1、Rec. ITU-R TF.1010-1 23 RECOMMENDATION ITU-R TF.1010-i RELATIVISTIC EFFECTS IN A COORDINATE TIME SYSTEM IN THE VICINITY OF THE EARTH (Question ITU-R 152/7) (1994-1997) The ITU Radiocommunication Assembly, considering that it is desirable to maintain coordination of standard frequency and time-signal
2、 emissions in the vicinity of 4 the Earth; b) geoid; c) that atomic clocks are subject to path-dependent second-order motional frequency shifts and position-dependent gravitational frequency shifts; d) well-defined procedures to account for relativistic effects in timing systems and time comparisons
3、 e) that, since time comparisons in non-inertial frames require special consideration and the CCDS has recommended an appropriate set of equations which provide a consistent set of measurements of UTC in the vicinity of the Earth; f) g) that universal coordinated time (UTC) is the official coordina
4、te time scale for the Earth defined on the rotating that the Consultative Committee for the Definition of the Second (CCDS) has recognized the need for that there is a growing trend to place accurate, stable clocks in Earth-bound orbits for time-keeping purposes; that there is a need for comparing f
5、requency standards in the vicinity of the Earth with an accuracy of lO-I4, recommends 1 that for calculating coordinate time intervals in the vicinity of the Earth (out to at least geosynchronous radius) to an accuracy of 1 ns (or lei4 of the integration time), the following procedures, based on the
6、 first order terms in the full general relativistic expressions, should be followed (some practical examples are given in Annex 1): 1.1 Clock transport in a rotating reference frame When transferring time from point P to point Q by means of a portable clock, the coordinate time accumulated during tr
7、ansport is: where: C: o: V: r: AE : - AU( F) : speed of light angular velocity of rotation of the Earth velocity of the clock with respect to the ground vector whose origin is at the centre of the Earth and whose terminus moves with the clock from P to Q equatorial projection of the area swept out d
8、uring the time transfer by the vector F as its terminus moves from P to Q gravitational potential difference (including the centrifugal potential) between the location of the clock at F and the geoid as viewed from an earth-fixed coordinate system, in agreement with the convention (Resolution A4, IA
9、U, 1992) that AU( I) is negative when the clock is above the geoid STD=ITU-R RECMN TF.LOLO-1-ENGL 1997 4855212 0531546 449 H 24 Rec. ITU-R TF.1010-1 ds : increment of proper time accumulated on the portable clock. The increment of proper time is the time accumulated on the portable standard clock as
10、 measured in the “rest frame” of the clock; that is, in the reference fiame travelling with the clock. AE is measured in an earth-fixed coordinate system. As the area AE is swept, it is taken as positive when the projection of the path of the clock on the equatorial plane moves eastward. When the he
11、ight h of the clock is less than 24 km above the geoid, AU( I) may be approximated by gh, where g is the total acceleration due to gravity (including the rotational acceleration of the Earth) evaluated at the geoid. This approximation applies to all aerodynamic and earthbound transfers. When h is gr
12、eater than 24 km, the potential difference AU( F) must be calculated to greater accuracy as follows: AU(?) = GMe/r + J2 GMea: (1 - 3cos2)/2r3 + oz 9 sin2/2 - Ug (2) where: al : equatorial radius of the Earth al = 6378.136km r : 8 : colatitude GM, : product of the Earths mass and the gravitational co
13、nstant GM, = 398600km3/s2 quadrupole moment coefficient of the Earth J2 = +1.083 x lW3 angular velocity of the Earth w = 7.292115 x le5 rads magnitude of the vector F 32 : w : Ug : potential (gravitational and centrifugal) at the geoid Ug = 62.63686 km2/s2. For time transfer at the 1 ns level of acc
14、uracy, this expression should not be used beyond a distance of about 50000 km from the centre of the Earth. 1.2 When transferring time from point P to point Q by means of a clock the coordinate time elapsed during the motion of the Clock transport in a non-rotating reference frame clock is: where: Q
15、 At = ds P 2 U(Y) - ug v +- 2 2c2 1+ C (3) U( F): gravitational potential at the location of the clock excluding the centrifugal potential v : velocity of the clock, both as viewed (in contrast to equation (i) from the geocentric non-rotating reference frame potential at the geoid (Ug/c2 = -6.9694 x
16、 10-lo), including the effect on the potential of the Earths rotational motion. Us : Note that AU( I) # U( F) - U, since U( ?)does not include the effect of the Earths rotation. This equation also applies to clocks in geostationary orbits but should not be used beyond a distance of about 50000 km fr
17、om the centre of the Earth. Rec. ITU-R TF.lO1O-1 25 1.3 From the viewpoint of a geocentric, earth-fixed, rotating frame, the coordinate time elapsed between emission and reception of an electromagnetic signal is: Electromagnetic signals in a rotating reference frame where: do : AU(?): potential at t
18、he point, ? , on the transmission path less the potential at the geoid (see equation (3), as AE: increment of standard length, or proper length, along the transmission path viewed from an earth-fixed coordinate system area circumscribed by the equatorial projection of the triangle whose vertices are
19、 - - - at the centre of the Earth at the point, P, of transmission of the signal at the point, Q, of reception of the signal. The area, AE, is positive when the signal path has an eastward component. The second term amounts to about one tenth nanosecond for an Earth-to-geostationary satellite-to-Ea
20、rth trajectory. In the third term, 20/c2 = 1.6227 x le6 ns/km2; this term can contribute hundreds of nanoseconds for practical values of AE. The increment of proper length, do, can be taken as the length measured using standard rigid rods at rest in the rotating system; this is equivalent to measure
21、ment of length by taking c/2 times the time (normalized to vacuum) of a two-way electromagnetic signal sent from P to Q and back along the transmission path. 1.4 From the viewpoint of a geocentric non-rotating (local inertial) frame, the coordinate time elapsed between emission and reception of an e
22、lectromagnetic signal is: Electromagnetic signals in a non-rotating reference frame I U(?) - ug C 2 Q 1 At = - C I& I + (5) where U( I) and U, are defined as in equation (3), and do is the increment of standard length, or proper length, along the transmission path. ANNEX 1 Examples Due to relativist
23、ic effects, a clock at an elevated location will appear to be higher in frequency and will differ in normalized rate from International Atomic Time (TAI) by: AU 2 - C where: AU: difference in the total potential (gravitational and the centrifugal potentials) c : velocity of light. STD=ITU-R RECMN TF
24、1010-1-ENGL 1777 m 4855212 0531548 211 = 26 Rec. ITU-R TF.1010-1 Near sea level this is given by: where: cp : geographical latitude g(q) : total acceleration at sea level (gravitational and centrifugal) g(cp) = (9.780 + 0.052 sin2 cp) m/s2 h: distance above sea level. Equation (6) must be used in c
25、omparing primary sources of the SI second with TAI and with each other. For example, at latitude 40, the rate of a clock will change by +1.091 x lWi3 for each kilometre above the rotating geoid. If a clock is moving relative to the Earths surface with speed V, which may have a component VE in the di
26、rection to the East, the normalized difference in frequency of the moving clock relative to that of a clock at rest at sea level is: where: o : angular rotational velocity of the Earth o = 7.292 x rads r : distance of the clock from the centre of the Earth (equatorial Earth radius = 6 378.136 km) c
27、 velocity of light c = 2.99792458 x lo5 km/s cp : geographical latitude. For example, if the clock is moving 270 m/s East at 40“ latitude at an altitude of 9 km, the normalized difference in frequency of the moving clock relative to that of a clock at rest at sea levei due to this effect is: -4.06
28、x 10-13 + 9.82 x - 1.072 x = -4.96 x The choice of a coordinate frame is purely a discretionary one, but to define coordinate time, a specific choice must be made. It is recommended that for terrestrial use an earth-fixed frame be chosen. In this frame, when a clock B is synchronized with a clock A
29、both clocks being stationary on the Earth) by a radio signal travelling from A to B, these two clocks differ in coordinate time by: where: cp: latitude h : longitude (the positive sense being toward East) p : path over which the radio signai travels from A to B. If the two clocks are synchronized b
30、y a portable clock, they will differ in coordinate time by: c 2C ) P where: V: portable clocks ground speed p : portable clocks path from A to B. - STD-ITU-R RECMN TF.LOLO-1-ENGL 1997 - 4855212 0531549 158 Rec. ITU-R TF.lO1O-1 27 This difference can also be as much as several tenths of a microsecond
31、 It is recommended that equations (8) or (9) be used as correction equations for long-distance clock synchronization. Since equations (8) and (9) are path dependent, they must be taken into account in any self-consistent coordinate time system. If a clock is transported from a point A to a point B
32、and brought back to A on a different path at infinitely low speed at h = O, its time will differ from that of a clock remaining at A by: where AE is the area defined by the projection of the round trip path on to the plane of the Earths equator. AE is considered positive if the path is traversed in
33、the clockwise sense viewed from the South Pole. For example since: 20/c2 = 1,6227 x le6 ns/km2 the time of a clock carried eastward around the Earth at infinitely low speed at h = O at the equator will differ from a clock remaining at rest by -207.4 ns. At the System (GPS) indicated height are all equivalent. level of correction, heights above sea level, above the rotating geoid and the Global Positioning
