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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