2. Relation between the Earth orientation parameters and linking angles
There are five Earth orientation parameters (EOP) that are routinely calculated from the radio interferometric observations: the components of polar motion (giving the position of the celestial pole in the terrestrial reference system), the celestial pole offsets (giving the displacement of the actual spin axis of the Earth from its ephemeris position in the celestial reference system in obliquity and longitude ) and the difference between Universal and Atomic time scales UT1-TAI (giving the angle between the actual zero meridian and its expected position, assuming the uniform angular rotation of the Earth). The accuracy is on the level of 1 milliarcsecond (mas), or better (Carter & Robertson 1993 ).
The new reduction of optical astrometry observations proposed by Vondrák (1991 ) yields the same EOP at roughly 5-day intervals since the beginning of the century; the expected accuracy of the Earth orientation in the Hipparcos reference frame is, according to Vondrák et al. (1992 ) on the level of 10 mas during the last two decades of the solution.
The relative orientation of the two celestial reference systems (Hipparcos and extragalactic) and their mutual rotation can be described by six parameters. The transformation of a column vector given in the system of Hipparcos as into the extragalactic system is given by the formula
where we assume the orientation angles to be small and linearly changing with time. The six parameters defining the orientation of the Hipparcos catalog with respect to extragalactic reference system are then the values of at a fixed epoch (, , ) and their time derivatives (, , ). Their relation to the EOP on the celestial sphere is graphically shown in Fig. 1.
The intersections of the axes X and Z of both reference systems (Extragalactic and Hipparcos) with the sphere are denoted as , respectively, the position of the spin axis, corrected for the standard precession-nutation (Lieske 1979, Seidelmann 1982 ) as R, and the terrestrial zero meridians used by the two techniques are marked VLBI and OA. The small angles between R and are measured by optical astrometry and VLBI, respectively, and expressed by means of celestial pole offsets and . The observed Universal time UT1 can easily be converted into the Greenwich mean sidereal time S, using the conventional formula (Aoki et al. 1982 ); their values as observed by the two techniques measure the angles between the respective zero meridian of a technique and the great circles , respectively. The orientation angles between the two reference frames H and E, as also displayed in the figure, are then given in terms of the differences of EOP as
Notice that two of the five EOP, polar motion components , are useless for the purpose since they are fully sensitive only to the terrestrial reference system, and almost entirely insensitive to the choice of the celestial reference system. They just measure the offsets of the axis z of the terrestrial system from the spin axis R, and as such can be used to determine the relation between the terrestrial reference systems used by the two techniques. From the third equation of (2) follows that the angle can be determined only if the misalignment of the two terrestrial reference systems in longitude, , is known. Unfortunately it is not the case - the origins of terrestrial longitudes of VLBI and optical astrometry are given by the adopted directions of the baselines and local verticals, respectively. The direct geodetic link between these two systems is practically impossible.
Having the series of EOP observed by the two techniques simultaneously in a sufficiently long time interval, we can derive the angles for a chosen epoch , and their rates , , . The value can obviously be determined only on the assumption that the longitude difference is constant.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998