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EART 160: Planetary Science
13 February 2008
Last Time
• Planetary Interiors
– Pressure and Temperature inside Planets
– Heat Sources
• Accretion
• Differentiation
• Radioactivity
Today
• Get your midterm reviews in
• Class Schedule Revision
• Planetary Interiors
– Cooling Mechanisms
• Conduction
• Convection
– Rheology
• Viscoelasticity
• Flexure
Revised Schedule
W 13 Feb
Terrestrial Planet Interiors -- Cooling
F 15 Feb
Terrestrial Planet Atmospheres -- Structure and Dynamics
M 18 Feb
Presidents Day -- no class
W 20 Feb
Terrestrial Planet Atmospheres -- Origin and Escape
HW 4
F 22 Feb
Jovian Planets
HW 5
M 25 Feb
Rings, Moons, and Tides
W 27 Feb
Icy Satellites
F 29 Feb
Kuiper Belt / Comets
M 03 Mar
Solar System Exploration
W 05 Mar
Extrasolar Planets, Astrobiology
F 07 Mar
Planets and Politics: NASA, Spacecraft Missions, Funding
M 10 Mar
LPSC – no class -- Work on your projects!
W 12 Mar
LPSC – no class -- Work on your projects!
F 14 Mar
LPSC – no class -- Work on your projects!
M 17 Mar
Course Review
W 19 Mar
Final Exam (4 – 7 PM)
HW 6
HW 7
Project due
Paper Discussions
F 15 Feb
Tobie et al. (2006) Nature
440, 61-64
Titan Atmosphere
Elena Amador
F 22 Feb
Guillot et al. (1999), Science
286, 72-77
Giant Planets
Ryan Cook
Gladman et al. (1997),
M 25 Feb
Science 277, 197-201 ?
Orbital Resonances
Aaron Masters
Porco et al. (2006) Science
W 27 Feb
311, 1393-1401
Enceladus
???
Oort Cloud Comets
Miljan Draganic
Astrobiology
Matthew
Kammerer
NASA
Shayna Kram
F 29 Feb
Levison et al. (2002),
Science 296, 2212-2215
Vogel 1999 Science 286 70
W 05 Mar
?
F 07 Mar
Something direct from
NASA?
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 rC p z
z
• Here k is the thermal diffusivity (=k/rCp) 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 
rH
6k
( R2  r 2 )
Here Ts is the surface temperature, R is the planetary radius, r is the density
• So the central temperature is Ts+(rHR2/6k)
• E.g. Earth R=6400 km, r=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
• Ra is the ratio of buoyancy forces to diffusive forces
• 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)
rg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  h
Viscoelasticity
• A Maxwellian material has a viscous term and an
elastic term.
s s
  
h E
• 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
Next Time
• Paper Discussion – Titan Atmosphere
– Tobie et al., 2006
• Planetary Interiors
– Elastic Flexure
• Planetary Atmospheres
– Structure
– Dynamics