EART162: PLANETARY INTERIORS

F.Nimmo EART162 Spring 10

Francis Nimmo

EART162: PLANETARY INTERIORS


Last Week• Tidal bulge amplitude depends on mass, position, rigidity of body,

and whether it is in synchronous orbit• Love numbers measure how much the body is deformed (e.g. by

rotation or tides)• Love numbers are reduced by central mass concentration and/or

rigidity• We can use the observed shape of a body to infer the fluid Love

number and thus its moment of inertia – as long as it is behaving like a fluid

• Tidal torques are responsible for orbital evolution e.g. orbit circularization, Moon moving away from Earth etc.

• Tidal strains cause dissipation and heating


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 you’ve encountered before


Case Study: Europa


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 1:2:4) (what about their orbital radii?)• Orbital eccentricities are higher than expected due to

this resonance (tidal heating)

Io Europa Ganymede Callisto


Basic parameters

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

Moon Europa

a (Rp) 60.3 9.4

Period (days) 27.3 3.55

Eccentricity 0.055 0.01

Radius (km) 1737 1565

Mean density (g/cc) 3.34 2.99

g (m s-2) 1.62 1.31

C/MR2 0.394 0.34


~100km

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

ridges

lenticulae bands

chaos


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


Bulk Properties

Europa

Io

Radius = 1560 km

Flyby (schematic)

Earth

r

2r

GMa

M=5x1022 kg

So bulk density = 3 g/cc

What does this tell us?What’s the surface gravity?What’s the pressure at the centre?


Shape

R

ca GM

Rh f

23

22

1 GM

Rh f

23

22

ca

cb

14

1

Planet Synch. Sat.Quantity

Only truefor fluid bodies!

• For Europa, a=1562.6 km, b=1560.3 km, c=1559.5 km• Thus (b-c)/(a-c)=0.26. So what?

• We can use (a-c)/R to obtain h2f=1.99. Now what?

2/1

22

15

5

21

3

2

fhMR

C Only truefor fluid bodies!

• Plug in the values, we get C/MR2=0.34. So what?


Interior Structure• Let’s assume a very simple two-layer model

R

fRm

gm

)1( 3334 gfRM m

Mass constraint:

MoI constraint:

)1(4.0 5534 gfRC m

Combining the two:)1(

)1(4.0

3

5

2 gf

gf

MR

C

• We know R (1560 km), M (5x1022 kg) and C/MR2 (0.34)

• Assuming a value for m, we can solve for f,g . . .


Interior structure (cont’d)• Assuming m=1000 kg m-3, we get f=0.87

and g=3.26 (see diagram)• The implied density of the interior (4260

kg m-3) is greater than low-pressure mantle silicates. Could the density simply be due to high pressures?

1565

1362

Kgz01

1

0

• Remember the simple equation of state (Week 3):

Where does this come from?

• Use K=200 GPa, g=1.3ms-2, =3300 kgm-3, this gives ~3400 kgm-3 at the centre

• What do we conclude from this?


Summary• Radius, mass give us bulk density and some constraint on the bulk

structure (mostly rock/metal, not ice)• Shape tells us that body is hydrostatic• For a hydrostatic body, we can use the observed flattening to

derive h2f and hence C/MR2

• C/MR2 allows us to make further inferences e.g. how thick the outer ice shell is, presence of an iron core

• Rather than use the shape, we could also use the observed gravity (e.g. J2) and the hydrostatic assumption. This gives us the same answer (as it should)


Interior Structure• Probably similar to Io, but

with a layer of ice (~100 km) on top

• We can’t 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) Ocean

Silicate mantle

~120km

Ice shell


Why do we think there’s an ocean?

Khurana et al. 2002

• Jupiter’s 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 Europa’s interior is

• The results are consistent with a shallow salty ocean > a few km thick

• Why couldn’t the conductive layer be deeper?


So What?

• Astrobiology (groan)

• Interesting physical problem – why hasn’t the ocean frozen?


How could we check the ocean exists?

• Equilibrium tide:• Tidal amplitude d is reduced by rigidity , depending

on the tidal Love number:

3

a

R

M

mRH

• What is the size of the equilibrium tide for Europa? (m/M=40,000, a/R=430)

• What is the size of the fluid diurnal tide?• How big would the diurnal tide be if there were no

ocean?

gRt HHhd

2192 1

2/5

(Note that this assumes a uniform density structure)


Europa’s Temperature Structure• What’s the surface

temperature?• If there were only

radioactive heat sources, how thick would the conductive ice shell be?

• Is the ocean convecting? • How long would the ocean

take to freeze?• Are there other heat sources

we’ve forgotten about?

ocean

ice

mantle

T

z


Tidal Dissipation

• 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?

JupiterSatellite

Eccentric orbitDiurnal tides can

be large e.g. ~30m on Europa

62

22

2

2

)~1(

4/259

a

R

M

me

Q

E

QR

EdP


Equilibrium Shell thickness

• Let’s put some numbers on this . . .• How reliable is the shell thickness derived?

• Is the shell really conductive? How might we tell?

Shell thickness

Hea

t flo

w

Heat production

Heat lossMantle

Equilibrium


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)

150km

• 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?


Equilibrium Shell Thickness

cond

ucti

on

conv

ecti

on

Equbm. Shell thickness

Hussmann et al. 2002

• What would happen if Europa’s mantle was like Io’s?• Is the shell actually in steady state?• How else might we measure the shell thickness?

• Why does convective heat transport decrease as shell thickness increases?

• Obtain equilibrium shell thickness 20-50 km


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

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

270 K100 K

elastic

viscous

190 K

Depth

Te

Temp.


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?


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

From Carr et al., Nature, 1998

40 km

icebergwater

h

tc

tc ~ 10 h


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?


Future Mission - JEO• Launch 2020, arrive 2025,

end of mission 2029• Will tour other satellites

before orbiting Europa• Nuclear powered,

radiation-shielded orbiter• Cassini-like instrument

package (+ altimeter)• Will get approval from

Congress (hopefully!) next year


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 . . .


Planning Ahead . . .• Week 9

– Tues 25th – Tides pt II– Thurs 27th – Case study I

• Week 10– Tues 1st – Case study II– Thurs 3rd – Revision lecture

• Final Exam – Mon 7th June 4:00-7:00 p.m.


Gravity field• MacCullagh’s formula tells us how the acceleration

varies with latitude ():

]1sin3[2

31sin3

2

)(3 224

2

22

42

J

r

GMR

r

GM

r

ACG

r

GMa

• So what’s the difference between the acceleration at the poles and at the equator?

Flyby again

Earthr

R

• Closest approach is 600 km (above the surface) and we measure an acceleration difference between pole and equator of 0.65 mm s-2. What is J2? And (C-A)?


Now we have J2 – what next?

• We really want C – how do we get it?– Measure the precession rate (C-A)/C, or . . .

– Assume hydrostatic

• Is hydrostatic assumption reasonable?

Mass deficit at poles

Mass excess atequator

223

425

2523

2 )1)((12

1

2

3

2

32

MaC

GMa

GM

aJ

What causes the flattening?

Here a is equatorial radius

• Plug in the values, we get C/Ma2=0.34. So what?


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?

EART162: PLANETARY INTERIORS

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