A chaotic orbit of a satellite around two planets question

[abejackson]

Homework Statement

If there's a satellite(not man-made although I wonder it would matter) is chaotically orbiting around two planets, will it ever cross the same orbital path twice or not?

Homework Equations

The Attempt at a Solution

I think if the system is chaotic, it will not ever cross the same orbital path but I really want to know why.

 

[gneill]

If you take the system as a whole (both planets, their primary, and all other perturbing influences), if the system returns to precisely the same conditions once, it will do so again and again on a regular basis, and the trajectory will be periodic and not chaotic.

This presumes that the system is not evolving in some way that could drive the satellite into repeating its trajectory a fixed number of times when particular resonances occur (suppose one of the bodies was shedding mass and resonances occur for specific ratios of masses in the system).

 

[D H]

abejackson is not asking whether the system will return to the same state. He is asking whether it will "cross the same orbital path twice or not?"

What if the motion is planar?

 

[gneill]

I'm not sure that I didn't interpret his intended meaning correctly Perhaps he will clarify and tell me I'm all wet.

 

[femmefatale8]

abejackson is not asking whether the system will return to the same state. He is asking whether it will "cross the same orbital path twice or not?"

What if the motion is planar?

I think what he's getting at is that if the orbit returns to it's precise initial conditions, then it will follow the exact path. Any minuscule change in any of the initial conditions (e.g., mass, velocity) will cause the orbit to continue on a different path.

Theoretically it may be possible to cross the same path, but I'm not sure it can be proved

 

[D H]

I know what gneill was talking about. The OP asked about the trajectory crossing itself, which to me means reaching the same point but with different velocities.

This certainly can happen if the motion is planar, and that the planar three body problem is chaotic is well-known.

 

[abejackson]

I guess this is a three-body problem, then? I think that the satellite would never cross the same orbital path but how would I explain this is someone asks me why?

 

[D H]

What do you mean by "crossing"? If you mean something akin to the intersection of two roads, the answer is yes. If you mean something akin to taking an on-ramp to a freeway, the answer is no.

 

[abejackson]

oh my mistake.
I meant the satellite will never be on the same orbital path twice.