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Home , Planetary system, Zodiacal light

Planetary 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 , 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 Course content

0. architecture: overview of and extrasolar , detectability, formation 1. Two-body problem: equation of motion, orbital elements, barycentric motion, Kepler's equation, perturbed 2. Small body forces: stellar radiation, optical properties, , Poynting-Robertson drag, planetocentric orbits, drag, Yarkovsky forces, gas drag, motion in protoplanetary disc, solar , 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, assist, patched conics, escape velocity, gravitational focussing, dynamical friction, Tisserand parameter, cometary dynamics, 5. Collisions: , 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, 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 (out to ~100,000 AU, ~0.5 pc)

• The Sun • Mass//evolution

• and their and ring systems • Terrestrial planets: , , , • Jovian planets: , , , • Dwarf planets: (, )

• Minor planets • : Belt, Trojans, Near Earth Asteroids • : ,

• • 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 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 , the plane of Earth’s )

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 and liquid under surface of

Spin-orbit resonances: The ’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 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), (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; scattered by this cloud is affected by planets; e.g., visible as the , 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: (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 ; thought comets to be captured asteroids, but origin of equatorial orbits I<10 is mystery.

7 Giant impacts Pluto satellite 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 Uranus and Neptune into Kuiper belt causing depletion and Late Heavy Bombardment

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How to detect extrasolar planets?

2-body motion: Effect on motion of parent both bodies orbit • Astrometric wobble centre of mass • Timing shifts • 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 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 star 51 Peg (G2 at 15pc) used radial velocity method to detect >0.45Mjupiter planet at 0.05AU near circular orbit (Mayor & Queloz 1995; Marcy & Butler 1997) =

Now 591 planets discovered using this method (see http://exoplanet.eu or http://exoplanets.org) and >5% of have planets

10 Planet discovery space

To interpret observed stats need to understand detection bias: e.g., -0.5 -0.5 instrument sensitivity 30apl (Mpl/MJ)(M*/Msun) m/s and survey duration

1% stars have HJs: tides Long period circularise : new! orbits, mass loss, formed further out Eccentric Jupiters then migrated around ~5% or scattered stars: origin of in? eccentricity?

Super- 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 e.g. γ Cephei has 1.7MJ eccentricity and secular resonances: pericentres at 2.1AU and 0.4Msun at 0 28AU with e=0.4 oscillate 34 about ϖb=ϖc and line of apsides precesses at -410/yr (Laughlin et al. 2005; Beauge et al. strongly perturbing 2006); also 3 body resonance (Rivera et al. 2010)? planet

Transit detection method

e.g., HD209458b discovered by rv; If orientation just right, star gets fainter transit lasts 3hrs every 3.5days when the planet passes in front of it confirming the planet and giving its mass, size, density

Space mission Kepler already detected 1Mearth 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 5-13Mjup 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

<2Mjup planet imaged at inner edge of debris disk around 200Myr A5V star (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

spiral 50 stirring Planet: 1Mjup, 5AU, e=0.1, I=5o

Disk: 20-60AU 0 Time: 100Myr of secular perturbations AU

-50 offset So disk structures tell warp us about planets

e.g., β Pic’s warped -50 0 50 100 AU 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 into resonances

• Collisional destruction of Mars-sized

Hot dust puzzle Star

Eta Corvi’s disk: a 150AU Kuiper belt, and dust at 1.5AU

Evident in both images and Hot dust at Cold dust 1/4 -1/2 spectrum, as Tdust=278.L* .r 1.5AU at 150AU

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?

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