Resonance In the Solar System
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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 ModelIsaac 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 |2Resonance
• Transition from ratios/ integer spacing to more physical
description, resonance plays a key role in celestial mechanicsResonance
• 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
periodsExamples • 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 eccentricityMy 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 problemKepler Problem
m1 m2 m1 m2 (r1 − r2 )
m1 r¨1 = G 2
=G 3
r12 r12
m1 m2 m1 m2 (r2 − r1 )
m2 r¨2 = G 2 = G 3
r12 r12
Center of Mass is stationary/ moves at constant velocityClassic treatment
r¨2 − r¨1 = r̈
r
r̈ + µ 3 = 0
r
G (m1 + m2 ) = µClassic 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
r̈ = (r̈ − r θ̇)r̂ + (r θ̇) θ̂,
r dt
one finds the solution:
p
r (θ) = ,
1 + e cos(θ)
h2
where p = µ.Elliptical Orbit
c
Figure: Axes of an ellipse, Eccentricity = aKepler’s Laws
1 The motion of m2 is an ellipse with m1 at one focus
dA h
2 dt = 2 = constant
Figure: Kepler’s 2nd LawKepler’s third law
• From Kepler’s second law, we have dA h
dt = 2 .
• area of ellipse = A = πab
A
• τ = dA
dt
4π 2 a3
3 τ2 = 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)
∂H
q̇ =
∂p
−∂H
ṗ =
∂qN-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
N X
i−1
X Gmi mj
U=−
|q1 − qj |
i=2 j=1N-Body Hamiltonian
• from Hamilton equations:
pi
q̇i = ∇pi H =
mi
n
X mj (qi − qj )
ṗi = ∇qi H = −Gmi
j6=i
|qi − qj |3Numerical 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
1 dt
by introducing a ”half time step” i + 2 of size 2.Derivation
New integrating scheme is now
dt
qi+ 1 = qi + ∇pi H
2 2
pi+1 = pi − dt∇qi+ 1 H
2
dt
qi+1 = qi+ 1 + ∇pi+1 H.
2 2Leapfrog Algorithm
• additional half time-step transforms Euler’s method to
symplectic integrator
• more stable over long integrations
• angular momentum is preserved explicitlyLeapfrog 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 SolutionLeapfrog Test
Figure: Numerical SolutionSo far...
• semi-major axis/ orbital period relationship necessary for
resonance
• appropriate integrating scheme
Unresolved...
• Initial conditions for Sun, Jupiter, asteroidsInitial Conditions
• Positions
• sun at origin
• Jupiter at aphelion
• asteroids at perihelion
• Velocities (from ṙ · ṙ )
2 2 1
v =µ −
r aModel
• 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 axisResults 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 systemReferences
1 Meteorites may follow a chaotic route to Earth, Wisdom,
Nature 315, 731-733 (27 June 1985)
2 The origin of the Kirkwood gaps - A mapping for asteroidal
motion near the 3/1 commensurability, Wisdom, Astronomical
Journal, vol 87, Mar. 1982
3 Numerical Investigation of Chaotic Motion in the Asteroid
Belt, Danya Rose, University of Sydney Honours Thesis,
November 2008
4 Motion of Asteroids at the Kirkwood Gaps, Makoto
Yoshikawa, Icarus, Vol. 87, 1990
5 The role of chaotic resonances in the Solar System, N. Murray
and M. Holman, Nature, vol. 410, 12 April 2001
6 Introduction to Celestial Mechanics, Jean Kovalevsky, D.
Reidel, 1967
7 Classical Mechanics, John R. TaylorYou can also read
