V. Shkodrov, M. Geffert, Testing the astrometric accuracy of the. 348. C. Pollas, J.-L. Heudier 2m ... L.D. Kovbasyuk free nutation (core ...... Sergei DIAKONOV.
Ism
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IT
" I International Astronomical Union Union Astronomique Internationale Proceedings of the 127th Colloquium of the International Astronomical Union Reference Systems held in Virginia Beach, VA, USA 14-20 October 1990 SAcolsio hiWS
Editea by ghes James A. H'1 Clayton A. Smith George H. !Yaplan
*awale-vit
9/6/91
United States Naval Observatory Washington, D.C. 1991
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Statement A per Dr. Kaplan Naval Observatory Code AA Dist Washington, DC 20392-5100
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Published by,the U.S. Naval Observatory
Washington, DC 20392 U.S.A.
1
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iii TABLE OF COUTEM? ix
Introduction Acknowledgements List of Participants PART 1. ORAL PAPERS
1. B. Guinot
x xi
Ir SJ ,I
.
,
"---Report of the Subgroup on'Time ,
3
2. J. Kovalevsky
+-Preliminary report of the work of the Subgroup on ,Coordinate Frames and Origins,
17
3. T. Fukushima
5 year time span and >200 one-day position estimates, the rates of change of right ascension and declination are generally less than 5 mas/century, giving upper limits on real motion.
1. Introduction A kinematically fixed celestial reference frame can be realized from VLBI (Very Long Baseline Interferometry) observations of compact extragalactic radio sources. Since most of these radio sources have optical counterparts (albeit faint) that can be measured in the conventional FK5 frame, this radio reference frame can anchor the stellar frame. The level of stability of the radio reference frame is thus of considerable importance. This paper discusses two aspects of stability: the global orientation of the frame and individual source positions relative to the frame. The current distribution of the data and progress in analysis are briefly reviewed. This paper is an extension of the work described in Robertson et al. (1986), Ma el al. (1986), Ma et al. (1990), Ma (1990) and Russell et al. (in press). A parallel and completely independent effort is described in Fanselow et al. (1984), Sovers et al. (1988), Sovers (1990), and Sovers (this volume). The comparison and convergence of these independent catalogs indicate that milliarrsecond (mas) accuracy has been achieved.
2. Data The 461,000 dual frequency Mark III observations used for this work can be divided into four categories from three geodetiL programs and the combined astrometric effort: 1) 206,000 observations from the Crustal Dynamics Project (CDP) of the National Aeronautics and Space Administration (NASA) acquired at irregular intervals beginning in 1979 and using many networks around the globe, 2) 210,000 observations from the IRIS program of the National Oceanic and Atmospheric Administration (NOAA) acquired at regular intervals beginning in 1980 using a small number of networks, 3) 29,000 observations from the
136
.. .
..
"....•. . ... ........... ,.
... . ..
44
.. .....
.
.
.
.
. . .
....
.
Figure . GLB677 sources on an Aitoff projection.
hr at right.
Navnet program of the US Naval Observatory (USNO) acquired at weekly intervals beginning in 1988 on a network normally including Hawaii, Alaska, West Virginia and Florida, and 4)16,000 observations acquired by the CDP, Naval Research Lboratory (NRL), and NOAA astrometric programs from various northern and southern hemisphere networks in 48 sessions since 1980. Geodetic data from small ( 16.5) of the Torino program on optical positions of extragalactic radio-sources (Chiumiento et al. 1987, Chiumiento et al. 1989) brought us to experiment with the 320 x 512 pixels R'7A CCD (1 square pixel = 30 /Lmon a side) attached to the 1.5 m Ritchey-Chretien of the University of Bologna. The CCD system operated at a temperature of about 120 °K and the telescope scale at the CCD chip is 0.51"/pixel. Several targets exist in our observing list such that 3 or more stars (besides the QSO) are measurable on our photographic plates within the CCD field (2'.7 x 4'.4). The plates are calibrated via CAMC stars (La Palma 1989); thus the COD frames can be tied to the CAMO (FK5) system by using the stars surrounding the QSO as the link stars. In the next section we will briefly describe the available COD observations. Then, we report on the internal precision of our CCD astrometry. Finally, a preliminary evaluation of the precision of CCD based QSO positions is given in section 5. There, we also compare the quality of our CCD astrometry to photographic errors we had previously obtained for the same targets.
235 2. Observations During four nights (Jan 24, Feb 5-7, 1990) partially devoted to this program, we obtained 34 CCD frames, B and V colors, of 10 radiosources of our main program. The exposure times ranged from 5 to 30 minutes. The February nights were during full Moon; thus about one third of the available exposures have an abnormally high sky background. The number of usable link stars in the frames varies from 3 to 10. 3. Reduction of the CCD frames As usual, the CCD frames are first corrected for dark current, bias, and pixel-topixel sensitivity variations (flat fielding). Then, the astrometric image processing is done using the PC-based software ROBIN developed at Torino Observatory (Lanteri 1990). ROBIN is a package specifically designed for astrometric reductions of both CCD and PDS images. ROBIN fits a bidimensional gaussian-like function to the star images, plus a linear (both in x and y coordinates) polynomial to take into account the sky background. The fits are performed on windows extracted from the frames, whose sizes vary with object magnitudes. ROBIN outputs the estimated x and y coodinates (in pixel units) of the objects successfully centered (i.e. the center of the bidimensional gaussian model) and their dispersions (o 's). Instrumental magnitudes of the same objects are also derived. Finally, these quantities are processed through our standard astrometric software. 4. Internal precision of CCD Astrometry For this preliminary study of the internal precision of CCD based positions, we used frames of comparable exposure times on the two QSO's 1148-001 and 1611+343 (Argue et al., 1984). Both targets have a B magnitude close to 17'.5. Ten and seven star-like images (besides the QSO's) were successfully centroided (see sec. III) in the fields of 1148-001 and 1611+343 respectively. The signal-to-noise ratios of the different images varied from S/N - 25 (for objects of B = 16') to S/N - 3 (for B = 20m). Also, through the estimated a's of the stars, we computed the FWHM of the PSF of each frame (FWHM = 2.4 x aY). A typical a during our observing nights was 1.6 pixel, which gives a typical seeing of = 2". The internal astrometric error was determined by comparison of the different exposures after bringing each of them into the same instrumental reference frame via a 6-constant least squares adjustment. The average error in one coordinate ranges from 0.02 pixel (C- 0".01) for the "bright" objects (< B >= 17m ) to about 0.15 pixel (= 0".07) for those close to the frame limit (< B >t-" 20m). Estimates of the photometric internal precision were calculated as well. The average errors vary from 0m.01 for the bright objects to 0m.15 for the faint ones.
236
5. CCD based QSO positions In order to evaluate the internal consistency of QSO positions, we selected test objects from our main list for which we have both plate and CCD series. Unfortunately, the CCD run covered a range in right ascension poorly covered by our reduced photographic material. Anyway, we had available 2 plates of 1611+343. Both plates were taken at the same telescope used for the CCD observations. One of the two plates covers a square field of about 65' on a side and contains 13 CAMC stars. On this plate we measured both the secondary (15 stars) and the tertiary (7 stars) reference frames. The secondary link stars were used to reduce the other plate which is only 9x12 cm. The magnitudes of the CAMC stars range from V=10 to V=13. Secondary link stars have magnitudes between 14' and 16' and the Tertiary stars cover the interval from 16.5' to 17.5'. Given the photographic tertiary frame, the three CCD frames available for 1611+343 were independently reduced. Then, the three positions were intercompared to evaluate the internal consistency. The resulting root-mean-square errors are ±0".038 in RA and ±0".047 in DEC. This result, although very preliminary, points to an improvement of about a factor of 10 over the traditional photographic technique at B c- 17'.5, as shown in Figure 1.
".40- A8 / o/
.20.
/
/, /0 *40/
/
4
. 40- Aa
0,
.20"
15
16
17
18
m
Figure 1. Photographic vs CCD astrometric precision 9 = Photographic;
,t
= CCD
237 REFERENCES Argue, A.N., De Vegt, C., Elsmore, B. et al.: 1984, Astron.Astrophys., 130, 191. Carlsberg Meridian Catalogue No. 4, La Palma 1989. Eds. Copenhagen Univ. Observatory, Royal Greenwich Observatory, Instituto y Obs. de San Fernando. Chiumiento, G., Lattanzi, M.G., Massone, G., Morbideli, R., Pannunzio, R., Sarasso, M.: in "Fundamentals of Astrometry", IAU Col. No. 100, Belgrade 1987, Ed. H.K. Eichhorn, Astrophys. Space Science, in press. Chiumiento, G., Lattanzi, M.G., Massone, G., Morbideli. R., Pannunzio, R., Sarasso, M.: 1990, Inertial Coordinate Systems on the Sky, Leningrad 1989, 501-502, Eds. J.H. Lieske and V.K. Abalakin, Kluwer Acad. Publ., Dordrecht. Lanteri, L.: 1990, Osserv. Astron. di Torino, InternalReport no. 16.
238
PRESENT STATUS OF WORK ON THE F15 EXTENSION
THOMAS E. CORBIN
HEINER SCHWAN
U.S. Naval Observatory 34th and Massachusetts Ave., NW Washington, D.C. 20392 U.S.A.
Astron. Rechen-Institut MHnchhofstr. 12-14 D-6900 Heidelberg 1 F.R.G.
ABSTRACT. The FK5 Extension, consisting of 3117 new fundamental stars selected from the FK4 Sup catalogue and the IRS list, will extend the FK5 system to about 9.5th visual magnitude. The construction of the FK5 Extension is briefly described and the main characteristics are given. 1. INTRODUCTION The Fifth Fundamental Catalogue (FK5) will consist of two parts, namely the "Basic FK5" (Fricke et al., 1988) and the "FK5 Extension". The Basic FK5 contains the classical 1535 fundamental stars already given in the FK4. Systematic and individual corrections to the mean positions and proper motions of the FK4 have been derived and the IAU(1976) System of Astronomical Constants has been introduced. The Basic FK5 defines the system of the new fundamental catalogue; it has been constructed with the aim to represent an inertial system as far as possible. Details are given in the introduction to the Basic FK5 as well as by Schwan (1987) and in the literature quoted there. One important shortcoming of the FK4 is the predominance of bright stars. Only about 100 stars are fainter than magnitude 6.5 and thus the FK4 is not well defined at magnitudes fainter than this. The extension of the optical system to fainter magnitudes was therefore an indispensable task in constructing the FK5. 2. SELECTION OF THE NEW FUNDAMENTAL STARS It was realized by Fricke (1973) that there are essentially two star lists from which the new fundamental stars could be selected: the International Reference Stars (IRS) for extending the system to about magnitude 9.5 and the FK4 Sup stars which had to fill a remaining gap in the magnitude distribution from about 5th to 7th magnitude. Mean positions and proper motions were determined for all IRS at the U.S. Naval Observatory (Corbin and Urban, 1990) and for all FK4 Sup stars at the Astronomisches Rechen-Institut (Schwan, 1987). On the
239 basis of the mean errors of the positions and proper motions and the distribution over the sky coupled with the distribution in magnitude we have selected 992 stars from the FK4 Sup and 2125 stars from the IRS, altogether 3117 new fundamental stars. These new fundamental stars represent the FK5 Extension and they are to define the FK5 system for fainter stars up to about 9.5th mag. It seems to be worth mentioning that the FK5 Extension includes 12 FK4 Sup stars not yet in the tape version of the bright stars in the FK5 Extension which has been distributed since 1988. 3. DERIVATION OF MEAN POSITIONS AND PROPER MOTIONS The FK5 Extension was primarily derived in the system of the FK4. All observations which could be used for the derivation of mean positions and proper motions had therefore to be referred to that system. This transformation was comparatively easy in the case of the FK4 Sup stars since most of them do not exceed seventh magnitude. On the basis of the FK4 stars in an observational catalogue the systematic relations Cat-FK4 were determined and the observed positions were directly transformed to the FK4 system. In the case of the IRS, in particular of the southern IRS, this simple procedure was not possible, since most of these stars are outside the limit of FK4 magnitudes. The systematic relations Cat-FK4, determined on the basis of the FK4 stars alone, could not be directly applied to the fainter stars. It was necessary to construct first an intermediate system which represents the FK4 system for fainter magnitudes. The observations which could be used for that purpose had either to be free of magnitude dependent errors or their systematic errors at faint magnitudes had to be determined. North of -30 degrees the catalogues observed with screens (which eliminate magnitude equations) could be used to derive such an intermediate system. There was, however, an insufficient number of appropriate catalogues for deriving a corresponding system south of -30 degrees. In that region an extrapolation of the magnitude equation from northern declinations to the southern region was necessary. This extrapolation could be performed by making use of southern catalogues observed with a moving-wire micrometer. Such catalogues have been found to have magnitude equations that are not declination dependent. The construction of an intermediate system extending the FK4 system to about magnitude 9.5 was the essential step in deriving the astrometric data for the IRS stars. This extended FK4 system could be used to reduce many other catalogues with observed faint stars to the FK4 (see also Corbin and Urban, (1990)). After having transformed all relevant observations to the FK4 system we have performed weighted least squares solutions for deriving the mean positions and proper motions from the various observed catalogue positions.
240
10175 .210
.9.0
11.4
nig1
t250
Ma.
Wo:
:0:0
1210
:2. a
.0
Fig. 1: FK5 Exteneion: Distribution of mean
.90
-214
11
epochs.
Ln 0 n
30
.U UI ARCSEC/C
SGACD
',MIO:* UUI ACbC/C
Fig. 3. FK5 Extension: Distribution of mean errors of me positioscy oin RA (left) and DEC (right), respectively; units: 0.01 arcsec
Fig.4. Distribution of apparent magnitudes of the whole FKS (Basic FK5 plus Extension)
0
*
I
1
241 4. MAIN CHARACTERISTICS OF THE FK5 EXTENSION The main characteristics of the FK5 Extension are presented in the four figures. In Fig. 1 we show the distribution of mean epochs in right ascension (left) and declination (right), respectively. There is a small dip in both distributions indicating that the FK4 Sup stars in the FK5Extension have, on the average, more recent mean epochs than the IRS. This is a consequence of the fact that the Sup stars were preferentially observed after 1955 when they had been proposed as candidates for a future extension of the fundamental system. The two subgroups can more or less also be identified in Fig. 2 and Fig. 3. The average mean epoch of the FK5 Extension is 1944. In Fig. 2 are given the distributions of the mean errors of mean positions in right ascension (left, multiplied with cos(delta)) and declination (right), respectively, and in Fig. 3 one finds the corresponding distribution of the proper motion errors. The FK4 Sup stars are a little more precise than the IRS. The overall precisions are 0.055 arcsec for the mean positions and 0.255 arcsec/cy for the proper motions. In Fig. 4 we present the distribution of apparent visual magnitudes of the whole FK5 (Basic plus Extension). Preliminary magnitudes were used in the star selection and also in Fig. 4. It is, however, unlikely that the final magnitudes will alter this distribution significantly. 5. PRESENTATION OF THE FK5 EXTENSION The FK5 Extension
will be given, as far as possible, in the same format
as the Basic FK5. We plan to publish the following data for each star: FK5 number, apparent visual magnitude, spectral type, position and proper motion for the epoch and equinox J2000 in accordance with the IAU (1976) System of Astronomical Constants, the corresponding values transformed to epoch and equinox B1950, mean epochs of observation, mean errors of position and proper motion at the mean epochs, and identifications with some other important star lists. Paralaxes and radial velocities will be given in the catalogue for all stars with significant foreshortening terms. We hope that a tape version of the FK5 around the beginning of the next year.
Extension can be made available
REFERENCES Corbin, T., Urban, S.E.: 1990, in IAU Symp. 141, Inertial Coordinate System on the Sky, eds. J.H. Lieske, V.K. Abalakin, p. 433 Fricke, W.: 1974, in IAU Symp. 61, New Problems in Astrometry, p. 23 Fricke, W., Schwan, H., Lederle, T., and collaborators: 1988, Fifth Fundamental Catalogue (FK5); Part 1: The Basic Fundamental Stars. VerOff. Astron. Rechen-Inst., Heidelberg, No. 32 Schwan, H.: 1990, in IAU Symp. 141, Inertial Coordinate System on the Sky, eds. J.H. Lieske, V.A. Abalakin, p. 371
242
8 PERSEI, A FUNDAMENTAL STAR AMONG THE RADIOSTARS
Suzanne DEBARBAT
Observatoire de Paris, DANOF/URA 1125 61 avenue de l'Observatoire 75014 Paris
ABSTRACT. Optical fluctuations of the radiostar Persei are seen from 13 campaigns performed with the astrolabe located at the Paris Observatory.
1.
INTRODUCTION
Among the radiostars, oPersel (Algol) - a fundamental star - was chosen by radioastronomers as a zero reference for right ascensions in radioastrometry. Since 1975 this fundamental star has been included in the observing programme performed by the "Astrolabe et systames de rddfrence" group in charge of the instrument at the Paris Observatory. The eight first campaigns published have been presented at the IAU Colloquium n* 100 (Belgrade 1987). The average of the mean square errors given were 0.004s in right ascension and 0.13" in declination, according to the FK4 and the constants in use at that time.
2.
DETERMINATIONS AND ERRORS
There are now thirteen campaigns available from 1975/76 to 1987/88 and they have been reduced in the FK5 system with the new fundamental constants according to the formulas established by Chollet (1984). Due to the fact that the group and the internal smoothing corrections (according to Ddbarbat et Guinot, 1970) are not yet available in the case of the FK5, the reduction have been performed for both FK4 and FK5. As an example of residuals, for the zenith distance, to which accuracy this quantity is obtainable when 12 transits (at east and at west) are observed, Table I gives the values for the 1983/1984 campaign (J 2000, FK4 and FK5),
243
Table I East residuals FK4
- 0.114" * 0.085" - 0.123" t 0.096"
12 transits
FK5
West residuals FK4 J 2000 FK5
+ 0.077" ± 0.078" + 0.093" t 0.081"
12 transits
J 2000
The residuals are not significantly different when the FK4 and the FK5 quantities are used. Also their mean square errors have the same order of magnitude. For each of the 13 campaigns (1975/1976-1987/1988), Aa andA6 have been derived, the calculation being made in two cases J 2000, FK4 and J 2000 FK5. The probable errors calculated for each campaign correspond to an average which is the same in the case of the FK4 and in the case of the FK5 : £ 0.0058s (right ascension), ± 0.116" (declination).
3.
FLUCTUATIONS IN RIGHT ASCENSION AND IN DECLINATION
As fluctuations appeared in both coordinates (fig.l and fig.2), smoothing curves (according to Vondrak 1969) have been determined with the same smoothing factor ; they are reported on figures 1 and 2. The corresponding mean square errors, together with the errors with which the curves are mathematically given, are in Table II.
Table II Mean square error Right ascension Declination
± 0.0042s ± 0.140"
Error of the curve ± 0.0012s ± 0.039"
The amplitudes for the fluctuations (0.010s in right ascension, 0.25" in declination) and the associated errors (0.0012s and 0.039") show that the optical variations appears to be real. The optical positions of this radiostar (which is also a multiple star), no longer used as a zero reference in right ascension, but still an object of interest for radioastronomers and double star specialists, must be compared with VLA and/or VLBI determinations for the same period.
244
aa
(Astrolabe - MS)
Figure I
ETA PERSEI
oSo
1975176
1987188 :980/81
-0.03
I
II 0.0
2200.0
?Wthode de Uondrak
66 (Astrolabe
4400.0
II.OE-00t3
-
FlS)
2442765,50
Figure 2
BETA PERSEI 0:70
-': 00 1975/76
1987/88
I
,
I
I
0.0 Wthode do Vondrak 4
I
I
,
2200.0 L.OE-0013
--
4400.0 jj
-
2442765.50
245
4.
CONCLUSION
a Persel, as a fundamental star among the radiostars will be used for the linkage of the "optical" and the 'radio" system of reference. a Persel represents an example of the problems to which the link will have to face due to the fluctuations this star is showing after 13 years of optical observations.
REFERENCES Chollet, F. 1984, Astron. and Astrophys., 132, 296-300. Ddbarbat, S. et Guinot, B., 1970, La Mdthode des hauteurs dgales en astronomie, Gordon and Breach, Ed. Vondrak, J., 1969, Bull. Astron. Inst. Czeh. 20, 349.
246
Improving the Reference Frame by Radio-and Optical Astrometry of Radio Stars
Chr. de Vegt N. Zacharias Hamburg Observatory, Germany K.J. Johnston Naval Research Laboratory, Washington D.C. USA R. Hindsley U.S.Naval Observatory, Washington D.C. USA ABSTRACT. A longterm program of precise radio - and optical astrometry of selected radio stars has been conducted in the last decade by our observatories using the VLA and astrographs on both hemispheres. Positions of 54 stars north of -26 deg. declination including 6 MASER stars have been obtained. The program status and some results from the southern hemisphere are reported. 1. Introduction The VLA has been used since the early '80 to determine precise astrometric parameters of selected radio stars in the primary extragalactic VLBI reference frame. For a recent detailed description of the observing program and previous results we refer to /l/ and further references therein. The main goal of the program is to provide a net of about 100 radio stars globally to link the present groundbased optical reference frame to the VLBI based extragalactic reference frame and to provide a similar link for the space based HIPPARCOS stellar net /2/. As most stars display radio emission at cm wavelengths only on the level of a few mJ, presently only the VLA can provide the necessary sensitivity although the accesible sky coverage is limited to > -26 deg. decl. 2. Status of Radio Work At present precise positions (±0.01-0.02 arcsec) have been obtained for 54 stars, including 6 MASER stars. Radio proper motions (±0.004 mas/yr) have been determined now for the stars HR1099 and UX Ari and second epoch observations have begun for additional stars.
247 The quoted accuracies may be improved finally to the 1 mas level by incorporating additional calibrator sources in the close vicinity of the stars. In addition work on radio parallaxes has been started for UX Ari. The diagram displays the distribution of the present radio stars. The lack of observations on the southern hemisphere is obvious, however, the coverage extends already far enough to the south to allow for a rotation solution in the comparision of the reference frames. 3. Status of Optical Observations Almost all radio stars of the present sample are optically brighter than visual magnitude 12 . Therefore they zan be tied to the optical fundamental reference frame as given by the IRS-catalog easily by use of high quality wide field astrographs on both hemispheres. Due to the favourable field size of these instruments which is at least 25 sq.deg. and corresponding number of IRSreference stars, the positions of the radio star can be determined with high systematic accuray ( 08M) deviations for reduction models with linear and quadratic terms of the rectangular coordinates. Only the reduction with third order polynomials resulted in errors of 6M (in c) and 7j (in 6) which correspond to 0.07 and 0.88 arcsec for a focal length of 16.5m.
DISCUSSION If we compare these deviations of 0.07/0.08 arcsec with the 0.08 arcsec internal error of the positions obtained from the CERGA plate, the Rozhen telescope can be used for further precise astrometric work. REFERENCES Geffert M.: 1986,Verbff. Bayer Kommis. Int. Erdmess.Bayer Astron.- Geod. Arb. n 48, 300
Akad.
Wiss.
351
CONSISTENT RELATIVISTIC VLBI THEORY WITH PICOSECOND ACCURACYt
M. SOFFEL I , J. MULLER 2, X. WU i , C. XU' 1 Theoretical Astrophysics
Uni. Tiibingen Auf der Morgenstelle 10 7400 Tibingen, FRG 2 Technical
Uni. Munich
Institut f. Astron. u. Physikal. Geodsiie Arcisstr. 21 8000 Munich 2, FRG
1. Introduction The accuracy of Very Long Baseline Interferometry (VLBI), representing one of the most important space techniques of modern geodesy, especially for the determination of the Earth's rotation parameters and baselines, is steadily increasing. Presently, delay residuals are of the order of 30 - 50 ps, corresponding to an uncertainty in length of about 1 centimeter e.g. in the determination of baselines or the position of the rotation pole. As has already been stressed by many authors, at this level of accuracy a relativistic formulation of the VLBI measuring process is indispensable (e.g. the gravitational time delay for rays getting close to the limb of the Sun amounts to 170 ns!). Starting with the work by Finkelstein et al. (1983) a series of papers has meanwhile been published on a relativistic VLBI theory (Soffel et al., 1986; Hellings, 1986; Zeller et al., 1986; Herring, 1989). However, possibly apart from Brumberg's treatment in his new monograph (Brumberg, 1990) all of these theories have one fatal drawback: they are not based upon some consistent theory of reference frames, which relates the global, barycentric coordinates, in which the measuring process is primarily formulated and in which positions and velocities of the bodies of the solar system are computed, with the local, geocentric coordinates, comoving with t Paper presented at the Eubanks Meeting at the U.S. Naval Observatory, Washington, D.C., October 12th, 1990. Work supported by the Deutsche Forschungsgemeinschaft (DFG) and the Volkswagen Stiftung; M.S. kindly acknowledges the receipt of a Heisenberg fellowship
352 the Earth, in which the geodetically meaningful baselines are defined. Furthermore, none of these theories (including Brumberg's (1990) treatment) have the accuracy of one picosec which seems desirable with respect to the achieved residual values. This article presents a relativistic VLBI theory with an accuracy of better than 1 ps. It is based upon a consistent theory of reference frames in the solar system which first has been introduced by Damour, Soffel and Xu (1990a-c) and complete at the first post-Newtonian level. For an accuracy of 1 ps our result differs from all results that have been published earlier. 2. Post-Newtonian VLBI Theory For The Group Delay We consider some radio signal being emitted from some remote source at barycentric coordinate time t o and position x0 . We consider two "light-rays", contained in the signal, which arrive at two VLBI antennas (called 1 and 2) at coordinate time tj and t2 . This barycentric coordinate time t is also called TCB. Let us denote the Euclidean unit vector from the source to the barycenter by k (k'k' = 1). Then the barycentric coordinate arrival time difference t 2 - tl to first post-Newtonian order is given by At =t-
=
[(t
-1
C
) - X2 (t2 )]. k + (At)gv,
3
(1)
where x,(t,) denotes the barycentric coordinate position of antenna i at coordinate time t,, (At)grav is the gravitational time delay, resulting from solving the equation for null geodesics (light rays) in some background metric describing the gravitational influence of the Sun and the planets. To sufficient accuracy (At)gav can be written as a sum over the contributions of the various massive bodies in the solar system. Now, for picosecond ac zuracy it is sufficient to consider the spherical part of the gravitational potential in (At)gav only. Taking earlier results from Richter and Matzner (1983) we estimate the contribution from the quadrupole moment of the Sun to the time delay to be much less than a picosecond (- 10-18 s). The effect from the angular momentum of the Sun (a gravitomagnetic effect of 1.5 post-Newtonian order) is of the same order, while the dominant post-post Newtonian terms aie expected to be less than about 0.5 picoseconds. For the solar contribution we can neglect the notion of the Sun about the barycenter and the usual "light-time equation" for the spherical field can be written in the form (e.g. Soffel, 1989) (A)
-92mC In
l,2!
- X2 k) k)
I2
(2)
where x, refers to t, and m.. _-- GM. /c 2 = 1.48 km. The time difference At can be
neglected in the In-term and writing X1-
,. + Ar,
353 we obtain (Finkelstein et al., 1983; Zeller et al., 1986): (At)E,,av
2,-o (1- ee k) + Ar,.- (ee - k)+ (A" )'/2rD - (e9 -Ar) /2, ] nclr( - e- k) + Ar 2 (ee - k)+ (Ar 2 )2 / 2 r. - (ee Ar2)2/2reJ (3) with
rq = IxeI =(xx%12
e-=x/r; 2
For baselines of - 6000nkm, the (Ar,) 2-terms are of order 3 x 10- 14 sec and can be neglected for picosec-accuracy. 2 For the gravitational time delay due to the Earth one finds (me = GMle/c = 0.44 cm) (At)'Dav,
-
- Arl kc Ar2 k] IAr 2I I-- -I 1ij~ln
(4)
if the motion of the Earth during signal propagation is neglected. Similarly, for any other planet A, if its motion is neglected, one obtains ( A t ) sAv
_
gr
M-
ln
-I
1 1lXA IXA2 XA2'
(5 )
-
k
where XA, = X, - XA. Note that the maximal gravitational time delays due to Jupiter, Saturn, Uranus and Neptune are of order 1.6(Jup), .6(Sat), .2(U) and .2(N) nanosec resprectively, but these values decrease rapidly with increasing angular distance from the limb of the planet. E.g. 10 arcmin from the center of the planet the gravitational time delay amounts only to about 60 picosec for Jupiter, 9 picosec for Saturn and about one picosec for Uranus. To consider the barycentric motion of the planet during signal propagation the position of the planet might be taken at the time of closest approach (e.g. Hellings, 1986); it is, however, unclear how good this correction for the planet's velocity really is. Let us define baselines at signal arriving time tj at antenna 1. Let the barycentric baseline b be defined as
b(tl) then a Taylor expansion of at-
-(b.
k)
X. 2(t 2 )
(6)
=_X (tl) - X2(tl), about tj yields (0(7t)
++ 0(•
(.,t)g,., +0o(4,t.
11-
k)2
0(c-")) 1
)]
(7)
354 all quantities now referring to barycentric coordinate time t1.We call this relation the "VLBI-delay equation", describing the barycentric coordinate time delay At entirely by quantities defined in the global system. We will now relate the various barycentric quantities with corresponding geocentric ones apart from the propagation vector k. This will remind us that the process of signal propagation from the source to the antennas cannot be formulated in the local, accelerated, geocentric system. We now write the time transformation in the form (Damour et al., 1990a-c) =
z(T) + e°(T)Xa + 0(3) dT'+ eo (T)Xa +0(3)
=
~ 1 T/ =c(T - To)4 -
U(za ) ±1 2
c JTo2
dT' + 1Ra(T)v .(T)Xa +0(3). c
Here, T = TCG is the geocentric coordinate time and X' the geocentric spatial coordinate. U is the external gravitational potential which does not include the contribution from the Earth. Replacing T' by t' in the integral and considering that R' is a slowly time dependent matrix we can relate At t 2 - tI with the corresponding local time interval AT = ATCG = T 2 - TI:
At = AT +
U(z)
2
+ 2
+ I Rv,.(T2)Xa(T2) -
) dt'
I
R'v ,(TI)Xa(TI) +
0(4).
With
,,()TX2(T
2)
(,,)+ 0(2),
_V .X - V ,V2 (b
where quantities on the right hand side now refer t(, T, we formally get the relation
At=ZATi -2rv - .B -f 3
C
j(v.
V2)(b
(!(z
k) i (a
2 d1'
Ai' 2 )(b k)
where B(TI) -- X(7Tl)
1
X2(7",)
, ()(1).
(8)
355 and
v2 - a,V RIX a 2
Ar ",2
v 2 is the geocentric velocity of antenna 2. Next, we will relate the barycentric baseline vector b(ti) appearing in the VLBI-delay equation (7), with the corresponding geocentric one B(T 1 ). Using the notation of Damour et al. (1990a-c) we find 4(t,)
-
X(t 2 ) = z(tl) - ZO(t 2 ) + Ra (X (T1 ) -
(,v~v4 +U(ze)6) -
-
with
() [,l~ 'l X2 de(T)
X (T2 ))
-
B+
(T
y-2, X a) (9)
,Xa)-
)] -
X (A.
X )
+ 0(4)
(10)
and A' (T)
e%(T) L- 1 a = dT 2
With Z%(t1) - z%(t2) -_--V'At- -a (At2, 2 2
X',(tl) - X,'(t2) :1 bl(t1) - (At)4(ti) and
XaT)d
X (T,)
-
Xa
X (T 2 )
BB (T,)-
(
1
X2)AT-
2(
d'
(d-x=
l
2
")
(AT) 2 ,
where 1 AT = At + -Iv,,
B + 0(3),
C2
(the integral in (8) is practically of order c - ') the desired relation between barycentric and geocentric baselines reads 7 fl b%= B' -
(
.B)
-
1
k , ,+
k
)
U(Z )6,k
Bk
with A
-
7(T 1 , X1') - C(,X2).
0(3)
(11)
356 Using equation (8), the VLBI-delay equation (7) for (At) and the relation (11) for the baselines we obtain the formal expression (B k = B'k' etc.): AT=
1 )- 1-(B.- k)k.- (v(D + v?.) + 1j(vo •B) -(B.) C2
c
,2 )(B
+ 1 (v.
k) - 1 (B . k)[k . (ve + v2)] 2 + 1 1B . k)U (ze)
+- yj(vo - k)(va) -B) + i(VE) B)(V2 . k) - 1k'A ' +
c
k) 2 k(aD + i 2 ) +
203
+',
dt'
(B k)(ae , Ar)
C3
(12)
+ (,t)rv + 0(4).
Keeping only terms with amplitudes greater than 1 picosec for baselines of the order of 6000 km, wc approximately find: AT= -
(B k)k. (v(;,+ v2) +
k) -
C22 C + l3(vD .v 2 )(B .k)
+
-(B
k)[k
(v
•(ve B) + v 2 )]))
2
a
(13)
k)(vq;. B) + -l-(ve B)(v 2 k) + (At)g,,,. .-(v
Finally, the geocentric coordinate time T can be related with proper time T as indicated by some (atomic) clock located at some VLBI station. Neglecting ,ll tidal effects on local clock rates for clocks at rest at the Earth's surface we find d-T
-
I - _I c
U"(X) + (
X X) 2
= 1-
Ugo(X),
(14)
where U p is the gravitational potential of the Earth and Q is the angular velocity of the Earth's rotation. This can be written in the form d- a-
1 - 1-Ugoo
-1 g()h dT,
(15)
where U', is the geopotential at the geoid, g(4') = (9.78027 + 0.05192 sin 2o) × 102 cm/s z is the latitude dependent gravity acceleration and h is the height above
357 the geoid. Instead of using this formula for ,he T + r relation, we split it into two parts, defining T* TT as proper time on the geoid:
d7-*=d(TT)
(1-
e2
o)
dT=_no dT = o d(TCG)
dr = 1 + -1g(4)h) dr*.
(16a) (16b)
The constant r.0 relating 7* with the geocentric coordinate time T has the numerical value K0 : 1 - 6.9 x 10- 1° . Finally we would like to adress the question of the orientation of spatial coordinates of the local geocentric system. This orientation is determined by the matrix R' (remember that in eqs.(12) and (13) B. k = B'k' with B' = RBa). There are two preferred choices for R' leading to geocentric coordinates which are either - fixed star oriented (kinematically non.rotating) - or locally inertial (dynamically non-rotating). In the first case of kinematically non-rotating coordinates we can take R' = 6'. Then the geodesic precession will be in the precession-nutation matrices as well as in the dynamical equations (e.g. for satellies orbiting the Earth); it will not appear in the group delay equations (12) or (13). On the other hand if dynamically non-rotating geocentric coordinates are chosen then the geodesic precession (secular and annual term) has to be included in the R' matrix. in this case the precession-nutation matrices (and dynamical equations) do not contain the geodesic precession.
References Brumberg, V.A., 1990, "Essential Relativistic Celestial Mechanics", Hilger, Bristol Brumberg, V.A , Kopejkin, S.N , 1988, in. "lRefcrence Systems", J K(,valevsky, I 1. Mueller and B.Kolaczek (eds.), Reidel, Dordrecht Brumberg, V.A., Kopejkin, S M., 1989, Nuovo Cim. [3103, 63 Blanchet,L., Damour,T , 1989, Ann Inst. Henri Polincar6. 50, 377 Damour,T., Soffel,M , Xu,C., 1990a, in: "Les Journ6es 1990", Systenes de Reference Spatio-Temporels, N.Capitaine (ed.), Observatoire de Paris
358 DamourT, SoffelM., Xu,C., 1990b, "General Relativistic Celestial Mechanics I. Method and Definition of Reference Systems", submitted for publication Damour,T., Soffel,M., Xu,C., 1990c, "Relativistic Celestial Mechanics and Reference Frames", Proc. of the IAU Colloqiurn 127, Virginia Beach, Oct., 14-.20, 1990 Finkelstein,A.M., Kreinovich,V.J., Pandey,S.N., 1983, Astrophys. Space Sci. 94, 233 Hellings, R.W., 1986, Astron. J. 91, 650 Herring, T., 1989, Mem'o to Jim Ryan, dated January, 30, 1989 based upcn previous notes by I.I.Shapiro Misner,C., Thorne,K.S., Wheeler,J.A., 1973, "Gravitation", Freeman, San Francisco Richter, G.W., Matzner, R.A., 1983, Phys. Rev. 28, 3007 Soffel,M., Ruder,H., Schneider,M., Campbell,J., Schuh,H., 1986, in: "Relativistic effects in Celestial Mechanics and Geodesy", J.Kovalevsky and V.Brumberg (eds.). Reidel, Dordrecht Soffel,M., 1989, "Relativity in Astrometry, Celestial Mechanics and Geodesy", Sprin-I ger, Heidelberg, Berlin, New York Will,C., 1981, "Theory and Experiment in Gravitational Phy~iLs", Camb:idgc lUniversity Press, Cambridge Zeller,G., Soffel,M., Ruder,H., Schneider,M., 1986, Ver6ff. der Bayr. K(omrn f.d. Intern. Erdmessung, Astronomisch-Geoditische Arbeiten, Heft Nr.48, pp 218-236
359
SYSTEMATIC CORRECTIONS TO THE FUNDAMENTAL CATALOGUE DUE TO THE PRECESSION ERROR AND THE EQUINOX CORRECTION
Mitsuru S6MA National Astronomical Observatory Mitaka, Tokyo 181, Japan
ABSTRACT. Fundamental catalogues of stars have systematic errors due to the precession error and the equinox correction. The formula for these errors to be applied to tile PK4 is presented.
1. Introduction F,.ndamental catalogues of stars have been compiled mainly from meridian observations. The adopted precession constant is obtained from analyses of stars' proper motions, and the adopt?d pozition al any epoch of the zero-point in right ascension (equinox) is obtained from analyses of positional observations of the Sun, Moon and planets in addition to a study of stars' proper motions. When a new fundamental catalogue is constructed, a decision is made whether the prcccssion constant and/or the position of the equinox should be changed or not.. In the case of the Fifth Fundamental Catalogue (FK5) (Fricke et al. 1988), the correction to Newcomb's precession constant obtained by Fricke (1977) and the correction to the position of the PK4 equinox derived by Fricke (1982) are adopted. In the course of investigating the transformation from FK4 system to FK5 system, S6ma and Aoki (1990) have obtained the forniula for the systematic correction to the FK14 due tc the errors in the precession constant and in the location of the equinox. In deriving the formula they assumed the equinox correction (error in the location of the equinox) to be expressed by a linear function of time, but the linearity carnot be assumed a prior. In this paper we d,'rive the for~nula for the systcniatic correction to the FK4 without assumtnig the linearity of th9 equinox correction and show that S6ma and Aoki's assunption is valid up to the 10 */century 2 .
2. Derivation of the Formula Ve will deal with the zystemat;c corrections to the F14 due to the errors in the piecession and in the locati'mn of the equiiox. For this purpose we will ignore tile systenlatic correct i,,i,n to the 1K4 at the epoel, of 131950.0. hi thi,, paper it is asSunled that tile 1, terims of aberration (tw elliptic part of Aberration due to the eccentricity of the Fartl ,s orbit) ate ahxeady retroved ro)n the po,.ioion,-, anid pioper iiAc.tlo.,s.
360
The position vector r and velocity v in this paper are related to the right ascension a,
declination 6, proper motions in right ascension p,, (in radians/tropical century) and in declination ph (in radians/tropical century), radial velocity V (in kin/sec), and parallax 7r (in radians) by the following formulae: cos6 sina \
Cos 6-C
os"'co cos 6sin a and sin 6 /\/
v=
o
nosoi21.094502Co
sin6cosa +r. Ii,,
/
cos 6
The 3 x 3 unit matrix is denoted by I.
The position vector r and velocity v of a star in the FK5 system are related to the position vector rl and velocity v, of the star in the FK4 system by the following equations: PIAu7(B1950.0, t)(r + vt)It=( = R3(-E(t))PNE wC(B1950.0, t)(r, 4- vit)It=, ( (1) d [PIAU76(B1950.0, t)(r + vt)Ilt=o = d [R3(-E(t))PNEwC(B1950.0, t)(rl + vIt)I=U, where the vectors r, v, rl, and vl are evaluated at the epoch of B1950.0, P(TI,T 2 ) = R3(-ZA)R2(OA)R3(-(A) is the 3 x 3 precession matrix for the equatorial coordinates from the epoch T1 to the epoch T based on either Ncwcomb's precession (subscript NEWC) or the IAU (1976) precession (subscript IAU76), R, is the standard 3 x 3 rotation matrix about the it, axis, E(t) is the equinox correction, and t is the time reckoned from B1950.0 in tropical centuries. The variables (A, ZA and OA denote the equatorial precession parameters (see Lieske et al. 1977). The above equations were given by Aoki et al. (1983) and their correctness has been confirmed by S6ina and Aoki (1990). Since P(B1950.0, t)I,;o = I
d
and
dt P(B1950.0,t)It_
00 _n
0 = n 0
0
(2)
0
we obtain from solving Eq. (1) the following: r = Mrl
v = Nrl + Mv,,
and
(3)
where cos Eo0
( NN =
P)sin E0
(nNN -
-n
N
7710
- E) cos EO
cos Eo + no
sinl E0
0 (4)
N
0
(InNIV
M' - E) cos
(-nN -
in' - k) sin E
nV sin EO
N_ n o -ns0 Sin 0
)
In the above equations EO and F are tle values of the equinox correction and its first derivative at B1950.0 derived by Fricke (1982), and m and n are the rates of general precession in right ascension and declination, respectively, based on the Newcomb precession (superscript 0) and the IAU (1976) precession (superscript N). The quantities m and are obtained from
m=-((A(TF,t) + ZA (T,
It[ 0
and
n =-(OA(7T,t))l(.O,
(5)
361 where T is evaluated at the epoch of B1950.0. The position vector sFK4(t) in the FK4 system with respect to the mean equinox of date is given by sF(4(t) = PNEWC(B1 9 50.0, t)(rI + vIt). (6) The position vector SFK5(t) in the FK5 system with respect to the mean equinox of date is given by SFK5(t)
= =
PIAU7G(B1950.0,t)(r+vt) PIAU7G(B!950.0, t)[(M + Nt)r, + Mtvi].
(7)
The difference between (6) and (7) is mainly due to the equinox correction at the epoch t. The systematic correction other than the equinox correction expressed as E + Et is
obtained by
SFKC5(t) R 3 (Eo + Et)SFK4(t)
[(+0.0003 -0.3170 ,+0.1234
=
+
-0.0415
(+0.0016
+0.0001 +0.0000
+/-1.1539
++00000
+
-0.0001
-+0.000,
+0.0016 +0.0007
010-t +0.0001
1 4
+0.0000 -2.1113' ×1-, +1.1539 +0.0247)
(+0.0053
\+2.1113
+
+0.0001 -5.9233
(+0.0059 \-0.0668
+
-0.1235\ +5.9231J x 10- 8 t 2 +0.0002/ +0.0991 -0.0657\ x 10- 8 t 3 4-0.0071 +0.0009 -0.0005 -0.0013/
+0.3170
-0.0032 \+0.0012
(+0.0001 -0.0004 -0.0006
-0.0247
40.0000/
+0.0034
-0.0018\
+0.0258 -0.0480 +0.0006 +0.0001 +0.0000
+0.0120) -0.0205
X 10" 6 t3
+0.0000
+0.0000 +0.0000/
x l0 6 t4
1 ,
(8)
where the difference is expanded to the polynomial of t. The rl-term is the regional systematic correction, and v 1-term is the correction depending on the star's velocity which contributes to the individual correction.
3. Comparison with the Previous Result Eq. (8) shows the systematic corrections to the FK4 due to the errors in precession and in the location of the equinox. The corresponding difference in accord with the discussion by
362 S6ma and Aoki (1990) is obtained from their Eqs. (12a) and (13): PNEwc(B1950.0, t)[(X2
-
XIXo'X 1 )Xo't
[(
2
+ (X
3 -
X2Xo0XI)Xo1 t3]rl
+PNEwc(B1950.0, t)(XIX0 t 2 + X 2 Xo't 3 )vI
+0.0002 40.3148 -0.1354
=
x 10-8t 2 4+0.0001 +5.9231 -5.9232 +0.0002/ (+0.0058 +0.1006 -0.0579 + |-0.0430 +0.0070 +0.0007) × 10-t 3 \-0.0746 -0.0007 -0.0014/
-0.3148 +0.1354
+
+
(+0.0017
0o0001)
+0.0001
+0.0016
+0.0002
+0.0000
+0.0007
+0.0001
X
+1.1539
-2.1112
-1.1539
+0.0000
+0.0247J x 10-t
(+0.0053
+0.0258
+0.0120
+0.0014
-0.0480
-0.0205/
+0.0002
-0.0007)
+0.0001 +0.0000
-0.0001 +0.0000
+0.0001 +0.0000
4
ri
2
-0.0247 +0.0000/ +0.0034 -0.0019
-0.0031
(+0.0001
1
x 1O-t
+0.0000
L+2.1112 +
-0.0001
x1l-1t
3
1
x l-
6t 4
v.(9)
Note that Eq. (12a) of S6ma and Aoki is with respect to the reference frame of B1950.0 and therefore the difference is multiplied by PNEWC to express the difference with respect to the frame of date. The difference between (8) and (9) is less than 10- 1°/century 2 which is equal to 2 x 10-5 arcsec/century 2 . Therefore the linearity assumption of the equinox correction is valid within this accuracy. The calculations were carried out on the FACOM M780/10S of the National Astronomical Observatory of Japan.
References Aoki, S., S6ma, M., Kinoshita, H., and Inoue, K. (1983) Astron. Astrophys. 128, 263-267 Fricke, W. (1982) Astron. Astrophys. 107, L13-L]6 Fricke, W. (1977) Ver6ff. Astron. Rechen-Inst. Hlezdelberg No. 28, Verlag G. Braun, Karlsruhe Fricke, W., Schwan, H., and Lederle, T. (1988) Ver6ff. Astron. Rechen-Inst. Heldelberg No. 32, Verlag G. Braun, Karlsruhe Lieske, J. H., Lederle, T., Fricke, W., and Morando, B. (1977) Astron. Astrophys. 58, 1-16 S6ma, M. and Aoki, S. (1990) 'Transformation from FK4 system to FK5 system', Astron. Astrophys. (in press)
363
INTERMEDIATE STAR REFERENCE SYSTEMS IN THE VICINITY OF RADIO SOURCES
V.V.TEL'NYUK-ADAMCHUK Astronomical Observatory of Kiev University Observatorna Str. 3 252053, Kiev, Ukraine I.I.KUMKOVA Institut of Applied Astronomy USSR Academy of Science Zdanovskaja-8 197042, Leningrad, USSR S.SADZAKOV Astronomical observatory Volgina-7 Yu-11050, Belgrade, Yugoslavia E.TOMA Institut of Astronomy Romania Academy of Sciences Cutitul de Argint 5 75212, Bucharest 28, Romania M.Yu.VOLIANSKA Astronomical Observatory of Odessa University Shevchenko's Park 270014, Odessa, Ukraine
ABSTRACT. In the framework of CONFOR program the formation of star lists of two intermediate reference star systems is being carried out. The first list, RRS2, contains meridian stars in the fields centered at extragalactic radio/optical sources.The second one is formed on the base of 12-14 magnitude stars. The observations are in progress now. The main purpose of this program is to form a base for investigation of mutual orientation of fundamental reference system and new ones.
In the frame of program CONFOR described earlier by Gubanov et al.(1989) the work on establishment of int2rmediate roferenco systems for photoraphic determinations of extragalactic radiosource positions is being carried out. This program has ns an aim the investiqotion of connection Wetveen radiointerferometric and optical coordinate frames. The work is
364
Fig.I. Distribilion of RS2 siars by Yicual rsa-,9iiLde
01683 1500
looo i
15 S0 S9.5
Fig.2. Displlaetent of radiosources irro the cenler of referer:e star
