three-body problem the most important unsolved problem in mathematics? a special case figure 8...

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