1. 1 Towards the relativistic theory of precession and nutation _ S.A. Klioner Lohrmann Observatory, Dresden Technical University Problems of Modern Astrometry, Moscow, 25 October 2007

2. 2 Relativity and Earth rotation: why to bother? Earth rotation is the only astronomical phenomenon - which is observed with a high accuracy and - which has no consistent relativistic model Modern theories of precession/nutation (IAU2000) are based on purely Newtonian theories with geodetic precession and nutation added in an inconsistent way Modern theories of rigid Earth nutation are intended to attain formal accuracy of 1  as

3. 3 Relativity and Earth rotation: one more reason The main relativistic effects are significantly larger: 0.019  per year from geodetic precession (3  10 -4 of general precession) 200  as from geodetic nutation 1–50  as from relativistic torques (different authors give different estimates) Geodynamical observations give important tests of General Relativity The best estimates of the PPN  using large range of angular distances from the Sun comes from geodetic VLBI data:(Eubanks, et al. 1998)

4. 4 Main goal of the project Derivation of a new consistent and improved precession/nutation series for a rigidly rotating multipole model of the Earth in the post-Newtonian approximation of general relativity using post-Newtonian definitions of : - potential coefficients - moment of inertia tensor dynamical equations in the GCRS correct relativistic time scales rigorous treatment of the geodetic precession and nutation

5. 5 Relativistic post-Newtonian theory The Earth rotates, but relative to what?

6. 6 Post-Newtonian theory: kinematics Two answer are possible in relativity: the stellar compass the inertial compass

7. 7 In Newtonian physics this are only two ways to get to the same absolute space: M. C. Escher Cubic space division, 1952

8. 8 Post-Newtonian theory: orientation of the BCRS It is assumed that our global reference system, BCRS, does not rotate with respect to remote quasars. This is a cosmological assumption to be verified by dynamical observations This is also an idealization: source structure is not negligible

9. 9 IAU (1991): … the orientation of the reference systems be chosen so that it shows no rotation with respect to distant celestial objects… No rotational matrix in the transformation between BCRS and GCRS Post-Newtonian theory: orientation of the GCRS The GCRS is kinematically non-rotating with respect to the BCRS

10. 10 Post-Newtonian theory: metric of the GCRS The GCRS is kinematically non-rotating with respect to the BCRS produces Coriolis forces: relativistic precession

11. 11 Post-Newtonian theory: relativistic precession The GCRS is kinematically non-rotating with respect to the BCRS produces Coriolis forces: relativistic precession geodetic precession Lense-Thirring precession Thomas precession

12. 12 Relativistic precession: experimental status Gravity Probe B the longest lasting experiment in modern history (1959-2008?) launched 20 April 2004 6.6 ’’/yr 42 mas/yr Frame dragging Geodetic prec. L (orbital) J (Earth) IM Peg gyro Francis Everitt No results yet…

13. 13 Relativistic precession: experimental status LLR: geodetic precession <1% (Newhall et al., 1996; … ) SLR: Lense-Thirring precession 2-10% ??? (Ciufolini, Pavlis, 2004) VLBI & Earth rotation: geodetic precession 30% (Krasinsky, 2006)

14. 14 Post-Newtonian equations of rotational motion - Post-Newtonian equations of rotational motion in the GCRS (Voinov, 1988; Damour, Soffel, Xu, 1993) - the angular momentum of the body reads is the energy-momentum tensor

15. 15 Post-Newtonian equations of rotational motion - Post-Newtonian equations of rotational motion in the GCRS (Voinov, 1988; Damour, Soffel, Xu, 1993) - the torque - M L and S L are the Blanchet-Damour multipole moments of the body - simplification of the torque can be neglected - G L and H L for l>2 are the gravitoelectric and gravitomagnetic tidal moments

16. 16 Further simplifications of the torque - Mass dipole M a vanishes in the GCRS - S L for l>1 can be neglected at the level of 0.1  as

17. 17 Further simplifications of the torque - All external bodies are assumed to be mass monopoles Newtonian term

18. 18 Post-Newtonian equations of rotational motion - Post-Newtonian equations of rotational motion in the GCRS (Voinov, 1988; Damour, Soffel, Xu, 1993) - The gravitomagnetic tidal moment H a defines the Coriolis forces: geodetic precession Lense-Thirring precession Thomas precession

19. 19 Post-Newtonian angular velocity? We have spin S a … What is about angular velocity???

20. 20 Newtonian physics: non-rigid body One introduces some rigidly rotating reference system 1. Principal axes of inertia one borrows the definition of the tensor of inertia from the rigid-body theory and the matrix is selected in such a way that the body can be considered “at rest on the average” in the rotating coordinates. condition defines the angular velocity and matrix

21. 21 Newtonian physics: non-rigid body 2. Tisserand axes “angular momentum relative to the rotating coordinates” (no immediate physical meaning the same functional form as the conserved spin in inertial coordinates) condition defines the Tisserand axes and the corresponding taking the angular velocity corresponding to that one has

22. 22 Post-Newtonian Tisserand axes Both - restricted rigid body models of Thorne & Gürsel, 1983 - post-Newtonian Tisserand axes of Klioner, 1996 give the same equation: - the post-Newtonian tensor of inertia has a definition as an integral over the volume of the body - but that definition plays no role for practical work: exactly as multipole moments of the Earth, the numerical value of the tensor of inertia is determined from observations

23. 23 Rigidly rotating multipoles Klioner, Soffel, Xu, Wu, 2001 (based on many previous results): - Post-Newtonian equations of rotational motion in the GCRS - Rigidly rotating multipoles: several assumptions on the multipole moments and the tensor of inertia

24. 24 STF approach to compute the torque 1. For any STF tensor: 2l +1 real numbers STF basis 2. For the multipole moments of the Earth: equivalent to the 2l +1 harmonic coefficients 3. For the tidal moments: functions of the ephemeris data

25. 25 STF approach to compute the torque 2. For the multipole moments of the Earth: equivalent to the 2l +1 harmonic coefficients with C lm and S lm defined from

26. 26 STF approach to compute the torque 4. The torque: numbers 5. This is equivalent to the classical formulation with Legendre polynomials for Newtonian tidal potentials

27. 27 STF approach to compute the torque 4. The torque:

28. 28 Legendre polynomials vs. STF tensors for the Newtonian torque  as years from J2000

29. 29 Numerical integration Fortran 95 code, about 15000 lines careful coding to avoid numerical instabilities and excessive round-off errors two numerical integrators: ODEX and ABM with dense output automatic accuracy check: forth and back integrations tuneable arithmetic: 64 bit, 80 bit, 128 bit (availability is hardware- and compiler-dependent) FMlib: arbitrary-precision arithmetic for precision-critical operations STF code was automatically generated by Mathematica baseline: 80 bit on Intel architecture gives errors <0.001  as for 150 years

30. 30 Numerical integration A number of relativistic issues are treated for the first time fully post-Newtonian torques with STF machinery rigorous treatment of geodetic precession/nutation rigorous treatment of the time scales (TT vs. TDB) correct relativistic scaling of constants and parameters (GM compatible with TT is NOT compatible with TDB!) Performance of the code: Newtonian torque with Legendre polynomials:4.0 sec per year Newtonian torque with STF tensors:3.3 sec per year Newtonian and geodetic torques:3.5 sec per year post-Newtonian torque:8.2 sec per year

31. 31 Definition of the Euler angles Two main reference systems: xyz in the BCRS is defined by the ecliptic and the equinox J2000:  - terrestrial system:

32. 32 Newtonian test bed: SMART97 Bretagnon, Rocher, Simon, Francou, 1996-1997 analytical theory of precession/nutation for rigid Earth detailed comparisons with numerical integrations Published series of SMART97: formal accuracy of 2  as Jean Louis Simon has provided the authors with the unpublished full version of SMART97: about 70000 Poisson terms for each of the three angles formal accuracy 0.01  as

33. 33 Differences between our integration and SMART97: 50 days years from J2000  as

34. 34 Differences between our integration and SMART97: 55 years years from J2000  as

35. 35 Differences between our integration and SMART97: 150 years years from J2000  as

36. 36 Comparison with Bretagnon’s results JD Bretagnon, Francou, Rocher, Simon, 1997: SMART97-numerical integration

37. 37 Effects of the post-Newtonian torque  as years from J2000

38. 38 Effects of the post-Newtonian torque  as years from J2000

39. 39 Effects of the post-Newtonian torque  as years from J2000

40. 40 Geodetic precession as a Coriolis torque 1.Geodetic precession is added to the torque: 2.Initial conditions of the numerical integration are changed to correct for the geodetic precession/nutation 3.The results are subtracted from the numerical solution corresponding to the dynamical version of SMART97 4. The published analytical solution for the geodetic precession/nutation was used to subtract the “known geodetic precession/nutation” from the result: 72 terms for  102 terms for  61 terms for  The results show the errors in our current models for geodetical p/n

41. 41 Geodetic precession as a Coriolis torque  as years from J2000

42. 42 Geodetic precession as a Coriolis torque  as years from J2000 detrended

43. 43 Geodetic precession as a Coriolis torque  as Known analytical solutions for geodetic precession/nutation are only valid if the kinematical and dynamical angles are defined with respect to the same plane in the BCRS. This is however not true in practice and  gets basically the same geodetic precession/nutation terms as . The approximate integral still holds:

44. 44 Geodetic precession as a Coriolis torque  as years from J2000 It is NOT sufficient just to add the geodetic precession/nutation: The frequencies of the Newtonian solution must be also corrected for geodetic precession/nutation!

45. 45 Next steps Clarify the meaning of the initial conditions in the relativistic context: a mixture of dynamically and kinematically non-rotating GCRS is used in SMART97 and IAU2000 The clarification of the initial conditions implies clear relativistic definitions of all involved planes and points, e.g. a clear relativistic definition of ecliptic or “image of the ecliptic” in the GCRS Relativistically meaningful values of the constants (related to the first two): the moments of inertia A, B, C the longitude of the principal axes of inertia in the terrestrial system

46. 46 Non-rigid Earth Violate the assumptions of the model of rigidly rotating multipoles the one by one and check what is important No way to proceed with the idea of “transfer functions” in relativity Generalize the approach of “extended SOS theory” into the post-Newtonian approximation of general relativity (e.g. Krasinsky, 2006; Getino, et al. 1991 -)

47. 47 Backup slides

48. 48 Newtonian physics: rigid body Arbitrary motion of a rigid body with a fixed point is a rotation around that fixed point described by an orthogonal matrix P ij (t). Consequences: Velocity distribution within a rigid body: A reference system where the body is at rest:

49. 49 Newtonian physics: rigid body Total angular momentum (spin) of the body is defined as Physical merit of this definition: for an isolated body From the distribution of the velocity inside the body is the Newtonian tensor of inertia

50. 50 Newtonian theory: equations of rotational motion Newtonian equations of rotational motion read L is a multi-index: are the multipole moments of the gravitational field of the body are the moments of the external gravitational potential at the “geocenter”

51. 51 Relativistic formulation - Post-Newtonian torque from “geodetic” precession - Post-Newtonian tidal moments from point masses machinery of STF tensors in the Newtonian limit equivalent to the Legendre polynomials

52. 52 Relativistic formulation - Post-Newtonian tidal moments from point masses (some notations)

53. 53 Why  and  are so similar? For an axially symmetric “Earth” one has: years from J2000 For triaxial Earth:  as per day

54. 54 Why  and  are so similar? Numerical integration and SMART give very similar years from J2000  as per day

55. 55 Why  and  are so similar? Numerical integration and SMART give very similar  

56. 56 Why  and  are so similar? years from J2000 detrended  as