Dynamical Time: TT, TDB

Dynamical time (formerly Ephemeris Time, ET) is the independent variable in the theories which describe the motions of bodies in the solar system. When using published formulae or tables that model the position of the Earth in its orbit, for example, or look up the Moon's position in a precomputed ephemeris, the date and time must be in terms of one of the dynamical time scales. It is a common but understandable mistake to use UTC directly, in which case the results will be over a minute out (at the time of writing).

It is not hard to see why such time scales are necessary. UTC would clearly be unsuitable as the argument of an ephemeris because of leap seconds. A solar-system ephemeris based on UT1 or sidereal time would somehow have to include the unpredictable variations of the Earth's rotation. TAI would work, but in principle the ephemeris and the ensemble of atomic clocks would eventually drift apart. In effect, the ephemeris is a clock, with the bodies of the solar system the hands from which the ephemeris time is read.

Only two of the dynamical time scales are of any great importance to observational astronomers, TT and TDB.

Terrestrial Time, TT, is the theoretical time scale of apparent geocentric ephemerides of solar system bodies. It applies to clocks at sea-level, and for practical purposes it is tied to Atomic Time TAI through the formula TT $=$ TAI $+$  $32^{\rm s}\hspace{-0.3em}.184$. In practice, therefore, the units of TT are ordinary SI seconds, and the offset of $32^{\rm s}\hspace{-0.3em}.184$ with respect to TAI is fixed. The SLALIB function sla_DTT returns TT$-$UTC for a given UTC (n.b. sla_DTT calls sla_DTT, and the latter must be an up-to-date version if recent leap seconds are to be taken into account).

Barycentric Dynamical Time, TDB, is a coordinate time, suitable for labelling events that are most simply described in a context where the bodies of the solar system are absent. Applications include the emission of pulsar radiation and the motions of the solar-system bodies themselves. When the readings of the observer's TT clock are labelled using such a coordinate time, differences are seen because the clock is affected by its speed in the barycentric coordinate system and the gravitational potential in which it is immersed. Equivalently, observations of pulsars expressed in TT would display similar variations (quite apart from the familiar light-time effects).

TDB is defined in such a way that it keeps close to TT on the average, with the relativistic effects emerging as quasi-periodic differences of maximum amplitude rather less than 2ms. This is negligible for many purposes, so that TT can act as a perfectly adequate surrogate for TDB in most cases, but unless taken into account would swamp long-term analysis of pulse arrival times from the millisecond pulsars.

Most of the variation between TDB and TT comes from the ellipticity of the Earth's orbit; the TT clock's speed and gravitational potential vary slightly during the course of the year, and as a consequence its rate as seen from an outside observer varies due to transverse Doppler effect and gravitational redshift. The main component is a sinusoidal variation of amplitude $0^{\rm s}\hspace{-0.3em}.0017$; higher harmonics, and terms caused by Moon and planets, lie two orders of magnitude below this dominant annual term. Diurnal (topocentric) terms, a function of UT, are $2\,\mu$s or less.

The IAU 1976 resolution defined TDB by stipulating that TDB$-$TT consists of periodic terms only. This provided a good qualitative description, but turned out to contain hidden assumptions about the form of the solar-system ephemeris and hence lacked dynamical rigour. A later resolution, in 1991, introduced new coordinate time scales, TCG and TCB, and identified TDB as a linear transformation of one of them (TCB) with a rate chosen not to drift from TT on the average. Unfortunately even this improved definition has proved to contin ambiguities. The SLALIB sla_RCC function implements TDB in the way that is most consistent with the 1976 definition and with existing practice. It provides a model of TDB$-$TT accurate to a few nanoseconds.

Unlike TDB, the IAU 1991 coordinate time scales TCG and TCB (not supported by SLALIB functions at present) do not have their rates adjusted to track TT and consequently gain on TT and TDB, by about $0^{\rm s}\hspace{-0.3em}.02$/year and $0^{\rm s}\hspace{-0.3em}.5$/year respectively.

As already pointed out, the distinction between TT and TDB is of no practical importance for most purposes. For example when calling sla_PRENUT to generate a precession-nutation matrix, or when calling sla_EVP or sla_EPV to predict the Earth's position and velocity, the time argument is strictly TDB, but TT is entirely adequate and will require much less computation.

The time scale used by the JPL solar-system ephemerides is called $T_{eph}$ and is numerically the same as TDB.

Predictions of topocentric solar-system phenomena such as occultations and eclipses require solar time UT as well as dynamical time. TT/TDB/ET is all that is required in order to compute the geocentric circumstances, but if horizon coordinates or geocentric parallax are to be tackled UT is also needed. A rough estimate of $\Delta {\rm T} = {\rm ET} - {\rm UT}$ is available via the function sla_DT. For a given epoch (e.g. 1650) this returns an approximation to $\Delta {\rm T}$ in seconds.