A new approach can include the use of the latest VLBI-based CRF solutions containing more radio sources with accurate positions, especially in the Southern Hemisphere, and the catalog OCARS Optical Characteristics of Astrometric Radio Sources, Malkin, b that contains the most complete photometric data for astrometric radio sources and thus provides the maximum choice for selection of optically bright radio sources.
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Although fainter sources will have much larger Gaia positional error, the number of these sources may compensate for such a loss of precision. An analysis based on the rotation covariance matrix analysis was performed by Mignard Another approach based on Monte Carlo simulation was used in the current study. It is supposed that faint galaxies can be expected to be unresolved objects for Gaia de Souza et al. Figure 4 illustrates the distribution of the optical magnitudes in the source set used in analysis. The median uncertainty of the gsfb source positions for different sets of sources selected with different upper limits for the optical magnitude is depicted in Figure 5.
One can see that moving to the optically fainter sources improves the overall position precision. Figure 5. Median errors of the gsfb positions depending on source selection based on the faint-end optical magnitude threshold. For the VLBI position errors, the uncertainty estimates given in the gsfb catalog were used. The Gaia position errors used during Monte Carlo simulation were estimated in the following way.
Figure 6. Expected Gaia position error according to de Bruijne et al.
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Solid line corresponds to the approximation function Equation 4 multiplied by 0. This approximation function is depicted in Figure 6. The result obtained with Monte Carlo simulation 10, iterations is shown in Figure 7. This result is generally similar to that obtained by Mignard using a different method, however the errors in the orientation angles are about two times smaller in the current work, in particular, due to larger number of sources used.
Figure 7. Simulated error in orientation angles depending on source selection based on the faint-end optical magnitude threshold. Surely, both results are too optimistic because they were obtained without taking into account the radio—optics position shift, such as the DARN mentioned in Section 3. The result of this test obtained by Monte Carlo simulation in the same way as the previous one is presented in Figure 8.
The obtained error in the orientation angles agrees with the value of 0. Figure 8. Simulated error in orientation angles depending on the additive error due to radio—optics position shift. Indeed, this is the first stage of selection of the best ICRF—GCRF link sources that should follow by estimation of their radioastrometric quality as discussed in Bourda et al.
However, it allows one to have several times more prospective link sources at this stage than was considered earlier: sources vs. Figure 9. One of the methods applied for this purpose during the Hipparcos mission was using radio stars.
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It proved to be the most accurate method among others considered for the orientation of the Hipparcos catalog Kovalevsky et al. The error in the orientation angles between the two frames obtained from VLBI observations of 12 radio stars was estimated to be about 0.
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Later, Boboltz et al. The position of radio stars were determined by means of phase referencing to close ICRF sources with an error of about 10 mas on average. Consequently, properly scheduled radio star observations using large regional or global VLBI networks can provide much smaller position errors and hence better accuracy of the link between optical and radio frames. It can be reasonably supposed that the Gaia positions will be brought to t 0 using its own proper motions. The situation with the ICRF positions is not so simple.
It is definitely not the case at the micro-arcsecond level of accuracy. Several works showed that astrometric VLBI observations are capable of revealing statistically meaningful apparent motions of the ICRF objects, see, e. The question of the nature of these motions is very complicated, and the consistency between various estimates is not satisfactory yet. The overall problem of source motions is beyond the scope of this paper. Here, only one systematic component of source motion pattern, namely galactic aberration in proper motions GA is discussed. The theory of this effect is considered in Kovalevsky , Kopeikin and Makarov , Liu et al.
The proper motion caused by the GA is given by Malkin :. The idea of this test was proposed by Malkin b and is extended here. Two catalogs of radio source positions were used for the simulation. The first catalog comprises ICRF2 sources of AGN type and with visual magnitude 18 m or brighter following the principles of the link source selection proposed by Bourda et al. The second catalog consists of gsfb sources discussed in Section 4.
All the existing catalogs of radio source positions are derived without accounting for the GA during data processing. Therefore, to bring the selected link source positions to the epoch t 0 the following equations should be used:. Then, we have two catalogs for each of four variants two initial catalogs and two t 0 epochs. The first catalog in each pair is merely the initial catalog. The second catalog contains positions of the same sources transferred to the epoch t 0 for the GA-induced proper motions. This catalog would correspond to the radio source positions computed with taking into account the GA effect during VLBI data processing.
Consequently the orientation angles between the two catalogs were computed. Results are presented in Table 2. Table 2. The first line of Table 2 corresponds to computations made in Malkin b. The three other lines are the test extension that allowed us to correct the earlier preliminary conclusion. More detailed test performed in this study has shown that the GA effect can be substantial.
This can be seen from both the significant value of the orientation angles and their uncertainties. The errors in the orientation angles depend on both the number of common sources in the catalogs and the time interval between the mean source observation epoch in the catalog let us call it catalog epoch and t 0. Mean epochs for two catalogs are shown in Figure In contrast, the gsfb catalog epoch is closer to A great milestone in the construction of the celestial reference frame is the Gaia mission, which will result in the GCRF with expected accuracy of a few tens of micro-arcsecond for final release in the early s.
Constructing of a single multi-frequency celestial reference frame based on the ICRF and GCRF is the primary task of fundamental astronomy for the next decade Gaume, The desired goal is to achieve mutual orientation between the two frames with an accuracy of 0. Currently, the International VLBI Service for Geodesy and Astrometry IVS is conducting a special program on observation of prospective link sources not having a sufficient number of observations in the framework of regular observing programs Le Bail et al. The two first criteria seem to be outdated.
Secondly, as shown in Section 5, using optically fainter sources up to 20 m provides more precise determination of the orientation angles. It is even more important that the use of more sources with reliable VLBI positions is necessary to mitigate the impact of most of the negative factors mentioned below, see Figure 8 and related text. As discussed, there are possibilities to multiply the number of link sources taking into account the substantial increase of the number of radio sources with reliably determined positions and the number of sources having photometry measurements.
The following problems were identified in the literature that dilute the accuracy of the link between radio and optical frames:. Most probably, the main factor that will deteriorate the accuracy of the link between the radio and optical frames, is the source structure. Moreover, the structure effects are often variable. Although many studies are devoted to this problem, see, Fey and Charlot ; da Silva Neto et al. Evidently, the most complete study is provided by Zacharias and Zacharias used in the current work. It should be noted that though the structure effect can reach several mas for an individual source, it can hardly be systematic and thus will be averaged over the sky during the computation of the orientation angles between the GCRF and ICRF.
However, supplementary observations and theoretical considerations are needed to quantify this effect more accurately. This method allows for a straightforward solution of the task. However, there are serious constraints on the accuracy of this method caused by the previously mentioned factors. Observations of radio stars can serve as an alternative equally accurate method.
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However, this method is also affected by some accuracy-limiting factors Malkin, a. Many radio stars comprise double or multiple systems, and thus their orbital motions must be accounted for. The accurate Gaia -derived orbits can be used for this purpose. Moreover, radio stars may have complex and variable structures, which might cause a time-dependent bias between the radio and optical positions.
Lestrade et al. Provided that several tens of radio stars have been observed, this factor should not significantly impact on the errors in the orientation angles.
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Combination of two methods of linking the GCRF to the ICRF should facilitate improvement of the systematic accuracy of the link between radio and optical frames. Finally, it should be noted that the preparation for the aligning of the Gaia catalog to ICRF3 planned for — is only an intermediate stage in construction of the multi-frequency celestial reference frame. Certainly, the main work on the link between radio and optical frames is to be done in the early s, after the final Gaia catalog is prepared.
In the framework of this activity, it appears to be very important to fast-track the plans on the ICRF improvements in the Southern Hemisphere Jacobs et al. As the Hipparcos experience has shown, it can be expected that the work on improving the link between radio and optical frames will be continued during a long period after completing the Gaia mission. The author confirms being the sole contributor of this work and approved it for publication. The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author would like to thank the two reviewers and the handling editor for their valuable help in improving the manuscript. Andrei, A. Gaia initial QSO catalogue: the variability and compactness indexes. Google Scholar. Capitaine Paris: Paris Observatory , 84— IAU Gen. Arias, E. Berghea, C. Boboltz, D. Very large array plus pie town astrometry of 46 radio stars.
Bouffet, R. Cambresy, F. Martins, E.
Nuss, and A. Palacios Montpelier, VT , — Bourda, G. Astrometric suitability of optically-bright ICRF sources for the alignment with the future Gaia celestial reference frame. VLBI observations of optically-bright extragalactic radio sources for the alignment of the radio frame with the future Gaia frame.
Source detection. Imaging candidate sources. A pattern of noncoincidence between radio and optical positions of international celestial reference frame sources. Gaia astrometric science performance - post-launch predictions. EAS Publ. Detection of galaxies with Gaia. Fey, A. VLBA observations of radio reference frame sources. Astrometric suitability based on observed structure.
Gaume, R. Gordon, D. Jacobs, C. Capitaine Paris , 51— Kopeikin, S. The 3D x,y,z position does not need to fall on the unit sphere. The RA, Dec will be inferred from the relations Corresponds to r cos dec cos ra :param y: The y position in 3 dimensions. Corresponds to r cos dec sin ra :param z: The z position in 3 dimensions. The connection between ra,dec and x,y,z are given by the following formulae May be a numpy array. Using plain arctan is slightly faster. It still gives the right answer, but we catch and ignore the warning here.
This convention is not enforced on construction, so this function exists to make it easy to convert if desired. The return value is an Angle object :param coord2: The CelestialCoord to calculate the distance to. This formula is perfectly stable here. But in this case, the cross product provides an accurate distance. Two points, c1 and c2, on the unit sphere define a great circle so long as the two points are not either coincident or antipodal. The current coordinate along with the two other coordinates form a spherical triangle on the sky. This function calculates the angle between the two sides at the location of the current coordinate.
Note that this returns a signed angle. It is negative if the direction is clockwise. If we treat each coord as a x,y,z vector, then we can use the following spherical trig identities: A x C.