Transcript Slide 1

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
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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:
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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
a
apocentre
empty focus
e is eccentricity
a is semi-major axis
ae
b
focus
pericentre
Newton (1687)
• Explained Kepler’s observations
by assuming an inverse square law
Gm1m2
for gravitation:
F
r2
Here F is the force acting in a straight line joining masses m1 and m2
separated 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
M
r
Angular frequency
w=2 p/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)
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:
I   mr   r dm
2
2
Note that I must be defined relative to a particular rotation axis
r
dm
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
L  Iw  mr w  m(GMr)
2
1/ 2
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
C is the MoI of the planet,
Lspin  C
is its spin angular frequency
R is its radius
• For a uniform planet, C = 0.4 MR2
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?
Moon
r
Earth
• What evidence do we have that
this story is correct?
• What is one problem with the
current rate of recession?
• What about energy conservation?
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=C2/2
C is moment of inertia,  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
a
R
• But a point on the far side
experiences an acceleration =
Gm/(a+R)2
a
3
• The net acceleration is 2GmR/a 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:
m
• Tides are reciprocal:
Tides (2)
– 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
• E.g. Lord Kelvin calculated the
rigidity of the Earth
• Kelvin as hero or villain?
(Glasgow, 1st scientific peer, buried next to Newton)
Tidal Amplitude
• For a uniform, fluid body the equilibrium tide H is
M is the body mass, m is the
3
given by
mass of the tide-raising body,
5 m R
 
H R  
2 Ma
R is the body radius, a is the
semi-major axis
• 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
Tidal bulge
torques cause the planet to
spin down
• 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
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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
3
M is the body mass, m is the
tide H is
5 m R
mass of the tide-raising body,
H
2
R
 
Ma
R is the body radius, a is the
semi-major axis
• How might we decide when the Roche limit is reached?
• An approximate answer for the Roche limit distance is
a
1/ 3
1/ 3
1/ 3
5  m 
5
aRoche      R   
2  M 
2
1/ 3
 r 

 r
 R 
• In reality, the typical Roche limit is
roughly twice the planet radius
The radius of the tideraising body (the planet) is
r and the densities of the
planet and satellite are r
and R, respectively.
Ring locations (1)
Jupiter
Saturn
Roche
limits
Roche
limits
How do we get satellites inside the Roche limit?
Ring locations (2)
Uranus
Roche
limits
Neptune
Roche
limits
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
2ae
over the course of each orbit
Tidal bulge
Fixed point on
satellite’s surface
a
Empty focus
Planet
a
This tidal pattern
consists of a static
part plus an oscillation
• 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
w1

w2
2:1, 3:2, 3:1, etc.
Jupiter System
Io
Europa
Callisto
Ganymede
• 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
J
G
I
E
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 . . .
Images from Voyager (1979)
and Galileo (1996)
Amirani lava flow, Io
500km
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
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Angular frequency
Angular momentum
Tides & diurnal tides
Roche limit
Synchronous satellite
Laplace resonance
GM=r3w2
I   mr   r dm
2
L  Iw
E= - GM/2r
2
End of lecture
More realistic orbits
• Mean motion n (=2p/period) of planet is independent
of e, depends on m (=G(m1+m2)) and a:
n a m
2
3
• Angular momentum per unit mass of orbiting body is
constant, depends on both e and a:
h  na 1  e
2
2
• Energy per unit mass of orbiting body is constant,
depends only on a:
m
E
2a
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
The Problem
Known recession
rate (gives us Q)
distance
• The Moon must only
have formed 1-2 Gyr
ago!
• Major embarrassment
for geophysicists
• Also used as an
argument by Creationists
Present day
Higher Q
in past
4.5 Gyr
ago
Constant Q
~1.5 Gyr
ago
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?
time
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!