The Gravitomagnetic effect measurement ( Creditis to Davide Lucchesi )

The Gravitomagnetic effect measurementThe Gravitomagnetic effect measurement

((Creditis to Davide LucchesiCreditis to Davide Lucchesi))

Table of ContentsTable of Contents

Gravitomagnetism and Lense–Thirring effect;Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR;The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB);The 2004 measurement and its error budget (EB); Difficulties in improving the present measurement Difficulties in improving the present measurement

with LAGEOS satellites only;with LAGEOS satellites only; Conclusions;Conclusions;

Gravitomagnetism and Lense–Thirring effectGravitomagnetism and Lense–Thirring effect

It is interesting to note that, despite the simplicity and beauty of the ideas of Einstein’s GR, the theory leads to very complicated non–linear equations to be solved: these are second–order–partial–differential–equations in the metric tensor g, i.e., hyperbolic equations similar to those governing electrodynamics.

However, we can find very interesting solutions, removing at the same time the mathematical complications of the full set of equations, in the so–called weak field and slow motion (WFSMWFSM) limit .

Under these simplifications the equations reduce to a form quite similar to those of electromagnetism.

€

Rμν − gμν R = −8

c 4π GTμν


This leads to the “ Linearized Theory of Gravity ”:

€

h ,βαβ = 0

gαβ =ηαβ + hαβ

Δh αβ = −16πG

c 4Tαβ

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

gauge conditions;

metric tensor;

field equations;

€

ηαβ =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

Flat spacetime metric:

and hαβ represents the correction due to spacetime curvature⎪

⎩

⎪⎨

⎧

=≡

−≡

αβαβα

α

αβαβαβ

η

η

hhh

hhh2

1

where

weak field means hαβ« 1; in the solar system 6

210−≤

Φ≅c

hαβ

where Φ is the Newtonian or “gravitoelectric” potential: SunSun RGM−=Φ


€

Δh αβ = −16πG

c 4Tαβ

νν π jA 4=Δare equivalent to Maxwell eqs.:

That is, the tensor potential plays the role of the electromagnetic vector potential Aν and the stress energy tensor Tαβ plays the role of the four–current jν.

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

Ο=

−=

Φ=

−4

20

200

2

4

ch

c

Ah

ch

ij

ll

represents the solution far from the source: (M,J)

r

GM−=Φ gravitoelectric potential;

lnk

knl

r

xJ

c

GA ε3= gravitomagnetic vector potential;

J represents the source total angular momentum or spin

αβh


Following this approach we have a field, the Gravitoelectric field produced by masses, analogous to the electric field produced by charges:

and a field, the Gravitomagnetic field produced by the flow of matter, i.e., mass–currents, analogous to the magnetic field produced by the flow of charges, i.e., by electric currents:

This is a crucial point and a way to understand the phenomena of GR associated with rotation, apparent forces in rotating frames and the origin of inertia in general.

t

A

cEG ∂

∂−Φ∇−=

rrr

21

ABG

rrr∧∇=


These phenomena have been debated by scientists and philosophers since Galilei and Newton times.

In classical physics, Newton’s law of gravitation has a counterpart in Coulomb’s law of electrostatics, but it does not have any phenomenon formally analogous to magnetism.

On the contrary, Einstein’s theory of gravitation predicts that the force generated by an electric current, that is Ampère’s law of electromagnetism, should have a formal counterpart force generated by a mass–current.


r

Je

A(r)

B(r) J

r

Jm

h(r)

BG(r)

Classical electrodynamics: Classical geometrodynamics:

The Analogy

G = c = 1


B’

J

BGS

This phenomenon is known as the “dragging of gyroscopes” or “inertial frames dragging”

The Analogy

This means that an external current of mass, such as the rotating Earth, drags and changes the orientation of gyroscopes,

and gyroscopes are used to define the inertial frames axes.

G = c = 1


In GR the concept of inertial frame has only a local meaning:

they are the frames where locally, in space and time,

the metric tensor (gαβ) of curved spacetime is equal to the

Minkowski metric tensor (ηαβ) of flat spacetime:

And a local inertial frame is ‘’rotationally dragged‘’ by mass-currents, i.e., moving masses influence and change the orientation

of the axes of a local inertial frame (that is of gyroscopes);

αβαβ η≅g


The main relativistic effects due to the Earth on the orbit of a satellite come from Earth’s mass M and angular momentum J.

2222222

222

2 sin2

12

1 φθθ drdrdrrcGM

dtcrcGM

ds −−⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ −=

Schwarzschild metric

which gives the field produced by a non–rotating massive sphere

dtdrc

GJdrdrdr

rc

GMdtc

rc

GMds φθφθθ 2

2222222

222

22 sin

4sin

21

21 +−−⎟

⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ −≅

Kerr metric

which gives the field produced by a rotating massive sphere

In terms of metric they are described by Schwarzschild metric and Kerr metric:


Secular effects of the Gravitomagnetic field:

( ) 23232sec 1

2

e

J

ac

G

dt

d

−=

Ω ⊕

( ) sec23232

sec

cos3cos1

6

dt

dII

e

J

ac

G

dt

d Ω−=

−−= ⊕ω

Rate of change of the ascending node longitude:

Rate of change of the argument of perigee:

(Lense–Thirring, 1918)

Angular momentum

These are the results of the frame–dragging effect or Lense–Thirring effect:

moving masses (i.e., mass–currents) are rotationally dragged by the angular momentum of the primary body (mass–currents)

Keplerian elements

semimajor axis;semimajor axis;

eccentricity;eccentricity;

inclination;inclination;

===Ω===

M

I

e

a

ω

longitude of the ascending node;longitude of the ascending node;

argument of perigee;argument of perigee;

mean anomaly;mean anomaly;

Orbital plane Equatorial plane

X Y

Z

LI

Ω

Ascending Node direction

Gravitomagnetism and Lense–Thirring effectGravitomagnetism and Lense–Thirring effectEpochOrbital InclinationRight Ascension of Ascending Node (R.A.A.N.)Argument of Perigee Eccentricity Mean Motion Mean Anomaly

The LT effect on

LAGEOS and LAGEOS II

orbit

( ) 232321

2

e

J

ac

GLT

−=Ω ⊕&

( ) Ie

J

ac

GLT cos

1

623232

−−= ⊕ω&

Rate of change of the ascending node longitude and of the argument of perigee :

LAGEOS:

⎩⎨⎧

+≅+≅Ω

yrmas

yrmasLageosLT

LageosLT

/0.32

/8.30

ω&

&

⎩⎨⎧

−≅+≅Ω

yrmas

yrmasLageosIILT

LageosIILT

/0.57

/6.31

ω&

&

LAGEOS II:

1 mas/yr = 1 milli–arc–second per year

30 mas/yr 180 cm/yr at LAGEOS and LAGEOS II altitude

⎪⎪⎩

⎪⎪⎨

⎧

⋅≅

⋅≅

⋅≅−

⊕

−−−

scmc

gscmJ

gscmG

10

1240

1238

109979250.2

10861.5

10670.6




The Corner Cubes The Corner Cubes

Corner cubes on the Moon:4 systems left on the Moon surface by

Apollo 11Apollo 14Apollo 15

Lunokhod

Apollo 15

The LAGEOS satellites and SLRThe LAGEOS satellites and SLR

LAGEOS and LAGEOS II satellites

LAGEOSLAGEOS (LALAser GEOGEOdynamic SSatellite)

yryr

yryr

ssP

I

e

kmkma

IILAGEOSLAGEOS

deg/438.0deg/214.0

deg/632.0deg/343.0

324,13500,13

65.529.109

014.0004.0

163,12270,12

−−+Ω

&

&

oo

LAGEOSLAGEOS, launched by NASA (May 4, 1976);

LAGEOS IILAGEOS II, launched by ASI/NASA (October 22, 1992);


Spherical in shape satellite: Spherical in shape satellite: DD = 60 cm; = 60 cm;

Passive satellite;Passive satellite;

Low Low area-to-massarea-to-mass ratio: ratio: A/mA/m = 6.95·10 = 6.95·10-4-4 m m22/kg/kg..;;

Outer portion: Outer portion: AlAl, , MMAA 117 kg; 117 kg;

Inner core: Inner core: CuBeCuBe, , LL = 27.5 cm, = 27.5 cm, dd = 31.76 cm, = 31.76 cm, MMBCBC 175 kg; 175 kg;

426 Corner Cubes Retro reflectors (422 silica + 4 germanium);426 Corner Cubes Retro reflectors (422 silica + 4 germanium);

cube–corner The CCR cover The CCR cover 42% of the satellite surface; 42% of the satellite surface; mm = 33.2 g; = 33.2 g; rr = 1.905 cm; = 1.905 cm;


• The LAGEOS satellites are tracked with very high accuracy through the powerful Satellite Laser Ranging (SLR) technique.

• The SLR represents a very impressive and powerful technique to determine the round–trip time between Earth–bound laser Stations and orbiting passive (and not passive) Satellites.

• The time series of range measurements are then a record of the motions of both the end points: the Satellite and the Station;

Thanks to the accurate modelling (of both gravitational and non–gravitational perturbations) of the orbit of these satellites approaching 1 cm in range accuracy we are able to determine their Keplerian elements with about the same accuracy.


GEODYN II range residuals

Accuracy in the data reduction

From January 3, 1993

The mean RMS is about 2 – 3 cm in range and decreasing in time.

This means that “real data” are scattered around the fitted orbit in such a way this orbit is at most 2 or 3 cm away from the “true” one with the 67% level of confidence.

LAGEOS range residuals (RMS)

Courtesy of R. Peron


• In this way the orbit of LAGEOS satellites may be considered as a reference frame, not bound to the planet, whose motion in the inertial space (after all perturbations have been properly modelled) is in principle known.

• Indeed, the normal points have typically precisions of a few mm, and accuracies of about 1 cm, limited by atmospheric effects and by variations in the absolute calibration of the instruments.

With respect to this external and quasi-inertial frame it is then possible to measure the absolute positions and motions of the ground–based stations, with an absolute accuracy of a few mm and mm/yr.


The motions of the SLR stations are due:

1.1. to plate tectonics and regional crustal deformations;to plate tectonics and regional crustal deformations;

2.2. to the Earth variable rotation;to the Earth variable rotation;

1. induce interstations baselines to undergo slow variations:v a few cm/yr;

2. we are able to study the Earth axis intricate motion:

2a. Polar Motion (Xp,Yp);

2b. Length-Of-Day variations (LOD);

2c. Universal Time (UT1);


Dynamic effects of Geometrodynamics

Today, the relativistic corrections (both of Special and General relativity) are an essential aspect of (dirty) – Celestial Mechanics as well as of the electromagnetic propagation in space:

1. these corrections are included in the orbit determination–and–analysis programs for Earth’s satellites and interplanetary probes;

2. these corrections are necessary for spacecraft navigation and GPS satellites;

3. these corrections are necessary for refined studies in the field of geodesy and geodynamics;




The 2004 measurement and its error budgetThe 2004 measurement and its error budget

Thanks to the very accurate SLR technique relative accuracy of about 2109 at LAGEOS’sLAGEOS’s altitude we are in principle able to detect the subtle Lense–Thirring relativistic precession on the satellites orbit.

For instance, in the case of the satellites node, we are able to determine with high accuracy (about 0.5 mas 0.5 mas over 15 days arcs) the total observed precessions:

Therefore, in principle, for the satellites node accuracy we obtain :

yrObserLageos /126o& +≅Ω yrObser

LageosII /231o& −≅Ω

%3.010031

245.0100 =≈

ΩΩ

LT&&δ

Which corresponds to a ‘’direct‘’ measurement of the LT secular precession

Over 1 year


Unfortunately, even using the very accurate measurements of the SLR technique and the latest Earth’s gravity field model, the uncertainties arising from the even zonal harmonics J2n and from their temporal variations (which cause the classical precessions of these two orbital elements) are too much large for a direct measurement of the Lense–Thirring effect.

( ) ( ) ( )[ ]IeIeee

IeC 4cos1891962cos2522081531081

1),( 222

22+++++

−=


Therefore, we have three main unknowns:

1.1. the precession on the node/perigee due to the LT effect: the precession on the node/perigee due to the LT effect: LTLT ;;

2.2. the the JJ22 uncertainty: uncertainty: JJ22;;

3.3. the the JJ44 uncertainty: uncertainty: JJ44;;

Hence, we need three observables in such a way to eliminate the first two even zonal harmonics uncertainties and solve for the LT effect. These observables are:

1.1. LAGEOS node: LAGEOS node: ΩΩLageosLageos;;

2.2. LAGEOS II node:LAGEOS II node: ΩΩLageosIILageosII;;

3.3. LAGEOS II perigee:LAGEOS II perigee: LageosIILageosII;;

LAGEOS II perigee has been considered thanks to its larger eccentricity ( 0.014) with respect to that of LAGEOS ( 0.004).


yrmas

kk

kk

kk LageosIILageosIILageosLageosIILageosIILageosLT /1.60576.318.30

21

21

21

&&&&&& +Ω+Ω=

−+

+Ω+Ω≅

The solutions of the system of three equations (the two nodes and LAGEOS II perigee) in three unknowns are:

k1 = + 0.295;

k2 = 0.350; ⎩⎨⎧

=PhysicsClassical

lativityGeneralLT 0

Re1μwhere

and

⎪⎪⎩

⎪⎪⎨

⎧

Ω

Ω

LageosII

LageosII

Lageos

ωδ

δ

δ

&

&

& are the residuals in the rates of the orbital elements

((Ciufolini, Il Nuovo Cimento, 109, N. 12, 1996Ciufolini, Il Nuovo Cimento, 109, N. 12, 1996))

((Ciufolini-Lucchesi-Vespe-Mandiello, Il Nuovo Cimento, 109, N. 5, 1996Ciufolini-Lucchesi-Vespe-Mandiello, Il Nuovo Cimento, 109, N. 5, 1996))

i.e., the predicted relativistic signal is a linear trend with a slope of 60.1 mas/yr


yrmas

K

K

K LageosIILageosLageosIILageosLT 1.486.318.30

3

3

3 Ω+Ω=

+

Ω+Ω≅

&&&& δδδδμ

Thanks to the more accurate gravity field models from the CHAMPCHAMP and GRACEGRACE satellites, we can remove only the first even zonal harmonic J2 in its static and temporal uncertainties while solving for the Lense–Thirring effect parameter LT.

In such a way we can discharge LAGEOS IILAGEOS II perigee, which is subjected to very large non–gravitational perturbations (NGPNGP).

The solution of the system of two equations in two unknowns is:

k3 = + 0.546

The GRACE mission: measurement of the gravity The GRACE mission: measurement of the gravity gradients of the Earthgradients of the Earth

A milligal is a convenient unit for describing variations in gravity over the surface of the Earth. 1 milligal (or mGal) = 0.00001 m/s2, which can be compared to the total gravity on the Earth's surface of approximately 9.8 m/s2. Thus, a milligal is about 1 millionth of the standard acceleration on the Earth's surface.

The GGM02 model is based on the analysis of 363 days of GRACE in-flight data, spread between April 4, 2002 and Dec 31, 2003.

”Tapley B. et al. GGM02 – An improved Earth gravity model from GRACE” – Jou. Of Geoedy (2008), DOI 10.1007,s00190-005-040z

Gravity Anomaly mGal

The long-term average distribution of the mass within the Earth system determines its mean or static gravity field. The motion of water and air, on time scales ranging from hours to decades, largely determines the time variations of Earth's gravity field. The mean and time variable gravity field of the Earth affect the motion of all Earth satellites. The motion of twin GRACE satellites are affected slightly differently since they occupy different positions in space. These differences cause small relative motions between these satellites, designated GRACE-A and GRACE-B. The distance changes are manifested as the change in time-of-flight of microwave signals between the two satellites, which in turn is measured as the phase change of the carrier signals. The influence of non-gravitational forces on the inter-satellite range is measured using an accelerometer, and the orientation of the spacecraft in space is measured using star cameras. The on-board GPS receivers provide geo-location and precise timing.

The EIGEN–GRACE02S gravity field model

Reigber et al., (2004)Reigber et al., (2004) Journal of Geodynamics Journal of Geodynamics.

C(2,0) 0.1433E-11 0.5304E-10 0.1939E-10 0.3561E-10

C(4,0) 0.4207E-12 0.3921E-11 0.2230E-10 0.1042E-09

C(6,0) 0.3037E-12 0.2049E-11 0.3136E-10 0.1450E-09

C(8,0) 0.2558E-12 0.1479E-11 0.4266E-10 0.2266E-09

C(10,0) 0.2347E-12 0.2101E-11 0.5679E-10 0.3089E-09

(EG02S) (EG02S–CAL) (EIGEN2S) (EGM96)

Formal and (preliminary) calibrated errors of EIGEN–GRACE02S:

37

9

7



a) Observed (and combined) residuals of LAGEOS and LAGEOS II nodes (raw data);

b) As in a) after the removal of six periodic signals: 1044 days; 905 days; 281 days; 569 days and 111 days;

The best fit line through these observed residuals has a slope of about:

= (47.9 6) mas/yr

i.e., 0.99 LT

c) The theoretical Lense–Thirring effect on the node–node combination: the slope is about 48.2 mas/yr;

The LT effect and the EIGEN–GEACE02S model: Ciufolini & Pavlis, 2004, Lett. to NatureCiufolini & Pavlis, 2004, Lett. to Nature

11 years analysis of the LAGEOS’s orbit

LageosIILageos K Ω+Ω=Ω 3

yrmasLT 2.48=


The error budgetThe error budget: : systematic effectssystematic effects

Perturbation

Even zonal 4%

Odd zonal 0%

Tides 2%

Stochastic 2%

Sec. var. 1%

Relativity 0.4%

NGP 2%

( )%LTμδμ

RSS (ALL) 5.3%( )

⎭⎬⎫

⎩⎨⎧

±±=⎭⎬⎫

⎩⎨⎧

±⎟⎠

⎞⎜⎝

⎛ ±=

Ω+Ω≅

10.0

05.012.099.0

10.0

05.0

2.48

69.47

2.483

yrmas

K III&& δδ

μ

But they allow for a 10% error in order to include underestimated and unknown sources of error

0 2 4 6 8 10 12 years

yr

masLT 2.48=

ΩI 0.545ΩII (mas)

Ω (m

as)

600

400

200

0

( ) yrmas69.47 ±≅μ

After the removal of 6 periodic terms

The LT effect and the EIGEN–GEACE02S model: Ciufolini & Pavlis, 2004, Lett. to Nature


• Review of such error budget was carried on later because of some criticism raised in the literature to the estimate performed by Ciufolini and Pavlis.

• In particular, the secular variations of the even zonal harmonics were suggested to contribute at the level of 11% of the relativistic precession over the time span of the measurement, i.e., over 11 years.

• Moreover, also the question of possible correlations between the various sources of error and the imprinting of the Lense–Thirring effect itself in the gravity field coefficients was raised.


Gravitomagnetism and Lense–Thirring effect;Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR;The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB);The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution;The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement Difficulties in improving the present measurement


Difficulties in improving the present measurement with Difficulties in improving the present measurement with LAGEOS satellites onlyLAGEOS satellites only

An interesting question is related to how far we can go with the LAGEOSLAGEOS satellites in order to test the gravitomagneticgravitomagnetic interaction.

The NASA NASA’s GPBGPB (see Fitch et al., 1995 for a review) space mission is in principle able to measure the gravitomagnetic field of the Earth to the 0.3% level (the first scientific results are expected in 2007), i.e., more than a factor 10 better than it is currently possible with the two LAGEOS’sLAGEOS’s.

It seems that the present gravimetric space missions (CHAMPCHAMP and GRACEGRACE) are not able to improve significantly the low degree coefficients of the Earth’s field (even using the GPSGPS data) to which the orbit of LAGEOSLAGEOS satellites is more sensitive.

Therefore, the 1%1% level probably represents an horizon for the Lense–ThirringLense–Thirring effect accuracy when using the node–only combination of the two laser–ranged satellites.

Also the forthcoming GOCEGOCE space mission will be less sensitive to the low degree components of the Earth’s field and not so great improvements are expected.


The use of the linear combination involving also LAGEOS IILAGEOS II argument of perigee as an additional observable as the great advantage of eliminating also the uncertainties of all the systematic gravitational effects with degree ℓ = 4 and order m = 0.

Using the present gravity field models the error budget from the even zonal harmonics uncertainties (starting from J6) fall down to less than 1% of the relativistic precession.

Unfortunately, the systematic errors from the non–gravitational perturbations increase with the use of LAGEOS IILAGEOS II argument of perigee, and a factor 10 improvement in the modelling of their subtle effects (in truth quite difficult to reach) is not enough to reduce their error contribution to the level of the gravitational perturbations.

Moreover, such a modelling of the NGPNGP requires an improvement also in the range accuracy of the SLRSLR technique. Again, a not easy task.

We need LARES !We need LARES ! Ciufolini et al., 1998(ASI), 2004(INFN)

yrmaskk LTLageosIILageosIILageos 1.6021 ≅+Ω+Ω &&&


Direct solar radiation + 946.42 1 + 15.75

Earth albedo 19.36 20 6.44

Yarkovsky–Schach effect 98.51 10 16.39

Earth–Yarkovsky 0.56 20 0.19

Neutral + Charged particle drag negligible negligible

Asymmetric reflectivity

Perturbation ( )yrmasNGPδμ ( )%LTNGP μδμ( )%.Mis


( ) LTLTi

iNGP μμδμδμ %24%63.236

1

2 ≈=≅ ∑=

k1 = + 0.295

k2 = 0.350

7 years 7 years simulationsimulation


LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude AYS

Lucchesi, Ciufolini, Andrés, Pavlis, Peron, Noomen and Currie, Plan. Space Science, 52, 2004

0 500 1000 1500 2000 2500 3000-10000

-5000

0

5000

10000

Residuals YS fit (Lucchesi et al., 2004) YS (Lucchesi, 2002)

LAGEOS II perigee rate (mas/yr)

Time (days)

EGM96 The plot (red line) represents the best–fit we obtained for the Yarkovsky–Schach perturbation assuming that this is the only disturbing effect influencing the LAGEOS II argument of perigee.

Initial Yarkovsky–Schach parameters:

AYS = 103.5 pm/s2 for the amplitude

= 2113 s for the CCR thermal inertia

Final Yarkovsky–Schach amplitude:

AYS = 193.2 pm/s2

i.e., about 1.9 times the pre–fit value.

With EGM96 in GEODYN II software and the LOSSAM model in the independent numerical simulation (red and blue lines).


Direct solar radiation + 946.42 1 + 15.75

Earth albedo 19.36 2 0.64

Yarkovsky–Schach effect 98.51 1 1.64

Earth–Yarkovsky 0.56 2 negligible

Neutral + Charged particle drag negligible negligible

Asymmetric reflectivity

Perturbation ( )yrmasNGPδμ ( )%LTNGP μδμ( )%.Mis


( ) LTi

iNGP μδμδμ %8.156

1

2 =≅ ∑=

k1 = + 0.295

k2 = 0.350

7 years 7 years simulationsimulation

Assuming an improvement by a factor of 10


The Largest (and Best Modelled) NGPNGP on LAGEOSLAGEOS and LAGEOS IILAGEOS II orbit is due to direct solar radiation pressure (SRPSRP):

sD

D

mc

ACa

Sun

SunSunRSun ˆ

2

⎟⎟⎠

⎞⎜⎜⎝

⎛Φ−=

⊕

⊕r 3.6·109 m/s2

CR represents the satellite radiation coefficient, about 1.12 for LAGEOS IILAGEOS II;

A/m represents the area–to–mass ratio of the satellites, about 7104 m2/kg;

ΦSun represents the solar irradiance at 1 AU, about 1380 W/m2;

c represents the speed of light, about 3108 m/s;

DSun represents the average Earth–Sun distance, i.e., 1 AU;

ŝ represents the Earth–Sun unit vector;

Where:

211104 smaSun−⋅≈

rδError in SRP SRP :



211104 smaSun−⋅≈

rδError in SRP SRP :

1/101/10 212104 smaSun−⋅≈

rδ

Such small accelerations are ‘’visible’’ in LAGEOSLAGEOS satellites residuals.

However, a factor of 10 corresponds to a 0.1% mismodelling of the SRPSRP, and this is presently unreachable because of the solar irradiance uncertainty, at the level of 0.3% (also CR) :

0.3% LTSRP %5≈


With the proposed LARESLARES (Ciufolini et al., 1998 (ASIASI), 2004 (INFNINFN)), which has a factor 2 smaller area-to-mass ratio and a larger eccentricity, we obtain:

0.3%

Therefore, the three elements combination is not competitive with the two nodes combination when applied to LARESLARES satellite.

We need LARESLARES and the node only combination to reach a 0.3% measurement of the Lense–ThirringLense–Thirring effect (Lucchesi&Rubincam, 2004Lucchesi&Rubincam, 2004).

LTNGP %3≈

Lageos conclusionLageos conclusion

The overall error budget of the 2004 measurement of the Lense–ThirringLense–Thirring effect seems reliable, about 5% at 1–sigma level;

The reviewed analysis confirmed this result

In particular, the impact of the even zonal harmonics secular trends is around 1% of the relativistic prediction;

We have underlined the difficulties in improving the present results using LAGEOSLAGEOS satellites only: the LARESLARES satellites is necessary in addition to the two LAGEOSLAGEOS for achieving the goal of a 0.3% measurement;

Documents

The Gravitomagnetic effect measurement ( Creditis to Davide Lucchesi )