-dimensional velocities, and of course all the forces acting on the objects, you can immediately say what those values will be for any given time in the future. For 2 objects, that gives you 12 variables – but you simplify this by taking advantage of conserved quantities like energy and angular momentum and end up solving for a single variable. For 3 objects with 18 variables, though, there aren’t enough constraints to work the same trick. Newton solved the “two-body problem” in 1687, but adding a third mutually gravitating body leads to a system of differential equations that has no analytic solution. Various famous mathematicians have described various aspects of this problem and pointed to classes of solutions through the 18th and 19th centuries, notably Euler, Hamilton, Lagrange, and Poincare, but others as well. Newton wrote:
>[an exact solution] exceeds, if I am not mistaken, the force of any human mind.
So does that mean we have no idea how a three-body system will behave? No! We still have a set of differential equations that you can use to solve the system numerically. In general, numerical solutions work through the idea that if you know the initial conditions and the forces acting on the system, then you increment the time just a little bit into the future to predict the conditions at that next small step. Then you increment it a little more and recalculate. The smaller the steps in time, the more accurate you will be but the more computationally expensive it is. People studying galaxy motions run these sorts of simulations for hundreds of thousands of objects. There are computational tricks that exist that I am not an expert in that make the simulations easier than the purely brute-force method I’ve described.
There is another issue though which is that small changes in initial conditions can lead to wildly different outcomes, which is what is meant by “chaos”. So a numerical simulation will eventually not accurately tell you the future state of every object in the system. And even if you compute it perfectly, small measurement error in the input leads to the same results. Poincare wrote:
>It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible…
Practically speaking, though, some configurations of stars are stable over timescales like “the current age of the universe”. If you have a tight inner binary star system and a third object orbiting that binary from far away, the third object will more or less see that inner system as a single object and can orbit it with a minimum of fuss. There can still be noticeable interactions in this setup, like the Kozai mechanism, but the system will persist for billions of years. If a three-body system is not in one of these mostly stable configurations, it can quickly disrupt and permanently kick out one of the three objects, leaving a tight binary behind. If this is going to happen, it tends to happen relatively early on (where “relative” again means relative to the lifetime of the stars or the age of the universe) because it only needs to happen once. We have even seen star systems with 5 members that has persisted until the death of one of the stars. The stable configurations are stable enough that we see many triple systems in the universe today, even billions of years after the stars form. Alpha Centauri is an example, with a tighter inner binary and a wide third object (Proxima Centauri) that is more or less stable and has persisted for billions of years.
It’s worth noting that our own solar system is a “multiple body problem”, with Jupiter exerting significant gravitational influence on the rest of the solar system. Earth’s orbit is, however, stable. Comets can have their orbits altered since they change at least a little bit every time they pass close to the Sun and lose material (or are destroyed altogether). Most of the asteroids that will hit something already have, but there are enough of them that “most” is not “all”. Most of that was settled out one way or the other during the early age of bombardment over the solar system’s first few hundred million years of existence, which included the impact that formed the Moon and presumably ones that knocked Venus upside down and Uranus on its side. By now, things have settled down and we have a fairly stable arrangement of asteroids and planets. The rate of impacts now is much, much lower. As a general rule of thumb, any system that’s billions of years old is pretty stable on human scales, or even evolutionary ones, by definition.
Comments
This is covered extensively in every classical mechanics textbook I’ve ever seen, so I’m not going to try to reproduce or explain the math itself (if you want to know that, it’s not hard to find), but I will provide a quick summary.
The “three-body problem” refers to the fact that there is no “analytic” solution to the equations of motion for 3 objects interacting gravitationally under Newtonian dynamics. An “analytic solution” is a single closed-form equation that you can enter initial conditions into and out pops the future dynamical state information at all future times. So if you know exactly the 3-dimensional location coordinates and the 3-dimensional velocities, and of course all the forces acting on the objects, you can immediately say what those values will be for any given time in the future. For 2 objects, that gives you 12 variables – but you simplify this by taking advantage of conserved quantities like energy and angular momentum and end up solving for a single variable. For 3 objects with 18 variables, though, there aren’t enough constraints to work the same trick. Newton solved the “two-body problem” in 1687, but adding a third mutually gravitating body leads to a system of differential equations that has no analytic solution. Various famous mathematicians have described various aspects of this problem and pointed to classes of solutions through the 18th and 19th centuries, notably Euler, Hamilton, Lagrange, and Poincare, but others as well. Newton wrote:
>[an exact solution] exceeds, if I am not mistaken, the force of any human mind.
So does that mean we have no idea how a three-body system will behave? No! We still have a set of differential equations that you can use to solve the system numerically. In general, numerical solutions work through the idea that if you know the initial conditions and the forces acting on the system, then you increment the time just a little bit into the future to predict the conditions at that next small step. Then you increment it a little more and recalculate. The smaller the steps in time, the more accurate you will be but the more computationally expensive it is. People studying galaxy motions run these sorts of simulations for hundreds of thousands of objects. There are computational tricks that exist that I am not an expert in that make the simulations easier than the purely brute-force method I’ve described.
There is another issue though which is that small changes in initial conditions can lead to wildly different outcomes, which is what is meant by “chaos”. So a numerical simulation will eventually not accurately tell you the future state of every object in the system. And even if you compute it perfectly, small measurement error in the input leads to the same results. Poincare wrote:
>It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible…
Practically speaking, though, some configurations of stars are stable over timescales like “the current age of the universe”. If you have a tight inner binary star system and a third object orbiting that binary from far away, the third object will more or less see that inner system as a single object and can orbit it with a minimum of fuss. There can still be noticeable interactions in this setup, like the Kozai mechanism, but the system will persist for billions of years. If a three-body system is not in one of these mostly stable configurations, it can quickly disrupt and permanently kick out one of the three objects, leaving a tight binary behind. If this is going to happen, it tends to happen relatively early on (where “relative” again means relative to the lifetime of the stars or the age of the universe) because it only needs to happen once. We have even seen star systems with 5 members that has persisted until the death of one of the stars. The stable configurations are stable enough that we see many triple systems in the universe today, even billions of years after the stars form. Alpha Centauri is an example, with a tighter inner binary and a wide third object (Proxima Centauri) that is more or less stable and has persisted for billions of years.
It’s worth noting that our own solar system is a “multiple body problem”, with Jupiter exerting significant gravitational influence on the rest of the solar system. Earth’s orbit is, however, stable. Comets can have their orbits altered since they change at least a little bit every time they pass close to the Sun and lose material (or are destroyed altogether). Most of the asteroids that will hit something already have, but there are enough of them that “most” is not “all”. Most of that was settled out one way or the other during the early age of bombardment over the solar system’s first few hundred million years of existence, which included the impact that formed the Moon and presumably ones that knocked Venus upside down and Uranus on its side. By now, things have settled down and we have a fairly stable arrangement of asteroids and planets. The rate of impacts now is much, much lower. As a general rule of thumb, any system that’s billions of years old is pretty stable on human scales, or even evolutionary ones, by definition.