USNO Circular 179 - U.S. Naval Observatory

56 MODELING THE EARTH’S ROTATION
1. True equator and equinox of t — azimuthal origin at the true equinox (Υ) of t
2. Celestial Intermediate Reference System (CIRS) — azimuthal origin at the Celestial Intermediate
Origin (CIO or σ) of t
3. Terrestrial Intermediate Reference System (TIRS) — azimuthal origin at the Terrestrial Intermediate
Origin (TIO or ϖ) of t
In this circular we will often refer to these reference systems by the symbols E Υ , Eσ, and Eϖ,
respectively; E denotes an equatorial system, and the subscript indicates the azimuthal origin. Eϖ
rotates with the Earth whereas the orientations of E Υ and Eσ change slowly with respect to local
inertial space. The transformation between E Υ and Eϖ is just a rotation about the z-axis (which
points toward the CIP) by GAST, the angular equivalent of Greenwich apparent sidereal time.
The transformation between Eσ and Eϖ is a similar rotation, but by θ, the Earth Rotation Angle.
These two transformations reflect different ways — old and new — of representing the rotation of
the Earth.
A short digression into terrestrial, i.e., geodetic, reference systems is in order here. These
systems all have their origin at the geocenter and rotate with the crust of the Earth. Terrestrial
latitude, longitude, and height are now most commonly given with respect to a reference ellipsoid,
an oblate spheroid that approximates the Earth’s overall shape (actually, that best fits the geoid,
a gravitational equipotential surface). The current standard reference elliposid for most purposes
is that of the World Geodetic System 1984 (WGS 84), which forms the basis for the coordinates
obtained from GPS. The WGS 84 ellipsoid has an equatorial radius of 6,378,137 meters and a
polar flattening of 1/298.257223563. For the precise measurement of Earth rotation, however, the
International Terrestrial Reference System (ITRS) is used, which was defined by the International
Union of Geodesy and Geophysics (IUGG) in 1991. The ITRS is realized for practical purposes by
the adopted coordinates and velocities 3 of a large group of observing stations. These coordinates are
expressed as geocentric rectangular 3-vectors and thus are not dependent on a reference ellipsoid.
The list of stations and their coordinates is referred to as the International Terrestrial Reference
Frame (ITRF). The fundamental terrestrial coordinate system is therefore defined in exactly the
same way as the fundamental celestial coordinate system (see Chapter 3): a prescription is given
for an idealized coordinate system (the ITRS or the ICRS), which is realized in practice by the
adopted coordinates of a group of reference points (the ITRF stations or the ICRF quasars). The
coordinates may be refined as time goes on but the overall system is preserved. It is important to
know, however, that the ITRS/ITRF is consistent with WGS 84 to within a few centimeters; thus
for all astronomical purposes the GPS-obtained coordinates of instruments can be used with the
algorithms presented here.
Our goal is to be able to transform an arbitrary vector (representing for example, an instrumental
position, axis, boresight, or baseline) from the ITRS (≈WGS 84≈GPS) to the GCRS. The three
equatorial reference systems described above — E Υ , Eσ, and Eϖ — are waypoints, or intermediate
stops, in that process. The complete transformations are:
3 The velocities are quite small and are due to plate tectonics and post-glacial rebound.