Resonance In the Solar
System
Steve Bache
UNC Wilmington
Dept. of Physics and Physical Oceanography
Advisor : Dr. Russ Herman
Spring 2012
Goal
• numerically investigate the dynamics of the asteroid belt
• relate old ideas to new methods
• reproduce known results
History
The role of science:
• make sense of the world
• perceive order out of apparent randomness
History
The role of science:
• make sense of the world
• perceive order out of apparent randomness
• the sky and heavenly bodies
Anaximander (611-547 BC)
• Greek philosopher, scientist
• stars, moon, sun 1:2:3
Figure: Anaximander’s Model
Pythagoras (570-495 BC)
• Mathematician, philosopher, started a religion
• all heavenly bodies at whole number ratios
• ”Harmony of the spheres”
Figure: Pythagorean Model
Tycho Brahe (1546-1601)
• Danish
astronomer,
alchemist
• accurate
astronomical
observations, no
telescope
• importance of
data collection
Johannes Kepler (1571-1631)
• Brahe’s assistant
• Used detailed data provided by Brahe
• Observations led to Laws of Planetary Motion
Johannes Kepler (1571-1631)
• Brahe’s assistant
• Used detailed data provided by Brahe
• Observations led to Laws of Planetary Motion
• orbits are ellipses
• equal area in equal time
• T 2 ∝ a3
Kepler’s Model
• Astrologer, Harmonices Mundi
• Used empirical data to formulate laws
Figure: Kepler’s Model
Isaac Newton (1642-1727)
• religious, yet desired a physical mechanism to explain Kepler’s
laws
• contributions to mathematics and science
• Principia
• almost entirety of an undergraduate physics degree
• Law of Universal Gravitation
~ 12 = −G m1 m2 r̂12 .
F
|r12 |2
Resonance
• Transition from ratios/ integer spacing to more physical
description, resonance plays a key role in celestial mechanics
Resonance
• Transition from ratios/ integer spacing to more physical
description, resonance plays a key role in celestial mechanics
•
Commensurability
The property of two orbiting objects, such as planets, satellites, or
asteroids, whose orbital periods are in a rational proportion.
Resonance
•
Commensurability
The property of two orbiting objects, such as planets, satellites, or
asteroids, whose orbital periods are in a rational proportion.
•
Resonance
Orbital resonances occur when the mean motions of two or more
bodies are related by close to an integer ratio of their orbital
periods
Examples
• Pluto-Neptune 2:3
• Ganymede-Europa-Io 1:2:4
Examples
Cassini division in Saturn’s rings
1:2 Resonance with Mimas
Kirkwood Gaps
Daniel Kirkwood (1886)
Kirkwood Gaps
• Commensurability in the orbital periods cause an ejection by
Jupiter
• explanation provided by Kirkwood, using 100 asteroids
• now thought to exhibit chaotic change in eccentricity
My Goal
• To create a simulation of the interactions of Jupiter, the Sun,
and ’test’ asteroids
• Integrate Newton’s equations of motion in MATLAB over a
large time span (≈ 1MY )
Requirements
1
an idea for what causes orbital resonance
2
an appropriate integrating scheme
3
initial conditions for all bodies being considered
Requirements
1
an idea for what causes orbital resonance
2
an appropriate integrating scheme
3
initial conditions for all bodies being considered
• Start with the Kepler problem
Kepler Problem
• The problem of two bodies interacting only by a central force
is known as the Kepler Problem
• Also known as the 2-body problem
Kepler Problem
m1 m2
m1 m2 (r1 − r2 )
=G
2
3
r12
r12
m1 m2
m1 m2 (r2 − r1 )
m2 r¨2 = G 2 = G
3
r12
r12
m1 r¨1 = G
Center of Mass is stationary/ moves at constant velocity
Classic treatment
r¨2 − r¨1 = r̈
r
r̈ + µ 3 = 0
r
G (m1 + m2 ) = µ
Classic treatment
Considering motion of m2 with respect to m1 gives:
r × r̈ = 0,
which, integrating once, gives
r × ṙ = h
This implies that
the motion in the two-body problem lies in a plane.
Treat this relative motion in polar coordinates (r,θ).
Polar form
Using,
r = rr̂
ṙ = rr̂ + r θ̇θ̂
1d 2
(r θ̇) θ̂,
r̈ = (r̈ − r θ̇)r̂ +
r dt
one finds the solution:
r (θ) =
where p =
h2
µ.
p
,
1 + e cos(θ)
Elliptical Orbit
Figure: Axes of an ellipse, Eccentricity =
c
a
Kepler’s Laws
1
The motion of m2 is an ellipse with m1 at one focus
2
dA
dt
=
h
2
= constant
Figure: Kepler’s 2nd Law
Kepler’s third law
h
• From Kepler’s second law, we have dA
dt = 2 .
• area of ellipse = A = πab
A
• τ = dA
dt
3
τ2 =
4π 2 a3
µ ,
or τ 2 ∝ a3 .
N-Body Problem
• no analytical
solutions for
N>2
• computational
methods →
Euler’s method,
Runge-Kutta
N-Body Problem
• no analytical
solutions for
N>2
• computational
methods →
Euler’s method,
Runge-Kutta
• need a better
method
System
• N bodies - Sun, Jupiter, asteroids
• centralized force
• kinetic and potential energies independent
• Hamiltonian system
Hamiltonian Formulation
H(q, p) = T (p) + U(q)
q̇ =
ṗ =
∂H
∂p
−∂H
∂q
N-Body Hamiltonian
• Hamiltonian is separable, i.e. H = H(q, p, t) = T (p) + U(q)
n
1 X pi2
T =
2
mi
i=1
U=−
N X
i−1
X
Gmi mj
|q1 − qj |
i=2 j=1
N-Body Hamiltonian
• from Hamilton equations:
q̇i = ∇pi H =
pi
mi
ṗi = ∇qi H = −Gmi
n
X
mj (qi − qj )
j6=i
|qi − qj |3
Numerical Scheme
• best approach → symplectic integrator
• designed for solutions to Hamiltonian systems
• preserves volume in phase space
Derivation
To derive the simplectic integrator to be used, compose Euler
method map
qi+1 = qi + dt∇pi H
pi+1 = pi − dt∇qi+1 H
with its adjoint
pi+1 = pi − dt∇qi H
qi+1 = qi + dt∇pi+1 H
by introducing a ”half time step” i +
1
2
of size
dt
2.
Derivation
New integrating scheme is now
qi+ 1 = qi +
2
dt
∇pi H
2
pi+1 = pi − dt∇qi+ 1 H
2
qi+1 = qi+ 1
2
dt
+ ∇pi+1 H.
2
Leapfrog Algorithm
• additional half time-step transforms Euler’s method to
symplectic integrator
• more stable over long integrations
• angular momentum is preserved explicitly
Leapfrog Algorithm
• additional half time-step transforms Euler’s method to
symplectic integrator
• more stable over long integrations
• angular momentum is preserved explicitly
• a simple test of the Leapfrog integrator →
Leapfrog Test
Figure: Theoretical Solution
Leapfrog Test
Figure: Numerical Solution
So far...
• semi-major axis/ orbital period relationship necessary for
resonance
• appropriate integrating scheme
Unresolved...
• Initial conditions for Sun, Jupiter, asteroids
Initial Conditions
• Positions
• sun at origin
• Jupiter at aphelion
• asteroids at perihelion
• Velocities (from ṙ · ṙ )
2 1
v =µ
−
r
a
2
Model
• Integrate orbits of the Sun, Jupiter, and five asteroids
• range of initial semi-major axes, e = 0.15
• initial postions
• Sun at origin
• Jupiter at aphelion
• asteroids at perihelion
• calculate eccentricities and semi-major axis
Results
Figure: 3:1 Resonance - 10K Jupiter Years - ∆t = 10.83 days
Results
Figure: 3:1 Resonance - 10K Jupiter Years - ∆t = 10.83 days
Results
Figure: 3:1 Resonance - 100K Jupiter Years - ∆t = 10.83 days
Results
Figure: 3:1 Resonance - 100K Jupiter Years - ∆t = 10.83 days
Results
Figure: 3:1 Resonance - 100K → 200K Jupiter Years - ∆t = 10.83 days
Results
Figure: 3:1 Resonance - 100K → 200K Jupiter Years - ∆t = 10.83 days
Further Abstraction
Conclusion
• resonances play a key role
• unite pre-scientific revolution → modern science
• increased computational power → insights into development
of solar system
References
1
2
3
4
5
6
7
Meteorites may follow a chaotic route to Earth, Wisdom,
Nature 315, 731-733 (27 June 1985)
The origin of the Kirkwood gaps - A mapping for asteroidal
motion near the 3/1 commensurability, Wisdom, Astronomical
Journal, vol 87, Mar. 1982
Numerical Investigation of Chaotic Motion in the Asteroid
Belt, Danya Rose, University of Sydney Honours Thesis,
November 2008
Motion of Asteroids at the Kirkwood Gaps, Makoto
Yoshikawa, Icarus, Vol. 87, 1990
The role of chaotic resonances in the Solar System, N. Murray
and M. Holman, Nature, vol. 410, 12 April 2001
Introduction to Celestial Mechanics, Jean Kovalevsky, D.
Reidel, 1967
Classical Mechanics, John R. Taylor