Resonance In the SolarSystem

Steve Bache

UNC WilmingtonDept. 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

• the sky and heavenly bodies

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)

• Danishastronomer,alchemist

• accurateastronomicalobservations, notelescope

• importance ofdata collection

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

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’slaws

• contributions to mathematics and science• Principia• almost entirety of an undergraduate physics degree• Law of Universal Gravitation

~F12 = −G m1m2

|r12|2r12.

Resonance

• Transition from ratios/ integer spacing to more physicaldescription, resonance plays a key role in celestial mechanics

• Commensurability

The property of two orbiting objects, such as planets, satellites, orasteroids, whose orbital periods are in a rational proportion.

• ResonanceOrbital resonances occur when the mean motions of two or morebodies are related by close to an integer ratio of their orbitalperiods

Resonance

• Transition from ratios/ integer spacing to more physicaldescription, resonance plays a key role in celestial mechanics

• Commensurability

The property of two orbiting objects, such as planets, satellites, orasteroids, whose orbital periods are in a rational proportion.

• ResonanceOrbital resonances occur when the mean motions of two or morebodies are related by close to an integer ratio of their orbitalperiods

Resonance

• Transition from ratios/ integer spacing to more physicaldescription, resonance plays a key role in celestial mechanics

• Commensurability

The property of two orbiting objects, such as planets, satellites, orasteroids, whose orbital periods are in a rational proportion.

• ResonanceOrbital resonances occur when the mean motions of two or morebodies are related by close to an integer ratio of their orbitalperiods

Examples

• Pluto-Neptune 2:3

• Ganymede-Europa-Io 1:2:4

ExamplesCassini division in Saturn’s rings1:2 Resonance with Mimas

Figure: Cassini Divison

Kirkwood Gaps

Daniel Kirkwood (1886)

Kirkwood Gaps

• Commensurability in the orbital periods cause an ejection byJupiter

• 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 alarge 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

• 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

• Start with the Kepler problem

Kepler Problem

• The problem of two bodies interacting only by a central forceis known as the Kepler Problem

• Also known as the 2-body problem

Kepler Problem

m1r1 = Gm1m2

r212= G

m1m2(r1 − r2)

r312

m2r2 = Gm1m2

r212= G

m1m2(r2 − r1)

r312

Center of Mass is stationary/ moves at constant velocity

Classic treatment

r2 − r1 = r

r + µr

r3= 0

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 + r θθ

r = (r − r θ)r +

[1

r

d

dt(r2θ)

]θ,

one finds the solution:

r(θ) =p

1 + e cos(θ),

where p = h2

µ .

Elliptical Orbit

Figure: Axes of an ellipse, Eccentricity = ca

Kepler’s Laws

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

2dAdt = h

2 = constant

Figure: Kepler’s 2nd Law

Kepler’s third law

• From Kepler’s second law, we have dAdt = h

2 .

• area of ellipse = A = πab

• τ = AdAdt

3 τ2 = 4π2a3

µ , or τ2 ∝ a3.

N-Body Problem

• no analyticalsolutions forN > 2

• computationalmethods →Euler’s method,Runge-Kutta

• need a bettermethod

N-Body Problem

• no analyticalsolutions forN > 2

• computationalmethods →Euler’s method,Runge-Kutta

• need a bettermethod

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

p =−∂H∂q

N-Body Hamiltonian

• Hamiltonian is separable, i.e. H = H(q, p, t) = T (p) + U(q)

T =1

2

n∑i=1

p2imi

U = −N∑i=2

i−1∑j=1

Gmimj

|q1 − qj |

N-Body Hamiltonian

• from Hamilton equations:

qi = ∇piH =pimi

pi = ∇qiH = −Gmi

n∑j 6=i

mj(qi − qj)

|qi − qj |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 Eulermethod map

qi+1 = qi + dt∇piH

pi+1 = pi − dt∇qi+1H

with its adjointpi+1 = pi − dt∇qiH

qi+1 = qi + dt∇pi+1H

by introducing a ”half time step” i + 12 of size dt

2 .

Derivation

New integrating scheme is now

qi+ 12

= qi +dt

2∇piH

pi+1 = pi − dt∇qi+1

2

H

qi+1 = qi+ 12

+dt

2∇pi+1H.

Leapfrog Algorithm

• additional half time-step transforms Euler’s method tosymplectic integrator

• more stable over long integrations

• angular momentum is preserved explicitly

• a simple test of the Leapfrog integrator →

Leapfrog Algorithm

• additional half time-step transforms Euler’s method tosymplectic 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 forresonance

• appropriate integrating scheme

Unresolved...

• Initial conditions for Sun, Jupiter, asteroids

Initial Conditions

• Positions• sun at origin• Jupiter at aphelion• asteroids at perihelion

• Velocities (from r · r)

v2 = µ

[2

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 developmentof 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 asteroidalmotion near the 3/1 commensurability, Wisdom, AstronomicalJournal, vol 87, Mar. 1982

3 Numerical Investigation of Chaotic Motion in the AsteroidBelt, Danya Rose, University of Sydney Honours Thesis,November 2008

4 Motion of Asteroids at the Kirkwood Gaps, MakotoYoshikawa, Icarus, Vol. 87, 1990

5 The role of chaotic resonances in the Solar System, N. Murrayand M. Holman, Nature, vol. 410, 12 April 2001

6 Introduction to Celestial Mechanics, Jean Kovalevsky, D.Reidel, 1967

7 Classical Mechanics, John R. Taylor