EART160 Planetary Sciences

1. EART160 Planetary Sciences

2. 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 • Lithospheric thickness tells us about thermal gradient • Bodies with atmospheres/hydrospheres have sedimentation and erosion – Earth, Mars, Venus, Titan

3. 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?

4. a Planetary Mass • The mass M and density r 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 . . . 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 w is the angular frequency of the orbiting satellite, equal to 2p/period. Note that the mass of the satellite is insignificant. 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

5. Bulk Densities • So for bodies with orbiting satellites (Sun, Mars, Earth, Jupiter etc.) M and r 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: Data from Lodders and Fegley, 1998

6. 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)

7. 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

8. Example: Venus • Bulk density of Venus is 5.24 g/cc • Surface composition of Venus is basaltic (extrusive mafic, iron-rich volcanic rock), suggesting peridotite (mafic, iron rich) 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?

9. a Pressures inside planets • Hydrostatic assumption (planet has no strength) • For a planet of constant density r (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

10. 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 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)

11. Real planets • Which planet is this? • Where does this information come from? • PREM – Preliminary Reference Earth Model • Notice the increase in mantle density with depth – is it a smooth curve? • How does gravity vary within the planet?

12. g vs. depth

13. Other techniques • There are other things we can do • 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)

14. 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

15. Three modes of heat loss • Conduction • Convection • Radiation

16. Conduction - Fourier’s Law T1>T0 T0 d F • Heat flow F 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) • 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? What would happen to the geotherm if the sun went out? milliWatt=10-3W

17. 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

18. a Diffusion Equation • The specific heat capacityCp is the change in temperature per unit mass for a given amount of energy: W=mCpDT • We can use Fourier’s law and the definition of Cp to find how temperature changes with time: F2 dz F1 • Here k is the thermal diffusivity (=k/rCp) and has units of m2s-1 • Typical values for rock/ice 10-6 m2s-1

19. a 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: • 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?

20. Deriving the Diffusion lengthscale • How long does it take a change in temperature to propagate a given distance? • Consider an isothermal body suddenly top cooled to To • The temperature change will propagate downwards a distance d in time t • After time t, F~k(T1-T0)/d • The cooling of the near surface layer involves an energy change per unit area DE~d(T1-T0)Cpr/2 • We also have Ft~DE • This gives us Temp. T0 T1 Initial profile d Depth Profile at time t

21. a 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? In which situation is more energy delivered? early later Total accretional energy = If all this energy goes into heat*, what is the resulting temperature change? * Is this a reasonable assumption? Earth M=6x1024 kg R=6400km so DT=30,000K Mars M=6x1023 kg R=3400km so DT=6,000K What do we conclude from this exercise?

22. 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 energy 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

23. 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, cooling is rapid (convection) (~ kyrs) • What happens once the magma ocean has solidified?

24. 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 numberRa exceeds a critical value, ~1000 Temp. Stagnant (conductive) lid Here r is density, g is gravity, a is thermal expansivity, DT is the temperature contrast, d is the layer thickness, k is the thermal diffusivity and h is the viscosity. Note that h is strongly temperature-dependent. Convecting interior Depth.

25. 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, h 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

26. 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?)

27. a Internal Heat Generation • Assume we have internal heating H (in Wkg-1) • Heat flow equation becomes: • 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:

28. Heat flux out of the surface • Surface flux a mix of heat from the mantle, plus some internally generated heat in the crust from radioactivity. • The bulk composition of the Earth and planets is of fundamental importance (changes H). • How can we relate the measured heat flux to the bulk planet properties?

29. Moon and Io • Present lunar heat flux: ~10mW/m2

30. The Moon concentrated radioactive elements in the upper ~50 km Incompatible elements including U, K, Th are trapped in the late stage liquids.

31. Heat flux out of the surface • Assume Fo = 10 mW/m2 • What is H? What is the implied content of radioactive elements? • ~7 x 10-10 W/kg (over 50 km) • Typical granite on Earth: ~10-9 W/kg • Granites contain ~20 ppm Th • Some lunar rocks ~10 ppm Th Fo Lunar crust U, K, Th rich layer 50 km Fm = 0 Lunar mantle

32. Io • Measured heat flux Fo is ~2 W/m^2 • Is this larger or smaller than the Moon and Earth? • What is the implied heat generation per unit volume (H) averaged over the whole satellite and how does it compare to that of granite and the lunar crust? Hand in the answer to this question by Oct. 25 to receive up to 3% extra credit on the midterm. Show all work for full credit and state all assumptions. NO COLLABORATION ALLOWED.

33. 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

34. Key Concepts • Bulk density • Self-compression • Bulk modulus • Hydrostatic assumption • Accretionary energy • Magma ocean • Conduction and convection • Rayleigh number • Viscosity • Thermal diffusivity • Diffusion lengthscale W=mCpDT

35. End of Lecture

36. Extra slides not used

37. Caveats • Exponential changes in heat producing elements with depth • Convection

38. a Example • Let’s take a spherical, conductive planet in steady state • In spherical coordinates, the diffusion equation is: • The solution to this equation is Here Ts is the surface temperature, R is the planetary radius, r is the density • So the central temperature is Ts+(rHR2/6k) • E.g. Earth R=6400 km, r=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?

39. Logistics • Term-paper outlines due today • Will provide feedback by email or by appointment • Term papers are due in two weeks • Please visit me for help • Must use published papers as references • Use websites/wiki for general information only • Mid-term next Thursday – 1 hour long • Covers everything up to 10/14 (except atmospheres) • Bring your questions to class on Tuesday • Homework 4 - not due for 2 weeks

40. One issue • Crust vs. lithosphere?

41. Io cooling • 26Al heating • Grav heating Rp R solid porous d crust mantle