EART160 Planetary Sciences

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EART160 Planetary Sciences
Francis Nimmo
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Last Week

• Volcanism happens because of higher temperatures,
  reduced pressure or lowered solidus
• Conductive cooling time t d2/k
• Planetary cooling leads to compression
• Elastic materials s E e

• Flexural parameter controls the lengthscale of
  deformation of the elastic lithosphere

• Lithospheric thickness tells us about thermal
  gradient
• Bodies with atmospheres/hydrospheres have
  sedimentation and erosion Earth, Mars, Venus,
  Titan

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This Week Interiors

• Mostly solid bodies (gas giants next week)
• How do we determine a planets bulk structure?
• How do pressure and temperature vary inside a
  planet?
• How do planets lose heat?
• See also EART 162

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Planetary Mass

• The mass M and density r of a planet are two of
  its most fundamental and useful characteristics
• These are easy to obtain if something (a
  satellite, artificial or natural) is in orbit
  round the planet, thanks to Isaac Newton . . .

Wheres this from?
Here G is the universal gravitational constant
(6.67x10-11 in SI units), a is the semi-major
axis (see diagram) and w is the angular frequency
of the orbiting satellite, equal to 2p/period.
Note that the mass of the satellite is not
important. Given the mass, the density can
usually be inferred by telescopic measurements of
the bodys radius R
a
ae
focus
e is eccentricity
Orbits are ellipses, with the planet at one focus
and a semi-major axis a
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Bulk Densities

• So for bodies with orbiting satellites (Sun,
  Mars, Earth, Jupiter etc.) M and r are trivial to
  obtain
• For bodies without orbiting satellites, things
  are more difficult we must look for subtle
  perturbations to other bodies orbits (e.g. the
  effect of a large asteroid on Mars orbit, or the
  effect on a nearby spacecrafts orbit)
• Bulk densities are an important observational
  constraint on the structure of a planet. A
  selection is given below

Object Earth Mars Moon Mathilde Ida Callisto Io Saturn Pluto
R (km) 6378 3390 1737 27 16 2400 1821 60300 1180
r (g/cc) 5.52 3.93 3.34 1.3 2.6 1.85 3.53 0.69 1.9
Data from Lodders and Fegley, 1998
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What do the densities tell us?

• Densities tell us about the different proportions
  of gas/ice/rock/metal in each planet
• But we have to take into account the fact that
  bodies with low pressures may have high porosity,
  and that most materials get denser under
  increasing pressure
• A big planet with the same bulk composition as a
  little planet will have a higher density because
  of this self-compression (e.g. Earth vs. Mars)
• In order to take self-compression into account,
  we need to know the behaviour of material under
  pressure.
• On their own, densities are of limited use. We
  have to use the information in conjunction with
  other data, like our expectations of bulk
  composition (see Week 1)

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Bulk composition (reminder)
Element C O Mg Si S Fe
Log10 (No. Atoms) 7.00 7.32 6.0 6.0 5.65 5.95
Condens. Temp (K) 78 -- 1340 1529 674 1337

• Four most common refractory elements Mg, Si, Fe,
  S, present in (number) ratios 110.90.45
• Inner solar system bodies will consist of
  silicates (Mg,Fe,SiO3) plus iron cores
• These cores may be sulphur-rich (Mars?)
• Outer solar system bodies (beyond the snow line)
  will be the same but with solid H2O mantles on top

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Example Venus

• Bulk density of Venus is 5.24 g/cc
• Surface composition of Venus is basaltic,
  suggesting peridotite mantle, with a density 3
  g/cc
• Peridotite mantles have an MgFe ratio of 91
• Primitive nebula has an MgFe ratio of roughly
  11
• What do we conclude?
• Venus has an iron core (explains the high bulk
  density and iron depletion in the mantle)
• What other techniques could we use to confirm
  this hypothesis?

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Pressures inside planets

• Hydrostatic assumption (planet has no strength)

• For a planet of constant density r (is this
  reasonable?)

• So the central pressure of a planet increases as
  the square of its radius
• Moon R1800km P7.2 GPa Mars R3400km P26 GPa

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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 materials
  density to change significantly depends on the
  bulk modulus of that material

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)

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Real planets

• Which planet is this?
• Where does this information come from?

• Notice the increase in mantle density with depth
  is it a smooth curve?
• How does gravity vary within the planet?

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Other techniques

• There are other things we can do (not covered
  here, see ES162 Planetary Interiors)
• We can make use of more gravitational information
  to determine the moment of inertia of a body, and
  hence the distribution of mass within its
  interior
• There are also other techniques
• Seismology (Earth, Moon)
• Electromagnetic studies (Earth, Moon, Galilean
  satellites)

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Temperature Structures

• 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
  planets interior and surface
• We can get some information on this gradient by
  measuring the elastic thickness (Week 3)
• The temperature gradient means that the planet
  will tend to cool down with time

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Conduction - Fouriers Law
T1gtT0
T0

• Heat flow F

d
F
T1

• Heat flows from hot to cold (thermodynamics) and
  is proportional to the temperature gradient
• Here k is the thermal conductivity (Wm-1K-1) and
  units of F are Wm-2 (heat flux is per unit area)
• Typical values for k are 2-4 Wm-1K-1 (rock, ice)
  and 30-60 Wm-1K-1 (metal)
• Solar heat flux at 1 A.U. is 1300 Wm-2
• Mean subsurface heat flux on Earth is 80 mWm-2
• What controls the surface temperature of most
  planetary bodies?

milliWatt10-3W
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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
• Units J kg-1 K-1
• Typical values rock 1200 J kg-1 K-1 , ice 4200 J
  kg-1 K-1
• Energy mass x specific heat capacity x temp.
  change
• E.g. if the temperature gradient near the Earths
  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?

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Energy of Accretion

• Lets 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?

In which situation is more energy delivered?
early
later
Total accretional energy
If all this energy goes into heat, what is the
resulting temperature change?
Is this a reasonable assumption?
Earth M6x1024 kg R6400km so DT30,000K Mars
M6x1023 kg R3400km so DT6,000K What do we
conclude from this exercise?
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Accretion and Initial Temperatures

• 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 planets interior
• Additional energy is released as differentiation
  occurs dense iron sinks to centre of planet and
  releases potential energy as it does so
• What about radioactive isotopes? Short-lived
  radio-isotopes (26Al, 60Fe) can give out a lot of
  heat if bodies form while they are still active
  (1 Myr after solar system formation)
• A big primordial atmosphere can also keep a
  planet hot
• So the rate and style of accretion (big vs. small
  impacts) is important, as well as how big the
  planet ends up

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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?

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Cooling a planet (contd)

• Planets which are small or cold will lose heat
  entirely by conduction
• For planets which are large or warm, the interior
  (mantle) will be convecting beneath a
  (conductive) stagnant lid (also known as the
  lithosphere)
• Whether convection occurs depends if the Rayleigh
  number Ra exceeds a critical value, 1000

Temp.
Stagnant (conductive) lid
Here r is density, g is gravity, a is thermal
expansivity, DT is the temperature contrast, d is
the layer thickness, k is the thermal diffusivity
and h is the viscosity. Note that h is strongly
temperature-dependent.
Convecting interior
Depth.
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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 -gt self-regulating
  system

Stagnant lid (cold, rigid)
Plume (upwelling, hot)
Sinking blob (cold)
The number of upwellings and downwellings depends
on the balance between internal heating and
bottom heating of the mantle
Image courtesy Walter Kiefer, Ra3.7x106, Mars
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Diffusion Equation

• The specific heat capacity Cp is the change in
  temperature per unit mass for a given amount of
  energy WmCpDT
• We can use Fouriers law and the definition of Cp
  to find how temperature changes with time

F2
dz
F1

• Here k is the thermal diffusivity (k/rCp) and
  has units of m2s-1
• Typical values for rock/ice 10-6 m2s-1

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Diffusion lengthscale (again)

• 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
• 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?
• d0.02m, k10-6 m2s-1 so t6 minutes
• 2. How long does it take for the molten Moon to
  cool?
• d1800 km, k10-6 m2s-1 so t100 Gyr.
• What might be wrong with this answer?

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Heat Generation in Planets

• Most bodies start out hot (because of
  gravitational energy released during accretion)
• But there are also internal sources of heat
• For silicate planets, the principle heat source
  is radioactive decay (K,U,Th at present day)
• For some bodies (e.g. Io, Europa) the principle
  heat source is tidal deformation (friction)
• Radioactive heat production declines with time
• Present-day terrestrial value 5x10-12 W kg-1
  (or 1.5x10-8 W m-3)
• Radioactive decay accounts for only about half of
  the Earths present-day heat loss (why?)

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Internal Heat Generation

• Assume we have internal heating H (in Wkg-1)
• From the definition of Cp we have HtDTCp
• So we need an extra term in the heat flow
  equation

• 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

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Example

• Lets take a spherical, conductive planet in
  steady state
• In spherical coordinates, the diffusion equation
  is

• The solution to this equation is

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 R6400 km, r5500 kg m-3, k3 Wm-1K-1,
  H6x10-12 W kg-1 gives a central temp. of
  75,000K!
• What is wrong with this approach?

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Summary

• Planetary mass and radius give us bulk density
• Bulk density depends on both composition and size
  
• Larger planets have greater bulk densities
  because materials get denser at high pressures
• The increase in density of a material is
  controlled by its bulk modulus
• Planets start out hot (due to accretion) and cool
• Cooling is accomplished (usually) by either
  conduction or convection
• Vigour of convection is controlled by the
  Rayleigh number, and increases as viscosity
  decreases
• Viscosity is temperature-dependent, so planetary
  temperatures tend to be self-regulating

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Key Concepts

• Bulk density
• Self-compression
• Bulk modulus
• Hydrostatic assumption
• Accretionary energy
• Magma ocean
• Conduction and convection
• Rayleigh number
• Viscosity
• Thermal diffusivity
• Diffusion lengthscale

WmCpDT
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End of Lecture
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Example - Earth

• Near-surface consists of a mechanical boundary
  layer (plate) which is too cold to flow
  significantly (Lecture 3)
• The base of the m.b.l. is defined by an isotherm
  (1400 K)
• Heat must be transported across the m.b.l. by
  conduction
• Lets assume that the heat transported across the
  m.b.l. is provided by radioactive decay in the
  mantle (true?)

By balancing these heat flows, we get
m.b.l.
d
interior
R
Here H is heat production per unit volume, R is
planetary radius
Plugging in reasonable values, we get m.b.l.
thickness d225 km and a heat flux of 16 mWm-2.
Is this OK?
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Deriving the Diffusion lengthscale

• How long does it take a change in temperature to
  propagate a given distance?
• Consider an isothermal body suddenly cooled at
  the top
• The temperature change will propagate downwards a
  distance d in time t

Temp.

• After time t, Fk(T1-T0)/d
• The cooling of the near surface layer involves an
  energy change per unit area DEd(T1-T0)Cpr/2
• We also have FtDE
• This gives us

T0
T1
Initial profile
d
Depth
Profile at time t
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• Io cooling
• 26Al heating
• Grav heating

Rp
R
solid
porous
d
crust
mantle