USNO Circular 179 - U.S. Naval Observatory

54 MODELING THE EARTH’S ROTATION
of about 1.5 mm/cen. The exact motion of ϖ depends, of course, on what polar motion, which is
unpredictable, actually turns out to be.
The two non-rotating origins, σ and ϖ, are called the Celestial Intermediate Origin (CIO) and
the Terrestrial Intermediate Origin (TIO). Both lie in the same plane — the equator of the CIP.
The Earth Rotation Angle, θ, is defined as the geocentric angle between these two points. The
angle θ is a linear function of Universal Time (UT1). The formula, given in the note to res. B1.8 of
2000, is simply θ = 2π (0.7790572732640 + 1.00273781191135448 DU), where DU is the number of
UT1 days from JD 2451545.0 UT1. The formula assumes a constant angular velocity of the Earth:
no attempt is made to model its secular decrease due to tidal friction, monthly tidal variations,
changes due to the exchange of angular momentum between the atmosphere and the solid Earth,
and other phenomena. These effects will be reflected in the time series of UT1–UTC or ∆T values
(see Chapter 2) derived from precise observations.
The expression given above for θ is now the “fast term” in the formula for mean sidereal time;
see eq. 2.12. It accounts for the rotation of the Earth, while the other terms account for the motion
of the equinox along the equator due to precession.
The plane defined by the geocenter, the CIP, and TIO is called the TIO meridian. For most
ordinary astronomical purposes the TIO meridian can be considered to be identical to what is often
referred to as the Greenwich meridian. The movement of this meridian with respect to a conventional
geodetic system is important only for the most precise astrometric/geodetic applications.
It is worth noting that the TIO meridian, and the zero-longitude meridians of modern geodetic
systems, are about 100 m from the old transit circle at Greenwich (Gebel & Matthews 1971).
The term “Greenwich meridian” has ceased to have a technical meaning in the context of precise
geodesy — despite the nice line in the sidewalk at the old Greenwich observatory. This has become
obvious to tourists carrying GPS receivers!
6.3 The Path of the CIO on the Sky
If we take the epoch J2000.0 as the starting epoch for evaluating the integral that provides the
position of the CIO, the only mathematical requirement for the initial point is that it lie on the
instantaneous (true) equator of that date — its position along the equator is arbitrary. By convention,
however, the initial position of the CIO on the instantaneous equator of J2000.0 is set so that
equinox-based and CIO-based computations of Earth rotation yield the same answers; we want the
hour angle of a given celestial object to be the same, as a function of UT1 (or UTC), no matter
how the calculation is done. For this to happen, the position of the CIO of J2000.0 must be at
GCRS right ascension 0 ◦ 0 ′ 00. ′′ 002012. This is about 12.8 arcseconds west of the true equinox of
that date.
Since the CIO rides on the instantaneous equator, its primary motion over the next few millenia
is southward at the rate of precession in declination, initially 2004 arcseconds per century. Its rate
of southward motion is modulated (but never reversed) by the nutation periodicities. Its motion
in GCRS right ascension is orders of magnitude less rapid; remember that the CIO has no motion
along the instantaneous equator, and the instantaneous equator of J2000.0 is nearly co-planar with
the GCRS equator (xy-plane). The motion of the CIO in GCRS right ascension over the next few
millenia is dominated by a term proportional to t 3 ; the GCRS right ascension of the CIO at the
beginning of year 2100 is only 0. ′′ 068; at the beginning of 2200 it is 0. ′′ 573; and at the beginning of
2300 it is 1. ′′ 941. Nutation does produce a very slight wobble in the CIO’s right ascension, but the
influence of the nutation terms is suppressed by several orders of magnitude relative to their effect