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Istituto Nazionale di AstrofisicaIstituto di Fisica dello Spazio Interplanetario Consiglio Nazionale delle RicercheIstituto di Scienza e Tecnologie della Informazione David M. Lucchesi Torino 4 – 7 September 2006 The Lense–Thirring effect measurement with LAGEOS satellites: David M. Lucchesi 1,2,3 Error Budget and impact of the time–dependent part of Earth’s gravity field 1.Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF), Roma, Italy 2.Istituto di Scienza e Tecnologie della Informazione (ISTI/CNR), Pisa, Italy 3.Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma II, Roma, Italy XVII SIGRAV Conference

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect It is interesting to note that, despite the simplicity and beauty of the ideas of Einstein’s GR, the theory leads to very complicated non–linear equations to be solved: these are second–order–partial–differential–equations in the metric tensor g, i.e., hyperbolic equations similar to those governing electrodynamics. Indeed, these equations have been solved completely only in a few special cases under particularly symmetry conditions. WFSM However, we can find very interesting solutions, removing at the same time the mathematical complications of the full set of equations, in the so–called weak field and slow motion (WFSM) limit. Under these simplifications the equations reduce to a form quite similar to those of electromagnetism.

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect This leads to the “ Linearized Theory of Gravity ”: gauge conditions; metric tensor; field equations; Flat spacetime metric: and h represents the correction due to spacetime curvature where weak field means h « 1; in the solar system where is the Newtonian or “gravitoelectric” potential:

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect are 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. represents the solution far from the source: (M,J) gravitoelectric potential; gravitomagnetic vector potential; J represents the source total angular momentum or spin

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect Following this approach we have a field, the Gravitoelectric field produced by masses, analogous to the electric field produced by charges: and a field, the Gravitomagnetic field produced by the flow of matter, i.e., mass–currents, analogous to the magnetic field produced by the flow of charges, i.e., by electric currents: This is a crucial point and a way to understand the phenomena of GR associated with rotation, apparent forces in rotating frames and the origin of inertia in general.

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect These phenomena have been debated by scientists and philosophers since Galilei and Newton times. In classical physics, Newton’s law of gravitation has a counterpart in Coulomb’s law of electrostatics, but it does not have any phenomenon formally analogous to magnetism. On the contrary, Einstein’s theory of gravitation predicts that the force generated by an electric current, that is Ampère’s law of electromagnetism, should have a formal counterpart force generated by a mass–current.

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect r JeJe A(r)B(r)A(r)B(r) J r JmJm h(r) B G (r) Classical electrodynamics:Classical geometrodynamics (WFSM): The Analogy G = c = 1

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect B ’’ J BGBG S This phenomenon is known as the “dragging of gyroscopes” or “inertial frames dragging” The Analogy This means that an external current of mass, such as the rotating Earth, drags and changes the orientation of gyroscopes, and gyroscopes are used to define the inertial frames axes. G = c = 1

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect In GR the concept of inertial frame has only a local meaning: they are the frames where locally, in space and time, the metric tensor (g ) of curved spacetime is equal to the Minkowski metric tensor ( ) of flat spacetime: And a local inertial frame is ‘’rotationally dragged‘’ by mass-currents, i.e., moving masses influence and change the orientation of the axes of a local inertial frame (that is of gyroscopes);

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect The main relativistic effects due to the Earth on the orbit of a satellite come from Earth’s mass M and angular momentum J . Schwarzschild metric which gives the field produced by a non–rotating massive sphere 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:

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Torino 4 – 7 September 2006 David M. Lucchesi Gravitomagnetism and Lense–Thirring effect Secular effects of the Gravitomagnetic field: Rate of change of the ascending node longitude: Rate of change of the argument of perigee: (Lense–Thirring, 1918) Angular momentum These are the results of the frame–dragging effect or Lense–Thirring effect: moving masses (i.e., mass–currents) are rotationally dragged by the angular momentum of the primary body (mass–currents)

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Torino 4 – 7 September 2006 David M. Lucchesi Keplerian elements semimajor axis; eccentricity;inclination; longitude of the ascending node; argument of perigee; mean anomaly; Orbital plane Equatorial plane X Y Z L I Ascending Node direction Gravitomagnetism and Lense–Thirring effect

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Torino 4 – 7 September 2006 David M. Lucchesi The LT effect on LAGEOS and LAGEOS II orbit Rate of change of the ascending node longitude and of the argument of perigee : LAGEOS: LAGEOS II: 1 mas/yr = 1 milli–arc–second per year 30 mas/yr 180 cm/yr at LAGEOS and LAGEOS II altitude Gravitomagnetism and Lense–Thirring effect

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR LAGEOS and LAGEOS II satellites LAGEOSLAGEOS LAGEOS (LAser GEOdynamic Satellite) LAGEOS LAGEOS, launched by NASA (May 4, 1976); LAGEOS II LAGEOS II, launched by ASI/NASA (October 22, 1992);

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR Spherical in shape satellite: D = 60 cm; Passive satellite; Low area-to-mass ratio: A/m = 6.95·10 -4 m 2 /kg.; Outer portion: Al, M A 117 kg; Inner core: CuBe, L = 27.5 cm, d = 31.76 cm, M BC 175 kg; 426 CCR (422 silica + 4 germanium); cube–corner The CCR cover 42% of the satellite surface; m = 33.2 g; r = 1.905 cm;

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR The LAGEOS satellites are tracked with very high accuracy through the powerful Satellite Laser Ranging (SLR) technique. The SLR represents a very impressive and powerful technique to determine the round–trip time between Earth–bound laser Stations and orbiting passive (and not passive) Satellites. The time series of range measurements are then a record of the motions of both the end points: the Satellite and the Station; Thanks to the accurate modelling (of both gravitational and non–gravitational perturbations) of the orbit of these satellites approaching 1 cm in range accuracy we are able to determine their Keplerian elements with about the same accuracy.

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR GEODYN II range residuals Accuracy in the data reduction From January 3, 1993 The mean RMS is about 2 – 3 cm in range and decreasing in time. This means that “real data” are scattered around the fitted orbit in such a way this orbit is at most 2 or 3 cm away from the “true” one with the 67% level of confidence. LAGEOS range residuals (RMS) Courtesy of R. Peron

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR In this way the orbit of LAGEOS satellites may be considered as a reference frame, not bound to the planet, whose motion in the inertial space (after all perturbations have been properly modelled) is in principle known. Indeed, the normal points have typically precisions of a few mm, and accuracies of about 1 cm, limited by atmospheric effects and by variations in the absolute calibration of the instruments. With respect to this external and quasi-inertial frame it is then possible to measure the absolute positions and motions of the ground–based stations, with an absolute accuracy of a few mm and mm/yr.

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR The motions of the SLR stations are due: 1. to plate tectonics and regional crustal deformations; 2. to the Earth variable rotation; 1. induce interstations baselines to undergo slow variations:v a few cm/yr; 2. we are able to study the Earth axis intricate motion: 2a. Polar Motion (X p,Y p ); 2b. Length-Of-Day variations (LOD); 2c. Universal Time (UT1);

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Torino 4 – 7 September 2006 David M. Lucchesi The LAGEOS satellites and SLR Dynamic effects of Geometrodynamics Today, the relativistic corrections (both of Special and General relativity) are an essential aspect of (dirty) – Celestial Mechanics as well as of the electromagnetic propagation in space: 1.these corrections are included in the orbit determination–and–analysis programs for Earth’s satellites and interplanetary probes; 2. these corrections are necessary for spacecraft navigation and GPS satellites; 3. these corrections are necessary for refined studies in the field of geodesy and geodynamics;

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget LAGEOS’s 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 Lense–Thirring relativistic precession on the satellites orbit. 0.5 mas For instance, in the case of the satellites node, we are able to determine with high accuracy (about 0.5 mas over 15 days arcs) the total observed precessions: Therefore, in principle, for the satellites node accuracy we obtain : Which corresponds to a ‘’direct‘’ measurement of the LT secular precession Over 1 year

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget 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 J 2n 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.

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget Therefore, we have three main unknowns: 1. the precession on the node/perigee due to the LT effect: LT ; 2. the J 2 uncertainty: J 2 ; 3. the J 4 uncertainty: J 4 ; 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).

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget The solutions of the system of three equations (the two nodes and LAGEOS II perigee) in three unknowns are: k 1 = + 0.295; k 2 = 0.350; where and 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 a slope of 60.1 mas/yr

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget CHAMP GRACE Thanks to the more accurate gravity field models from the CHAMP and GRACE satellites, we can remove only the first even zonal harmonic J 2 in its static and temporal uncertainties while solving for the Lense–Thirring effect parameter LT. LAGEOS II NGP In such a way we can discharge LAGEOS II perigee, which is subjected to very large non–gravitational perturbations (NGP). The solution of the system of two equations in two unknowns is: k 3 = + 0.546

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Torino 4 – 7 September 2006 David M. Lucchesi The EIGEN–GRACE02S gravity field model Calibration is based on subset solution inter–comparisons (preliminary result). Reigber et al., (2004) Journal of Geodynamics Reigber et al., (2004) Journal of Geodynamics. C(2,0)0.1433E-110.5304E-10 0.1939E-100.3561E-10 C(4,0)0.4207E-120.3921E-11 0.2230E-100.1042E-09 C(6,0)0.3037E-120.2049E-11 0.3136E-100.1450E-09 C(8,0)0.2558E-120.1479E-11 0.4266E-100.2266E-09 C(10,0)0.2347E-120.2101E-11 0.5679E-100.3089E-09 (EG02S) (EG02S–CAL) (EIGEN2S) (EGM96) Formal and (preliminary) calibrated errors of EIGEN–GRACE02S: 37 9 7 The 2004 measurement and its error budget

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget a)Observed (and combined) residuals of LAGEOS and LAGEOS II nodes (raw data); b)As in a) after the removal of six periodic signals: 1044 days; 905 days; 281 days; 569 days and 111 days; The best fit line through these observed residuals has a slope of about: = (47.9 6) mas/yr i.e., 0.99 LT c)The theoretical Lense–Thirring effect on the node–node combination: the slope is about 48.2 mas/yr; Ciufolini & Pavlis, 2004, Lett. to Nature The LT effect and the EIGEN–GEACE02S model: Ciufolini & Pavlis, 2004, Lett. to Nature 11 years analysis of the LAGEOS’s orbit

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget The error budget: systematic effects Perturbation Even zonal4% Odd zonal0% Tides2% Stochastic2% Sec. var. 1% Relativity0.4% NGP2% RSS (ALL)5.3% But they allow for a 10% error in order to include underestimated and unknown sources of error 0 2 4 6 8 10 12 years I 0.545 II (mas) (mas) 600 400 200 0 After the removal of 6 periodic terms The LT effect and the EIGEN–GEACE02S model: Ciufolini & Pavlis, 2004, Lett. to Nature

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Torino 4 – 7 September 2006 David M. Lucchesi The 2004 measurement and its error budget We are interested in reviewing such error budget because of some criticism raised in the literature to the estimate performed by Ciufolini and Pavlis. In particular, the secular variations of the even zonal harmonics were suggested to contribute at the level of 11% of the relativistic precession over the time span of the measurement, i.e., over 11 years. Moreover, also the question of possible correlations between the various sources of error and the imprinting of the Lense–Thirring effect itself in the gravity field coefficients was raised.

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution LAGEOSIIILARES The Error budget analysis of the 2004 measurement of the Lense–Thirring effect by Ciufolini & Pavlis is substantially the same as that of the LAGEOSIII/LARES experiment; Ciufolini & Pavlis emphasized that these two space mission have been carefully studied in the past; LAGEOSIIILARES LAGEOS II Ciufolini & Pavlis highlighted the differences between the LAGEOSIII/LARES satellite and the LAGEOS II satellite; Ciufolini & Pavlis have explicitly computed the errors of the even zonal harmonics uncertainties, the largest source of error; renormalizedLAGEOS II Ciufolini & Pavlis have simply ‘’ renormalized ’’ the other errors to the LAGEOS II case without a detailed analysis of each perturbation to their 11 years analysis; even zonal harmonics secular variations Ciufolini & Pavlis reanalyzed the errors from the even zonal harmonics secular variations with a careful data analysis, only after the criticisms of Iorio to their error budget; However, Ciufolini & Pavlis have not been able to explain their results from the physical point of view; Some aspects of the Ciufolini & Pavlis Error Budget estimate

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Perturbation Even zonal4% Odd zonal0% Tides2% 1% ? Stochastic (…)2% Sec. var. 1% Relativity0.4% NGP2% RSS (ALL)5.3% Ciufolini & Pavlis (Nature, 2004) Perturbation Even zonal4% Odd zonal0.005% Tides0.1% Stochastic (Inc.)2% (0.6%) Sec. var. 0.8% Relativity0.4% NGP0.4% RSS (ALL)4.1% This Study RSS (not ALL)4.9% 2% = (tides + sec. var. + NGP)

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics uncertainties Even zonal harmonics uncertainties: EIGEN-GRACE02S gravity model Contribution to the combined nodes from the harmonics with degree ℓ 4. The covariance matrix was not available. Root Sum Square of the J ℓ errors give:3% LT Sum Absolute values of the J ℓ errors give:4% LT upper bound In agreement with Ciufolini and Pavlis estimate, see also L. Iorio: gr-qc/0408031.

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Odd zonal harmonics uncertainties Odd zonal harmonics uncertainties: EIGEN-GRACE02S gravity model Contribution to the combined nodes from the harmonics with degree ℓ 3. The covariance matrix was not available. Odd zonals (J 3 error) over 11 years In agreement with Ciufolini and Pavlis estimate.

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Torino 4 – 7 September 2006 David M. Lucchesi LAGEOS Comparison between the most significant solid and ocean tides on LAGEOS node amplitude (mas) after a 9 years time span as function of the initial phase of the various tidal signals considered. 9 years A represents the full amplitude of the tidal perturbation on the satellite node, while is the residual unmodelled amplitude after a time span of about 9 years. Solid tides Ocean tides The reviewed EB and the J-dot contribution The error budget: systematic effects Solid and Ocean tides uncertainties Solid and Ocean tides uncertainties: Iorio, Celest. Mech. & Dyn. Astr., vol. 79, 2001 Pavlis – Iorio, Int. J. Mod. Phys., vol. D 11, 2002

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Solid and Ocean tides uncertainties Solid and Ocean tides uncertainties: Ciufolini and Pavlis fitted the combined residuals with a secular trend + various periodic terms. With this procedure they obtained a maximum 2% variation of the slope with respect to the relativistic prediction. They assumed that 1% of this variation was produced by the tides mismodelling They assumed that 1% of this variation was produced by the tides mismodelling. The other 1% was due to the unmodelled trends in the even zonal harmonics. Moreover, they included in this 2% also the contribution of the NGP. This study Not in agreement with Ciufolini and Pavlis estimate Not in agreement with Ciufolini and Pavlis estimate.

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Stochastic errors Stochastic errors: Ciufolini and Pavlis considered the following effects: seasonal variations of the Earth gravity field; drag and observation biases; random errors; LAGEOS measurement uncertainty (random and systematic) in the inclination of LAGEOS satellites; They estimated an error of about: In the present study we directly consider only the measurement errors of the inclination

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Stochastic errors Stochastic errors: Inclination errors A 3 cm accuracy in the orbit determination with arcs of 15 days length translates into a 0.5 mas accuracy in the orbit orientation in space over the same time span. Over 11 years this give an error of about: This study

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects General Relativity errors General Relativity errors: geodetic precession (de Sitter effect) for an Earth gyroscope in the field of the Sun; change of a satellite ascending node longitude; Obliquity of the ecliptic; This study In agreement with Ciufolini and Pavlis estimate The geodetic precession is presently measured with an accuracy of about 0.7%

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Torino 4 – 7 September 2006 David M. Lucchesi NGP Error Budget NGP Error Budget: (over a 9 years time span) K 3 = + 0.546 The reviewed EB and the J-dot contribution The error budget: systematic effects

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects This study Ciufolini and Pavlis fitted the combined residuals with a secular trend + various periodic terms. With this procedure they obtained a maximum 2% variation of the slope with respect to the relativistic prediction. They have not separately estimated the NGP error budget. Has previously highlighted, they included in this 2% error also tides and secular variations of the even zonal harmonics. Not in agreement with Ciufolini and Pavlis estimate Not in agreement with Ciufolini and Pavlis estimate. NGP Error Budget NGP Error Budget: They estimated an error of about:

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution Ciufolini & Pavlis, 2004, Lett. to Nature The LT effect and the EIGEN–GEACE02S model: Ciufolini & Pavlis, 2004, Lett. to Nature 1S) Two–frequency fit of the Lense– Thirring effect. The two signals have periods of 569 and 1044 days (the nodal periods of the LAGEOS satellites). The observed Lense–Thirring effect is 47 mas/yr, corresponding to 0.97 of the general relativistic prediction. The RMS of the post–fit residuals of the combined nodal longitudes is about 11 mas. 2S) Ten–frequency fit of the Lense– Thirring effect. The ten signals have periods of 1044, 905, 281, 222, 522, 569, 111, 284.5, 621 and 182.6 days. The observed Lense–Thirring effect is 47.8 mas/yr, corresponding to 0.99 of the general relativistic prediction. The RMS of the post–fit residuals is about 5.5 mas. Two–frequency Fit Ten–frequency Fit The error budget: systematic effects NGP Error Budget

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : We are able to remove the impact, on the relativistic measurement, due to the first even zonal harmonic, J 2, uncertainties, both in its static and time dependent contributions. Therefore, with regard to the even zonal harmonics secular variations, the errors arise from the values and/or the mismodeling/unmodeling of the coefficients with degree ℓ 4, that is from: In particular a lumped or effective coefficient could be defined:

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : CHAMP and GRACE space missions CHAMPGRACE The characteristic of the new models from CHAMP and GRACE is to improve the gravity field knowledge with a limited amount of data, i.e., of time, and in particular in the medium and short wavelengths; t 0 The reference epoch t 0 for the gravity field coefficients determination (static part) is the middle epoch of the analyzed time span of satellites data; IERS SLR In order to obtain the gravity field solution for a different epoch, the static coefficients are propagated using IERS standard values for their rates (estimated using SLR observations):

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : 1% LAGEOS 11 yr That is, either assumingfor the secular variation, or several times such a nominal value, or even assuming, they have always obtained a 1% modification of the slope of the integrated nodes residuals of the two LAGEOS satellites with respect to the prediction of general relativity over a time span of about 11 yr. In particular, Ciufolini&Pavlis have always found from their analyses an impact of the uncertainties of even zonal harmonics secular variations of about: Ciufolini – Pavlis, Letters to Nature, vol. 421, 2004 Ciufolini – Pavlis, New Astronomy, vol. 10, 2005

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : from the nodal rate 1%1 yr 11 yr Lorenzo Iorio highlighted that, because of the quadratic effect of the secular variations of the even zonal harmonics on the satellites node, the 1% error was valid only for a 1 yr analysis, therefore Iorio concluded that over 11 yr the error budget is about 11% of the relativistic effect from the coefficients with degree ℓ 4. Indeed, if Iorio, New Astronomy, vol. 10, 2005 we obtain for the node quadratic effect

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Lense–Thirring effect Even zonal harmonics uncertainties Even zonal harmonics secular variations uncertainties Mismodeled periodic effects Node “signal” vs perturbations/errors: Because of the J 2 effects cancellation with the nodes only combination Reference epoch of the gravity field determination

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Impact on the LT effect slope LT effect quadratic effect distorted LT effect linear trend fit from the fit 100% mismodeling of the quadratic effect

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Impact on the LT effect slope LT effect quadratic effect distorted LT effect linear trend fit 100% mismodeling of the quadratic effect When the vertex of the parabola is centered in the middle of the time span of the orbit analysis, the positive variation of the slope before that middle epoch is compensated by the negative variation of the slope after that epoch. Therefore, the secular variations have no impact on the slope of the relativistic effect. 400% mismodeling of the quadratic effect

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : How this new approach has been reached Let us now suppose that the coefficient due to nature has not been modeled in the orbit determination program. This means that the static value of the J 4 coefficient, which was determined by the GRACE mission for the period 2002/2003, will be applied to all the time span of the two LAGEOS satellites orbit analysis (red–dashed line). Therefore, there will be a discrepancy during the satellites orbit estimation-and-analysis, i.e., during each arc (about two weeks in the case of the Lense–Thirring effect derivation), between the dynamical model included in the program and the SLR observations, i.e., the normal points data. True evolution Discrepancy This discrepancy is responsible for one more error in the value of the even zonal harmonic coefficient in addition to the static error due to its uncertainty. This error has a dynamical origin, because it is time–dependent due to the presence of the secular variation.

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution True evolution Discrepancy Over a time span t the dynamic error is given by: and this quantity has a linear impact on the node Eq: In the case of the combined nodes, the resultant error over the analyzed time span becomes: and this error compensates exactly the impact of the parabola on the slope of the relativistic Lense–Thirring effect.

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : The parabolic term: A pure quadratic effect whose role is to change the slope of the Lense–Thirring effect (Iorio’s argument) An additional (linear in time) contribution to the slope of the Lense–Thirring effect A constant term, that is a bias in the fitting procedure

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Parabolic term: Dynamic error: Resultant impact on the Lense–Thirring effect slope:

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Parabolic term: Dynamic error: We stress that the role of the dynamic error is not to cancel the parabola, but to cancel its impact on the Lense–Thirring effect slope, that is, its effect is to shift the symmetry axis of the parabola, i.e., the epoch t 0, to the value t max /2. The resultant effect is to change the position of the symmetry axis, the vertex and the focus of the parabola, but not its concavity. Symmetry axis: Lucchesi, Int. Journ. of Modern Phys. D, vol. 14, No. 12, 2005

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Previously we have highlighted that: That is, either assuming for the secular variation, or several times such a nominal value, or even assuming, they have always obtained a 1% modification of the slope of the integrated nodes residuals of the two LAGEOS satellites with respect to the prediction of general relativity over a time span of about 11 yr. Ciufolini&Pavlis have always found from their analyses an impact of the uncertainties of even zonal harmonics secular variations of about 1% of LT. The question is: WHY they have not found a null impact on the Lense–Thirring effect slope from the even zonal harmonics secular variations?

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Answer: one more possible error must be considered. I.Because a given gravity field is determined for a given epoch t 0 (the middle epoch of the gravity field determination time span), if this epoch is not the same of the reference epoch used in the orbit determination program (T 0 ), a constant error will appear in all the zonal harmonics coefficients that we know are time–dependent. II.Therefore, if the two epochs are different, we need to propagate the gravity field coefficients that we know are time–dependent, say J 2, J 4, J 6, J 8, …, to the reference epoch of the orbit determination program (or vice versa), otherwise a constant bias (static–like error) will appear for each of these time–dependent coefficients. III.Each of these additional errors will be negative, and therefore they will decrease the slope of the Lense–Thirring regression line, if the gravity field epoch is greater than T 0, and positive (hence increasing the slope) if the gravity field epoch is smaller than T 0. IV.This error will change the b coefficient of the parabola and will shift away its symmetry axis from the time t max /2.

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : V.Moreover, because these static errors are independent from the values that we use for the secular variations in the orbit determination program, but they only depend from the impact that the “true” values of these coefficients (say nature) have had during the propagation time, they produce a constant error in the slope of the Lense–Thirring effect independently from the adopted values of the secular variations in the orbit analysis. VI.This is very important and represents the key to explain the 1% error in the recovery of the slope of the Lense–Thirring effect in all the simulations performed by Ciufolini&Pavlis. In fact, Ciufolini&Pavlis have correctly propagated to the epoch J2000 the J 2 and J 4 coefficients determined with the EIGEN–GRACE02S solution for the middle epoch of February 14, 2003. However, they have not propagated back the other coefficients with degree ℓ 6. Therefore, the constant error that they produce in the recovery of the slope of the Lense– Thirring effect is of the order of:

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Torino 4 – 7 September 2006 David M. Lucchesi The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget Even zonal harmonics secular variations Error Budget : Therefore, we obtain: where: comes from their fit of the quadratic effect comes from IERS standard comes from the discrepancy of the two reference epochs The result is very close to the 1% error claimed by Ciufolini&Pavlis. The discrepancy is probably due to the effects of the periodic perturbations that are also present in the integrated node residuals obtained with the EIGEN–GRACE02S gravity field model. Lucchesi, Int. Journ. of Modern Phys. D, vol. 14, No. 12, 2005

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only LAGEOS gravitomagnetic An interesting question is related to how far we can go with the LAGEOS satellites in order to test the gravitomagnetic interaction. NASAGPB LAGEOS’s The NASA’s GPB (see Fitch et al., 1995 for a review) space mission is in principle able to measure the gravitomagnetic field of the Earth to the 0.3% level (the first scientific results are expected in 2007), i.e., more than a factor 10 better than it is currently possible with the two LAGEOS’s. CHAMPGRACE GPSLAGEOS It seems that the present gravimetric space missions (CHAMP and GRACE) are not able to improve significantly the low degree coefficients of the Earth’s field (even using the GPS data) to which the orbit of LAGEOS satellites is more sensitive. 1%Lense–Thirring Therefore, the 1% level probably represents an horizon for the Lense–Thirring effect accuracy when using the node–only combination of the two laser–ranged satellites. GOCE Also the forthcoming GOCE space mission will be less sensitive to the low degree components of the Earth’s field and not so great improvements are expected.

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only LAGEOS II The use of the linear combination involving also LAGEOS II argument of perigee as an additional observable as the great advantage of eliminating also the uncertainties of all the systematic gravitational effects with degree ℓ = 4 and order m = 0. Using the present gravity field models the error budget from the even zonal harmonics uncertainties (starting from J 6 ) fall down to less than 1% of the relativistic precession. LAGEOS II Unfortunately, the systematic errors from the non–gravitational perturbations increase with the use of LAGEOS II argument of perigee, and a factor 10 improvement in the modelling of their subtle effects (in truth quite difficult to reach) is not enough to reduce their error contribution to the level of the gravitational perturbations. NGP SLR Moreover, such a modelling of the NGP requires an improvement also in the range accuracy of the SLR technique. Again, a not easy task. We need LARES ! Ciufolini et al., 1998(ASI), 2004(INFN)

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only 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 dragnegligible negligible Asymmetric reflectivity Perturbation k 1 = + 0.295 k 2 = 0.350 7 years simulation

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude A YS Lucchesi, Ciufolini, Andrés, Pavlis, Peron, Noomen and Currie, Plan. Space Science, 52, 2004 EGM96 The plot (red line) represents the best– fit we obtained for the Yarkovsky– Schach perturbation assuming that this is the only disturbing effect influencing the LAGEOS II argument of perigee. Initial Yarkovsky–Schach parameters: A YS = 103.5 pm/s 2 for the amplitude = 2113 s for the CCR thermal inertia Final Yarkovsky–Schach amplitude: A YS = 193.2 pm/s 2 i.e., about 1.9 times the pre–fit value. With EGM96 in GEODYN II software and the LOSSAM model in the independent numerical simulation (red and blue lines).

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only Direct solar radiation+ 946.42 1+ 15.75 Earth albedo 19.36 2 0.64 Yarkovsky–Schach effect 98.51 1 1.64 Earth–Yarkovsky 0.56 2 negligible Neutral + Charged particle dragnegligible negligible Asymmetric reflectivity Perturbation k 1 = + 0.295 k 2 = 0.350 7 years simulation Assuming an improvement by a factor of 10

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only NGPLAGEOSLAGEOS II SRP The Largest (and Best Modelled) NGP on LAGEOS and LAGEOS II orbit is due to direct solar radiation pressure (SRP): 3.6·10 9 m/s 2 LAGEOS II C R represents the satellite radiation coefficient, about 1.12 for LAGEOS II; A/m represents the area–to–mass ratio of the satellites, about 7 10 4 m 2 /kg; Sun represents the solar irradiance at 1 AU, about 1380 W/m 2 ; c represents the speed of light, about 3 10 8 m/s; D Sun represents the average Earth–Sun distance, i.e., 1 AU; ŝ represents the Earth–Sun unit vector; Where: SRP Error in SRP :

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Torino 4 – 7 September 2006 David M. Lucchesi Difficulties in improving the present measurement with LAGEOS satellites only SRP Error in SRP : 1/10 LAGEOS Such small accelerations are ‘’visible’’ in LAGEOS satellites residuals. SRP However, a factor of 10 corresponds to a 0.1% mismodelling of the SRP, and this is presently unreachable because of the solar irradiance uncertainty, at the level of 0.3% (also C R ) : 0.3% LARESASIINFN With the proposed LARES (Ciufolini et al., 1998 (ASI), 2004 (INFN)), which has a factor 2 smaller area-to-mass ratio and a larger eccentricity, we obtain: 0.3% LARES Therefore, the three elements combination is not competitive with the two nodes combination when applied to LARES satellite. LARES Lense–ThirringLucchesi&Rubincam, 2004 We need LARES and the node only combination to reach a 0.3% measurement of the Lense–Thirring effect (Lucchesi&Rubincam, 2004).

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Torino 4 – 7 September 2006 David M. Lucchesi Table of Contents Gravitomagnetism and Lense–Thirring effect; The LAGEOS satellites and SLR; The 2004 measurement and its error budget (EB); The reviewed EB and the J-dot contribution; Difficulties in improving the present measurement with LAGEOS satellites only; Conclusions;

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Torino 4 – 7 September 2006 David M. Lucchesi Conclusions Lense–Thirring 1.From the results of the present analysis the overall error budget of the 2004 measurement of the Lense–Thirring effect seems reliable, about 5% at 1–sigma level; 2.In particular, the impact of the even zonal harmonics secular trends is around 1% of the relativistic prediction; 3.However, the error budget of the 2004 measurement is not characterized by a clear separation of the individual contributions to the resultant error budget, in particular in the periodic effects; 4.We have tried to better quantify such contributions to the error budget, in particular: i) we have explained from the physical point of view the 1% error due to the secular trends of the even zonal harmonics; ii) we have estimated the contribution of the main tidal signals as well as of the NGP;

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Torino 4 – 7 September 2006 David M. Lucchesi Conclusions LAGEOSLARES LAGEOS 5.We have underlined the difficulties in improving the present results using LAGEOS satellites only: the LARES satellites is necessary in addition to the two LAGEOS; 6.LARESGPB 6.LARES will be competitive with GPB goal: a 0.3% measurement; 7.LARESfundamental physics geodesy and geodynamics 7.LARES will be very useful not only for fundamental physics in space but also for space geodesy and geodynamics; LARESGPB Lense–Thirring LAGEOS 8.Waiting to LARES mission (?) and GPB results (2007), it will be very useful a new measurement of the Lense–Thirring effect with the two LAGEOS satellites together with the measurement of the first even zonal harmonics coefficients, those at which the two satellites orbit are most sensitive to, in such a way to obtain a reliable error budget at a few % level; finis

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