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GravityAn Introduction
Reiner RummelInstitute of Advanced
Study
(IAS)
Technische Universität München
Lecture
One
5th ESA Earth Observation Summer School
2-13 August 2010, ESA-ESRIN, Frascati
/ Italy
micro-g
environment:
e.g. boiling
water
fundamental physics:
e.g. equivalence
principle
earth
sciencese.g. gravitational
field
of earth
moon
and planets
gravitation
and space
science
global map
of gravity
anomalies
Lecture
One:Theoretical
basics
of gravitation
as applied
to the
earth[gravitational
law, properties, mathematical
representation]The
„language“
for
the
two
other
lecturesLecture
Two:The
role
of the
earth‘s
gravitational
field
in earth
sciences[gravity
anomalies, geoid
as a reference, temporal variations]Lecture
Three:Principles
of satellite
gravimetry; in their
logic
derived
from
free
fall testsin a laboratory
on earth[the
orbit, principles
of GRACE, ESA‘s
mission
GOCE and satellite
gradiometry]
the
plan
Newton‘s
law
of gravitation:
Newton‘s
second law:
gravitational
acceleration:AB
A
B
AF
ABe
introduction
to gravitation
2 3A B A B
A AB A BAB A B
m m m mF G e G x xx x
IAA AmF a
2B
A ABAB
AIA
mm
ma G e
Fundamental properties
of Newton‘s
law
of gravitation:• central
force• action
= reaction• inverse
square
distance• superposition
of all partial forces• instantaneous
introduction
to gravitation
from
single
mass, to many
masses, to a continuum
2B
A AB BAB
a G e d
Aa
is
a vector
field
in spacewith
the
following
properties:
i.e. there
exists
a gravitational
potential Vand in outer
space, we
get:
introduction
to gravitation
2B
A AB BAB
a G e d
0A A Aa curl free a V
20 0Aa source free V
Aa
is
a vector
fieldwith
the
following
properties:
i.e. there
exists
a gravitational
potential Vand in outer
space, we
get:
introduction
to gravitation
2B
A AB BAB
a G e d
0A A Aa curl free a V
20 0Aa source free V
2 2 2
2 2 2 0V V Vx y z
BA B
AB
V G d
2
2
Vx
example: satellite
orbit
a nice
application: a satellite
orbit
0 0
" "
: ;
A A Ax a V perturbations
and initial conditions x x
sphere flattened
sphere real earth
a nice
application: a satellite
orbit
Kelplerian
ellipse precessing
ellipse precessing
ellipseplus „gravitational
code“
example: satellite
orbit
example: satellite
orbit
What
about
the
attraction
of sun, moon
and planets?
Answer: They
determine
the
earth‘s
orbit
about
the
sun
Tides
are
acceleration
relative
to the
earth‘s
center
of mass
Marshak, 2005
aa
zz
gg
on the
surface
of the
rotating
earth
one
measures:gravity= gravitation
+ centrifugal
acceleration
gravitation
and gravity
g a zandW V Z
10 0 spherical
Earth10-3 flattening
& centrifugal
acceleration
10-4 mountains, valleys, ocean
ridges, subduction10-5 density
variations
in crust
and mantle
10-6 salt
domes, sediment
basins, ores10-7 tides, atmospheric
pressure
10-8 temporal variations: oceans, hydrology10-9 ocean
topography, polar motion
10-10 general
relativity
gravity
(in laboratory
at TU München)9.807 246 72 m/s2
stat
iona
ryva
riabl
esize
of gravity
signals
level
surfacegeoid
plumb
line
gravity
vector
0 . W W const
geometry
of the
earth‘s
gravity
field
<1m to 2m
)
gravity
related
quantities
-150.00 -100.00 -50.00 0.00 50.00 100.00 150.00
-50.00
0.00
50.00
-100.00
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
80.00
map
with
geoid
heightsrelative to the
GRS80 ellipsoid
gravity
related
quantities
…almost a Fourier series
Laplace
equation
(PDE)
solution
in Cartesian
coordinates
(after
determination):
solution
in spherical
coordinates
(after
determination):
series
representation
of gravitational
field
2 2 2
2 2 2 0V V Vx y z
2 20, , exp exp
0
A kk
V x y z V k z c i kx y
with z
1
00 0
1 1
00
, , cos sin
cos sin
n n
A nm nm nmn m
n n
nm nm nm nmm n m n m
RV r V P C m S mr
R RV C P m S P mr r
20000[ ]max
kmkmn
series
representation
of gravitational
field
500km
333km
250km
200km
spectral
representation
of the
earth‘s
gravitational
field:triangular
plot
of the
spherical
harmonic
coefficients ;nm nmC S
log
C00
A00 C11S11 C10
C20
C30
C40
S22 S21
S31S33 S32
S41S42S44 S43
C21 C22
C31
C41
C32 C33
C42 C43 C44
C50S51S52S54 S53 C51 C52 C53 C54 C55S55
zonalsectorial sectorialtesseral tesseral
series
representation
of gravitational
field
surface
spherical
harmonic
functions: cos,
sinnm nm
mY P
m
scale
origin
orientation
Degree n
Order m
CC00
SS21
SS11
SS22 CC21
CC11CC10
nm nmS C
series
representation
of gravitational
field
in analogy
to signal
processing: characteristics
of signal
and noiseHere: degree
variances
correspond
to power spectral
density
signaldegree
variance
2 2
0
( )
n
n nm nmm
c C S2 1
nn
caven
series
representation
of gravitational
field
signal
and error
degree
variances
(dimensionless)
10 31.6 10 /nc n
2n
error
series
representation
of gravitational
field
power „law“
by
WM Kaula
Degree
Deg
ree
Sta
ndar
dD
evia
tion
inG
eoid
Hei
ght[
m]
10
10
20
20
30
30
40
40
50
50
10-6 10-6
10-5 10-5
10-4 10-4
10-3 10-3
Atmosphere dailyECMWFOceandailyMITHydrologymonthlyEuropeGRACE ErrorPrediction
rapid time variable geoid
signals (RMS)
[ to be divided by the earth radius in order to arrive at dimensionless units]
series
representation
of temporal variations
of gravitational
field
Degree
Deg
ree
Sta
ndar
dD
evia
tion
inG
eoid
Hei
ght[
m]
10
10
20
20
30
30
40
40
50
50
10-6 10-6
10-5 10-5
10-4 10-4
10-3 10-3
Ocean semi-annualOcean annualGRACE Error PredictionAtmosphere annual
“slow”
geoid
time variable signals (RMS)[ to be divided by the earth radius in order to arrive at dimensionless units]
series
representation
of temporal variations
of gravitational
field
δV δr
V δrr
V1r
n2r
n
disturbancepotential orgeoid
gravity
disturbancesorgravity
anomalies
gravity
gradientsortorsion
balance
three
levels
of gravity
quantities
on earth
and in space
1
00 0
1 1
00
, , cos sin
cos sin
n n
A nm nm nmn m
n n
nm nm nm nmm n m n m
RV r V P C m S mr
R RV C P m S P mr r
various
gravity
quantities
on earth
and in space
Gravity
model
EGM08, D/O1000
h = 0km
δV
δVr
δVrr
Gravity
model
EGM08, D/O1000
satellitealtitude: r
earth‘sSurface: R δV δr
V δrr
V1R
n2R
n
δV δr
V δrr
V1r
n2r
n
1nRr
2nRr
3nRr
three
levels
of gravity
quantities
on earth
and in space
various
gravity
quantities
on earth
and in space
Gravity
model
EGM08, D/O1000
h = 0km
h = 250km
h = 400km
δVr
Gravity
model
EGM08, D/O1000
T rT rrT1R
n2R
n
T rT rrT1r
n2r
n
1nRr
2nRr
3nRr
Gravity
model
EGM08, D/O1000
x y
x
z
y
z
Vik
[E]
−0.5 0 0.5
various
gravity
quantities
on earth
and in space
satellitealtitude
r
earth‘ssurface
R
satellitegradiometry
satellite-to-satellitetrackinghigh-low
satellite-to-satellitetrackinglow-low
disturbancepotential orgeoid
gravity
disturbancesorgravity
anomalies
gravity
gradientsortorsion
balance
δV δr
V δrr
V1R
n2R
n
δV δr
V δrr
V1r
n2r
n
1nRr
2nRr
3nRr
summary
of lecture
One1.
Newton‘s
law
of gravitation
describes
all its
relevant properties
such asinverse
square
distance, principle
of superposition, its
stationary
partbeing
vorticity
free, and source
free
outside
the
earth
(Laplace
equation)2. Gravity
is
the
sum
of gravitation
and the
centrifugal
part3. Satellite
orbits
are
essentially
described
by
gravitation4. Tides
are
an accelertation
(a force) relative to the
earth‘s
center
of mass5. The
global gravitational
field
is
represented
as a series
of spherical
harmonics,being
a solution
of Laplace
partial differential equation
(Dirichlet) 6. Spherical
harmonics
on a sphere
are
analogous
to a Fourier
series
in a plane7. Therefore
there
exists
a closed
theory
of „signal
and noise
processing“8. With
increasing
distance from
the
earth
sphere
the
series
coefficients
aredampening
out per degree
n like
(R/R+h)n+1
9. With
each
radial derivative
of the
gravitational
potential the
series
coefficientsare
amplified
per degree
n like
(n+1)10. The
strategy
of satellite
missions
GRACE and GOCE rests
on the
principleof compensating
the
dampering
effect
by
amplification
(see
8. and 9.)