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EART 160: Planetary Science 13 February 2008 Last Time • Planetary Interiors – Pressure and Temperature inside Planets – Heat Sources • Accretion • Differentiation • Radioactivity Today • Get your midterm reviews in • Class Schedule Revision • Planetary Interiors – Cooling Mechanisms • Conduction • Convection – Rheology • Viscoelasticity • Flexure Revised Schedule W 13 Feb Terrestrial Planet Interiors -- Cooling F 15 Feb Terrestrial Planet Atmospheres -- Structure and Dynamics M 18 Feb Presidents Day -- no class W 20 Feb Terrestrial Planet Atmospheres -- Origin and Escape HW 4 F 22 Feb Jovian Planets HW 5 M 25 Feb Rings, Moons, and Tides W 27 Feb Icy Satellites F 29 Feb Kuiper Belt / Comets M 03 Mar Solar System Exploration W 05 Mar Extrasolar Planets, Astrobiology F 07 Mar Planets and Politics: NASA, Spacecraft Missions, Funding M 10 Mar LPSC – no class -- Work on your projects! W 12 Mar LPSC – no class -- Work on your projects! F 14 Mar LPSC – no class -- Work on your projects! M 17 Mar Course Review W 19 Mar Final Exam (4 – 7 PM) HW 6 HW 7 Project due Paper Discussions F 15 Feb Tobie et al. (2006) Nature 440, 61-64 Titan Atmosphere Elena Amador F 22 Feb Guillot et al. (1999), Science 286, 72-77 Giant Planets Ryan Cook Gladman et al. (1997), M 25 Feb Science 277, 197-201 ? Orbital Resonances Aaron Masters Porco et al. (2006) Science W 27 Feb 311, 1393-1401 Enceladus ??? Oort Cloud Comets Miljan Draganic Astrobiology Matthew Kammerer NASA Shayna Kram F 29 Feb Levison et al. (2002), Science 296, 2212-2215 Vogel 1999 Science 286 70 W 05 Mar ? F 07 Mar Something direct from NASA? 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 • 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 Conduction - Fourier’s Law T >T 1 (T1 T0 ) dT k • Heat flow F F k d dz 0 T0 F d T1 • 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-1K-1 (rock, ice) and 3060 Wm-1K-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? Diffusion Equation • We can use Fourier’s law and the definition of Cp to find how temperature changes with time: F2 dz F1 T k 2T 2T k 2 2 t rC p z z • Here k is the thermal diffusivity (=k/rCp) and has units of m2 s-1 • Typical values for rock/ice 10-6 m2s-1 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 T 0 t 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: 2 d ~ kt • 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? Internal Heating • Assume we have internal heating H (in Wkg-1) • From the definition of Cp we have Ht=DTCp • 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 rH 6k ( R2 r2 ) 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? 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, h increases, Ra decreases, rate of cooling decreases -> self-regulating system Stagnant lid (cold, rigid) Plume (upwelling, hot) Sinking blob (cold) rgDTd Ra kh Image courtesy Walter Kiefer, Ra=3.7x106, Mars 3 Viscosity • Ra controls vigor of convection. Depends inversely on viscosity, h . • Viscosity depends on Temperature T, Pressure P, Stress s, Grain Size d. h Ae A – pre-exponential constant V – Activation Volume n – Stress Exponent E PV RT s d n m E – Activation Energy R – Gas Constant m – Grain-size exponent Viscosity relates stress and strain rate s h Viscoelasticity • A Maxwellian material has a viscous term and an elastic term. s s h E • If h is high, we get an elastic behavior. If h 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 Next Time • Paper Discussion – Titan Atmosphere – Tobie et al., 2006 • Planetary Interiors – Elastic Flexure • Planetary Atmospheres – Structure – Dynamics