USNO Circular 179 - U.S. Naval Observatory

48 PRECESSION & NUTATION
S1 = sin (ɛ)
S2 = sin (−∆ψ)
S3 = sin (−ɛ − ∆ɛ)
then the nutation matrix can also be written:
⎛
N(t) =
⎜
⎝
C1 = cos (ɛ)
C2 = cos (−∆ψ) (5.20)
C3 = cos (−ɛ − ∆ɛ)
C2 S2C1 S2S1
−S2C3 C3C2C1 − S1S3 C3C2S1 + C1S3
S2S3 −S3C2C1 − S1C3 −S3C2S1 + C3C1
⎞
⎟
⎠ (5.21)
Where high accuracy is not required, coordinates corrected for nutation in right ascension and
declination can be obtained from
αt ≈ αm + ∆ψ (cos ɛ ′ + sin ɛ ′ sin αm tan δm) − ∆ɛ cos αm tan δm
δt ≈ δm + ∆ψ sin ɛ ′ cos αm + ∆ɛ sin αm (5.22)
where (αm, δm) are coordinates with respect to the mean equator and equinox of date (precession
only), (αt, δt) are the corresponding coordinates with respect to the true equator and equinox of
date (precession + nutation), and ɛ ′ is the true obliquity. Note the tan δm factor in right ascension
that makes these formulas unsuitable for use close to the celestial poles.
The traditional formula for the equation of the equinoxes (the difference between apparent and
mean sidereal time) is ∆ψ cos ɛ ′ , but in recent years this has been superceded by the more accurate
version given in eq. 2.14.
5.4.3 Alternative Combined Transformation
The following matrix, C(t), combines precession, nutation, and frame bias and is used to transform
vectors from the GCRS to the Celestial Intermediate Reference System (CIRS). The CIRS is defined
by the equator of the CIP and an origin of right ascension called the Celestial Intermediate Origin
(CIO). The CIO is discussed extensively in Chapter 6. There, the CIRS is symbolized Eσ; it is
analogous to the true equator and equinox of date, but with a different right ascension origin.
The matrix C(t) is used in the sense
r CIRS = C(t) r GCRS
(5.23)
and the components of C(t), as given in the IERS Conventions (2003) and The Astronomical
Almanac, are
⎛
⎞
C(t) = R3(−s)
⎜
⎝
1 − bX 2 −bXY −X
−bXY 1 − bY 2 −Y
X Y 1 − b(X 2 + Y 2 )
⎟
⎠ (5.24)
where X and Y are the dimensionless coordinates of the CIP in the GCRS (unit vector components),
b = 1/(1 + Z), Z = √ 1 − X 2 − Y 2 , and s is the CIO locator, a small angle described in Chapter 6.
All of these quantities are functions of time. R3 is a standard rotation around the z axis; see
“Abbreviations and Symbols Frequently Used” for a precise definition.