# Will a body ever escape in a 3 body problem?

1. Feb 21, 2015

### dynawics

I am asking this question because I found it in a lecture by Stephen Wolfram which you can see at this link at about 54:10:

Apparently the question cannot be answered with current methods of physics- Wolfram describes it as "undecidable". But what is the question? When he says "escape" does he mean escape velocity? Does he mean that it is impossible to determine the escape velocity of a body in a three body problem? Or is it something else?

Thanks.

2. Feb 21, 2015

### jbriggs444

Google for three body problem, e.g. http://en.wikipedia.org/wiki/Three-body_problem.

There is no way to write down a formula for the future position of the three bodies in the general case. That's been known since 1887. One can write programs to simulate the motion and predict the positions, but as one tries to predict farther and farther into the future, the required accuracy becomes larger and larger. No matter how good the hardware, there will be a future time beyond which that hardware cannot make usefully accurate predictions. That is assuming one knows the starting positions, masses and velocities with perfect accuracy.

The orbital patterns have a sensitive dependence on initial conditions. In order to make a prediction, one has to known the initial data precisely. The farther futureward one tries to predict, the more accurate the input data must be. Eventually quantum mechanical effects intrude and sufficiently accurate measurements are impossible.

It is possible without violating conservation of momentum or energy or any of the rules of classical Newtonian physics for the three bodies to orbit around each other in such a way that one of the three is eventually ejected -- escaping toward infinity. What Wolfram is likely saying is that there is no general way to predict when (or if) this will happen.

My understanding is that, barring collisions and certain known special cases, such an escape it is almost certain. But I could be wrong on that.

3. Feb 21, 2015

### dynawics

How then is it possible to know that it is possible for a body in such a system to escape if we can never prove when or if it does so? Are there special cases in which we can prove that one body escapes and thus affirm this possibility- but then in other cases we simply lack the methods to prove if or when this possibility will occur?

4. Feb 22, 2015

### jbriggs444

We can run simulations and see ejection events where one object escapes to infinity. My understanding is that no special setup is required. We can run simulations and see that capture events occur. Since the laws of physics are time-reversible, those also demonstrate that ejections are possible.

5. Apr 16, 2016

### Ezequiel Meza

There are many properties of an initial condition that allow us know what will happen with certain approximation before running a simulation: Virial Theorem!

Initial conditions means, mases, positions and velocities. From this we obtain initial Potential Energy, Kinetic Energy, Angular Impetu. When there is a lot of Kinetic Energy chances are that ejection will occur because gravity will not be enough to retain the bodies in orbit. When velocities are too small then Kinetic Energy is not enough to prevent gravity to cause a collision between two bodies. Remember that velocity is a vector that means that its direction also determines the output, for example, no matter how high speed two bodies have if they travel one toward the other they will collide. Also, there is a minimum speed at which if the three bodies are directed in opposite direction of the center of mass they will escape even their speed gradually decreases reaching a zero speed at infinitum. If with this minimum speed they are directed in other direction one collision or ejection would occur.

6. Apr 16, 2016

### sophiecentaur

Dang me. I was just going to suggest that! Grrrr.