Satellite geodesy. Basic concepts. I1.1a = geocentric latitude φ = geodetic latitude r = radial distance, h = ellipsoidal height a = semi-major axis, b = semi-minor axis z = axis of rotation, flattening = (a-b)/a. C.C.Tscherning, University of Copenhagen, Meridian plane h r b a Z X-Y φ

Coordinate-systems Example: Frederiksværk φ=56 0, λ=12 0, h= 50 m C.C.Tscherning,

h=H+N=Orthometric height + geoid height along plumb-line =H N +ζ=Normal height + height anomaly, along plumb-line of gravity normal field Geoid and mean sea level C.C.Tscherning, Ellipsoid Earth surface N H Geoid: gravity potential constant

Gravity potential, Kaula Chap. 1. Attraction (force): Direction from gravity center of m to M. With m = 1 (unitless), then acceleration C.C.Tscherning,

Gradient of scalar potential, V, C.C.Tscherning,

Volume distribution, ρ(x,y,z) V fulfills Laplace equation C.C.Tscherning,

Spherical coordinates Geocentric latitude Longitude, λ, r = distance to origin. C.C.Tscherning,

Laplace in spherical coordinates C.C.Tscherning,

Spherical harmonics Define: C.C.Tscherning,

Orthogonal basis functions Generalizes Fourier-series from the plane C.C.Tscherning,

Centrifugal potential On the surface of the Earth we also measure the centrifugál acceleration, C.C.Tscherning, r

Normal potential, U Good approximation to potential of ideal Earth Reference ellipsoid is equipotential surface, U=U 0, ideal geoid. It has correct total mass, M. It has correct centrifugal potential Knowledge of the series development of the gravity potential can be used to derive the flattening of the Earth ! C.C.Tscherning,

If we know the orbit of the satellite and measure distances (or distance differences) to the satellite, then we may determine our own position: GPS, Doppler. Satellite Geodesy: distances or ranges C.C.Tscherning, Us S1S1 S2S2 S3S3

Known orbit. One or two way ranging. Satellite altimetry PRARE Synthetic Aperture Radar Satellite Geodesy: ranging to the Earth C.C.Tscherning, Envisat Ocean or ice ERS1

Laser ranging Optical directions Enables orbit determination Enables gravity potential estimation Satellite Geodesy: from ground to satelite C.C.Tscherning, LAGEOS

We measure the position of a satellite from other satellites: SST Enables calculation of position and velocity Enables calculation of Kinetic Energy Satellite to satellite tracking C.C.Tscherning, S4S4 S1S1 S2S2 S3S3

We measure range rates between two satellites Gives (approximately) difference in Kinetic energy Satellite to satellite tracking, Low-Low C.C.Tscherning, S4S4 S1S1 S2S2 S3S3 S4S4 GRACE Range-rate

Measurement of acceleration differences inside satellite (GOCE). Determines second-order derivatives of potential. SST and gradiometry C.C.Tscherning, Mass 1 Mass 2

Points on the surface of the Earth (and Planets) and changes (geodynamics, climate changes, sea-level) Geoid/quasi-geoid determines state of sea-level if no currents – tides – salinity or temperature variations Determines tides, (geostrophic) currents etc. If geoid known Position of the Earth in Inertial system, Earth rotation, pole-position, Earth elastic parameters. Changes of gravity (due to crustal uplift or hydrosphere changes) Task of Geodesy: to determine C.C.Tscherning,