USNO Circular 179 - U.S. Naval Observatory

PRECESSION & NUTATION 41
an intermediate step in the computations, or the final step where only very low accuracy is needed
(10 arcseconds or worse) and nutation can be ignored.
Thus, a precession transformation is applied to celestial coordinates to convert them from
the mean equator and equinox of J2000.0 to the mean equator and equinox of another date, t.
Nutation is applied to the resulting coordinates to transform them to the true equator and equinox
of t. These transformations should be understood to be inherently geocentric rotations and they
originate in dynamical theories. Generally we will be starting with celestial coordinates in the
GCRS, which are obtained from basic ICRS data by applying the usual algorithms for proper
place. 4 As discussed in Chapter 3, the ICRS is not based on a dynamically defined equator and
equinox and so neither is the GCRS. Therefore, before we apply precession and nutation — and
if we require a final accuracy of better than 0.02 arcsecond — we must first apply the frame bias
correction (section 3.5) to transform the GCRS coordinates to the dynamical mean equator and
equinox of J2000.0. Schematically,
GCRS
frame bias
⇓
mean equator & equinox of J2000.0
precession
⇓
mean equator & equinox of t
nutation
⇓
true equator & equinox of t
Mathematically, this sequence can be represented as follows:
r true(t) = N(t) P(t) B r GCRS
where rGCRS is a direction vector with respect to the GCRS and rtrue(t) is the equivalent vector
with respect to the true equator and equinox of t. N(t) and P(t) are the nutation and precession
rotation matrices, respectively. The remainder of this chapter shows how to compute the elements
of these matrices. B is the (constant) frame-bias matrix given in section 3.5.
The transformation from the mean equator and equinox of J2000.0 to the mean equator and
equinox of t is simply
(5.4)
r mean(t) = P(t) r mean(J2000.0)
4 Computing proper place involves adjusting the catalog place of a star or other extra-solar system object for proper
motion and parallax (where known), gravitational light deflection within the solar system, and aberration due to the
Earth’s motions. For a solar system object there are comparable adjustments to its position vector taken from a
barycentric gravitational ephemeris. See section 1.3. In conventional usage, an apparent place can be considered to
be a proper place that has been transformed to the true equator and equinox of date. The details of the proper
place computations are beyond the scope of this circular but are described in detail in many textbooks on positional
astronomy, and in Hohenkerk et al. (1992), Kaplan, et al. (1989), and Klioner (2003).
(5.3)