Resonance in the Solar System

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 • the sky and heavenly bodies

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 • orbits are ellipses • equal area in equal time • T 2 ∝ a3

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 ~ m1m2 F12 = −G 2 ˆr12. |r12| • 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

Resonance

• Transition from ratios/ integer spacing to more physical description, resonance plays a key role in celestial mechanics • 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

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. • Transition from ratios/ integer spacing to more physical description, resonance plays a key role in celestial mechanics

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

Figure: Cassini Divison 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 ) • Start with the Kepler problem

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

m1m2 m1m2(r1 − r2) m1r¨1 = G 2 = G 3 r12 r12 m1m2 m1m2(r2 − r1) m2r¨2 = G 2 = G 3 r12 r12

Center of Mass is stationary/ moves at constant velocity Classic treatment

r¨2 − r¨1 = ¨r r ¨r + µ = 0 r 3 G(m1 + m2) = µ Classic treatment

Considering motion of m2 with respect to m1 gives:

r × ¨r = 0,

which, integrating once, gives

r × r˙ = 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 = rˆr r˙ = rˆr + rθ˙θˆ 1 d ¨r = (¨r − rθ˙)ˆr + (r 2θ˙) θ,ˆ 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 = a Kepler’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 Law Kepler’s third law

dA h • From Kepler’s second law, we have dt = 2 . • area of ellipse = A = πab A • τ = dA dt

2 4π2a3 2 3 3 τ = µ , or τ ∝ a . • need a better method

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 q˙ = ∂p −∂H p˙ = ∂q N-Body Hamiltonian

• Hamiltonian is separable, i.e. H = H(q, p, t) = T (p) + U(q)

n 1 X p2 T = i 2 mi i=1

N i−1 X X Gmi mj U = − |q1 − qj | i=2 j=1 N-Body Hamiltonian

• from Hamilton equations:

pi q˙ i = ∇pi H = mi n X mj (qi − qj ) p˙ i = ∇qi H = −Gmi 3 j6=i |qi − qj | 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 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∇q 1 H i+ 2 dt qi+1 = qi+ 1 + ∇pi+1 H. 2 2 • a simple test of the Leapfrog integrator →

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 (fromr ˙ · r˙)

2 1 v 2 = µ − r a 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 - 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 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. Taylor