Simulations of The Three-Body Problem:Finding Chaos in the Cosmos

Craig Huber and Leah ZimmermanSan Francisco State UniversityMathematics 490Professor Goetz

Early Studies of Planetary Motion Ptolemy (2nd century C.E.) of the Early Greeks

developed a geocentric scheme for the solar system. Earth remains stationary in the center while the other

planets rotate around it. Had to use epicycles (planets move around a circle

whose center moves around another circle centered on the earth) to explain their motion.

Nicolaus Copernicus (1473-1543) proposed a heliocentric (Sun-centered) scheme.

Earth

Sun

Johannes Kepler (1571-1630) interpreted Tycho Brahe’s (1546-1601) accurate observational data. He came up with three laws: All planets move in elliptical orbits having the Sun at

one focus. A line joining any planet to the Sun sweeps out equal

areas in equal times. The square of the period of any planet about the Sun

is proportional to the cube of the planet’s mean distance from the Sun.

Kepler’s laws are empirical!

Early Studies of Planetary Motion

Later, Sir Isaac Newton derived Kepler’s Laws from his Law of Universal Gravitation. Unified the previously separate terrestrial and

celestial mechanics. A particle is attracted to any other with a force

directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Early Studies of Planetary Motion

F -F221

rmmGF

The Three-Body Problem The N-body problem in n dimensions has

(n2+3n+2)/2 classical integrals of motion: n from the total momentum, n from the position of the center of mass, and n(n-1)/2 from the total angular momentum, and finally, 1 from the energy.

Because 2nN-1 integrals of motion were necessary to integrate the N-body problem, there are in general not enough classical integrals if N>2. Already the Newtonian 3-body problem in n=2 contains all the complexity a dynamical system can have.

Solving the Three-Body Problem Instead of solving the three-body problem

analytically, we wrote a software program to simulate three masses under mutual gravitational attraction.

Assumption 1: Each mass moves linearly in a small amount of time (~ 0.01 year for planets in our solar system).

Assumption 2: The three masses are restricted to move in a coplanar fashion (Lagrange).

Pseudocode

221

221

212

2112 )()( yyxx

mGmrmGmF

The attractive force between two masses in orbit is proportional to the both the masses and inversely proportional to the square of the distance between them.

Where G is Newton’s gravitational constant:

2

31110726.6

skgmG

And the acceleration of mass 1 due to mass 2 is found directly

2212

21

221 yyxx

Gma

PseudocodeOnce the accelerations are broken down into their x and y components, the velocities and positions can be found by recursive elementary physics calculations.

The velocity of mass 1 in the x-direction is

The position of mass 1 in the x-direction, where xo is the position of mass 1 one iteration prior.

tavv xox 11

2111 2

1 tatvxx xxo

A Real World Example: The Earth, Moon, and SunFig. 1(a) The Moon (red) orbiting the

Earth (green) which is orbiting the Sun (blue), in the x-y plane.

(b) x-x’ phase plot.(c) y-y’ phase plot.

(a)

(b) (c)

A Real World Example: The Earth, Venus, and SunFig. 2(a) Venus (red) and the Earth

(green) orbiting the Sun (blue), in the x-y plane.

(b) x-x’ phase plot.(c) y-y’ phase plot.

(a)

(b) (c)

Question: does the presence of other planets in our solar system perturb the motion of the earth?Answer: Yes!Especially the more massive planets,

like Jupiter, affect the orbit of the earth in small ways.

EarthSun Jupiter

Perturbations in the Earth’s orbit due to the presence of Jupiter.

Fig. 2

Earth’s orbit with Jupiter present (T=1 yr, ∆t=.01 yr).

Fig. 3

Earth’s orbit without Jupiter present (T=1 yr, ∆t=.01 yr).

Question: What would happen if Jupiter were a little closer to the earth?Answer:As Jupiter’s initial radius from the Sun

approaches Earth’s radius from the Sun, the Earth’s orbit becomes a toroid.

At one point, if Jupiter gets too close to the Earth, their orbits become unstable.

Chaos due to the presence of Jupiter (with Jupiter closer)

Fig. 3

Jupiter at an initial distance of 1.1 AU (T=11 yrs, ∆t=.01 yr).

Fig. 4

Jupiter at an initial distance of 1.105 AU (T=11 yrs, ∆t=.01 yr).

http://www.physics.sfsu.edu/~lzimmer/chaos/EJS1.avi http://www.physics.sfsu.edu/~lzimmer/chaos/EJS2.avi

Chaos due to the presence of Jupiter (MJ is greater)

Fig. 3

Mass of Jupiter is 100 times its original, now at 31,700 Earth masses (T=11 yrs, ∆t=.01 yr).

Fig. 4

The mass of Jupiter is now 1,000 its original mass (T=11 yrs, ∆t=.01 yr).

http://www.physics.sfsu.edu/~lzimmer/chaos/EJS3.avi http://www.physics.sfsu.edu/~lzimmer/chaos/EJS4.avi

Chaos for fun

Chaos for fun

ReferencesDiacu, Florin and Holmes, Philip. Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, Princeton, New Jersey. 1996.

 Halliday, David, Kenneth Krane, and Robert Resnick. Physics vol. I, 4th Ed. John Wiley & Sons, Inc., New York. 1992.

 Leimanis, E. and Minorsky, N. Dynamics and Nonlinear Mechanics. John Wiley & Sons, Inc., New York. 1958.

 Marion, Jerry and Stephen Thornton. Classical Dynamics of Particles and Systems, 3rd Ed. Harcourt Brace & Company, Fort Worth. 1995.

 Peterson, Ivars. Newton’s Clock: Chaos in the Solar System. W. H. Freeman and Company, New York. 1993.