PTYS 411 Evolution of Planetary Surfaces Gravity and Topography PYTS 411 – Gravity and Topography Quick History – The Shape of the World Pythagoras.
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Transcript PTYS 411 Evolution of Planetary Surfaces Gravity and Topography PYTS 411 – Gravity and Topography Quick History – The Shape of the World Pythagoras.
PTYS 411
Evolution of Planetary Surfaces
Gravity and Topography
PYTS 411 – Gravity and Topography
Quick History – The Shape of the World
Pythagoras (~550 BC)
Eratosthenes (~250 BC)
Speculation that the Earth was a sphere
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
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Quick History – Gravity
Galileo Galilei (~1600 AD)
Isaac Newton (1687)
Accurately determined g
All objects fall at the same rate
1 gal = 1 cm s-2, g = 981 gals
Universal law of gravitation
Derived to explain Kepler’s third law
Led to the discovery of Neptune
F=
GMm
r2
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
so g =
GM
r2
PYTS 411 – Gravity and Topography
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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
R
Integrate shells of material to add up their
contribution to pressure
ΔR
Planets are flattened by rotation and represented by ellipsoids
Central pressure = ½ ρ g RT
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
f=
a-c
a
fEarth =
1
298.257
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Moment of Inertia
Analogous to mass for linear systems
Momentum
Energy
Response to
force
Linear
Rotational
P=mv
L=Iω
E = ½ m v2
E = ½ I ω2
F m
‘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
v
t
C = I about the rotation axis
A = I about an equatorial axis
H=
C-A
C
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
I
t
PYTS 411 – Gravity and Topography
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
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
IEarth = 0.33 - big core
IMars= 0.36 - smaller core (closer to homogeneous)
Typically two solutions
For Mars I=0.3662 MR2
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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 w 2
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
PYTS 411 – Gravity and Topography
Thanks to Isamu Matsuyama
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PYTS 411 – Gravity and Topography
Polar wander driven by Tharsis?
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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
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
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PYTS 411 – Gravity and Topography
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
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Planetary Shape Continued
Planets are flattened by rotation
f=
a-c
a
Hydrostatic approximation can tell us how much
Gravity at equator adjusted by centrifugal acceleration
Gravity at pole unaffected by rotation
2a coslatitude
gp
ge
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
Melosh, 2011
PYTS 411 – Gravity and Topography
Fossil bulges can exist
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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
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PYTS 411 – Gravity and Topography
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
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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
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Histograms of planetary elevation - hypsograms
Melosh 2011
PYTS 411 – Gravity and Topography
Earth’s bimodal topography is caused by plate tectonics
Venus has a near-Gaussian distribution
Titan (preliminary) appears to have very little relief
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PYTS 411 – Gravity and Topography
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
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PYTS 411 – Gravity and Topography
Moon also has two terrain types
Lunar fossil bulge is a mystery
Anorthosite highlands
Basalt flooding lowlands
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
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PYTS 411 – Gravity and Topography
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?
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Corrections to Observations
Before we can start interpreting gravity anomalies we need to make sure we’re
comparing apples to apples…
g
GM
g
r
h
FA
Free-Air correction
r
r r 2
Assume there’s nothing but vacuum between observer and
2 gh
reference ellipsoid
g FA
Just a distance correction
r
PYTS 411 – Gravity and Topography
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
g B 2 Gh
Ref.
Ellipsoid
Ref.
Ellipsoid
Bouguer
Free-Air
More complicated corrections for terrain, tides etc… also exist
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GRAIL mission solves
the lunar farside
gravity problem.
Free Air
Zuber et al., 2013
Bouguer
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Compensation
Simple view of topography
Crust
Supported by lithospheric strength
Large positive free-air anomaly
Bouguer correction should get rid of this
Mantle
Anomalies due to topography are much weaker than expected though
Due to compensation
Airy Isostasy
Pratt Isostasy
Compensation achieved by mountains having
roots that displace denser mantle material
gh1 ρu = gr1 (ρs – ρu)
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
PYTS 411 – Gravity and Topography
Uncompensated
Strong positive free-air anomaly
Zero or weak negative Bouguer anomaly
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Compensated
Weak positive free-air anomaly
Strong negative Bouguer anomaly
PYTS 411 – Gravity and Topography
-ve Bouguer
(subsurface deficits)
0 Bouguer
(Topography only)
+ve free
air
(strength
)
Crust
0 free air
(isostasy)
-ve free
air
(strength
)
Mantle
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+ve Bouguer
(subsurface excesses)
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Lunar gravity
Mountains
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
Positive free-air anomalies
Support by a rigid lithosphere
Zuber et al., 2013
Free Air
Only the center ring was
flooded with mare lavas
Flexure
South pole Aitken Basin
Appears fully
compensated
Older
Bouguer
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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
PYTS 411 – Gravity and Topography
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
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PYTS 411 – Gravity and Topography
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
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PYTS 411 – Gravity and Topography
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
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PYTS 411 – Gravity and Topography
Crustal thickness of different
areas
But many features are
uncompensated….
So Bouguer anomaly doesn’t
translate directly into crustal
thickness
Zuber et al., 2000
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PYTS 411 – Gravity and Topography
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
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PYTS 411 – Gravity and Topography
Crustal thickness maps show lunar crustal dichotomy
Zuber et al., 1994
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PYTS 411 – Gravity and Topography
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
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