Simulations of The ThreeBody Problem:
Finding Chaos in the Cosmos
Craig Huber and Leah Zimmerman
San Francisco State University
Mathematics 490
Professor 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.
Sun
Earth
Early Studies of Planetary
Motion

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.
F
-F
m1m2
F G 2
r
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 3body 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
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.
Gm1m2
Gm1m2
F12 

2
r
( x1  x2 ) 2  ( y1  y2 ) 2
Where G is Newton’s gravitational constant:
m3
G  6.726 10
kg  s 2
And the acceleration of mass 1 due to mass 2 is found directly
11
Gm2
a12  
x1  x2 2   y1  y2 2
Pseudocode
Once 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
vx1  vo  ax1t
The position of mass 1 in the x-direction, where xo is the
position of mass 1 one iteration prior.
1
x1  xo  vx1t  a x1t 2
2
A Real World Example: The
Earth, Moon, and Sun
Fig. 1
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)
(a)
(b)
(c)
A Real World Example: The
Earth, Venus, and Sun
Fig. 2
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)
(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.
Earth
Sun
Jupiter
Perturbations in the Earth’s orbit
due to the presence of Jupiter.
Fig. 2
Fig. 3
Earth’s orbit with Jupiter
present (T=1 yr, ∆t=.01 yr).
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)
http://www.physics.sfsu.edu/~lzimmer/chaos/EJS1.avi
http://www.physics.sfsu.edu/~lzimmer/chaos/EJS2.avi
Fig. 3
Fig. 4
Jupiter at an initial distance of
1.1 AU (T=11 yrs, ∆t=.01 yr).
Jupiter at an initial distance of
1.105 AU (T=11 yrs, ∆t=.01 yr).
Chaos due to the presence of
Jupiter (MJ is greater)
http://www.physics.sfsu.edu/~lzimmer/chaos/EJS3.avi
http://www.physics.sfsu.edu/~lzimmer/chaos/EJS4.avi
Fig. 3
Fig. 4
Mass of Jupiter is 100 times its
original, now at 31,700 Earth
masses (T=11 yrs, ∆t=.01 yr).
The mass of Jupiter is now
1,000 its original mass (T=11
yrs, ∆t=.01 yr).
Chaos for fun
Chaos for fun
References
Diacu, 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.