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 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 T2 ∝ 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 ˆr12. 2|r12| 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 m1m2 m1r¨1 = G r122 m1m2(r1 − r2) = G r132 m1m2(r2 − r1) r132 m1m2 m2r¨2 = G r122 = G Center of Mass is stationary/ moves at constant velocity Classic treatment r¨ − r¨ = ¨r <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">1</li></ul>r¨r + µ = 0 r3 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 ˙ˆ r˙ = rˆr + rθθ <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>1 d 2<ul style="display: flex;"><li style="flex:1">˙</li><li style="flex:1">˙</li><li style="flex:1">ˆ</li></ul>¨r = (¨r − rθ)ˆr + (r θ) θ, r dt one ﬁnds the solution: pr(θ) = ,1 + e cos(θ) h2 µwhere p = .Elliptical Orbit caFigure: Axes of an ellipse, Eccentricity = Kepler’s Laws 1 The motion of m2 is an ellipse with m1 at one focus dA dt h22<ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">= constant </li></ul>Figure: Kepler’s 2nd Law Kepler’s third law dA dt h2•<ul style="display: flex;"><li style="flex:1">From Kepler’s second law, we have </li><li style="flex:1">= . </li></ul>••area of ellipse = A = πab Aτ = dA dt 4π2a3 µ3 τ2 = , or τ2 ∝ a3. N-Body Problem •no analytical solutions for N > 2 •computational methods → Euler’s method, Runge-Kutta

Resonance in the Solar System

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