Transcript EART 160: Planetary Sciences

EART 160: Planetary Science
06 February 2008
Last Time
• Planetary Surfaces
– Summary
• Planetary Interiors
– Terrestrial Planets and Icy Satellites
– Structure and Composition: What all is inside?
– Exploration Geophysics: How can we tell?
Today
•
•
•
•
Homework 3 graded
Projects – Have you got a topic yet?
Midterm Friday! – details
Paper Discussion: Stevenson (2001)
– Mars Magnetic Field
• Planetary Interiors
– Pressure and Temperature
– Heat Sources and Cooling Mechanisms
– Rheology
Histogram
3
Mean:
St. Dev.:
35
8
2
1
Bin
More
50
45
40
35
30
25
20
15
10
5
0
0
Freq.
4
Homework Issues
• Please talk to me if you have difficulties
– Before class is not usually a good time
– No more Monday due dates
• Units:
– Pressure:
1 Pa = 1N m-2 = 1 kg m-1 s-2
– Energy:
1 J = 1 kg m2 s-2
– Power:
1 W = 1 J s-1
• Stress
– Tectonic stress is not the Lithostatic Pressure
– Normal stress is the Pressure + normal component of Tectonic
– Shear stress is the tangential component of Tectonic
Midterm Exam
• Closed-book
– I will provide a formula sheet
– You may provide an 8.5” × 11” sheet of paper with
whatever you want on it; hand it in with your test.
– Formulae won’t help you if you don’t understand
them!
• Several short-answer questions, descriptive
• 3 quantitative problems, pick 2 to answer
– Similar to Homework, but less involved
– Show your work!
• Review Session? What say ye?
Exam Topics
• Orbital Mechanics
– Kepler’s Laws, Newton’s Laws
– Conservation of Energy, Momentum, Angular Momentum
– Escape Velocity
• Solar System Formation
–
–
–
–
Composition of the Solar Nebula
Jeans Collapse
Accretion and Runaway Growth
Frost Line
• Meteorites and Asteroids
– Chondrites: Remnants from Early Solar System
– Role of collisions
– Radiometric Dating
Impacts
• Crater size depends on impactor size, impact
velocity, surface gravity
• Crater morphology changes with increasing size
• Simple vs. complex crater vs. impact basin
• Depth:diameter ratio
• Crater size-frequency distribution can be used to
date planetary surfaces
• Energetics, Global effects due to impacts
• Atmospheres and geological processes can
affect size-frequency distributions
Volcanism
• Solidus & liquidus
• Magmatism when solidus crosses adiabat
– Higher temperatures, reduced pressure or lowered
solidus
• Volcanism when buoyant magma erupts
• Conductive cooling time t = d2/k
• Magma composition controls style of volcanism
• Flow controlled by viscosity
– Viscous materials s = h de/dt
Tectonics
• Planetary cooling leads to compression
• Hooke’s law and Young’s modulus
– Elastic materials s = E e
• Contraction and cooling
• Byerlee’s law
• Styles of tectonicsm: compression,
extension, shear
Gradation
• Erosion on planets with atmospheres
– Aeolian, Fluvial, Glacial
• Mass Wasting, Sputtering everywhere.
• Valley networks, gullies and outflow
channels
Planets are like Ogres
Compositional Layers
1. Core: Metal
2. Mantle: Dense silicate
rock (peridotite)
3. Crust: thin silicate rock
(basalt)
4. Ocean: liquid layer
5. Atmosphere: gas layer
ON an icy satellite, the ocean will be
beneath the icy mantle.
Other ice phases are denser than
water. May have ice – ocean -- ice
Mechanical Layers
1. Inner Core: solid metal
2. Outer Core: liquid metal
3. Lower Mantle: High
viscosity silicate
4. Aesthenosphere:
ductile upper mantle
5. Lithosphere: Brittle
uppermost mantle and
crust
Actual Planetary Interiors
Venus
Mercury
Earth
Mars
Moon
Only Earth has an layered core
The Moon has a TINY core (why?)
Icy satellites may have liquid oceans
beneath the ice shell
High-Pressure Ices beneath that.
Interior of Europa -- NASA
Io
Ganymede
Stevenson et al., 2001 Nature
Pressures inside planets
dP
  g
dr
• Hydrostatic assumption
(planet has no strength)
• For a planet of constant
density 
(is this reasonable?)
4
r
g (r )  Gr  g 0
3
R
 r  2G 2 2  r 
1
P(r )  g 0R1  2  
R  1  2 
2
3
 R 
 R 
2
2
• So the central pressure of a planet increases as the square of its radius
• Moon: R=1800km, P=7.2 GPa
• Mars: R=3400km, P=26 GPa
Pressures inside planets
• The pressure inside a planet controls how materials behave
• E.g. porosity gets removed by material compacting and flowing,
at pressures ~ few MPa
• The pressure required to cause a material’s density to change
significantly depends on the bulk modulus of that material
d
dP


K
The bulk modulus K controls the
change in density (or volume) due to
a change in pressure
• Typical bulk modulus for silicates is ~100 GPa
• Pressure near base of mantle on Earth is ~100 GPa
• So change in density from surface to base of mantle should be
roughly a factor of 2 (ignoring phase changes)
Real planets
• Notice the increase in mantle density with
depth – is it a smooth curve?
• How does gravity vary within the planet?
Phase Transitions
• Under pressure,
minerals transform to
different crystal
structure
• How do we detect this?
• Transition zone can
sore a LOT of water!
• How do the depths
change on other
planets?
Temperature
• Planets generally start out hot (see below)
• But their surfaces (in the absence of an
atmosphere) tend to cool very rapidly
• So a temperature gradient exists between the
planet’s interior and surface
• We can get some information on this gradient by
measuring the elastic thickness, Te
• The temperature gradient means that the planet
will tend to cool down with time
Heat Sources
• Accretion and Differentiation
– U = Eacc
– Eacc = m Cp DT
– Cp: specific heat
• Radioactive Decay
– E= H m
– H ~ 5x10-12 W kg-1
– K, U, Th today
– Al, Fe early on
• Tidal Heating in some
satellites
Specific Heat Capacity Cp
• The specific heat capacity Cp tells us how much energy
needs to be added/subtracted to 1 kg of material to make its
temperature increase/decrease by 1K
• Energy = mass x specific heat capacity x temp. change
E  mC p DT
• Units: J kg-1 K-1
• Typical values: rock 1200 J kg-1 K-1 , ice 4200 J kg-1 K-1
• E.g. if the temperature gradient near the Earth’s surface is
25 K/km, how fast is the Earth cooling down on average?
(about 170 K/Gyr)
• Why is this estimate a bit too large?
– Atmosphere insulates
Energy of Accretion
• Let’s assume that a planet is built up like an onion,
one shell at a time. How much energy is involved in
putting the planet together?
early
In which situation is
more energy delivered?
later
3 GM 2
Total accretional energy =
5 R
If all this energy goes into heat*,
what is the resulting temperature change?
3 GM
* Is this a reasonable
DT 
5 CpR
assumption?
Earth M=6x1024 kg R=6400km so DT=30,000K
Mars M=6x1023 kg R=3400km so DT=6,000K
What do we conclude from this exercise?
If accretion occurs by lots of small
impacts, a lot of the energy may be
lost to space
If accretion occurs by a few big
impacts, all the energy will be
deposited in the planet’s interior
So the rate and style of accretion (big
vs. small impacts) is important, as well
as how big the planet ends up
Cooling a planet
• Large silicate planets (Earth,
Venus) probably started out molten
– magma ocean
• Magma ocean may have been
helped by thick early atmosphere
(high surface temperatures)
• Once atmosphere dissipated, surface will have cooled rapidly
and formed a solid crust over molten interior
• If solid crust floats (e.g. plagioclase on the Moon) then it will
insulate the interior, which will cool slowly (~ Myrs)
• If the crust sinks, then cooling is rapid (~ kyrs)
• What happens once the magma ocean has solidified?
Cooling
• Radiation
– Photon carries energy out into space
– Works if opacity is low
– Unimportant in interior, only works at surface
• Conduction
– Heat transferred through matter
– Heat moves from hot to cold
– Slow; dominates in lithosphere and boundary layers
• Convection
– Hot, buoyant material carried upward, Cold, dense
material sinks
– Fast! Limited by viscosity of material
Running down the stairs with
buckets of ice is an effective
way of getting heat upstairs.
-- Juri Toomre
Conduction - Fourier’s Law T >T
1
(T1  T0 )
dT
k
• Heat flow F F  k
d
dz
0
T0
F
d
T1
• Heat flows from hot to cold (thermodynamics) and is
proportional to the temperature gradient
• Here k is the thermal conductivity (W m-1 K-1) and units
of F are W m-2 (heat flux is power per unit area)
• Typical values for k are 2-4 Wm-1K-1 (rock, ice) and 3060 Wm-1K-1 (metal)
• Solar heat flux at 1 A.U. is 1300 W m-2
• Mean subsurface heat flux on Earth is 80 mW m-2
• What controls the surface temperature of most planetary
bodies?
Diffusion Equation
• We can use Fourier’s law and the
definition of Cp to find how temperature
changes with time:
F2
dz
F1
T
k  2T
 2T

k 2
2
t C p z
z
• Here k is the thermal diffusivity (=k/Cp) and has units of m2 s-1
• Typical values for rock/ice 10-6 m2s-1
In steady-state, the heat produced inside the planet
exactly balances the heat loss from cooling. In this
situation, the temperature is constant with time
T
0
t
Diffusion length scale
• How long does it take a change in temperature
to propagate a given distance?
• This is perhaps the single most important
equation in the entire course:
2
d ~ kt
• Another way of deducing this equation is just by
inspection of the diffusion equation
• Examples:
– 1. How long does it take to boil an egg?
d~0.02m, k=10-6 m2s-1 so t~6 minutes
– 2. How long does it take for the molten Moon to cool?
d~1800 km, k=10-6 m2s-1 so t~100 Gyr.
What might be wrong with this answer?
Internal Heating
• Assume we have internal heating H (in Wkg-1)
• From the definition of Cp we have Ht=DTCp
• So we need an extra term in the heat flow equation:
T
 2T H
k 2 
t
z
Cp
• This is the one-dimensional, Cartesian thermal diffusion
equation assuming no motion
• In steady state, the LHS is zero and then we just have
heat production being balanced by heat conduction
• The general solution to this steady-state problem is:
T  a  bz 
H
2kC p
z
2
Example
• Let’s take a spherical, conductive planet in steady state
• In spherical coordinates, the diffusion equation is:
T
1   2 T  H
 k 2 r
0

t
r r  r  C p
• The solution to this equation is
T ( r )  Ts 
H
6k
( R2  r2 )
Here Ts is the surface temperature, R is the planetary radius,  is the density
• So the central temperature is Ts+(HR2/6k)
• E.g. Earth R=6400 km, =5500 kg m-3, k=3 Wm-1K-1,
H=6x10-12 W kg-1 gives a central temp. of ~75,000K!
• What is wrong with this approach?
Convection
• Convective behaviour is governed by the Rayleigh
number Ra
• Higher Ra means more vigorous convection, higher
heat flux, thinner stagnant lid
• As the mantle cools, h increases, Ra decreases, rate of
cooling decreases -> self-regulating system
Stagnant lid (cold, rigid)
Plume (upwelling, hot)
Sinking blob (cold)
gDTd
Ra 
kh
Image courtesy Walter Kiefer, Ra=3.7x106, Mars
3
Viscosity
• Ra controls vigor of convection. Depends
inversely on viscosity, h .
• Viscosity depends on Temperature T, Pressure
P, Stress s, Grain Size d.
h  Ae
A – pre-exponential constant
V – Activation Volume
n – Stress Exponent
E  PV
RT
s d
n
m
E – Activation Energy
R – Gas Constant
m – Grain-size exponent
Viscosity relates stress and strain rate
s  he
Viscoelasticity
• A Maxwellian material has a viscous term and an
elastic term.

s s
e  
h 
• If h is high, we get an elastic behavior.
If h is low, we get a viscous behavior.
• Depends also on the rate of stress.
Materials are elastic on a short timescale,
viscous on a long one.
• There are other types of viscoelasticity, but
Maxwell is the simplest
Elastic Flexure
• The near-surface, cold parts of a planet (the
lithosphere) behaves elastically
• This lithosphere can support loads (e.g. volcanoes)
• We can use observations of how the lithosphere
deforms under these loads to assess how thick it is
• The thickness of the lithosphere tells us about how
rapidly temperature increases with depth i.e. it
helps us to deduce the thermal structure of the
planet
• The deformation of the elastic lithosphere under
loads is called flexure
• EART163: Planetary Surfaces
Flexural Stresses
load
Crust
Elastic plate
Mantle
• In general, a load will be supported by a
combination of elastic stresses and buoyancy forces
(due to the different density of crust and mantle)
• The elastic stresses will be both compressional and
extensional (see diagram)
• Note that in this example the elastic portion includes
both crust and mantle
Flexural Parameter
• Consider a load acting
on an elastic plate:
w
load
m
Te

• The plate has a particular elastic thickness Te
• If the load is narrow, then the width of deformation is
controlled by the properties of the plate
• The width of deformation  is called the flexural
parameter and is given by
   3g ( 

ETe3
m   w )(1
2

)
1
4
E is Young’s modulus, g is gravity and n is Poisson’s ratio (~0.3)
• If the applied load is much wider than , then the
load cannot be supported elastically and must be
supported by buoyancy (isostasy)
• If the applied load is much narrower than , then
the width of deformation is given by 
• If we can measure a flexural wavelength, that
allows us to infer  and thus Te directly.
• Inferring Te (elastic thickness) is useful because Te
is controlled by a planet’s temperature structure

Example
10 km
• This is an example of a profile
across a rift on Ganymede
• An eyeball estimate of  would
be about 10 km
• For ice, we take E=10 GPa,
D=900 kg m-3 (there is no
overlying ocean), g=1.3 ms-2
Distance, km
•
•
•
•
If =10 km then Te=1.5 km
A numerical solution gives Te=1.4 km – pretty good!
So we can determine Te remotely
This is useful because Te is ultimately controlled by the
temperature structure of the subsurface
Te and temperature structure
• Cold materials behave elastically
• Warm materials flow in a viscous fashion
• This means there is a characteristic temperature
(roughly 70% of the melting temperature) which
defines the base of the elastic layer
Depth
•E.g. for ice the base of the elastic layer 270 K
110 K
190 K
is at about 190 K
• The measured elastic layer thickness is
1.4 km
1.4 km (from previous slide)
elastic
• So the thermal gradient is 60 K/km
• This tells us that the (conductive) ice
viscous
shell thickness is 2.7 km (!)
Temperature
Te in the solar system
• Remote sensing observations give us Te
• Te depends on the composition of the material (e.g.
ice, rock) and the temperature structure
• If we can measure Te, we can determine the
temperature structure (or heat flux)
• Typical (approx.) values for solar system objects:
Body
Te (km)
Body
Te
30
dT/dz
(K/km)
15
Earth
(cont.)
Venus
(450oC)
30
Mars
(recent)
Europa
100
5
2
40
Moon
15
(ancient)
Ganymed 2
dT/dz
(K/km)
15
30
40
Next Time
• Paper Discussion – Stevenson (2001)
• Planetary Interiors
– Cooling Mechanisms
– Rheology: How does the material deform?
– Magnetism