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 cubeoctahedron dodecahedronicosahedron

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 important Orbits 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 Models Fixed 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?

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

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 Helene Dione Longitude 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 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 byand 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

Websites NASA Solar System Exploration JPL Solar System Dynamics JPL Cassini Royal Astronomical Society