N15. Lucchesi- "fundamental physics with lageos satellites"
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Talk of the "International Workshop on Paolo Farinella (1953-2000): the Scientists, the man", Pisa, 14-16 June 2010

Talk of the "International Workshop on Paolo Farinella (1953-2000): the Scientists, the man", Pisa, 14-16 June 2010

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N15. Lucchesi- "fundamental physics with lageos satellites" Presentation Transcript

  • 1. 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
  • 2. 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
  • 3. 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 3
  • 4. 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 4
  • 5. The age of “Dirty Celestial Mechanics  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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 5
  • 6. The age of “Dirty Celestial Mechanics 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 6
  • 7. The age of “Dirty Celestial Mechanics 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 7
  • 8. 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 8
  • 9. 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 9
  • 10. The LAGEOS satellite and Space Geodesy 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 10
  • 11. The LAGEOS satellite and Space Geodesy 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 11
  • 12. The LAGEOS satellite and Space Geodesy The Satellite Laser Ranging (SLR) loop: EARTH SLR SLR station: – tracking system; – Earth reference system (ITRF, …); – models (trajectory, refraction, …); – range data; POD LAGEOS ORBIT Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 12
  • 13. The LAGEOS satellite and Space Geodesy The Satellite Laser Ranging (SLR) loop: 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 13
  • 14. The LAGEOS satellite and Space Geodesy The Satellite Laser Ranging (SLR) loop: 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 14
  • 15. 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 15
  • 16. 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 16
  • 17. Fundamental Physics with LAGEOS satellites 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 17
  • 18. Fundamental Physics with LAGEOS satellites 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 18
  • 19. Fundamental Physics with LAGEOS satellites 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! Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 19
  • 20. Fundamental Physics with LAGEOS satellites 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 20
  • 21. Fundamental Physics with LAGEOS satellites 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
  • 22. Fundamental Physics with LAGEOS satellites 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 22
  • 23. 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 23
  • 24. 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); Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 24
  • 25. The Lense-Thirring effect and its measurements 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 25
  • 26. The Lense-Thirring effect and its measurements 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” Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 26
  • 27. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 27
  • 28. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 28
  • 29. The Lense-Thirring effect and its measurements 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
  • 30. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 30
  • 31. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 31
  • 32. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 32
  • 33. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 33
  • 34. 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 34
  • 35. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 35
  • 36. The Lense-Thirring effect and its measurements The LT effect on LAGEOS and LAGEOS II orbit 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 36
  • 37. The Lense-Thirring effect and its measurements 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   Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 37
  • 38. The Lense-Thirring effect and its measurements The LT effect on LAGEOS and LAGEOS II orbit 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 38
  • 39. The Lense-Thirring effect and its measurements 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). Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 39
  • 40. The Lense-Thirring effect and its measurements The LT effect on LAGEOS and LAGEOS II orbit 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) Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 40
  • 41. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 41
  • 42. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 42
  • 43. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 43
  • 44. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 44
  • 45. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 45
  • 46. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 46
  • 47. The Lense-Thirring effect and its measurements 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 47
  • 48. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 48
  • 49. The Lense-Thirring effect and its measurements 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
  • 50. The Lense-Thirring effect and its measurements 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 50
  • 51. 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 51
  • 52. 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 52
  • 53. Thermal models and Spin modeling 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 53
  • 54. Thermal models and Spin modeling 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 54
  • 55. Thermal models and Spin modeling 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 55
  • 56. Thermal models and Spin modeling 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 56
  • 57. Thermal models and Spin modeling 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 57
  • 58. Thermal models and Spin modeling 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 58
  • 59. Thermal models and Spin modeling 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 59
  • 60. Thermal models and Spin modeling 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 60
  • 61. Thermal models and Spin modeling Comparison between Farinella et al. and Andres et al.: LAGEOS II Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 61
  • 62. 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 62
  • 63. 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. Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 63
  • 64. New Results The search for Yukawa–like 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 (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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 64
  • 65. New Results The search for Yukawa–like interactions • 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; Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 65
  • 66. New Results The search for Yukawa–like interactions 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
  • 67. 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 67
  • 68. 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
  • 69. 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 Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 69
  • 70. 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
  • 71. 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 ! Dipartimento di Matematica: Pisa 15 June, 2010 David M. Lucchesi 71