Planetary system dynamics Mathematics tripos part III / part III Astrophysics Lecturer: Dr Mark Wyatt Schedule: Lent 2015 – Mon Wed Fri 10am MR12, 24 lectures, start Fri 16 Jan, end Wed 11 Mar Problems: My office is Hoyle 38 at the Institute of Astronomy, or email [email protected] Examples sheets: 4 examples sheets, handed out around Mon 19 Jan, 2 Feb, 16 Feb, 2 Mar Examples classes: 3-5pm in HCR (IoA) on Tue 3 Feb, 17 Feb, 3 Mar (*Ryle Meeting Room), 28 Apr Main textbook Other useful textbooks Course content 1. Two body problem 2. Small body dynamics 3. Three body problem 4. Close approaches 5. Collisions 6. Disturbing function 7. Secular perturbations 8. Resonant perturbations Planetary system dynamics Course content 0. Planetary system architecture: overview of Solar System and extrasolar systems, detectability, planet formation 1. Two-body problem: equation of motion, orbital elements, barycentric motion, Kepler's equation, perturbed orbits 2. Small body forces: stellar radiation, optical properties, radiation pressure, Poynting-Robertson drag, planetocentric orbits, stellar wind drag, Yarkovsky forces, gas drag, motion in protoplanetary disc, minimum mass solar nebula, settling, radial drift 3. Three-body problem: restricted equations of motion, Jacobi integral, Lagrange equilibrium points, stability, tadpole and horseshoe orbits 4. Close approaches: hyperbolic orbits, gravity assist, patched conics, escape velocity, gravitational focussing, dynamical friction, Tisserand parameter, cometary dynamics, Galactic tide 5. Collisions: accretion, coagulation equation, runaway and oligarchic growth, isolation mass, viscous stirring, collisional damping, fragmentation and collisional cascade, size distributions, collision rates, steady state, long term evolution, effect of radiation forces 6. Disturbing function: elliptic expansions, expansion using Legendre polynomials and Laplace coefficients, Lagrange's planetary equations, classification of arguments 7. Secular perturbations: Laplace coefficients, Laplace-Lagrange theory, test particles, secular resonances, Kozai cycles, hierarchical systems 8. Resonant perturbations: geometry of resonance, physics of resonance, pendulum model, libration width, resonant encounters and trapping, evolution in resonance, asymmetric libration, resonance overlap 1 Components of the Solar System Material gravitationally bound to the Sun (out to ~100,000 AU, ~0.5 pc) • The Sun • Mass/luminosity/evolution • Planets and their moons and ring systems • Terrestrial planets: Mercury, Venus, Earth, Mars • Jovian planets: Jupiter, Saturn, Uranus, Neptune • Dwarf planets: Pluto (Ceres, Eris) • Minor planets • Asteroids: Asteroid Belt, Trojans, Near Earth Asteroids • Comets: Kuiper Belt, Oort Cloud • Dust • Zodiacal Cloud The planets – overview/mass Mass Distance Sun 300000Mearth 0.0046AU Mercury 0.06 Mearth 0.39 AU Venus 0.82 M 0.72 AU earth Terrestrial Earth 1.0 Mearth 1.0 AU planets Mars 0.11 Mearth 1.5 AU Jupiter 318 Mearth 5.2 AU Saturn 98 Mearth 9.5 AU Jovian planets Uranus 15 Mearth 19.2 AU Neptune 17 Mearth 30.1 AU Dwarf planet Pluto 0.002 Mearth 39.5 AU 24 -6 11 1 Mearth = 6 x 10 kg = 3x10 Msun , 1 AU = 1.5 x 10 m 2 The planets - orbits Aphelion Perihelion ae Orbits defined by: 1.5 • Semimajor axis, a (tper=a ) • Eccentricity, e 2a • Inclination, I (relative to the ecliptic, the plane of Earth’s orbit) a, AU e I, deg • Evenly spaced, orbiting in Mercury 0.39 0.206 7.0 same direction in same plane (Sun’s rotation axis inclined by Venus 0.72 0.007 3.4 7.3o) with nearly circular orbits Earth 1.0 0.017 0.0 Mars 1.5 0.093 1.9 • La Grande Inequalite (JS near Jupiter 5.2 0.048 1.3 5:2 resonance) and NP in 3:2 Saturn 9.5 0.054 2.5 resonance Uranus 19.2 0.047 0.8 • System is stable for >4.5Gyr, Neptune 30.1 0.009 1.8 though Mercury’s orbit evolves Pluto 39.5 0.249 17.1 chaotically on such timescales Other examples of resonances Mean motion resonances: Jupiter’s satellites in 4:2:1 resonance causes strong tides and vulcanism on Io and liquid water under surface of Europa Spin-orbit resonances: The Moon’s rotation period = orbital period, synchronous rotation, means Moon keeps same face to us (caused by tidal evolution) 3 Secular interactions between planets Secular interactions between the planets cause the obliquity and eccentricity of Earth’s orbit to vary on 100,000 yr timescales This changes the insolation of upper atmosphere And is reflected in global temperature changes measured in ice cores Minor planets in the inner solar sytem • The Asteroid Belt is the Jupiter 20,000-strong belt of rocky asteroids orbiting 2-3.5 AU from the Sun (green) • Some asteroids in the Earth region (Near Earth Asteroids in red) that originate in AB until orbits become chaotic • Another family of asteroids are the Jupiter Trojans at ± 60o from Jupiter at L4 and L5 points (blue, other planets also have Trojans) 4 Minor planets: dynamical structures Kirkwood gaps in the distribution of asteroids at mean motion resonances with Jupiter; Yarkovsky forces move ~100m sized asteroids into these unstable regions where they may be perturbed into Earth-crossing orbits Orbital distribution of KBOs: resonant (e.g., Pluto in 3:2 with Neptune), classical (low e,I, outer edge 47AU), scattered disc (high e, but perihelia near Neptune), detached (e.g., Sedna with perihelion at 44AU) Minor planets: mutual collisions Asteroid orbits are Nesvorny (2003) In last few years clustered into Hirayama found evidence of there is evidence of (1918) families created in families created dust created in the break-up Gyr-ago of when medium-sized collision in asteroid large asteroids asteroids collided just ~1Myr ago belt 5 Minor planets: size distribution Asteroids also collide with planets Asteroid belt’s size distribution is that of a and moons, and crater counts give collisional cascade that extends from size distribution and imply more 1000km objects down to micron-sized massive population in past (e.g., dust, and is reason many are rubble piles most Moon craters from Late Heavy Bombardment epoch 3.8Ga) Dust: Zodiacal cloud PR drag moves dust from AB toward the Zodiacal cloud structure is Sun; sunlight scattered by this cloud is affected by planets; e.g., visible as the zodiacal light, and its brighter behind Earth because of thermal emission is the brighest thing in a coorbiting clumpy ring of the IR sky; some dust is accreted by Earth resonantly trapped particles Sun Earth 6 Kuiper Belt: origin of comets • Belt of comets orbiting the Sun >30AU; discovered 1992, now ~1000 known • Scattered by giant planets until they reach inner SS, or Jupiter ejects them, or collide with a planet (origin of H2O?) • Few km nucleus of frozen gases and embedded dust released when heated at perihelion of eccentric orbit • Long period comets originate in Oort Cloud 1000-100,000AU; perturbed by Galactic tides Circumplanetary material: captured minor planets Most of the giant planets’ satellites are irregulars: small Mars has two 6-10km (2-200km) and on eccentric (~0.4) inclined (~400) satellites: Phobos (will more often retrograde orbits filling a large fraction of spiral into Mars in few Hill sphere; origin in capture from passing asteroids/ Myr) and Deimos; thought comets to be captured asteroids, but origin of equatorial orbits I<10 is mystery. 7 Giant impacts Pluto satellite Charon is half its diameter Moon (0.012 M , 3.3g/cm3) earth (in mutually synchronous rotation, keeping formed from Earth crust same face to each other), and two stripped in collision 50Myr 60-165km satellites in 6:4:1 orbital period after Earth formed ratio, all thought to have collisional origin Uranus’ tilted spin axis, Mercury’s high density, Mars’ hemispheric dichotomy Giant impacts also explain: Kuiper Belt - evolution Missing mass problem: total mass now is 0.05-0.3Mearth but 100x that required to form Pluto and KBO binaries Currently favoured model starts with a more massive Kuiper belt outside a more compact planetary system Planetary system becomes unstable after 800Myr scattering Uranus and Neptune into Kuiper belt causing depletion and Late Heavy Bombardment 8 How to detect extrasolar planets? 2-body motion: Effect on motion of parent star both bodies orbit • Astrometric wobble centre of mass • Timing shifts • Radial velocity method M Effect on flux from parent star pl • Planetary transits • Gravitational microlensing (rather flux from another star) Direct detection • Direct imaging M* Other techniques • Disc structures Methods using motion of parent star Astrometric wobble = in plane of sky Angular scale is -3 2x10 (apl/d*)(Mpl/MJ)(Msun/M*) arcsec so Jupiter around 1Msun at 10 pc is 1mas [hard] Timing shifts = out of sky plane ms radio pulsar timing variation is Δt = 3apl(Mpl/Mearth)(Msun/Mstar) ms so Earth around 1M gives a 3 ms shift sun Radial velocity = out of sky plane Stellar radial velocity semi-amplitude is -0.5 -0.5 30apl (Mplsini/MJ)(M*/Msun) m/s so Jupiter around 1Msun gives 13 m/s 9 Pulsar Planets First extrasolar planets detected around 6.2 ms pulsar PSRB1257+12 (Wolszczan & Frail 1992) a, AU M, M I e earth The planets are small, A 0.19 0.02 - 0 coplanar, low eccentricity B 0.36 4.3 530 0.019 (Konacki & Wolszczan 2003) C 0.47 3.9 470 0.025 Gravitational interactions between B and C near 3:2 resonance (Malhotra 1992) detected so orbital planes and masses derived Possible fourth planet or asteroid belt beyond C (Wolszczan et al. 2000) Radial Velocity Planets First extrasolar planet around main sequence star 51 Peg (G2 at 15pc) used radial velocity method to detect >0.45Mjupiter