Physics
P01 - Space-Time symmetries P02 - Fundamental constants P03 - Relativistic reference frames P04 - Equivalence Principle P05 - General Relativity P06 - Astrometry, VLBI, Pulsar Timing P07 - Atomic physics for clocks P08 - Astronomy and GNSS P09 - Quantum non-locality and
decoherence
Misje kosmiczne związane z badaniami efektów relatywistycznych
Gravity Probe-B – badanie efektu Lense-Thirringa
LAGEOS I, II, III – różne efekty GPS – różne efekty LISA – zbadanie fal grawitacyjnych STEP – test zasady równoważności BepiColombo – perihelium
Merkurego
Possible detection of the gravity field disturbance with help of gradiometer on the Galileo orbit and higher
Janusz B. Zieliński 1, Robert R. Gałązka 2, Roberto Peron3
1/ Space Research Centre, Polish Academy of Sciences, POLAND2/ Institute of Physics, Polish Academy of Sciences, POLAND3/ Instituto di Fisica dello Spazio Interplanetario, Istituto Nazionale di Astrofisica, ITALY
Scientific and Fundamental Aspects of the Galileo ProgrammeToulouse,1-4 October 2007
Introductory remarks
Temporal variations of the gravity field exist in the local inertial space around the Earth
Gradiometry – the differential measurement of the gravitational acceleration
GNSS – the most precise tool for position measurements in space and time
Is it possible to combine Gradiometry + GNNS for the determination of cg ?
Gravity field of the Earth
EIGEN-GRACE02S 150 × 150 from GRACE mission
2 0
cossincos1n
n
mnmnmnm
n
PmSmCra
rGM
V
)(/ Galscm978
zW
yW
xW
rddW
dHdW
g 2
2
222
2
2
22
22
2
2
zW
yzW
xzW
zyW
yW
xyW
zxW
yxW
xW
zg
yg
xg
Expression for the vertical component of the gradient Trr
sinsincos nm
n
0m
nmnm
E
N
2n
1n
E2rr PmSmC
aGM
ra
2n1nr1
T
Eötvös unit of the gravity acceleration gradient 1 EU = 10-9 m s-2 / m
50 100 150 200 250 300 350-1.5
-1
-0.5
0
0.5
1
1.5
longitude West in deg.of arc
grad
ient
in E
U
Gradient profiles along the parallel 0, long. 0 - 360, N=250
200 km
400 km
600 km800 km
1000 km
Gradient Trr profiles along equator, model n,m = 250
1
1.52
2.53
3.54
4.55
050
100150
200250
300350
400-1.5
-1
-0.5
0
0.5
1
1.5
Height levels: 200-1000 km
Gradient evolution with height - model n=250
long.diff., units=1 deg.arc
grad
ient
in E
U
Gradient Trr evolution with height from 200 km to 1000 km,
model n,m = 250
1
23
45
0
100
200
300
400-0.3
-0.2
-0.1
0
0.1
0.2
Height levels: 1000-5000km
Gradient evolution with height - model n=250
long.diff., units=1 deg.arc
grad
ient
in E
U
11.5
22.5
33.5
44.5
5
050
100150
200250
300350
400-4
-3
-2
-1
0
1
2
3
4
x 10-3
Height levels: 7500-17500 km
Gradient evolution with height - model n=250
long.diff., units=1 deg.arc
grad
ient
in E
U
12
34
5
0
100
200
300
400-1.5
-1
-0.5
0
0.5
1
1.5
x 10-4
Height levels: 20000-24000 km
Gradient evolution with height - model n=250
long.diff., units=1 deg.arc
grad
ient
in E
U
1
23
45
0
100
200
300
400-4
-2
0
2
4
x 10-5
Height levels: 27000-40000 km
Gradient evolution with height - model n=120
long.diff., units=1 deg.arc
gra
die
nt
in E
U
Upward continuation procedure
UCrtTrtT 11rr21rr ),(),(
t
Earth's rotation
r1Trr(ti,r1)
r2
Trr(ti,r2)
t1
t2
P0
(f ixed direction in space)
Fig 8a. Newtonian propagation of the rotating Earth's gravitational field
Lense-Thirring precession
mΩ 2
cLT
gradzyx
,,
0.042”/y
t
Earth's rotation
r1Trr(ti,r1)
r2Trr(ti,r2)
t1
t2
P0
Fig 8b. Einstein's propagation of the rotating Earth's gravitational field
Upward Continuation in ECIR(Earth Centered Inertial Reference Frame)
tdt
TdUCrtTrtT rr
12rr21rr
),(),(
dtTd rr ),(),( 1irr2irrrr rtTrtTT - rate of change of the
with
t = δTrr = Trr(t2,r1)۞UC –Trr(t1,r2)
and
cg=(r2 – r1)/Δt
or
cg =
dtTd rr
g
12
crr
t
dt
Td
T
rr rr
rr
12 .
)(
50 100 150 200 250 300 350-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
longitude West in deg.of arc
grad
ient
in E
U
Einstein's shadow function
GPS altitude
Galileo altitude (Newton)Galileo altitude(Einstein)
/\t
g
12
crr
t
For GPS-Galileo case
For r2 – r1 ≈ 3000 km and cg=c Δt ≈ 0.01 s ≈ 0.15 a.s. ≈ 18 m for Galileo orbit
Period of the signal ≈ 12 hours and the amplitude 1*10-4 EU. It means that from the bottom to the peak of the signal we have about 6 hours. With the linear approximation we can tell that for 1 s we get the 0.5*10-8 EU change of the gradient. As we are interested in the ±0.001 s accuracy in the determination of the signal arrival time it means that equivalent accuracy in the measurement should be ± 0.5*10-11 EU.
GOCE Mission (ESA)
Circular orbit, mean altitude ≈ 250 km, i = 96.50 , launch spring 2008
To accurately measure the Earth's gravity field, the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite is equipped with a core instrument called the Electrostatic Gradiometer, which consists of three pairs of identical ultra-sensitive accelerometers, mounted on three mutually orthogonal 'gradiometer arms'.
GO
CE
gradiometer
Length of Baseline for an accelerom
eter pair: 0.5 m
Accelerom
eter noise: < 3 m
EU
= 3 * 10-12
s-2
Experimental activity at IFSI-INAF
Since many years the Experimental Experimental GravitationGravitation group (head V. Iafolla) is active in the field of gravitation physics with a number of projects:
GravimetrySupport to satellite missionsGeophysicsFundamental physics
ISA (Italian Spring Accelerometer)
High sensitivity three axes accelerometer
ISA accelerometer
BepiColombo GEOSTAR
STEP accelerometer
sensitivity 18-18 g ~ to 10-17 m s-2
Expected development in gradiometry
GOCE 10-3 EU IFSI 10-4 EU Paik 10-5 EU STEP 10-8 EU
If we have the accurate theoretical model of the curve that should be fitted by measurements then only one term of the zero order has to be determined. The accuracy of this term is roughly described as
M0 = ± σ0 /√n
where σ0 is the standard deviation of the measurement and n is the number of measurements.
Supposing that the measurement is done with the frequency 1 Hz, during 24 hours we have 86400 measurements and during 12 days more than one million. With the individual measurement error ±10-8 EU and 12 days measurement interval we can get close to the desired accuracy ± 10-11
Conclusions
It seems that concept for the determina-tion of the velocity of the gravitational signal, using the rotating Earth as the signal generator, and GNNS plus gradio-metry as detector, is realistic, but of course not easy. It should provide the motivation for the development of the gradiometry technology and could widen the spectrum of scientific applications of GNSS.
Thank you for your attention