Does the 3-body problem have stable solutions?

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In summary, the conversation discusses the stability of three bodies with non-negligible mass and whether there are any stable configurations. It also mentions a calculation and a mathematical equation that can determine stability, as well as the possibility of a planet orbiting one of the stars within its "hill sphere."
  • #1
kjknohw
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Given 3- bodies of non-negleble mass, are there stable configurations or not.


for example,
Will a planet (with non-zero mass) orbiting 1000 au away from two stars 0.1 au apart eventually leave the system if you give it 10^9999999... years; or is the system stable for a truly infinete time?
assume: Newtonian physics with point masses. (no gravitational waves no time dialation, etc.)

Note: this is a more of a mathmatical question than physical.

If no stable system exists, is there any way to estimate the time it will take for the system to destabalize.
 
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  • #2
kjknohw said:
Given 3- bodies of non-negleble mass, are there stable configurations or not.

Yes.


Take a look at

http://groups-beta.google.com/group/rec.arts.sf.science/msg/05738a5d682962cc

and

http://groups-beta.google.com/group/rec.arts.sf.science/msg/1fd8079db42c1137

Three bodies orbiting around their common center of mass in an equilateral triangle will be stable if

(m[1]+m[2]+m[3])^2 -27*(m[1]*m[3]+m[3]*m[2]+m[1]*m[2]) >= 0

Sources

Volume 5 of of "What's Happening in the Mathematical Sciences" by Barry Cipra.

and a personal (rather messy) computerized calculation (see the second link). Basically you start with Hamilton's equations, you linearize them, and you wind up with a 12x12 eigenvalue problem. If all of the eigenvalues have negative real parts, the (linearized) system is stable.

Home computers with the right software are good enough nowadays to solve this symbolically. (It helps a lot that the matrix is very sparse).
 
  • #3
one way to put it is also to make 2 of the stars orbit around their comon center and then let those 2 orbig with the third star around the comon center, this solution is the one nature prefer cause the chance of 3 stars bieng created at the right spot with the right distance with the right mass are slim. if u have a planet then it will have to orbit in the hill sphere of one of the stars. The most stable one i can think of is to orbit the third and "lonly" star. its hills sphere will probebly be bigger.
hill sphere radius
r=a(m/(3M))^(1/3)
the hill sphere radius (r) for a body with mass (m) orbiting a heavier body with mass M at a distance of a is approximently to that.
 

1. What is the 3-body problem?

The 3-body problem is a classical mechanics problem that involves predicting the motion of three bodies, such as planets or stars, that are affected by each other's gravitational pull.

2. Why is the 3-body problem difficult to solve?

The 3-body problem is difficult to solve because it involves solving a set of nonlinear differential equations, which do not have analytical solutions. This means that the motion of the three bodies cannot be predicted using simple equations and requires complex mathematical calculations.

3. What are stable solutions in the 3-body problem?

Stable solutions in the 3-body problem refer to a set of initial conditions where the three bodies are able to maintain a stable orbit around each other without any significant changes in their positions or velocities over time.

4. Can the 3-body problem have stable solutions?

Yes, the 3-body problem can have stable solutions, but they are rare and require very specific initial conditions. Most solutions in the 3-body problem are chaotic, meaning that small changes in the initial conditions can lead to drastically different outcomes.

5. Why is the 3-body problem important in science?

The 3-body problem is important in science because it has implications in many fields such as astronomy, physics, and engineering. It helps us understand the complex interactions between multiple bodies in our universe and has practical applications in predicting the trajectories of spacecraft and satellites.

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