Planetary system dynamics Mathematics tripos part III / part III Astrophysics Lecturer: Dr Mark Wyatt Schedule: Lent 2014 Mon Wed Fri 10am MR9, 24 lectures, start Fri 17 Jan, end Wed 12 Mar Problems: My office is Hoyle 38 at the Institute of Astronomy, or email wyatt@ast.cam.ac.uk Examples sheets: 4 examples sheets, handed out around Mon 20 Jan, 3 Feb, 17 Feb, 3 Mar Examples classes: 2.00-3.30pm in Hoyle Committee Room (IoA) on Tue 4 Feb, 18 Feb, 4 Mar, 6 May 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 Main textbook Other useful textbooks 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 300000M earth 0.0046AU Mercury 0.06 M earth 0.39 AU Venus 0.82 M earth 0.72 AU Earth 1.0 M earth 1.0 AU Mars 0.11 M earth 1.5 AU Jupiter 318 M earth 5.2 AU Saturn 98 M earth 9.5 AU Uranus 15 M earth 19.2 AU Neptune 17 M earth 30.1 AU Pluto 0.002 M earth 39.5 AU 1 M earth = 6 x 10 24 kg = 3x10-6 M sun, 1 AU = 1.5 x 10 11 m Terrestrial planets Jovian planets Dwarf planet 2
The planets - orbits Aphelion Orbits defined by: Semimajor axis, a (t per =a 1.5 ) Eccentricity, e Inclination, I (relative to the ecliptic, the plane of Earth s orbit) 2a ae Perihelion a, AU e I, deg Mercury 0.39 0.206 7.0 Venus 0.72 0.007 3.4 Earth 1.0 0.017 0.0 Mars 1.5 0.093 1.9 Jupiter 5.2 0.048 1.3 Saturn 9.5 0.054 2.5 Uranus 19.2 0.047 0.8 Neptune 30.1 0.009 1.8 Pluto 39.5 0.249 17.1 Evenly spaced, orbiting in same direction in same plane (Sun s rotation axis inclined by 7.3 o ) with nearly circular orbits La Grande Inequalite (JS near 5:2 resonance) and NP in 3:2 resonance System is stable for >4.5Gyr, though Mercury s orbit evolves 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 20,000-strong belt of rocky asteroids orbiting 2-3.5 AU from the Sun (green) Jupiter 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 ± 60 o 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 clustered into Hirayama (1918) families created in the break-up Gyr-ago of large asteroids Nesvorny (2003) found evidence of families created when medium-sized asteroids collided just ~1Myr ago In last few years there is evidence of dust created in collision in asteroid belt 5
Minor planets: size distribution Asteroid belt s size distribution is that of a collisional cascade that extends from 1000km objects down to micron-sized dust, and is reason many are rubble piles Asteroids also collide with planets and moons, and crater counts give size distribution and imply more massive population in past (e.g., most Moon craters from Late Heavy Bombardment epoch 3.8Ga) Dust: Zodiacal cloud PR drag moves dust from AB toward the Sun; sunlight scattered by this cloud is visible as the zodiacal light, and its thermal emission is the brighest thing in the IR sky; some dust is accreted by Earth Zodiacal cloud structure is affected by planets; e.g., brighter behind Earth because of a coorbiting clumpy ring of 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 H 2 O?) 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 (2-200km) and on eccentric (~0.4) inclined (~40 0 ) more often retrograde orbits filling a large fraction of Hill sphere; origin in capture from passing asteroids/ comets Mars has two 6-10km satellites: Phobos (will spiral into Mars in few Myr) and Deimos; thought to be captured asteroids, but origin of equatorial orbits I<1 0 is mystery. 7
Giant impacts Moon (0.012 Mearth, 3.3g/cm3) formed from Earth crust stripped in collision 50Myr after Earth formed Giant impacts also explain: Pluto satellite Charon is half its diameter (in mutually synchronous rotation, keeping same face to each other), and two 60-165km satellites in 6:4:1 orbital period ratio, all thought to have collisional origin Uranus tilted spin axis, Mercury s high density, Mars hemispheric dichotomy 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? Effect on motion of parent star Astrometric wobble Timing shifts Radial velocity method Effect on flux from parent star Planetary transits Gravitational microlensing (rather flux from another star) Direct detection Direct imaging 2-body motion: both bodies orbit centre of mass M * M pl Other techniques Disc structures Methods using motion of parent star Astrometric wobble = in plane of sky Angular scale is 2x10-3 (a pl /d * )(M pl /M J )(M sun /M * ) arcsec so Jupiter around 1M sun at 10 pc is 1mas [hard] Timing shifts = out of sky plane ms radio pulsar timing variation is Δt = 3a pl (M pl /M earth )(M sun /M star ) ms so Earth around 1M sun gives a 3 ms shift Radial velocity = out of sky plane Stellar radial velocity semi-amplitude is 30a pl -0.5 (M pl sini/m J )(M * /M sun ) -0.5 m/s so Jupiter around 1M sun gives 13 m/s 9
Pulsar Planets First extrasolar planets detected around 6.2 ms pulsar PSRB1257+12 (Wolszczan & Frail 1992) The planets are small, coplanar, low eccentricity (Konacki & Wolszczan 2003) a, AU M, M earth I e A 0.19 0.02-0 B 0.36 4.3 53 0 0.019 C 0.47 3.9 47 0 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 >0.45M jupiter planet at 0.05AU near circular orbit (Mayor & Queloz 1995; Marcy & Butler 1997) = HOT JUPITER Now 545 planets discovered using this method (see http://exoplanet.eu or http://exoplanets.org) and >5% of stars have planets 10
Planet discovery space To interpret observed stats need to understand detection bias: e.g., instrument sensitivity 30a pl -0.5 (M pl /M J )(M * /M sun ) -0.5 m/s and survey duration 1% stars have HJs: tides circularise orbits, mass loss, formed further out then migrated or scattered in? Eccentric Jupiters around ~5% stars: origin of eccentricity? Long period Jupiters: new! Super-Earths common (30-50%?): cores of evaporated Jupiters or massive Earths? Planet eccentricity distribution Giant planets at few AU have eccentric orbits: mean 0.32, up to 0.92 (compared with <0.05 for the Solar System) Theories for origin of high eccentricities range from: planet-planet scattering planet-disk interactions, scattering by passing stars perturbations of companion stars 11
Multiple bodies: dynamical interactions There are 176 systems with multiple planets, and 57 planets in multiple stellar systems e.g., GJ876 planets in 2:1 mean motion eccentricity and secular resonances: pericentres oscillate 34 0 about ϖ b =ϖ c and line of apsides precesses at -41 0 /yr (Laughlin et al. 2005; Beauge et al. 2006); also 3 body resonance (Rivera et al. 2010)? e.g. γ Cephei has 1.7M J at 2.1AU and 0.4M sun at 28AU with e=0.4 strongly perturbing planet Transit detection method If orientation just right, star gets fainter when the planet passes in front of it e.g., HD209458b discovered by rv; transit lasts 3hrs every 3.5days confirming the planet and giving its mass, size, density Space mission Kepler already detected 1M earth planets (3538 planet candidates, 238 of which confirmed, see keplerscience.arc.nasa.gov) 12
Transit Timing Variations (TTV) Transits are precise clock meaning perturbations detectable from other planets or satellites (e.g., Nesvorny & Beauge 2010; Veras et al. 2011), and TTVs used to confirm planet and constrain planet masses; e.g., Kepler 9 has two transiting planets close to 2:1 resonance (also imply additional planet; Holman et al. 2010) Direct imaging of outer planetary systems Four planets imaged around 60Myr A star HR8799 with masses 5-13M jup at 14-68AU (Marois et al. 2010) Are outer planetary systems common, how do they relate to inner planetary systems, resonance required for stability (Fabrycky & Murray-Clay 2010), formation? 13
Planet interacting with debris disk <2M jup planet imaged at inner edge of debris disk around 200Myr A5V star Fomalhaut (Kalas et al. 2008, 2013) But emission spectrum is circumplanetary dust not planet, so rings or irregular satellite swarm (Kennedy & Wyatt 2011)? Also, planet s orbit is eccentric, crossing the disk, which would rapidly destroy it unless planet is young or low mass 0Myr 5Myr Planets inevitably affect disk structure Planet: 1M jup, 5AU, e=0.1, I=5 o 50 stirring spiral Disk: 20-60AU AU 0 Time: 100Myr of secular perturbations So disk structures tell us about planets -50 offset warp e.g., β Pic s warped edge-on disk was used to predict a planet that was later imaged 2003 2010 14
Clumps: planet migration or collision Mid-IR image shows clump at 52AU (Telesco et al. 2005) 52AU Origin could be: Outward migration of planet which trapped planetesimals into resonances Collisional destruction of Mars-sized protoplanet Hot dust puzzle Star Eta Corvi s disk: a 150AU Kuiper belt, and dust at 1.5AU Evident in both images and spectrum, as T dust =278.L * 1/4.r -1/2 Collisions would have depleted any asteroid belt over 1Gyr age of star, so where does hot dust come from? Recent collision, or comets scattered in during an epoch similar to the Late Heavy Bombardment? Hot dust at 1.5AU Cold dust at 150AU 15