astm001/mth724u solar system carl murray / nick cooper lecture 2: structure of the solar system

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ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

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Page 1: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

ASTM001/MTH724U

SOLAR SYSTEM

Carl Murray / Nick Cooper

Lecture 2: Structure of the Solar System

Page 2: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Books That Changed the World

Page 3: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Almagest

Page 4: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Copernicus’ De Revolutionibus Orbium

Celestium

Page 5: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s Harmonices Mundi

Page 6: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Five Convex Regular Polyhedra

tetrahedron

cube octahedron

dodecahedron

icosahedron

Page 7: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s Model of Planetary Spacing

• Each planet moves in a shell separated from next by regular polyhedron

• Six planets separated by five shells

• Thickness of shell is important

• Ordering of polyhedra is importantOrbits of Jupiter, Saturn and

Mars

Page 8: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s Model

Page 9: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Galileo’s Dialogue

Page 10: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The “Retrograde” Path of Mars

Apparent motion of Mars, June – November 2003

Page 11: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s First Law

The planets move in ellipses with the Sun at one focus

Page 12: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s Second Law

A line drawn from the Sun to a planet sweeps out equal areas in equal times

Page 13: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Kepler’s Third Law

It is most certain and most exact that the proportion between the periods of any two planets is precisely three halves the proportion of the mean distance

J. Kepler, 15 May 1618

The square of the orbital period of a planet is proportional to the cube of its semi-major axis

Page 14: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Daphnis making waves in the Keeler Gap

Keeler Gap

‘Slow lane’

‘Fast lane’ Arrows show direction of motion of

ring particles relative to Daphnis

Page 15: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Newton’s Universal of Gravitation

Any two bodies attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them

Page 16: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Newton’s Laws of Motion

• Bodies remain in a state of rest or uniform motion unless acted upon by a force

• The force experienced by a body is equal to the rate of change of momentum

• To every action there is an equal and opposite reaction

Page 17: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Orbit Determination

Orbit ModelsFixed Ellipse 2-body Point-masses Orbital

elements Precessing Ellipse 2-body Oblate primary Orbital elements Full Equations of Motion n-body Oblate primary Position/velocity

Choose an appropriate mathematical model for the orbit.

The model is defined by a set of parameters.

The numerical values of the model parameters are initially unkown.

Use the model to estimate the observed quantities.

Iteratively solve for the set of parameter values which generates a satisfactory match between the estimated and actual observations.

Page 18: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Titius’ 1766 Translation of Bonnet’s Contemplation de la

Nature

Page 19: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Titius-Bode ‘Law’ of Planetary Distance

The distance of a planet from the Sun obeys a geometric progression.

Page 20: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Titius-Bode ‘Law’ of Planetary Distance

Page 21: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Bode’s Law for Uranian Satellites?

Page 22: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Bode’s Law for Uranian Satellites?

Page 23: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Uniqueness of Uranian System

Actual system:

Generate 100,000 sets of 5 satellites and calculate best fit for each set

Page 24: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Saturn System (as of 1997)

4:3

2:1

2:1

Page 25: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

The Geometry of Orbital Resonance

2:1 Resonance, Stable configuration:

2:1 Resonance, Unstable configuration:

Page 26: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in the Saturn System

• Saturn Ring Features (gaps, edge waves, density waves)

• Janus : Epimetheus (co-orbital - horseshoe motion)• Dione : Helene : Polydeuces (co-orbital - tadpole motion)• Tethys : Telesto : Calypso (co-orbital - tadpole motion)

• Mimas : Tethys (4:2)• Titan : Hyperion (4:3)• Enceladus : Dione (2:1)• Mimas : Anthe (10:11), Mimas : Methone (14:15), Mimas :

Aegaeon (7:6)

• Most regular satellites are in synchronous rotation (like The Moon). Hyperion (shown in the movie) is an exception.

Hyperion

Page 27: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Polydeuces

Polydeuces

Helene

Dione

Long

itude

lag

(deg

)

Long

itude

lag

(deg

)Y (km)

X (km)

Saturn

Page 28: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in Saturn’s Ring System

Page 29: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in the Jupiter System

Page 30: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in the Uranus System

5:3 near-resonance between Cordelia and Rosalind

Anomalously high inclination of Miranda (4.22 deg) suggests existence of resonances in the past

Currently no known resonances between the major satellites

Page 31: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in Uranus’ Ring System

24:25 resonance between Cordelia and epsilon ring inner edge

14:13 resonance between Ophelia and epsilon ring outer edge

Page 32: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in Neptune’s Ring System

42:43 resonance between Galatea and Adams ring

Page 33: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in the Planetary System

Jupiter-Saturn near 5:2 resonance

Neptune-Pluto 3:2 resonance

Page 34: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Spin-orbit Resonance in the Planetary System

Mercury 3:2 spin-orbit resonance

QuickTime™ and a decompressor

are needed to see this picture.

Pluto-Charon 1:1:1 spin-orbit resonance

Page 35: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Resonance in the Asteroid Belt

Page 36: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Trojan Asteroids

Page 37: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Preference for Commensurability

Two orbits are commensurate when

For orbits in the solar system

Let ratio be bounded by

and

Let

Page 38: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Preference for Commensurability

A: Enceladus-Dione

B: Mimas-Tethys

C: Titan-Hyperion

D: Io-Europa

E: Europa-Ganymede

F: Neptune-Pluto

Page 39: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

High phase-angle Cassini image of Saturn

Page 40: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

High phase-angle Cassini image of Saturn

2006-258

Page 41: ASTM001/MTH724U SOLAR SYSTEM Carl Murray / Nick Cooper Lecture 2: Structure of the Solar System

Websites

NASA Solar System Explorationhttp://solarsystem.nasa.gov/index.cfm

JPL Solar System Dynamicshttp://ssd.jpl.nasa.gov

JPL Cassinihttp://saturn.jpl.nasa.gov/

Royal Astronomical Societyhttp://www.ras.org.uk