# The time scales TAI and EAL

### International Atomic Time TAI

TAI (Temps Atomique International) is calculated by the International Bureau for Weights and Measures (BIPM) and published monthly in the "Circular T" in the form of time differences to other free atomic time scales. The difference between TAI and TA(PTB) on February 1, 2015, for example, was 2005,7 ns.

The starting point of TAI was fixed such that on January 1, 1958, at 0:00 TAI approximately agreed with the corresponding instant of the astronomical time scale UT1. From then to the end of 1999, 1,325,376,000 atomic seconds elapsed. However, after this period, the earth had not yet completed its 15,340th revolution; 31.3 seconds were lacking (rotational angle 0.13°). Due to this systematic deviation between TAI and the time scale proportional to the earth's rotation, TAI is not used in everyday life to date events. Instead, the time scale UTC is used.

### The calculation of TAI and EAL

The method of calculating TAI was modified several times in response to the changing requirements and the quality of the clocks involved. The calculation is based on the comparisons of the time scales realized at the individual time institutes among one another as well as on comparisons of the individual (nearly 400 in total) clocks within each institute. The BIPM applies a specific calculation method (ALGOS) to first determine the rate instabilities of all clocks involved. The smaller the instability, the greater the assigned statistic weight with which a clock contributes to averaging the clock rates. The rate itself and the specified uncertainty of the clock are not taken into account.

The ALGOS algorithm limits the maximum relative statistic weight of a clock. This limitation enhances the reliability of the time scale even if it implies that the quality of the best clocks is not fully utilized. The universal mean thus formed is called EAL (Échelle Atomique Libre, free atomic time scale) and can in some respect be compared with the TA scales of the individual time institutes.

The scale unit of EAL is not in accord with the SI second. This was already found in the early 1970s through comparison measurements against the primary clock CS1 of PTB and later confirmed by the National Research Council, Canada, and the National Bureau of Standards, USA. Since the beginning of 1977, TAI has therefore been obtained from EAL by means of frequency adjustments. The relative deviation between TAI and EAL is in the range of 10-13.

For this purpose, EAL is compared with the primary clocks of the time institutes in France, Japan, Russia, USA and with CS1, CS2, CSF1 and CSF2 of PTB. TAI is adjusted in infrequent small steps of less than relativly 1·10-15 because otherwise the instability of the time scale would be increased. Instead, the BIPM monthly publishes the calculated mean deviation between the scale unit of TAI and the SI second as it was realized with the primary clocks involved. In June 2013, for example, this deviation was 0.0·10-15 and was known with an uncertainty of 0.3·10-15.

### TAI and the gravitational time dilatation

When time scale information is transferred from one reference frame to another, the transformation laws of the General Relativity must be taken into account. TAI is defined in a geocentric reference system. Its scale unit is the SI second as realised at the mean sea level of the rotating earth. The corresponding gravitational equipotential surface is also known as the geoid.

Due to the relativistic time dilatation caused by the earth's gravitational potential, the SI second could only be realised by atomic clocks at sea level if no corrections were applied. In order to compensate for the gravitational time dilatation, the rates of atomic clocks located at an altitude h above sea level are corrected by a relative amount of -1.09.10-16 (h/m). The PTB clocks for example are located at heights of h = 78 until 79 m so that the relative rate correction is about -8.6.10-15. This takes into account that the atomic second intervals realised by the PTB clocks are shorter by 8.6.10-15 than the SI second produced by a caesium clock located on the geoid.