1. Construction of the Non-Rigid Earth Rotation Series
V.V. Pashkevich
Central (Pulkovo) Astronomical Observatory
of the Russian Academy of Science
St.Petersburg
Space Research Centre of the Polish Academy of Sciences
Warszawa
2007

2. The last years a lot of attempts to derive a high-precision theory of the non-rigid Earth rotation was carried out. For these purposes used the different transfer functions, which usually applied to the nutation in longitude and in obliquity of the rigid Earth rotation with respect to the ecliptic of date.
Aim:
Construction of a new high-precision non-rigid Earth rotation series (SN9000), dynamically adequate to the DE404/LE404 ephemeris over 2000 years, which is expressed as a function of the three Euler angles with respect to the fixed ecliptic plane and equinox J

3. S T E P S:
The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.

4. S T E P - 1:
Numerical
integration of
the differential
equations
Discrepancies:
numerical
solution minus
SMART97
Initial conditions
from SMART97

5. Problem expressed by the Rodrigues – Hamilton parameters:
S T E P - 1:
Numerical
integration of
the differential
equations
Discrepancies:
numerical
solution minus
SMART97
Initial conditions
from SMART97
LAGRANGE DIFFERENTIAL EQUATIONS OF THE SECOND KIND:
VECTOR OF THE GEODETIC ROTATION:
Problem expressed by the Rodrigues – Hamilton parameters:

6. Fig.1 Numerical solution for Rigid Earth rotation minus solution SMART97 in the longitude of the ascending node of the Earth equator.
Kinematical (relativistic) case Dynamical (Newtonian) case
Secular terms of… Secular terms of…
smart97 (as)
d (as)
smart97(as)
7.00
6.89 T T T T T2 T2 T2 T2 T3 T3 T3 T3 T4 T4 T4 T4 T5 T5 T5 T5 T6 T6

7. Fig.2 Numerical solution for Rigid Earth rotation minus solution SMART97 in the angle of the proper rotation of the Earth.
Kinematical (relativistic) case Dynamical (Newtonian) case
Secular terms of… Secular terms of…
smart97 (as)
d(as)
d (as)
6.58
6.53 T T T T2 T2 T2 T2 T3 T3 T3 T3 T4 T4 T4 T4 T5 T5 T5 T5 T6 T6

8. Fig.3 Numerical solution for Rigid Earth rotation minus solution SMART97 in the inclination angle.
Kinematical (relativistic) case Dynamical (Newtonian) case
Secular terms of… Secular terms of…
 smart97(as)
d(as)
1.42
1.39 T T T T T2 T2 T2 T2 T3 T3 T3 T3 T4 T4 T4 T4 T5 T5 T5 T5 T6 T6

9. S T E P S:
The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.

10. S T E P S:
The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.
Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ).

11. S T E P - 1:
Numerical
integration of
the differential
equations
Discrepancies:
numerical
solution minus
SMART97
Initial conditions
from SMART97
S T E P - 2

12. S T E P S - 1 and 2:
Numerical
integration of
the differential
equations
Discrepancies:
numerical
solution minus
SMART97
Initial conditions
from SMART97
Precession and
GMST terms of
SMART97
Calculation of the
secular terms
by the LSQ method
Set of nutation
terms of
SMART97
Computation of the
new precession and
GMST parameters
Removal of the secular
trends from the
discreapancies
Calculation of the
periodical terms
by the SA method
High-precision
series S9000
Construction of the
new nutation series

13. Fig.4. The numerical solution of the rigid Earth rotation minus S9000 after formal removal of the secular trends in the proper rotation angle.
Kinematical case Dynamical case

14. Fig.5. Differences between S9000 and SMART97.
Kinematical case Dynamical case

15. S T E P S:
The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.
Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ).

16. S T E P S:
The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.
Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ).
The new high-precision non-rigid Earth rotation series (SN9000), which expressed in the function of the three Euler angles, is constructed by using the method (P.Bretagnon, P.M.Mathews, J.-L.Simon: 1999) and the transfer function (Mathews, P. M., Herring, T. A., and Buffett B. A., 2002).

17. Expressions for Euler angles:
S T E P - 3:
Expressions for Euler angles:

18. Classical A l g o r i t h m

19. Classical A l g o r i t h m
Expressions for the non-rigid Earth nutations in longitude and obliquity:

20. TRANSFER FUNCTIONS
J.M.Wahr ,1981; V. Dehant and P. Defraigne, 1997:
T. Shirai and T. Fukushima, 2001:

21. TRANSFER FUNCTIONS
P.Bretagnon, P.M. Mathews, J.-L. Simon,1999:
P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:

22. P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:
TRANSFER FUNCTIONS
P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:
Geophisical model includes :
The effects Electromagnetic coupling
The Ocean effects
Mantle inelasticity effects
Atmospheric effects
Change in the global Earth dynamical flattening and in the core flattening

23. Algorithm of Bretagnon et al. 1999
1.The rigid Earth angular velocity vector:

24. Expressions for the Earth’s angular velocity vector:
The coefficients of the developments of p, q in case of the rigid Earth:

25. The components of the Rigid Earth angular velocity vector:

26. The components of the coefficients of the developments of p, q
in case of the rigid Earth:

27. The amplitudes of the prograde and retrograde components:
Complex transformation of the componets of the angular velocity vector:

28. Algorithm of Bretagnon et al. 1999
1.The rigid Earth angular velocity vector:
2.The non-rigid Earth angular velocity vector is obtained by:

29. The components of the coefficients of the developments of p, q
in case of the non-rigid Earth:

30. The prograde and retrograde components of the coefficients
of the developments of p in case of the non-rigid Earth:

31. The prograde and retrograde components of the coefficients
of the developments of q in case of the non-rigid Earth:

32. The sin coefficients of the developments
for the non-rigid Earth angular velocity vector:

33. The cos coefficients of the developments
for the non-rigid Earth angular velocity vector:

34. Algorithm of Bretagnon et al. 1999
1.The rigid Earth angular velocity vector:
2.The non-rigid Earth angular velocity vector is obtained by:
3.The derivatives of Euler angles for the non-rigid Earth rotation:

35. Common form of the periodic part of Euler angles:
Cascade method:

36. Iterative solution of the algorithm of Bretagnon et al. 1999
Details:
Iterative solution of the algorithm of Bretagnon et al. 1999
1 iteration

37. Iterative solution of the algorithm of Bretagnon et al. 1999
Details:
Iterative solution of the algorithm of Bretagnon et al. 1999
n iteration
Iterations are repeated until the absolute value of the difference
between iterations K-1 and K exceedes some DEFINITE values 

38. DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999;
Results:
Table 1. Comparison of different solutions Argument (λ3 +D-F ) with a 18.6 year period
Solution
Ψ(sin)
μas
Ψ(cos)
θ (sin)
θ (cos)
φ (sin)
φ (cos)
S9000+Wahr 1981
-431 35
-395
S9000+DD
-396
S9000+SF
-2701
-1269
-2479
S9000+BMS
-3351
-1486
-3075
S9000+MHB
-3386
-1505
-3106
DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999;
SF-Shirai and Fukushima, 2001; MHB- Mathews et al., 2002.

39. DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999;
Results:
Table 1. Comparison of different solutions Argument (λ3 +D-F ) with a 18.6 year period
Solution
Ψ(sin)
μas
Ψ(cos)
θ (sin)
θ (cos)
φ (sin)
φ (cos)
SMART97+Wahr 1981
-431 35
-395
SMART97+DD
-396
SMART97+SF
-2701
-1269
-2479
SMART97+BMS
-3351
-1486
-3075
SMART97+MHB
-3386
-1505
-3106
DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999;
SF-Shirai and Fukushima, 2001; MHB- Mathews et al., 2002.

40. Table 2. Comparison of different solutions For some arguments
Argu-ment, Period (days)
Ψ(sin)
μas
Ψ(cos)
θ (sin)
θ (cos)
φ (sin)
φ (cos)
SMN
SN9000
λ3 +D-F
2λ3
182.62
2λ3+2D
13.66
2λ3 +2D-2F λ3
365.24
3λ3
121.75
SMN=SMART97+MHB; SN9000=S9000+MHB;
MHB- Mathews et al., 2002.

41. Fig.6. The differences between SN9000 and SMN after removal of the secular terms. (non-rigid Earth rotation)
Dynamical case
μas Δφ Δθ Δψ
YEARS

42. Fig.7. The differences between S9000 and SMART after removal of the secular terms. (rigid Earth rotation)
Dynamical case
μas Δφ Δθ Δψ
YEARS

43. Fig.8. The differences between SN9000 and SMN after removal of the secular terms. (non-rigid Earth rotation)
Kinematical case
μas Δφ Δθ Δψ
YEARS

44. Fig.9. The differences between S9000 and SMART after removal of the secular terms. (rigid Earth rotation)
Kinematical case
μas Δφ Δθ Δψ
YEARS

45. Fig.10. The differences between S9000 and SN9000.
Dynamical case
μas Δφ Δθ Δψ
YEARS

46. Fig.11. The differences between S9000 and SN9000.
Kinematical case
μas Δφ Δθ Δψ
YEARS

47. Fig.12. The differences between SMART97 and SMN.
Dynamical case
μas Δφ Δθ Δψ
YEARS

48. Fig.13. The differences between SMART97 and SMN.
Kinematical case
μas Δφ Δθ Δψ
YEARS

49. Fig.14. The discrepancies between S9000 and SN9000 minus discrepancies between SMART97 and SMN.
Dynamical case
μas Δφ Δθ Δψ
YEARS

50. Fig.15. The discrepancies between S9000 and SN9000 minus discrepancies between SMART97 and SMN.
Kinematical case
μas Δφ Δθ Δψ
YEARS

51. Kinematical solution of the rigid Earth rotation= Dynamical solution of the rigid Earth rotation + Geodetics corrections Dynamical solution of the non-rigid Earth rotation= Dynamical solution of the rigid Earth rotation + TRANSFER FUNCTIONS Original Kinematical solution of the non-rigid Earth rotation= Dynamical solution of the rigid Earth rotation + TRANSFER FUNCTIONS + Geodetics corrections Kinematical solution of the non-rigid Earth rotation= Kinematical solution of the rigid Earth rotation + TRANSFER FUNCTIONS

52. CONCLUSION
The exact expressions for the algorithm of Bretagnon et al. (1999) are obtained.
The new semi-analytical solution of the non-rigid Earth rotation SMN (SMART97+MHB2002) is derived.
The high-precision non-rigid Earth rotation series SN9000, which is expressed as a function of the three Euler angles and is dynamically adequate to the ephemerides DE404/LE404 over 2000 years, has been constructed.

53. R E F E R E N C E S
P.Bretagnon and G.Francou. Planetary theories in rectangular and spherical variables // Astronomy and Astrophys., 202, 1988, pp. 309–-315.
P.Bretagnon, G.Francou, P.Rocher, J.L.Simon. SMART97: A new solution for the rotation of the rigid Earth // Astron. Astrophys. , 1998, 329, pp
P.Bretagnon,P.M.Mathews,J.-L.Simon. Non Rigid Earth Rotation // in Proc. Journees 1999: Motion of Celestial Bodies, Astrometry and Astronomical Reference Frames Les Journees 1999 \& IX. Lohrmann - Kolloquium, (Dresden, September 1999), p
V.A..Brumberg, P.Bretagnon. Kinematical Relativistic Corrections for Earth’s Rotation Parameters // in Proc. of IAU Colloquium 180, eds. K.Johnston, D. McCarthy, B. Luzum and G. Kaplan, U.S. Naval Observatory, 2000, pp. 293–302.
V.Dehant and P.Defraigne. New transfer functions for nutations of a non-rigid Earth // J. Geophys. Res., 1997,102, pp
Mathews, P. M., Herring, T. A., and Buffett B. A.. Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's Interior // J. Geophys. Res., 2002, 107, B4, /2001JB
V.V.Pashkevich, G.I.Eroshkin and A. Brzezinski. Numerical analysis of the rigid Earth rotation with the quadruple precision // Artificial Satellites, Vol. 39, No. 4, Warszawa, 2004, pp. 291–304.
V. V. Pashkevich and G. I. Eroshkin. Spectral analysis of the numerical theory of the rigid Earth rotation // in Proc. of “Journees 2004”, Fundamental Astronomy: New concepts and models for high accuracy observations» (Observatoire de Paris, September 2004.), pp
V.V.Pashkevich and G.I.Eroshkin. Choice of the optimal spectral analysis scheme for the investigation of the Earth rotation problem // in Proc. of “Journees 2005”, Earth dynamics and reference systems: five years after the adoption of the IAU 2000 Resolutions (Space Research Centre of Polish Academy of Sciences, Warsaw, Poland, September 2005), pp
V.V.Pashkevich and G.I.Eroshkin. Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation // Artificial Satellites, Vol. 40, No. 4, Warszawa, 2005, pp. 251–260.
T.Shirai and T.Fukushima. Construction of a new forced nutation theory of the nonrigid Earth // The Astron. Journal, 121, 2001, pp
J.M.Wahr. The forced nutationsof an elliptical, rotating, elastic and oceanless Earth // Geophys. J. R. Astron. Soc., 1981, 64, pp

54. A C K N O W L E D G M E N T S
The investigation was carried out at the
Central (Pulkovo) Astronomical
Observatory of the Russian Academy
of Sciences
and the
Space Research Centre
of the Polish Academy of Sciences,
under a financial support of the
agreement cooperation between
the Polish and Russian Academies
of Sciences, Theme No 31.