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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