1. 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.

2. Isaac Newton Principia 1687

3. Perturbations Is the orbit of the Earth stable? Orbits of comets Anders Lexell

4. . Alexis Clairaut

5. . Albert Einstein

6. Einstein’s General Relativity Curvature of spacetime

7. Map projections

8. Post Newtonian approximation.

9. 2-dimensional example

10. .

12. OJ287 light variations

13. OJ 287 A Binary Black Hole System Sillanpää et al. 1988, Lehto & Valtonen1996, Sundelius et al. 1997

15. Black hole – Accretion disk collision Ivanov et al. 1998

17. New outbursts: Tuorla monitoring

19. Solution of the timing problem. Level II

20. Post Newtonian terms.

21. 1. order Post Newtonian term

22. 2. Order Post Newtonian term

23. Radiation term

24. Spin – orbit term

25. Quadrupole term

26. Parameters

27. Conclusion The no-hair theorem is confirmed Black holes are real General Relativity is the correct theory of gravitation

28. Pierre-Simon, Marquis de Laplace Proof of stability of the solar system, 1787 Lagrange 1781

29. Leonhard Euler 1760: Restricted problem 1748 & 1772: Prize of Paris Academy of Sci.

30. Joseph-Louis Lagrange Lagrangian points 1772 Prize of 1764, 1772

31. Carl Gustav Jacobi

32. Johann Peter Gustav Lejeune Dirichlet Solution of the three-body problem?

33. Henri Poincare Deterministic chaos, Prize of King Oscar of Sweden 1889 Stability in question

34. Karl Sundman A converging series solution of the three- body problem 1912

35. Carl Burrau Ernst Meissel and the Pythagorean problem 1893, Burrau 1913Ernst Meissel and the Pythagorean problem

36. Burrau’s solution of the Pythagorean problem First close encounters

37. Numerical integration by computer Interplay: Exchange of pairs

38. Final stages of the Pythagorean triple system Ejection loops

39. Victor Szebehely and the solution of the Burrau’s three-body problem Escape

40. Cambridge 1971-1974

41. Three-Body Group Aarseth Saslaw Heggie

42. 25000 three-body orbits

45. Escape cone

46. Density of escape states

49. Monaghan’s calculation corrected

51. Barbados 2000-2001 Re-evaluation of Monaghan’s conjecture

53. Heggie: Detailed balance

55. UWI St. Augustine 2001-2006 Stability limit

56. Stability of triple systems M. Valtonen, A. Mylläri University of Turku, Finland V. Orlov, A. Rubinov St. Petersburg State University, Russia

57. Idea of new criterion Perturbing acceleration from the third body to the inner binary Change of semi-major axis of inner binary where m B is the mass of inner binary and n is the mean motion. Integrate over full cycle of the inner orbit:

58. Idea of new criterion The final formula for stability criterion for comparable masses (triple stars):

59. 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 = a in /a ex.

60. 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 = a in /a ex.

61. 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 = a in /a ex.

62. 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 = a in /a ex.

63. 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 = a in /a ex.

64. 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 = a in /a ex.

65. 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).

66. Long-time orbit integrations Jacques Laskar 1989, 150,000 terms, 200M yr Chaotic but confined ?

67. Climate cycles Milankovitch 1912 Adhemar 1842 Croll 1864

68. Three-body chaos

70. Arrow of Time Albert Einstein & Arthur Eddington Eddington was the first to coin the phrase "time arrow"

72. 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.

73. 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

74. Demonstration Reversing arrow of time by making entropy decrease

75. James Clerk Maxwell Maxwell's demon Information Entropy Claude Elwood Shannon

76. 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

77. Orbits are not reversible 3-body scattering

78. Kolmogorov - Sinai Entropy olmo Andrey Kolmogorov Yakov Sinai

79. 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