Gravitomagnetism and other physics with the LAGEOS satellites

1. Roberto Peron IFSI-INAF Email: roberto.peron@ifsi-roma.inaf.it Gravitomagnetism and other physics with the LAGEOS satellites

2. Introduction (motivations)

3. Introduction (motivations) 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!

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

5. LAGEOS satellites and SLR

6. LAGEOS and SLR General relativity (geometrodynamics) implies a continuous feedback between geometry and mass-energy (nonlinearity) Practical needs often force to “hold on something” • TEST MASS • No electric charge • Gravitational bounding energy negligible with respect to rest mass-energy • Angular momentum negligible • Sufficiently small to neglect tidal effects

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

8. LAGEOS and SLR Cassini A test mass in the outer solar system

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

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

11. LAGEOS and SLR

12. LAGEOS and SLR 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) ilrs.gsfc.nasa.gov CCR Matera MLRO

13. LAGEOS and SLR International Laser Ranging Service (ILRS) network ilrs.gsfc.nasa.gov

14. LAGEOS and SLR Main space geodetic networks ilrs.gsfc.nasa.gov

15. Gravitomagnetism (in weak-field and slow-motion)

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

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

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

19. Gravitomagnetism 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

20. Gravitomagnetism Obtain a solution Celestial mechanics tools • Osculating ellipse (Keplerian elements) • Perturbation first-order analysis (Lagrange and Gauss equations) • Periodic effects • Secular effects ( t)

21. Gravitomagnetism Lagrange perturbation equations

22. Gravitomagnetism Gauss perturbation equations

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

24. Parameter estimation

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

26. Parameter estimation Procedure sketch

27. Models

28. Models 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

29. Models Gravitational • 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 Non-gravitational

30. Models Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

31. Models Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

32. Models The Earth is not a sphere! Spherical harmonics classification Spherical harmonics expansion

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

34. Models Some geopotential models

35. Models Geoid (EIGEN-GRACE02S) 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

36. Models Gravity anomalies (EIGEN-GRACE02S) The gravity anomalies are the difference between the real gravity field and that of a reference body (rotation ellipsoid)

37. Models The degree variance is useful when comparing various geopotential solutions Its behaviour is well described by the so-called Kaula’s rule A similar rule seems to be valid also for the Moon and the other terrestrial planets

38. Models EGM96 and Kaula’s rule Earth geopotential degree variance is well approximated by Kaula’s rule

39. Models Signal-to-noise ratio for EGM96 The signal-to-noise ratio indicates how well the signal is recognizable from the noise

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

41. Models Comparison between various geopotential models The various geopotential solutions differ strongly in the uncertainty associated to the harmonic coefficients

42. Models Direct solar radiation pressure It is due to reflection-diffusion-absorption of solar photons from the spacecraft surface • The strongest among the non-gravitational perturbations • Well modeled for LAGEOS (though the CR estimate could be biased due to some other not modeled signal)

43. Models Total SoLar Irradiance 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

44. Models Yarkovsky-Schach effect It is due to infrared radiation anisotropically emitted from the satellite (warmed by the Sun) • Effective on argument of perigee behaviour • Difficult modelization (the acceleration depends on S)

45. Models Spin evolution of LAGEOS Farinella–Vokhroulicky–Barliermodel

46. Models Spin evolution of LAGEOS II Farinella–Vokhroulicky–Barlier model

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

48. Models Pole motion IERS data (EOP 05 C04)

49. Models Length of Day variation IERS data (EOP 05 C04)

50. Data analysis (extracting physics)