1、 Recommendation ITU-R TF.2018(08/2012)Relativistic time transfer in the vicinity of the Earth and in the solar systemTF SeriesTime signals and frequency standards emissionsii Rec. ITU-R TF.2018 Foreword The role of the Radiocommunication Sector is to ensure the rational, equitable, efficient and eco
2、nomical use of the radio-frequency spectrum by all radiocommunication services, including satellite services, and carry out studies without limit of frequency range on the basis of which Recommendations are adopted. The regulatory and policy functions of the Radiocommunication Sector are performed b
3、y World and Regional Radiocommunication Conferences and Radiocommunication Assemblies supported by Study Groups. Policy on Intellectual Property Right (IPR) ITU-R policy on IPR is described in the Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of Resolution ITU-R 1. Forms to be u
4、sed for the submission of patent statements and licensing declarations by patent holders are available from http:/www.itu.int/ITU-R/go/patents/en where the Guidelines for Implementation of the Common Patent Policy for ITU-T/ITU-R/ISO/IEC and the ITU-R patent information database can also be found. S
5、eries of ITU-R Recommendations (Also available online at http:/www.itu.int/publ/R-REC/en) Series Title BO Satellite delivery BR Recording for production, archival and play-out; film for television BS Broadcasting service (sound) BT Broadcasting service (television) F Fixed service M Mobile, radiodet
6、ermination, amateur and related satellite services P Radiowave propagation RA Radio astronomy RS Remote sensing systems S Fixed-satellite service SA Space applications and meteorology SF Frequency sharing and coordination between fixed-satellite and fixed service systems SM Spectrum management SNG S
7、atellite news gathering TF Time signals and frequency standards emissions V Vocabulary and related subjects Note: This ITU-R Recommendation was approved in English under the procedure detailed in Resolution ITU-R 1. Electronic Publication Geneva, 2012 ITU 2012 All rights reserved. No part of this pu
8、blication may be reproduced, by any means whatsoever, without written permission of ITU. Rec. ITU-R TF.2018 1 RECOMMENDATION ITU-R TF.2018 Relativistic time transfer in the vicinity of the Earth and in the solar system (2012) Scope The purpose of this Recommendation is to establish common convention
9、al algorithms and procedures to be used in comparing clocks on the surface of the Earth and on platforms far from the Earth but within the solar system. These expressions are explicitly determined in the general relativity theory that is presently accepted to form the basis of space-time reference s
10、ystems. It is envisioned that these algorithms and procedures would be used for comparisons of clocks on Earth satellites, interplanetary spacecraft, and on the surfaces of solar system bodies. The ITU Radiocommunication Assembly, considering a) that it is desirable to maintain coordination of stand
11、ard time and frequency on platforms operating in the vicinity of the Earth and in the solar system; b) that accurate means of transferring time and frequency are required to meet the future needs of timekeeping, navigation, science, and communication systems in the vicinity of the Earth and in the s
12、olar system; c) that clocks are subject to path-dependent time and frequency variations due to their motion and to the gravitational potential in which they operate; d) that the conceptual foundation for the transfer of time and frequency should be clearly outlined; e) that procedures for the transf
13、er of time and frequency in the vicinity of the Earth and across celestial bodies and spacecraft in the solar system require the use of mathematical algorithms that account for relativistic effects; f) that requirements for precision and accuracy for the transfer of time and frequency in the vicinit
14、y of the Earth and in the solar system depend on the specific application, recommends that the mathematical algorithms that account for relativistic effects in the transfer of time and frequency as provided in Annex 1 should be used, as appropriate. 2 Rec. ITU-R TF.2018 Annex 1 Objective The purpose
15、 of this Recommendation is to raise the level of awareness for the need to address the effects of relativity in timekeeping, navigation, science, and communication systems. The Recommendation recalls the basic concepts and procedures to be applied in the analysis of such systems. No attempt is made
16、to describe the details of any particular system. Rather, it is intended that the information presented here may serve as a convenient reference and a point of departure for specific applications. One important application of this Recommendation is the comparison of the times registered by clocks on
17、 spacecraft in orbit around the Earth, in interplanetary space, and on planetary surfaces with the times recorded by clocks on Earth. An appropriate timescale for terrestrial measurements is Coordinated Universal Time (UTC). Therefore, the objective might be to relate the times registered by clocks,
18、 wherever they may be in the vicinity of the Earth and in the solar system, with the times of clocks on Earth that measure UTC. The following discussion is based on the IERS Conventions (2010), the ITU-R Handbook on Satellite Time and Frequency Transfer and Dissemination (2010), Nelson, Metrologia (
19、2011), and Petit and Wolf, Metrologia (2005). Users may consult those publications and references cited therein for further details. Relativistic framework The relativistic framework for space-time reference systems has been defined by Resolutions of international scientific organizations. The most
20、important are: 1) IAU Resolution A4 (1991) defines the Geocentric Celestial Reference System (GCRS) and the Barycentric Celestial Reference System (BCRS) and their time coordinates. IAU Resolution B1 (2000) further refines the BCRS definition. 2) IUGG Resolution 2 (2007) defines the Geocentric Terre
21、strial Reference System (GTRS), along with the International Terrestrial Reference System (ITRS). The nomenclature used in this document follows the one used in the past ITU-R Recommendations and may be related to the IAU/IUGG framework as follows: in this Recommendation the GCRS is termed the Earth
22、-Centered Inertial (ECI) coordinate system, the GTRS (in practice, the ITRS) is termed the Earth-Centered Earth-Fixed (ECEF) coordinate system, and the BCRS is termed the barycentric coordinate system. Definitions Proper time Proper time is the actual reading of a clock or the local time in the cloc
23、ks own frame of reference. Coordinate time Coordinate time t is the independent variable in the equations of motion of material bodies and in the equations of propagation of electromagnetic waves. It is a mathematical coordinate in the four-dimensional space-time of the coordinate system. For a give
24、n event, the coordinate time has the same value everywhere. Coordinate times are not measured; rather, they are computed from the proper times of clocks. Rec. ITU-R TF.2018 3 Space-time interval The relation between coordinate time and proper time depends on the clocks position and state of motion i
25、n its gravitational environment and is derived by integration of the space-time interval. In the comparison of the proper times of two clocks, the coordinate time is ultimately eliminated. Thus the relativistic transfer of time between clocks is independent of the coordinate system. The coordinate s
26、ystem may be chosen arbitrarily on the basis of convenience. In general, the space-time interval is described by: jiijjjvvdxdxgdxdtcgdtcgdxdxgds += 0220022 (1) where: g: components of the metric. A Greek index assumes the range 0, 1, 2, 3 and a Latin index assumes the range 1, 2, 3. A repeated index
27、 implies summation on that index. The metric depends upon the gravitational potentials and upon the angular velocity and linear acceleration of the reference frame. Upon a transformation of the coordinates, the space-time interval remains invariant. Thus the metric gtransforms as a second order cova
28、riant tensor. The general expression for the relationship between proper time and the coordinates of the chosen coordinate system, comprising the coordinate time x0 ct and the spatial coordinates xi, is given by: 220220022 =+= dcdxdxgdxdtcgdtcgdsjiijjj(2) where: : the proper time. Thus dt = d for a
29、clock at rest in an inertial frame of reference, for which dxi= 0 and g00= 1, g0 j= 0, and gi j= i j. The elapsed coordinate time corresponding to the measured proper time as registered by a clock along a path between points A and B is: +=BAjjBAjijiijdddxggcdddxddxggggcgt00000002001111(3) For an ele
30、ctromagnetic signal, the space-time interval is: 22200 020jijjijds g c dt g c dt dx g dx dx=+ + =(4) The speed of light is c in every inertial frame of reference. The elapsed coordinate time of propagation along a path between points A and B is: 00 000 000011 1BBij jij jijAAgg gt g dx dx dxcgcg= + +
31、(5) The expression i j gi j +g0 ig0 j / (g00) represents the metric of three-dimensional space and jiijdxdxd = represents the increment of three-dimensional distance. 4 Rec. ITU-R TF.2018 Time scales Atomic time scales The fundamental scale of time based on atomic clocks is International Atomic Time
32、 (TAI), which is calculated at the BIPM from a weighted average of the readings of atomic clocks in timing laboratories distributed around the world. It is a continuous reference timescale without steps. The atomic timescale for civil timekeeping is Coordinated Universal Time (UTC), which differs fr
33、om TAI by an integral number of seconds. In 2011, UTC = TAI 34 s. UTC is disseminated every month in BIPM Circular T in the form of the differences from individual laboratory realizations UTC(k). Coordinate time scales Geocentric Coordinate Time (TCG) is the coordinate time in a coordinate system wi
34、th origin at the Earths center (ECI or ECEF). Terrestrial Time (TT) is another coordinate time that is rescaled from TCG so that it has approximately the same rate as the proper time of a clock at rest on the geoid. The geoid is the surface of constant gravity potential, which is closely approximate
35、d by mean sea level. The relationship between TCG and TT is defined such that dTT/dTCG 1 LG, where LG 6.969 290 134 1010 60.2 s/d as discussed below following equation (18). The value of LGis a defined constant. Consequently, ()01TTTTLLTCGLTTTCGGGG=()()001TTTTLLTCGTCGLTTTCGGGG= (6) where: TCG0and TT
36、0: correspond to JD 2443144.5 TAI (1977 January 1, 0h). A practical realization of TT is TT = TAI + 32.184 s. (7) Barycentric Coordinate Time (TCB) is the coordinate time in a coordinate system with origin at the solar system barycenter. The coordinate time difference between TCB and TCG is a transf
37、ormation that depends on both time and position. To order 1 / c2() () ()ttcdtvUcTCGTCBttEEEext+=0R121r1222(8) where: R(t) = Exx: time-dependent position vector with respect to the geocenter x: barycentric position of the observer and xeand vedenote the barycentric position and velocity of the Earths
38、 center of mass. Rec. ITU-R TF.2018 5 This equation may be expressed in the form )(1)()()(200 EECcTCBPTCBPTCBTCBLTCGTCBxxv+= (9) where: LC= 1.480 826 867 41 108 1.28 ms/d. In this expression, P represents a series of periodic terms. The last term is diurnal at the surface of the Earth, with an ampli
39、tude smaller than 2.1 s. An alternative formulation to equation (9) is (IERS Conventions (2010), Chapter 10) )(11)()()(200EEBCcLTTPTTPTTTTLTCGTCBxxv+= (10) where: TT and LB 1.550 519 768 108 1.34 ms/d is the time argument. The value of LBis a defined constant. Periodic terms denoted by P(TT) have a
40、maximum amplitude of around 1.6 ms and can be evaluated by the “FB” analytical model (Fairhead and Bretagnon, 1990). Alternatively, P(TT) P(TT0) may be provided by a numerical time ephemeris such as TE405 (Irwin and Fukushima, 1999), which provides values with an accuracy of 0.1 ns from 1600 to 2200
41、. A series, HF2002, providing the value of LC(TT TT0) + P(TT) P(TT0) as a function of TT over the years 1600-2200 has been fit (Harada and Fukushima, 2003) to TE405. This fit differs from TE405 by less than 3 ns over the years 1600-2200 with an rms error of 0.5 ns. The difference between TCB and TT
42、is: +=+=RvEGBcPLTCBLTTTCGTCGTCBTTTCB21)1()( (11) The transformation from TCB to TCG consists of an average offset in rate dTCG/dTCB 1 LCand periodic terms. The transformation from TCG to TT is an exact offset in rate dTT/dTCG 1 LG. Thus the transformation from TCB to TT has an average offset in rate
43、. dTT/dTCB = (dTT/dTCG)dTCG/dTCB = (1 LG)(1 LC) (12) From the definition of LB , (1 LG)(1 LC) (1 LB), so that equation (12) may be expressed as dTT/dTCB = (1 LB) to within a few parts in 1018. Similarly to TT, Barycentric Dynamical Time (TDB) is another coordinate time in a barycentric system rescal
44、ed to have approximately the same rate as TT. The relationship between TCB and TDB is defined such that dTDB/dTCB 1 LB. Relativistic effects on clocks In the following discussion, the transformation between the proper time of an ideal clock (one that exactly realizes the SI second) and the coordinat
45、e time in the geocentric and barycentric coordinate systems is considered. 6 Rec. ITU-R TF.2018 Earth-Centered Inertial coordinate system The coordinate time associated with an Earth-Centered Inertial (ECI) coordinate system is Geocentric Coordinate Time (TCG). Through terms of order 1 / c2, the com
46、ponents of the metric tensor in this coordinate system are g00= 1 2 U / c2, g0 j= 0, and gi j= (1 + 2 U / c2) i j, where U is the gravitational potential. The elapsed TCG in the ECI coordinate system corresponding to the elapsed proper time as registered by a clock moving along a path between points
47、 A and B with velocity v is given by: +=dcUctBA222v12111 (13) The Earths potential at radial distance r, geocentric latitude , and longitude may be expressed as an expansion in spherical harmonics as: ()+=20sincos)(sin1),(nnmnmnmnmnEmSmCPrRrGMrU +=mKmJPrRPrRJrGMmnmnnnmnmnEnnEnnsincos)(sin)(sin1212(1
48、4) where: GM: the gravitational constant of the Earth RE: the equatorial radius of the Earth The factors Pn(sin ): the Legendre polynomials of degree n the factors Pnm(sin ): the associated Legendre functions of degree n and order m. The geocentric latitude is related to the geographic latitude by tan = (1 f 2) tan , where f is the flattening. For practical applications it may be sufficient to include only the first oblateness correction and approximate the gravitational potential as )sin31()(sin222222+=rRJrGMPrRJrGMUEE(15) 1) Clock at rest on the geoid For a clock at rest on the surface of
