geoid by using an ensemble of atomic clocks spread over the surface and low orbital space of the Earth.

TAI is conventionally related to the Terrestrial Time (TT) and the Universal Coordinate Time (UTC) by the two following relations:
T T (T AI) = T AI + 32.184s . (4)

and UT C = T AI −leap seconds , (5) leap seconds being added by the International Earth Rotation Service at irregular intervals to compensate for the Earth’s rotation irregularities.

Mainly, leap seconds are used to allow UTC to closely track UT1, which effectively represents the Earth rotation.

It is thus more convenient to consider a regular time-scale like TT.

Moreover due to its IAU1991 definition, TT is related to TCG by a fixed linear function as follows:
T CG −T T = LG × (JD −2443144.5) × 86400 , (6) where LG = 6.969290134×10−10 and JD is TAI measured in Julian days.
2.3.

The TDB-TCB transformation The TDB has a more tumultuous history.

It was introduced by IAU1976 in order to remain close to TT up to periodic variations.

However such a definition was flawed because in that case TDB cannot be a linear function of TCB.

Consequently, the relativistic equations derived in TCB cannot be simply adapted to TDB.

In practice, Caltech/JPL, Harvard/CfA, and RAS/IAA ephemeris programs have used a relativistic coordinate time whose mean rate is automatically adjusted to the mean rate of TT by way of the ephemeris fits.

As such, those times, often referred to as Teph, can be related to TCB by the following equation with the notation of equation 1:
dTeph
T CB = 1 + 1
c2 α(T CB) (7) and then from (Irwin and Fukushima 1999), T CB ≡ 1 1 −LB (Teph −Teph0). (8) Differences between Teph and TCB have been estimated by numerical quadrature method of equation 7 which is equivalent to equation 10 limited at the 1/c2 terms.

Chebychev polynomials of the solutions were provided to users and corrections were brought to the analytical representation of TDB done few years earlier by Fairhead and Bretagnon (1990).

This situation changed with the recommendation B3 of IAU2006 which made TDB a fixed linear function of TCB as follows:
T DB = T CB −LB(JD −T0) × 86400 + T DB0, (9) where T0 = 2443144.5003725, LB = 1.550519768 × 10−8, T DB0 = −6.55 × 10−5s and JD is the TCB Julian date.

TCB value (like TT and TCG ones) is T0 for the event 1977 January 1 00h 00m 00s TAI at the geocenter.

It increases by one for each 86400s of TCB.

The use of a TDB planetary ephemeris as defined in equation (9) has no significant impact for normal users: values of masses and initial conditions of solar system objects being the same at the common level of accuracy.
2.4.

Implementation in INPOP08 From equations (1), (6) and (9), we have (Klioner, 2007):
d (T T −T DB)
c2 α � (1 + LB −LG) −LG
dT DB = � LB + 1
+ 1
c4 β (10)
One can notice that the values of α and β do not change when using quantities (GM, positions, velocities and accelerations of bodies) expressed in TDB instead of TCB units.

It is then straightforward to construct a time ephemeris T T −T DB = f(T DB) (noted ∆T DB = f(T DB)).

This is the choice done for INPOP08.

The central part of our implementation into INPOP is to numerically integrate Eq. (10) together with the equations of motions of all bodies.

Because the right member of (10) does not depend on ∆T DB, the equation (10) is not strictly an ordinary differential equation.

In the computation of α, A enumerates all massive bodies of the Solar System, that is, the Sun, the planets, Pluto, the Moon and all the 303 asteroids.

In β (which is divided by c4, and so is less important than α), A and B enumerate all bodies except the 298 “small” asteroids.

For the acceleration term aA, all the interactions are taken into account, including newtonian interactions, relativistic corrections, figure and tide effects.