ASTM001/MTH724U
SOLAR SYSTEM
Carl Murray / Nick Cooper
Lecture 2: Structure of the Solar System
Books That Changed the World
The Almagest
Copernicus’ De Revolutionibus Orbium
Celestium
Kepler’s Harmonices Mundi
The Five Convex Regular Polyhedra
tetrahedron
cube octahedron
dodecahedron
icosahedron
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
Kepler’s Model
Galileo’s Dialogue
The “Retrograde” Path of Mars
Apparent motion of Mars, June – November 2003
Kepler’s First Law
The planets move in ellipses with the Sun at one focus
Kepler’s Second Law
A line drawn from the Sun to a planet sweeps out equal areas in equal times
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
Daphnis making waves in the Keeler Gap
Keeler Gap
‘Slow lane’
‘Fast lane’ Arrows show direction of motion of
ring particles relative to Daphnis
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
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
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.
Titius’ 1766 Translation of Bonnet’s Contemplation de la
Nature
The Titius-Bode ‘Law’ of Planetary Distance
The distance of a planet from the Sun obeys a geometric progression.
The Titius-Bode ‘Law’ of Planetary Distance
Bode’s Law for Uranian Satellites?
Bode’s Law for Uranian Satellites?
Uniqueness of Uranian System
Actual system:
Generate 100,000 sets of 5 satellites and calculate best fit for each set
The Saturn System (as of 1997)
4:3
2:1
2:1
The Geometry of Orbital Resonance
2:1 Resonance, Stable configuration:
2:1 Resonance, Unstable configuration:
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
Polydeuces
Polydeuces
Helene
Dione
Long
itude
lag
(deg
)
Long
itude
lag
(deg
)Y (km)
X (km)
Saturn
Resonance in Saturn’s Ring System
Resonance in the Jupiter 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
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
Resonance in Neptune’s Ring System
42:43 resonance between Galatea and Adams ring
Resonance in the Planetary System
Jupiter-Saturn near 5:2 resonance
Neptune-Pluto 3:2 resonance
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
Resonance in the Asteroid Belt
Trojan Asteroids
Preference for Commensurability
Two orbits are commensurate when
For orbits in the solar system
Let ratio be bounded by
and
Let
Preference for Commensurability
A: Enceladus-Dione
B: Mimas-Tethys
C: Titan-Hyperion
D: Io-Europa
E: Europa-Ganymede
F: Neptune-Pluto
High phase-angle Cassini image of Saturn
High phase-angle Cassini image of Saturn
2006-258
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