PTYS 411 Evolution of Planetary Surfaces Gravity and Topography

PTYS 411

Evolution of Planetary Surfaces

Gravity and TopographyGravity and Topography

PYTS 411 – Gravity and Topography 2

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 – The Shape of the World


Galileo Galilei (~1600 AD) Accurately determined g All objects fall at the same rate 1 gal = 1 cm s-2, g = 981 gals

Quick History – Gravity

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


Most of what follows assumes hydrostatic equilibrium i.e. increasing pressure with depth balances self gravity Much of what follows assumes constant density

Hydrostatic Equilibrium

ΔR

R

Total Radius RT

Constant density ρ

Integrate shells of material to add up their contribution to pressure

Central pressure = ½ ρ g RT

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

Linear Rotational

Momentum P = m v L = I ω

Energy E = ½ m v2 E = ½ I ω2

Response to force t

vmF

t

I

‘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 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 wSpin energy = ½ I w2

i.e. Spin energy = (½ L2) / I

Lowest energy = highest IC is the largest angular momentumSo 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

Paleo-poles 90° from Tharsis Expected, as it would be very difficult to move Tharsis off the equator

Perron et al. 2007


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


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

Planetary Shape Continued

gp

ge

latitudea cos2

Melosh, 2011


Fossil bulges can exist


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

Geoid

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


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?

Measuring Gravity with Spacecraft




Before we can start interpreting gravity anomalies we need to make sure we’re comparing apples to apples…

Corrections to Observations

Free-Air correction Assume there’s nothing but vacuum between observer and

reference ellipsoid Just a distance correction r

ghg

hr

GM

rr

r

gg

FA

FA

2

2


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

hGgB 2Ref.

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

Bouguer

Zuber et al., 2013


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

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

Compensation

Crust

Mantle


Uncompensated

Strong positive free-air anomaly

Zero or weak negative Bouguer anomaly

Compensated

Weak positive free-air anomaly

Strong negative Bouguer anomaly


Crust

Mantle

0 Bouguer(Topography only)

+ve Bouguer(subsurface excesses)

-ve Bouguer(subsurface deficits)

0 free air(isostasy)

-ve free air

(strength)

+ve free air

(strength)


Free Air

Bouguer

Zuber et al., 2013

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

Lunar gravity


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

Topography BouguerFree Air

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


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

Interpreting Bouguer Anomalies as Crustal Thickness Variations


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

Crustal ThicknessZuber et al., 2000

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


A common occurrence with large impact basins Lunar mascons (near-side basins holding the

mare basalts) Utopia basin on Mars

Initially isostatic

+ve Bouguer0 free-air

Sediment/lava fill basinNow 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

Documents

PTYS 411 Evolution of Planetary Surfaces Gravity and Topography