Gravity and Topography

Quick History – The Shape of the World Pythagoras (~550 BC) Speculation that the Earth was a sphere Eratosthenes (~250 BC) Calculation of Earth’s size Shadows at Syene vs. none at Alexandria Angular separation and distance converted to radius Estimate of 7360km – only ~15% too high Invention of the telescope Jean Picard (1671) – length of 1° of meridian arc Radius of 6372 Km – only 1km off! Length of 1° changes with latitude Controversy of prolate vs. oblate spheroids Pierre Louis Maupertuis - Survey 1736-1737 Equatorial degrees are smaller Earth is an oblate spheroid

Quick History – Gravity Galileo Galilei (~1600 AD) Accurately determined g All objects fall at the same rate 1 gal = 1 cm s-2, g = 981 gals Isaac Newton (1687) Universal law of gravitation Derived to explain Kepler’s third law Led to the discovery of Neptune Henry Cavendish (1798) Attempt to measure the Earth’s density Measured G as a by-product Found Earth~5500 kg/m3 > rocks Density must increase with Depth Nineteenth century Everest and Bouguer both find mountains cause deflections in gravity field Deflections less than expected Airy and Pratt propose isostasy via different mechanisms

Hydrostatic Equilibrium Most of what follows assumes hydrostatic equilibrium i.e. increasing pressure with depth balances self gravity Much of what follows assumes constant density Total Radius RT Constant density ρ Integrate shells of material to add up their contribution to pressure Central pressure = ½ ρ g RT R ΔR Planets are flattened by rotation and represented by ellipsoids i.e. a = b ≠ c Triaxial ellipsoids can be used: a ≠ b ≠ c ... but only for a few irregular bodies Planetary flattening described by: f for a perfectly fluid Earth 1/299.5 Difference due to internal strength Perhaps a relict of previously faster spin f for Mars ~ 1/170 – much more flattened

Analogous to mass for linear systems Moment of Inertia Analogous to mass for linear systems Linear Rotational Momentum P = m v L = I ω Energy E = ½ m v2 E = ½ I ω2 Response to force ‘I’ can be integrated over entire bodies, usually I = k MR2 For solid homogeneous spheres I = 0.4 MR2 …but planets are ellipsoids, so I depends on what axis you choose C = I about the rotation axis A = I about an equatorial axis Dynamical ellipticity: Obtained from satellite orbits, Hearth = 1 / 305.456 Or precession rates (usually requires a lander e.g. pathfinder on Mars) Oblateness of the gravity field (J2) depends on (C-A) / MR2 So H/J2 gives C / MR2 i.e. you can’t figure this out from the gravity field alone

For Mars I=0.3662 MR2 E.g. for a simple core-mantle sphere For solid homogeneous spheres I = 0.4 MR2 If extra mass is near the center (e.g. core of a planet) then I < 0.4 IEarth = 0.33 - big core IMars= 0.36 - smaller core (closer to homogeneous) Knowledge of the moment of inertia can give us clues about the internal structure E.g. Mariner 10’s flyby of Mercury revealed the large iron core E.g. for a simple core-mantle sphere Typically two solutions For Mars I=0.3662 MR2

Response to loads Planets spin around the axis of greatest moment of inertia Lowest energy configuration Moment of Inertia can change Mantle convection Plate tectonics Ice ages Building volcanoes Impact basins Spin re-aligns - angular momentum is conserved The planet moves – spin vector remains pointing in the same direction Mass excesses move towards the equator, mass deficits to the poles Angular Momentum = L = I w Spin energy = ½ I w2 i.e. Spin energy = (½ L2) / I Lowest energy = highest I C is the largest angular momentum So spinning around the shortest axis is the lowest energy state

Thanks to Isamu Matsuyama

Polar wander driven by Tharsis? Very large volcanic construct On present day equator Several km of overlapping lava flows Lithosphere shape and Tharsis compete Fossil bulge wants to stay on the equator Tharsis wants to move to the equator Matsuyama et al. 2006

Ocean shorelines postulated on Mars Reorientation of Mars would change the equilibrium shape of the body Shorelines would be warped out of shape Deviations of shoreline from a constant elevation can be explained by polar wander Perron et al. 2007 Paleo-poles 90° from Tharsis Expected, as it would be very difficult to move Tharsis off the equator

Low density ‘loads’ move towards the pole Mass removal from impact basins E.g. the asteroid Vesta Rising plumes (must be lower density to rise) E.g. Enceladus Enceladus south pole Geologic evidence for extension Rising diapir could explain bulging of surface South pole location explained by polar wander

Planetary Shape Continued gp Planets are flattened by rotation Hydrostatic approximation can tell us how much Gravity at equator adjusted by centrifugal acceleration Gravity at pole unaffected by rotation Dynamical flattening not equal real flattening Objects are not in hydrostatic equilibrium Solid planets have some strength to maintain their shape Ellipsoids are too simple to represent planetary shapes ge Melosh, 2011

Fossil bulges can exist

Geoid Real planets are lumpy, irregular, objects Deviations of the equipotential surface from the ellipsoid make up the geoid Expressed in meters – range on Earth from ~ -100 to +100 meters Earth’s geoid corresponds to mean sea level This is the definition of a flat surface – but it has high and low points Topography is measured relative to the geoid

Geoid undulates slowly over long distances i.e. it contains only very long wavelength features Shorter wavelength structure in the gravity field are called gravity anomalies Plumb lines point normal to the geoid Lithospheric mass excesses Cause positive geoid anomaly E.g. Subducting slab Lithospheric mass deficit Causes negative geoid anomaly Mantle plumes Topography measured relative to geoid Use geoid to convert planetary radius to topography Topography and geoid height are usually correlated Ratio of topography and geoid heights called the admittance

Histograms of planetary elevation - hypsograms Melosh 2011

Earth’s bimodal topography is caused by plate tectonics Venus has a near-Gaussian distribution Titan (preliminary) appears to have very little relief

Martian topography also appears bimodal Can be corrected with a center of mass/center of figure offset Bimodal topography is not diagnostic of plate tectonics Earth’s bimodality could also be removed if all the continents were in one hemisphere

Moon also has two terrain types Anorthosite highlands Basalt flooding lowlands Lunar fossil bulge is a mystery Moon is more oblate than expected given its current slow spin Bulge ‘frozen-in’ from previous faster spin? No. Early eccentric orbit can explain bulge Some influence from lithospheric strength must occur here… Lunar center of figure offset Tidal distortion of moon with solidifying magma ocean …but there’s no thick crust on the near-side

Measuring Gravity with Spacecraft Gravity measured in Gals 1 gal = 1 cm s-2 Earth’s gravity ranges from 976 (polar) to 983 (equatorial)  gal Gravity anomalies (deviations from expected gravity) are measured in mgal i.e. in roughly parts per million for the Earth Gravitational anomalies Only really addressable with orbiters Surface resolution roughly similar to altitude Anomalies cause along-track acceleration and deceleration Changes in velocity cause doppler shift in tracking signal Convert Earth line-of-sight velocity changes to change in g Downward continue to surface to get surface anomaly What about the far side of the Moon?

Corrections to Observations Before we can start interpreting gravity anomalies we need to make sure we’re comparing apples to apples… Free-Air correction Assume there’s nothing but vacuum between observer and reference ellipsoid Just a distance correction

More complicated corrections for terrain, tides etc… also exist Bouguer correction Assume there’s a constant density plate between observer and reference ellipsoid Remove the gravitation attraction due to the mass of the plate If you do a Bouguer correction you must follow up with a free-air correction Ref. Ellipsoid Ref. Ellipsoid Bouguer Free-Air More complicated corrections for terrain, tides etc… also exist

GRAIL mission solves the lunar farside gravity problem. Free Air Zuber et al., 2013 Bouguer

Airy Isostasy Pratt Isostasy Vening Meinesz Compensation Crust Mantle Simple view of topography Supported by lithospheric strength Large positive free-air anomaly Bouguer correction should get rid of this Anomalies due to topography are much weaker than expected though Due to compensation Crust Mantle Airy Isostasy Compensation achieved by mountains having roots that displace denser mantle material gh1 ρu = gr1 (ρs – ρu) Pratt Isostasy Compensation achieved by density variations in the crust g D ρu = g (D+h1) ρ1 = g (D+h2) ρ2 etc.. Vening Meinesz Flexural Model that displaces mantle material Combines flexure with Airy isostasy

Uncompensated Compensated Strong positive free-air anomaly Weak positive free-air anomaly Zero or weak negative Bouguer anomaly Strong negative Bouguer anomaly

(subsurface deficits) (subsurface excesses) -ve Bouguer (subsurface deficits) 0 Bouguer (Topography only) +ve Bouguer (subsurface excesses) +ve free air (strength) Crust 0 free air (isostasy) Mantle -ve free air (strength)

Lunar gravity Free Air Bouguer Mountains Positive free-air anomalies Support by a rigid lithosphere Mascons First extra-terrestrial gravity discovery Very strong positive anomalies Uplift of denser mantle material beneath large impact basins Later flooding with basalt Bulls eye pattern – multiring basins Only the center ring was flooded with mare lavas Flexure South pole Aitken Basin Appears fully compensated Older Free Air Zuber et al., 2013 Bouguer

Free Air Topography Bouguer Local structure visible E.g. Korolev Crater – low density annulus with dense center within peak ring Small craters in Free-Air but not Bouguer so uncompensated Free Air Topography Bouguer Zuber et al., 2013

Local structure visible Gradient of Bouguer Anomaly reveals long linear features within lunar crust Thought to be dikes permitted by global expansion of a few km (pre-Nectarian to Nectarian) Andrews-Hanna et al., 2013

Interpreting Bouguer Anomalies as Crustal Thickness Variations Assume this… Topography is compensated Crustal density is constant Bouguer anomalies depend on Density difference between crust and mantle Thickness of crust Negative anomalies mean thicker crust Positive anomalies mean thinner crust Choose a mean crustal thickness or a crust/mantle density difference -ve Bouguer +ve Bouguer

Crustal Thickness Free Air Zuber et al., 2000 Crustal Thickness Tharsis Large free-air anomaly indicates it is uncompensated But it’s too big and old to last like this Flexurally supported? Crustal thickness Assume Bouguer anomalies caused by thickness variations in a constant density crust Need to choose a mean crustal thickness Isidis basin sets a lower limit Free Air

Crustal thickness of different areas But many features are uncompensated…. So Bouguer anomaly doesn’t translate directly into crustal thickness Zuber et al., 2000

Sediment/lava fill basin Now flexurally supported A common occurrence with large impact basins Lunar mascons (near-side basins holding the mare basalts) Utopia basin on Mars Initially isostatic +ve Bouguer 0 free-air Sediment/lava fill basin Now flexurally supported +ve Bouguer +ve free-air

Crustal thickness maps show lunar crustal dichotomy Zuber et al., 1994

Things have come a long way in 214 yrs Planets are mostly spheres distorted by rotation Moments of inertia can tell you the internal structure Extra lumpiness comes from surface and buried geologic structures Gravity fields are also ‘lumpy’ Lumpiness due to surface effects can be removed Sub-surface structure can be investigated