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.
Isaac Newton
Principia 1687
Perturbations
• Is the orbit of the Earth stable?
• Orbits of cometsAnders 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 SystemSillanpää 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
-1 -0.5 0.5 1
0.1
0.2
0.3
0.4
0.5
Stability of triple systems
M. Valtonen, A. MylläriUniversity of Turku, Finland
V. Orlov, A. Rubinov St. Petersburg State University, Russia
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:
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 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
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 reversible3-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