EART160 Planetary Sciences

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## EART160 Planetary Sciences

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**EART160 Planetary Sciences**Francis Nimmo**Last Week**• Giant planets primarily composed of H,He with a ~10 Me rock-ice core which accreted first • They radiate more energy than they receive due to gravitational contraction (except Uranus!) • Clouds occur in the troposphere and are layered according to condensation temperature • Many (~300) extra-solar giant planets known • Many are close to the star or have high eccentricities – very unlike our own solar system • Nebular gas probably produced inwards migration**This Week – Orbits and Gravity**• Kepler’s laws • Newton and inverse square law • Orbital period, angular momentum, energy • Tides • Roche limit**Orbital Mechanics**• Why do we care? • Probably the dominant control on solar system architecture: • Why are satellites synchronous? • Why does Saturn have rings? • Why is Io volcanically active? • Why is the Moon moving away from the Earth?**Kepler’s laws (1619)**• These were derived by observation (mainly thanks to Tycho Brahe – pre-telescope) • 1) Planets move in ellipses with the Sun at one focus • 2) A radius vector from the Sun sweeps out equal areas in equal time • 3) (Period)2 is proportional to (semi-major axis a)3 ae a b apocentre pericentre focus empty focus e is eccentricity a is semi-major axis**Newton (1687)**• Explained Kepler’s observations by assuming an inverse square law for gravitation: Here F is the force acting in a straight line joining masses m1 and m2separated by a distance r; G is a constant (6.67x10-11 m3kg-1s-2) • A circular orbit provides a simple example (but it is also true for elliptical orbits): Period T Centripetal acceleration = rw2 Gravitational acceleration = GM/r2 So GM=r3w2 (this is a useful formula to be able to derive) So (period)2 is proportional to r3 (Kepler) Centripetal acceleration M r Angular frequency w=2 p/T**Angular Momentum (1)**• Regular momentum = mv • Angular momentum is momentum when object is moving in a non-straight line (e.g. a circle) • For a point mass m moving in a circle with radius r and angular frequency w the angular momentum L = mr2w • This can also be written L=I w where I=mr2 is the moment of inertia of the point mass • For a distribution of masses, the moment of inertia is: r dm Note that I must be defined relative to a particular rotation axis**Angular Momentum (2)**• Angular momentum (=Iw) is conserved (classic example is an ice skater) in the absence of external torques • Orbital angular momentum L is also conserved Where does the final equality come from? • For non-circular orbits, the angular momentum also depends on eccentricity e • In some cases, a planet’s spin angular momentum is also important • For a uniform planet, C = 0.4 MR2 • C is the MoI of the planet, • is its spin angular frequency R is its radius**Example – Earth-Moon system**• The Moon is moving away from the Earth (due to tides, see below) – how do we know this? • What happens to the angular momentum of the Moon as it moves away from the Earth? • What happens to the spin rate of the Earth as the Moon moves further away? • What evidence do we have that this story is correct? • What is one problem with the current rate of recession? • What about energy conservation? Moon r Earth**Energy**• Example for circular orbits - results are the same for elliptical orbits. • Gravitational energy per unit mass Eg=-GM/r why the minus sign? • Kinetic energy per unit mass Ev=v2/2=r2w2/2=GM/2r • Total sum Eg+Ev=-GM/2r (for elliptical orbits, -GM/2a) • Energy gets exchanged between k.e. and g.e. during the orbit as the satellite speeds up and slows down • But the total energy is constant, and depends only on the distance from and mass of the primary (independent of eccentricity) • Energy of rotation (spin) of a planet is Er=CW2/2 C is moment of inertia, W angular frequency • Energy can be exchanged between orbit and spin, like momentum, but spin energy is usually negligible.**a**Tides (1) • Body as a whole is attracted with an acceleration = Gm/a2 • But a point on the far side experiences an acceleration = Gm/(a+R)2 a R m • The net acceleration is 2GmR/a3 for R<<a • On the near-side, the acceleration is positive, on the far side, it’s negative • For a deformable body, the result is a symmetrical tidal bulge:**E.g. Lord Kelvin calculated the rigidity of the Earth**• Kelvin as hero or villain? (Glasgow, 1st scientific peer, buried next to Newton) Tides (2) • Tides are reciprocal: • The planet raises a tidal bulge on the satellite • The satellite raises a tidal bulge on the planet • The amplitude of the bulge on a body depends on its radius, and the masses of both bodies & their distance • The amplitude is reduced if the body is rigid**Tidal Amplitude**M is the body mass, m is the mass of the tide-raising body, R is the body radius, a is the semi-major axis • For a uniform, fluid body the equilibrium tide H is given by • Does this make sense? (e.g. the Moon at 60RE, M/m=81) • For a rigid body, the tide may be reduced due to the elasticity of the planet • Note that the tidal amplitude is a strong function of distance • Also note that tides are reciprocal – Moon raises tides on Earth; Earth raises tides on Moon**Tidal Torques**Synchronous distance • Friction in the primary leads to a phase-lag • Phase lag makes torques • If the satellite is outside the synchronous point, the torques cause the planet to spin down Tidal bulge • Conservation of angular momentum: as the planet spins down, the satellite speeds up and moves outwards • The rate of recession depends on how fast energy is dissipated in the primary (due to friction) • If sat. is inside the synchronous point (or its orbit is retrograde), the sat. moves inwards and the planet spins up.**Tidal torques (cont’d)**• From the satellite’s point of view, the planet is in orbit and generates a tide on the satellite which will act to slow the satellite’s rotation. • Because the tide raised by the planet on the satellite is large, so is the torque. • This is why most satellites rotate synchronously with respect to the planet they are orbiting (sat. orbital period = sat. rotation period) Tidal torque Primary Satellite Tidal bulge**Tidal Torques**• Examples of tidal torques in action • Almost all satellites are in synchronous rotation • Phobos is spiralling in towards Mars (why?) • So is Triton (towards Neptune) (why?) • Pluto and Charon are doubly synchronous (why?) • Mercury is in a 3:2 spin:orbit resonance (not known until radar observations became available) • The Moon is currently receding from the Earth (at about 3.5 cm/yr), and the Earth’s rotation is slowing down (in 150 million years, 1 day will equal 25 hours). How do we know this?**Summary**• Tides generate torques (this is why almost all satellites are phase-locked to the primary) • Dissipation in the primary normally causes the primary to spin down, and the satellite to move out • Rate at which energy is dissipated controls the satellite recession rate**Roche Limit (1)**• If a satellite gets too close to a planet, it will be pulled apart by tidal forces (e.g. comet SL-9) • The distance from the planet that this happens is called the Roche limit • It determines where planetary rings are found**a**Roche Limit (2) • If a fluid body gets too close to a planet, it will be pulled apart by the tidal stresses • The distance at which this happens is the Roche Limit • For a uniform, fluid body the size of the equilibrium tide His • How might we decide when the Roche limit is reached? M is the body mass, m is the mass of the tide-raising body, R is the body radius, a is the semi-major axis • An approximate answer for the Roche limit distance is The radius of the tide-raising body (the planet) is r and the densities of the planet and satellite are rr and rR, respectively. • In reality, the typical Roche limit is roughly twice the planet radius**Ring locations (1)**Jupiter Saturn Roche limits Roche limits How do we get satellites inside the Roche limit?**Ring locations (2)**Uranus Neptune Roche limits Roche limits**2ae**Tidal bulge Fixed point on satellite’s surface a Empty focus Planet This tidal pattern consists of a static part plus an oscillation a Diurnal Tides (1) • Consider a satellite which is in a synchronous, eccentric orbit • Both the size and the orientation of the tidal bulge will change over the course of each orbit • From a fixed point on the satellite, the resulting tidal pattern can be represented as a static tide (permanent) plus a much smaller component that oscillates (the diurnal tide) N.B. it’s often helpful to think about tides from the satellite’s viewpoint**Diurnal tides (2)**• The amplitude of the diurnal tide is 3e times the static tide (does this make sense?) • E.g. For Io, static tide (bulge) is about 8 km, diurnal tide is about 300 m • Why are diurnal tides important? • Stress – the changing shape of the bulge at any point on the satellite generates time-varying stresses • Heat – time-varying stresses generate heat (assuming some kind of dissipative process, like viscosity or friction). • We will see that diurnal tides dominate the behaviour of some of the Galilean satellites**Tidal Dissipation**• The amount of tidal heating depends on eccentricity • Normally, this dissipation results in orbit circularization and a reduction in e and tidal heating • But what happens if the eccentricity is continually being pumped back up? Large amounts of tidal heating can result. • Orbital resonances can lead to eccentricity increasing: w1 w2 2:1, 3:2, 3:1, etc.**Callisto**Europa Ganymede Io J G I E Jupiter System • Io, Europa and Ganymede are in a Laplace resonance • Periods in the ratio 1:2:4 • So the eccentricities of all three bodies are continually pumped up**Peale, Cassen and Reynolds**• The amount of tidal heating depends very strongly on distance from the primary (as well as e) • Io is the closest in, so one would expect heating to be most significant there • Peale, Cassen and Reynolds realized that Io’s eccentricity was so high that the amount of tidal dissipation generated would be sufficient to completely melt the interior • They published their prediction in 1979 • Two weeks later . . .**Amirani lava flow, Io**500km Images from Voyager (1979) and Galileo (1996)**Tidal Heating**• Io is the most volcanically active body in the solar system • Tidal heating decreases as one moves outwards • Europa is heated strongly enough to maintain a liquid water ocean beneath a ~10 km thick ice shell • Ganymede is not heated now, but appears to have had an episode of high tidal heating in the past • Enceladus is (presumably) tidally heated, but Mimas (closer to Saturn, and higher eccentricity) is not. Why? Cassini image of plume coming off S pole of Enceladus**Other Examples . . .**• Tidal processes are ubiquitous across the solar system, and there are lots of other interesting stories: • Mercury in a 3:2 spin:orbit resonance • Triton was a captured object which had its orbit tidally circularized (laying waste to the Neptune system as it did so) • Many of the satellites of Uranus and Saturn appear to have undergone tidal heating at some time in their history • Extra-solar planets (“hot Jupiters”) are in circular orbits due to tidal torques • Et cetera ad nauseam**Summary**• Elliptical orbits (Kepler’s laws) are explained by Newton’s inverse square law for gravity • In the absence of external torques, orbital angular momentum is conserved (e.g. Earth-Moon system) • Orbital energy depends on distance from primary • Tides arise because gravitational attraction varies from one side of a body to the other • Tides can rip a body apart if it gets too close to the primary (Roche limit) • Tidal torques result in synchronous satellite orbits • Diurnal tides (for eccentric orbit) can lead to heating and volcanism (Io, Enceladus)**Key Concepts**GM=r3w2 • Angular frequency • Angular momentum • Tides & diurnal tides • Roche limit • Synchronous satellite • Laplace resonance E= - GM/2r**More realistic orbits**• Mean motion n (=2p/period) of planet is independent of e, depends on m (=G(m1+m2)) and a: • Angular momentum per unit mass of orbiting body is constant, depends on both e and a: • Energy per unit mass of orbiting body is constant, depends only on a:**The Moon**• Phase-locked to the Earth (its rotation rate was slowed by torques from tides raised by the Earth) • Has moonquakes which repeat once every month in the same – presumably triggered by tidal stresses Image taken by Galileo (the spacecraft, not the man)**Lunar Recession**• The Apollo astronauts left laser reflectors on the surface (as well as seismometers) • So we can measure the rate at which the Moon is receding due to tidal torques: ~4 cm per year • As a result, the Earth is spinning down, by about 2s per 100,000 years (conservation of angular momentum) Apollo 14 laser reflectometer McDonald Observatory, Texas**Known recession**rate (gives us Q) Present day distance Higher Q in past ~1.5 Gyr ago 4.5 Gyr ago The Problem • The Moon must only have formed 1-2 Gyr ago! • Major embarrassment for geophysicists • Also used as an argument by Creationists Constant Q time What is the solution? • The Earth’s Q must have been higher (i.e. less dissipation) in the past • What controls dissipation in the Earth?**The Solution (cont’d)**• Bulk of the dissipation occurs in the oceans • What controls dissipation in the oceans? • Bathtub effect – sloshing gets amplified if the driving frequency equals the resonant frequency of the basin. • What controls the resonant frequency?**Plate Tectonics!**• Resonant frequency of an ocean basin is controlled by its length • So as continental drift occurs, the length of the ocean basins changes and so does the amount of dissipation • There will also be an effect from sea-level changes – much of the dissipation occurs on shallow continental shelves • So in the past, when the continental configuration was different, oceanic dissipation was smaller and the Moon retreated more slowly So the evolution of the Moon’s orbit is controlled by plate tectonics!