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Does the 3-body problem have stable solutions?

  1. Aug 12, 2005 #1
    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.
     
  2. jcsd
  3. Aug 14, 2005 #2

    pervect

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    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).
     
  4. Aug 16, 2005 #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.
     
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