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Determination of the relative orientation
between radio and dynamical reference
frames using VLBI observations of spacecraft
By N. V. Shuyg i n a
Institute of Applied Astronomy, Russian Academy of Sciences, 8 Zhdanovskaya st.,
St.Petersburg, 197042, RUSSIA
The experience of determination of the link between radio sources and dynamical reference
frames is presented. To obtain this connection we used a series of very long baseline interfer­
ometry (VLBI) observations of the spacecraft Phobos­2 and angularly nearby quasars. Orbit
determination of the spacecraft was carried out by numerical integration of relativistic equations
of motion taking into account all major planet perturbations. The coordinates of perturbing
planets were calculated by the DE200/LE200 ephemerides. The dynamical reference frame is
determined by equator and equinox of these ephemerides. The resulting spacecraft orbit was
fitted to the VLBI and tracing (range and Doppler frequency shift) data by means of weighted
least squares procedure with the uncertainties at the level of a few kilometers. After precise
orbit determination using VLBI measurements only we obtained three rotation angles between
two systems. The radio and ephemeris dynamical frames were found to be coincident in all three
axis to less than 100 milliarcseconds.
1. Introduction
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 helio­
centric 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.
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 0 00 :0001. 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 Phobos­2 to link radio sources
and dynamical reference frames.
2. Method
The relative orientation of the radio and dynamical reference frames can be described
by a sequence of three rotations
~ k 1 = R 1 (ff 1 )R 2 (ff 2 )R 3 (ff 3 ) ~ k; (2.1)
where ~ k is barycentric unit position vector of quasar in the radio reference frame, ~ k 1
represents the same vector in the dynamical frame, R i (i = 1; 2; 3) are well­known matrices
of rotation about x, y, z axis respectively.
The quasar coordinates are considered to know exactly from some extragalactic source
1

2 N.V. Shuygina: Radio and dynamical reference frames
Reference Connection Accuracy Authors
frames method (arcsec)
FK4--RSS 1. Photographic quasar 0.1 Gubanov et al. (1990)
observations Kumkova et al. (1980)
2. VLBI radio stars 0.01(ff)
observations 0.06(ffi) Lestrade (1988)
FK4--PS Methods of meridian 0.05(ff)
astrometry 0.01(ffi) Standish (1986)
FK4--SS Lunar occultations 0.02(ff)
of stars 0.01(ffi) Jordi & Rosello (1987)
PS--RSS Differential VLBI
spacecraft observations 0.02(ff)
on the background of 0.02(ffi) Newhall et al. (1986)
quasars
Table 1. Examples of linking different reference frames with their accuracies.
catalogue Sovers et al. (1988). Thus calculated time delay of a quasar Ü q may be written
as a function of the rotation angles ff i (in parentheses we represent parameters to be
obtained during differential correction process only)
cÜ q = f(ff 1 ; ff 2 ; ff 3 ): (2.2)
The time delay for a spacecraft Ü s , described as a function of time t and initial apparatus
coordinates ~r 0 and ~ —
r 0 , is
cÜ s = f(t; ~r 0 ; ~ —
r 0 ): (2.3)
The calculated differential time delay as the difference between the quasar and the space­
craft delays is therefore given by
c\DeltaÜ c = f(t; ~r 0 ; ~ —
r 0 ; ff 1 ; ff 2 ; ff 3 ): (2.4)
The observed interferometry time delay \DeltaÜ o ,
c\DeltaÜ o = f(t; ~r 0 + \Delta~r 0 ; ~ —
r 0 + \Delta ~ —
r 0 ; ff 1 + \Deltaff 1 ; ff 2 + \Deltaff 2 ; ff 3 + \Deltaff 3 ); (2.5)
is represented by a function of the parameters and their first­order corrections. After
linearization process we obtain an equation of condition in the form
@(c\DeltaÜ q )
@ff 1
\Deltaff 1 + @(c\DeltaÜ q )
@ff 2
\Deltaff 2 + @(c\DeltaÜ q )
@ff 3
\Deltaff 3 +
@(c\DeltaÜ s )
@~r 0
\Delta~r 0 + @(c\DeltaÜ s )
@ ~ —
r 0
\Delta ~ —
r 0 \Gamma c(\DeltaÜ o \Gamma \DeltaÜ c ) = w: (2.6)
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 meas­
urements. 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.

N.V. Shuygina: Radio and dynamical reference frames 3
Quasar Quasar Right Ascension Declination
number name
1 P2320\Gamma035 23 h 23 m 31 s :95381 \Gamma03 ffi 17 0 05 00 :0220
\Sigma0 s :00004 \Sigma0 00 :00016
2 P0019+058 00 h 22 m 32 s :44128 +06 ffi 08 0 04 00 :2715
\Sigma0 s :00004 \Sigma0 00 :00016
3 P0106+01 01 h 08 m 38 s :77113 +01 ffi 35 0 00 00 :3200
\Sigma0 s :00004 \Sigma0 00 :00010
4 P0202+14 02 h 04 m 50 s :41399 +15 ffi 14 0 11 00 :0450
\Sigma0 s :00004 \Sigma0 00 :00010
Table 2. Coordinates of the quasars.
Type of Number of Time A priori
observations observations interval accuracy
Differential 03.11.1988
VLBI 13 20.01.1989 0.1­0.4 ns
Range 86 10 m
20.10.1988
Doppler frequency 23.01.1989
shift 83 2 mm s \Gamma1
Table 3. Information on observations of the spacecraft Phobos­2.
3. Observations
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 Phobos­2
received within the joint experiment ``Phobos'' in 1988--1989. During 150 days from
``Deep Space Network''stations 13 differential interferometry observations of apparatus
Phobos­2 on the background of 4 quasars were received. Barycentric coordinates of this
quasars are given in Table 2.
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.
4. Results
Orbit determination of the spacecraft was performed by numerical integration of the
relativistic heliocentric equations of motion, taking into account perturbations of all ma­
jor 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

4 N.V. Shuygina: Radio and dynamical reference frames
Type of Number of RMS residuals
observations observations
Range 86(3) 24.610 m
VLBI 13 0.454 ns
Table 4. The RMS residuals.
Baseline Number of Date of the (o \Gamma c) ns
quasar observation
Goldstone--Canberra 1 1988.10.20 \Gamma6:267
Goldstone--Madrid 1 1988.11.03 \Gamma6:065
Goldstone--Canberra 1 1988.11.05 \Gamma1:899
Goldstone--Canberra 2 1988.11.19 \Gamma5:631
Goldstone--Madrid 3 1988.12.19 +5:068
Goldstone--Canberra 3 1988.12.20 +1:969
Goldstone--Madrid 3 1988.12.26 +5:352
Goldstone--Madrid 3 1989.01.06 \Gamma0:772
Goldstone--Madrid 3 1989.01.09 \Gamma1:432
Goldstone--Madrid 3 1989.01.13 \Gamma5:337
Goldstone--Madrid 4 1989.01.17 +1:923
Goldstone--Canberra 3 1989.01.18 +1:194
Goldstone--Canberra 4 1989.01.20 \Gamma6:592
Table 5. Resulting (o \Gamma c) for VLBI observations.
first step we define a precise orbit of the spacecraft or in other words dynamical reference
frame.
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 pro­
cedure we have got the following values of rotation angles
ff 1 = 0:103 00
\Sigma 0:074 00
ff 2 = 0:025 00 \Sigma 0:024 00 (4.7)
ff 3 = 0:003 00 \Sigma 0:011 00 ;
with correlation between them r ff 1 ff 2
= 0:66, r ff 2 ff 3
= 0:14, r ff 1 ff 3
= 0:64. The final
individual residuals for all VLBI measurements are illustrated in Table 5.
5. Conclusion
In this paper we have analysed all available differential VLBI data of the spacecraft
Phobos­2 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 reas­
onable 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.
The author expresses sincere thanks to Dr G.A.Krasinskii for the attention he paid to
this work and to the Russian Foundation of Fundamental Investigations for the travel
grant which enables the participation in the conference.

N.V. Shuygina: Radio and dynamical reference frames 5
REFERENCES
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(IAU Symposium 141), eds Lieske J.H. & Abalakin V.K., p.85.
Jordi C. & Rosello G., 1987, MNRAS, 225, 723.
Hellings R.W., 1986, AJ, 92, 769.
Kumkova I.I., 1980, in Reference coordinate systems for Earth dynamics (IAU Colloquium 56),
eds Gaposhkin E.M. & Kolaczek B., p.369.
Lestrade J.­F., 1988, AJ, 96, 1746.
Moritz H. & Mueller I.I., 1987, Earth rotation, theory and observation, (Ungar).
Newhall X.X., Preston R.A. & Esposito,P.B., 1986, in Astrometric techniques (IAU Symposium
109), eds Eichorn H.K. & Leackock R.J., p.789.
Sovers O.J. et al., 1988, AJ, 95, 1647.
Standish E.M., 1986, in Astrometric techniques (IAU Symposium 109), eds Eichorn H.K. &
Leackock R.J., p.677.