Planetary system dynamics Part III Mathematics / Part III Astrophysics Lecturer: Prof. Mark Wyatt (Dr. Amy Bonsor on 9,11 Oct) Schedule: Michaelmas 2017 Mon, Wed, Fri at 10am MR11, 24 lectures, start Fri 6 Oct, end Wed 29 Nov Queries: 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 9 Oct, 23 Oct, 6 Nov, 20 Nov Examples classes: Group A - MR11 from 2-4pm on Tue 31 Oct, Wed 15 Nov, Tue 28 Nov, Tue 16 Jan; Group B MR15 from 2-4pm on Thu 2 Nov, 16 Nov, 30 Nov, 18 Jan (Groups TBD) 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: 3. Three-body problem: 4. Close approaches: 5. Collisions: 6. Disturbing function: 7. Secular perturbations: 8. Resonant perturbations: 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 restricted equations of motion, Jacobi integral, Lagrange equilibrium points, stability, tadpole and horseshoe orbits hyperbolic orbits, gravity assist, patched conics, escape velocity, gravitational focussing, dynamical friction, Tisserand parameter, cometary dynamics, Galactic tide 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 elliptic expansions, expansion using Legendre polynomials and Laplace coefficients, Lagrange's planetary equations, classification of arguments Laplace coefficients, Laplace-Lagrange theory, test particles, secular resonances, Kozai cycles, hierarchical systems 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: e.g., Pluto, Ceres, Eris Minor planets Asteroids: Asteroid Belt, Trojans, Near Earth Asteroids Comets: Kuiper Belt, Oort Cloud Dust Zodiacal Cloud 2
The planets overview/mass Mass Distance Sun 300,000 M earth 0.0046 au 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 Terrestrial planets Jovian planets Dwarf planet 1 M earth = 6 x 10 24 kg = 3x10-6 M sun 1 au = 1.5 x 10 11 m The planets - orbits planet Orbits defined by 6 variables: Semimajor axis, a (t per =a 1.5 ) Eccentricity, e Inclination, I Longitude of pericentre, ϖ Longitude of ascending node, Ω True anomaly, f (relative to the ecliptic, the plane of Earth s orbit) Evenly spaced, near circular orbits in same direction and plane (Sun s rotation off by 7.3 o ) JS near 5:2 resonance; NP in 3:2 resonance System stable for >4.5Gyr, though Mercury s orbit chaotic 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 Sun 3
Minor planets in the inner solar sytem Asteroid Belt contains >20,000 rocky asteroids orbiting 2-3.5 au (green) Jupiter Near Earth Asteroids (red) start in AB until orbits become chaotic Jupiter Trojan asteroids at ± 60 o L4 and L5 points (blue); other planets also have Trojans Mars Kuiper Belt: origin of comets Comet belt >30au; discovered 1992, ~1000 known Scattered by planets until reach inner SS, or ejected by Jupiter, or collide with planet (origin of H 2 O?) Few km nucleus of frozen gas and dust released when heated at perihelion Long period comets originate in Oort Cloud 1000-100,000au; perturbed by Galactic tides 4
Planet Nine Inferred from clustering of perihelia of distant KBOs (Batygin & Brown 2014) Explained by secular and resonant perturbations of 10Mearth planet at 700au (BB16; Beust 2016) Dust: Zodiacal cloud Radiation forces drag dust from AB and comets into the Sun, so Earth sits in dust disk called the Zodiacal cloud Zodiacal light = sunlight scattered by dust IR sky dominated by its thermal emission Some dust is accreted by Earth 5
Circumplanetary material Regular satellites are like mini-planetary systems, and rings analogous to planetesimal belts Most giant planets satellites are irregulars (captured asteroids/ comets): small (2-200km), on eccentric (~0.4) inclined (~40 0 ) or retrograde orbits filling Hill sphere How to detect extrasolar planets? Effect on motion of parent star Astrometric wobble Timing shifts Radial velocity method Effect on flux from star(s) Planetary transits Gravitational microlensing Direct detection Direct imaging Other techniques Disk structures 2-body motion: both bodies orbit centre of mass M * M pl 6
Methods using motion of parent star Astrometric wobble = in plane of sky 2x10-3 (a pl /au)(pc/d * )(M pl /M J )(M sun /M * ) arcsec 1 arcsec = angle subtended by 1au at 1pc Timing shifts = out of sky plane Δt = 3(a pl /au)(m pl /M earth )(M sun /M star ) ms Radial velocity = out of sky plane 30(a pl /au) -0.5 (M pl sini/m J )(M * /M sun ) -0.5 m/s Pulsar Planets First extrasolar planets: 6.2ms pulsar PSRB1257+12 (Wolszczan & Frail 1992) Small, coplanar, low eccentricity (Konacki & Wolszczan 2003) B and C near 3:2 resonance (Malhotra 1992) pinpoints orbital planes and masses 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 Asteroid belt beyond C (Wolszczan et al. 2000)? 7
Radial Velocity Planets First main sequence star planet: 51 Peg (G2 at 15pc) from RV, >0.45M jupiter at 0.05au near circular (Mayor & Queloz 1995) = HOT JUPITER Now 728 planets discovered using this method (see http://exoplanet.eu or http://exoplanets.org) Transit detection method If orientation just right, star dims as planet passes in front e.g., HD209458b transit lasts 3hrs every 3.5days; with RV get planet mass, size, density Kepler already detected <1M earth planets (>4600 planet candidates, many of which confirmed, see keplerscience.arc.nasa.gov) 8
Direct imaging of outer planetary systems Four 5-13M jup planets imaged 14-68au around 60Myr A star HR8799 (Marois et al. 2010) Are outer planetary systems common, relation to inner planets, formation? Planet discovery space RV detection bias (grey): 30(a pl /au) -0.5 (M pl /M J )(M * /M sun ) -0.5 m/s + 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? 9
Protoplanetary disks (<10Myr) Stars are born with massive disks of gas and dust that last a few Myr Dust motion affected by gas drag, concentrating dust into dense regions resulting in dust collisions and growth into planetesimals and planets? High resolution imaging shows structure of the disks on planetary system scales Debris disks (>10Myr) Infrared emission of Fomalhaut is brighter than the star: thermal emission from cold (70K) dust heated by star Imaging shows emission from eccentric 130au dust ring with sharp inner edge, the system s Kuiper belt (Kalas et al. 2013) Flux density (Jy) 70K Wavelength (µm) Fractional luminosity = dust luminosity / stellar luminosity ~ 10-4 10
Exocomets Some stars have asteroid belts, but few have dust within a few au η Corvi One example is η Corvi, which has a 152au Kuiper belt, and hot dust at ~1au Collisions would deplete asteroid belt over 1Gyr, so hot dust from comets scattered in from outer belt that sublimate at ~20au KIC8462852 Bizarre light-curve claimed by some as evidence of alien mega-structure: More likely: fragments of a comet (like Shoemaker-Levy 9) passing in front of star 11