Resonance In the Solar System


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1 Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012
2 Goal numerically investigate the dynamics of the asteroid belt relate old ideas to new methods reproduce known results
3 History The role of science: make sense of the world perceive order out of apparent randomness
4 History The role of science: make sense of the world perceive order out of apparent randomness the sky and heavenly bodies
5 Anaximander ( BC) Greek philosopher, scientist stars, moon, sun 1:2:3 Figure: Anaximander s Model
6 Pythagoras ( BC) Mathematician, philosopher, started a religion all heavenly bodies at whole number ratios Harmony of the spheres Figure: Pythagorean Model
7 Tycho Brahe ( ) Danish astronomer, alchemist accurate astronomical observations, no telescope importance of data collection
8 Johannes Kepler ( ) Brahe s assistant Used detailed data provided by Brahe Observations led to Laws of Planetary Motion
9 Johannes Kepler ( ) 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
10 Kepler s Model Astrologer, Harmonices Mundi Used empirical data to formulate laws Figure: Kepler s Model
11 Isaac Newton ( ) 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.
12 Resonance Transition from ratios/ integer spacing to more physical description, resonance plays a key role in celestial mechanics
13 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.
14 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
15 Examples PlutoNeptune 2:3 GanymedeEuropaIo 1:2:4
16 Examples Cassini division in Saturn s rings 1:2 Resonance with Mimas
17 Kirkwood Gaps Daniel Kirkwood (1886)
18 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
19 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 )
20 Requirements 1 an idea for what causes orbital resonance 2 an appropriate integrating scheme 3 initial conditions for all bodies being considered
21 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
22 Kepler Problem The problem of two bodies interacting only by a central force is known as the Kepler Problem Also known as the 2body problem
23 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
24 Classic treatment r 2 r 1 = r r + µ r r 3 = 0 G(m 1 + m 2 ) = µ
25 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 twobody problem lies in a plane. Treat this relative motion in polar coordinates (r,θ).
26 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.
27 Elliptical Orbit Figure: Axes of an ellipse, Eccentricity = c a
28 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
29 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.
30 NBody Problem no analytical solutions for N > 2 computational methods Euler s method, RungeKutta
31 NBody Problem no analytical solutions for N > 2 computational methods Euler s method, RungeKutta need a better method
32 System N bodies  Sun, Jupiter, asteroids centralized force kinetic and potential energies independent Hamiltonian system
33 Hamiltonian Formulation H(q, p) = T (p) + U(q) q = H p ṗ = H q
34 NBody 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
35 NBody 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
36 Numerical Scheme best approach symplectic integrator designed for solutions to Hamiltonian systems preserves volume in phase space
37 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 of size dt 2.
38 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.
39 Leapfrog Algorithm additional half timestep transforms Euler s method to symplectic integrator more stable over long integrations angular momentum is preserved explicitly
40 Leapfrog Algorithm additional half timestep transforms Euler s method to symplectic integrator more stable over long integrations angular momentum is preserved explicitly a simple test of the Leapfrog integrator
41 Leapfrog Test Figure: Theoretical Solution
42 Leapfrog Test Figure: Numerical Solution
43 So far... semimajor axis/ orbital period relationship necessary for resonance appropriate integrating scheme Unresolved... Initial conditions for Sun, Jupiter, asteroids
44 Initial Conditions Positions sun at origin Jupiter at aphelion asteroids at perihelion Velocities (from ṙ ṙ) [ 2 v 2 = µ r 1 ] a
45 Model Integrate orbits of the Sun, Jupiter, and five asteroids range of initial semimajor axes, e = 0.15 initial postions Sun at origin Jupiter at aphelion asteroids at perihelion calculate eccentricities and semimajor axis
46 Results Figure: 3:1 Resonance  10K Jupiter Years  t = days
47 Results Figure: 3:1 Resonance  10K Jupiter Years  t = days
48 Results Figure: 3:1 Resonance  100K Jupiter Years  t = days
49 Results Figure: 3:1 Resonance  100K Jupiter Years  t = days
50 Results Figure: 3:1 Resonance  100K 200K Jupiter Years  t = days
51 Results Figure: 3:1 Resonance  100K 200K Jupiter Years  t = days
52 Further Abstraction
53 Conclusion resonances play a key role unite prescientific revolution modern science increased computational power insights into development of solar system
54 References 1 Meteorites may follow a chaotic route to Earth, Wisdom, Nature 315, (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 Numerical Investigation of Chaotic Motion in the Asteroid Belt, Danya Rose, University of Sydney Honours Thesis, November Motion of Asteroids at the Kirkwood Gaps, Makoto Yoshikawa, Icarus, Vol. 87, The role of chaotic resonances in the Solar System, N. Murray and M. Holman, Nature, vol. 410, 12 April Introduction to Celestial Mechanics, Jean Kovalevsky, D. Reidel, Classical Mechanics, John R. Taylor
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