**2.3 Time scales**

6 Will-be-set-by-IN-TECH

both aspects of light propagation in curved space-times and cosmological assumptions like the Hubble law (velocity-redshift linear relation). Therefore they belong to another type of

The time scales like the SI (International System of Units) *atomic second* are based on frequency standards for *microwave* atomic clocks based on isotopes like cesium (133*Cs*) and rubidium

<sup>87</sup>*Rb*) and with frequencies of the order of GigaHertz (109*Hz*). While the national standard agencies (National Institute of Standards and Technology NIST in USA, National Physical Laboratory NPL in United Kingdom, Paris Observatory in France, Physikalisch-Technische Bundesanstalt PTB in Germany, Istituto Nazionale di Ricerca Metrologica INRIM in Italy) maintain an accuracy of 1 nanosecond per day (1*ns* = 10−<sup>9</sup> *s*), many primary cesium atomic clocks using laser cooled atomic fountains have an inaccuracy less than 100 picoseconds per day (1*ps* = 10−<sup>12</sup> *s*) with the best ones approaching 10 ps per day (Bize S. et al, 2005; Parker

If atomic clocks operating on different quantum transitions are considered as ideal clocks in general relativity, then they measure the same proper time (and not a coordinate time) along their trajectory (Guinot B., 1997). See Ref. (Reynaud S. et al, 2009) and its bibliography for the experiments on the *universality of clock rates* (relative frequency ratios between different clocks are constant at a level of the order of 10−<sup>16</sup> per year). See also Ref. (Perlick V., 1987, 1994) for another general relativistic effect, the *second clock effect*, according to which two clocks synchronized at the same point, then separated and finally rejoined remain synchronized in

A new family of *optical* atomic clocks in the region of 1015*Hz* is developing quickly with the help of optical frequency-combs for direct optical frequency measurements. They allow one to reach a fractional frequency inaccuracy of better than 10−<sup>17</sup> (corresponding to better than 1 ps per day) (Gill P., 2005; Rosenbad T. et al, 2008; Ludlow A.D. et al, 2008; Chou C.W. et al 2010a) and will become relevant for metrology in the near future. Moreover optical clocks allow to verify the "time dilation effect" for relative speeds of less than 10 m/s or for a change

In Ref. (Arias E.F., 2005) there is a review of time metrology with a comparison of various time scales, the use of GPS receiver for time transfer (see also Ref. (Petit G. et al, 2005)) and on the dissemination and access to the international time scales. See also Refs. (Lemonde P. et al, 2001; SIGRAV 2006) for the status of atomic clocks in space near the Earth or on spacecrafts

The Atomic Clock Ensemble in Space (ACES) mission of the European Space Agency ESA (ACES 2010; Cacciapuoti L. et al, 2007, 2008; Blanchet L. et al, 2000), to be launched in 2015, aims to put a new microwave atomic clock (PHARAO, Projet d'Horloge Atomique par Refroidissement d'Atome en Orbite) together with an active hydrogen maser (SHM, Space active Hydrogen Maser) on the International Space Station (ISS; height 400 Km, rotation period 90 min, inclination angle 51.6*o*). The two clocks will generate an on-board timescale with an expected frequency instability and inaccuracy at the 10−<sup>16</sup> level. There will be a frequency comparison between the space clocks and ground clocks using microwave links: in particular ACES will give the first precision measurement of the gravitational redshift of the geoid, namely of the 1/*c*<sup>2</sup> deformation of Minkowski light-cone caused by the geo-potential.

Riemannian space-times like Einstein's ones but not in Weyl space-times.

in height near the Earth's surface of less than 1 meter (Chou C.W., 2010b).

metrology.

T.E., 2010).

inside the Solar System.

(

**2.2 Atomic clocks and ACES**

The fundamental conceptual time scale is the *SI atomic second* whose definition is (Resolution 1, 1956)

*The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. This definition refers to a cesium atom at rest at a temperature of* 0*<sup>o</sup> K.*

It gives a precise and constant rate of time measurement for *observers local to the apparatus on the surface of the Earth, i.e. on the rotating geoid* (for them it is a unit of *proper time* (Guinot B., 1997)), in which such seconds are counted.

However from 1971 the conventional practical high-precision atomic time standard is a coordinate time (Guinot B., 1997), the *International Atomic Time TAI*. TAI is defined as a suitable weighted average of the SI time kept by over 200 atomic clocks (mainly cesium clocks) in about 70 national laboratories worldwide (Circular T263, 2009). The comparison of the clocks is done using GPS signals and two-way satellite time and frequency transfer.

The next step is to connect TAI to the time scales based the *Earth Rotation*, which were used in astronomical applications as telescope pointing, depended on the geographical location of the observer and were based on observing celestial bodies crossing the meridian every day. Two such scales are:

*Greenwich sidereal time*is the hour angle of the equinox measured with respect to the Greenwich meridian.

*Local sidereal time* is the local hour angle of the equinox or the Greenwich sidereal time plus the longitude (east positive) of the observer, expressed in time units. Sidereal time appears in two forms, *mean* (GAST Greenwich Apparent mean Sidereal Time or LMST Local Mean Sidereal Time) and *apparent* (LAST, Local Apparent Sidereal Time), depending on whether the mean or true equinox is the reference point. The position of the mean equinox is affected only by *precession* while the true equinox is affected by both *precession and nutation*. Let us remember that the equinox is a direction in space along the nodal line defined by the intersection of the ecliptic (the plane of the Earth's orbit) and equatorial planes. The difference between true and mean sidereal time is the *equation of the equinoxes*, which is a complex periodic function with a maximum amplitude of about 1sec. Of the two forms, apparent sidereal time is more relevant to actual observations, since it includes the effect of nutation. Greenwich (or local) apparent sidereal time can be operationally obtained from the right ascensions of celestial objects transiting the Greenwich (or local) meridian.

Nowadays *Universal Time UT* is the generic timescale based on Earth's rotation. It is determined by Very Long Baseline Interferometry (VLBI) observations of distant quasars with an accuracy of microseconds. There are various variants of UT. The most used is UT1, based on VLBI observations of quasars, on Lunar Laser Ranging (LLR), on determination of GPS satellite orbits. UT1 is the same everywhere on Earth and is proportional to the rotation angle of the Earth with respect to distant quasars.

An approximate version of UT1 is the *Coordinate Universal Time UTC*. It is an atomic time scale and the international standard for *civil time*. It is a hybrid time scale (ITUR, 2007), which uses SI atomic seconds on the geoid (it usually has 86 400 SI seconds per day), but subject to occasional 1 second adjustments (the so-called *leap second*) to keep it within 0.9 seconds from UT1 (*UT*1 ≈ *UTC* + *DUT*1 with *DUT*1 ≈ ±0.1*sec*) and to have *TAI* = *UTC* + �*AT* (�*AT* is an integer number of leap seconds).

Other civil times given in Ref. (Moyer T.D., 2003) are

*GPS Master Time*: it is an atomic time for GPS receiving station on Earth and for GPS satellites - *TAI* = *GPS* + *const*..

*ST Station Time*: it is an atomic time at a Deep Space Network (DSN) tracking station on Earth. It is assumed *UTC or GPS* <sup>=</sup> *ST* <sup>+</sup> *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* (*<sup>t</sup>* <sup>−</sup> *to*) + *<sup>c</sup>* (*<sup>t</sup>* <sup>−</sup> *to*)2.

As UT is slightly irregular in its rate, astronomers introduced Ephemeris Time and then replaced it with Terrestrial Time TT.

The *Ephemeris Time Teph* = *ET*, replacing an old barycentric dynamical time TDB, is a relativistic coordinate time based on high-precision ephemerides, which are lists of instantaneous positions of the centers of mass of Sun, Moon and planets with respect to (equatorial rectangular 3-coordinates of) BCRS for any date and time between 1600 and 2001, developed at the Jet Propulsion Laboratory (JPL) and denoted DE405/LE405 (Kaplan G.H., 2005; Standish E.M., 1998). Lunar rotation angles are also provided. The DE405 coordinate system has been aligned to the ICRS. The JPL ephemerides are computed by an N-body numerical integration carried out in BCRS.

*Terrestrial Time TT*, which is an astronomical time scale used for geocentric and topo-centric ephemerides. The "standard epoch" for modern astrometric reference data, designated J2000.0 is expressed as a TT instant: J2000.0 means 2000 January 1, 12*<sup>h</sup>* TT at geo-center (Julian date JD 24515450 TT; J2000.0 is shorthand for the celestial reference system defined by the mean dynamical equator and equinox of J2000.0) (Kaplan G.H., 2005). TT is an idealized form of TAI (*TT* = *TAI* + 32*s*.184). TT runs at the same rate as a time scale based on the SI second on the surface of the Earth.

As shown in Eq.(2.6) of Ref. (Kaplan G.H., 2005) we have *Teph* ≈ *TDB* ≈ *TT* + *F*(*T*), where *F*(*T*) is a given function of the number T of Julian centuries of TT from J2000.0 (*T* = (*JD*(*TT*) − 2451545.0)/36525).

See also Ref.(Moyer T.D., 2003), p.18, where the following chains of transformations are defined

*Teph* → *TAI* (→ *UT*1) → *UTC*, *GPS* → *ST* (ST = time scale of a tracking station on the Earth),

*Teph* → *TAI* → *GPS* → *ST* (ST = time scale at an Earth satellite)

Let us remark that the astronomical universal time UT1 is defined by using the new earth precession-nutation theory denoted IAU2000A (relating the International Celestial Reference Frame ICRF to the International Terrestrial Reference Frame ITRF from 2003), which has been replaced in 2009 with a more dynamically consistent precession model denoted IAU2006 (Coppola V., 2009; IAU, 2006).

According to IAU2006 UT1 is *linear* in the Earth rotation angle *θ*, a geocentric angle (such that ˙ *θ* = *ωearth* is the average angular velocity of rotation of the Earth) with a *Non-Rotating Origin NLO* in the equatorial plane orthogonal to the *Celestial Intermediate Pole CIP* from the axes centered in the *Celestial Intermediate Origin CIO* with no instantaneous rotation around 8 Will-be-set-by-IN-TECH

UT1 (*UT*1 ≈ *UTC* + *DUT*1 with *DUT*1 ≈ ±0.1*sec*) and to have *TAI* = *UTC* + �*AT* (�*AT* is

*GPS Master Time*: it is an atomic time for GPS receiving station on Earth and for GPS satellites

*ST Station Time*: it is an atomic time at a Deep Space Network (DSN) tracking station on Earth.

As UT is slightly irregular in its rate, astronomers introduced Ephemeris Time and then

The *Ephemeris Time Teph* = *ET*, replacing an old barycentric dynamical time TDB, is a relativistic coordinate time based on high-precision ephemerides, which are lists of instantaneous positions of the centers of mass of Sun, Moon and planets with respect to (equatorial rectangular 3-coordinates of) BCRS for any date and time between 1600 and 2001, developed at the Jet Propulsion Laboratory (JPL) and denoted DE405/LE405 (Kaplan G.H., 2005; Standish E.M., 1998). Lunar rotation angles are also provided. The DE405 coordinate system has been aligned to the ICRS. The JPL ephemerides are computed by an N-body

*Terrestrial Time TT*, which is an astronomical time scale used for geocentric and topo-centric ephemerides. The "standard epoch" for modern astrometric reference data, designated J2000.0 is expressed as a TT instant: J2000.0 means 2000 January 1, 12*<sup>h</sup>* TT at geo-center (Julian date JD 24515450 TT; J2000.0 is shorthand for the celestial reference system defined by the mean dynamical equator and equinox of J2000.0) (Kaplan G.H., 2005). TT is an idealized form of TAI (*TT* = *TAI* + 32*s*.184). TT runs at the same rate as a time scale based on the SI second on

As shown in Eq.(2.6) of Ref. (Kaplan G.H., 2005) we have *Teph* ≈ *TDB* ≈ *TT* + *F*(*T*), where *F*(*T*) is a given function of the number T of Julian centuries of TT from J2000.0 (*T* = (*JD*(*TT*) −

See also Ref.(Moyer T.D., 2003), p.18, where the following chains of transformations are

*Teph* → *TAI* (→ *UT*1) → *UTC*, *GPS* → *ST* (ST = time scale of a tracking station on the

Let us remark that the astronomical universal time UT1 is defined by using the new earth precession-nutation theory denoted IAU2000A (relating the International Celestial Reference Frame ICRF to the International Terrestrial Reference Frame ITRF from 2003), which has been replaced in 2009 with a more dynamically consistent precession model denoted IAU2006

According to IAU2006 UT1 is *linear* in the Earth rotation angle *θ*, a geocentric angle (such

*θ* = *ωearth* is the average angular velocity of rotation of the Earth) with a *Non-Rotating Origin NLO* in the equatorial plane orthogonal to the *Celestial Intermediate Pole CIP* from the axes centered in the *Celestial Intermediate Origin CIO* with no instantaneous rotation around

*Teph* → *TAI* → *GPS* → *ST* (ST = time scale at an Earth satellite)

an integer number of leap seconds).

replaced it with Terrestrial Time TT.

numerical integration carried out in BCRS.


the surface of the Earth.

(Coppola V., 2009; IAU, 2006).

2451545.0)/36525).

defined

Earth),

that ˙

Other civil times given in Ref. (Moyer T.D., 2003) are

It is assumed *UTC or GPS* <sup>=</sup> *ST* <sup>+</sup> *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* (*<sup>t</sup>* <sup>−</sup> *to*) + *<sup>c</sup>* (*<sup>t</sup>* <sup>−</sup> *to*)2.

the Earth axis to the axes centered in the *Terrestrial Intermediate Origin TIO* rotating with the Earth (in IAU2000A CIO and TIO were called CEO, Celestial Ephemeris Origin, and TEO, Terrestrial Ephemeris Origin, respectively).

The lengths of the sidereal (*θ*) and UT1 seconds, and the value of ˙ *θ*, are not precisely constant when expressed in a uniform time scale such as TT. The accumulated difference in time measured by a clock keeping SI seconds on the geoid from that measured by the rotation of the Earth is �*T* = *TT* − *UT*1. The long-term trend is for �*T* to increase gradually because of the tidal deceleration of the Earth's rotation, which causes UT1 to lag increasingly behind TT. In predicting the precise times of topo-centric phenomena, like solar eclipse contacts, both TT and UT1 come into play, and this requires assumptions about the value of �*T* at the time of the phenomenon. Alternatively, the circumstances of such phenomena can be expressed in terms of an *imaginary system of geographic meridians* that rotate uniformly about the Earth's axis (�*T* is assumed zero, so that UT1 = TT) rather than with the real Earth; the real value of �*T* then does not need to be known when the predictions are made. The zero-longitude meridian of the uniformly rotating system is called the *ephemeris meridian*.

Finally the astronomical conventions IUA2000 (Soffel M.H. et al, 2003) for the description of the Solar System (BCRS) and of the space near the Earth (GCRS) introduced the following two theoretical time scales not taken by any real clock but connected with Post-Newtonian solutions of Einstein's equations in special harmonic gauges with Sun, Earth, Moon, planets as matter.

*Barycentric Coordinate Time* - *tB* = *TCB* - it advances at a rate 1.55 10−<sup>8</sup> faster with respect to SI seconds on the surface of the Earth and is the time coordinate in BCRS.

*Geocentric Coordinate Time* - *tG* = *TCG* - it advances at a rate 6.97 10−<sup>10</sup> faster with respect to SI seconds on the surface of the Earth and is the time coordinate in GCRS. The connection to the terrestrial time is assumed to be *TT* = *TCG* − *LG* (*TCG* − *to*) with a constant rate *d TT dtG* <sup>=</sup> <sup>1</sup> <sup>−</sup> *LG* with *LG* <sup>=</sup> 6.969290134 10−10, while the transformation connecting TCB and TCG is given in the next Section.

Let us notice that the discussion whether it is better to use primary conventions based on atomic clocks or to revert to astronomical conventions is still open (Finkleman D. et al, 2011) and will be discussed again in 2012. For a recent update on the problem of time see Ref. (McCarthy D.D., 2009). At this stage it is difficult to say which point of view will become more relevant in the near future: how to compare astronomic precisions connected to VLBI and LLR with theoretical problems of atomic clocks like whether an atomic fountain clock can be approximated with a mass-point with a well defined proper time?
