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 a 3
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 F 12 = G m 1m 2 r 12 2 ˆr 12.
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 m 1 r 1 = G m 1m 2 r 2 12 m 2 r 2 = G m 1m 2 r 2 12 = G m 1m 2 (r 1 r 2 ) r12 3 = G m 1m 2 (r 2 r 1 ) r12 3 Center of Mass is stationary/ moves at constant velocity
Classic treatment r 2 r 1 = r r + µ r r 3 = 0 G(m 1 + m 2 ) = µ
Classic treatment Considering motion of m 2 with respect to m 1 gives: r r = 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 = rˆr ṙ = rˆr + r θˆθ r = ( r r θ)ˆr + [ ] 1 d r dt (r 2 θ) ˆθ, one finds the solution: r(θ) = p 1 + e cos(θ), where p = h2 µ.
Elliptical Orbit Figure: Axes of an ellipse, Eccentricity = c a
Kepler s Laws 1 The motion of m 2 is an ellipse with m 1 at one focus 2 da dt = h 2 = constant Figure: Kepler s 2nd Law
Kepler s third law From Kepler s second law, we have da dt = h 2. area of ellipse = A = πab τ = A da dt 3 τ 2 = 4π2 a 3 µ, or τ 2 a 3.
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) q = H p ṗ = H q
N-Body Hamiltonian Hamiltonian is separable, i.e. H = H(q, p, t) = T (p) + U(q) T = 1 2 n p 2 i m i i=1 U = N i 1 i=2 j=1 Gm i m j q 1 q j
N-Body Hamiltonian from Hamilton equations: q i = pi H = p i m i ṗ i = qi H = Gm i n j i m j (q i q j ) q i q j 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 q i+1 = q i + dt pi H with its adjoint p i+1 = p i dt qi+1 H p i+1 = p i dt qi H q i+1 = q i + dt pi+1 H by introducing a half time step i + 1 2 of size dt 2.
Derivation New integrating scheme is now q i+ 1 2 = q i + dt 2 p i H p i+1 = p i dt qi+ 1 2 H q i+1 = q i+ 1 + dt 2 2 p i+1 H.
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 v 2 = µ r 1 ] 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