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Resonance in the Solar System

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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

  • 2
  • 1

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

  • ꢀ
  • ꢁ

1 d

2

  • ˙
  • ˙
  • ˆ

¨r = (¨r − rθ)ˆr +

(r θ) θ,

r dt

one finds the solution:

p

r(θ) =

,

1 + e cos(θ)

h2

µ

where p =

.

Elliptical Orbit

ca

Figure: Axes of an ellipse, Eccentricity =

Kepler’s Laws

1 The motion of m2 is an ellipse with m1 at one focus

dA dt h

2

2

  • =
  • = constant

Figure: Kepler’s 2nd Law

Kepler’s third law

dA dt h

2

•

  • From Kepler’s second law, we have
  • = .

••

area of ellipse = A = πab

A

τ =

dA dt

4π2a3

µ

3 τ2 =

, or τ2 ∝ a3.

N-Body Problem

•

no analytical solutions for

N > 2

•

computational methods → Euler’s method, Runge-Kutta

Recommended publications
  • The Asteroidal Belt and Kirkwood Gaps—I. a Statistical Study

    The Asteroidal Belt and Kirkwood Gaps—I. a Statistical Study

    Pramfina, Vol. 8, No. 5, 1977, pp. 438-446i © Printed in India. The asteroidal belt and Kirkwood gaps--I. A statistical study R PRATAP Physical Research Laboratory, Ahmedabad 380009 MS received 2 February 1977 Abstract. In this paper we have made a spectral analysis study of matter distri- bution in the asteroidal belt. We have Fourier analysed this distribution and obtained the autocorrelation and power spectrum, and have identified the ratios from the resonance theory. We have shown that the Kirkwood gaps cannot be satisfactorily interpreted as due to mere resonance between the asteroid and Jupi(er orbital motions. We propose that they may be regarded as a consequence of density waves generated in the g~s disc in the ecliptic plane in the ncighbourhood of the Sun. We have also shown that the process is non-Marcovian and hence cannot be subjected to a hydrodynamical analysis. Keywords. D~nsity waves; Non-Marcovian; Kirkwood gaps. 1. Introduction It was observed by Daniel Kirkwood (1867) that the astetoidal belt had a ring struc- ture, and since then, resonance theory was invoked to explain the rings of minimal matter-gaps. The total mass distribution in these rings has been estimated to be about 0.1 M@. Recently Ovenden (1972) tried to estimate as to what would have b~n the mass in this region at the origin of the solar system. He invoked a "principle of least interaction action" and using this, he calaulated the semi- major axes of the various planets, trying to examine the physical contem of Titius-Bode's law.
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  • The Philosophy of Mathematics: a Study of Indispensability and Inconsistency

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  • Proceedings of the Indiana Academy of Science

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  • Narrative and the Making of Contested Landscapes in Postwar American Astronomy

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