EART160 Planetary Sciences
Francis Nimmo
Last Week
• Volcanism happens because of higher temperatures,
reduced pressure or lowered solidus
• Conductive cooling time t = d2/k
• Planetary cooling leads to compression
• Elastic materials s = E e
• Flexural parameter controls
the lengthscale of deformation
of the elastic lithosphere
   3g ( 


2
m   w )(1 ) 
ETe3
• Lithospheric thickness tells us about thermal gradient
• Bodies with atmospheres/hydrospheres have
sedimentation and erosion – Earth, Mars, Venus, Titan
1
4
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 . . .
3 2
GM  a 
Where’s this from?

Here G is the universal gravitational
a
ae
constant (6.67x10-11 in SI units), a is the
semi-major axis (see diagram) and  is the
focus
angular frequency of the orbiting satellite,
e is eccentricity
equal to 2p/period. Note that the mass of the
satellite is not important. Given the mass, the
density can usually be inferred by telescopic Orbits are ellipses, with the planet at
one focus and a semi-major axis a
measurements of the body’s radius R
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
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
Data from Lodders and Fegley, 1998
Saturn Pluto
0.69
~1.9
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 selfcompression (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)
Element
C
O
Mg Si
S
Fe
5.65
5.95
1340 1529 674
1337
Log10 (No. 7.00
Atoms)
7.32 6.0
Condens.
Temp (K)
--
78
6.0
• 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
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)
dP
  g
dr
• For a planet of constant density  (is this reasonable?)
4
r

g (r )  pGr  g 0
3
R
2
2
 r  2pG 2 2  r 
1
P(r )  g 0R1  2  
R  1  2 
2
3
 R 
 R 
• 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
The bulk modulus K controls the
d dP

change in density (or volume) due

K
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
T0
•
•
•
•
•
•
•
T1>T0
(T1  T0 )
dT
d
F
Heat flow F  k
k
d
dz
T1
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)
milliWatt=10-3W
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?
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
W  mC p T
• 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?
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

In which situation is
more energy delivered?
later
2
3
GM
Total accretional energy =
5 R
If all this energy goes into heat*, what is the resulting temperature change?

3 GM
T 
5 CpR
Earth M=6x1024 kg R=6400km so T=30,000K
Mars M=6x1023 kg R=3400km so T=6,000K
What 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 radioisotopes (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
Temp.
Here  is density, g is gravity,  is thermal
expansivity, T is the temperature contrast, d is
the layer thickness, k is the thermal diffusivity
and  is the viscosity. Note that  is strongly
temperature-dependent.
Stagnant
(conductive) lid
Depth.
gTd
Ra 
k
3
Convecting
interior
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
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
Image courtesy Walter Kiefer, Ra=3.7x106, Mars
Diffusion Equation
F2
dz
F1
• The specific heat capacity Cp is the change in
temperature per unit mass for a given amount
of energy: W=mCpT
• We can use Fourier’s law and the definition of
Cp to find how temperature changes with time:

T
k  2T
 2T

k 2
2
t C p z
z
• Here k is the thermal diffusivity (=k/Cp) and has
units of m2s-1
• Typical values for rock/ice 10-6 m2s-1
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:
d ~ kt
2
• 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, k=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?

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:
T
 2T H
k 2 
t
z
Cp
• 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:

T  a  bz 
H
2kC p
z
2
Example
• Let’s take a spherical, conductive planet in steady state
• In spherical coordinates, the diffusion equation is:
T
1   2 T  H
 k 2 r
0

t
r r  r  C p
• The solution to this equation is

T ( r )  Ts 
H
6k
( R2  r2 )
Here Ts is the surface temperature, R is the planetary radius,  is the density
• 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?
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
dP
  g
dr
d ~ kt
2
W=mCpT
dT
F k
dz
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?)
d
m.b.l.
interior
R
By balancing these heat flows, we get
3kT

d
HR
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?
Deriving the Diffusion lengthscale
• 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
Temp.
T0
T1
Depth
Initial
profile
d
Profile
at time t
• 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 d 2 ~ kt
• Io cooling
• 26Al heating
• Grav heating
d
crust
mantle
R
Rp
solid
porous