USNO Circular 179 - U.S. Naval Observatory

MODELING THE EARTH’S ROTATION 57
Equinox-Based Transformation
ITRS or WGS 84
polar motion
⇓
Eϖ — Terrestrial Intermediate Ref. System
Greenwich apparent sidereal time
⇓
E Υ — true equator & equinox
equinox-based rotation for
nutation + precession + frame bias
⇓
GCRS
CIO-Based Transformation
ITRS or WGS 84
polar motion
⇓
Eϖ — Terrestrial Intermediate Ref. System
Earth Rotation Angle
⇓
Eσ — Celestial Intermediate Ref. System
CIO-based rotation for
nutation + precession + frame bias
⇓
GCRS
which are equivalent. That is, given the same input vector, the same output vector will result
from the two procedures. In the CIO-based transformation, the three sub-transformations (for
polar motion, Earth Rotation Angle, and nutation/precession/frame bias) are independent. That
is not true for the equinox-based method, because apparent sidereal time incorporates precession
and nutation. Each of the two methods could be made into a single matrix, and the two matrices
must be numerically identical. That means that the use of the CIO in the second method does not
increase the precision of the result but simply allows for a mathematical redescription of the overall
transformation — basically, a re-sorting of the effects to be taken into account. This redescription
of the transformation provides a clean separation of the three main aspects of Earth rotation, and
recognizes that the observations defining modern reference systems are not sensitive to the equinox.
It thus yields a more straightforward conceptual view and facilitates a simpler observational analysis
for Earth-rotation measurements and Earth-based astrometry.
These transformations are all rotations that pivot around a common point, the geocenter. Although
developed for observations referred to the geocenter, the same set of rotations can be applied
to observations made from a specific point on the surface of the Earth. (This follows from the assumption
that all points in or on the Earth are rigidly attached to all other points. The actual
non-rigidity — e.g., Earth tides — is handled as a separate correction.) In such a case, the computations
for parallax, light-bending, aberration, etc., must take into account the non-geocentric
position and velocity of the observer. Then the final computed coordinates are referred, not to the
GCRS, but rather to a proper reference system (in the terminology of relativity) of the observer.
We return to the more familiar problem mentioned at the beginning of the chapter: the computation
of local hour angle. In the usual equinox-based scheme, the apparent place 4 of the star or
planet is expressed with respect to the true equator and equinox of date (E Υ ). The local hour angle
is just h = GAST – α Υ + λ, where GAST is Greenwich apparent sidereal time, α Υ is the apparent
4 See footnote 1 on page 41.