USNO Circular 179 - U.S. Naval Observatory

PRECESSION & NUTATION 45
Where high accuracy is not required, the precession between two dates, t1 and t2, not too far
apart (i.e., where |t2 − t1| ≪ 1 century), can be approximated using the rates of change of right
ascension and declination with respect to the mean equator and equinox of date. These rates are
respectively
m ≈ 4612.16 + 2.78 T
n ≈ 2004.19 − 0.86 T (5.13)
where the values are in arcseconds per century and T is the number of centuries between J2000.0
and the midpoint of t1 and t2. If the dates are expressed as Julian dates, T = ((t1 + t2)/2 −
2451545.0)/36525. Then, denoting the celestial coordinates at t1 by (α1, δ1) and those at t2 by
(α2, δ2),
α2 ≈ α1 + τ (m + n sin α1 tan δ1)
δ2 ≈ δ1 + τ (n cos α1) (5.14)
where τ = t2 − t1, expressed in centuries. These formulas deteriorate in accuracy at high (or low)
declinations and should not be used at all for coordinates close to the celestial poles (how close
depends on the accuracy requirement and the value of τ) .
5.4.2 Formulas for Nutation
Nutation is conventionally expressed as two small angles, ∆ψ, the nutation in longitude, and ∆ɛ, the
nutation in obliquity. These angles are measured in the ecliptic system of date, which is developed
as part of the precession formulation. The angle ∆ψ is the small change in the position of the
equinox along the ecliptic due to nutation, so the effect of nutation on the ecliptic coordinates of
a fixed point in the sky is simply to add ∆ψ to its ecliptic longitude. The angle ∆ɛ is the small
change in the obliquity of the ecliptic due to nutation. The true obliquity of date is ɛ ′ = ɛ + ∆ɛ.
Nutation in obliquity reflects the orientation of the equator in space and does not affect the ecliptic
coordinates of a fixed point on the sky.
The angles ∆ψ and ∆ɛ can also be thought of as small shifts in the position of the celestial
pole (CIP) with respect to the ecliptic and mean equinox of date. In that coordinate system,
and assuming positive values for ∆ψ and ∆ɛ, the nutation in longitude shifts the celestial pole
westward on the sky by the angle ∆ψ sin ɛ, decreasing the pole’s mean ecliptic longitude by ∆ψ.
Nutation in obliquity moves the celestial pole further from the ecliptic pole, i.e., southward in
ecliptic coordinates, by ∆ɛ. (Negative values of ∆ψ and ∆ɛ move the pole eastward and northward
in ecliptic coordinates.)
The effect of nutation on the equatorial coordinates (α,δ) of a fixed point in the sky is more
complex and is best dealt with through the action of the nutation matrix, N(t), on the equatorial
position vector, r mean(t). Where high accuracy is not required, formulas that directly give the
changes to α and δ as a function of ∆ψ and ∆ɛ are available and are given at the end of this
subsection.
The values of ∆ψ and ∆ɛ are obtained by evaluating rather lengthy trigonometric series, of the
general form
∆ψ =
N
i=1
(Si + ˙ SiT ) sin Φi + C ′
i cos Φi