THREE-BODY PROBLEM

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					THREE-BODY PROBLEM
The gravitational three-body problem has been called the
oldest unsolved problem in mathematical physics.
      The most important unsolved problem in mathematics?
                  A special case figure 8 orbit:
         http://www.santafe.edu/~more/figure8-3.loop.gif
           http://www.santafe.edu/~more/rot8x.loop.gif
           http://www.santafe.edu/~moore/gallery.html
         Isaac Newton
Principia 1687
              Perturbations
• Is the orbit of the Earth stable?
                                      Anders
• Orbits of comets                    Lexell
    Alexis Clairaut
.
.   Albert Einstein
  Einstein’s General Relativity
• Curvature of spacetime
Map projections
 Post Newtonian approximation
• .
2-dimensional example
• .
OJ287 light variations
                OJ 287
A Binary Black Hole System
Sillanpää et al. 1988, Lehto & Valtonen1996,
Sundelius et al. 1997
   Black hole – Accretion disk
            collision
• Ivanov et al. 1998
New outbursts: Tuorla
     monitoring
Solution of the timing problem.
             Level II
      Post Newtonian terms
• .
1. order Post Newtonian term
2. Order Post Newtonian term
Radiation term
Spin – orbit term
Quadrupole term
Parameters
               Conclusion
• The no-hair theorem is confirmed
• Black holes are real
• General Relativity is the correct theory of
  gravitation
 Pierre-Simon, Marquis de Laplace
• Proof of stability of the solar system, 1787
• Lagrange 1781
             Leonhard Euler
• 1760: Restricted problem
• 1748 & 1772:
Prize of Paris
Academy of Sci.
     Joseph-Louis Lagrange
• Lagrangian points 1772
• Prize of 1764, 1772
Carl Gustav Jacobi
   Johann Peter Gustav Lejeune
            Dirichlet
• Solution of the three-body problem?
            Henri Poincare
Deterministic chaos, Prize of King Oscar of
  Sweden 1889
Stability in question
            Karl Sundman
• A converging series solution of the three-
  body problem 1912
             Carl Burrau
• Ernst Meissel and the Pythagorean
  problem 1893, Burrau 1913
       Burrau’s solution of the
        Pythagorean problem
• First close encounters
Numerical integration by computer
• Interplay: Exchange of pairs
  Final stages of the Pythagorean
            triple system
• Ejection loops
 Victor Szebehely and the solution
of the Burrau’s three-body problem
• Escape
Cambridge 1971-1974
        Three-Body Group
• Aarseth   Saslaw   Heggie
25000 three-body orbits
Escape cone
Density of escape states
• Monaghan’s calculation corrected
       Barbados 2000-2001
• Re-evaluation of Monaghan’s conjecture
Heggie: Detailed balance
  UWI St. Augustine 2001-2006
• Stability limit

             0.5

             0.4


             0.3

             0.2


             0.1


 -1   -0.5         0.5   1
   Stability of triple systems


        M. Valtonen, A. Mylläri
     University of Turku, Finland
         V. Orlov, A. Rubinov
St. Petersburg State University, Russia
                 Idea of new criterion

               Perturbing acceleration
        from the third body to the inner binary



      Change of semi-major axis of inner binary



                          where
mB is the mass of inner binary and n is the mean motion.
       Integrate over full cycle of the inner orbit:
           Idea of new criterion

The final formula for stability criterion for
    comparable masses (triple stars):
               Testing of new criterion

The stability region for equal-mass three-body problem
    and zero initial eccentricities of both binaries.
                Here ζ = cos i, η = ain/aex.
            Testing of new criterion

The stability region for unequal-mass three-body
 problem (mass ratio is 1:1:10) and zero initial
         eccentricities of both binaries.
            Here ζ = cos i, η = ain/aex.
                Testing of new criterion

The stability region for equal-mass three-body problem
and non-zero initial eccentricity of outer binary (e=0.5).
                Here ζ = cos i, η = ain/aex.
                Testing of new criterion

The stability region for equal-mass three-body problem
and non-zero initial eccentricity of outer binary (e=0.9).
                Here ζ = cos i, η = ain/aex.
             Testing of new criterion

 The stability region for unequal-mass three-body
problem (mass ratio is 1:1:0.1) and non-zero initial
       eccentricity of outer binary (e=0.9).
             Here ζ = cos i, η = ain/aex.
             Testing of new criterion

 The stability region for unequal-mass three-body
problem (mass ratio is 1:1:10) and non-zero initial
       eccentricity of outer binary (e=0.9).
             Here ζ = cos i, η = ain/aex.
                Conclusions
1. The new stability criterion was suggested for
hierarchical three-body systems. It is based on
the theory of perturbations and random walking
of the orbital elements of outer and inner
                    binaries.
2. The numerical simulations have shown that a
criterion is working very well in rather wide
range of mass ratios (two orders at least).
   Long-time orbit integrations
• Jacques Laskar 1989, 150,000 terms,
  200M yr
• Chaotic but confined ?
         Climate cycles
Milankovitch 1912
Adhemar 1842
Croll 1864
Three-body chaos
            Arrow of Time
• Albert Einstein & Arthur Eddington




             Eddington was the first to coin the phrase "time arrow"
     Different Arrows of time?
• According to Roger Penrose, we now
  have up to seven perceivable arrows of
  time, all asymmetrical, and all pointing
  from past to future.
     BOLTZMANN'S ENTROPY
       AND TIME'S ARROW
• Given that microscopic physical laws are
  reversible, why do all macroscopic events
  have a preferred time direction?
• S = k log W
           Demonstration
• Reversing arrow of time by making
  entropy decrease
           James Clerk Maxwell

• Maxwell's demon

 Information Entropy
  Claude Elwood Shannon
              Common view
• …chaotic behavior …, which can be observed
  already in systems consisting of only a few
  particles, will not have a unidirectional time
  behavior in any particular realization. Thus if we
  had only a few hard spheres in a box, we would
  get plenty of chaotic dynamics and very good
  ergodic behavior, but we could not tell the time
  order of any sequence of snapshots.
J. L. Lebowitz,
          38 PHYSICS TODAY SEPTEMBER 1993
     Orbits are not reversible
3-body scattering
   Kolmogorov - Sinai Entropy
• olmo

                    Andrey Kolmogorov
          Problem solved?
• Time goes forward in the direction of
  increasing entropy
• In macroscopic systems, the entropy is
  Boltzmann entropy + von Neumann
  entropy
• In microscopic systems, it is Kolmogorov –
  Sinai entropy

				
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posted:4/30/2013
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