Tides
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Gravitational forces of the Sun and
Moon deform the Earth’s shape ⇒
tides in the oceans, atmosphere, and
solid earth
Tidal effect of the Moon:
– Earth and Moon are coupled by
gravitational attraction: each one
rotates around the center of mass of
the pair.
– The rotation of the Earth around that
center of mass induces a centrifugal
acceleration directed away from the
Moon
– The Moon produces a gravitational
attraction on the Earth
– The resulting force (centrifugal
acceleration + gravitational attraction)
is responsible for the tides
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Tidal effect of the Sun: same principle
but 45% smaller effect because of
larger Earth-Sun distance
400
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Ocean tides: effect of ocean
surface, amplitude of largest
component = several meters
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Solid Earth tides: effect on the
solid Earth surface, amplitude,
amplitude of largest component
~10-50 cm
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By-product of ocean tides: ocean
tide loading = elastic deformation
of the Earth crust due to variations
of ocean water column: up to 1520 cm near the coast
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0
-100
-200
-300
The ocean tides for harmonic M2 (period of 12 hours and 25 minutes) . The color
represent the amplitude and the contour lines indicate the phase lag of the tides with
a spacing of 60 degrees. (Doc. H.G. Scherneck)
Day of year, 1999
304
302
303
301
300
299
298
295
297
293
294
292
291
290
288
289
286
285
284
283
282
281
279
280
277
275
-400
276
Earth rotation (24 hr) combined
with Moon revolution (~27 days)
=> major tidal component is semidiurnal (M2 = 12 hr 25 min)
200
274
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300
water height (cm)
Tides
Water height variations in Brest (France) measured by a tide gauge (October 1999)
Example: ocean tide
loading in Brittany, France
Measuring gravity
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XVIIth century: pendulum clocks had to be
tuned when moved from Paris (49N) to
Cayenne (5N) ⇒ first gravity measurements
made with a pendulum using:
T = 2!
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l
g
T = period
l = wire length
absolute measurement
Absolute measurements: Acceleration of a
mass in free fall
Relative measurements:
– Extension of a spring (w.r.t. a reference
position)
– Levitation of a metal mass in an
electromagnetic field = supraconducting
gravimeters
⇒ Need for reference sites where absolute gravity
is known
relative measurement
Relative gravity measurements
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Mobile masses attached to springs:
– Stable: Measurement of the extension
of the spring (Scintrex)
– Unstable: Measurement of the
displacement to apply to the spring to
bring it back to an equilibrium
position (LaCoste & Romberg)
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Mechanical properties of springs
depend on temperature ⇒ thermostat
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Perfect leveling necessary
Elasticity of springs vary with age ⇒
instrumental drift
– Complex, function of age,
transportation, etc.
– ~ linear for spring-based gravimeters
– Specific to each gravimeter
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Precision ~ 0.01 mGal
LaCoste & Romberg gravimeter
Absolute gravity measurements
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Most common technique:
– Glass prism in free fall
– Atomic clock => timing of the fall
– G = 8H / (DT2 –Dt2)
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Precision ~1 µGal [~ 3 mm]
Transportable, but not easily portable, unlike
relative gravimeters.
Precision of ground-based
gravity measurements
Satellite
orbitography
• Orbit of artificial satellites are perturbed by variations of the
gravity field.
• Therefore, precise measurements of their trajectory ⇒ gravity
field
• “Geodetic” satellites and ground tracking network ⇒
estimation of precise orbit ⇒ restitution of gravity field
• Satellite trajectory derived from Satellite Laser Ranging (SLR)
Tracking a satellite with a network of SLR stations
Starlette, a geodetic satellite
Launched in 1975, 48 cm diameter, 47 kg
SLR at the Goddard Geophysical and Astronomical Observatory. The two
laser beams are coming from the network standard SLR station, MOBLAS-7
(MOBile LASer) and the smaller TLRS-3 (Transportable Laser Ranging
System) during a collocation exercise.
A global gravity field from space…
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Current version of global gravity field = GRIM5 [21 satellites, data since 1971, precision
3 mGals]
Advantage = global coverage
Recent space missions
• Obectives are to improve:
– Temporal resolution:
atmospheric mass redistribution, ocean
circulation, sea level changes and the
visco-elastic response of the Earth's
lithosphere to past and present loads
– Spatial resolution from space
• Plus atmospheric research
• CHAMP
• GRACE
CHAMP
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CHAllenging Minisatellite Payload
German (GFZ-Potsdam) small
satellite mission launched in July
2000.
Included instruments:
magnetometer, accelerometer, GPS
receiver, laser retro reflector and
ion drift meter.
Low altitude, near polar orbit
Continuous GPS satellite-tosatellite tracking ability
On-board measurements of nongravitational orbit perturbations
GRACE
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Gravity Recovery And Climate
Experiment
Launched in March 2002 by
NASA/DLR.
Two identical spacecrafts flying about
220 km apart in a polar orbit 500 km
above the Earth.
Included instruments: K-Band ranging
system, accelerometer, GPS receiver,
laser retro-reflector, star camera, coarse
Earth and Sun sensor, ultra stable
oscillator, and center of mass trim
assembly.
Gravity field found by highly accurate
measurements of the distance between
the 2 satellites using GPS and a
microwave ranging system
Satellite altimetry and the geoid
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Direct measurement of the ocean
surface using satellites
Satellite carries radar ⇒ ocean –
satellite range
Ground tracking system ⇒ ellipsoid –
satellite distance
Difference = ocean – ellipsoid
distance = dynamic topography
Contains:
–
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Precision:
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Oceanographic effects: waves,
currents, tides
Gravimetric effects = the geoid
SEASAT (1979) = 10 cm
TOPEX-POSEIDON (1992) = 5 cm
JASON (launched in 2002) < 5 cm
Advantages: precision, coverage
Satellite altimetry
Seafloor topography derived from Seasat altimetric measurements
Satellite altimetry
and the geoid
• Long wavelength → mantle convection
• Short wavelength → ocean floor topography
• http://topex.ucsd.edu/marine_grav/mar_grav.html
What have we learned?
• The Earth’s gravity is the result of its mass, its rotation,
and its (ellipsoidal) shape.
• The Earth’s gravity field is associated with a potential.
• The geoid is the particular equipotential surface that best
fits the mean sea level (= the horizontal)
• Relationship between geoid and gravity
• Gravity and geoid “height” vary:
– As a function of mass distribution (in space and time)
– As a function of position (e.g. altitude)
• Direct measurements:
– Gravity: up to 1 µGal
– Geoid height : up to 5 cm