three-body problem the most important unsolved problem in mathematics? a special case figure 8...
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THREE-BODY PROBLEM
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
The gravitational three-body problem has been called the oldest unsolved problem in mathematical physics.
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Isaac Newton
Principia 1687
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Perturbations
• Is the orbit of the Earth stable?
• Orbits of cometsAnders Lexell
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.Alexis Clairaut
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. Albert Einstein
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Einstein’s General Relativity
• Curvature of spacetime
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Map projections
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Post Newtonian approximation
• .
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2-dimensional example
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• .
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OJ287 light variations
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OJ 287
A Binary Black Hole SystemSillanpää et al. 1988, Lehto & Valtonen1996,
Sundelius et al. 1997
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Black hole – Accretion disk collision
• Ivanov et al. 1998
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New outbursts: Tuorla monitoring
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Solution of the timing problem. Level II
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Post Newtonian terms
• .
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1. order Post Newtonian term
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2. Order Post Newtonian term
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Radiation term
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Spin – orbit term
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Quadrupole term
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Parameters
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Conclusion
• The no-hair theorem is confirmed• Black holes are real• General Relativity is the correct theory of
gravitation
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Pierre-Simon, Marquis de Laplace
• Proof of stability of the solar system, 1787
• Lagrange 1781
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Leonhard Euler
• 1760: Restricted problem
• 1748 & 1772:
Prize of Paris
Academy of Sci.
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Joseph-Louis Lagrange
• Lagrangian points 1772
• Prize of 1764, 1772
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Carl Gustav Jacobi
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Johann Peter Gustav Lejeune Dirichlet
• Solution of the three-body problem?
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Henri Poincare
Deterministic chaos, Prize of King Oscar of Sweden 1889
Stability in question
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Karl Sundman
• A converging series solution of the three-body problem 1912
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Carl Burrau
• Ernst Meissel and the Pythagorean problem 1893, Burrau 1913
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Burrau’s solution of the Pythagorean problem
• First close encounters
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Numerical integration by computer
• Interplay: Exchange of pairs
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Final stages of the Pythagorean triple system
• Ejection loops
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Victor Szebehely and the solution of the Burrau’s three-body problem• Escape
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Cambridge 1971-1974
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Three-Body Group
• Aarseth Saslaw Heggie
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25000 three-body orbits
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Escape cone
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Density of escape states
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• Monaghan’s calculation corrected
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Barbados 2000-2001
• Re-evaluation of Monaghan’s conjecture
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Heggie: Detailed balance
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UWI St. Augustine 2001-2006
• Stability limit
-1 -0.5 0.5 1
0.1
0.2
0.3
0.4
0.5
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Stability of triple systems
M. Valtonen, A. MylläriUniversity of Turku, Finland
V. Orlov, A. Rubinov St. Petersburg State University, Russia
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Idea of new criterion
Perturbing accelerationfrom 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:
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Idea of new criterion
The final formula for stability criterion for comparable masses (triple stars):
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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Long-time orbit integrations
• Jacques Laskar 1989, 150,000 terms, 200M yr
• Chaotic but confined ?
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Climate cycles
Milankovitch 1912
Adhemar 1842
Croll 1864
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Three-body chaos
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Arrow of Time
• Albert Einstein & Arthur Eddington
Eddington was the first to coin the phrase "time arrow"
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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.
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BOLTZMANN'S ENTROPYAND TIME'S ARROW
• Given that microscopic physical laws are reversible, why do all macroscopic events have a preferred time direction?
• S = k log W
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Demonstration
• Reversing arrow of time by making entropy decrease
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James Clerk Maxwell
• Maxwell's demon
Information Entropy
Claude Elwood Shannon
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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
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Orbits are not reversible3-body scattering
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Kolmogorov - Sinai Entropy
• olmo
Andrey Kolmogorov
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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