• Gravitational forces of the Sun andMoon deform the Earth’s shape ⇒tides in the oceans, atmosphere, andsolid earth

• Tidal effect of the Moon:– Earth and Moon are coupled by

gravitational attraction: each onerotates around the center of mass ofthe pair.

– The rotation of the Earth around thatcenter of mass induces a centrifugalacceleration directed away from theMoon

– The Moon produces a gravitationalattraction on the Earth

– The resulting force (centrifugalacceleration + gravitational attraction)is responsible for the tides

• Tidal effect of the Sun: same principlebut 45% smaller effect because oflarger Earth-Sun distance

Tides

Tides• Earth rotation (24 hr) combined

with Moon revolution (~27 days)=> major tidal component is semi-diurnal (M2 = 12 hr 25 min)

• Ocean tides: effect of oceansurface, amplitude of largestcomponent = several meters

• Solid Earth tides: effect on thesolid Earth surface, amplitude,amplitude of largest component~10-50 cm

• By-product of ocean tides: oceantide loading = elastic deformationof the Earth crust due to variationsof ocean water column: up to 15-20 cm near the coast

The ocean tides for harmonic M2 (period of 12 hours and 25 minutes) . The colorrepresent the amplitude and the contour lines indicate the phase lag of the tides with

a spacing of 60 degrees. (Doc. H.G. Scherneck)

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Day of year, 1999

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Water height variations in Brest (France) measured by a tide gauge (October 1999)

Example: ocean tideloading in Brittany, France

Measuring gravity• XVIIth century: pendulum clocks had to be

tuned when moved from Paris (49N) toCayenne (5N) ⇒ first gravity measurementsmade with a pendulum using:

• Absolute measurements: Acceleration of amass in free fall

• Relative measurements:– Extension of a spring (w.r.t. a reference

position)– Levitation of a metal mass in an

electromagnetic field = supraconductinggravimeters

⇒ Need for reference sites where absolute gravityis known

g

lT !2= T = period

l = wire length

relative measurement

absolute measurement

Relative gravity measurements• Mobile masses attached to springs:

– Stable: Measurement of the extensionof the spring (Scintrex)

– Unstable: Measurement of thedisplacement to apply to the spring tobring it back to an equilibriumposition (LaCoste & Romberg)

• Mechanical properties of springsdepend on temperature ⇒ thermostat

• 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

• Precision ~ 0.01 mGal

LaCoste & Romberg gravimeter

Absolute gravity measurements• Most common technique:

– Glass prism in free fall– Atomic clock => timing of the fall– G = 8H / (DT2 –Dt2)

• Precision ~1 µGal [~ 3 mm]• Transportable, but not easily portable, unlike

relative gravimeters.

Precision of ground-basedgravity measurements

Satelliteorbitography

• Orbit of artificial satellites are perturbed by variations of thegravity field.

• Therefore, precise measurements of their trajectory ⇒ gravityfield

• “Geodetic” satellites and ground tracking network ⇒estimation of precise orbit ⇒ restitution of gravity field

• Satellite trajectory derived from Satellite Laser Ranging (SLR) SLR at the Goddard Geophysical and Astronomical Observatory. The twolaser beams are coming from the network standard SLR station, MOBLAS-7

(MOBile LASer) and the smaller TLRS-3 (Transportable Laser RangingSystem) during a collocation exercise.Tracking a satellite with a network of SLR stations

Starlette, a geodetic satelliteLaunched in 1975, 48 cm diameter, 47 kg

A global gravity field from space…• 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, oceancirculation, sea level changes and thevisco-elastic response of the Earth'slithosphere to past and present loads

– Spatial resolution from space• Plus atmospheric research• CHAMP• GRACE

CHAMP• CHAllenging Minisatellite Payload• German (GFZ-Potsdam) small

satellite mission launched in July2000.

• Included instruments:magnetometer, accelerometer, GPSreceiver, laser retro reflector andion drift meter.

• Low altitude, near polar orbit• Continuous GPS satellite-to-

satellite tracking ability• On-board measurements of non-

gravitational orbit perturbations

GRACE

• Gravity Recovery And ClimateExperiment

• Launched in March 2002 byNASA/DLR.

• Two identical spacecrafts flying about220 km apart in a polar orbit 500 kmabove the Earth.

• Included instruments: K-Band rangingsystem, accelerometer, GPS receiver,laser retro-reflector, star camera, coarseEarth and Sun sensor, ultra stableoscillator, and center of mass trimassembly.

• Gravity field found by highly accuratemeasurements of the distance betweenthe 2 satellites using GPS and amicrowave ranging system

Satellite altimetry and the geoid• 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:

– Oceanographic effects: waves,currents, tides

– Gravimetric effects = the geoid• Precision:

– 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 altimetryand 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