1. Compatibility of the IERS earth rotation representation and its relation to the NRO conditions Athanasios Dermanis Department of Geodesy and Surveying The Aristotle University of Thessaloniki

2. Department of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Earth Rotation: Relation of Terrestrial to Celestial Reference System Celestial Reference System: Terrestrial Reference System: Mathematical model: = orthogonal rotation matrix = earth rotation parameters

3. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 To every orthogonal rotation matrix R(t) corresponds a unique rotation vector: defined by Notation: [a  ] is the antisymmetric matrix with axial vector a

4. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 R = QDW: Separation of earth rotation in 3 parts: D = Diurnal rotation Number of independent parameters needed:3 (geometric description) 6 (dynamic description – state vector) 9 parameters NRO conditions: Q = Precession-Nutation s = s (g,F) = s (x P,y P ) s = s(d,E) = s(X,Y) Classical model: IERS model (IAU 2000): OSU Report Nr. 245, 1977: W = Polar motion 5 parameters 7 parameters reduced to 5 by 2 NRO conditions

5. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Characteristics of the IERS earth rotation representation Q D W Precession Nutation Diurnal Rotation around Polar Motion from theory from observations R high frequency terms removed from precession-nutation CIP Consequences on model-compatible rotation vector Rotation vector not aligned to common 3 rd axis of intermediate systems Magnitude not equal to rate of diurnal rotation angle

6. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation

7. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation Involving 2 intermediate reference systems: Find a representation of the same separated form a the IERS representation Intermediate Celestial: Intermediate Terrestrial:

8. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation Involving 2 intermediate reference systems: Find a representation of the same separated form a the IERS representation Subject to the following (natural) compatibility conditions: Intermediate Celestial: Intermediate Terrestrial: 2 directional conditions: 1 magnitude condition:

9. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors

10. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors Relative rotation vectors:

11. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors of Intermediate Celestial with respect to Celestial Relative rotation vectors: Defined by:

12. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors of Intermediate Celestial with respect to Celestial Relative rotation vectors: of Intermediate Terrestrial with respect to Intermediate Celestial Defined by:

13. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors of Intermediate Celestial with respect to Celestial Relative rotation vectors: of Intermediate Terrestrial with respect to Intermediate Celestial of Terrestrial with respect to Intermediate Terrestrial Defined by:

14. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Decomposition of the rotation vector in 3 relative rotation vectors of Intermediate Celestial with respect to Celestial Relative rotation vectors: of Intermediate Terrestrial with respect to Intermediate Celestial of Terrestrial with respect to Intermediate Terrestrial Defined by:

15. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 In the Intermediate Celestial reference system The compatibility conditions

16. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 In the Intermediate Celestial reference system = Celestial Pole (direction of diurnal rotation), e.g. CEP, CIP = Compatible Celestial Pole (CCP) = Compatible rotation vector (derived from rotation matrix R) The compatibility conditions

17. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 7 parameters instead of 3 minimum required = 4 conditions needed ! The compatibility conditions

18. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 7 parameters instead of 3 minimum required = 4 conditions needed ! The compatibility conditions

19. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 2 direction conditions: 7 parameters instead of 3 minimum required = 4 conditions needed ! The compatibility conditions

20. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 2 direction conditions: 1 magnitude condition: 7 parameters instead of 3 minimum required = 4 conditions needed ! The compatibility conditions

21. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 2 direction conditions: 1 magnitude condition: Missing 4th condition: 7 parameters instead of 3 minimum required = 4 conditions needed !  s = arbitrary ! The compatibility conditions

22. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 2 direction conditions: 1 magnitude condition: Missing 4th condition: 7 parameters instead of 3 minimum required = 4 conditions needed !  s = arbitrary ! 4th condition = arbitrary definition of origin of diurnal rotation angle The compatibility conditions

23. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) :

24. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : 2 direction conditions:

25. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : 2 direction conditions: 1 magnitude condition:

26. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : The 4 independent compatibility conditions 2 direction conditions: 2 NRO conditions: 2 direction conditions: 1 magnitude condition:

27. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : The 4 independent compatibility conditions 2 direction conditions: 2 NRO conditions: 2 direction conditions: 1 magnitude condition:

28. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : The 4 independent compatibility conditions 2 direction conditions: 2 NRO conditions: 2 direction conditions: 1 magnitude condition:

29. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : The 4 independent compatibility conditions 2 direction conditions: 2 NRO conditions: 2 direction conditions: 1 magnitude condition:

30. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 The NRO conditions in relation to the compatibility conditions The 2 NRO (Non Rotating Origin) conditions: CEO (Celestial Ephemeris Origin) : TEO (Terrestrial Ephemeris Origin) : The 4 independent compatibility conditions 2 direction conditions: 2 NRO conditions: 2 direction conditions: 1 magnitude condition: magnitude condition satisfied !

31. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Explicit form of the 4 compatibility conditions

32. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Explicit form of the 4 compatibility conditions Direction conditions:

33. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Explicit form of the 4 compatibility conditions NRO conditions: Direction conditions:

34. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Explicit form of the 4 compatibility conditions NRO conditions: Direction conditions: Direction conditions + NRO conditions : When , E, d [and s(E,d)] are known then F, g [and s(F,g)] are uniquely determined !

35. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Construct a compatible separated model from observations only Analyze observations using a 3 parameter model:

36. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Construct a compatible separated model from observations only Analyze observations using a 3 parameter model: Compute rotation vector components, magnitude & directions (CCP components):

37. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Construct a compatible separated model from observations only Analyze observations using a 3 parameter model: Compute rotation vector components, magnitude & directions (CCP components): Compute precession-nutation and polar motion angles:

38. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Construct a compatible separated model from observations only Analyze observations using a 3 parameter model: Compute rotation vector components, magnitude & directions (CCP components): Compute precession-nutation and polar motion angles: Determine s and s from NRO conditions:

39. The Aristotle University of ThessalonikiDepartment of Geodesy and Surveying Athanasios DermanisJournées des Systèmes de Référence Spatio-Temporels – Warsaw 2005 Construct a compatible separated model from observations only Analyze observations using a 3 parameter model: Compute rotation vector components, magnitude & directions (CCP components): Compute precession-nutation and polar motion angles: Compute diurnal rotation angle: Determine s and s from NRO conditions: