USNO Circular 179 - U.S. Naval Observatory

52 MODELING THE EARTH’S ROTATION
space. What we need is an appropriate azimuthal 1 origin — a point in the moving equatorial plane,
which is orthogonal to the CIP.
6.2 Non-Rotating Origins
The reference point that we define must be such that the rate of change of the Earth’s rotation
angle, measured with respect to this point, is the angular velocity of the Earth about the CIP. As
the CIP moves, the point must move to remain in the equatorial plane; but the point’s motion must
be such that the measured rotation angle is not contaminated by some component of the motion
of the CIP itself.
The concept of a “non-rotating origin” (NRO) on the equator can be applied to any rotating
body. The NRO idea was first described by Bernard Guinot (Guinot 1979, 1981) and further
developed by Nicole Capitaine and collaborators (Capitaine et al. 1986; Capitaine 1990; Capitaine
& Chollet 1991; Capitaine et al. 2000; Capitaine 2000). The condition on the motion of such a
point is simple: as the equator moves, the point’s instantaneous motion must always be orthogonal
to the equator. That is, the point’s motion at some time t must be directly toward or away from the
position of the pole of rotation at t. Any other motion of the point would have a component around
the axis/pole and would thus introduce a spurious rate into the measurement of the rotation angle
of the body as a function of time. The point is not unique; any arbitrary point on the moving
equator could be made to move in the prescribed manner. For the Earth, the difference between
the motion of a non-rotating origin and that of the equinox on the geocentric celestial sphere is
illustrated in Fig. 6.1.
As illustrated in the figure, the motion of the non-rotating origin, σ, is always orthogonal to the
equator, whereas the equinox has a motion along the equator (the precession in right ascension).
How do we specify the location of a non-rotating origin? There are three possibilities, outlined
in the Formulas section of this chapter. In the most straightforward scheme, one simply uses the
GCRS right ascension of σ obtained from a numerical integration (the GCRS is the “geocentric
ICRS”). Alternatively, the position of σ can be defined by a quantity, s, that is the difference
between the lengths of two arcs on the celestial sphere. Finally, one can specify the location of σ
with respect to the equinox, Υ: the equatorial arc Υσ is called the equation of the origins. Whatever
geometry is used, the position of σ ultimately depends on an integral over time, because the
defining property of σ is its motion — not a static geometrical relationship with other points or
planes. The integral involved is fairly simple and depends only on the coordinates of the pole and
their derivatives with respect to time. The initial point for the integration can be any point on the
moving equator at any time t0.
1 The word “azimuthal” is used in its general sense, referring to an angle measured about the z-axis of a coordinate
system.