Table 1: Examples of linking different reference frames with their accuracies.
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N. V. Shuygina
Institute of Applied Astronomy, Russian Academy of Sciences, 8 Zhdanovskaya st., St.Petersburg, 197042, RUSSIA
In order to specify positions of astronomical objects it is necessary to have a reference frame. Celestial reference system may be defined kinematically (through the positions of extragalactic radio sources - EGRS), or dynamically (through the geocentric or heliocentric motions of artificial satellites, the Moon, and planets). Although we may have a conceptual ideal reference system, each realization of reference frame will be somewhat different, and there will be a need to determine the transformations from one reference frame to another (Moritz & Mueller (1987)). Table 1 shows several conventional quasi-inertial reference systems with examples of possible connection.
Table 1: Examples of linking different reference frames with their accuracies.
Nowadays the most accurate reference frame is constructed on the basis of EGRS
positions defined by methods of radio interferometry with very long baselines
(VLBI). The internal accuracy of such systems is of the order of 
. To
obtain the connection between the frame mentioned and practically used
dynamical frame, differential VLBI observations of spacecraft and angularly
nearby quasars are used (Newhall et al. (1986)). We have analysed such
measurements of the spacecraft Phobos-2 to link radio sources and dynamical
reference frames.
The relative orientation of the radio and dynamical reference frames can be described by a sequence of three rotations
where 
is barycentric unit position vector of quasar in the radio
reference frame, 
represents the same vector in the dynamical frame,
are well-known matrices of rotation about 
, 
, 
axis
respectively.
The quasar coordinates are considered to know exactly from some extragalactic
source catalogue Sovers et al. (1988). Thus calculated time delay of a quasar 
may be written as a function of the rotation angles 
(in parentheses
we represent parameters to be obtained during differential correction process
only)
The time delay for a spacecraft 
, described as a function of time 
and initial apparatus coordinates 
and 
, is
The calculated differential time delay as the difference between the quasar and the spacecraft delays is therefore given by
The observed interferometry time delay 
,
is represented by a function of the parameters and their first-order corrections. After linearization process we obtain an equation of condition in the form
Using theoretical expressions for a quasar (Hellings (1986)) and a spacecraft time delays we get directly the condition equations. The system of equations 2.6 being written for all VLBI observations allows us to calculate orientational parameters and corrections to the position and velocity of the spacecraft. But in an orbit determination process these data are to be supplemented with another type of observations, for example, range measurements. Thus processing of the real observations divides into two parts: firstly from all radar and interferometry data we determined a precise orbit of the spacecraft, and then from the differential VLBI measurements only we obtained orientation parameters system of interest.
In order to obtain a mutual orientation of radio source and dynamical reference frames we used a complete set of all available VLBI observations of the spacecraft Phobos-2 received within the joint experiment ``Phobos'' in 1988-1989. During 150 days from ``Deep Space Network''stations 13 differential interferometry observations of apparatus Phobos-2 on the background of 4 quasars were received. Barycentric coordinates of this quasars are given in Table 2.
Table 2: Coordinates of the quasars.
These observations were supplemented with 175 radar data (time delay and Doppler frequency shift) received during the same time interval from 3 stations on the territory of the former SU. The information on all observations is presented in Table 3.
Table 3: Information on observations of the spacecraft Phobos-2.
Orbit determination of the spacecraft was performed by numerical integration of the relativistic heliocentric equations of motion, taking into account perturbations of all major planets and Schwarzschild's terms due to the Sun. For the calculation of coordinates of perturbing planets and the Moon the DE200/LE200 ephemerides were used. So the dynamical reference frame under consideration is determined by equator and equinox of these ephemerides.
The orbit was computed using a linearized weighted least squares estimation algorithm. Root mean square (RMS) residuals given in Table 4 indicate how well the resulting orbit is fitted to various observations and the convergence of the iteration process. So on the first step we define a precise orbit of the spacecraft or in other words dynamical reference frame.
Table 5: Resulting (
) for VLBI observations.
After that, using differential VLBI observations only one can obtain three rotation angles of radio sources and dynamical reference frames. By means of least squares procedure we have got the following values of rotation angles
with correlation between them 
,
, 
. The final
individual residuals for all VLBI measurements are illustrated in
Table 5.
In this paper we have analysed all available differential VLBI data of the spacecraft Phobos-2 to get a connection between the dynamical and radio sources reference frames. The calculations can be considered as preliminary ones but we have just confirmed that even so small a number of such measurements gives us the opportunity to obtain reasonable values of rotation angles. We hope that future projects and also the increase of observational accuracy permit us to achieve more precise and reasonable results.
