n15. lucchesi- "fundamental physics with lageos satellites"

International Workshop on International Workshop on Paolo Paolo FarinellaFarinella (1953(1953--2000): 2000):

the the scientistscientist and the manand the man

Fundamental Physics with LAGEOS Fundamental Physics with LAGEOS satellites and Paolo's legacysatellites and Paolo's legacy

David M. LucchesiDavid M. LucchesiIstituto Istituto di di Fisica dello Spazio Interplanetario Fisica dello Spazio Interplanetario (IFSI/INAF) (IFSI/INAF)

Via Via Fosso Fosso del del CavaliereCavaliere, 100, 00133 Roma, Italy, 100, 00133 Roma, Italy

Istituto Istituto di di Scienza Scienza e e Tecnologie della Informazione Tecnologie della Informazione (ISTI/CNR) (ISTI/CNR) Via G. Via G. MoruzziMoruzzi, 1, 56124 Pisa, Italy, 1, 56124 Pisa, Italy

Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 2

The age of “Dirty” Celestial Mechanics;

The LAGEOS satellite and Space Geodesy;

Fundamental Physics with LAGEOS satellites;

The Lense-Thirring effect and its measurements;

Thermal models and Spin modeling;

New results;

Table of ContentsTable of Contents


The age of “Dirty” Celestial Mechanics

The lectures of Giuseppe (Bepi) Colombo at the “Scuola Normale Superiore” in Pisa (1976/1978) have had a great impact on Paolo and there probably started his interest for “Dirty” Celestial Mechanics.

Indeed, with the advent of the space age, after the Sputnik-1 firsts radio beeps on 4 October 1957, it was clear that the known and small corrections — at that time — of the non–gravitational forces to the larger and purely conservative gravitational forces have begun to play, since that time, a different and increasing role in terms of their subtle and complex perturbative effects, especially with the increasing of the accuracy of the tracking systems of the Earth’s artificial satellites.

The Sputnik:

m = 83.6 kg

D = 58 cm

P = 96.2 min.

Спутник


The age of “Dirty Celestial Mechanics

Indeed, the non–gravitational perturbations acceleration depends on the area–to–mass ratio of the body on which they act, they are therefore negligible (with a few and important exceptions) for the natural bodies, because they are characterized by a small value of such a parameter, but they are significant for the artificial ones.

In the first year of the lectures of Bepi Colombo, the LAGEOS satellites was launched by NASA on May 4, 1976.

LAGEOSLAGEOS (LALAser GEOGEOdynamic SSatellite):

a = 12,270 km

e = 0.0044

I = 109°.9

P = 13,500 s

R = 30 cm

m = 407 kg

4 26.94 10A m kgm

2 21.26 10A m kgm

Sputnik



Therefore, when the orbital tracking is carried out by a very accurate technique, such that of the Satellite Laser Ranging (SLR), the need to model better and better disturbing effects of non–gravitational origin such as atmospheric drag, direct solar radiation and thermal thrust effects, become more and more important.

Paolo, together with a few other, e.g., D.P. Rubincam, has been a real master in all this.

The ability and capabilities of Paolo of using both the formalism of the classical Hamiltonian mechanics as well as that characteristic of the non–conservative forces, is well known and clearly evident from its publications, both in the field of the planetary sciences and in space geodesy.

Moreover, he not only understood very well the physics and the mathematics of a given problem, but also the data with their analysis and interpretation.


• L. Anselmo, B. Bertotti, P. Farinella, A. Milani & A.M. Nobili.Orbital perturbations due to radiation pressure for a spacecraft of complex shape. Celestial Mechanics 29, p.27 1983.

• L. Anselmo, P. Farinella, A. Milani & A.M. Nobili.Effects of the Earth reflected sunlight on the orbit of the LAGEOS satellite. Astronomy and Astrophysics 117, p.3 1983.

• F. Barlier, M. Carpino, P. Farinella, F. Mignard, A. Milani & A.M. Nobili.Non-gravitational perturbations on the semimajor axis of LAGEOS. Annales Geophysicae 4, A, 3, p.193 1986.

• M. Carpino, P. Farinella, A. Milani & A.M. Nobili.Sensitivity of LAGEOS to changes in Earth’s (2,2) gravity coefficients.Celestial Mechanics 39, p.1 1986.


A first list of publications on the LAGEOS satellite:



These early publications contain “in nuce” all physics around the LAGEOS satellites that has evolved during the next 25 years:

they contain the analysis and study of the non–gravitational perturbations (direct solar radiation pressure and Earth’s albedo) acting on the satellite;

their impact on the satellite orbit (semimajor axis);

the difficulties in modeling their subtle effects on complex in shape satellites ( drag–free satellites and onboard accelerometers);

they finally contain what we can learn on the Earth’s structure and figure from their studies, such as the gravity field coefficients;







New results;



The LAGEOS satellite and Space Geodesy

The SLR represents a very impressive and powerful technique to determine the round–trip time between Earth–bound laser Stations and orbiting satellites equipped with retro-reflectors mirrors.

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 modeling (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.



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.

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 is in principle known (after all perturbations have been properly modeled).

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.



POD

SLR

LAGEOS

EARTH

ORBIT

The Satellite Laser Ranging (SLR) loop: SLR science products:

a) Terrestrial Reference Frame

• geocenter motion and scale

• station coordinates

b) Earth Orientation Parameters

• polar motion (Xp,Yp)

• Length of Day (LOD) variations

• universal time UT1

c) Centimeter accuracy orbits

• calibration (GPS,PRARE,DORIS)

• orbit determination (geodetic, CHAMP, GRACE, laser altimeter)

d) Geodynamics

• global tectonic plate motion

• regional and crustal deformation

e) Earth gravity field

• static medium to long wavelength components

• time variations in long wavelength components

f) Fundamental Physics



POD

SLR

LAGEOS

EARTH

ORBIT

SLR station:

– tracking system;

– Earth reference system (ITRF, …);

– models (trajectory, refraction, …);

– range data;

The Satellite Laser Ranging (SLR) loop:



POD

SLR

LAGEOS

EARTH

ORBIT

LAGEOS:

– mass, radius;

– physical characteristics (A,B,C,

optical and infrared coefficients, electric and magnetic properties, …);

– models (radiation pressure, thermal, spin, …) for the POD;




POD

SLR

LAGEOS

EARTH

ORBIT

Precise Orbit Determination (POD):

– dynamical models (gravitational and non-gravitational perturbations);

– SLR data (normal points);

– differential correction procedure and state-vector adjustment (plus other parameters);









New results;


Fundamental Physics with LAGEOS satellites

Dynamic effects of Geometrodynamics

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

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

these corrections are necessary for spacecraft navigation and GPS satellites;

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



Dynamic effects of Geometrodynamics

A very significant example about the importance of such aspects is the Superior Conjunction Experiment (SCE) performed with the CASSINI spacecraft in 2002:

by B. Bertotti, L. Iess, P. Tortora, Letters to Nature, 425, p.3, 2003.

51 2.1 2.3 10 @1

The post newtonian parameter measures the curvature of spacetime per unit of mass:

= 1 in Einstein general relativity and = 0 in Newtonian gravity.

The bending and delay of the photons in their round-trip path from the Earth to the spacecraft and back are proportional to + 1.

The ESA BepiColomboBepiColombo mission to Mercury aims to improve such result by a factor of 10 with a dedicated SCE during the cruise phase.

with an improvement of a factor of 50 in accuracy



In 1988, when I asked to Paolo my degree THESIS, he suggested to me several different topics, some in the field of planetary sciences and other in that of space geodesy.

In particular, with regard to space geodesy he proposed a thesis on the albedo perturbations on LAGEOS semimajor axis and, concerning the importance of the studies on LAGEOS–like satellites, he soon highlighted the possibilities of using two LAGEOS satellites for measuring the Earth’s gravitomagnetic field.

Paolo was talking of the LAGEOS III proposal of I. Ciufolini to ASI and NASA for the measurement of the Lense–Thirring effect on the orbit of two LAGEOS satellites in supplementary orbital configuration.

Paolo was involved in that proposal and he was working mainly on the long–term effects of some non–gravitational perturbations on the nodes of the two LAGEOS satellites: the nodes are the observable in this experiment.



Paolo gave me also a popular science article that he wrote on this argument:

“Un altro LAGEOS darà ragione a Mach?”

L’Astronomia, n. 76, p.15, 1988.

This is one of the most interesting and also beautiful aspects of Paolo’s research activity.

Indeed, he has always immediately translated in science popularization articles the studies in which he was involved with the objective to

communicate SCIENCE to everybody.

Paolo was a true open mind person!



A second (not complete) list of Paolo publications during these years:

• G. Afonso, F. Barlier, M. Carpino, P. Farinella, F. Mignard, A. Milani & A.M. Nobili.Orbital effects of LAGEOS’ seasons and eclipses. Annales Geophysicae 7 (5), p.501 1989.

• P. Farinella, A.M. Nobili, F. Barlier & F. Mignard.Effects of thermal thrust on the node and inclination of LAGEOS.Astronomy and Astrophysics 234, p.546 1990.

• F. Mignard, G. Afonso, F. Barlier, M. Carpino, P. Farinella, A. Milani & A.M. Nobili.LAGEOS: Ten years of quest for the non-gravitational forces. Advances in Space Research 10, 3, p.221 1990.

• D. Lucchesi & P. Farinella. Optical properties of the Earth’s surface and long-term perturbations of LAGEOS’ semimajor axis. Journal of Geophysical Research 97, p.7121 1992.

• I. Ciufolini, P. Farinella, A.M. Nobili, D. Lucchesi & L. Anselmo.Results of a joint ASI-NASA study on the LAGEOS gravitomagnetic experiment and the nodal perturbations due to radiation pressure and particle drag effects.Il Nuovo Cimento B 108(2), p.151 1993.



The attention of Paolo to fundamental physics was not only focused on LAGEOS satellites, but also to a dedicated space mission for the measurement of the gravitational constant. A third list of publications:

• P. Farinella, A. Milani & A.M. Nobili.The measurement of the gravitational constant in an orbiting laboratory. Astrophysics and Space Science 73, p.417 1980.

• A.M. Nobili, A. Milani & P. Farinella. Testing Newtonian gravity in space. Physics Letters A, 120, 9, p.437 1987.

• A.M. Nobili, A. Milani & P. Farinella.The orbit of a space laboratory for the measurement of G. Astronomical Journal 95, p.576 1988.

• A.M. Nobili, A. Milani, E. Polacco, I.W. Roxburgh, F. Barlier, K. Aksnes, C.W.F. Everitt, P. Farinella, L. Anselmo & Y. Boudon.The NEWTON mission – A proposed manmade planetary system in space to measure the gravitational constant. ESA Journal 14, p.389 1990.



Which science measurements we can perform, in the field of the Earth, with LAGEOS’s and other dedicated satellites?

1. relativistic effects on the orbital elements (LT effect, PPN, G-dot, …);

2. gyroscope precession (DS and LT effects);

3. Einstein’s Equivalence Principle;

4. special relativity (MM and KT experiments);

5. …;

Despite the small gravitational radius of the Earth and its slow rotation, today technology allow the measurement of a paramount of relativistic effects:

1. LAGEOS–like satellites and/or dedicated drag–free satellites;

2. Gravity Probe B (GPB) satellite;

3. Galileo Galilei (GG), MicroScope and STEP satellites, GReAT (capsula);

4. OPTIS satellite;








New results;


The Lense-Thirring effect and its measurements

Einstein’s theory of General Relativity (GR) states that gravity is not a physical force transmitted through space and time but, instead, it is a manifestation of spacetime curvature.

Three main ideas have inspired Einstein to GR:

1.1. firstfirst, there is Einstein Equivalence Principle (EEP), 1911, one of th, there is Einstein Equivalence Principle (EEP), 1911, one of the best e best

tested principles in physics, presently with an accuracy of aboutested principles in physics, presently with an accuracy of about 1 part in t 1 part in

10101313 ((BaeBaeler ler et al., 1999);et al., 1999);

2.2. secondsecond, there is the idea of , there is the idea of RiemannRiemann that space that space —— by telling mass how to by telling mass how to

move move —— must itself be affected by mass, i.e., the space geometry must must itself be affected by mass, i.e., the space geometry must be a be a

participant in the world of physics (participant in the world of physics (RiemannRiemann, 1866);, 1866);

3.3. thirdthird, there is Mach, there is Mach’’s Principle, i.e., the acceleration relative to absolute s Principle, i.e., the acceleration relative to absolute

space of Newton is properly understood when it is viewed as an aspace of Newton is properly understood when it is viewed as an acceleration cceleration

relative to distant stars (Mach, 1872);relative to distant stars (Mach, 1872);



Consequences of these ideas:

1. The geometrical structure of GR

•• Spacetime Spacetime is a is a Lorentian Lorentian manifold, that is a 4manifold, that is a 4––dimensional pseudodimensional pseudo––

RiemannianRiemannian manifold with signature + 2 (or manifold with signature + 2 (or –– 2), or, equivalently, a 2), or, equivalently, a

smooth manifold with a continuous (and covariant) metric tensor smooth manifold with a continuous (and covariant) metric tensor field field gg ::

0det

ggg symmetric tensor;

non–degenerate tensor;

TcGG 48

where G is Einstein tensor and T the stress–energy tensor;

G = gravitational constant;

c = speed of light;

2. The field equations of GR

dxdxgds 2

invariant



Practically, the field equations of GR connect the metric tensor g with the density of mass–energy T and its currents:

mass–energy T “tells” geometry g how to “curve”

geometry g “tells” (from the field equation) mass–energy T how to “move”



In the Weak Field and Slow Motion (WFSM) limit we obtain the “Linearized Theory of Gravity ”:

TcGh

hg

h

4

,

16

0 gauge conditions;

metric tensor;

field equations;

1000010000100001

Flat spacetime metric

and h represents the correction due to spacetime curvature

hhh

hhh21

where

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

2 10

c

h

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



TcGh 416 jA 4are equivalent to Maxwell eqs.:

That is, the tensor potential plays the role of the electromagnetic vector potential A and the stress energy tensor T plays the role of the four-current j.

4

20

200

2

4

chcAh

ch

ij

ll

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

rGM

gravitoelectric potential;

lnk

knl

rxJ

cGA 3 gravitomagnetic vector potential;

J represents the source total angular momentum or spin

h



B’

J

BG S

G = c = 1

BF

)'( GBSF

)(21

BN

' GBSN

21

3

ˆˆ3r

rrB

3

ˆˆ321

rJrrJBG

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

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



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

2222222

222

2 sin2121 drdrdrrcGMdtc

rcGMds

Schwarzschild metric

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

dtdrcGJdrdrdr

rcGMdtc

rcGMds 2

2222222

222

22 sin4sin2121

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:



Schwarzschild metric describe the effects produced by the Gravitoelectric field, while Kerr metric retain the effects produced by the Gravitomagnetic field.

The two fields produce both periodic and secular effects on the orbit of a satellite;

These orbital effects may be computed with the perturbative methods characteristic of Celestial Mechanics (small perturbations):

1. Lagrange equations;

2. Gauss equations;

perturbation potential

perturbation acceleration



Secular effects of the Gravitoelectric field: (Schwarzschild, 1916)

2252

23

sec 13

eacGM

dtd

sec

2212252

23

sec

11

3dtde

eac

GMdt

dM

• Rate of change of the argument of perigee:

• Rate of change of the mean anomaly:

Mass

Schwarzschild, Math.-Phys. Tech., 1916



Secular effects of the Gravitomagnetic field:

23232sec 1

2

e

JacG

dtd

sec232252

sec

cos3cos1

6dtdII

e

JacG

dtd

• Rate of change of the ascending node longitude:

• Rate of change of the argument of perigee:

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)

(Lense–Thirring, 1918)

Lense-Thirring, Phys. Z, 19, 1918



The LT effect on LAGEOS and LAGEOS II orbit

232321

2

e

JacG

LT

I

e

JacG

LT cos1

6232252

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

LAGEOS:

yrmasyrmas

LageosLT

LageosLT

/0.32/8.30

yrmasyrmas

LageosIILT

LageosIILT

/0.57/6.31

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



Thanks to the very accurate SLR technique relative accuracy of about 2109 at LAGEOS’s altitude we are in principle able to detect the subtle 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/yr) the total observed precessions:

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

yrObserLageos /126 yrObser

LageosII /231

%6.110031

5.0100

LT

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




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.

22

223

22

4222

2

1

14sin7

85

1

cos23

e

eI

aRJJ

e

Ia

RnClass

IIeCI

aRJJ

e

Ia

RnClass2

22

4222

22

cos51),(4sin7

2565

1

cos5143

IeIee

eIeC 4cos1891962cos252208153108

1

1),( 22222




The LAGEOS/LAGEOS III experiment (1987 Proposal to ASI/NASA by I. Ciufolini)

LAGEOS inclination: I1 = 109.9° LAGEOS III inclination: I3 = 180° - I1 = 70.1°

1 3 0class class

1 3 12obs LT LT LT

cosclass I



nClassLageos Jyrmas 2/45



Previous multi–satellite gravity field models:

GEM–L2 (Lerch et al., J. Geophs. Res, 90, 1985) (J2/J2) 106

JGM–3 (Lerch et al., J. Geophys. Res, 99, 1994) (J2/J2) 107

EGM–96 (Lemoine et al., NASA TM-206861, 1998) (J2/J2) 7108 (also with LAGEOS II data)

LT

LT

LT

Therefore, starting from 1995, the situation was favourable for a first detection of the LT effect




The larger errors were concentrated in the J2 and J4 coefficients:

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: LageosLageos;;2.2. LAGEOS II node:LAGEOS II node: LageosIILageosII;;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).



yrmaskk

kkkk LageosIILageosIILageosLageosIILageosIILageos

LT /1.60576.318.3021

21

21

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

Re1where

and

LageosII

LageosII

Lageos

are the residuals in the rates of the orbital elements

(Ciufolini, Il Nuovo Cimento, 109, N. 12, 1996)(Ciufolini-Lucchesi-Vespe-Mandiello, Il Nuovo Cimento, 109, N. 5, 1996)

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




Results

2.2–year

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

The plot has been obtained after fitting and removing 13 tidal signals and also the inclination residuals.

From the best fit (dashed line) we obtained:

From November 1992 to December 1994, using GEODYN II and JGM–3.

2.03.1



Results

The plot has been obtained after fitting and removing 10 periodical signals.

From the best fit we obtained:

3.1–year

Ciufolini, Chieppa, Lucchesi, Vespe, (Class. Quant. Gravity, 1997):

From November 1992 to December 1995, using GEODYN II and JGM–3.

2.01.1



Results

They fitted (together with a straight line) and removed four small periodic signals, corresponding to:

LAGEOS and LAGEOS II nodes periodicity (1050 and 575 days),LAGEOS II perigee period (810 days),

and the year periodicity (365 days).

From the best fit they obtained:

4–year

Ciufolini, Pavlis, Chieppa, Fernandes–Vieira, (Science 279, 1998):

From January 1993 to January 1997, using GEODYN II and EGM–96.

03.010.1



Results

0 500 1000 1500 2000-50

050

100150200250300350

= 68 +- 2 mas/yr GR = 60.07 mas/yr

= 1.13 +- 0.04

I+

0.29

5 II-0

.35

II (m

as)

Time (days)

Lucchesi (2001), PhD Thesis (Nice University and OCA/CERGA):

4.7–year

Without removing and fitting any periodical signal.

From the best fit has been obtained:

From January 1993 to August 1997, using GEODYN II and EGM–96.

04.013.1



ResultsCiufolini, Pavlis, Peron, Lucchesi, (2001): Preliminary result (unpublished)

From January 1993 to January 2000, using GEODYN II and EGM–96.

7-year

1

for the first time,

but with a large rms

We obtained:

7–year



ResultsCiufolini, Pavlis, Peron and Lucchesi, (2002):

7.3–year

Preliminary result (unpublished)

Four small periodic signalscorresponding to: LAGEOS and LAGEOS II nodes periodicity (1050 and 575 days),LAGEOS II perigee period (810

days), and the year periodicity (365 days), have been fitted (together with a straight line) and removed with some non–gravitational signals.

From the best fit has been obtained:

From January 1993 to April 2000, using GEODYN II and EGM–96.

02.000.1



The CHAMP mission with its satellite orbiting in a near polar orbit and the two twin

satellites again in a near polar orbit of the GRACE mission, are expected to deeply

improve our knowledge of the Earth’s gravity field, both in its static (the long–to–medium

wavelengths harmonics) and temporal dependence, and indeed they do it with their

preliminary solutions.

Two orders–of–magnitude improvement are expected at the longer wavelengths.

This suggests:

The CHAMP and GRACE gravity field solutions

1.1. the potential of a the potential of a LenseLense––Thirring Thirring measurement that might reach a deeper measurement that might reach a deeper accuracy;accuracy;

2.2. the possibility to release LAGEOS II perigee, which is subjectedthe possibility to release LAGEOS II perigee, which is subjected to large to large unmodelled unmodelled nonnon––gravitational forces and to the odd gravitational forces and to the odd zonal zonal harmonics harmonics uncertainties;uncertainties;

3.3. the use of the nodethe use of the node––node only combination (node only combination (JJ22 free solution);free solution);

4.4. of course, the quality of the of course, the quality of the LenseLense––ThirringThirring measurement still rest on the measurement still rest on the estimated errors of the low degree even estimated errors of the low degree even zonal zonal harmonics and in their temporal harmonics and in their temporal variations;variations;



The EIGEN–GRACE02S gravity field model

A medium–wavelength gravity field model has been calculated from 110 days of GRACE tracking data, called EIGEN–GRACE02S (in the period 2002/2003).

The solution has been derived solely from GRACE intersatellite observations and is independent from oceanic and continental surface gravity data which is of great importance for oceanographic applications, as for example the precise recovery of sea surface topography features from altimetry.

This model that resolves the geoid with on accuracy of better than 1 mm at a resolution of 1000 km half–wavelength is about one order of magnitude more accurate than recent CHAMP derived global gravity models and more than two orders of magnitude more accurate than the latest pre–CHAMP satellite–only gravity models.

Reigber et al., 2004. An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN–GRACE02S, Journal of Geodynamics.

http://op.gfz-potsdam.de/grace/index_GRACE.html

http://www.csr.utexas.edu


The Lense-Thirring effect and its measurementsThe EIGEN–GRACE02S gravity field model

1 mm accuracy at a resolution of about 1000 km half–wavelength Error and difference–amplitudes as a function of spatial resolution in terms of geoid heigths

Reigber et al., 2004



Results with GRACE model EIGEN-GRACE02SThe error budget

Perturbation

Even zonal 4%

Odd zonal 0%

Tides 2%

Stochastic 2%

Sec. var. 1%

Relativity 0.4%

NGP 2%

%LT

RSS (ALL) 5.4%

10.005.0

12.099.010.005.0

2.4869.47

2.483

yrmasC III

RSS (SAV + NGP) 9.6% represents a more conservative estimate

Indeed, Ciufolini and Pavlis claimed a 10% error allowing for unknown and unmodelled error sources

0 2 4 6 8 10 12 years

yrmas

LT 2.48

I 0.545II (mas)

(m

as)

600

400

200

0 yrmas69.47

Ciufolini & Pavlis, 2004, Letters to Nature11 years analysis








New results;


Thermal models and Spin modeling

Non–gravitational perturbations: 7 years analysis

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 yrmasNGP %LTNGP %.Mis

yrmaskk LTLageosIILageosIILageos 1.6021

LTLTi

iNGP %24%63.236

1

2

k1 = + 0.295

k2 = 0.350



Non–gravitational perturbations: 7 years analysis

Perturbation LAGEOS LAGEOS II Mis. % LTNGP % Solar radiation 7.80 32.44 1 0.21 Earth’s albedo 0.98 1.46 20 0.08

Yarkovsky–Schach 7.83103 0.36 20 0.08 Earth–Yarkovsky 7.35102 1.47 20 0.30

Neutral + Charged drag negligible negligible

yrmasC LTLageosIILageos 1.483 C3 = + 0.546

yrmas yrmas

LTLTi

iNGP %4.0%38.05

1

2



These results on the non–gravitational effects and their modeling on the two LAGEOS satellites are the outcome of my PhD Thesis (2001):

“ Effets des Forces non-gravitationnelles sur les

Satellites LAGEOS:

Impact sur la Détermination de l’Effet Lense-Thirring “

“Effects of Non-Gravitational Forces acting on

LAGEOS Satellites:

Impact on the Lense-Thirring Effect Determination”

1° Supervisor: Francois Barlier

Supervisor: Paolo Farinella

Supervisor: Anna M. Nobili

Per correr miglior acqua alza le vele

ormai la navicella del mio ingegno,

che lascia dietro a sé mar sì crudele;

Dante Alighieri (Divina Commedia)

In memory of PaoloFor best rushing water set the sails

by now the vessel of my genius,

that leaves behind itself a so cruel sea


Thermal models and Spin modelingTherefore, the Yarkovsky–Schach effect plays a crucial role when using LAGEOS satellites for GR tests in the field of the Earth, in particular if we are interested in the argument of pericenter as a physical observable.

Incident

Sun Light

Earth

Case of maximum perturbation: both the spin–axis and the Sun are contained in the orbital plane of the satellite.

The satellite sense of revolution is assumed to be clock–wise. The larger arrow represents the recoil acceleration produced by the imbalance of the temperature distribution across the satellite surface and directed along the satellite spin–axis, away from the colder pole.

As soon as the satellite is in full sun light, i.e., in the absence of eclipses, the along–track acceleration at a given point of the orbit is compensated by an equal and opposite acceleration in the opposite point of the orbit, giving a resultant null acceleration over one orbital revolution.

When eclipses occur the finite thermal inertia of the satellite produces a smaller acceleration during the shadow transition, giving rise to a non null along–track acceleration and long–term effects in the satellite semimajor axis.

nTa 2



A non complete lists of publications on the thermal effects and spin

Afonso, G., Barlier, F., Carpino, M., Farinella, P., et al., Orbital effects of LAGEOS seasons and eclipses, Ann. Geophysicae 7, 501-514, 1989.

Bertotti, B., Iess., L., The Rotation of LAGEOS, J. Geophys. Res. 96, No. B2, 2431-2440, February 10, 1991.

Farinella, P., Nobili, A. M., Barlier, F., and Mignard, F., Effects of the thermal thrust on the node and inclination of LAGEOS, Astron. Astrophys. 234, 546-554, 1990.

Farinella, P., Vokrouhlichý, D., Barlier, F., The rotation of LAGEOS and its long-term semimajor axis decay: A self-consistent solution, J, Geophys. Res., 101, 17861-17872, 1996a.

Farinella, P., Vokrouhlichý, D., Thermal force effects on slowly rotating, spherical artificial satellites – I. Solar heating. Planet. Space Sci., 44, 12, 1551-1561, 1996b

Habib, S., Holz, D. E., Kheyfetz, A., et al., Spin dynamics of the LAGEOS satellite in support to a measurement of the Earth’s gravitomagnetism, Phys. Rev. D, 50, 6068-6079, 1994.

Metris, G. and Vokrouhlický, D., Thermal force perturbation of the LAGEOS orbit: the albedoradiation part, Planet. Space Sci., 44, 6, 611-617, 1996.

Metris, G., Vokrouhlický, D., Ries, J. C., Eanes, R. J., Nongravitational effects and the LAGEOS eccentricity excitations, J. Geophys. Res. 102, NO. B2, 2711-2729, February 10, 1997.

Metris, G., Vokrouhlický, D., Ries, J. C., Eanes, R. J., LAGEOS Spin Axis and Non-Gravitational Excitations of its Orbit, Adv. Space Res., 23, 721-725, 1999.


Ries, J. C., Eanes R. J., and Watkins, M. M., Spin vector influence on LAGEOS ephemeris, presented at the Second Meeting of IAG Special Study Group 2.130, Baltimore, 1993.

Rubincam, , D. P., LAGEOS Orbit Decay Due to Infrared radiation From Earth, J. Geophys. Res., 92, No. B2, 1287-1294, 1987b.

Rubincam, D. P., Yarkovsky Thermal Drag on LAGEOS, J, Geophys. Res., 93, No. B11, 13,805-13,810, 1988.

Rubincam, , D. P., Drag on the LAGEOS Satellite, J, Geophys. Res., 95, No. B4, 4881-4886, 1990.

Scharroo, R., Wakker, K. F., Ambrosius, B. A. C., and Noomen, R., On the along-track acceleration of the LAGEOS satellite, J. Geophys. Res. 96, 729-740, 1991.

Slabinski, V. J., LAGEOS acceleration due to intermittent solar heating during eclipses periods. Paper 3.9 presented at the 19th meeting of the Division on Dynamical Astronomy, American Astronomical Society, Gaithersburg, Maryland, July 1988 (Abstract in Bull. Am. Astron. Soc. 20, 902, 1988).

Slabinski, V. J., A Numerical Solution for LAGEOS Thermal Thrust: the Rapid-Spin case, Celest. Mech., 66, 131-179, 1997.

Vokrouhlicky, D. and Farinella, P., Thermal force effects on slowly rotating, spherical artificial satellites. II. The Earth IR heating, Planet. Space Sci., 1996.

Andres, J., I., Noomen, R., Bianco, G., Currie, D., and Otsubo, T., 2003. The Spin Axis Behaviour of the LAGEOS Satellites. Journ. Geophys. Res., 109, B06403, 2004.


A non complete lists of publications on the thermal effects and spin


• Rubincam, D. P., Yarkovsky Thermal Drag on LAGEOS, J, Geophys. Res., 93, No. B11, 13,805-13,810, 1988.

• Afonso, G., Barlier, F., Carpino, M., Farinella, P., et al., Orbital effects of LAGEOS seasons and eclipses, Ann. Geophysicae 7, 501-514, 1989.

• Scharroo, R., Wakker, K. F., Ambrosius, B. A. C., and Noomen, R., On the along-track acceleration of the LAGEOS satellite, J. Geophys. Res. 96, 729-740, 1991.

• Slabinski, V. J., A Numerical Solution for LAGEOS Thermal Thrust: the Rapid-Spin case, Celest. Mech., 66, 131-179, 1997.

• Metris, G., Vokrouhlický, D., Ries, J. C., Eanes, R. J., Nongravitational effects and the LAGEOS eccentricity excitations, J. Geophys. Res. 102, NO. B2, 2711-2729, February 10, 1997.


Crucial papers for the Yarkovsky effect modeling:

Crucial papers for the Spin modeling:

• Bertotti, B., Iess., L., The Rotation of LAGEOS, J. Geophys. Res. 96, No. B2, 2431-2440, February 10, 1991.

• Farinella, P., Vokrouhlichý, D., Barlier, F., The rotation of LAGEOS and its long-term semimajoraxis decay: A self-consistent solution, J, Geophys. Res., 101, 17861-17872, 1996a.

• Andres, J., I., Noomen, R., Bianco, G., Currie, D., and Otsubo, T., 2003. The Spin Axis Behaviourof the LAGEOS Satellites. Journ. Geophys. Res., 109, B06403, 2004.



Bertotti and Iess model (1991):

They fit the observational data for LAGEOS spin period with a:

• model for the magnetic torque;

• model for the gravitational torque;

• and an initial southward orientation for the spin direction;

Farinella et al. model (1996):

They generalized the Bertotti and Iess model and compared their results with the along–track residuals of both LAGEOS satellites:

• they confirm the correctness of the initial southern orientation for the spin;

• they considered other possible contributions to the torque (further computations by David V.);

• they compute a long-term evolution of the spin for both satellites;



Andres et al. (2005):

They fit the observational data for LAGEOS spin period and orientation with a:

• model for the magnetic torque;

• model for the gravitational torque;

• and the additional torques (offset and asymmetric reflectivity)proposed by Farinella et al. (1996)

They result is the LOSSAM model (LAGEOS Spin Axis Model), presently the best model based on averaged equations.


Thermal models and Spin modelingComparison between Comparison between FarinellaFarinella et al. and Andres et al.: LAGEOS IIet al. and Andres et al.: LAGEOS II








New results;


New Results

The search for The search for YukawaYukawa––like interactionslike interactions• The extensions of the Standard Model (SMSM) in order to find a Unified Theory of Fields (UTOFUTOF), such as the String Theory (STST) or the M–Theory (MTMT), naturally leads to violations of the Weak Equivalence Principle (WEPWEP) and of the Newtonian Inverse Square Law (NISLNISL).

• Tests for Newtonian gravity and for a possible violation of the WEPWEP are strongly related and represent a powerful approach in order to validate Einstein theory of General Relativity (GRGR) with respect to proposed alternative theories of gravity and to tune – from the experimental point of view – gravity itself into the realm of quantum physics.

• Moreover, New Long Range Interactions (NLRINLRI) may be thought as the residual of a cosmological primordial scalar field related with the inflationary stage (dilatonscenario);

• Twentyfive years ago, the hypothesis of a fifth–force of nature has thrust scientists to a strong experimental investigation of possible deviations from the gravitational inverse–square–law.


New Results

The search for The search for YukawaYukawa––like interactionslike interactions• In fact, the deviations from the usual 1/r law for the gravitational potential would lead to new weak interactions between macroscopic objects.

• The interesting point is that these supplementary interactions may be either consistent with Einstein Equivalence Principle (EEPEEP) or not.

• In this second case, non–metric phenomena will be produced with tiny, but significant, consequences in the gravitational experiments.

• The characteristic of such very weak interactions, which are predicted by several theories, is to produce deviations for masses separations ranging through several orders of magnitude, starting from the sub–millimeter level up to the astronomical scale:

scale distances between 10scale distances between 1044 m m ──10101515 m have been tested during last 25 years m have been tested during last 25 years with null results for a possible violation of NISL and for the Wwith null results for a possible violation of NISL and for the WEPEP


New Results

The search for The search for YukawaYukawa––like interactionslike interactions• These very weak NLRINLRI are usually described by means of a Yukawa–like potential with strength and range :

• This Yukawa–like parameterization seems general (at the lowest order interaction and non-relativistic limit):

c

mK

MK

G

erMGV

ryuk

2

2

1

1

1

1

M1 = Mass of the primary source;

m2 = Mass of the secondary source;

G = Newtonian gravitational constant;

r = Distance;

= Strength of the interaction; K1,K2 = Coupling strengths;

= Range of the interaction; = Mass of the light-boson;

ħ = Reduced Planck constant; c = Speed of light

─ scalar field with the exchange of a spin–0 light boson;

─ tensor field with the exchange of a spin–2 light boson;

─ vector field with the exchange of a spin–1 light boson;


New Results

The search for The search for YukawaYukawa––like interactionslike interactionsConstraints on a Yukawa interaction from a possible gravitational NISLNISL violation

LaboratoryLake

Tower

Earth-LAGEOS

LAGEOS-Lunar

Lunar Precession Planetary

Reference: Coy, Fischbach, Hellings, Standish, & Talmadge (2003)

Long Range Limits (Courtesy of Prof. E. Fischbach)

meters

Composition independent experimentsThe region above

each curve is ruled out at the 95.5% confidence level


New Results

Orbital effects of a Orbital effects of a YukawaYukawa––like interactionlike interaction

rS e

rMG

rV 1

rerrMG

gAr

r ˆ112

r

erra

aMG

1

2

2

The perturbed two–body problem:

Interacting potential between the two source masses

Interacting acceleration between the two source masses

Disturbing radial acceleration

a = orbit semimajor axis


New Results

Orbital effects of a Orbital effects of a YukawaYukawa––like interactionlike interaction

Satellite pericenter shift (LAGEOS IILAGEOS II)

r

erra

aMG

1

2

2

fe

earcos1

1 2

Behavior of LAGEOS IILAGEOS IIpericenter rate perturbed by a Yukawa–like interaction as a function of the range .

As we can see, the pericenterrate peaks for a value of the range of about 6081 km, very close to 1 Earth radii.

The peak value is about 1.27394×10-4 rad/s in unit of .

Rkm 1081,6sraddtd Peak

4

2

1027394.1

2

0

2

2 cos121 dff

enae

In unit of

dtd

srad

cm

2a


New Results

We analyzed LAGEOS and LAGEOS II orbit over a 12 years time spanusing GEODYN II (NASA/GSFC) code, but we did not:

• modeled the relativistic effects;

• modeled the thermal thrust effects;

• adjust empirical accelerations;

• adjust radiation coefficient;

We used the EGM96 and the EIGENGRACE02S gravity field models and look for the total relativistic precession in the orbital elements residuals. In particular we focused on the:

• argument of pericenter: Einstein, de Sitter and Lense-Thirring precessions

• ascending node longitude: de Sitter and Lense-Thirring precessions

Lucchesi D., Peron R., 2010


New Results

LAGEOS II argument of pericenter secular drift

Total expected relativistic precession: 3305.64rel mas yr

3306.58fit mas yr

We fitted for a linear trend plus three periodic effects related with the Yarkovsky-Schach effect.

Best fit: error 0.03%

The total fit error is less than 0.2% and is equivalent to a 99.8% measurement of the PPN parameters combination.

100 0.03%fit rel

rel

4

21.27394 10 0.2%

Peakrel

d rad sdt

128 10

85 1010 Present limits:

3 2

22 5 2

3 2 231rel

G Mc a e

Where: 1 are the Parametrized

Post Newtonian (PPN) parameters of GR

Lucchesi D., Peron R., 2010


Conclusions

In order to further verify Einstein theory of general relativity we need to improve our models of the non-gravitational perturbations, with particular care of:

• all the effects related with solar radiation effects (thermal …);

• spin model using non-averaged equations in the slow rotation regime;

Thank you for your attention

and especially to Paolo,

a dear friend,

a kind person

and an extraordinary scientist !

n15. lucchesi- "fundamental physics with lageos satellites"

Education