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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 ^r12: jr12j • 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 = r^r r_ = r^r + rθ_θ^ 1 d ¨r = (¨r − rθ_)^r + (r 2θ_) θ;^ 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 = − jq1 − qj j i=2 j=1 N-Body Hamiltonian • from Hamilton equations: pi q_ i = rpi H = mi n X mj (qi − qj ) p_ i = rqi H = −Gmi 3 j6=i jqi − qj j 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 + dtrpi H pi+1 = pi − dtrqi+1 H with its adjoint pi+1 = pi − dtrqi H qi+1 = qi + dtrpi+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 + rpi H 2 2 pi+1 = pi − dtrq 1 H i+ 2 dt qi+1 = qi+ 1 + rpi+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 - 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 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.
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