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Planetary system dynamics Part III Mathematics / Part III Astrophysics

Michaelmas 2018

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

Lecturer: Prof. Mark Wyatt Schedule: Mon, Wed, Fri at 10am MR11, 24 lectures, start Fri 5 Oct, end Wed 28 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 8 Oct, 22 Oct, 5 Nov, 19 Nov Examples classes: MR11 from 2-4pm on Tue 30 Oct, 13 Nov, 27 Nov, 15 Jan Website: http://www.ast.cam.ac.uk/~wyatt

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

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Components of the Solar System

•  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

Material gravitationally bound to the Sun (out to ~100,000 au, ~0.5 pc)

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The planets – overview/mass

Sun 300,000 Mearth 0.0046 au

Mercury 0.06 Mearth 0.39 au

Venus 0.82 Mearth 0.72 au

Earth 1.0 Mearth 1.0 au

Mars 0.11 Mearth 1.5 au

Jupiter 318 Mearth 5.2 au

Saturn 98 Mearth 9.5 au

Uranus 15 Mearth 19.2 au

Neptune 17 Mearth 30.1 au

Pluto 0.002 Mearth 39.5 au

1 Mearth = 6 x 1024 kg = 3x10-6 Msun 1 au = 1.5 x 1011 m

Terrestrial planets

Jovian planets

Dwarf planet

Mass Distance

The planets - orbits Orbits defined by 6 variables: •  Semimajor axis, a (tper=a1.5) •  Eccentricity, e •  Inclination, I •  Longitude of pericentre, ϖ •  Longitude of ascending node, Ω •  True anomaly, f

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, near circular orbits in same direction and plane (Sun’s rotation off by 7.3o) •  JS near 5:2 resonance; NP in 3:2 resonance •  System stable for >4.5Gyr, though Mercury’s orbit chaotic

planet

Sun

(relative to the ecliptic, the plane of Earth’s orbit)

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Minor planets in the inner solar sytem

•  Asteroid Belt contains >20,000 rocky asteroids orbiting 2-3.5 au (green)

•  Near Earth Asteroids (red) start in AB until orbits become chaotic

•  Jupiter Trojan asteroids at ± 60o L4 and L5 points (blue); other planets also have Trojans

Jupiter

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 H2O?)

•  Long period comets originate in Oort Cloud 1000-100,000au; perturbed by Galactic tides

•  Few km nucleus of frozen gas and dust released when heated at perihelion

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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

Zodiacal light = sunlight scattered by dust

IR sky dominated by its thermal emission

Some dust is accreted by Earth

Radiation forces drag dust from AB and comets into the Sun, so Earth sits in dust disk called the Zodiacal cloud

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Circumplanetary material

Most giant planets’ satellites are irregulars (captured asteroids/comets): small (2-200km), on eccentric (~0.4) inclined (~400) or retrograde orbits filling Hill sphere

Regular satellites are like mini-planetary systems, and rings analogous to planetesimal belts

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

How to detect extrasolar planets?

M*

Mpl

2-body motion: both bodies orbit centre of mass

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Astrometric wobble = in plane of sky Timing shifts = out of sky plane Radial velocity = out of sky plane

Methods using motion of parent star

2x10-3(apl/au)(pc/d*)(Mpl/MJ)(Msun/M*) arcsec 1 arcsec = angle subtended by 1au at 1pc

Δt = 3(apl/au)(Mpl/Mearth)(Msun/Mstar) ms

30(apl/au)-0.5(Mplsini/MJ)(M*/Msun)-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 Asteroid belt beyond C (Wolszczan et al. 2000)?

a, au M, Mearth I e A 0.19 0.02 - 0 B 0.36 4.3 530 0.019 C 0.47 3.9 470 0.025

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Radial Velocity Planets

First main sequence star planet: 51 Peg (G2 at 15pc) from RV, >0.45Mjupiter 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

Kepler already detected <1Mearth planets (>4600 planet candidates, many of which confirmed, see keplerscience.arc.nasa.gov)

e.g., HD209458b transit lasts 3hrs every 3.5days; with RV get planet mass, size, density

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Direct imaging of outer planetary systems

Four 5-13Mjup 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(apl/au)-0.5(Mpl/MJ)(M*/Msun)-0.5 m/s + survey duration

1% stars have HJs: tides circularise orbits, mass loss, formed further out then migrated or scattered in?

Super-Earths common (30-50%): cores of evaporated Jupiters or massive Earths?

Eccentric Jupiters around ~5% stars: origin of eccentricity?

Long period Jupiters: new!

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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)

Wavelength (µm)

Flux

den

sity

(Jy)

70K

Fractional luminosity = dust luminosity / stellar luminosity ~ 10-4

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Exocomets

Some stars have asteroid belts, but few have dust within a few au 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

η Corvi

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