Francis Nimmo

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EART162 PLANETARY INTERIORS

• Francis Nimmo

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Last Week

• Tidal bulge amplitude depends on mass, position,
  rigidity of body, and whether it is in
  synchronous orbit
• Tidal Love number is a measure of the amplitude
  of the tidal bulge compared to that of a uniform
  fluid body
• Tidal torques are responsible for orbital
  evolution e.g. orbit circularization, Moon moving
  away from Earth etc.
• Tidal strains cause dissipation and heating
• Orbits are described by mean motion n, semi-major
  axis a and eccentricity e.
• Orbital angular momentum is conserved in the
  absence of external torques if a decreases, so
  does e

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This Week

• Case Study we will do a whole series of
  calculations, which give you an idea of roughly
  what planetary scientists actually do
• We may not be as precise or sophisticated as the
  real thing, but the point is that you can get a
  very long way with order of magnitude / back of
  the envelope calculations!
• This should also serve as a useful reminder of
  many of the techniques youve encountered before

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Case Study Europa
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Galilean Satellites

• Large satellites orbiting Jupiter
• Europa is roughly Moon-size (1500 km radius)
• 3 inner satellites are in a Laplace resonance
  (periods in the ratio 124) (what about their
  orbital radii?)
• Orbital eccentricities are higher than expected
  due to this resonance (tidal heating)

Callisto
Europa
Ganymede
Io
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Surface Observations

• Only lightly cratered (surface age 60 Myr)
• Surface heavily deformed

100km
lenticulae
bands
ridges
chaos
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What is it like?

• Cold ( 120K on average)
• Rough heavily tectonized
• Young surface age 60 Myrs
• Icy, plus reddish non-ice component, possibly
  salts?
• Trailing side darker and redder, probably due
  preferential implantation of S from Io
• Interesting it has an ocean, maybe within a few
  km of the surface, and possibly occasionally
  reaching the surface

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Bulk Properties
Europa
Io
Radius 1560 km
M5x1022 kg
So bulk density 3 g/cc What does this tell
us? Whats the surface gravity? Whats the
pressure at the centre?
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Gravity field

• MacCullaghs formula tells us how the
  acceleration varies with latitude (f)

• So whats the difference between the acceleration
  at the poles and at the equator?

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Now we have J2 what next?
What causes the flattening?

• We really want C how do we get it?
• Measure the precession rate a (C-A)/C, or . . .
• Assume hydrostatic
• Is hydrostatic assumption reasonable?

Here a is equatorial radius

• Plug in the values, we get C/Ma20.35. So what?

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Interior Structure

• Lets assume a very simple two-layer model

Mass constraint
MoI constraint
Combining the two

• We know R (1560 km), M (5x1022 kg) and C/MR2
  (0.346)
• Assuming a value for rm, we can solve for f,g . .
  .

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Interior structure (contd)

• Assuming rm1000 kg m-3, we get f0.89 and g2.82
  (see diagram)
• The implied density of the interior (3820 kg m-3)
  is greater than low-pressure mantle silicates.
  Could the density simply be due to high pressures?

1565
1393
3.82
1.00

• Remember the simple equation of state (Week 3)

Where does this come from?

• Use K200 GPa, g1.3ms-2, r03300 kgm-3, this
  gives r3400 kgm-3 at the centre
• What do we conclude from this?

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Summary

• Radius, mass give us density and some constraint
  on the bulk structure (mostly rock/metal, not
  ice)
• J2 (from gravity) gives us C-A
• Hydrostatic assumption gives us C from J2
• C/Ma2 allows us to make further inferences e.g.
  how thick the outer ice shell is, presence of an
  iron core

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Interior Structure

• Probably similar to Io, but with a layer of ice
  (100 km) on top
• We cant tell the difference between ice and
  water due to density alone
• Magnetometer data strongly suggest ocean at least
  a few km thick (see later)
• Thickness of solid ice shell not well known (see
  later)

Ice shell
Ocean
120km
Silicate mantle
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Why do we think theres an ocean?

• Jupiters varying field induces a current and a
  secondary magnetic field inside Europa
• Galileo detected this secondary field
• The amplitude of the secondary field depends on
  how conductive Europas interior is

• The results are consistent with a shallow salty
  ocean gt a few km thick
• Why couldnt the conductive layer be deeper?

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So What?

• Astrobiology (groan)
• Interesting physical problem why hasnt the
  ocean frozen?

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How could we check the ocean exists?

• Equilibrium tide
• Tidal amplitude d is reduced by rigidity m,
  depending on the Love number

• What is the size of the equilibrium tide for
  Europa? (m/M40,000, a/R430)
• What is the size of the fluid diurnal tide?
• How big would the diurnal tide be if there were
  no ocean?

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Europas Temperature Structure
T

• Whats the surface temperature?
• If there were only radioactive heat sources, how
  thick would the conductive ice shell be?
• Is the ocean convecting? Whats the temperature
  gradient?
• How long would the ocean take to freeze?
• Are there other heat sources weve forgotten
  about?

ice
ocean
z
mantle
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Tidal Dissipation
Eccentric orbit
Diurnal tides can be large e.g. 30m on Europa
Satellite
Jupiter

• Recall from Week 8, dissipation per unit volume
• How much power is being dissipated in the ice?
  What about in the mantle?
• What is the effect of the dissipation on the ice
  shell thickness?

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Equilibrium Shell thickness
Heat flow
Heat production
Heat loss
Mantle
Equilibrium
Shell thickness

• Lets put some numbers on this . . .
• How reliable is the shell thickness derived?
• Is the shell really conductive? How might we
  tell?

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Convection!

• Maybe the ice shell is convecting?
• How thick would the ice shell have to be for
  convection to occur?
• Congratulations you have just written a Nature
  paper! (Because you have just constrained the ice
  shell thickness)

• What kind of topography would you expect to be
  associated with the convection?
• If the ice shell is convecting, what happens to
  the equilibrium shell thickness argument?

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Equilibrium Shell Thickness

• Why does convective heat transport decrease as
  shell thickness increases?
• Obtain equilibrium shell thickness 20-50 km

• What would happen if Europas mantle was like
  Ios?
• Is the shell actually in steady state?
• How else might we measure the shell thickness?

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Flexural models

• Wavelength of deformation gives rigidity of ice
  (can be converted to elastic thickness Te see
  Week 4)
• Rigidity can be converted to shell thickness
  (assuming a conductive temperature structure)
• tc 2-3 Te

Temp.
270 K
100 K
190 K
Te
elastic
Depth

• What determines the temperature at which the ice
  ceases to behave elastically?

viscous
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Flexure and gravity

• There seem to be a wide range of elastic
  thicknesses on Europa, from 0.1-6 km. Why?
• What constraints do these values place on the
  shell thickness?

• What sort of gravity anomaly would you see at the
  surface associated with this feature?
• What about at 100 km altitude?
• What if it were compensated?

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Icebergs

• Icebergs and the edges of chaos regions stand
  a few 100 m higher than the matrix
• What does this observation imply about the
  thickness of the ice blocks? (Another Nature
  paper in the bag!)
• Do chaos regions really involve liquid water?
• Rotation and translation of blocks suggest a
  liquid matrix

h
iceberg
water
tc
tc 10 h
40 km
From Carr et al., Nature, 1998
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What about seismology?

• What would the velocities of P and S waves be on
  Europa?
• What would the potential sources of seismicity
  be?
• How would you use them to measure the shell
  thickness?
• What other remote-sensing techniques can you
  think of to constrain the shell thickness?

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Where does all that deformation come from, anyway?

• How much stress do we need to get deformation?
• What are the sources of stress we can think of?

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Conclusions

• Planetary science is not that hard
• A few observations can go a very long way
• The uncertainties are so large that simple
  approaches are perfectly acceptable
• Combining surface observations with simple
  calculations is the right way to proceed

• Although sometimes it can get you into trouble .
  . .

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Planning Ahead . . .

• Week 9
• Tues 27th case study I
• Thurs 29th no lecture
• Week 10
• Tues 3rd case study II
• Thurs 5th revision lecture
• Final Exam Tues 10th June 800-1100 a.m.

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Basic parameters

• Note higher eccentricity and greater degree of
  mass concentration than the Moon