Exploring Planetary Interiors: Methods & Techniques

1. EART162: PLANETARY INTERIORS

2. Course Overview • How do we know about the interiors of (silicate) planetary bodies? Their structure, composition and evolution. • Techniques to answer these questions • Cosmochemistry • Orbits and Gravity • Geophysical modeling • Seismology • Case studies – examples from this Solar System

3. Course Outline • Week 1 – Gravity • Week 2 – Gravity (cont’d), moments of inertia • Week 3 – Material properties, equations of state • Week 4 – Isostasy and flexure • Week 5 – Heat generation and transfer • Week 6 – Midterm; Seismology • Week 7 – Fluid dynamics and convection • Week 8 – Magnetism and planetary thermal evolution • Week 9 – Case studies • Week 10 – Recap. and putting it all together; Final

4. Logistics • Website: http://people.ucsc.edu/~igarrick/EART162 • Set text – Turcotte and Schubert, Geodynamics, 3rd edition (2014) • Supplemental resource: MIT Open Courseware for Essentials of Geophysics • Prerequisites – some knowledge of calculus expected • Grading – based on weekly homeworks (33%), midterm (33%), final (33%). • Homeworks due by 5pm (10% penalty per day) • Office hours –Tu/We 3:00-4:00 (A137 E&MS) or by appointment (email: igarrick@ucsc.edu) • Questions? - Yes please!

5. Expectations • Homework typically consists of 3 questions • Shouldn’t take more than ~1 hour per question on average • Midterm/finals consist of short (compulsory) and long (pick from a list) questions • Showing up and asking questions are usually routes to a good grade • Plagiarism – see website for policy.

6. Gravity • Governs orbits of planets and spacecraft • Largely controls accretion, differentiation and internal structure of planets • Spacecraft observations allow us to characterize structure of planets: • Bulk density (this lecture) • Moment of inertia (next week)

7. r F F m2 m1 R M a Gravity • Newton’s inverse square law for gravitation: • Hence we can obtain the acceleration g at the surface of a planet: Here F is the force acting in a straight line joining masses m1 and m2separated by a distance r; G is a constant (6.67x10-11 m3kg-1s-2) g has units of gals (=0.01m/s2) • We can also obtain the gravitational potential U at the surface (i.e. the work done to get a unit mass from infinity to that point): What does the negative sign mean? Conversely: (Very important! - examples)

8. Properties of U and g • Let S be an equipotenial of U. The rate of change of U along that surface must therefore be zero. The rate of change of U is proportional to g. Therefore, g has no component along S, and is perpendicular to S. • Fluids cannot support shear stresses. Forces along the surface of a fluid result in flow, until the in-surface component is eliminated. In steady state a fluid only has normal forces acting on it. Therefore, fluids conform to equipotential surfaces.

9. a Poisson’s and Laplace’s equation • Would like to learn about the whole volume of a body, as a whole, but we are stuck with the surface observable of gravity. • Gauss’ theorem (divergence theorem) can help! It relates a surface integral to an integral over the whole volume. Divergence of the vector field = source or sink Flux through surface

10. Right hand side of Gauss’ theorem Vector field on a surface S

11. Poisson and Laplace’s equation • After some algebra: • This holds when you are (or surface S is) examining a point INSIDE the total mass distribution. • S outside the total mass distribution:

13. Non-uniqueness • It can be shown that if U(r) is well measured on a boundary or surface), U can be uniquely determined everywhere. • BUT: The internal density distribution CANNOT be uniquely determined. • Can you measure the gravity field of the Moon at high altitudes, and know it at the surface? Why bother with low altitude measurements? • Can you measure the magnetic field at the surface of the Earth and predict the field at the core mantle boundary?

14. Downward continuation • If I know U at some high altitude, I can extrapolate to a lower one, at the risk of blowing up instrument noise, and not real measurements of small scale surface features.

15. Solving Laplace’s equation • Spherical harmonics are solutions to Laplace’s equation: essentially Fourier analysis on a sphere. Let a 1D function be represented by a sum of harmonic functions of increasing frequency, and let a and b be the weights of those functions.

16. 2D Cartesian example • JPEG compression (Discrete cosine transformation)

17. Separation of Variables • Solving for U: • Let U be the product of three functions with radial, longitudinal, and latitudinal dependence.

18. Spherical Harmonic Representation of U • For co-latitude θ (pole = 0°) and longitude φ on a sphere with radial coordinate r, multiply GM by: • Pl,m – Associated Legendre polynomials (with argument cos(θ)). • l (= “degree”) is like a frequency (wavelength). • m is the “order” or structure of that frequency. • Use (1/r)l+1 when the potential is external to the sphere (for this class). • Works for any function actually (e.g. topography). cos(θ) is the argument of P

19. S.H. examples • Which correspond to rotational flattening, or tides? • What is l = 0? • When was l = 2 for Earth measured? • Which one is “quadrupole”? • Which one is easiest to measure during a flyby?

20. a Now, 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 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

21. 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

22. 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 most materials get denser under increasing pressure • So 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 i.e. its equation of state. We’ll deal with this in a later lecture. • 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.

23. Mass of the Earth? • M was not known, only GM. • G must be determined experimentally. • Schiehallion mountain experiment (1774): Plumb bob feels pull of mountain and Earth. Deflection near mountain gives you relative masses of the Earth and mountain, if you can measure the density of the mountain. • Cavendish experiment (1797)

24. a Escape velocity and impact energy M • Now back to gravity . . . • Gravitational potential r R • How much kinetic energy do we have to add to an object to move it from the surface of the planet to infinity? • The velocity required is the escape velocity: • Equally, an object starting from rest at infinity will impact the planet at this escape velocity • Earth vesc=11 km/s. How big an asteroid would cause an explosion equal to a 1 MT nuclear bomb?

25. a 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 per unit mass? 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?

26. Equal total mass Uniform density r2 r1 r1<r2 a Differentiation • Which situation has the lower potential energy? • Consider a uniform body with two small lumps of equal volume DV and different radii ra,rb and densities ra,rb • Which configuration has the lower potential energy? rb,rb rb,ra PEleft=(g0DV/R)(ra2ra+rb2rb) ra,ra ra,rb PEright=(g0DV/R)(rb2ra+ra2rb) R Surface gravity g0 We can minimize the potential energy by moving the denser material closer to the centre (try an example!) Does this make sense?

27. Differentiation (cont’d) • So a body can lower its potential energy (which gets released as heat) by collecting the densest components at the centre – differentiation is energetically favoured • Does differentiation always happen? This depends on whether material in the body can flow easily (e.g. solid vs. liquid) • So the body temperature is very important • Differentiation can be self-reinforcing: if it starts, heat is released, making further differentiation easier, and so on

28. End of Lecture • Next week – more on using gravity to determine internal structures

29. Planetary Crusts • Remote sensing (IR, gamma-ray) allows inference of surface (crustal) mineralogies & compositions: • Earth: basaltic (oceans) / andesitic (continents) • Moon: basaltic (lowlands) / anorthositic (highlands) • Mars: basaltic (plus andesitic?) • Venus: basaltic • In all cases, these crusts are distinct from likely bulk mantle compositions – indicative of melting • The crusts are also very poor in iron relative to bulk nebular composition – where has all the iron gone? How can we tell?

30. Moon ratios Garrick-Bethell et al. 2011

31. Timing Accretion • One of the reasons samples are so valuable is that they allow us to measure how fast planets accrete and differentiate. • Oldest U: basaltic achondrites, e.g. Angrites • Oldest igneous rock: 3 My after SS formation • We can also do this using short-lived radioisotopes e.g. 26Al (thalf=0.7 Myr), 60Fe (thalf=2.6 Myr), 182Hf (thalf=9 Myr) • Processes which cause fractionation (e.g. melting, core formation) can generate isotopic anomalies if they happen before the isotopes decay. • Core formation finished as rapidly as 3 Myr (Vesta) and as slowly as ~30 Myr (Earth).

32. Differentiated mantle Undiff. planet Core forms 182Hf (lithophile) Early core formation: excess 182W in mantle Late core formation: no excess 182W 182W (siderophile) Core forms Hf-W system • 182Hf decays to 182W, half-life 9 Myrs • Hf is lithophile, W is siderophile, so abundances time core formation (related to accretion process) Kleine et al. 2002

33. Remote Sensing • Again, restricted to surface (mm-mm). Various kinds: • Spectral (usually infra-red) reflectance/absorption – gives constraints on likely mineralogies e.g. Mercury, Europa • Neutron – good for sensing subsurface ice (Mars, Moon) • Gamma-ray – gives elemental abundances (especially of naturally radioactive elements K,U,Th) • Energies of individual gamma-rays are characteristic of particular elements

34. K/U ratios • Potassium (K) and uranium (U) behave in a chemically similar fashion, but have different volatilities: K is volatile, U refractory • So differences in K/U ratio tend to arise as a function of temperature, not chemical evolution • K/U ratios of most terrestrial planet surfaces are rather similar (~10,000) • What does this suggest about the bulk compositions of the terrestrial planets? • K/U ratio is smaller for the Moon – why? • K/U ratio larger for the primitive meteorites – why? K/U From S.R. Taylor, Solar System Evolution, 1990

35. 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 2.3:1 • 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?