1. EART 160: Planetary Science 13 February 2008

2. Last Time Planetary Interiors –Pressure and Temperature inside Planets –Heat Sources Accretion Differentiation Radioactivity

3. Today Get your midterm reviews in Class Schedule Revision Planetary Interiors –Cooling Mechanisms Conduction Convection –Rheology Viscoelasticity Flexure

4. Revised Schedule W 13 FebTerrestrial Planet Interiors -- Cooling F 15 FebTerrestrial Planet Atmospheres -- Structure and Dynamics M 18 FebPresidents Day -- no class W 20 FebTerrestrial Planet Atmospheres -- Origin and EscapeHW 4 F 22 FebJovian PlanetsHW 5 M 25 FebRings, Moons, and Tides W 27 FebIcy Satellites F 29 FebKuiper Belt / CometsHW 6 M 03 MarSolar System Exploration W 05 MarExtrasolar Planets, Astrobiology F 07 MarPlanets and Politics: NASA, Spacecraft Missions, FundingHW 7 M 10 MarLPSC – no class -- Work on your projects! W 12 MarLPSC – no class -- Work on your projects! F 14 MarLPSC – no class -- Work on your projects! M 17 MarCourse ReviewProject due W 19 MarFinal Exam (4 – 7 PM)

5. Paper Discussions F 15 Feb Tobie et al. (2006) Nature 440, 61-64Titan AtmosphereElena Amador F 22 Feb Guillot et al. (1999), Science 286, 72-77Giant PlanetsRyan Cook M 25 Feb Gladman et al. (1997), Science 277, 197-201 ?Orbital ResonancesAaron Masters W 27 Feb Porco et al. (2006) Science 311, 1393-1401Enceladus??? F 29 Feb Levison et al. (2002), Science 296, 2212-2215Oort Cloud CometsMiljan Draganic W 05 Mar Vogel 1999 Science 286 70 ?Astrobiology Matthew Kammerer F 07 Mar Something direct from NASA?NASAShayna Kram

6. 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?

7. Cooling Radiation –Photon carries energy out into space –Works if opacity is low –Unimportant in interior, only works at surface Conduction –Heat transferred through matter –Heat moves from hot to cold –Slow; dominates in lithosphere and boundary layers Convection –Hot, buoyant material carried upward, Cold, dense material sinks –Fast! Limited by viscosity of material Running down the stairs with buckets of ice is an effective way of getting heat upstairs.-- Juri Toomre

8. Conduction - Fourier’s Law Heat flow F T 1 >T 0 T1T1 T0T0 F d Heat flows from hot to cold (thermodynamics) and is proportional to the temperature gradient Here k is the thermal conductivity (W m -1 K -1 ) and units of F are W m -2 (heat flux is power per unit area) Typical values for k are 2-4 Wm -1 K -1 (rock, ice) and 30- 60 Wm -1 K -1 (metal) Solar heat flux at 1 A.U. is 1300 W m -2 Mean subsurface heat flux on Earth is 80 mW m -2 What controls the surface temperature of most planetary bodies?

9. Diffusion Equation Here  is the thermal diffusivity (=k/  C p ) and has units of m 2 s -1 Typical values for rock/ice 10 -6 m 2 s -1 F1F1 F2F2 zz We can use Fourier’s law and the definition of C p to find how temperature changes with time: In steady-state, the heat produced inside the planet exactly balances the heat loss from cooling. In this situation, the temperature is constant with time

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

11. Internal Heating Assume we have internal heating H (in Wkg -1 ) From the definition of C p we have Ht=  TC p So we need an extra term in the heat flow equation: 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:

12. Example Let’s take a spherical, conductive planet in steady state In spherical coordinates, the diffusion equation is: The solution to this equation is So the central temperature is T s +(  HR 2 /6k) E.g. Earth R=6400 km,  =5500 kg m -3, k=3 Wm -1 K -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

13. Convection Convective behaviour is governed by the Rayleigh number Ra Ra is the ratio of buoyancy forces to diffusive forces 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=3.7x10 6, Mars Stagnant lid (cold, rigid) Plume (upwelling, hot) Sinking blob (cold)

14. Viscosity Ra controls vigor of convection. Depends inversely on viscosity, . Viscosity depends on Temperature T, Pressure P, Stress , Grain Size d. A – pre-exponential constantE – Activation Energy V – Activation VolumeR – Gas Constant n – Stress Exponentm – Grain-size exponent Viscosity relates stress and strain rate

15. Viscoelasticity A Maxwellian material has a viscous term and an elastic term. If  is high, we get an elastic behavior. If  is low, we get a viscous behavior. Depends also on the rate of stress. Materials are elastic on a short timescale, viscous on a long one. There are other types of viscoelasticity, but Maxwell is the simplest

16. Next Time Paper Discussion – Titan Atmosphere –Tobie et al., 2006 Planetary Interiors –Elastic Flexure Planetary Atmospheres –Structure –Dynamics