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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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.
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.