RECOMMENDATION ITU-R TF * Relativistic effects in a coordinate time system in the vicinity of the Earth

Transcription

1 Re. ITU-R TF RECOMMENDATION ITU-R TF * Relativisti effets in a oodinate time system in the viinity of the Eath (uestion ITU-R 15/7) ( ) The ITU Radioommuniation Assembly, onsideing a) that it is desiable to maintain oodination of standad fequeny and time-signal emissions in the viinity of the Eath; b) that univesal oodinated time (UTC) is the offiial oodinate time sale fo the Eath defined on the otating geoid; ) that atomi loks ae subjet to path-dependent seond-ode motional fequeny shifts and position-dependent gavitational fequeny shifts; d) that the Consultative Committee fo Time and Fequeny (CCTF, fomely CCDS) has eognized the need fo well-defined poedues to aount fo elativisti effets in timing systems and time ompaisons; e) that, sine time ompaisons in non-inetial fames equie speial onsideation and the CCDS has eommended an appopiate set of equations whih povide a onsistent set of measuements of UTC in the viinity of the Eath; f) that thee is a gowing tend to plae auate, stable loks in Eath-bound obits fo timekeeping puposes; g) that thee is a need fo ompaing fequeny standads in the viinity of the Eath with an auay of 10 14, eommends 1 that fo alulating oodinate time intevals in the viinity of the Eath (out to at least geosynhonous adius) to an auay of 1 ns (o of the integation time), the following poedues, based on the fist ode tems in the full geneal elativisti expessions, should be followed (some patial examples ae given in Annex 1): 1.1 Clok tanspot in a otating efeene fame When tansfeing time fom point P to point by means of a potable lok, the oodinate time aumulated duing tanspot is: t = ds 1 P U( ) V ω A E (1) * Radioommuniation Study Goup 7 made editoial amendments to this Reommendation in 003 in aodane with Resolution ITU-R 44.

2 Re. ITU-R TF : ω : V : : speed of light angula veloity of otation of the Eath veloity of the lok with espet to the gound veto whose oigin is at the ente of the Eath and whose teminus moves with the lok fom P to A E : equatoial pojetion of the aea swept out duing the time tansfe by the veto as its teminus moves fom P to U( ): ds : gavitational potential diffeene (inluding the entifugal potential) between the loation of the lok at and the geoid as viewed fom an eath-fixed oodinate system, in ageement with the onvention (Resolution A4, IAU, 199) that U( ) is negative when the lok is above the geoid inement of pope time aumulated on the potable lok. The inement of pope time is the time aumulated on the potable standad lok as measued in the est fame of the lok; that is, in the efeene fame tavelling with the lok. A E is measued in an eath-fixed oodinate system. As the aea A E is swept, it is taken as positive when the pojetion of the path of the lok on the equatoial plane moves eastwad. When the height h of the lok is less than 4 km above the geoid, U( ) may be appoximated by gh, whee g is the total aeleation due to gavity (inluding the otational aeleation of the Eath) evaluated at the geoid. This appoximation applies to all aeodynami and eathbound tansfes. When h is geate than 4 km, the potential diffeene U( ) must be alulated to geate auay as follows: U( ) = GM e / J GM e a 1 (1 3 os θ) / 3 ω sin θ / U g () a 1 : : θ : GM e : J : equatoial adius of the Eath a 1 = km magnitude of the veto olatitude podut of the Eath s mass and the gavitational onstant GM e = km 3 /s quadupole moment oeffiient of the Eath ω : U g : J = angula veloity of the Eath ω = ad/s potential (gavitational and entifugal) at the geoid U g = km /s. Fo time tansfe at the 1 ns level of auay, this expession should not be used beyond a distane of about km fom the ente of the Eath.

3 Re. ITU-R TF Clok tanspot in a non-otating efeene fame When tansfeing time fom point P to point by means of a lok the oodinate time elapsed duing the motion of the lok is: t = ds 1 P U( ) Ug v (3) U( ): gavitational potential at the loation of the lok exluding the entifugal potential v : U g : veloity of the lok, both as viewed (in ontast to equation (1)) fom the geoenti non-otating efeene fame potential at the geoid (U g / = ), inluding the effet on the potential of the Eath s otational motion. Note that U( ) U( ) U g, sine U( ) does not inlude the effet of the Eath s otation. This equation also applies to loks in geostationay obits but should not be used beyond a distane of about km fom the ente of the Eath. 1.3 Eletomagneti signals in a otating efeene fame Fom the viewpoint of a geoenti, eath-fixed, otating fame, the oodinate time elapsed between emission and eeption of an eletomagneti signal is: t = 1 dσ 1 P U( ) ω A E (4) dσ : U( ): A E : inement of standad length, o pope length, along the tansmission path potential at the point,, on the tansmission path less the potential at the geoid (see equation (3)), as viewed fom an eath-fixed oodinate system aea iumsibed by the equatoial pojetion of the tiangle whose veties ae: at the ente of the Eath at the point, P, of tansmission of the signal at the point,, of eeption of the signal. The aea, A E, is positive when the signal path has an eastwad omponent. The seond tem amounts to about one tenth nanoseond fo an Eath-to-geostationay satellite-to-eath tajetoy. In the thid tem, ω/ = ns/km ; this tem an ontibute hundeds of nanoseonds fo patial values of A E. The inement of pope length, dσ, an be taken as the length measued using standad igid ods at est in the otating system; this is equivalent to measuement of length by taking / times the time (nomalized to vauum) of a two-way eletomagneti signal sent fom P to and bak along the tansmission path.

4 4 Re. ITU-R TF Eletomagneti signals in a non-otating efeene fame Fom the viewpoint of a geoenti non-otating (loal inetial) fame, the oodinate time elapsed between emission and eeption of an eletomagneti signal is: t = 1 U( ) dσ 1 P whee U( ) and U g ae defined as in equation (3), and dσ is the inement of standad length, o pope length, along the tansmission path. Ug (5) Annex 1 Examples Due to elativisti effets, a lok at an elevated loation will appea to be highe in fequeny and will diffe in nomalized ate fom Intenational Atomi Time (TAI) by: U U : : diffeene in the total potential (gavitational and the entifugal potentials) veloity of light. Nea sea level this is given by: g ( ϕ) h (6) ϕ : g(ϕ) : geogaphial latitude total aeleation at sea level (gavitational and entifugal) g (ϕ) = ( sin ϕ) m/s h : distane above sea level. Equation (6) must be used in ompaing pimay soues of the SI seond with TAI and with eah othe. Fo example, at latitude 40, the ate of a lok will hange by fo eah kilomete above the otating geoid. If a lok is moving elative to the Eath s sufae with speed V, whih may have a omponent V E in the dietion to the East, the nomalized diffeene in fequeny of the moving lok elative to that of a lok at est at sea level is: 1 V g( ϕ) h 1 ω os ϕ V E (7)

5 Re. ITU-R TF ω : angula otational veloity of the Eath ω = ad/s : distane of the lok fom the ente of the Eath (equatoial Eath adius = km) : ϕ : veloity of light = km/s geogaphial latitude. Fo example, if the lok is moving 70 m/s East at 40 latitude at an altitude of 9 km, the nomalized diffeene in fequeny of the moving lok elative to that of a lok at est at sea level due to this effet is: = The hoie of a oodinate fame is puely a disetionay one, but to define oodinate time, a speifi hoie must be made. It is eommended that fo teestial use an eath-fixed fame be hosen. In this fame, when a lok B is synhonized with a lok A (both loks being stationay on the Eath) by a adio signal tavelling fom A to B, these two loks diffe in oodinate time by: t B t A = ω dλ os ϕ (8) p ϕ : λ : latitude longitude (the positive sense being towad East) p : path ove whih the adio signal tavels fom A to B. If the two loks ae synhonized by a potable lok, they will diffe in oodinate time by: t B t A ds U( ) V ω dλ p os ϕ (9) p V : potable lok s gound speed p : potable lok s path fom A to B. This diffeene an also be as muh as seveal tenths of a mioseond. It is eommended that equations (8) o (9) be used as oetion equations fo long-distane lok synhonization. Sine equations (8) and (9) ae path dependent, they must be taken into aount in any self-onsistent oodinate time system. If a lok is tanspoted fom a point A to a point B and bought bak to A on a diffeent path at infinitely low speed at h = 0, its time will diffe fom that of a lok emaining at A by: t = ω AE whee A E is the aea defined by the pojetion of the ound tip path on to the plane of the Eath s equato. A E is onsideed positive if the path is tavesed in the lokwise sense viewed fom the South Pole.

6 6 Re. ITU-R TF Fo example sine: ω / = 1, ns/km the time of a lok aied eastwad aound the Eath at infinitely low speed at h = 0 at the equato will diffe fom a lok emaining at est by 07.4 ns. At the level of oetion, heights above sea level, above the otating geoid and the Global Positioning System (GPS) indiated height ae all equivalent.