EART160 Planetary Sciences

Francis Nimmo

Last Week• Volcanism happens because of higher temperatures,

reduced pressure or lowered solidus

• Conductive cooling time t = d2/• Planetary cooling leads to compression• Elastic materials = E

3

2

1

4

3 ( )(1 )e

m w

ET

g

• 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

This Week – Interiors

• Mostly solid bodies (gas giants next week)• How do we determine a planet’s bulk structure?• How do pressure and temperature vary inside a

planet?• How do planets lose heat?

• See also EART 162

Planetary Mass• The mass M and density 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 . . .

23aGM Where’s this from?

Here G is the universal gravitational constant (6.67x10-11 in SI units), a is the semi-major axis (see diagram) and is the angular frequency of the orbiting satellite, equal to 2/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 body’s radius R

a ae

focus

e is eccentricity

Orbits are ellipses, with the planet at one focus and a semi-major axis a

Bulk Densities• So for bodies with orbiting satellites (Sun, Mars, Earth,

Jupiter etc.) M and 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 spacecraft’s 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

(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

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)

Bulk composition (reminder)

• Four most common refractory elements: Mg, Si, Fe, S, present in (number) ratios 1:1:0.9:0.45

• 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

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

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 Mg:Fe ratio of 9:1• Primitive nebula has an Mg:Fe ratio of roughly 1:1• 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?

Pressures inside planets• Hydrostatic assumption (planet has no strength)

gdr

dP

R

rgrGrg 03

4)(

2

222

2

2

0 13

21

2

1)(

R

rR

G

R

rRgrP

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

• So the central pressure of a planet increases as the square of its radius

• Moon R=1800km P=7.2 GPa Mars R=3400km P=26 GPa

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 material’s density to

change significantly depends on the bulk modulus of that material

K

dPd

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)

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?

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)

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 planet’s 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

Conduction - Fourier’s Law

• Heat flow F

T1>T0

T1

T0

F d

dz

dTk

d

TTk

)( 01

• 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?

milliWatt=10-3W

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 Earth’s 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?

TmCW p

Energy of Accretion• Let’s 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?

early later

In which situation is more energy delivered?

Total accretional energy =

R

GM 2

5

3

If all this energy goes into heat*, what is the resulting temperature change?

RC

GMT

p5

3

Earth M=6x1024 kg R=6400km so T=30,000KMars M=6x1023 kg R=3400km so T=6,000KWhat do we conclude from this exercise?

* Is this a reasonable assumption?

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 planet’s 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

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?

Cooling a planet (cont’d)• 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

3Tdg

Ra

Here is density, g is gravity, is thermal expansivity, T is the temperature contrast, d is the layer thickness, is the thermal diffusivity and is the viscosity. Note that is strongly temperature-dependent.

Temp.

Dep

th.

Stagnant (conductive) lid

Convectinginterior

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, increases, Ra decreases, rate

of cooling decreases -> self-regulating system

Image courtesy Walter Kiefer, Ra=3.7x106, Mars

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

Diffusion Equation

• Here is the thermal diffusivity (=k/Cp) and has units of m2s-1

• Typical values for rock/ice 10-6 m2s-1

F1

F2

z

• The specific heat capacity Cp is the change in temperature per unit mass for a given amount of energy: W=mCpT

• We can use Fourier’s law and the definition of Cp to find how temperature changes with time:

2

2

2

2

z

T

z

T

C

k

t

T

p

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?d~0.02m, =10-6 m2s-1 so t~6 minutes

– 2. How long does it take for the molten Moon to cool? d~1800 km, k=10-6 m2s-1 so t~100 Gyr.What might be wrong with this answer?

td ~2

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

Earth’s present-day heat loss (why?)

Internal Heat Generation• Assume we have internal heating H (in Wkg-1)

• From the definition of Cp we have Ht=TCp

• So we need an extra term in the heat flow equation:

pC

H

z

T

t

T

2

2

• 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:2

2 zbzaTpC

H

Example• Let’s take a spherical, conductive planet in steady state• In spherical coordinates, the diffusion equation is:

01 2

2

pC

H

r

Tr

rrt

T

• The solution to this equation is

)()( 226 rRTrT kH

s

• So the central temperature is Ts+(HR2/6k)

• E.g. Earth R=6400 km, =5500 kg m-3, k=3 Wm-1K-1, H=6x10-12 W kg-1 gives a central temp. of ~75,000K!

• What is wrong with this approach?

Here Ts is the surface temperature, R is the planetary radius, is the density

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

Key Concepts• Bulk density • Self-compression• Bulk modulus• Hydrostatic assumption• Accretionary energy• Magma ocean• Conduction and

convection• Rayleigh number• Viscosity• Thermal diffusivity• Diffusion lengthscale

td ~2

dTF k

dz

W=mCpT

gdr

dP

End of Lecture

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• Let’s assume that the heat transported across the m.b.l. is

provided by radioactive decay in the mantle (true?)

dBy balancing these heat flows, we get

HR

Tkd

3

Here H is heat production per unit volume, R is planetary radius

Plugging in reasonable values, we get m.b.l. thickness d=225 km and a heat flux of 16 mWm-2. Is this OK?

m.b.l.

interior

R

Deriving the Diffusion lengthscale

• After time t, F~k(T1-T0)/d

• The cooling of the near surface layer involves an energy change per unit area E~d(T1-T0)Cp/2

• We also have Ft~E• This gives us

T0 T1

• 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

Initial profile

Profile at time t

Temp.

Depth d

td ~2

• Io cooling

• 26Al heating

• Grav heatingsolid

porous

R Rp

crust

mantle

d