PTYS 411 Evolution of Planetary Surfaces Gravity and Topography.
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Transcript of 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
PYTS 411 – Gravity and Topography 3
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
PYTS 411 – Gravity and Topography 4
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
PYTS 411 – Gravity and Topography 5
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
PYTS 411 – Gravity and Topography 6
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
PYTS 411 – Gravity and Topography 7
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
PYTS 411 – Gravity and Topography 8
Thanks to Isamu Matsuyama
PYTS 411 – Gravity and Topography 9
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
PYTS 411 – Gravity and Topography 10
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
PYTS 411 – Gravity and Topography 11
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
PYTS 411 – Gravity and Topography 12
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
PYTS 411 – Gravity and Topography 13
Fossil bulges can exist
PYTS 411 – Gravity and Topography 14
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
PYTS 411 – Gravity and Topography 15
PYTS 411 – Gravity and Topography 16
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
PYTS 411 – Gravity and Topography 17
PYTS 411 – Gravity and Topography 18
Histograms of planetary elevation - hypsograms
Melosh 2011
PYTS 411 – Gravity and Topography 19
Earth’s bimodal topography is caused by plate tectonics Venus has a near-Gaussian distribution Titan (preliminary) appears to have very little relief
PYTS 411 – Gravity and Topography 20
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
PYTS 411 – Gravity and Topography 21
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
PYTS 411 – Gravity and Topography 22
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
PYTS 411 – Gravity and Topography 23
PYTS 411 – Gravity and Topography 24
PYTS 411 – Gravity and Topography 25
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
PYTS 411 – Gravity and Topography 26
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
PYTS 411 – Gravity and Topography 27
GRAIL mission solves the lunar farside gravity problem.
Free Air
Bouguer
Zuber et al., 2013
PYTS 411 – Gravity and Topography 28
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
PYTS 411 – Gravity and Topography 29
Uncompensated
Strong positive free-air anomaly
Zero or weak negative Bouguer anomaly
Compensated
Weak positive free-air anomaly
Strong negative Bouguer anomaly
PYTS 411 – Gravity and Topography 30
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)
PYTS 411 – Gravity and Topography 31
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
PYTS 411 – Gravity and Topography 32
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
PYTS 411 – Gravity and Topography 33
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
PYTS 411 – Gravity and Topography 34
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
PYTS 411 – Gravity and Topography 35
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
PYTS 411 – Gravity and Topography 36
Crustal thickness of different areas
But many features are uncompensated….
So Bouguer anomaly doesn’t translate directly into crustal thickness
Zuber et al., 2000
PYTS 411 – Gravity and Topography 37
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
PYTS 411 – Gravity and Topography 38
Crustal thickness maps show lunar crustal dichotomy
Zuber et al., 1994
PYTS 411 – Gravity and Topography 39
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