Relationship between CIS and CTS

The transition from the space-fixed equatorial system (CIS) to the conventional terres- trial system (CTS) is realized through a sequence of rotations that account for

− nutation

− Earth rotation including polar motion.

These can be described with matrix operations. For a point on the celestial sphere, described through its position vectorr, we can write

rCTS= SNP rCIS. (2.16)

The elements of the rotation matrices must be known with sufficient accuracy for each observation epoch. These rotations are now considered in more detail.

(a) Precession and Nutation

Earth’s axis of rotation and its equatorial plane are not fixed in space, but rotate with respect to an inertial system. This results from the gravitational attraction of the Moon and the Sun on the equatorial bulge of Earth. The total motion is composed of a mean secular component (precession) and a periodic component (nutation) (Fig. 2.5).

a _b ecliptic equator E_N P_N 18.6 years 23.◦5

✗

Figure 2.5. Precession and nutation; Earth’s rotation axisP_SP_N describes a conic about the ecliptic polesE_S, E_N

The position and orientation of the equatorial plane and the first point of Aries, ✗, is called mean equator and mean equinox, respectively, when only the influence of precession is considered. When nutation is taken into account, they are called true equator and true equinox. The respective star coordinates are termed mean positions or true positions. Mean positions can be transformed from the reference epoch t0 (J2000.0) to the required observation epocht using the precession matrix

P = R3(−z)R2(θ)R3(−ζ) (2.17)

with three rotations (2.3) by the angles−z, θ, −ζ

z = 0◦.6406161 T + 0◦.0003041 T2_{+ 0}◦_{.0000051 T}3

θ = 0◦.5567530 T − 0◦.0001185 T2_{− 0}◦_{.0000116 T}3 _(2.18) ζ = 0◦.6406161 T + 0◦.0000839 T2_{+ 0}◦_{.0000050 T}3_.

T = (t − t0) is counted in Julian centuries of 36525 days.

The transformation from the mean equator and equinox to the instantaneous true equator and equinox for a given observation epoch is performed with the nutation matrix

N = R1(−ε − ,ε)R3(−,ψ)R1(ε) (2.19)

where

ε obliquity of the ecliptic, ,ε nutation in obliquity,

,ψ nutation in longitude (counted in the ecliptic), and

ε = 23◦2621.448 − 46.815 T − 0.00059 T2+ 0.001813 T3. (2.20) In 1980 the International Astronomical Union (IAU) adopted a nutation theory (Wahr, 1981) based on an elastic Earth model. ,ψ is computed using a series expansion involving 106 coefficients and,ε using one of 64 coefficients. The principal terms are

,ψ = −17.1996 sin . − 1.3187 sin(2F − 2D + 2.) − 0.2274 sin(2F + 2.) (2.21) ,ε = 9.2025 cos . + 0.5736 cos(2F − 2D + 2.) + 0.0977 cos(2F + 2.)

(2.22) with

. mean ecliptic longitude of the lunar ascending node, D mean elongation of the Moon from the Sun,

F = λM− .

withλ_M the mean ecliptic longitude of the Moon. By applying the transformations (2.17) and (2.19) we obtain true coordinates

rT = (XT, YT, ZT) (2.23)

in the instantaneous true equatorial system.

More details can be found in Seidelmann (ed.) (1992) and McCarthy (2000). The IAU decided at its 24th General Assembly in 2000 to replace the IAU 1976 Precession Model and the IAU 1980 Theory of Nutation by the Precession-Nutation Model IAU 2000, beginning on January 1, 2003. Two versions of the model exist (Capitaine, et al., 2002):

The IAU 2000A model contains 678 luni-solar terms and 687 planetary terms and provides directions of the celestial pole in the geocentric celestial reference system (GCRS) with an accuracy of 0.2 mas. The abridged model IAU 2000B includes 80

luni-solar terms and a planetary bias. The difference between both models is not greater than 1 mas after about 50 years.

(b) Earth Rotation and Polar Motion

For the transition from an instantaneous space-fixed equatorial system to a conventional terrestrial reference system we need three further parameters. They are called Earth Rotation Parameters (ERP) or Earth Orientation Parameters (EOP), namely

GAST the Greenwich apparent sidereal time xp, yp the pole coordinates.

GAST is also expressed as the difference UT1-UTC, cf. [2.2.2]. Unlike precession (2.17) and nutation (2.19), Earth rotation parameters cannot be described through theory but must be determined through actual observations by an international time and latitude service. Since the beginning of the last century until about 1980, this service was based mainly on astronomical observations (see [12.4.2]). On January 1, 1988 the International Earth Rotation Service (IERS) (Boucher et al., 1988) took over this task. The principle observation techniques now used are laser ranging to satellites and to the Moon [8.5.5] and Very Long Baseline Interferometry (VLBI) [11.1.2].

Fig. 2.6 shows the geometric situation for the transformation. The Earth-fixed system is realized through the conventional orientation of a Cartesian (X, Y, Z)CT system. TheZCT-axis is directed toward the conventional terrestrial pole CTP, and the XCT-axis toward the mean Greenwich meridian. The relative position of the instantaneous true pole with respect to the conventional terrestrial pole CTP is usually described through the pole coordinatesx_p,y_p(e.g. Mueller, 1969; Schödlbauer, 2000).

GAST true equator mean meridian Greenwich true instantaneous pole CTP conventional equator ZCT Z_T yp xp M X_T

✗

The relative orientation of theXCT-axis depends directly on Earth rotation and is determined through the apparent (= true) Greenwich Sidereal Time GAST (cf. [2.2.2]). The symbolθ is often used to denote GAST. The matrix which transforms the instan- taneous space-fixed system to the conventional terrestrial system is

S = R2(−xp)R1(−yp)R3(GAST) (2.24) where

R3(GAST) = 

_{− sin(GAST) cos(GAST) 0}cos(GAST) sin(GAST) 0

0 0 1



 (2.25)

and, because of small angles,

R2(−xp)R1(−yp) =   10 01 x0p −xp 0 1    10 01 −y0_p 0 y_p 1   =   10 01 −yxp_p −xp yp 1   . (2.26) For most practical purposes, the pole of the instantaneous true space-fixed equatorial system can be considered to be identical to the so-called Celestial Ephemeris Pole (CEP). The CEP is defined to be the reference pole for the computation of polar motion and nutation and is free of the quasi diurnal nutation terms with respect to Earth’s crust and inertial space (Seidelmann (ed.), 1992). The observed differences between the CEP and the conventional precession-nutation model are named celestial pole offsetsdψ, dε. They reach a few milliarcseconds and are published by the IERS,

see [11.1.2].

The CEP will be replaced by the Celestial Intermediate Pole (CIP) along with the introduction of the IAU 2000 precession-nutation model. Accordingly, for the rigorous definition of sidereal rotation of Earth, based on the early concept of the “non-rotating origin” (Guinot, 1979) the Terrestrial Ephemeris Origin (TEO), and the Celestial Ephemeris Origin (CEO), defined on the equator of the CIP, will be introduced. This implies that UT1 be linearly proportional to the Earth rotation angle. The CIP coincides with the CEP in the low-frequency domain (periods larger than two days). The reason for the adoption of the CIP is to clarify the difference between nutation and polar motion at high frequencies (periods less than two days). For details see e.g. Capitaine et al. (2000); Capitaine, et al. (2002).

The IERS will introduce the new system in 2003. The old system, however, will continue to be used, and the IERS will continue to provide all necessary data until further notice.