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Presentation Transcript

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.

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

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?

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.

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 . . .

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

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?

- 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!

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