Francis Nimmo

1
ESS 298 OUTER SOLAR SYSTEM

• Francis Nimmo

Io against Jupiter, Hubble image, July 1997
2
Last Week

• Solar system characteristics and formation Hill
  sphere, snow line, timescales
• Keplers laws and Newtonian orbits
• Tides
• Synchronous rotation
• Dissipation / heating
• Circularization and outwards migration

3
Galilean Satellites

• This week
• Preliminaries, common themes
• Io
• Callisto
• Next week
• Ganymede
• Europa

Callisto
Europa
Ganymede
Io
4
Why are they important?

• Life (!?) or sub-surface oceans, at any rate
• Relatively large (2000 km), geologically active
• Very complicated orbital relationships
• Some processes look familiar e.g. plate tectonics
  (?)
• Future exploration target (maybe)
• Why Galilean? He discovered them (telescope,
  1610)
• Subsequent exploration Voyagers Pioneers
  (1970s), Galileo (ended 2003), JIMO??

5
Voyagers 1 and 2

• A brilliantly successful series of fly-bys
  spanning more than a decade
• Close-up views of all four giant planets and
  their moons
• Both are still operating, and collecting data on
  solar/galactic particles and magnetic fields

Voyagers 1 and 2 are currently at 90 and 75 AU,
and receding at 3.5 and 3.1 AU/yr Pioneers 10
and 11 at 87 and 67 AU and receding at 2.6 and
2.5 AU/yr
The Death Star (Mimas)
6
Galileo
antenna

• More modern (launched 1989) but the high-gain
  antenna failed (!) leaving it crippled
• Venus-Earth-Earth gravity assist
• En route, it observed the SL9 comet impact into
  Jupiter
• Arrived at Jupiter in 1995 and deployed probe
  into Jupiters atmosphere
• Very complex series of fly-bys of all major
  Galilean satellites
• Deliberately crashed into Jupiter Sept 2003
  (why?)
• Main source of results

7
Where are they?
8
Laplace Resonance (1)

• Periods of IoEuropaGanymede are in ratio 124
• This means that successive conjunctions occur at
  the same point on the orbit
• So the eccentricities get pumped up to much
  higher values than if the satellites were not in
  a resonance

One of the conjunctions occurring due to the
Laplace resonance. Note that there is never a
triple conjunction.

• High eccentricities mean higher dissipation in
  the satellites and a tendency for the orbits to
  contract (see earlier)
• This tendency is counteracted by dissipation in
  Jupiter, which tends to cause the orbits to
  expand (like the Moon)
• The system is currently (roughly) in equilibrium

9
Compositions/Formation

• Surface compositions mainly water ice (except
  for Io), plus contaminants (spectroscopy)
• Ios surface is silicates sulphur
• Interiors discussed in detail later, but
  roughly equal mix of water ice, silicates and
  iron (how do we know?)
• How did they form?
• Presumably accreted while Jupiter was forming
• Lateral temperature gradient in nebula
• May have been earlier satellites that didnt
  survive (why?)
• Energy of accretion 0.6 Gms/Rs per unit mass 2
  MJ/kg this is enough to heat ice through 1000
  K. Why might this present a problem?
• Satellites subsequently evolved to their
  present-day positions

10
Composition (contd)

• Callisto has lower reflectivity and shallower
  absorption bands, indicating a higher non-ice
  component
• Ganymede and Callisto show slight differences
  between leading and trailing hemispheres (why?)
• Non-ice materials are probably hydrated minerals
  (clays)

Earth-based reflectance spectra, from Johnson, in
New Solar System
11
Differentiation

• Potential energy of a homogeneous satellite is
  reduced if the densest components sink to the
  centre differentiation is energetically
  favoured
• Differentiation is opposed by rigidity of body,
  so differentiation is favoured at higher
  temperatures
• As differentiation proceeds, energy is released,
  driving further differentiation potential
  runaway
• Heat released may generate thermal expansion and
  form a source of stress
• Sinking materials may undergo phase changes
  leading to volume changes and either expansion or
  contraction
• Not all Galilean satellites appear to have
  differentiated

12
Ice phase diagram
Ice I
Water
Ice V

• The key point is that because of the densities
  involved, we would expect to find liquid water
  around the ice I ice V interface (200 km
  depth). Why is this important?

13
Internal Structures (1)

• Because the satellites are rotating, they are
  flattened (oblate)
• This means that they do not act as a point mass
  the perturbations to the gravity field can be
  identified by tracking spacecraft on a close
  approach
• Potential V at a distance r for axisymmetric body
  is given by

• So the coefficients J2, J4 etc. can be determined
  from spacecraft observations (higher order terms
  require closer approaches why?)
• We can relate J2,J4 . . . to the internal
  structure of the satellite

14
Internal Structures (2)
C

• Mean density and J2 are especially useful
• It turns out that we can rewrite J2 in terms of
  the differences in moments of inertia of the
  planet (look at the diagram )

R
A

• What we would really like is C/MR2 (why?)
• If we can observe the precession of the planet,
  that gives us (C-A)/C and thus C given J2 (where
  can we do this?)
• Otherwise, we can assume that the planet has no
  strength (hydrostatic) and use theory to infer C
  from J2 (is this OK?)
• In practice, flybys of the Galilean satellites
  were usually equatorial (why?), so we determine
  the equivalent equatorial term to J2 which is
  called C22 the analysis is similar

15
Internal Structures (3)

• How do we know?
• Mean density
• Moment of inertia, derived from J2(C-A)/MR2 and
  hydrostatic assumption (is this likely?)
• Other observations (magnetometer)
• Expectations of likely components (silicates,
  ices, iron)
• Tradeoffs we only have two observations (J2 and
  r) and have more than two unknowns. Means the
  results are non-unique

Fe core
Fe-FeS core
Contours of Europa ice shell thickness giving
correct mean density for indicated core radius
and rock density. Bold line is MoI constraint.
From Anderson et al., Science, 1998
16
Magnetometer (1)

• Jupiters magnetic pole is offset from its
  rotation pole
• So as Jupiter rotates (10 hour period),
  satellites experience a time-varying magnetic
  field

• A time varying magnetic field induces eddy
  currents in a conductor
• These currents generate a secondary (induced)
  magnetic field
• The amplitude of the secondary magnetic field
  tells us about the conductor, in particular its
  conductivity and thickness

17
Magnetometer (2)

• Strong induced signatures have been detected at
  Europa, Ganymede and Callisto, indicating a layer
  of high conductivity
• A relatively near-surface ocean at least a few km
  thick satisfies these observations
• The direction of the induced signal depends on
  the orbital geometry but permanent (static)
  signals have also been detected at Ganymede and
  (possibly) Io
• These static fields are presumably generated by
  convection within an iron core, just like the
  Earth
• We can combine the magnetometer constraints with
  the geodetic constraints to infer internal
  structures . . .

18
Io liquid iron core (dynamo), silicate mantle
(partially molten?). No volatiles why not?
Ganymede liquid iron core (dynamo), silicate
mantle and 800 km thick ice shell containing an
ocean (presumably at the I-III/V boundary)
Europa core and mantle similar to Ganymede, but
ice shell much thinner (100-200 km) and mostly
liquid (magnetic induction signature)
Callisto has not differentiated completely (?).
An ice layer 300km thick, containing an ocean
and overlying a mixture of rock-ice. NB the
hydrostatic assumption is particularly dodgy here
why?
19
Ice Rheology (1)

• Under applied stress, ice will deform
• At low stresses and strains, elastically
  (recoverable)
• At low temperature and/or high strain rate,
  brittle
• At high temperature and/or low strain rate,
  ductile

stress
brittle
elastic
depth
ductile

• A good measure of its tendency to deform in a
  ductile fashion is the homologous temperature
  (ThT/Tmelt) (in K)
• Rock at Earth surface Th0.2
• Ice at Galilean satellite surface Th0.4
• Ice in Antarctica/Mars Th0.8
• So ice at the surface of the Galilean satellites
  behaves more like rock than ice on Earth

20
Ice Rheology (2)

• Ductile deformation is important because it
  controls convection, topographic relaxation and
  tidal dissipation (see later)
• But ice deformation is complicated and involves
  multiple mechanisms (see Goldsby and Kohlstedt
  JGR 2001)
• Each mechanism obeys the same equation

Here is strain rate, A is a constant, s is
stress, gs is grain size, T is temperature, Q is
activation energy, R is the gas constant and n
and p are constants. A Newtonian rheology has n1
and a grain-size independent rheology has p0.
Increasing stress / strain rate
Grain-boundary sliding (ngt1, grain-size
dependent) Actually two mechanisms, slower one
dominates
Diffusion creep (n1, grain-size dependent)
Dislocation creep (ngt1,p0)
21
Orbital Evolution

• Recall dissipation in primary drives satellite
  outwards
• Dissipation in satellite drives satellite inwards
  and circularizes orbit
• Possible scenario
• Io causes dissipation in Jupiter, moves outwards
  until . . .
• It encounters the 21 resonance with Europa the
  two bodies then move outwards in step until . . .
• They encounter the 21 resonance with Ganymede
• There are alternative scenarios
• The present-day configuration involves a balance
  between dissipation in primary (outwards) and
  dissipation in satellites (inwards)

22
Hypothetical orbital history
from Peale, Celest. Mech. Dyn. Ast. 2003
Note that we dont actually know whether the
orbits are currently expanding or
contracting Also note that during capture into
resonance, eccentricities are transiently excited
to high values so what?
23
Estimating Q

• Recall that the rate of outwards motion of a
  satellite depends on planetary dissipation Qp
  (see Week 1).
• If we assume that Io formed 4.5 Gyr B.P., and has
  been moving outwards ever since, we get a lower
  bound on Jupiters Q of 105 (why a lower bound?)
• This value is typical of gas giants, but is much
  higher than for silicate bodies (102)
• The Earths Q is anomalously high (12) because
  the current continental configuration means
  oceanic tides are close to resonance lots of
  dissipation
• Well calculate the rate of dissipation in a
  second

24
Tidal Deformation Recap.

• Satellite in synchronous rotation period of
  rotation equals orbital period
• Eccentric orbit (due to Laplace resonance)
  amplitude and direction of tidal bulge changes,
  so surface experiences changing stresses and
  strains
• These diurnal tidal strains lead to friction and
  thus tidal dissipation (heating)

Diurnal tides can be large e.g. 30m on Europa
Satellite
Jupiter
Eccentric orbit
25
Tidal Heating (1)

• Recall diurnal tidal amplitude goes as
  in the limit when rigidity dominates (
  )
• So strain goes as
• Energy stored per unit volume stress x strain
• In an elastic body, stress strain x m
  (rigidity)
• So total rate of work goes as me2H2Rs/
• For tide raised on satellite HRs(mp/ms)(Rs/a)3
• From the above, we expect the energy stored E to
  go as

Note that here we have used
26
Tidal heating (2)

• From the definition of Q, we have
• Weve just calculated the energy stored E, so
  given Qs and n we can thus calculate the heating
  rate dE/dt
• The actual answer (for uniform bodies) is
• But the main point is that you should now
  understand where this equation comes from
• Example Io
• We get 80 mW/m2, about the same as for Earth (!)
• This is actually an underestimate why?

27
How do we calculate Q?

• We can get estimates/bounds on Q by considering
  orbital evolution of some satellites (see Week 1)
• For solid bodies, we assume a viscoelastic
  rheology
• Such a body has a rigidity m, a viscosity h and a
  characteristic relaxation (Maxwell) timescale
  tmh/m
• The body behaves elastically at timescales ltlttm
  and in a viscous fashion at timescales gtgt tm

• Dissipation is maximized when timescale tm

Tobie et al. JGR 2003
28
Calculating Q (contd)

• Ice has rigidity 109 Pa and viscosity 1014 Pa
  s, so the Maxwell time is 105s which is
  comparable to the orbital period, so we expect
  dissipation in the ice shells
• Silicates m1011 Pa, h1021 Pa s, so less
  dissipation
• But silicate viscosity decreases significantly if
  melting occurs, which will lead to an increase in
  dissipation, and thus a feedback effect
• This runaway situation was first identified by
  Peale et al. (1979), who predicted massive
  volcanism on Io two weeks before it was observed
  for the first time
• A similar feedback effect may also occur in ice
  (see previous diagram)

29
Tidal Energy and Stress

• Tidal stresses and heating decrease markedly with
  distance
• Radiogenic heating is dominant in Callisto and
  Ganymede (now), secondary in Europa, and
  insignificant for Io

H is static tidal bulge for a fluid body, 3eH
gives peak-to-peak diurnal tidal amplitude, dW/dt
is tidal dissipation rate for a uniform body with
Jupiters mass1.899x1027 kg, k3/2 and Q100,
dWR/dt is radiogenic heat production within
silicate portion of body assuming a heating rate
of 3.5x10-12 W/kg, EeH/Rs gives the approximate
stresses due to diurnal tides with E10 GPa,
C/msRs2 gives the normalized moment of inertia
(Anderson et al. 1996,1998b,2001a,b) and 3Gms/5Rs
gives the energy delived during homogeneous
accretion. A uniform body has a normalized MoI of
0.4.
30
Non-synchronous rotation (1)

• From the satellites point of view, the planet
  travels in the opposite direction round the sky
  to the satellite itself
• The tidal bulge always lags the planets motion
• In an eccentric orbit the amplitude of the tidal
  bulge varies and is largest at the periapse
• The result of the varying bulge is a varying
  torque, which turns out to be positive i.e. it
  should increase the satellites rotation rate
  slightly above synchronous

Eccentric orbit
satellite
Apoapse
Periapse
planet
Torque opposes spin Smaller
Torque increases spin Larger
31
Non-synchronous rotation (2)

• For an eccentric satellite, the net tidal torque
  should lead to non-synchronous rotation
• But the torque may be balanced by a frozen-in
  mass asymmetry, leading to synchronous rotation
• A frozen-in mass asymmetry requires a relatively
  rigid body

(See Greenberg and Weidenschilling, Icarus 1984)
Mass torque
Tidal torque

• Both the rigidity of the satellite and Q depend
  on its internal structure, so there are potential
  feedbacks between orbital evolution and rotation
  state
• Internal structure Orbital behaviour

32
Impact Cratering

• Main source of impact craters in outer solar
  system is comets
• Synchronously rotating satellite will be
  preferentially cratered on its leading hemisphere
  (think raindrops)
• So distribution of impact craters on surface can
  be used to test whether NSR has occurred
• Density of impact craters can be used to infer
  surface age
• Obtaining absolute surface ages requires the
  impact rate to be derived, from a combination of
  current and historical astronomical observations,
  and models. Uncertainties are currently large.
• Note that the impact rate will increase for
  satellites closer to the primary (effect of
  gravitational focusing)

33
Cratering Statistics
Zahnle et al. Icarus 1997
Expected curves if NSR is not occurring
Absolute ages have been revised upwards since
(Zahnle et al. Icarus 2003)
34
Cratering Statistics - Results

• Io no impact craters observed (!) so surface is
  very young (lt 1 Myr)
• Europa few impact craters, surface age 50 Myr.
  Not enough craters to detect if NSR is happening
• Ganymede bimodal surface, ages 2 Gyr and 4
  Gyr (uncertainties large). Spatial distribution
  flatter than expected, suggests NSR has occurred.
• Callisto very ancient surface, 4.5 Gyr.
  Spatial distribution very flat, but may be
  because crater population is saturated everywhere
  (i.e. one crater is destroyed for every new one
  produced)

35
Thermal Orbital Evolution

• We would like to be able to answer the question
  how have the satellites orbits and interiors
  evolved over solar system history?
• This is difficult because
• Observations are severely limited (e.g. cratering
  evidence is not much use on Io or Europa)
• Important parameters (such as Q) are uncertain
• The theoretical problem is difficult. Why?

• Feedbacks. Orbital evolution, NSR and tidal
  dissipation all depend on Q, but Q is dependent
  on the internal structure of the satellite, which
  depends on tidal dissipation . . .
• Coupling. The satellites cant be treated as
  isolated objects, because of the Laplace
  resonance. So you have to model their evolution
  simultaneously . . .

36
Thermal Orbital Evolution (contd)

• Nonetheless, progress is being made, both on the
  observational and the theoretical front. Well
  discuss examples of both later in the course.

This is an example of Europas shell thickness
evolution with time, from Hussmann and Spohn,
Europas Ice Shell Meeting, LPI, 2004. The
periodicity arises because Io and Europas
eccentricities change over time, which changes
the dissipation in Europas ice shell and thus
the shell thickness. In this model the shell is
convecting.
37
Summary

• Tides are important in determining spin state,
  orbital evolution and heating of satellite
• Ice rheology is complicated
• Near-surface, it will behave like rock on Earth
• At depth, it will flow at a geologically rapid
  rate
• Cratering observations can provide us with
  relative surface ages, but absolute ages are
  subject to large uncertainties
• Satellite internal structures are constrained by
  a mixture of observations (C/MR2, mean density,
  magnetometer) and reasonable expectations

38
Io
39
Basic Parameters

• Note the likely structural similarities with the
  Moon

40
Whats it like?

• Volcanically very active (see later)
• Cold surface temperature about 130K, but
  variable (due to volcanism)
• Very tenuous atmosphere (volcanic)
• Sulphur-rich surface deduced from ground-based
  spectroscopic observations (different colours are
  different sulphur allotropes)
• Very hostile environment (for people or
  spacecraft) charged particles accelerated by
  Jupiters large magnetic field
• Not clear whether Io has an internal magnetic
  field (Kivelson et al. JGR 2001) interactions
  with Jupiters field make identification difficult

41
Landforms

• Three main types Volcanoes, Mountains and
  Paterae (irregular depressions, similar to
  calderas)

200km
350km
flow
patera
Low-sun angle shadow measurements give mountain
elevations of up to 4km. Lobate flows are large
landslides. Mountains show no signs of volcanic
activity and appear to be fault-bounded.
volcano
42
Lava flows
Amirani lava flow, Io
500km

• Dark flows are the most recent (still too hot for
  sulphur to condense on them)
• Flows appear relatively thin, suggesting low
  viscosity

500km
Comparably-sized lava flow on Venus (Magellan
radar image)
43
Time-Variability

• Changes detected from Voyager to Galileo missions
  and within Galileo mission

Lava flow erupted at Prometheus between Voyager
and Galileo missions (Davies JGR 2003)
April 1997
July 1999
Sept 1997
Pillan
Galileo images of overlapping deposits at Pillan
and Pele
400km
Pele
44
Volcanic activity (1)
Voyager image of eruption plume, approximately
300 km high
Fire fountain(?)

• Galileo image of Tvashtar, apparently in the
  process of erupting
• The CCD was overloaded by the eruption, but it
  has been interpreted as a fire-fountain 1.5 km
  high

45
Volcanic activity (2)
Galieo nightside image of Pele, SSI clear filter.
Radebaugh et al. 2004
Erta Ale lava lake, Ethiopia. Lake is 100m across.

• Images suggest molten magma immediately beneath
  the surface (at least in some places)
• Volcanic activity erupts about 1 tonne / second
  sulphur into the atmosphere, some of which may
  end up on Europa (contaminants have been detected
  there)

46
Ground-based observations

• Have the advantage of longer observation periods
  and better spectral resolution than spacecraft
• Spatial resolution is also getting much better
  thanks to adaptive optics and Hubble
• The sequence below shows a hot spot which flares
  up to equal the brightness of Loki (spot 2) over
  a few days

1 arcsecond
From Macintosh et al., Icarus 2003
July 12 1998
July 28 1998
Aug 4 1998
Keck interferometer
47
Energetics (1)

• We can measure the power output of Io by looking
  at its infra-red spectrum
• Heat flux is appx. 2.5 W m-2 .This is 30 times
  the Earths global heat flux.

• Assume low rigidity ( ) why?. To
  balance the heat being produced requires Qs90.
  Is this reasonable? What does it imply about
  viscosity?
• Where does the power ultimately come from?
• A heat loss of 2.5 Wm-2 over 4.5 Gyr is
  equivalent to 0.03 of Jupiters rotational energy

48
Energetics (2)

• How do we get 2.5 Wm-2 out of the ground?
• A conductive layer (or convective stagnant lid)
  would need to be 1 km thick. Reasonable?
• What about magma transport (advection)?
• Silicate magma generates 5 GJ/m3 on cooling
  1000K and solidifying
• A resurfacing rate of 1 cm/yr can account for
  the observed surface heat flux
• This resurfacing rate is also consistent with
  estimates based on impact craters and IR cooling
  models
• So Io is unique among the solar system in that
  its heat flux is dominated by advection

49
Interior Structure
After Anderson et al., JGR 2001

• Lacks outer ice layer (in contrast to other
  Galilean satellites). Why?
• Even though sulphur is abundant at the surface,
  the bulk of the interior must be silicates/iron
  from simple cosmochemistry

Silicates 3500 kg m-3
1821 km
700 km
Fe-FeS 5150 kg m-3
Remember these structures are non-unique the
ones shown assume plausible but not necessarily
correct densities.

• Io likely has a crust, but we cant detect it
  with current data
• We cant tell (directly) whether the core or the
  mantle are partially or completely liquid.
• Ios k21.29. What is this telling us? (rigidity
  or mass concn.)

50
Interior Structure(?)

• Rigid lid is required by high mountains and is a
  result of rapid burial of cooled surface material
• Bulk of dissipation occurs in partially molten
  mantle
• Magma percolates through mantle and ascends
  through cold lid in discrete fractures i.e. dikes
• Erupted material cools by radiation and is
  re-buried

After Moore, Icarus, 2001
51
Consequences of resurfacing

• Burial leads to large compressive stresses due to
  change in radius
• Stress E DR/R 100 MPa for 2 km burial
• Easily large enough to initiate faulting
• Additional compressive stresses may arise from
  reheating the base of the crust

DR
After McKinnon et al., Geology 2001
stereo
Low-angle (why?) thrust faulting is probably
responsible for many of the mountain ranges seen
on Io
550 km
10km
Schenk and Bulmer, Science 1998
52
Eruption Spectra

• Recall Wiens law lmax a 1/T
• So infra-red spectra give temperature information

• Single temperature curve provides poor fit
• Two-temperature curve provides much better fit
• Short-wavelength hump requires temperatures
  gt1400K
• So silicate volcanism must be involved
• Voyager could not resolve this issue
• Time-evolution gives cooling history

Davies, JGR 2003
53
Plumes

• Whats the exit velocity?
• How do speeds like this get generated?
• Most likely explanation is sulphur geysers
  decompression of sulphur leads to phase change
  and volatile release, driving flow

Loki
Pele
500 K
Constant entropy (adiabatic)
Energy available per unit mass is given by change
in enthalpy (internal energy PV term). Typical
enthalpy changes 100 kJ/kg, which results in
velocities of 400 m/s
Liq.
Vap.
Pressure decreases
LV
200 K
SV
0 K
Entropy (J kg-1 K-1)
After Smith et al., Nature 1979
54
Callisto
55
Basic Parameters
Anderson et al. JGR 2001
Anderson et al. Icarus 2001

• Note the lower density and the fact that Callisto
  is more centrally concentrated than Io (see
  later)

56
Geological Observations

• Very heavily cratered probably saturated
• No obvious non-crater landforms tectonically
  dead
• Some impact basins very large e.g. Valhalla
• Also several crater chains (catenae). How did
  they form? Why are they useful?

1500km
600km
57
Mass Wasting

• Lobate features associated with steep crater
  walls
• Triggered by impacts or devolatilization?
• Plot in similar parameter space to terrestrial
  landslides, despite different materials and
  gravity why?

From Moore et al. Icarus 1999
58
Degradation / Sublimation

• Callisto systematically lacks small (lt1km)
  craters relative to Ganymede
• Craters show significant degradation on Callisto
• This may be due to the presence of a highly
  volatile ice (e.g. CO2) which is subliming over
  time
• Evidence for (thin) atmospheric CO2 supports this
  hypothesis

Moore et al. Icarus 1999
59
Internal Structure

• Two interesting inferences
• It has an ocean
• It is only partly differentiated
• Where do these inferences come from?

Probable ocean location

• Ocean detected with magnetometer data
• Partial differentiation is the only way to fit
  the MoI and density data (see )

Anderson et al. Icarus 2001. Two layer model of
Callisto showing inner and outer shell densities
which match observations
60
An Ocean?

• We can (potentially) detect such an ocean because
  it allows the shell to flex more than it would do
  if it were overlying a solid interior
• Thermal evolution of an ocean will be controlled
  by balance between heat added (from below) and
  heat transported to the surface
• Present-day chondritic heat flux 5 mW/m2
• Heat flux k DT/z (k3 W/mK, DT100 K)
• So equilibrium conductive shell thickness 60 km
• This seems reasonable but what happens if the
  ice shell starts to convect?

61
An Aside on Convection (1)

• Convective vigour (and whether it occurs) is
  governed by the Rayleigh number
• Convection initiates for Ra gt 1000
• Is Callisto convecting?
• So the answer is probably yes
• This creates a problem Convective heat transport
  is much more efficient than conduction, and so we
  would expect any ocean to have frozen long ago
• How much heat is transported by convection?

r is density, a thermal expansivity, DT
temperature drop across the layer, k thermal
diffusivity, h viscosity, d layer thickness
Where does this come from?
62
An Aside on Convection (2)

• For a temperature-dependent viscosity material, a
  stagnant lid develops on top of a roughly
  isothermal, convecting interior
• The viscosity is given by hoexp(-gT-To) where
  ho is the reference viscosity at To and g is a
  constant (K-1) set by the rheology

Convection

• The stagnant lid thickness d is given by
• And so the heat flux across the stagnant lid is

Note that this heat flux is independent of shell
thickness and DT
63
Convecting ice shells (contd)

• For likely parameters, we get a convective heat
  flux of
• 70 (1014 Pa s /h)1/3 mWm-2
• This value is independent of shell thickness and
  exceeds the radiogenic contribution if h lt 3x1017
  Pa s (which would result in the ocean freezing)
• Tidal contribution to heating is negligible
• Most likely way of maintaining an ocean is by
  increasing the viscosity. Possibilities
• Antifreeze e.g. NH3 lowers temperature of ocean
  (and convecting ice) (see Spohn and Schubert
  Icarus 2003)
• Silicate particles in ice increase its viscosity
• Very large ice grains (?)
• Non-Newtonian convection less efficient (?)(Ruiz,
  Nature 2001)

64
Partially differentiated?

• Partial differentiation implies that the interior
  of Callisto never got above 270K (why?)
• 1) How do we stop melting during accretion?
• Accretion energy 0.6 GM2/R 1.7 MJ/kg
• This would give rise to 850 K temperature
  increase
• The nebular temperatures might also cause melting
• 2) How do we stop melting thereafter?
• Chondritic heating 3.5pW/kg now, x3 over 4.5
  Gyr
• Total 1.5 MJ/kg 750 K temperature increase

• Possible answers (or maybe it is
  differentiated?)
• 1) Accrete Callisto slowly (so that the energy
  can radiate)
• 2) Get rid of the heat rapidly enough to avoid
  deep melting (but slowly enough so that the
  shallow ocean survives)

65
Slow Accretion (?)

• If we assume that satellites accrete from small
  bodies, the temperature rise of the satellite is
  determined by the accretion rate (slower rate
  colder temperature)
• Canup and Ward (A.J. 2002) postulate an accretion
  disk round Jupiter which is supplied at a low
  rate, resulting in a low density, low disk
  temperatures and slow formation timescale (gt105
  yrs) of the satellites
• These characteristics would all help to generate
  a partially undifferentiated Callisto
• The low disk density also means that the
  satellites can survive disk torques which move
  them towards Jupiter
• Is it reasonable to assume that accretion
  involved only small objects, and not large
  collisions?

66
Removing heat

• A rock-silicate mixture will tend to separate
  over time as the rock heats the surrounding ice
• Areas with a higher rock fraction will have a
  higher viscosity and thus a lower heat flux
• Near-surface cold ice will retain its rock and
  act as an insulator for any underlying ocean

• A shallow ocean but absence of deep melting is
  probably a consequence of
• 1) the pressure-dependence of ice melting
  temperature
• 2) accretion leads to radially increasing
  temperatures

67
Orbital evolution

• Recall dissipation in satellite leads to
  circularization
• Assume no torque from primary, so momentum
  conserved
• In this case, it can be shown that
• We have previously calculated (see Io), and
  so we can obtain and circularization
  timescale te -e/ directly

Why?
At the present day, this gives us (8 )
Myr. For a solid rock-ice mixture, 15 and
100 so te12 Gyr. But, if there really is
an ocean present, then dissipation will be
amplified, Qs reduced and te reduced, leading to
potential problems . . .
68
Summary

• Ios silicate volcanic activity is driven by
  tidal heating of a partially molten mantle
  feedback between temperature, viscosity and
  heating
• Callisto, by contrast, has experienced no
  significant tidal heating over its history
• Nonetheless, Callisto has an ocean, probably as a
  result of incorporating antifreeze e.g. NH3
• How did it develop an ocean and yet (apparently)
  retain an undifferentiated interior ?!
• Next time Europa and Ganymede