5 Rotation of the Earth

The rotation of the Earth is ideally represented in geocentric coordinates by a transformation matrix $\cal M$ between a terrestrial reference system and the celestial reference system. In practice, the terrestrial system is the International Terrestrial Reference System (ITRS) defined by the International Union for Geodesy and Geophysics (IUGG 1992) and represented physically by the International Terrestrial Reference Frame (ITRF), which is a catalogue of positions and velocities of point marks on the Earth. The latest version is ITRF-2000 (Altamimi et al. 2002). The celestial system is, of course, the ICRS.

Now, just as was traditionally done, the transformation is made in two steps. First, the correction for polar motion is described by a matrix $\cal R$transforming ITRS into an Intermediate Reference System defined by the Celestial Intermediate Pole (CIP), the corresponding celestial equator, and the Celestial Ephemeris Origin (CEO). Then, a precession-nutation matrix $\cal N$ transforms the intermediate system into the ICRS. The difference with the previous procedure lies only in the definition of the intermediate system. One has, as before:

$${\cal M} = {\cal N} \times {\cal R}.$$

The pole is no longer the CEP, but the CIP. The difference is how the separation is done between polar motion and nutation. The CIP is defined in such a way that all nutation terms with periods smaller than 2 days are included in the polar motion. In addition, the subdiurnal tidal polar motion is taken into account by a model. In practice, the change that will occur on January 1, 2003, will be transparent to the user: the nutation theory and the published polar motion will conform to this change. The true equator, which was defined with respect to the CEP, will now be defined with respect to the CIP. Actually, the difference is of the order of a few tens of microarcseconds (Souchay 2000).

The other difference is in the choice of the CEO instead of the mean equinox on the equator. This choice has at least three advantages:

1.

the difficulties with the equinox, already mentioned in the Introduction, are removed;

2.

the CEO is defined in such a way that its motion on a fixed celestial sphere has no component along the equator. This means that the instantaneous movement of the CEO is always at right-angles to the instantaneous equator (Guinot 1979); and

3.

the angle $\theta$ (called Stellar Angle or Earth Rotation Angle) measured on the equator between the CEO and the longitude origin in the ITRF is such that it yields UT1 through a strictly linear relationship.

The position of the CEO on the equator is defined by an integral that involves only the path followed by the precessing-nutating pole since the reference epoch (Capitaine et al. 1986). This can be computed from the precession-nutation model and/or observations. Either numerical integration or an approximate formula given by Capitaine et al. (2000) can be used (see Sect. 7.1). The CEO motion depends only very slightly upon the pole used as the input

Stellar Angle, or Earth Rotation Angle, is the replacement for Greenwich Apparent Sidereal Time (GAST). The GAST origin was the equinox, which, in contrast to the CEO, had components of motion along the equator; these arose because the equator and ecliptic are moving relative to one another. Consequently, the relationship between GAST and UT1 included terms due to precession and nutation. The precession terms appeared in the formula that links Greenwich Mean Sidereal Time (GMST) and UT1, while the nutation terms appear in the formula GAST= GMST $+\Delta \psi \cos \varepsilon$ as the equation of the equinoxes. Now, the Stellar Angle does not depend on precession or nutation (see Sect. 7.3).

Remarks:

1.

When introduced by Guinot (1979), the CEO was called a non-rotating origin. This term is potentially confusing given that the CEO is moving, simply because the equator, on which it is located, is itself moving. The "non-rotating'' description refers to the absence of a component of motion along the equator; the instantaneous motion of the CEO is defined to be about an axis in the equatorial plane, such that there is no rotation about the z-axis. Note that this is a kinematical definition. It can be hard to visualize because there is no geometrical definition and, hence, a diagram defining where the CEO is at any given time cannot be drawn. The kinematical definition states how the CEO moves, not where it is. This motion does not depend on the choice of the system in which the coordinates of the pole of rotation are expressed.

2.

As pointed out by Fukushima (2001), the position of the CEO has a zig-zag secular motion across the ICRF sky over long periods of time (tens of thousands of years). However, it is the only choice of origin that gives an Earth rotation angle formula free from precession-nutation terms; all the other options listed by Fukushima contain such "crosstalk'' effects to a greater or lesser degree. The small motion of the CEO is due to the choice of the constant of integration and the x-axis of the ICRF being near the equinox of J2000.0. Moreover, the slow secular motion of the CEO has no obvious unfavorable consequences. Note that the quantity s (formula 8), analogous to the precession in right ascension, has a secular component mainly due to a t³ term, which reaches only 40 mas in 2100, and that the CEO is independent of the celestial reference system adopted.