Francis Nimmo

ES 290Q: OUTER SOLAR SYSTEM

Io against Jupiter,Hubble image,July 1997

Last Week• Solar system characteristics and formation –

Hill sphere, “snow line”, timescales

• Kepler’s laws and Newtonian orbits

• Tides– Synchronous rotation– Dissipation / heating– Circularization and orbital migration

32an 22 1 enah a

E2

3

a

R

M

mRH

~1

2/52 h

Galilean Satellites• This week

– Preliminaries, common themes

– Io

– Callisto

• Next week– Ganymede

– Europa

Io Europa Ganymede Callisto

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 (1970’s), Galileo (ended 2003), Europa Orbiter??

Galileo• 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 Jupiter’s atmosphere• Very complex series of fly-bys of all major

Galilean satellites

• Deliberately crashed into Jupiter Sept 2003 (why?)

• Main source of results

antenna

Where are they?

5 10 15 20 25 30Distance (Rj)

a

(106 km)

P

(days)e ms

(1020 kg)

Rs

(km)

(Mg m-3)

Io 422 1.769 .0041 893 1821 3.53

Europa 671 3.552 .010 480 1565 2.99

Ganymede 1070 7.154 .0015 1482 2634 1.94

Callisto 1883 16.69 .007 1076 2403 1.85

Laplace Resonance (1)• Periods of Io:Europa:Ganymede are in ratio 1:2:4

• 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

JI EG

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

Compositions/Formation• Surface compositions – mainly water ice (except for

Io), plus “contaminants” (spectroscopy)• Io’s 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 didn’t 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

Composition (cont’d)• 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

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 (?)

Ice phase diagram

• 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?

Ice I

Water

Ice V

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

)()(1 4

4

42

2

2 Pr

RJP

r

RJ

r

GMV

• 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

Internal Structures (2)• 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 ):

C

A

R

22 MR

ACJ

• 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

Internal Structures (3)

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

• 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 ) and have more than two unknowns. Means the results are non-unique

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?

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)

Io – liquid iron core (dynamo), silicate mantle (partially molten?). No volatiles – why not?

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

• A good measure of its tendency to deform in a ductile fashion is the homologous temperature (Th=T/Tmelt) (in K)

– Rock at Earth surface Th~0.2

– Ice at Galilean satellite surface Th~0.4

– Ice in Antarctica/Mars Th~0.8

• So ice at the surface of the Galilean satellites behaves more like rock than ice on Earth

brittle

elastic

ductile

stressdepth

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 2:1 resonance with Europa; the two bodies then move outwards in step until . . .

– They encounter the 2:1 resonance with Ganymede

• There are alternative scenarios• The present-day configuration involves a balance

between dissipation in primary (outwards) and dissipation in satellites (inwards)

Hypothetical orbital history

distance (schematic)

tim

e

Io Europa Ganymede

2:1 Io:Europa

2:1 Europa:Ganymede

Note that we don’t actually know whether the orbits are currently expanding or contractingAlso note that during capture into resonance, eccentricities are transiently excited to high values – so what?

from Peale, Celest. Mech.Dyn. Ast. 2003

How fast does it happen?• The speed of orbital evolution is governed by the rate at

which energy gets dissipated (in primary or satellite)• Since we don’t understand dissipation very well, we

define a parameter Q which conceals our ignorance:

• Where E is the energy dissipated over one cycle and E is the peak energy stored during the cycle. Note that low Q means high dissipation!

EEQ 2

• It can be shown that Q is related to the phase lag arising in the tidal torque problem we studied earlier: /1~Q

How fast does it happen(2)?• The rate of outwards motion of a satellite is governed by the

dissipation factor in the primary (Qp)

naa

R

m

m

Q

ka p

p

s

p

5

23

Here mp and ms are the planet and satellite masses, a is the semi-major axis, Rp is the planet radius and k2 is the Love number. Note that the mean motion n depends on a.

• Does this equation make sense? Recall

• Why is it useful? Mainly because it allows us to calculate Qp. E.g. since we can observe the rate of lunar recession now, we can calculate Qp. This is particularly useful for places like Jupiter.

• We can derive a similar equation for the time for circularization to occur. This depends on Qs (dissipation in the satellite).

3

a

R

m

mRH p

p

sp

Estimating Q• Recall that the rate of outwards motion of a satellite

depends on planetary dissipation Qp

• If we assume that Io formed 4.5 Gyr B.P., and has been moving outwards ever since, we get a lower bound on Jupiter’s Q of ~105 (why a lower bound?)

• This value is typical of gas giants, but is much higher than for silicate bodies (~102)

• The Earth’s Q is anomalously high (~12) because the current continental configuration means oceanic tides are close to resonance – lots of dissipation

• We’ll calculate the rate of dissipation in a second

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)

JupiterSatellite

Eccentric orbit

Diurnal tides can be large e.g. 30m on Europa

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(rigidity)

• So total rate of work goes as e2H2Rs/

• For tide raised on satellite H=Rs(mp/ms)(Rs/a)3

• From the above, we expect the energy stored E to go as

Gm

R

gR s

s

s2

4

3

38

2

19~

~/eH1~

sReH ~/

2~

a

Gm

a

ReE ps

s

252

~~

Note that here we have used

Tidal heating (2)• From the definition of Q, we have

• We’ve 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?

sQ

nE

dt

dE

a

Gm

a

R

Q

ne

dt

dE ps

ss

252

~4

63

0041.0,100,40~ eQss

How do we calculate Q?• For solid bodies, we assume a viscoelastic rheology• Such a body has a rigidity , a viscosity and a

characteristic relaxation (Maxwell) timescale m=• The body behaves elastically at timescales <<m and in a

viscous fashion at timescales >> m

2)(1 n

nQ

m

m

Tobie et al. JGR 2003

• Dissipation is maximized when timescale ~ m:

Calculating Q (cont’d)• 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 ~1011 Pa, ~1021 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)

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

Body H (m)

3eH (m)

dW/dt (1012 W)

dWR/dt

(1012 W)

EeH/Rs

(MPa)

C/msRs2 3Gms/5R

(MJ kg-1)

Io 7802 312 8900 0.31 0.57 0.3679(4) 1.96

Europa 1966 60 8.1 0.13 0.13 0.346(5) 1.23

Ganymede 1258 5.7 0.074 0.29 0.007 0.311(3) 2.25

Callisto 220 4.6 0.015 0.31 0.006 0.355(4) 1.79

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 Jupiter’s mass=1.899x1027 kg, k=3/2 and Q=100, 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 E=10 GPa, C/msRs

2 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.

Non-synchronous rotation (1)• From the satellite’s point of view, the planet travels in the

opposite direction round the sky to the satellite itself

• The tidal bulge always lags the planet’s 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 satellite’s rotation rate slightly above synchronous

Torque opposes spinSmaller

Torque increases spinLarger

Eccentric orbit

Periapse Apoapseplanet

satellite

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)

Cratering Statistics

Furrowed terrainGrooved terrain

Zahnle et al.Icarus 1997

Absolute ages have been revised upwards since (Zahnle et al. Icarus 2003)

Expected curves if NSR is notoccurring

Cratering Statistics - Results• Io – no impact craters observed (!) so surface is very

young (< 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)

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?1) 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 . . .

2) Coupling. The satellites can’t be treated as isolated objects, because of the Laplace resonance. So you have to model their evolution simultaneously . . .

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

Io

Basic Parameters

• Note the likely structural similarities with the Moon

Io Moon

a (Rp) 5.9 60.3

Period (days) 1.77 27.3

Eccentricity 0.004 0.055

Radius (km) 1821 1737

Mean density (g/cc) 3.53 3.34

g (m s-2) 1.80 1.62

C/MR2 0.378 0.394

Heat flow (mWm-2) ~2500 ~25

What’s 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 Jupiter’s large magnetic field

• Not clear whether Io has an internal magnetic field (Kivelson et al. JGR 2001) – interactions with Jupiter’s field make identification difficult

Landforms• Three main types: Volcanoes, Mountains and Paterae

(irregular depressions, similar to calderas)

350km

volcano

patera

200km

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.

flow

Lava flows

• Dark flows are the most recent (still too hot for sulphur to condense on them)

• Flows appear relatively thin, suggesting low viscosity

500km

Amirani lava flow, Io

500km

Comparably-sized lava flow on Venus(Magellan radar image)

Time-Variability• Changes detected

from Voyager to Galileo missions and within Galileo mission

April 1997 Sept 1997 July 1999

400kmPele

PillanGalileo images of overlapping deposits at Pillan and Pele

Lava flow erupted at Prometheus between Voyager and Galileo missions (Davies JGR 2003)

Volcanic activity (1)

• 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

Voyager image of eruption plume, approximately 300 km high

Fire fountain(?)

Volcanic activity (2)

• 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)

Galieo nightside image of Pele,SSI clear filter. Radebaugh et al. 2004

Erta Ale lava lake, Ethiopia. Lake is 100m across.

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

July 12 1998 July 28 1998 Aug 4 1998

From Macintosh et al., Icarus 20031 arcsecond

Keck interferometer

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

Earth’s global heat flux.

a

Gm

a

R

Q

ne

dt

dE ps

ss

252

~4

63

• Assume low rigidity ( ) – why?. To balance the heat being produced

requires Qs=90. 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 Jupiter’s rotational energy

1~ s

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

Interior Structure• 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

1821 km

700 kmFe-FeS5150 kg m-3

Silicates3500 kg m-3

After Anderson et al.,JGR 2001

• Io likely has a crust, but we can’t detect it with current data• We can’t tell (directly) whether the core or the mantle are

partially or completely liquid.

• Io’s k2=1.29. What is this telling us? (rigidity or mass concn.)

Remember these structures are non-unique: the ones shown assume plausible but not necessarily correct densities.

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

Solid lid

Partially-molten mantle

Solidmantle

~500

km~5

0km

After Moore, Icarus, 2001

Consequences of resurfacing• Burial leads to large compressive stresses

due to change in radius

• Stress ~ E R/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

R

After McKinnon et al.,Geology 2001

Low-angle (why?) thrust faulting is probably responsible for many of the

mountain ranges seen on Io

Schenk and Bulmer, Science 1998

550 km10km

stereo

Eruption Spectra• Recall Wien’s law – max 1/T

• So infra-red spectra give temperature information

Davies, JGR 2003

• Single temperature curve provides poor fit

• Two-temperature curve provides much better fit

• Short-wavelength “hump” requires temperatures >1400K

• So silicate volcanism must be involved• Voyager could not resolve this issue• Time-evolution gives cooling history

Plumes• What’s 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

Pele

Loki

250km

After Smith et al., Nature 1979

200 K

500 K

0 K

Liq. Vap.

L+V

S+V

Entropy (J kg-1 K-1)

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

Constant entropy (adiabatic)

Pressuredecreases

Callisto

Basic Parameters

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

Io Callisto

a (Rp) 5.9 26.3

Period (days) 1.77 16.7

Eccentricity 0.004 0.007

Radius (km) 1821 2400

Mean density (g/cc) 3.53 1.85

g (m s-2) 1.80 1.24

C/MR2 0.378* 0.355+

* Anderson et al. JGR 2001 + Anderson et al. Icarus 2001

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?

600km

1500km

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 M

oore et al. Icarus 1999

Degradation / Sublimation• Callisto systematically lacks small

(<1km) 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 hypothesisMoore et al. Icarus 1999

Internal Structure

Probable ocean location

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

• Two interesting inferences:– It has an ocean

– It is only partly differentiated

• Where do these inferences come from?

• Ocean detected with magnetometer data

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

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 T/z (k~3 W/mK, T~100 K)• So equilibrium conductive shell thickness ~ 60 km• This seems reasonable – but what happens if the ice

shell starts to convect?

An Aside on Convection (1)• Convective vigour (and whether it occurs) is governed by the

Rayleigh number:

• Convection initiates for Ra >~ 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?

3Tdg

Ra

Where does this come from?

is density, thermal expansivity, T temperature drop across the layer, thermal diffusivity, viscosity, d layer thickness

Paskm

dRa

14

39

1020010

An Aside on Convection (2)

3/43/1 )( TdRa

3/4

3/1

g

kFNote that this heat flux is independent of shell thickness and T

T

z

Stagnant lid• For a temperature-dependent viscosity

material, a stagnant lid develops on top of a roughly isothermal, convecting interior

• The viscosity is given by oexp(-[T-To]) where o is the reference viscosity at To and is a constant (K-1) set by the rheology

Convection

• The stagnant lid thickness is given by

• And so the heat flux across the stagnant lid is

Convecting ice shells (cont’d)• For likely parameters, we get a convective heat flux of

~70 (1014 Pa s /)1/3 mWm-2

• This value is independent of shell thickness and exceeds the radiogenic contribution if < 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)

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)

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 (>105 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?

Removing heat

• 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

Nagel et al. Icarus 2004

temperature

Rock fraction

Meltling tem

p.

• 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

temperature

Rock fraction

Meltling tem

p.

ocean

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 e= -e/ directly:

eE

Ee

2

Why?

eeE

n

Q

R

a

m

m ss

sp

se

~

63

45

At the present day, this gives us (8 ) Myr. For a solid rock-ice mixture, ~ 15 and ~ 100 so e~12 Gyr.But, if there really is an ocean present, then dissipation will be amplified, Qs reduced and e reduced, leading to potential problems . . .

ssQ~

s~ sQ

Summary• Io’s 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

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

Q

eT ABT Tidal torque: Mass torque:

(See Greenberg and Weidenschilling, Icarus 1984)

• 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

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:

Diffusion creep (n=1, grain-size dependent)

Grain-boundary sliding(n>1, grain-size dependent)Actually two mechanisms, slower one dominates

Dislocation creep(n>1,p=0)

Increasing stress / strain rate

RTQps

n egA / Here is strain rate, A is a constant, 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 n=1 and a grain-size independent rheology has p=0.

Thermal & Orbital Evolution (cont’d)• Nonetheless, progress is being made, both on the

observational and the theoretical front. We’ll discuss examples of both later in the course.

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

Magnetometer (1)

Khurana et al. 2002

•Jupiter’s 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

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 . . .