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Coordinate Times and Time Transformations Comparision of “Old” and “New” Concepts: Coordinate Times and Time Transformations Gerard Petit Bureau International des Poids et Mesures, Pavillon de Breteuil 92312 Sevres Cedex France, e-mail: [email protected] Foreword: The first three parts of this paper are largely based on the paper “The new IAU’2000 Conventions for coordinate times and time transformations” presented at the Journ´ees Syst`emes de R´ef´erenceSpatio-temporels, Brussels, September 2001. The fourth part is largely based on draft parts of the IERS Conventions 2000 (http://maia.usno.navy.mil/conv2000.html), interpreted here under the sole responsibility of the author. Abstract: At its 2000 General Assembly, the International As- tronomical Union has adopted a set of Resolutions that provide a consistent framework for defining the barycentric and geocentric celestial reference systems at the first post-Newtonian level. This extends and completes the IAU’1991 framework defined in Resolution A4 at the 1991 General Assembly. This paper describes in some detail the two Resolutions that define time coordinates and allow to realize time transformations. The application of the IAU’1991 and IAU’2000 framework to several fields of space geodesy and astrometry is discussed. 1 Introduction At its 1991 General Assembly the International Astronomical Union (IAU) ex- plicitly adopted the general theory of relativity as the theoretical framework for the definition and realization of space-time reference frames (IAU, 1991). Barycentric and geocentric coordinate time scales and the relativistic transfor- mations between them were defined, together with procedures for their realiza- tion. In section 2, we recall the content of the IAU 1991 resolution A4 dealing with the definition of reference systems, time coordinates and time transfor- mations, and we expose some of the limitations of this framework. For many applications in space geodesy, it is sufficient to discuss within the IAU’1991 framework, with the IAU’2000 framework providing an extension which is nec- essary for a few applications. It is therefore important to have in mind the basis of the IAU’1991 framework. The IAU Working Group on Relativity in Celestial Mechanics and Astrometry (RCMA), since 1994, and the BIPM/IAU Joint Committee on relativity for space-time reference systems and metrology (JCR), from 1997 to 2001, have worked to provide an extension of the IAU’1991 framework at the first post- Newtonian level (Soffel, 2000; Petit, 2000). This work resulted in a set of Resolutions passed at the IAU 2000 General Assembly. The complete text of the Resolutions may be found in IAU publications (http://danof.obspm.fr/IAU-resolutions/Resol-UAI.htm) and a complete explanatory supplement may be found in (Soffel et al., 2002). The new IAU’2000 framework is precisely specified in Resolutions B1.3(2000) ”Definition of barycen- IERS Technical Note No. 29 19 Comparison of “Old” and “New” Concepts: Time Session 4.1 tric celestial reference system and geocentric celestial reference system” and B1.4(2000) ”Post-Newtonian potential coefficients”. In Section 3 of this paper, we briefly present the two Resolutions which deal with time transformations and the definition of coordinate times, namely Res- olutions B1.5(2000) ”Extended relativistic framework for time transformations and realization of coordinate times in the solar system” in section 3.1 and B1.9(2000) ”Re-definition of Terrestrial Time TT” in section 3.2, and discuss at which level time transformations and coordinate times are affected by the IAU’2000 framework. Finally section 4 discusses the application of the IAU’1991 and IAU’2000 frame- work to several fields of space geodesy and astrometry which are based on time and frequency measurements. 2 The IAU’1991 framework and its limitations The IAU resolution A4 (1991) contains nine recommendations, the first five of which are directly relevant to our discussion. In the first recommendation, the metric tensor for space-time coordinate sys- tems (t, x) centered at the barycenter of an ensemble of masses is recommended in the form 2U(t, x) g = −1 + + O(c−4), 00 c2 −3 g0i = O(c ), (1) 1 + 2U(t, x) g = δ + O(c−4). ij ij c2 where c is the speed of light in vacuum (c = 299792458 m/s), U is the sum of the gravitational potentials of the ensemble of masses and of a tidal potential generated by bodies external to the ensemble, the latter potential vanishing at the barycenter. The algebraic sign of U is taken to be positive. This recommen- dation recognizes that space-time cannot be described by a single coordinate system. The recommended form of the metric tensor can be used not only to describe the barycentric reference system (BRS) of the whole solar system (which is called BCRS where C stands for Celestial since the IAU’2000 Reso- lutions), but also to define the geocentric reference system (GRS) centered in the center of mass of the Earth, which is now called GCRS. In analogy to the GRS, a corresponding reference system may be defined for any other body of the Solar system. In the second recommendation, the origin and orientation of the space coor- dinate grids for the solar system (BRS) and for the Earth (GRS) are defined. Notably it is specified that the space coordinate grids of these systems should show no global rotation with respect to a set of distant extragalactic objects. It also specifies that the SI (International System of units) second and the SI meter should be the physical units of proper time and proper length in all coordinate systems. It states in addition that the time coordinates should be derived from an Earth atomic time scale. The third recommendation defines TCB (Barycentric Coordinate Time) and TCG (Geocentric Coordinate Time) – the time coordinates of the BRS and GRS, respectively. The recommendation also defines the origin of the times scales (their reading on 1977 January 1, 0h 0m 0s T AI (JD = 2443144.5 T AI) must be 1977 January 1, 0h 0m 32.184s) and declares that the units of measure- ments of the coordinate times of all reference systems must be coincide with 20 IERS Technical Note No. 29 Comparison of “Old” and “New” Concepts: Time Session 4.1 the SI second and SI meter. The relationship between TCB and TCG is given by a full 4-dimensional transformation Z t 2 −2 vE i i −4 TCB − TCG = c + Uext(t, xE(t)) dt + vErE + O(c ), (2) t0 2 i i where xE and vE are the barycentric coordinate position and velocity of the i i i i geocenter, rE = x − xE with x the barycentric position of the observer, and Uext(t, xE(t)) is the Newtonian potential of all solar system bodies apart from the Earth evaluated at the geocenter. In the fourth recommendation another time coordinate, Terrestrial Time (TT ), is defined for the GRS. It differs from TCG by a constant rate only −10 TCG−TT = LG ×(JD−2443144.5)×86400,LG ≈ 6.969291×10 , (3) so that the unit of measurement of TT agrees with the SI second on the geoid. TT represents an ideal form of T AI, the divergence between them being a consequence of the physical defects of atomic clocks. The fifth recommendation states that the former dynamical barycentric time T DB may still be used where discontinuity with previous work is deemed to be undesirable. Limitations to the IAU’1991 framework Because of the form of the metric (1) in the IAU’1991 framework, time trans- formations and the realization of coordinate times in the barycentric system are not specified at the c−4 level, i.e. at a level of a few parts in 1016 in rate. The new IAU’2000 framework allows to remove this limitation. Nevertheless, within the IAU’1991 approximation, constants LB and LC were introduced in notes to the Recommendation 3 (1991) to express the mean rates between time scales as TCB − T DB = LB × (JD − 2443144.5) × 86400, −8 LB ≈ 1.550505 × 10 , and i i 2 TCB − TCG = LC × (JD − 2443144.5) × 86400 + vErE/c + P, −8 LC ≈ 1.480813 × 10 , (4) where P represents periodic terms. Since JD is not specified to be a particular time scale, these constants were not properly defined so that confusion appeared in their usage. This point was adressed in Resolution B1.5(2000) by defining < T CG/T CB >= 1 − LC and < T T/T CB >= 1 − LB, where <> means a sufficiently long term average taken at the geocenter. The actual computation of LC and LB requires the integration of solar system ephemerides and the specification of an averaging duration, and this process may be applied to the utmost accuracy, after a choice of ephemerides and averaging duration. For −8 example Irwin and Fukushima (1999) determined LC = 1.48082686741×10 ± 2 × 10−17. However, no completely unambiguous definition may be provided for LB and LC because they always depend on the ephemerides and time span used for their computation. Therefore the use of these constants is not advised to formulate time transformations when it would require knowing their value with an uncertainty of order 1 × 10−16 or less. Another problem arising from the situation before IAU’1991 is that the bary- centric dynamical time TDB did not have a good definition. This could have been corrected by turning a specific value of LB into a defining constant thus providing in retrospect a definition of TDB. It was not felt necessary to ad- dress explicitly this point in a recommendation. Prior to IAU’1991, authors IERS Technical Note No. 29 21 Comparison of “Old” and “New” Concepts: Time Session 4.1 had developped analytical formulas to transform TT (known as TDT prior to 1991) to TDB (e.g.
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