7 UT1 and stellar angle

Let us now analyze in detail the concepts introduced above.

7.1 Celestial ephemeris origin

For the concept of the CEO, consider a rigid celestial sphere, with reference ICRF Cartesian axes OXYZ with $\Sigma_{0}$ being the intersection of OX with the celestial sphere, as in Fig. 2. Let P(t) be the instantaneous celestial pole, now called the Celestial Intermediate Pole (CIP) at time t with Cartesian coordinates $O\xi \eta \zeta$ such that $O\zeta$ is along OP and $\sigma$ is the point on the equator of P where $O\xi$ pierces the sphere. The condition is imposed that for any infinitesimal displacement of P there is no rotation around $O\zeta$. Then $\sigma$ would be the non-rotating origin on the moving equator of P.

The motion of point P between dates t₀ and t is determined by evaluating the quantity s (t)

$$s = [\sigma N] - [\Sigma_{0} N] - ([\sigma _{0} N_{0}] - [\Sigma_{0} N_{0}]) ,$$ (4)

Figure 2: Definition of the CEO on ICRF with motion of the pole from t0 (P0) to t (P).

where quantities in brackets are arcs measured on inclined great circles shown in Fig. 2, $\sigma_{0} = \sigma (t_{0})$, N₀ and N are the ascending nodes of the equators at t₀ and t in the equator of the ICRS. Then s is given by Capitaine et al. (1986, Sect. 5.2):

$$s = -\int_{t_0}^t {{X \dot{Y} - Y \dot{X}} \over {1+Z}} {\rm d}t - ([\sigma _{0} N_{0}] - [\Sigma_{0} N_{0}]) ,$$ (5)

where X, Y and Z are the coordinates of the pole and the dots denote their time derivative. In polar coordinates, if D is the angle $(\vec{OZ,OP})$ and E is the angle of the plane $\vec{OZP}$ with respect to the principal plane $\vec{OZX}$, the equation is

$$s = \int_{t_0}^t (\cos D - 1) {\dot E} {\rm d}t -([\sigma_{0} N_0] - [\Sigma_{0} N_0]).$$ (6)

The convention may be adopted that $\sigma_0$ is on the true equator at epoch t₀ and $[\sigma _{0} N_{0}] = [\Sigma_{0} N_{0}]$. Then $[\sigma _{0} N_{0}]$ is the constant s(t₀) in the integral. Then,

$$s = \int_{t_0}^t (\cos D - 1) {\dot E} {\rm d}t.$$ (7)

 

 

Table 1: Additional terms in $\sigma (t)$ and $\Delta \sigma (t)$ for Eqs. (8) and (9) in $\mu$as units (Capitaine et al. 2002).

$\alpha _k$	Ck	$\alpha _i$	Di
$\Omega$	-2641	l'	-6
$2\Omega$	-63	l' + 2L	+2
$\Omega + 2L$	-12	l	-3
$\Omega - 2L$	+11	2 L'+ 2L	+2
2 L	+5	L +2 L'	+1
$\Omega +2 L'$	-2
$\Omega -2 L'$	+2
$3\Omega$	+2
$l' + \Omega$	+1
$l' - \Omega$	+1

For the period 1975-2025, the development of s(t) including all terms exceeding $0.5~\mu$as is, in $\mu$as (Capitaine et al. 2000):

$$-{XY\over 2} + 2004 +3812t - 121 t^2 - 72~574 t^3$$

$$+ \sum_{k} C_{k}\sin \alpha_{k} + 2t\sin \Omega + 4t\sin {2\Omega}$$

$$+ 744t^2 \sin \Omega + 57 t^2 \sin {2L}$$

$$+ 10 t^2 \sin 2L'-9t^2 \sin {2\Omega}$$ (8)

where t is expressed in centuries since J2000.0 (t₀=0) and where $\alpha _k$ and C_k are given in Table 1. To extend the development for the periods 1900-1975 and 2025-2100 for terms exceeding $0.5~\mu$as, add in $\mu$as:

$$\Delta s(t) +28 t^4 + 15 t^5 -22t^3 \cos \Omega$$

$$- t^3 \cos {2L}+ \sum_{i} D_i t^2 \sin {\alpha _i},$$ (9)

where $\alpha _i$ and D_i are given in Table 1. In this table and in Eqs. (8) and (9), the notations, referred to the ICRF axes and the corresponding fixed ecliptic are:

$\Omega$: mean longitude of the node of the lunar orbit;

L: mean longitude of the Sun;

L': mean longitude of the Moon;

l: mean anomaly of the Sun ( $l= L-\varpi$) where $\varpi$ is the mean longitude of the Sun's perigee;

l': mean anomaly of the Moon ( $l'= L'-\varpi ^\prime$), $\varpi ^\prime$ being the mean longitude of the Moon's perigee.

At present, the uncertainty of these expressions is smaller than 1 $\mu$as and is insensitive to expected modifications of the nutation-precession theory to about this level. A maximum error of 3 $\mu$as could be reached in 2100 (Capitaine et al. 2002).

7.2 Terrestrial ephemeris origin

For Earth orientation, the concept of the CEO must be applied to both the celestial and the terrestrial reference systems. So, a Terrestrial Ephemeris Origin (TEO) was introduced. The definitions are similar. The formula giving the shift of TEO is

$$s' = \int_{t_0}^t (\cos g - 1) {\dot F} {\rm d}t,$$ (10)

where F and g are the polar coordinates of the Earth's pole at time t, and $\dot F$ is the time derivative of F. Because of the very small magnitude of the polar displacements, this is a very small correction (smaller than 0.1 mas next century) that must however be evaluated from the measurement of the polar motion, when $\mu$as accuracies are sought.

7.3 Stellar angle

In the ICRS system the equivalent of sidereal time is the hour angle of the celestial ephemeris origin (CEO), and hence is subject only to the motions of that origin. It is called the Stellar Angle or Earth Rotation Angle.

Figure 3: Definition of the stellar angle.

The Stellar Angle in Fig. 3 is defined by the angle $\theta$ measured along the equator of the Celestial Intermediate Pole (CIP) between the TEO and the CEO positively in the retrograde direction. The definition of the CEO ensures that the derivative $\dot\theta$ is strictly equal to the instantaneous angular velocity $\omega$ of the Earth around the CIP. Thus, $\theta$ represents rigorously the sidereal rotation of the Earth around its axis and is such that UT1 is linearly proportional to the angle. The definition of UT1 by its relationship with mean sidereal time is proportional to $\theta$, but has additional terms that take account of precession. Thus, with the CEO the definition of UT1 is such that the time derivative of UT1 is proportional to $\omega$.

The numerical relationship between the Stellar Angle and UT1 has been derived (Capitaine et al. 2000) to be consistent with the conventional relationship between GMST and UT1. This provides the definition of UT1 for the ICRF. To an accuracy of 1 $\mu$as the relationship is

$$\theta (T_{\rm u}) 2 \pi (0.779~057~273~2640$$

$$+1.002~737~811~911~354~48~ T_{\rm u} ),$$ (11)

where

$$T_{\rm u} = ( \rm {Julian \;UT1 \;date} - 2~451~545.0 ) .$$

7.4 ITRS to ICRS matrix

The coordinate transformation from a unit vector $\vec{G}_{\rm T}$ of the ITRS to a unit vector $\vec{G}_{\rm S}$ of the GCRS at date t can be given as

$$\vec{G}_{\rm S} = {\cal PN}(t) \times {\cal R}(t) \times {\cal W}(t)\times \vec{G}_{\rm T}.$$ (12)

In this equation, the fundamental components ( ${\cal PN}(t), {\cal R}(t)$, and  ${\cal W}(t)$) are the transformation matrices arising from the motion of the CIP in the ICRF due to precession-nutation expressions, from the rotation of the Earth around the CIP, and from polar motion, respectively. They are given in the similar manner to that given in the IERS Conventions 1996 (McCarthy 1996):

$${\cal W}(t)= {\cal R}_3 (-s') \times {\cal R}_1 (-v) \times {\cal R}_2 (u),$$

$${\cal R}(t)= {\cal R}_3 (-\theta),$$

$${\cal PN}(t)= {\cal R}_3 (-E) \times {\cal R}_2 (-D) {\cal R}_3 (E)\times {\cal R}_3 (s).$$ (13)

In these equations, u and v are the coordinates of the CIP in the ITRF, s' is the shift of TEO defined by Eq. (10), and $\theta$ is the Stellar Angle. The quantities D, E and s are defined in Sect. 7.1. We have used the notation ${\cal R}_i$, i=1,2,3 for the matrix rotation respectively around the axes x,y,z.

${\cal PN}(t)$ can also be given in an equivalent form directly from the coordinates X and Y as:

$${\cal PN}(t) = \left(\begin{array}{ccc} 1-aX^2& aXY & X\\ -aXY &1-aY^2 & Y\\ -X &-Y &1-a(X^2+Y^2) \end{array}\right) ~ {\cal R}_3(s).$$ (14)

with

$$X=\sin D \cos E$$

$$Y=\sin D \sin E$$

$$a = {{1} \over {(1+ \cos D)}}\cdot$$ (15)

The latter can also be written with sufficient accuracy as

$$a= {1\over 2} + {1 \over 8} {(X^2 + Y^2)}.$$

7.5 The IAU 2000 precession-nutation series

In addition to the new definitions presented in the previous sections, the IAU has adopted a new model for precession and nutation based on the work of Mathews et al. (2002). It is presented in the form of series for nutation in longitude $\Delta \psi$ and obliquity $\Delta \epsilon$. There are two versions: the IAU 2000A series provides the position of the CIP in the GCRF with an accuracy of 0.1 mas (IERS 2002). A shorter version, accurate to 1 mas, is the IAU 2000B series.

To derive the position ($X,\;Y,\;Z$) of the CIP, the following procedure can be applied (IERS 2002). One has:

$$\bar{X} + \xi_0 -\bar{Y} {\rm d} \alpha_0$$

$$\bar{Y} + \eta_0 +\bar{X} {\rm d} \alpha_0$$ (16)

where $\xi_0 = -0\hbox{$.\!\!^{\prime\prime}$ }016\;617$ and $\eta_0 = -0\hbox{$.\!\!^{\prime\prime}$ }006\;819$ are the celestial pole offsets at the basic epoch (J2000.0) and d$\alpha_0$ is the right ascension of the mean equinox, and where:

$$\bar{X} \sin \omega \sin \psi$$

$$\bar{Y} -\sin \varepsilon _0 \cos \omega + \cos \varepsilon _0 \sin \omega \cos \psi .$$ (17)

In these equations, $\varepsilon_0$ is the obliquity of the ecliptic at J2000. One can compute $\omega$ and $\psi$ from the precession-nutation series by:

$$\omega \omega_{\rm A} + \Delta \psi \sin \epsilon_{\rm A} \sin \chi_{\rm A} + \Delta \epsilon \cos \chi_{\rm A} ,$$

$$\psi \psi_{\rm A} + {{\Delta \psi \sin \epsilon_{\rm A} \cos \chi_{\rm A} - \Delta \epsilon \sin \chi_{\rm A}} \over {\sin \omega_{\rm A}}}\cdot$$ (18)

Here, $\omega_{\rm A} ,\;\psi_{\rm A} ,\;\epsilon_{\rm A}$ and $\chi_{\rm A}$ are the precession quantities of the model given as polynomials of time.