1995A&A...304..653W

Astron. Astrophys. 304, 653-661 (1995)

ASTRONOMY AND
ASTROPHYSICS

Relativistic theory for clock syntonization and the realization of geocentric coordinate times
P. Wolf1•2 and G. Petit1 1Bureau International des Poids et Mesures, Pavilion de Breteuil, F-92312 Sevres Cedex, France 2 Queen Mary and Westfield College, School of Mathematical Sciences, Mile End Road, London El 4NS, Great Britain
Received 21 April 1995 I Accepted 19 May 1995

Abstract. General relativity predicts that two ideal clocks that move with a relative velocity, and are submitted to different gravitational fields will in general be observed to run at different rates. Similarly the rate of a clock with respect to the coordinate time of some space-time reference system is dependent on the velocity of the clock in that reference system and on the gravitational fields it is submitted to. For the syntonization of clocks and the realization of coordinate times (like TAI) this rate shift has to be taken into account with an uncertainty that should be below the frequency stability of the clocks in question, i.e. all terms that are larger than the instability of the clocks should be corrected for. Presently the best atomic clocks reach frequency stabilities of 1 x 10-15 , with new designs promising an improvement by several orders of magnitude. We present a theory for the calculation ofthe relativistic rate shiftfor clocks in the vicinity of the Earth, including all terms larger than one part in 1018. This, together with previous work on clock synchronization (Petit & Wolf 1994), amounts to a complete relativistic theory for the realization of coordinate time scales at picosec.ond synchronization and 10-18 syntonization accuracy, which should be sufficient to accommodate future developments in time transfer and clock technology.
Key words: relativity - reference systems - time
1. Introduction
At its 1991 General Assembly, the International Astronomical Union (IAU) explicitly 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 transformations between them were defined, together with procedures for their realization.
Coordinate time scales in the vicinity of the Earth can be realized by a weighted average of atomic clocks operating on the Earth and on terrestrial satellites. For this purpose the clocks
Send offprint requests to: P. Wolf (BIPM)

have to be synchronized and syntonized using a relativistic theory at the level of accuracy required.
A relativistic theory for the synchronization of clocks in the vicinity of the Earth has recently been developed to picosecond accuracy (Petit & Wolf 1994), which should be sufficient for present and near-future practical requirements which are at the level of :::::: 100 ps with an expected improvement by one order of magnitude.
A well kn.own prediction of Einstein's general theory of relativity states that two ideal clocks that move with a relative velocity, and are submitted to different gravitational fields will in general be observed to run at different rates, a prediction that has been verified experimentally with a relative uncertainty of 7 x 10-5 using two hydrogen masers (Vessot 1980). This implies that the rate of a clock with respect to the coordinate time of some space-time reference system is dependent .on the velocity of the clock in that reference system and on the gravitational fields to which it is submitted. If the clock is to be used for the realization of coordinate time this rate shift has to be taken into account at a level of accuracy which should be below the frequency stability of the clock in question, i.e. the clock has to be syntonized with respect to the coordinate time including all terms that are larger than the instability of the clock.
Presently hydrogen masers, the most stable atomic clocks, reach frequency stabilities of 1 x 10-15 (square root of the Allan variance) for averaging times of :::::: 10 000 s (Vessot et al. 1992; Busca 1993; Cutler 1993). For longer averaging times the best stabilities are displayed by caesium clocks and are below one part in 1014 (Cutler 1993). Several laboratories are currently developing caesium fountain standards which, in the near future, are expected to provide stabilities in the low 10-16 range (De Marchi 1993; Maleki 1993; Santarelli et al. 1994). In the longer term the advent of cooled hydrogen masers and trappedion standards may yield stabilities of a few parts in 1017 and 1018 respectively (Rolston & Phillips 1991; ltano 1991; Maleki 1993). Present and expected clock stabilities are summarized in Fig. I.
In this paper we present a relativistic theory for the syntonization of clocks in the vicinity of the Earth (within a geo-

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P. Wolf & G. Petit: Relativistic theory for clock syntonization

-12

Cs Fountain-1993

LEGEND:

-13

'

, I , ✓

-

MEASURED

.... ._r JPL LITS-1993 - - - • EXPECTED

~

Caes;um Fountaln-1995

z

0
~

-14

>

UJ

. ._

I

H-MASER-1991

.....

C

SCM0-1991

z -15

<

..J

.<.J

c., -16

0

..J

-17

origin (atmospheric pressure changes, movements of the Earth's crust, polar motion etc.).
The space-time metric for the geocentric reference system, including terms of order c-4, has been derived in Brumberg & Kopejkin (1988) and Kopejkin (1988). The relationship between TCG and the proper time of a clock in the vicinity of the Earth is given by Brumberg & Kopejkin (1990), Klioner (1992) and Brumberg et al. (1993). In these papers tidal terms of order 10-11 , including the response of an elastic Earth are given at an accuracy sufficient for our purpose. However, effects of oceanic tides and of non-tidal origin, which can contribute up to a few parts in 1017 , are not considered. Furthermore, the accuracy of the expressions given for the geopotential is limited at 10- 14 with no information on possible improvements.

2. Syntonization in a relativistic context

-18

10

103

106

-r/s

Fig, 1. Present and expected clock stabilities (from Maleki (1993)). SCMO stands for Superconducting Cavity stabilised Maser Oscilator and JPL LITS for the Jet Propulsion Laboratory Linear Ion Trap Standard. By combined performance is meant a combined time scale using an SCMO and a trapped ion standard

centric sphere of 300 000 km radius) including all terms larger than one part in 1018 . Outside this sphere the effect of the lunar quadrupole moment may exceed 1 x 10- 18 and should be
included.
On the surface of the Earth a relativistic frequency shift of 1 x 10- 18 corresponds to a change in altitude of~ 1 cm. Therefore, effects on syntonization due to the size of the clocks cannot be neglected when such accuracies are required and should be accounted for when using the theory provided in this paper.
Section 2 clarifies the concept of syntonization in a relativistic context. In Sect. 3 we present the individual relativistic terms affecting the rate of a clock with respect to coordinate time based on the post-Newtonian approximation of general relativity. Evaluation of the terms of order c 4 shows that these amount to a few parts in 1019 in the vicinity of the Earth and are hence negligible. We provide explicit expressions for syntonization with respect to Geocentric Coordinate Time (TCG) to be applied to clocks on the surface of the Earth (Sect. 3.1) and on board terrestrial satellites (Sect. 3.2). The effects that need to be taken into account when using these expressions at the 10- 11 and 10-18 accuracy level respectively are discussed. Terrestrial Time (TT), the ideal form of International Atomic Time (TAI), can be obtained from TCG via a transformation (IAU 1991) which is presented, together with its limitations, in Sect. 4. When determining the relative rate of two distant clocks one might be interested in time varying effects only (i.e. effects that influence the observed frequency stability). These are investigated to an accuracy of 10-18 in Sect. 5. We consider tidal variations of the gravitational field as well as those of non-tidal

When using the concept of syntonization in a relativistic context, certain ambiguities appear which can lead to confusion and misunderstanding. It is therefore essential to first clarify the different meanings of the expression as used in time metrology within a relativistic framework.
For the realization of coordinate time scales (like TAI) it is necessary to syntonize clocks with respect to the coordinate time in question, i.e. to determine the rate of a clock A with respect to an ideal coordinate time of some space-time reference frame. For example, using a geocentric non-rotating frame with TCG as coordinate time (as defined by the IAU (1991)), in the postNewtonian approximation, the proper time TA of the clock A is related to TCG by

drA

U (w)

+

v2
2

_4

dTCG = l - c2 +O(c ),

(1)

where (cTCG,wk) are coordinates in the geocentric frame with w representing the triplet wk. The potential at the position of the clock U(w) is the sum of the Earth's potential and the tidal potentials of external bodies, and vis the coordinate speed of the clock in the geocentric, non-rotating frame. Note that this rate depends entirely on the chosen reference frame. It is a coordinate quantity which cannot be obtained directly from measurement, but must be calculated theoretically using the definition of the reference frame in question with the appropriate metric
equation. When using repeated time transfers employing the conven-
tion of coordinate synchronization (Allan & Ashby 1986; Petit & Wolf 1994) for the determination of the relative rate of two clocks A and B the resulting rate predicted by theory is simply

dTA

dTA dTCG
=------

(2)

dra dTCG dra '

with dr/ dTCG given in (1). This is a combination ofcoordinate dependent quantities and therefore entirely dependent on the chosen reference frame and the convention of synchronization.
The value of the quantity (2) is not to be confused with the result obtained by an observer measuring the relative frequency

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of two signals emitted by A and B, each signal being locked to the clocks in A and B. This quantity, [/Alo/[!Blo, is a measurable quantity which is specific to the measuring observer 0. It is possible to relate the proper received frequency [/Alo (the frequency of the signal measured by the observer O using an ideal clock) at the time of reception tr to the proper emitted frequency [!AlA (the frequency measured by an observer at A) at the time of emission te by the formula

[/Alo = drA ( 1 + kA • ilio(tr)/c)

(3)

[/AlA dro 1 + kA • 'WA(te)/c '

where, in the formulation of the Doppler effect, the Newtonian
approximation is used with kA being the unit vector along the
propagation path, kA = (wA(te) - Wo(tr))/\wA(te) - wo(tr)\. Applying this formula to A and B, it is clear that [!Alo/UBlo differs from the value given by (2) by the Doppler terms which are dependent on the relative motion of the clocks and the observer. In addition the two expressions differ in terms of order
c-3 and higher, which are not represented in (3) but can reach
10-15_

3. Syntonization with respect to TCG
In the general theory of relativity it is usually impossible to define space-time coordinates x°' that have a globally constant relation to measurable quantities (so called proper quantities). Instead this relationship is location dependent and defined by the space-time metric. For two events with an infinitesimal, time like separation such a relationship is given by:

Substituting (5) into (4) and solving for dr/dt gives the relationship between the proper time of a clock and coordinate time, i.e. an expression for the rate of the clock with respect to coordinate time,
(6)
where vi = dxi / dt is the coordinate velocity of the clock. For a geocentric coordinate system with TCG as coordinate
time and non-rotating spatial coordinates, the components of
the space-time metric up to order h~b are given, for example,
by Brumberg et al. (1992). The fourth order term h~ is derived in Kopejkin (1988). Substituting these results into (6) we find that in the vicinity of the Earth all terms in c-4 amount to a few parts in 1019 or less. In particular the h~/ 2c4 term and terms due to the geodesic precession (in h~}vi /c4), which require the specification of coordinate conditions (harmonic, standard postNewtonian etc...) and the state of rotation of the frame (kinematically or dynamically non-rotating), are below the 10-18 limit. The choice of coordinate conditions and of the state of rotation (in the above sense) of the frame is therefore not significant for syntonization at an accuracy of 10-18.
For this reason only the h~ component of the metric tensor is required for our purpose. It is reproduced from Bromberg et al. (1992) in Eq. (7) below.

ds2 = -c2dr2 = go,,0(x>-)dx°'dx13 ,

(4)

assuming a summation over repeated indices with ds representing the relativistic line element and T the proper time elapsed between the two events as measured by a clock whose worldline passes through both. Here and throughout the paper greek indices run from Oto 3 and latin ones from 1 to 3. Space-time coordinates are denoted by x°' with x 0 = ct (t = coordinate time) and spatial coordinates xi. The go,13(x>-) are coordinate dependent components of the space-time metric tensor.
In the post-Newtonian approximation, valid for weak gravi-
tational fields and low velocities (e2 ~ U/ c2 ~ v2/ c2 « 1), the
components of the metric tensor can be expressed by a power series based on small corrections to the Minkowski metric of special relativity in terms of en (see for example Brumberg 1991),

where (cTCB,xk) (TCB = Barycentric Coordinate Time) are coordinates in the barycentric frame with (xkxk)112 =rand :z: denoting the triplet xk with the subscript E refering to the Earth's centre of mass. UE(w) and U(:z:) are the Newtonian gravitational potentials of the Earth and of external masses respectively, and Qk is the correction for the non-geodesic barycentric motion of the Earth.
The second term in (7), arising from the interaction of the Earth's quadrupole moments and the external masses [given explicitly e.g. in Bromberg & Kopejkin (1990)1 gives rise to a correction of less than a few parts in 1019 in the vicinity of the Earth and can be neglected for our purposes.
Hence the rate of a clock with respect to coordinate time (TCG) in the vicinity of the Earth, including all terms larger than one part in 1018, is

dr dTCG =1 -

1
c2

[
UE(w)

+

v2
2

_
+ U(:z:E

+

w)

- U(:z:E) - U ,k (XE)Wk] .

(8)

(5)

where the (- + ++) sign convention has been adopted. The

Orders of magnitude of the individual terms in (8), and their

h~J / en are of order en and 8ii is the Kronecker symbol (bij = 1 calculation at the required accuracy, are considered in detail in

for i = j; bij = 0 otherwise).

the following sections.

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Table 1. Effects on syntonization with respect to TCG of clocks on the Earth's surface; Orders of magnitude and uncertainties of the corrections

Effect

Order of Uncertainty magnitude

Earth's gravitational potential Centrifugal potential (v2/2/c2)
Volcanic and coseismic (highly localised) External masses (Moon, Sun) Solid Earth tides Ocean tides

7 X 10-JO 1 X 10- 12
< 10-16
10-11
10-11
10-11

10-11
< 10-18
?
< 10-18 < 10-18 < 10-18

3.1. Clocks on the Earth's suiface
The limiting factor for syntonization with respect to coordinate time, of a clock on the surface of the Earth is the inaccuracy in the determination of the Earth's gravitational potential. Currently this uncertainty is ::;:: 1 m2 s-2 for the total (gravitational + centrifugal) potential on the geoid, Wo, (Bursa et al. 1992; Bursa 1993) which is equivalent to a 10cm error in radial distance. This corresponds to an uncertainty of::;:: 1 x 10-17 in (8). In this section we therefore only consider effects whose influence on the terms in (8) is larger than one part in 1017. These are summarized in Table 1 together with orders of magnitude and uncertainties of the associated corrections.
The gravitational potential of the Earth, UE(w), can be expressed as a series expansion in spherical harmonics. However, owing to mass irregularities, such a series cannot be be considered convergent at the surface of the Earth (Moritz 1961). Nonetheless, due to the predominantly ellipsoidal shape of the Earth, one can use the first two terms ofthis series expansion as a first approximation (Allan & Ashby 1986; CCIR 1990; IGioner 1992). Thus,

UE(w)G=M-E-+ GMEp3atJ2 (1-3cos2 0)+ ... ,

(9)

p

2

where G is the Newtonian gravitational constant, ME is the mass of the Earth, p = (wkwk)112, a1 and J2 are, respectively, the equatorial radius and the quadrupole moment coefficient of the Earth (a1 = 6378136.3 m, J2 = 1, 0826 x 10-3) and 0 is the geocentric colatitude of the point of interest.
Substituting (9) into the second term of(8) gives terms which can amount to::;:: 7 x 10-10 and::;:: 8 x 10-13 for points on the surface of the Earth.
Considering the third term in (8), one can see that with

v =wpsin0,

(10)

for a clock fixed on the surface of the Earth (where w represents
the angular velocity of rotation of the Earth) this term is equiva-
lent to the centrifugal potential divided by c1-. Its magnitude can
reach 1.2 x 10-12.
The effect on this term of the movement of the pole (!10)
o- and variations in the length of day (Llw) are of order 1 1s and
smaller and are treated in more detail in Sect. 5.

The surface obtained when setting UE(w) = Wo-(wp sin 0)2
/2 in (9) differs from the ellipsoid of the Earth model by less than 10 m. Therefore, an estimate of the accuracy of (9) can be obtained by considering the maximal difference between the geoid and the reference ellipsoid. This can amount to ::;:: 100 m (Vanicek & Krakiwsky 1986), so expression (9) for the Earth's gravitational potential should not be used if accuracies better than one part in 1014 are required.
For improved accuracy the second and third term in (8) should not be computed separately using (9) and (10). Instead, their combined effect should be determined using

TVrE(W ) + (wpsin0)2 --W.0 - 1H gdH-- W.0 - -gH

(11)

2

o

where g is the Earth's gravitational + centrifugal acceleration, and H is the height above the geoid. A value of g averaged between O and H, g, obtained from a gravimetric model, can
be used instead of computing the integral when the required accuracy in (8) is of order 10-15 .
Using a geodetic GPS (Global Positioning System) receiver and a geoid model the height above the geoid, H, can be obtained with an accuracy of the order of 10 m. This allows the determination of the total (gravitational and centrifugal) effect on the clock with an accuracy of::;:: (1-2) x 10-15 . Similar accuracy can also be obtained by using a topographic map for the determination of H.
When higher accuracy is required, precise levelling should be used. Levelling measurements are referred to a zero-level reference point which can be compared to mean sea level using a tidal gauge. This level differs from the geoid by what is known as Sea Surface Topology (Ssn which can amount to ±0.7 m (Torge 1989). The SST can be determined with an accuracy of ::;:: 0.1 m (Torge 1989) using oceanographic methods and satellite altimetry which induces an uncertainty of::;:: 1 x 10-17 in (8). The uncertainty in the potential on the geoid, Wo, which is of order 1 m2 s-2 (Bursa et al. 1992; Bursa 1993), contributes another part in 1017 . The sum ofthe gravitational and centrifugal potential differences between mean sea level and an arbitrary point far from the coast can be obtained by geometrical levelling with simultaneous gravimetric measurements. The accumulated uncertainty when using modern levelling techniques and gravimetry is below (0.5 ✓D/km)mm (Kasser 1989), where
D is the distance between the reference point and the point of
interest, and does therefore not exceed a few centimeters even over large distances. In many countries levelling networks have been established at accuracies of ::;:: (2 ✓D /km) mm for primary points, the use of which would again induce errors at the centimetric level. Alternatively. levelling can be achieved with accuracies of order 10cm (for distances of::;:: 100km) using differential GPS (Milbert 1992).
Therefore the constant part of the total potential at any point on the Earth's surface can be determined with an ultimate uncertainty less than 2.5 m2 s-2 using a tidal gauge and good geometrical levelling. The main contributions to this uncertainty are inaccuracies in the determination of W0 and the SST. This limits the evaluation of (8) at the level of (2-3) x 10-17 , which is

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the limit for syntonization of clocks with respect to coordinate time (TCG or TT) on the surface of the Earth.
Additionally, account has to be taken of the time varying part of the potential on the surface of the Earth caused by the gravitational attraction of external masses (tidal effects) and changes in the Earth's own gravitational field (non-tidal effects).
A number of effects give rise to relativistic rate shifts which are larger than one part in 1018 and are of a periodic nature. For clock comparisons using time transfers with a synchronization accuracy of one picosecond, such terms are negligible if their period is sufficiently short to prevent their amplitude in the time domain from exceeding this limit. They might, however, be of· interest when using frequency transfers and are therefore included in this study.
At 10-17 accuracy non-tidal effects are highly localized and can be neglected in general. They relate mainly to movements of the Earth's crust caused by volcanic and coseismic processes. The resulting change in the second term of (8) can amount to ~ 10-16 on time scales ranging from a few days to one year (Torge 1989). From gravimetric measurements Ervin & McGinnis (1986) have inferred local elevation changes in the Mississippi embayment of up to 15 cm, caused by the surface loading associated with changes in river stage. The resulting change in the second term of (8) is ~ 1.5 x 10-17. In Sect. 5. we present a more detailed treatment of non-tidal effects at the 10-18 accuracy level.
The third, fourth and fifth terms of (8) represent the effect of external masses, mainly the Moon and Sun. The Newtonian potential of external bodies can be expressed in the spherical approximation by,

Evaluation ofexpression (13) for the Moon and the Sun gives a correction in (8) which is smaller than 4 x 10-17. Contributions from other planets and higher order terms in (8) add corrections which are smaller than 1 x 10-18 . The effect of oceanic tides
o- can amount to 9 x 1 18 for the M2 lunar tide in a few regions,
and roughly twice this value for the total tide (Scherneck 1994). It is discussed in more detail in Sect. 5.

3.2. Clocks on board terrestrial satellites
The accuracy of syntonization of satellite clocks with respect to TCG is limited by uncertainties in the geopotential model and the orbit determination. Solid Earth tides, ocean tides, polar motion and changes in atmospheric pressure may give rise to corrections of some parts in 1018 for low flying satellites, but can be neglected at altitudes exceeding 4000 km. The tidal potentials of external masses become more important with increasing altitude and so require an exact expression, rather than a series expansion as in (13), for their evaluation. Table 2 lists all relevant effects together with orders of magnitude and uncertainties of the associated corrections.
The rate of a clock with respect to TCG is given by Eq. (8). The geopotential in the second term can be expressed as a series expansion in spherical harmonics

+ L L UE(w) =G-M-E, [ 1 00 n Pnm(cos0) ( a 1 )n

p

n=2 m=O

p

X (Cnm cos(m).) + Bnm sin(m).))] ,

(14)

(12)
where TA is the coordinate distance between the point of interest and the centre of mass of the body A, MA is the mass of body A, the summation is over all celestial bodies apart from the Earth, and multipole terms are neglected as their effect on the surface of the Earth does not exceed 10-18. Note that it is not essential whether TA is expressed in geocentric or barycentric coordinates, as the induced error in Eq. (8) is several orders of magnitude smaller than 10-18.
Substituting (12) into the third, fourth and fifth terms of (8), expanding the third term in a Taylor series and using Love numbers to characterize the response of the Earth to the tidal potential (the solid Earth tide) gives,

where 0 and ). are the geocentric colatitude and longitude of the satellite, Pnm(cos 0) are associated Legendre polynomials, and the Cnm and Bnm are coefficients determined by fitting the data from satellite observations. For the latest model (GEM-TI) these coefficients are given with an accuracy ofa few parts in 109 up to degree and order 50 (Lerch et al. 1992; IERS 1992). For low altitudes (< 4000 km) this results in syntonization uncertainties which exceed one part in 1018, but decrease with increasing altitude. The effects of solid Earth, oceanic, and pole tides can be included in the model as small variations of the coefficients. The effect of atmospheric pressure variations may amount to 1-2 parts in 1018 for altitudes< 4000km but corrections to an accuracy of 10-18 are possible (see Sect. 5).
The third, fourth and fifth terms in (8) characterize the effect of external masses on the rate of a clock relative to TCG. It is not practical to use the Taylor expansion of the third term as in the previous section, since at higher altitudes a large number of terms would be needed to achieve the required 10-18 accuracy. Instead, substituting (12) into (8) for the external potentials and differentiating the fifth term, we obtain

u -U(xE + w) - -U(xE) - -

k ,k(XE)W

where Tk = x~ - xi, TEA= (TkT~A) 112, and where k2 and h2
= aretheLovenumbers.FormostEarthmodels(l+k2-h2) 0.69

L = GMA [-1- - _1_ + T~f:E]

(15)

(Farrell 1972).

A'FE

TpA TEA TEA

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Table 2. Effects on syntonization with respect to TCG of clocks on board terrestrial satellites; Orders of magnitude and uncertainties of the corrections where h represents the altitude of the satellite

Effect Earth's gravitational potential

Order of magnitude
< 6 X 10-lO

2nd order Doppler (v2/2/c2) External masses: Moon (at h = 300000km) Sun Venus

< 3 X 10- 10
4 X 10-lJ} 4 X 10- 14 6 X 10- 18

Uncertainty
Few 10- 18 (GEM-T3)
< 10-18 at h > 4000km
Few 10- 18 (5 cm orbit uncertainty)
< 10-18 at h > 10000km < .10- 18 at h > 10000km
< 10-18

Solid Earth tides } Ocean tides Polar motion Atmospheric pressure

10-18 (at low altitudes)

for the total tidal potential. Here r:A = r: - r! is the vector from the centre of mass of body A to the point of interest with magnitude TPA = (r:Ar:A) 112, the subscript E stands for the Earth, and the summation is carried out over all celestial bodies except the Earth. At the required accuracy either barycentric or geocentric coordinates may be used.
The maximal magnitude of the tidal terms in (8) is ~ 4 x 10-13 for the moon,~ 4 x 10-14 for the sun and~ 6 x 10-18 for Venus. The effects of other planets and asteroids are negligible. The constraints on the knowledge of the planetary ephemerides are ~ 117 m for the Earth-moon distance, ~ 1200 km for the Earth-sun distance and ~ 106 km for the distance to Venus, which present no difficulties for modern astrometry.
Syntonization of satellite clocks with respect to TCG is limited mainly by orbitography errors. For syntonization to 10-18 the required accuracies are of order 1 cm on position and 1x 10-5 m s- 1 on velocity for a satellite at 1000 km altitude. For satellites in higher orbits the constraints are less severe, being about 40 cm and 3 x 10-5 m s- 1 for a geostationary one.
The data necessary for orbit determination can be obtained by satellite ranging from a number of ground stations on the Earth using timing measurements of electromagnetic signals. For this purpose the clocks at the different stations have to be synchronized and syntonized, so the problems of satellite orbitography and time scale realization are not entirely independent. However, the accuracy required for station clock synchronization and syntonization for the realization of centimetric orbits is of order one microsecond and parts in 1011 respectively, values which can be achieved even with terrestrial methods. This is why the two problems can be separated.
At present satellite laser ranging produces measurements at a precision of a few millimetres, with an accuracy of roughly one centimeter (Degnan 1993), the limiting error source being

uncertainty in the atmospheric propagation delay. The satellite orbit is determined from the ranging measurements using an orbital model, which introduces further inaccuracies. Differences between orbits obtained using different ranging techniques and models for the Topex/Poseidon mission (altitude 1200 km) are typically a few centimetres (Yunck 1994; Nouel 1994; Schutz 1994) (consistent with an uncertainty of ~ 1 x 10-5 m s-1 in velocity) which is an indication of the accuracy of the orbits obtained. Beutler et al. (1994) estimate the uncertainty of precise GPS ephemerides (altitude 20000km) to be~ 15cm (~ 1 x 10-5 m s- 1 in velocity).
We conclude that syntonization of satellite clocks with respect to TCG is at present limited by uncertainties in the geopotential model (GEM- T3) and the satellite orbits to an accuracy of a few parts in 1018 at low altitudes, with a decrease ofthis limit below 1 x 10-18 with increasing altitude(> 10 000 km), which is an order of magnitude better than the uncertainty for clocks on the Earth's surface. Therefore it seems likely that future time will be provided from space.
4. Transformation to TT
TCG is related to TT by a relativistic transformation, so any clock syntonized with respect to TCG can also be syntonized with respect to TT. In this case the accuracy of syntonization may be limited by the uncertainty of the parameters participating in the transformation.
The IAU defined TT as a geocentric coordinate time scale differing from TCG by a constant rate, the scale unit of TT being chosen so that it agrees with the SI second on the geoid (IAU 1991). TT is an ideal form of the International Atomic Time

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Table 3. Time varying effects on the Earth's surface for the determination of the relative rate of two clocks; orders of magnitude and uncertainties of the corrections

Effect
Volcanic and coseismic (highly localised)
Geodynamic and man-made (localised and long-term > 1 year)
External masses (Moon, Sun) Solid Earth tides Ocean tides Polar motion (long-term~ 430 days) Atmospheric pressure

Order of magnitude
< 10-16 < 10-16
10-11
10-11
10-17
10-18
10 - 1 8

Uncertainty
? ?
< 10-18 < 10-18 < 10-18 < 10-18 < 10-18

TAI, apart from a constant offset, and can be obtained from TCG by the transformation

dTT

dTCG =I-Lg,

(16)

where Lg= W0 /c2 = 6.9692903 x 10-10 ± 1 x 10-11.
It follows that at present the accuracy of syntonization with respect to TT is limited to~ 1 x 10-17 by uncertainties in the determination of the potential on the geoid W0 , even for clocks on board terrestrial satellites.
This limit is inherent in the definition ofTT and can therefore only be improved by reducing the uncertainty of W0. If highly stable clocks on terrestrial satellites are to be used for the realization of TT at accuracies exceeding this limit it may prove necessary to change this definition. One possibility would be to tum Lg into a defining constant with a fixed value. This would also provide a relativistic definition of the geoid (Bjerhammar 1985; Soffel et al. 1988).

5. Small time varying effects
Sections 3 and 4 treat the syntonization of clocks with respect to some ideal coordinate time (TCG or TT). We found that for clocks on the Earth's surface the uncertainty of this syntonization is limited at 10-17 by uncertainties in the geopotential. For this reason several time varying effects with amplitudes below this level (ocean tides, polar motion, atmospheric pressure etc...) were neglected. In this section we will consider the case where only the stability of the relative rate between two clocks is of interest. Then only time varying effects need be considered which, as shown below, can be calculated to 10-18 accuracy even for clocks on the Earth's surface. These effects are summarized in Table 3, together with orders of magnitude and uncertainties of the associated corrections.
The correction due to external masses (Moon and Sun) and the solid Earth tide can be calculated (see Eq. (13)) with an uncertainty of~ 4 x 10-19 due to the uncertainty in the determination of the Love numbers (Farrell 1972).
The effect of the ocean tide, including the associated loading, can be obtained using global oceanic tide models (Schwiderski 1983) and the loading deformation coefficients of

Farrell (1972). The results of one such calculation (Schemeck 1994) describe the total effect of the oceanic tides on the potential at the surface of the Earth in a 1° x 1° grid covering most of the globe. For the M2 lunar tide, the correction is 9 x 10-18 in a few regions with a maximum of twice this value for the total tide, the uncertainty of these values being ~ 2 x 10-19 (IERS 1992).
Atmospheric pressure variations may cause rate shifts of a few parts in 1018 . The direct gravitational potential of the atmosphere can be calculated adapting a method employed by Merriam (1992), in order to obtain the effect on the gravitational potential rather than the gravitational acceleration. The contribution to the potential at some point P, due to a thin col-
umn of air of infinitesimal area at azimuth a and a geocentric
angular distance ¢ from the point of interest can be calculated as a function of the pressure and temperature at its base. Using the ideal gas law and assuming hydrostatic equilibrium and an isothermal atmosphere gives for the potential of one column

U(¢,a)

-G-P-oa-f=sin-¢-d-¢-d-a

(17)

1RTo Zmax
x

e-z/H

112 dz

o [p2 + (a1 + z)2 - 2p(_a1+ z)cos ¢]

where Po and To are the pressure and the temperature at the base of the column, R is the specific gas constant for dry air (R = 287.05 Jkg- 1K- 1), Zmax is the height ofthe atmosphere (Zmax ~ 50 km), and H is the scale height of the atmosphere. Typically, H varies from about 8 km near the surface to about 7 km in the stratosphere (Merriam 1992). Summing these functions over the globe using surface pressure and temperature data and small increments da and d¢ gives the total gravitational potential of the atmosphere. The additional change caused by the associated atmospheric loading can be calculated using the above method and the surface load Love numbers k~ and h~ (Farrell 1972). Alternatively, a simple regression formula by Rabbel & Zschau (1985) can be used, giving the radial displacement of a point on the Earth's surface by

Llp/mm = (-0.35p - 0.55p)/mbar

(18)

where p is the pressure variation at the point of interest and p
the average of the pressure variation in the surrounding area of

© European Southern Observatory • Provided by the NASA Astrophysics Data System

1995A&A...304..653W

660

P. Wolf & G. Petit: Relativistic theory for clock syntonization

2000 km radius with the pressure values set equal to zero over which implies that the specification ofcoordinate conditions and

ocean areas. Rabbel & Zschau (1985) estimate the uncertainty of the state of rotation of the reference system is not necessary.

this expression to be less than 1 mm. For pressure variations of

Syntonization with respect to terrestrial time (TI), an ideal

lOmbar on a global scale (corresponding to seasonal changes) form of TAI, is limited to 1 x 10-17 by uncertainties in the

the effect on the rate of a clock on the Earth's surface can reach determination of the potential on the geoid, Wo, inherent to its

2 parts in 1018 due to the direct potential (Eq. (17)) but the mag- definition.

nitude of the displacement effect remains below 10- 18. Local

For clocks on the Earth's surface, syntonization with respect

pressure changes ((anti)cyclones) can cause displacements of to TCG or TI is limited to (2-3)·x 10- 17 by uncertainties in the
o- up to 2.5 cm, corresponding to a correction of 2.7 x 1 18, but determination of the geopotential at the location of the clock.

have a negligible direct potential. Finally the secondary poten-

These sections were concerned with the syntonization of

tial due to the deformation of the Earth can be neglected as the clocks with respect to some ideal coordinate time scale, i.e. the

appropriate surface load Love numbers k~ are at least a factor 4 determination ofthe rate between a physical time scale provided

smaller than the corresponding h~ which are used to calculate by a clock and the ideal coordinate time of some reference sys-

the displacement effect (Farrell 1972).

tem using corrections which are provided by theory. In Sect. 5

For a clock on the Earth's surface the centrifugal potential we consider the syntonization of two distant clocks, i.e. the

is given by (wpsin0)2 /2. Differentiating this expression with determination of the relative rate of two physical clocks. Cor-

respect tow and to 0, dividing by c?- and including Love numbers rections due to time varying effects that could affect the stability

allows the calculation of the total correction due to polar motion of this rate are provided to an accuracy of 10-18 .

6.0 and variations in the angular velocity of the Earth 6.w,

At present the stability of atomic clocks is approaching

= 6.(wp sin 0)2

(1 + k2 - h2) 2c2

(1 + k2 - h2)

~ x (~w2p2sin206.0+wp2sin2 06.w) .(19)

10-16 (Maleki 1993) with further improvements expected in the near future. For comparisons of these highly stable clocks over large distances, and their application in experimental relativity, geodesy, geophysics etc.., a sufficiently accurate relativistic the-

ory for their syntonization, like the one presented in this paper,

Because the spherical harmonic contribution of the centrifugal seems indispensable.

potential is restricted to degree two (Hinderer et al. 1982) the

Combined with a previous paper (Petit & Wolf 1994) the re-

total effect can be obtained using the classical Love numbers sults obtained here amount to a complete relativistic theory for
= k 2, h 2 with (1 + k2 - h2) 0.69. Maximum values for 6.0 the realization of a geocentric coordinate time scale with uncer-
and 6.w are 2.4 x 10-6 rad and 7 x 10-12 rad s- 1 respectively tainties in synchronization and syntonization of one picosecond

(Torge 1989) which result in corrections ofup to 2 x 10-18 and and 10-18 respectively.

1.6 x 10-19 for the first and second terms in (19). Finally we mention long term effects of a geodynamic na-
ture, and give some examples of highly localised effects of volcanic, coseismic and man made origin which might have to be taken into account at certain sites.

Acknowledgements. The authors are grateful to Dr. C. Thomas and Prof. B. Guinot for helpful discussions, to Dr. H.G. Scherneck for providing data on ocean tides, and to Dr. D.A. Blackburn for his help in improving the manuscript.

On tectonic plate boundaries, geodynamic effects may give

rise to corrections of up to 10- 16 over a period of several years. For example in northern Iceland elevation changes of 1 m (cor- References

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Volcanic and coseismic activities observed in Hawaii and California caused elevation changes of the order of 1 m (Torge 1989) over periods up to several months.
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