N15. Lucchesi- "fundamental physics with lageos satellites"

International Workshop on
Paolo Farinella (1953-2000):
the scientist and the man
Fundamental Physics with LAGEOS
satellites and Paolo's legacy
David M. Lucchesi
Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF)
Via Fosso del Cavaliere, 100, 00133 Roma, Italy
Istituto di Scienza e Tecnologie della Informazione (ISTI/CNR)
Via G. Moruzzi, 1, 56124 Pisa, Italy

Table of Contents
 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;

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

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, Спутник
after the Sputnik-1 firsts radio beeps on 4
October 1957, it was clear that the known The Sputnik:
and small corrections — at that time — of the
m = 83.6 kg
non–gravitational forces to the larger and
purely conservative gravitational forces have D = 58 cm
begun to play, since that time, a different and P = 96.2 min.
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 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.

LAGEOS (LAser GEOdynamic Satellite):
LA GEO
a = 12,270 km
e = 0.0044 A
 6.94 10 4 m 2 kg
I = 109°.9 m
P = 13,500 s A
 1.26 10 2 m 2 kg
R = 30 cm m
Sputnik
m = 407 kg

 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.


A first list of publications on the LAGEOS satellite:

• 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.


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;


Table of Contents
 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.


The Satellite Laser Ranging (SLR) loop: SLR science products:
EARTH SLR 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,
POD laser altimeter)

d) Geodynamics
LAGEOS • global tectonic plate motion
• regional and crustal deformation
ORBIT f) Fundamental Physics
e) Earth gravity field
• static medium to long wavelength components
• time variations in long wavelength components

The Satellite Laser Ranging (SLR) loop:

EARTH SLR SLR station:
– tracking system;
– Earth reference system (ITRF, …);
– models (trajectory, refraction, …);
– range data;

POD

LAGEOS

ORBIT



EARTH SLR 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

LAGEOS

ORBIT



EARTH SLR 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);

POD

LAGEOS

ORBIT


Table of Contents
 New results;


Fundamental Physics with LAGEOS satellites
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:

 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:
with an improvement of a
  1   2.1  2.3 105 @1 factor of 50 in accuracy

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

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

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.
Dipartimento di Matematica: Pisa 1990. 2010
ESA Journal 14, p.389 15 June, David M. Lucchesi 21

Which science measurements we can perform, in the field of
the Earth, with LAGEOS’s and other dedicated satellites?
Despite the small gravitational radius of the Earth and its slow rotation, today
technology allow the measurement of a paramount of relativistic effects:

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. …;

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;

Table of Contents
 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. first, there is Einstein Equivalence Principle (EEP), 1911, one of the best
tested principles in physics, presently with an accuracy of about 1 part in
1013 (Baeler et al., 1999);

2. second, there is the idea of Riemann that space — by telling mass how to
move — must itself be affected by mass, i.e., the space geometry must be a
participant in the world of physics (Riemann, 1866);

3. third, there is Mach’s Principle, i.e., the acceleration relative to absolute
space of Newton is properly understood when it is viewed as an acceleration
relative to distant stars (Mach, 1872);

Consequences of these ideas:
1. The geometrical structure of GR
• Spacetime is a Lorentian manifold, that is a 4–dimensional pseudo–
Riemannian manifold with signature + 2 (or – 2), or, equivalently, a
smooth manifold with a continuous (and covariant) metric tensor field g :

 g  g  symmetric tensor;
 ds 2  g dx dx 

det g   0

non–degenerate tensor;
invariant

2. The field equations of GR

where G is Einstein tensor and T the stress–energy tensor;
G
G  8 T
4 
G = gravitational constant;
c c = speed of light;


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 ”:

 1 0 0 0 
h  ,   0 gauge conditions;  

 0 1 0 0 
 g    h  
metric tensor; 0 0 1 0 
  
h  16 G T  0 0 0  1
field equations;  

 c4 Flat spacetime metric
 1
h  h   h and h represents the correction due to spacetime
where  2 
h  h    h curvature
  


weak field means h« 1; in the solar system  h  2  10 6
c
where  is the Newtonian or “gravitoelectric” potential:    GM Sun R Sun


G
h  16 4 T are equivalent to Maxwell eqs.: A  4 j
c

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

 00  represents the solution far from the source: (M,J)
 h 4 2
c
 GM
 0l Al  gravitoelectric potential;
h  2 2 r
 c


 
h ij   c 4
Al 
G J n x k l gravitomagnetic vector potential;
 nk
3
 c r

J represents the source total angular momentum or spin



B BG
’ S

 J

G=c=1

     1   
F  (  ') B F  ( S  ) BG
2
    1 
N   ' B N  S  BG
2
   

    3r r   
  B 
ˆ ˆ 
 1 
   BG 

ˆˆ
 J  3r r  J 
r3 2 r3

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
Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi frames axes 29

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.

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

Schwarzschild metric
 2GM  2 2  2GM  2
ds 2  1  c dt  1 
2 2 2 2
dr  r d  r sin d
2

 rc 2   rc 2 

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

Kerr metric
 2GM  2 2  2GM  2 4GJ
ds 2  1  c dt  1 
2 2 2 2 2 2
dr  r d  r sin d  2 sin ddt
 rc 2   rc 2  rc

which gives the field produced by a rotating massive sphere


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;  perturbation  potential
2. Gauss equations;  perturbation  acceleration


Secular effects of the Gravitoelectric field: (Schwarzschild, 1916)
Mass
• Rate of change of the argument of perigee:

32
d 3GM  
 2 52
dt sec c a 1  e2 

• Rate of change of the mean anomaly:
32
dM 3GM   d
  1  e2
dt sec
2 52
c a 1  e 2 12 dt sec

Schwarzschild, Math.-Phys. Tech., 1916

Secular effects of the Gravitomagnetic field: (Lense–Thirring, 1918)
Angular
• Rate of change of the ascending node longitude: momentum
d 2G J

dt sec 
c 2a3 1  e2  32

• Rate of change of the argument of perigee:

d 6G J d
 2 52 cos I  3 cos I
dt sec c a 1  e2  32
dt sec

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, Phys. Z, 19, 1918

The Lense-Thirring effect and  measurements
its 8
G  6.670  10 cm s g 1
3 2


The LT effect on LAGEOS and LAGEOS II orbit  5.861  10 40 cm 2 gs 1
J

Rate of change of the ascending node longitude and of the2.9979250  1010 cm s
argument of perigee:
c 

  2G J 6G J
 LT 
 LT   cos I
c 2a3 1  e2  32

c 2a5 2 1  e2 
32

LAGEOS: LAGEOS II:


 Lageos  30.8mas / yr 
 LageosII  31.6mas / yr
LT LT
 Lageos  LageosII

 LT  32.0mas / yr 
 LT  57.0mas / yr

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

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

The LT effect on LAGEOS and LAGEOS II orbit
Thanks to the very accurate SLR technique  relative accuracy of about 2109 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:


 Obser  126 / yr
Lageos

 Obser  231 / yr
LageosII

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

 0 .5
100  100  1.6%

 LT 31
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.

   
3  R  cos I
2 
  5  R 2 1  3 e2 

 Class
  n   J 2  J 4  

 2
 7 sin I  4 2
 
2  a  1  e2 2   
 8  a 
 
1  e2
2
 
 



 Class
  n 
2

3  R  1  5 cos 2 I  J


 5  R  2
 J4  
2

 7 sin I  4
C (e, I )

 

  
4  a  1  e2
2
  

2
 256  a 
 1  5 cos 2 I 
 


1
C (e, I )  108  153e  208  252e cos 2I  196  189e cos 4I 
2 2 2

1  e  2 2


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°


 class  cos I   class
1class  3  0

   
 obs  1LT  3LT    21LT  



Previous multi–satellite gravity field models:
GEM–L2 (Lerch et al., J. Geophs. Res, 90, 1985)  (J2/J2)  106


 Class  450mas / yr   J 2 n  
  LT
Lageos

JGM–3 (Lerch et al., J. Geophys. Res, 99, 1994)  (J2/J2)  107


 Class  45mas / yr   J 2 n  
  LT
Lageos

EGM–96 (Lemoine et al., NASA TM-206861, 1998)  (J2/J2)  7108 (also with LAGEOS II data)


 Class  32mas / yr  J 2 n  
  LT
Lageos

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. the precession on the node/perigee due to the LT effect: LT ;
2. the J2 uncertainty: J2;
3. the J4 uncertainty: J4;

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. LAGEOS node: Lageos;
2. LAGEOS II node: LageosII;
3. LAGEOS II perigee: LageosII;

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


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

  
 Lageos  k1 LageosII  k 2 LageosII   
 Lageos  k1 LageosII  k 2 LageosII
 LT  
30.8  31.6k1  57k 2 60.1mas / yr

k1 = + 0.295; 1 General Re lativity
where  LT 
k2 =  0.350; 0 Classical Physics

 Lageos are the residuals in the rates of the orbital elements

 
and  LageosII
  i.e., the predicted relativistic signal is a linear trend with a
 LageosII
 slope of 60.1 mas/yr

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

Results

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

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

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:

2.2–year   1.3  0.2


Results

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

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

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

From the best fit we obtained:

  1.1  0.2

3.1–year

Results

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

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

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   1.10  0.03

Results

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

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

350
+ GR
 = 68 - 2 mas/yr  = 60.07 mas/yr
I+0.295II-0.35 II (mas)

300
+
 = 1.13 - 0.04
250
Without removing and fitting any
200
periodical signal.
150
100 From the best fit has been obtained:
50
0
-50   1.13  0.04
0 500 1000 1500 2000
Time (days)

4.7–year

Results
Ciufolini, Pavlis, Peron, Lucchesi, (2001): Preliminary result (unpublished)

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

We obtained:

 1

for the first time,
but with a large rms

7–year

7-year

Results
Ciufolini, Pavlis, Peron and Lucchesi, (2002): Preliminary result (unpublished)

From January 1993 to April 2000, using GEODYN II and EGM–96.
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),
have been fitted (together with a
straight line) and removed with
some non–gravitational signals.

From the best fit has been obtained:

7.3–year   1.00  0.02

The CHAMP and GRACE gravity field solutions
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:
1. the potential of a Lense–Thirring measurement that might reach a deeper
Lense–
accuracy;
2. the possibility to release LAGEOS II perigee, which is subjected to large
unmodelled non–gravitational forces and to the odd zonal harmonics
non–
uncertainties;
3. the use of the node–node only combination (J2 free solution);
node– (J
4. of course, the quality of the Lense–Thirring measurement still rest on the
Lense–
estimated errors of the low degree even zonal harmonics and in their temporal
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 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

Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi Reigber et al., 2004 49

Results with GRACE model EIGEN-GRACE02S
The error budget
11 years analysis Ciufolini & Pavlis, 2004, Letters to Nature

Perturbation   LT % I  0.545II (mas)
Even zonal 4% 600 mas
 LT  48.2

 (mas)
yr
Odd zonal 0% 400
Tides 2%
200
  47.9  6 mas yr
Stochastic 2%
0
Sec. var. 1% 0 2 4 6 8 10 12
Relativity 0.4% years

NGP 2%  
 I  C3 II  47.9  6  0.05 0.05
     0.99  0.12   
RSS (ALL) 5.4% 48.2 mas yr  48.2  0.10 0.10

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


Table of Contents
 New results;


Thermal models and Spin modeling
Non–gravitational perturbations: 7 years analysis

  
 Lageos  k1 LageosII  k 2 LageosII   LT 60.1 mas yr
k1 = + 0.295
k2 =  0.350

Perturbation  NGP mas yr  Mis .%   NGP  LT % 
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   

6
 NGP   i 2  23.63%  LT  24%  LT
i 1

Non–gravitational perturbations: 7 years analysis

 
 Lageos  C3 LageosII   LT 48.1 mas yr C3 = + 0.546

 
mas yr  mas yr 
Perturbation LAGEOS LAGEOS II Mis. %  NGP  LT  %
Solar radiation 7.80 32.44 1 0.21
Earth’s albedo 0.98 1.46 20 0.08
Yarkovsky–Schach 7.83103 0.36 20 0.08
Earth–Yarkovsky 7.35102 1.47 20 0.30
Neutral + Charged drag negligible negligible  

5
 NGP   i 2  0.38%  LT  0.4%  LT
i 1


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 Per correr miglior acqua alza le vele
ormai la navicella del mio ingegno,
Supervisor: Paolo Farinella
che lascia dietro a sé mar sì crudele;
Supervisor: Anna M. Nobili Dante Alighieri (Divina Commedia)
In memory of Paolo
For best rushing water set the sails
by now the vessel of my genius,
that leaves behind itself a so cruel sea

Therefore, 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.
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
Incident recoil acceleration produced by the imbalance of
the temperature distribution across the satellite
Earth surface and directed along the satellite spin–axis,
away from the colder pole.
Sun Light 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
2T acceleration in the opposite point of the orbit,

a giving a resultant null acceleration over one
n 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.

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 albedo
radiation 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.


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

Crucial papers for the Yarkovsky effect modeling:
• 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 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 semimajor
axis 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 Behaviour
of 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.


Comparison between Farinella et al. and Andres et al.: LAGEOS II


Table of Contents
 New results;


New Results
The search for Yukawa–like interactions
• The extensions of the Standard Model (SM) in order to find a Unified Theory of
SM
Fields (UTOF), such as the String Theory (ST) or the M–Theory (MT), naturally
UTOF ST MT
leads to violations of the Weak Equivalence Principle (WEP) and of the Newtonian
WEP
Inverse Square Law (NISL).
NISL

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

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

• 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
• 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 (EEP) or not.
EEP

• 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 104 m ─1015 m have been tested during last 25 years
with null results for a possible violation of NISL and for the W EP


New Results
• These very weak NLRI are usually described by means of a Yukawa–like
potential with strength  and range :
 G M 1  r  M1 = Mass of the primary source;
V yuk   e
 r m2 = Mass of the secondary source;

 1  K1 K 2 
     G = Newtonian gravitational constant;
 G  M 1 m2 
  r = Distance;
 
 

 c  = 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

• This Yukawa–like parameterization seems general (at the lowest order
interaction and non-relativistic limit):
─ 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
Constraints on a Yukawa interaction from a possible gravitational NISL violation
Long Range Limits (Courtesy of Prof. E. Fischbach)
Composition independent experiments
The region above
each curve is ruled
out at the 95.5%
confidence level
Lake
Tower
Laboratory

Earth-LAGEOS

LAGEOS-Lunar

Lunar Precession
Planetary

meters
Dipartimento di Matematica: Pisa 15 June,Fischbach, Hellings, Standish,
Reference: Coy, 2010 David M. Lucchesi & Talmadge (2003) 66

New Results
Orbital effects of a Yukawa–like interaction

The perturbed two–body problem:

G M Interacting potential between the
V r  S 
    1    e r 
r
 two source masses

   G M  
r
r   
Ar  g       2  1   1  e rˆ Interacting acceleration between
r    
  the two source masses

2 r
G M a  r   Disturbing radial acceleration
  2     1  e
a r   a = orbit semimajor axis


New Results
Orbital effects of a Yukawa–like interaction

Satellite pericenter shift (LAGEOS II)
II
G M a 
2
r 
    2     1  e 
r
r

a 1  e2 
a r   1  e cos f

d dt 2
rad s  1 1  e2 Behavior of LAGEOS II

     cos f df
 2
2 ena pericenter rate perturbed by a
0
Yukawa–like interaction as a
a function of the range .
2
 As we can see, the pericenter
rate peaks for a value of the
range  of about 6081 km,
In unit of  very close to 1 Earth radii.
 cm The peak value is about
1.27394×10-4 rad/s in unit of .
Peak
d
 1.27394  10 4   rad s   6,081km  1R
dt 2 15
Dipartimento di Matematica: Pisa June, 2010 David M. Lucchesi 68

New Results Lucchesi D., Peron R., 2010

We analyzed LAGEOS and LAGEOS II orbit over a 12 years time span
using 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


New Results Lucchesi D., Peron R., 2010

LAGEOS II argument of pericenter secular drift

Total expected relativistic precession: rel  3305.64 mas yr
We fitted for a linear
trend plus three periodic

 fit  3306.58 mas yr effects related with the
Yarkovsky-Schach effect.
 
 fit  rel
100  0.03% Best fit: error  0.03%

rel
The total fit error is less
than 0.2% and is
equivalent to a 99.8%
measurement of the PPN
parameters combination.
Peak
d
dt 2
 1.27394 104   rad s  0.2%rel
   8 1012
5 8
Present limits:   10  10
32
3 G M  2  2   
rel  2 5 2  2 
Where:     1 are the Parametrized
     Post Newtonian (PPN) parameters of GR
Dipartimento di Matematica: Pisa 15 c a 2010e   David M. 
June, 1  3 Lucchesi 70

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"

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