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The Lense–Thirring effect measurement with LAGEOS satellites:

XVII SIGRAV Conference. The Lense–Thirring effect measurement with LAGEOS satellites:. Error Budget and impact of the time–dependent part of Earth’s gravity field. David M. Lucchesi 1,2,3. Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF), Roma, Italy

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The Lense–Thirring effect measurement with LAGEOS satellites:

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  1. XVII SIGRAV Conference The Lense–Thirring effect measurement with LAGEOS satellites: Error Budget and impact of the time–dependent part of Earth’s gravity field David M. Lucchesi1,2,3 • Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF), Roma, Italy • Istituto di Scienza e Tecnologie della Informazione (ISTI/CNR), Pisa, Italy • Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma II, Roma, Italy

  2. 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;

  3. 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. 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.

  4. Gravitomagnetism and Lense–Thirring effect  This leads to the “ Linearized Theory of Gravity ”: Flat spacetime metric: gauge conditions; metric tensor; field equations; 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:

  5. 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

  6. 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.

  7. 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.

  8. h(r) BG(r) A(r) B(r) r r J  Jm Je Gravitomagnetism and Lense–Thirring effect The Analogy Classical electrodynamics: Classical geometrodynamics (WFSM): G = c = 1

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

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

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

  12. Angular momentum Gravitomagnetism and Lense–Thirring effect Secular effects of the Gravitomagnetic field: Rate of change of the ascending node longitude: (Lense–Thirring, 1918) Rate of change of the argument of perigee: 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)

  13. Z Orbital plane Equatorial plane semimajor axis; eccentricity; inclination; I longitude of the ascending node; argument of perigee; mean anomaly;   Y X Ascending Node direction L Gravitomagnetism and Lense–Thirring effect Keplerian elements

  14. Gravitomagnetism and Lense–Thirring effect 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

  15. 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;

  16. The LAGEOS satellites and SLR LAGEOS (LAser GEOdynamic Satellite) LAGEOS and LAGEOS II satellites LAGEOS, launched by NASA (May 4, 1976); LAGEOS II, launched by ASI/NASA (October 22, 1992);

  17. 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 m2/kg.; • Outer portion: Al, MA 117 kg; • Inner core: CuBe, L = 27.5 cm, d = 31.76 cm, MBC  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;

  18. 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.

  19. The LAGEOS satellites and SLR GEODYN II range residuals Accuracy in the data reduction LAGEOS range residuals (RMS) 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. Courtesy of R. Peron From January 3, 1993

  20. The LAGEOS satellites and SLR • 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 (after all perturbations have been properly modelled) is in principle known. 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.

  21. The LAGEOS satellites and SLR The motions of the SLR stations are due: • to plate tectonics and regional crustal deformations; • 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.PolarMotion (Xp,Yp); 2b.Length-Of-Day variations (LOD); 2c.UniversalTime (UT1);

  22. 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: • 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;

  23. 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;

  24. The 2004 measurement and its error budget 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. 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 : Over 1 year Which corresponds to a ‘’direct‘’ measurement of the LT secular precession

  25. 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 harmonicsJ2n 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.

  26. The 2004 measurement and its error budget Therefore, we have three main unknowns: • the precession on the node/perigee due to the LT effect: LT; • the J2 uncertainty: J2; • 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: • LAGEOS node: Lageos; • LAGEOS II node: LageosII; • 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).

  27. 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: k1 = + 0.295; k2 =  0.350; where are the residuals in the rates of the orbital elements and i.e., the predicted relativistic signal is a linear trend with a 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)

  28. The 2004 measurement and its error budget Thanks to the more accurate gravity field models from the CHAMP and GRACE satellites, we can remove onlythe first even zonal harmonic J2 in its static and temporal uncertainties while solving for the Lense–Thirring effect parameter LT. In such a way we can discharge LAGEOS II perigee, which is subjected to very large non–gravitational perturbations (NGP). The solution of the system of two equations in two unknowns is: k3 = + 0.546

  29. (EG02S) (EG02S–CAL) (EIGEN2S) (EGM96) C(2,0) 0.1433E-11 0.5304E-10 0.1939E-10 0.3561E-10 C(4,0) 0.4207E-12 0.3921E-11 0.2230E-10 0.1042E-09 C(6,0) 0.3037E-12 0.2049E-11 0.3136E-10 0.1450E-09 C(8,0) 0.2558E-12 0.1479E-11 0.4266E-10 0.2266E-09 C(10,0) 0.2347E-12 0.2101E-11 0.5679E-10 0.3089E-09 37 9 7 The 2004 measurement and its error budget The EIGEN–GRACE02S gravity field model Formal and (preliminary) calibrated errors of EIGEN–GRACE02S: Calibration is based on subset solution inter–comparisons (preliminary result). Reigber et al., (2004) Journal of Geodynamics.

  30. The 2004 measurement and its error budget The LT effect and the EIGEN–GEACE02S model:Ciufolini & Pavlis, 2004, Lett. to Nature 11 years analysis of the LAGEOS’s orbit • Observed (and combined) residuals of LAGEOS and LAGEOS II nodes (raw data); • 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;

  31. I  0.545II (mas) 600 400 200 0 Perturbation Even zonal 4% Odd zonal 0% Tides 2% Stochastic 2% Sec. var. 1% Relativity 0.4% NGP 2% (mas) 0 2 4 6 8 10 12 years The 2004 measurement and its error budget The LT effect and the EIGEN–GEACE02S model:Ciufolini & Pavlis, 2004, Lett. to Nature The error budget: systematic effects After the removal of 6 periodic terms RSS (ALL) 5.3% But they allow for a  10% error in order to include underestimated and unknown sources of error

  32. 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.

  33. 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;

  34. The reviewed EB and the J-dot contribution Some aspects of the Ciufolini & Pavlis Error Budget estimate • 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; • 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; • 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; • 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;

  35. Perturbation Even zonal 4% Odd zonal 0% Tides 2%  1% ? Stochastic (…) 2% Sec. var. 1% Relativity 0.4% NGP 2% Perturbation Even zonal 4% Odd zonal 0.005% Tides 0.1% Stochastic (Inc.) 2% (0.6%) Sec. var. 0.8% Relativity 0.4% NGP 0.4% RSS (not ALL) 4.9% 2% = (tides + sec. var. + NGP) The reviewed EB and the J-dot contribution The error budget: systematic effects Ciufolini & Pavlis (Nature, 2004) This Study RSS (ALL) 5.3% RSS (ALL) 4.1%

  36. The reviewed EB and the J-dot contribution The error budget: systematic effects 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 Jℓ errors give: 3% LT Sum Absolute values of the Jℓ errors give: 4% LT upper bound In agreement with Ciufolini and Pavlis estimate, see also L. Iorio: gr-qc/0408031.

  37. The reviewed EB and the J-dot contribution The error budget: systematic effects 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 (J3 error) over 11 years In agreement with Ciufolini and Pavlis estimate.

  38. Solid tides Ocean tides d W d W LAGEOS LAGEOS II Solid Mis. % A (mas) A (mas) (mas) (mas) - - 18.6 1.5 1982.2 19.53 1079.4 10.64 - - K1 0.5 1744.4 0.39 398.0 0.07 Ocean - - K1 3.8 156.6 0.27 35.7 0.11 - Sa 6.7 ¾ 37.7 ¾ 20.6 The reviewed EB and the J-dot contribution Iorio, Celest. Mech. & Dyn. Astr., vol. 79, 2001 Pavlis – Iorio, Int. J. Mod. Phys., vol. D 11, 2002 The error budget: systematic effects Solid and Ocean tides uncertainties: 9 years 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. 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.

  39. The reviewed EB and the J-dot contribution The error budget: systematic effects Solid and Ocean tides uncertainties: 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 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. Not in agreement with Ciufolini and Pavlis estimate.

  40. The reviewed EB and the J-dot contribution The error budget: systematic effects Stochastic errors: Ciufolini and Pavlis considered the following effects: • seasonal variations of the Earth gravity field; • drag and observation biases; • random errors; • 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

  41. The reviewed EB and the J-dot contribution The error budget: systematic effects 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

  42. The reviewed EB and the J-dot contribution The error budget: systematic effects 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; The geodetic precession is presently measured with an accuracy of about 0.7% This study In agreement with Ciufolini and Pavlis estimate

  43. The reviewed EB and the J-dot contribution The error budget: systematic effects NGP Error Budget: (over a 9 years time span) K3 = + 0.546

  44. The reviewed EB and the J-dot contribution The error budget: systematic effects NGP Error Budget: 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. They estimated an error of about: Not in agreement with Ciufolini and Pavlis estimate.

  45. The reviewed EB and the J-dot contribution The error budget: systematic effects NGP Error Budget 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. Two–frequency Fit Ten–frequency Fit 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.

  46. The reviewed EB and the J-dot contribution The error budget: systematic effects 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, J2, 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:

  47. The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget: CHAMP and GRACE space missions • 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; • The reference epoch t0 for the gravity field coefficients determination (static part) is the middle epoch of the analyzed time span of satellites data; • 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):

  48. The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget: In particular, Ciufolini&Pavlis have always found from their analyses an impact of the uncertainties of even zonal harmonics secular variations of about: 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, Letters to Nature, vol. 421, 2004 Ciufolini – Pavlis, New Astronomy, vol. 10, 2005

  49. quadratic effect The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget: 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 from the nodal rate we obtain for the node Iorio, New Astronomy, vol. 10, 2005

  50. Because of the J2 effects cancellation with the nodes only combination Reference epoch of the gravity field determination The reviewed EB and the J-dot contribution The error budget: systematic effects Even zonal harmonics secular variations Error Budget: Node “signal” vs perturbations/errors: Mismodeled periodic effects Lense–Thirring effect Even zonal harmonics secular variations uncertainties Even zonal harmonics uncertainties

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