Satellite geophysics. 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, 1900. flattening = (a-b)/a.
C.C.Tscherning, University of Copenhagen, 2013-10-25 1
Meridian planehrb
a
Z
X-Yφ
Coordinate-systems
Example:Frederiksværk φ=560, λ=120, h= 50 m
C.C.Tscherning, 2013-10-25.
h=H+N=Orthometric height + geoid height along plumb-line =HN+ζ=Normal height + height anomaly, along plumb-line of gravity normal field
Geoid and mean sea level
C.C.Tscherning, 2013-10-25.
Ellipsoid
Earth surface
NH Geoid: gravity potential constant
4
GEOID
5
Coordinate-systems and time.
NON INERTIAL SYSTEM
CTS:Conventional
Terrestrial System
Mean-rotationaxis1900.
Greenwich
X
Y- Rotates withthe Earth
Z
Gravity-centre
6
POLAR MOTION
• Approximatively circular• Period 430 days (Chandler period)• Main reason: Axis of Inertia does not co-inside
with axis of rotation.• Rigid Earth: 305 days: Euler-period.
8
Ch. 3, Transformation CIS - CTS
• Precession• Nutation• Rotation+• Polar movement
Sun+Moon
CISCTS rSNPr
Gravity potential, Kaula Chap. 1.
• Attraction (force):
• Direction from gravity center of m to M.• With m = 1 (unitless), then acceleration
2rmMkF
C.C.Tscherning, 2013-10-25.
22 rrkMa
Gradient of scalar potential, V,
C.C.Tscherning, 2013-10-25.
mass-point),,(,rMkzyxV
zVyVxV
Va
Volume distribution, ρ(x,y,z)
• V fulfills Laplace equation
dr
k
dzdydxzzyyxx
zyxk
zyxV
Earth
Earth
222 )'()'()'(
),,()',','(
C.C.Tscherning, 2013-10-25.
masses outside,02
2
2
2
2
2
Vz
Vy
Vx
V
Spherical coordinates
• Geocentric latitude• Longitude, λ, r = distance to origin.
C.C.Tscherning, 2013-10-25.
drddrdxdydzd
rzryrx
2cos :measure-Volume
sinsincoscoscos
Laplace in spherical coordinates
C.C.Tscherning, 2013-10-25.
order. m degree, n ),(sin)( sinor cos)( ,or )(
),()()( :Solutioncos
1
)(coscos
11
1
2
2
2
22
nm
nn
PmmrrrR
rRV
V
VrVr
rrV
Spherical harmonics
• Define:
C.C.Tscherning, 2013-10-25.
n
nmnmnm
n
mnn
n
nm
rVCkMrV
mmmm
PrarV
),,(),,(
then0,||sin
0,cos)(sin),,(
0
||1
Orthogonal basis functions
• Generalizes Fourier-series from the plane
C.C.Tscherning, 2013-10-25.
nmnm
nm
ijnm
PP
jminkMC
ddrVrV
and CC
thenmalized,(fully)nor bemay Functionsor for 0
jm i,nfor
cos),,(),,(
nmnm
180
0
90
90
16
Gravity model database.
Spherical harmonic coefficients:http://icgem.gfz-potsdam.de/ICGEM/
CCT, Nov. 2013. (CCT)
Centrifugal potential
• On the surface of the Earth we also measure the centrifugál acceleration,
space). inertial(in velocity rotational
)(21
cos21
222
222
yxV
rVW
C.C.Tscherning, 2013-10-25.
r
Normal potential, U• Good approximation to potential of ideal Earth• Reference ellipsoid is equipotential surface, U=U0, 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 !
Equator)at (gravity ,/
......)72
231(3/)2/1(
32
cos2
....)(sin)()(sin)(1
2
2
222
404
4202
2
egam
fmmffJ
rPraJP
raJU
C.C.Tscherning, 2013-10-25.
Anomalous potential,T• T=W-U, • same mass and gravity center.• Makes all quantities small,gives base for linearisation.
C.C.Tscherning, 2013-10-25.
ocean. on theanomaly height height Geoid/ :anomalyHeight
.height point within gravity normalr2- :anomalyGravity
.height lellipsoida with P,in gravity normal minusgravity
:edisturbancGravity
TH
TrTgg
hrTgg
QP
PP
20
Gravity.
Source: DTU-Space. Ole Andersen.
CCT, Nov. 2013. (CCT)
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