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Roberto Peron IFSI-INAF Email: roberto.peron@ifsi-roma.inaf.it

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Near-Earth space (spacetime) is a good place to perform tests on theories about gravitation Earth Schwarzschild radius 1 cm The effects searched for are by now relevant for current technology: think about GPS!

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What do we need in order to perform good science? SchwarzschildKerr A theory: Schwarzschild, Kerr (gravitomagnetism) exact solutions sufficiently general to be descriptive and predictive weak fieldslow motionPPN formalism Contat points with experiment: weak field and slow motion, PPN formalism test masses A probe: test masses

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General relativity (geometrodynamics) implies a continuous feedback between geometry and mass-energy (nonlinearity) Practical needs often force to “hold on something”

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The smallness of a test mass depends on the scale under consideration LAGEOS and SLR

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A test mass in the outer solar system LAGEOS and SLR

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A future test mass pretty close to the Sun LAGEOS and SLR

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The LAGEOS satellite are probably the closest to the ideal concept of a test mass LAGEOS and SLR

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LAGEOSLAGEOS II –COSPAR ID76039019207002 –Launch date4 May 197622 October 1992 –Diameter60 cm MMass406.965 kg405.38 kg –Retroreflectors426 CCR aSemimajor axis1.2286 · 10 7 m1.2155 · 10 7 m eEccentricity0.00450.0135 IInclination109.84°52.64° –Perigee height5.86 · 10 6 m5.62 · 10 6 m PPeriod225 min223 min nMean motion4.654 · 10 −4 s −1 4.696 · 10 −4 s −1 Node rate0.34266° d −1 -0.62576641° d −1 Perigee rate-0.21338° d −1 0.44470485° d −1

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Satellite Laser Ranging (SLR) A laser pulse from a ground station is sent to the satellite, where it is reflected back in the same direction from optical elements called Cube Corner Retroreflectors (CCR) The precision of this technique is noteworthy ( 1 mm) Matera MLRO CCR ilrs.gsfc.nasa.gov

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LAGEOS and SLR

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ilrs.gsfc.nasa.gov LAGEOS and SLR

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Moving (rotating) masses: what do they do? - Spacetime Kerr metricweak-field Kerr metric in weak-field (it describes in an approximate way the spacetime around a rotating mass)

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Moving (rotating) masses: what do they do? - Spacetime weak field Lorentz gauge Gravitomagnetic potential Gravitomagnetic field Defined by analogy with electromagnetic case

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Moving (rotating) masses: what do they do? - Geodesics Slow-motion Gravitoelectric field Gravitomagnetic contribution Thus mass-energy currents influence the motion of test masses: Gravitomagnetism Thus mass-energy currents influence the motion of test masses: Gravitomagnetism

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Spherically symmetric rotating mass- energy distribution (J is the angular momentum associated to the distribution) A gyroscope in a gravitomagnetic field precesses Dragging of inertial frames

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Obtain a solutionCelestial mechanics tools Osculating ellipse (Keplerian elements) Perturbation first-order analysis (Lagrange and Gauss equations) Periodic effects Secular effects ( t) Periodic effects Secular effects ( t)

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Lagrange perturbation equations

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Gauss perturbation equations

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Secular effects on longitude of ascending node and argument of perigee J. Lense and H. Thirring, 1918 LAGEOSLAGEOS II 30.6631.51 31.31-57.35 Values in mas 1 mas = 2.8 ∙ 10 -5 °

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Differential correction procedure Corrections to the models parameters Residuals Observation equations Least-squares (normal equations) Partials Covariance matrix

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Parameter estimation

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The analysis of experimental data to obtain the properties of a physical system requires models System dynamics Measurement procedure (Reference frame) The availability of good experimental data implies taking out a lot of “noise” in order to reach the phenomenology of interest – many orders of magnitude, in case of relativistic effects

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Geopotential (static part) Solid Earth and ocean tides / Other temporal variations of geopotential Third body (Sun, Moon and planets) de Sitter precession Deviations from geodetic motion Other relativistic effects Direct solar radiation pressure Earth albedo radiation pressure Anisotropic emission of thermal radiation due to visible solar radiation (Yarkovsky-Schach effect) Anisotropic emission of thermal radiation due to infrared Earth radiation (Yarkovsky-Rubincam effect) Asymmetric reflectivity Neutral and charged particle drag Gravitational Non-gravitational

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CauseFormulaAcceleration (m s -2 ) Earth’s monopole2.8 Earth’s oblateness1.0 ∙ 10 -3 Low-order geopotential harmonics 6.0 ∙ 10 -6 High-order geopotential harmonics 6.9 ∙ 10 -12 Perturbation due to the Moon 2.1 ∙ 10 -6 Perturbation due to the Sun 9.6 ∙ 10 -7 General relativistic correction 9.5 ∙ 10 -10 Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

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CauseFormulaAcceleration (m s -2 ) Atmospheric drag3 ∙ 10 -12 Solar radiation pressure3.2 ∙ 10 -9 Earth’s albedo radiation pressure 3.4 ∙ 10 -10 Thermal emission1.9 ∙ 10 -12 Dynamic solid tide3.7 ∙ 10 -8 Dynamic ocean tide 0.1 of the dynamic solid tide 3.7 ∙ 10 -9 Reference system: non-rigid Earth nutation (fortnightly term) 0.002 arsec in 14 days3.5 ∙ 10 -12 Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

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The Earth is not a sphere! Spherical harmonics expansion Spherical harmonics classification Zonal Tesseral Sectorial

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Quadrupole perturbation (l = 2, m = 0) to first order

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Some geopotential models ModelData typeMaximum degree JGM-3Combined70 GRIM5-S1Satellite95 GRIM5-C1Combined120 OSU89A/BCombined360 EGM96Combined360 EIGEN-2Satellite120 EIGEN-GRACE02SSatellite150 GGM02SSatellite160

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The geoid is a gravitational equipotential surface, taken as reference surface (“sea level”); It differs in general from a rotation surface, like the reference ellipsoid Models

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The gravity anomalies are the difference between the real gravity field and that of a reference body (rotation ellipsoid) Models

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The degree variance is useful when comparing various geopotential solutions Its behaviour is well described by the so-called Kaula’s rule

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Earth geopotential degree variance is well approximated by Kaula’s rule Models

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The signal-to-noise ratio indicates how well the signal is recognizable from the noise Models

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Though similar in behaviour, Earth and Moon gravity potentials differ in the way the power is distributed at the various wavelengths Models

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The various geopotential solutions differ strongly in the uncertainty associated to the harmonic coefficients Models

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Direct solar radiation pressure It is due to reflection-diffusion-absorption of solar photons from the spacecraft surface

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The energy flux from the Sun varies with a periodicity of about 11 years (Solar Cycle); plot from www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant Models

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Yarkovsky-Schach effect It is due to infrared radiation anisotropically emitted from the satellite (warmed by the Sun)

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Farinella–Vokhroulicky–Barlier model Models

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Farinella–Vokhroulicky–Barlier model Models

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“Inertial” reference frame ICRF Precession Nutation Length of Day Pole motion ITRF

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IERS data (EOP 05 C04) Models

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IERS data (EOP 05 C04) Models

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Tracking data: Normal Point from ILRS Software: Geodyn II (NASA GSFC) Arc length: 15 days Estimate: state vector and other model parameters Further information: residuals RESIDUAL = OBSERVED - CALCULATED

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FeatureModel Geopotential (static part)EGM96, EIGEN-2, GGM01S, EIGEN-GRACE02S (among others) Geopotential (tides)Ray GOT99.2 Third bodyJPL DE-403 Relativistic correctionsPPN Direct solar radiation pressureCannonball Albedo radiation pressureKnocke–Rubincam Yarkovsky–Rubincam effectGEODYN model Spin axisFarinella–Vokhroulicky–Barlier Stations positions (ITRF)ITRF2000 Ocean loadScherneck with GOT99.2 tides Pole motionIERS EOP Earth rotationIERS EOP

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ElementValueUncertainty a12265733.6258 m0.1616 m e0.0051376777090.000000008907 I109.891430986°0.000001159° 314.057527507°0.000001163° 45.867206979°0.000158387° M212.542258784°0.000157151° LAGEOS initial conditions estimate for 15-day arc with epoch 3 January 1993 (MJD 48990)

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ElementValueUncertainty a12160019.9315 m0.0066 m e0.0132775894980.000000010922 I52.622724892°0.000001051° 68.835933790°0.000001258° 244.234411803°0.000061024° M51.909462952°0.000063071° LAGEOS II initial conditions estimate for 15-day arc with epoch 3 January 1993 (MJD 48990)

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Data analysis

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Data analysis – Residuals

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The recent geopotential models make critical in the error budget only the uncertainty associated with C 20 (Earth quadrupole)

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Ciufolini, Pavlis and Peron, New Astron. 11, 527 (2006) Data analysis – Lense- Thirring

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A peculiar error estimate is associated to each analysis Data analysis – Lense- Thirring

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In common practice often empirical accelerations are included in the modelization setup to take into account small not modeled perturbations Fit improving Estimation biases risk Constant part ∼ 10 −12 m/s 2 Possible correlation between C R and empirical accelerations coefficients

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Data analysis – Estimation biases

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Lucchesi et al., Plan. Space Sci. 52, 699 (2004) Data analysis – Estimation biases

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Lucchesi et al., Plan. Space Sci. 52, 699 (2004) Data analysis – Yarkovsky- Schach effect

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The reported secular trend for the integrated residuals of the longitude of the LAGEOS ascending node abruptly changes from 0.026 mas d −1 to 0.24 mas d −1 Data analysis – Unexpected results

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The reported secular trend for the integrated residuals of the longitude of the LAGEOS II ascending node abruptly changes from -0.0079 mas d −1 to -0.53 mas d −1 Data analysis – Unexpected results

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Geopotential harmonics coefficients change in time: Tides Secular variations (e.g. postglacial rebound) Mass transport (e.g. oceans ↔ atmosphere) This variation seems to be due to an abrupt change in the quadrupole rate (Cox and Chao, Science 297, 831 (2002); Ciufolini, Pavlis and Peron, New Astron. 11, 527 (2006)) The causes are uncertain: Mantle? Tides? In any case, this implies a net mass transfer from polar to equatorial regions Geopotential harmonics coefficients change in time: Tides Secular variations (e.g. postglacial rebound) Mass transport (e.g. oceans ↔ atmosphere) This variation seems to be due to an abrupt change in the quadrupole rate (Cox and Chao, Science 297, 831 (2002); Ciufolini, Pavlis and Peron, New Astron. 11, 527 (2006)) The causes are uncertain: Mantle? Tides? In any case, this implies a net mass transfer from polar to equatorial regions

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D. W. Sciama, The Unity of the Universe, Faber & Faber, 1959 C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman and Co., 1973 I. Ciufolini and J. A. Wheeler, Gravitation and inertia, Princeton University Press, 1995 B. Bertotti, P. Farinella, and D. Vokrouhlický, Physics of the Solar System — Dynamics and Evolution, Space Physics, and Spacetime Structure, Kluwer Academic Publishers, 2003 A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987 O. Montenbruck and E. Gill, Satellite Orbits — Models, Methods and Applications, Springer, 2000 C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativity 9, (2006), 3, http://www.livingreviews.org/lrr-2006-3http://www.livingreviews.org/lrr-2006-3 D. McCarthy, and G. Petit (eds.), IERS Conventions (2003), IERS, 2004

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