Evidence of the $Gaia$--VLBI position differences being related to radio source structure
Ming H. Xu, Susanne Lunz, James M. Anderson, Tuomas Savolainen, Nataliya Zubko, Harald Schuh
AAstronomy & Astrophysics manuscript no. gaia_crf_edr3_R1 © ESO 2021February 1, 2021
Evidence of the
Gaia –VLBI position differences being related toradio source structure
Ming H. Xu , , , Susanne Lunz , James M. Anderson , Tuomas Savolainen , , Nataliya Zubko , and Harald Schuh , Aalto University Metsähovi Radio Observatory, Metsähovintie 114, 02540 Kylmälä, Finland; e-mail: [email protected] Aalto University Department of Electronics and Nanoengineering, PL15500, FI-00076 Aalto, Finland DeutschesGeoForschungsZentrum (GFZ), Potsdam, Telegrafenberg, 14473 Potsdam, Germany Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Straße des 17. Juni 135, 10623, Berlin, Germany Finnish Geospatial Research Institute, Geodeetinrinne 2, FIN-02430 Masala, FinlandReceived ***; accepted ***
ABSTRACT
Context.
We report the relationship between the
Gaia –VLBI position di ff erences and the magnitudes of source structure e ff ects inVLBI observations. Aims.
Because the
Gaia –VLBI position di ff erences are statistically significant for a considerable number of common sources, weattempt to discuss and explain these position di ff erences based on VLBI observations and available source images at cm-wavelengths. Methods.
Based on the derived closure amplitude root-mean-square (CARMS), which quantifies the magnitudes of source structuree ff ects in the VLBI observations used for building the third realization of the International Celestial Reference Frame, the arc lengthsand normalized arc lengths of the position di ff erences are examined in detail. The radio jet directions and the directions of the Gaia –VLBI position di ff erences are investigated for a small sample of sources. Results.
Both the arc lengths and normalized arc lengths of the
Gaia and VLBI positions are found to increase with the CARMSvalues. The majority of the sources with statistically significant position di ff erences are associated with the sources having extendedstructure. Radio source structure is the one of the major factors of these position di ff erences, and it can be the dominate factor for anumber of sources. The vectors of the Gaia and VLBI position di ff erences are parallel to the radio-jet directions, which is confirmedwith stronger evidence. Key words. galaxies: active / galaxies: jets / astrometry / reference systems / radio continuum: galaxies
1. Introduction
The International Celestial Reference Frame (ICRF) wasadopted as the Fundamental Celestial Reference Frame for as-tronomy in January 1998 by the International AstronomicalUnion (IAU) (Ma et al. 1998). The ICRF is realized by the po-sitions of distant radio sources, mostly active galactic nuclei(AGNs), based on the astrometric / geodetic very-long-baselineinterferometry (VLBI) observations coordinated by the Interna-tional VLBI Service for Geodesy and Astrometry (IVS; Schuh& Behrend 2012; Nothnagel et al. 2017, please also refer to theIVS website ), and relies on the VLBI technique for its main-tenance and improvement. As o ffi cially adopted by the IAU inJanuary 2019, the third realization of the ICRF (ICRF3; Charlotet al. 2020) is established based on 40 years of VLBI observa-tions and, for the first time, at three di ff erent radio frequenciesindependently. The radio source positions in the ICRF3 have ac-curacies at the sub-milliarcsecond levels. The European SpaceAgency mission Gaia (Gaia Collaboration et al. 2016) has re-leased position estimates and other astrometric parameters forthe celestial objects with optical G magnitudes <
21 mag basedon the observations during the 22 months since July 2014 (DR2;Gaia Collaboration et al. 2018a). The color-dependent calibra-tion is possible based on the
Gaia
DR2 and leads to improve-ments in the astrometric solution thereafter (Lindegren et al. https://ivscc.gsfc.nasa.gov/index.html https://sci.esa.int/web/gaia Gaia
Early Data Release 3 (EDR3; Gaia Collaborationet al. 2020) has made the data available based on the first 34months of its observations.A good overall agreement between radio and optical po-sitions was achieved for the cross-matched common objects(Mignard et al. 2016; Gaia Collaboration et al. 2018b); the me-dian arc length between the source positions from
Gaia andVLBI is ∼ Gaia
DR2 andthe ICRF3. However, the distribution of the arc lengths betweenradio and optical positions normalized by their uncertainties,called normalized arc length hereafter, deviates from the ex-pected Rayleigh distribution with σ =
1. The most obvious de-viations in that distribution are the long tail spreading to verylarge normalized arc lengths and the significant deficit of valuesin the bins around the expected peak. In the
Gaia
DR1, therewere only a few percent of sources with normalized arc lengths > Gaia
DR2, the number of such sources even increases to larger than10 percent (Petrov & Kovalev 2017a; Gaia Collaboration et al.2018b; Petrov et al. 2019; Makarov et al. 2019). By deselectingobjects mostly based on the optical properties, Makarov et al.(2019) still found 20 percent of sources having normalized arclengths >
3. The factors causing these position di ff erences be-tween optical and radio are still unclear, even though there area variety of possible astrophysical explanations (Makarov et al.2019; Plavin et al. 2019a; Kovalev et al. 2020). For instance,Kovalev et al. (2017) and Petrov et al. (2019) suggested that the Article number, page 1 of 13 a r X i v : . [ a s t r o - ph . GA ] J a n & A proofs: manuscript no. gaia_crf_edr3_R1 main cause of the position di ff erences is optical structure, theoptical jets at the mas scales. Understanding these position dif-ferences is very important because (1) it will lead to a betterselection of the common sources for aligning the optical frameto the radio frame; (2) the number of sources with statisticallysignificant position di ff erences can continue to increase in future Gaia data releases, which would allow more and more small po-sition di ff erences be detected at the 3 σ confidence level; and (3)the position di ff erences may tell something important about theastrophysics of the AGNs.We examine the position di ff erences between Gaia andVLBI from the radio side. As demonstrated in the imaging sur-vey of radio sources (Charlot 1990a; Fey & Charlot 1997), thecelestial reference frame (CRF) sources commonly have angularstructure at the mas scales at cm-wavelengths . Source structureis time and frequency dependent, and it is not modeled in thedata analysis of building the ICRF3. The position estimates andtheir uncertainties in the ICRF3 are based on global least-squarefitting (LSQ) and are thus not able to characterize the impactsof the systematical position variations over the 40 years due tosource structure. For example, the position uncertainties fromLSQ are likely underestimated in the presence of systematic er-rors. Our previous study has used the same VLBI observationsas for the creation of the ICRF3 to quantify the magnitude of ef-fects of source structure on VLBI observables for each individualsource (Xu et al. 2019). In this paper, we apply the results to in-vestigate the relationship between the Gaia –VLBI position dif-ferences and source structure at the cm-wavelengths. We then at-tempt to explain and discuss these position di ff erences based onthe radio images from the Monitoring Of Jets in Active galacticnuclei with VLBA Experiments (MOJAVE; Lister et al. 2018).The paper is structured as follows. We introduce in Sect. 2how the arc lengths of position di ff erences, the normalized arclengths, and the quantitative values of measuring structure ef-fects are derived. We describe in Sect. 3 the results from the ex-amination of arc lengths, normalized arc lengths, optical G mag-nitudes, and redshifts with respect to source structure. In Sect.4, the following topics are discussed: (1) source structure and itsquantification; (2) the impact of frequency dependence of sourcestructure; (3) the large position di ff erences that are statisticallysignificant; (4) the magnitudes of the position di ff erences; and(5) the directions of the position di ff erences. We make the con-clusion in Sect. 5.
2. Data
Gaia and VLBI
We used the right ascension and declination estimates, their un-certainties and the correlations between these two coordinates inthe ICRF3 , which contains 4536 sources observed by astromet-ric / geodetic VLBI at S / X band. The median uncertainties of rightascension and declination reported in the ICRF3 are 0.155 masand 0.217 mas, respectively. We used the
Gaia
DR2 and EDR3 to get the five astrometric parameters (source position, propermotion, and parallax), their uncertainties, the correlations be-tween them, and the optical magnitude.Even though the cross match between radio and optical cat-alogs basically relies on the position coincidence, other criteria Refer to the images of CRF sources at https://gea.esac.esa.int/archive/ are needed to reduce the risk of false match. Lindegren et al.(2018) applied constraints on the other three astrometric param-eters and the number of observations, and masked out the regionnear the Galactic plane, as shown in Eq. (13) of the publica-tion. Petrov & Kovalev (2017b) used the concept of probabilityof false association as a function of Gaia source density on aregular grid and the possible area defined by the positions andthe uncertainties at radio and optical wavelengths for each po-tential match. We combined these two methods to identify thecommon objects between the ICRF3 and the
Gaia
DR2, whichgives 2970 sources (Lunz et al. 2019, please refer to the poster ).Based on the Gaia
EDR3 and the ICRF3, we identified 3142common sources, the same number of matched sources as foundby the
Gaia team in the on-going analysis (François Mignard,private communication).For each common source, we calculated the arc length be-tween the
Gaia and VLBI positions, ρ , by ρ = (cid:113) ( ∆ α cos δ ) + ∆ δ , (1)where ∆ α and ∆ δ are the di ff erences of right ascension and dec-lination in the Gaia data and the ICRF3, respectively, and δ isthe declination. The normalized arc length, X ρ , is defined andcalculated by X ρ = ρ/σ ρ , (2)where σ ρ is the uncertainty of ρ based on the full 2 × σ pos,max , was computedfor both Gaia and VLBI by σ pos,max = (cid:34) ( σ α cos δ ) + σ δ + (cid:114)(cid:16) ( σ α cos δ ) − σ δ (cid:17) + (2C αδ σ α cos δσ δ ) (cid:35) , (3)where σ α and σ δ are the uncertainties of right ascension anddeclination, respectively, and C αδ is the correlation coe ffi cient ofthe two coordinates.Based on the Gaia
DR2 and the ICRF3, there are 732 sourceswith X ρ > Gaia
EDR3, 804 sources have X ρ > We adopted the log closure amplitude root-mean-square(CARMS) values from Table 2 in Xu et al. (2019) to quantify themagnitude of source structure e ff ects for each individual source .Due to the missing data for calibration and the insensitiv-ity of the parameters of geodetic concern, visibility amplitudesfrom interferometry were not used for most of the geodetic VLBIobservations. However, they carry valuable information aboutsource angular structure, which causes structure e ff ects in groupdelays up to hundreds of picoseconds (Charlot 1990b; Xu et al.2016). By forming quadrangles with four baselines, one can geta ratio of the four amplitude observables to cancel out exactlythe station-based errors, which is called closure amplitude andprovides information about the intrinsic source structure. For an The complete table is available through the CANFAR data DOI at: .Article number, page 2 of 13ing H. Xu et al.: Position di ff erences between Gaia and VLBI ideal point-like source, all the baselines will observe the sameamplitude within the thermal noise, giving closure amplitudesclose to unity; for an extended source, the closure amplitudesdeviate from unity. The CARMS value of a source is defined tobe the root-mean-square (rms) of log closure amplitudes at theX-band based on the basic weighting scheme (See Eqs. (2)–(4)and (6)–(8) in Xu et al. (2019)). In addition to the study in Xuet al. (2019), please also refer to its supporting material , wherethe closure phase and closure amplitude plots are available fortens of sources to demonstrate the source structure e ff ects andcompare with their CARMS values.The CARMS values are available for 3417 radio sources inthe ICRF3 and were derived from the astrometric / geodetic VLBIobservations from 1979 to 2018, the same dataset establishingthe ICRF3. They are in the range 0.03–1.63, and the mean andmedian values are 0.31 and 0.24, respectively. The CARMS val-ues generally classify the CRF sources into three categories:1. CARMS < > > ff erent source cate-gories in the ICRF catalogs. For instance, the 39 special handlingsources in the second realization of the ICRF (Fey et al. 2015),which have variations in the time series of VLBI position esti-mates at the mas or sub-mas levels, have the median CARMSof 0.60, while the median value for the ICRF3 defining sources,used for defining the axis directions of the ICRF3, is 0.25. Re-cently, these CARMS values were used to select radio sourceswith minimum structure to assess the quality of group delays inthe broadband VLBI system (Xu et al. 2020a).For the 3142 common sources from the Gaia
EDR3, theCARMS values are available for 2460 sources, 78 percent; themean and median CARMS values are 0.30 and 0.24, respec-tively, which are at the same level as those of the 3417 sources.We examined the source position estimates based on both the
Gaia
DR2 and EDR3, but we will focus on the results from theEDR3 in our study.
We used the Optical Characteristics of Astrometric RadioSources catalog (OCARS; Malkin 2018) to search for the red-shifts. The OCARS conveniently provides the redshifts for ra-dio sources by collecting them in the literature. Among the 2460sources, we got the redshifts for 2198 sources, ∼
89 percent. Theyare in the range 0.01–5.06 with mean and median values of 1.28and 1.18, respectively.
3. Results ρ We examined the arc lengths ρ between the VLBI and Gaia source position estimates with respect to the CARMS values. Ta-ble 1 shows the mean and median values of ρ and σ ρ for di ff erentranges of CARMS values. The median ρ steadily increases from ∼ ∼ ρ increases more significantly from ∼ ∼ ρ begins to arise when CARMS (cid:39) ρ arises significantly, above 1.0 mas, when CARMS (cid:39) ff ects.When CARMS < ρ have mean valuesof ∼ ∼ ∼ ∼ Gaia in the near future, as happenedfrom the DR2 to the EDR3. However, when CARMS > < ff erences between VLBI and Gaia , which are statisti-cally very significant, whereas the sources with minimum struc-ture tend to have smaller position di ff erences, which are statisti-cally insignificant.The mean ρ is always larger than the median due to a smallfraction of sources having considerably larger ρ than the rest ofsources in each group. The di ff erences between the mean andmedian ρ increase with the CARMS values. X ρ We examined the normalized arc lengths X ρ with respect to theCARMS values. The statistics of X ρ are shown in Table 1. Adependence of X ρ on the CARMS values is revealed. Figure 1shows the three distributions of X ρ for the 2460 common sources(top), the sources with CARMS < > X ρ >
3. For the 207 radio sources with little struc-ture, the distribution of X ρ shown in the middle panel is closeto the expected Rayleigh distribution; however, for the 556 radiosources with CARMS > X ρ clearly de-viates from the Rayleigh distribution — half of the sources have X ρ > X ρ >
10. The probability of hav-ing statistically significant position di ff erences is doubled for theradio sources with extended source structure (CARMS > X ρ >
10 (red open bins). The mean and median CARMS values for all2460 sources are 0.30 and 0.24, respectively; for the 147 sourceswith X ρ >
10 these values are 0.52 and 0.48. About 60 percentof the sources with X ρ >
10 have CARMS > > ff erences and the sources with ex-tended structure.The di ff erences of the CARMS values for the sources withvarious ranges of ρ and X ρ are shown in Table 2. For di ff erentmagnitudes of ρ , the mean and median CARMS values for thesources with X ρ > X ρ ; these values for the sources with X ρ in therange of 3 to 4 are larger than for the sources with X ρ <
3. On av-erage, the di ff erence in the CARMS values is ∼ ff er-ences. There are only a slight increase in the mean and medianCARMS values as ρ increases for X ρ >
4. One should be cau-tious when interpreting the results in Table 2, because they willchange with better uncertainties of source positions available infuture
Gaia data releases. With the significant improvement inposition uncertainties expected from the
Gaia observations, thesources with current X ρ ≤ X ρ >
3, as happened forthe
Gaia
DR2 compared to the
Gaia
DR1 and for
Gaia
EDR3compared to the
Gaia
DR2. Meanwhile, the arc lengths ρ for the1813 sources with X ρ ≤ Article number, page 3 of 13 & A proofs: manuscript no. gaia_crf_edr3_R1
Fig. 1.
Histogram of X ρ for the 2490 sources (top), the sources with CARMS < > X ρ >
10 are accounted in the last bin. The blue curves show the Rayleigh distributions with unit standard deviation. The totalnumber of sources in each of the three samples is shown in black on the top-right of each panel, and the number of the sources with X ρ > X ρ =
3. The remarkable di ff erences in the distributions of X ρ between these three groups of sources arethe numbers of sources in the last bins, X ρ > ff erences between Gaia and VLBI
Table 1.
Arc lengths ρ and normalized arc lengths X ρ with respect to CARMS CARMS N src ρ [mas] X ρ σ ρ [mas]Mean Median Mean Median Mean Median < ≥ Gaia
DR2 and EDR3. The arc lengthsof these 1813 sources are all smaller than 4.0 mas; the num-ber of the sources with ρ ≥ Gaia andthe ICRF3 will be identified from the future
Gaia data releases.Since the ICRF3 sources were systematically included in the
Gaia quasar list, those missing sources in the
Gaia
EDR3 areprobably too faint in optical and it is unlikely to have signifi-cantly more matches from
Gaia . As shown in Table 1, the un-certainties of ρ have mean and median values of about 0.3 masand 0.2 mas, which allow the large ρ , for instance larger than4.7 mas, be confidently detected but are not able to fully iden-tify the sources with ρ < Gaia data release is available to identify more sources with small ρ and large X ρ , the mean and median CARMS values will thus de-crease for the sources with ρ < X ρ >
4. We wouldexpect to have the CARMS values steady increasing with respectto ρ in the future Gaia data releases, as we see that ρ increaseswith CARMS in Table 1. In the following investigation, we setthe limit of X ρ = ff erences. Fig. 2.
Histogram of the CARMS values for the 2460 radio sources(filled gray bars) and for the 147 radio sources with X ρ > / G magnitude and redshift We examined optical G magnitude and redshift z with respect tothe CARMS values to investigate if there is any potential cor-relation between the CARMS values and the optical properties.Table 3 shows the statistics of G and z . Both the mean and me-dian magnitudes generally decrease with respect to the CARMSvalues; the di ff erence in G between the sources with CARMS < > z andCARMS in the table. The correlation between CARMS and both G and z seems to be significant.We further examined G and z in more detail. This investiga-tion can be biased, because the uncertainties of Gaia positionsdepend on G , as shown in Gaia Collaboration et al. (2018b). Thestatistics of arc lengths and normalized arc lengths with respectto G can be dramatically changed when new position estimateswith improved uncertainties are available from Gaia in the nearfuture. We nevertheless attempt to address it based on the
Gaia
EDR3.Table 4 shows the statistics of arc lengths, the major axesof the error ellipses of the
Gaia positions and the VLBI posi-tions, the CARMS values, and z with respect to di ff erent opti-cal G magnitudes for the 2028 sources with X ρ ≤
4. As we ex-pect, the G and the z are positively correlated for these sources— when object is further away, it appears dimmer. The di ff er-ences of the mean CARMS values at various ranges of G are nolarger than 0.06 and those of the median values are no larger than0.08. There is a small decrease in the CARMS values when G in-creases, which demonstrates that when a source locates fartheraway the scale of its structure may decrease. The magnitudes of ρ gradually increase with respect to G , however, the position un- Article number, page 5 of 13 & A proofs: manuscript no. gaia_crf_edr3_R1
Table 2.
CARMS values with respect to ρ and X ρ ρ [mas] if ( X ρ > ≥ X ρ > X ρ ≤ N src Mean Median N src Mean Median N src Mean Median < . . . . . . ≥ . . . . . . . . . . . . all 432 0.46 0.42 215 0.32 0.27 1813 0.26 0.22 Table 3.
Optical G magnitude and redshift z CARMS Optical G magnitude [mag] Redshift zN src Mean Median N z Mean Median < ≥ Gaia and VLBI also vastly increase. Since theratio of the arc lengths to its uncertainties is always at the samelevel for di ff erent ranges of G , it is not possible from the resultto conclude that there is dependence of ρ on G .Table 5 shows the statistics of the same quantities as Table 4but for the 432 sources with X ρ >
4. The arc lengths increase bya factor of ∼
10 from G <
15 mag to G ≥
20 mag. This apparentdependence of ρ on G , however, is mainly due to the high corre-lation between the Gaia position uncertainties and G , as shownin the Table. Because the Gaia position uncertainties get worsedramatically as G becomes higher, a uniformed threshold of X ρ ,which is 4 in the study, forces only the sources with large enougharc lengths to be selected at the higher optical magnitudes. Asdiscussed before, these statistics will be changed with new posi-tion estimates available from the future Gaia data releases.By comparing the results in Tables 4 and 5, the major di ff er-ences of these two groups of sources are found to be CARMSand z . The CARMS values of the sources with X ρ > ∼ X ρ ≤
4; the mean andmedian z values are smaller by 0.21 and 0.32, respectively. Therelationship between G and z for these two groups of sources areshown in Fig. 3. The sources with X ρ ≤ z steady in-creasing over G , while the sources with X ρ > z when G > ff erencesmay also be associated with, for instance, some weak but nearby(small z ) optical objects. Fig. 3.
Mean redshift values with respect to the optical G magnitudesfor the 2198 sources with their redshifts available. The blue curve is forthe sources with X ρ ≤
4, and the red curve is for the sources with X ρ > G are shown in the first column in Table 4. The sourceswith X ρ > Z at G > z at G (cid:39) . X ρ ≤
4. The statistics are shown inTables 4 and 5.Article number, page 6 of 13ing H. Xu et al.: Position di ff erences between Gaia and VLBI
Table 4.
Statistics of the 2028 sources with X ρ ≤ G [mag] N src ρ [mas] σ pos,max [mas] CARMS z Mean Median
Gaia
VLBI Mean Median N z Mean Median < ≥ Note . The values in the fifth and sixth columns are the mean σ pos,max for Gaia and VLBI position estimates, respectively.
Table 5.
Statistics of the 432 sources with X ρ > G [mag] N src ρ [mas] σ pos,max [mas] CARMS z Mean Median Gaia VLBI Mean Median N z Mean Median < ≥ Note . The values in the fifth and sixth columns are the mean σ pos,max for Gaia and VLBI position estimates, respectively.
4. Discussion
The CRF sources have radio emission with angular scales at maslevels over the sky, called source structure. It causes structuredelays up to hundreds of picoseconds as shown in modeling byCharlot (1990b) and in actual observations by Xu et al. (2016).Based on the CONT14 observations , Anderson & Xu (2018)suggested that source structure is the major contributor to errorsin the astrometric / geodetic VLBI. Since these e ff ects in VLBIgroup delays have not been modeled in the VLBI data analysis,based on which the ICRFs were built and maintained, the sourcepositions from VLBI change over time due to both the di ff er-ent observing geometry between antennas and sources and thevarying structure. For a large fraction of CRF sources, the struc-ture e ff ects can change their positions at the level of 0.5 mas, asshown in the position time series of 39 well-observed sources https://ivscc.gsfc.nasa.gov/program/cont14/ (Ma et al. 2009, see the plots in the IERS Technical Note 35 ).The number of sources a ff ected by the structure e ff ects will dra-matically increase when we consider the position di ff erences be-tween Gaia and VLBI down to the levels of ∼ ff ects in amplitude observables.For a source with CARMS = = = − + + + ff er-ent frequencies by di ff erent antenna arrays during di ff erent time Article number, page 7 of 13 & A proofs: manuscript no. gaia_crf_edr3_R1 periods compared to the observations for the ICRF3 and theCARMS values, we cannot expect an exact proportional relationbetween the CARMS values and the scales of the MOJAVE im-ages. However, they are already of great help to demonstrate thedi ff erences between the CARMS values smaller and larger than0.3. The two sources 0048 −
097 (CARMS = + = +
784 (CARMS = + = Gaia and VLBI are alsoshown in the plots. It is obvious in the plots that the
Gaia -VLBIposition di ff erences are typically parallel to the jet directions,which has already been reported by Kovalev et al. (2017) andPetrov et al. (2019) and will be discussed in Sec. 4.5.There are four remarks concerning CARMS. First, it was cal-culated based on actual VLBI observations rather than based onthe maps of radio sources. Once the CARMS is large, the sourceshould have extended structure; but if the source has extendedstructure, it does not necessarily have a large CARMS value dueto insu ffi cient observations in terms of ( u , v ) coverage to capturethe structure. However, the great advantage of using actual VLBIobservations is that it quantifies the magnitude of structure ef-fects over the whole time period of 40 years. Second, CARMS isbased on (log) closure amplitudes, which are not sensitive to theabsolute source position. Therefore, only the relative structure,i.e., the relative positions and the relative fluxes between the mul-tiple components, is defined by CARMS; if a source with com-pact structure changes its position on the sky, the CARMS valuecannot tell that change. Third, since there was no attempt to doproper weighting for di ff erent sizes of quadrangle and select anindependent set of closure amplitudes for each individual sourcein deriving CARMS values, it becomes di ffi cult to tell a sourcewith a medium CARMS value, 0.25–0.30, as having structureto what extent. Fourth, CARMS was derived from the X-bandobservations only, while the ICRFs are based on the ionosphere-free delays through the linear combination of the group delaysat the S / X band. The structure e ff ects in the S-band observationsthus are ignored in this study. Even though the contribution ofthe structure e ff ects at the S-band is scaled down by a factor of ∼ < X ρ > ff ects is still missing in astromet-ric / geodetic VLBI data analysis after it has been discussed forseveral decades. The practical problems are to continuouslymake images for hundreds of sources and for each source manytimes if structure changes. The main challenge is that the imagesfor modeling structure e ff ects have to be registered over timefor each source in order to maintain a stable CRF at high accu-racy. The next generation of geodetic VLBI, known as VGOS(Niell et al. 2007; Petrachenko et al. 2009), requires to regis-ter the images of each source at the four di ff erent bands in therange of 3.0–14.0 GHz (Xu et al. 2020b). Otherwise, only therelative structure e ff ects can be reduced, and the misalignmentof the images at di ff erent epochs or at di ff erent frequency bandsdue to core shift, discussed in the next section, inevitably leadsto source position variations. Due to the limitation in imagingresolutions, identifying the reference points in structure / imagesis di ffi cult for the accuracy levels better than 0.1 mas. Therefore,aligning the images and investigating core shift are very crucialin order to mitigate these systematic e ff ects. Source structure is frequency-dependent due to two factors: (1)the steep spectrum of the extended jet causing the sources tohave larger scales at lower frequencies; and (2) synchrotron self-absorption causing changes in the optical depth along the jet.The latter factor leads to changes in the position of the core,where the optical depth is unity, depending on the observing fre-quency. This e ff ect, so called core shift, was predicted by Bland-ford & Königl (1979). When the observing frequency increases,it causes the position of the core to move towards the jet base.Core shift was first measured for the source 1038 + ∼ + k ν − β , where k isa source-dependence core shift parameter — it can be variableover time according to the study of Plavin et al. (2019b), ν is theobserving frequency, and β is an astrophysical parameter. So far, β is measured to be close to 1 (Lobanov 1998; Sokolovsky et al.2011), which agrees with the prediction under the condition ofthe equipartition between jet particle and magnetic field energydensities (Blandford & Königl 1979).The impact of core shift on astrometric positions measuredby VLBI was discussed by Porcas (2009), using a simple modelof a point-source core. Based on the median core shift between2.3 GHz and 8.4 GHz, 0.44 mas, from Kovalev et al. (2008), thecore position is shifted by 0.166 mas at the frequency of 8.4 GHzand varies by 0.014 mas over the frequency band of 8.2–8.9 GHzused in most of geodetic VLBI observations. The position shiftof 0.166 mas can cause visibility phase variations of several de-grees over the band, which are canceled out exactly by the ad-ditional phase variations due to the position shifts of 0.014 masover the band. It was shown that given β =
1, group delays ofobservations on a point-like source refer to a fixed point at the jetbase at any frequency and at any time, no matter whether k variesor not over time. It is therefore believed that core shift will notcontribute to the position di ff erences between Gaia and VLBI,given that β (cid:39) β (cid:39) ff ects onsource structure: (1) moving the absolute position of the core to-wards the jet base when the frequency increases; and (2) chang-ing the relative positions between the core and the jet compo-nents in structure. Apparently, the discussion of Porcas (2009)investigated the first e ff ect only. The truth is again that almost allthe CRF sources have structure at the mas scales, which changesover time. In the previous discussion, the relative positions be-tween the core and the jets will also be changed by amount of0.014 mas over the band to the opposite direction of the abso-lute position shift of the core. The cancellation of the across-band phase variations in the point-source case breaks down forextended sources. Therefore, core shift can influence the posi-tion estimates determined from VLBI group delays. In this con-text, even though there may be no real connection between themagnitude of core shift and the scales of source structure, the Article number, page 8 of 13ing H. Xu et al.: Position di ff erences between Gaia and VLBI
Fig. 4.
MOJAVE images of four sources, 0048 −
097 (CARMS = +
581 (CARMS = + = +
738 (CARMS = ff erent frequencies , this assumption may introducesystematic errors. Based on their position di ff erences between the Gaia
EDR3 and the ICRF3, the
Gaia positions are thus located at the blue dots.The error bars are the 3 σ uncertainties of right ascension and declination from Gaia and VLBI. It is also conspicuous that when X ρ > Gaia position vectors favor the directions along and opposite to the jets, as shown by (Kovalev et al. 2017; Petrov et al. 2019). The ρ and X ρ values are shown on the upper-left corner of each plot. These four plots demonstrate how the scales of the structure look like in terms of di ff erentCARMS values. Nevertheless, we should mention that the CARMS values and the ICRF3 are based on VLBI observations at the frequency bandaround 8.4 GHz over 40 years, while these images were made from observations at 15.35 GHz during the short periods shown at the top of eachplot. The jet components always become more prominent at the lower frequency bands. The images were convolved with a circular beam of 0.3mas as indicated by the black circle in the bottom left corner, about 40% of the typical MOJAVE beam size. Overlay contours are shown at tenlevels of peak percentage specified in the bottom of plots. impact of core shift will correlate with structure e ff ects — ex-tended sources with large source structure e ff ects tend to havelarger core shift e ff ect in the position di ff erences between radioand optical than the sources with minimum structure. Furtherstudies are needed to verify this assumption. ρ > 4.0 mas and X ρ > 4 There are 75 sources with ρ > X ρ >
4. Among them,53 sources have CARMS > > ≤ z available, and 15 sources have z < z of these 20 sources is 0.25, which is only one fifth of the me-dian z of the 2198 sources with known z . A small fraction of Article number, page 9 of 13 & A proofs: manuscript no. gaia_crf_edr3_R1 these sources seem to be weak but nearby optical objects. It isimportant to investigate this further.
With an improvement in
Gaia position estimates in the near fu-ture, the number of the sources with X ρ > ff erences, for in-stance ρ > G <
18 mag, and the mean semi-major axis of the error ellipsesof the
Gaia positions for these 615 sources is already smallerthan 0.1 mas. In this sample of 615 sources, 181 sources have X ρ >
4, 2 /
5. About 74 percent of these 181 sources have ρ < ρ is ∼ ρ for the majority of the sources with X ρ > ff ects and core shift. For the434 sources with X ρ ≤
4, the median ρ is ∼ Gaia and VLBI for the whole ensemble of commonsources.If we assume that the median uncertainty of the
Gaia sourcepositions at higher optical magnitudes is ∼ Gaia andVLBI positions will agree with each other within their uncer-tainties for the 3 / ρ for these sourceswill be at the level of 0.24 mas. There will be 2 / ff erences with a median ρ of0.8 mas.Based on about 2000 evenly distributed sources over the skywith position di ff erences of ∼ Gaia frame with respect to the ICRF3 may be achieved atthe level of ten microarcseconds ( µ as); it is su ffi cient enough todetect systematic position di ff erences between Gaia and VLBIat the level of hundreds of µ as. Several hundreds of sources withwell-detected position di ff erences at such levels will provide in-valuable information to investigate the physical properties of ra-dio sources. Source structure and core shift are expected to cause the de-rived source positions from VLBI to shift towards the jets. If theVLBI-to-
Gaia position vectors are opposite to the directions ofthe radio jets, as shown for the source 1928 +
738 in the bottom-right panel of Fig. 4, the position di ff erences can be explainedby source structure e ff ects or core shift. However, it seems tobe di ffi cult to explain these position vectors along the jets, asshown for the source 1803 +
784 in the bottom-left panel, by thee ff ects of radio source structure and core shift. The recent studieshave demonstrated that the VLBI-to- Gaia position vectors favorthe directions both along and opposite to the jets (Kovalev et al.2017; Petrov et al. 2019), and more sources have these positionvectors along the jets than opposite to the jets. The presence ofparsec-scale optical jet structure in the directions of radio jets isproposed to explain the phenomenon in these studies.We compared the directions of the VLBI-to-
Gaia positionvectors and of the radio jets based on the MOJAVE data. The jetdirections were calculated as the median values of the jet posi- tion angles for the multiple jets of each individual source in theMOJAVE project. These jet position angles were robustly deter-mined from multiple-epoch measurements by MOJAVE (Listeret al. 2018). Figure 5 shows the 208 sources with the uncertain-ties of both the jet position angles and the VLBI-to-
Gaia posi-tion directions smaller than 30 degrees in gray dots and the 81sources with those uncertainties smaller than 12 degrees in reddots. About 88 percent of these 81 sources have the VLBI-to-
Gaia position vectors parallel to the jet directions within 25 de-grees and 96 percent within 45 degrees. It enhances the alreadyknown results from Kovalev et al. (2017) and Petrov et al. (2019)with stronger evidence. The majority of the sources have the di-rections of the position vectors along the jets and a significantfraction of sources have those vectors opposite to the jets, alsoconfirmed by this small sample of well-determined jet positionangles.However, we address several cases where the jet positionangles can be determined in the opposite direction. Figure 6shows the images of source 0743 − ff erentepochs. It has two compact components separated by ∼ ff set of 180 de-grees. The source has ρ = X ρ = ff erence between its Gaia and VLBIpositions can be explained by its radio source structure. As wecan see, in the right-hand plot, if we move the VLBI position tothe next component to the upper-left, then the
Gaia position fitsvery well the core.We further discuss two cases, sources 0923 +
392 and0429 + ff erences between VLBI and Gaia , which can be ex-plained by their radio structure. Their MOJAVE images areshown in Fig. 7 with their relative positions between VLBI and
Gaia illustrated. The figure demonstrates that the source posi-tions from geodetic VLBI are dominated by the positions of thepeak fluxes, whereas the optical positions from
Gaia are locatedclose to the cores. The separations between the cores and thejets for the CRF sources are typically at the mas level as demon-strated in Figs. 4 and 6 and up to tens of mas as shown in Fig.7. We should emphasize that for a significant number of sourcesthe VLBI position seems to be that of a jet component rather thanthe core. Without absolute position information in the MOJAVEimages, however, we have no knowledge of where the VLBI po-sition really is. Since the VLBI position to the core in the MO-JAVE images is so large if it locates at di ff erent jet componentsfor the cases like these two sources, phase referencing observa-tions can determine the positions of the jet components with suf-ficient accuracy, which will allow us to locate the VLBI positionwithin the image. This will eventually help to understand wherethe Gaia position locates. One also should notice from Fig. 7that since the cores of these two sources are not the brightestcomponents, without spectral index images it will be di ffi cult toidentify them from radio images, which can lead to a shift of 180degrees in determining jet position angles.To conclude, our study suggests that radio source structure isone of the major factors causing the position di ff erences and that Article number, page 10 of 13ing H. Xu et al.: Position di ff erences between Gaia and VLBI
Fig. 5.
Angles of the VLBI-to-
Gaia position vectors with respect to the jet directions as a function of the jet position angles based on theMOJAVE data. The error bars shown are the combined uncertainties from the formal errors of the two directions. There are 327 sources withrobust multi-epoch and multi-jet position angles, cross-matched from the 3142 sources. Out of them, 208 sources have both the uncertainties ofthe VLBI-to-
Gaia position directions and the median jet directions smaller than 30 degrees and are shown as gray dots. There are 81 sources withthose uncertainties smaller than 12 degrees, shown as red dots. For these 81 sources, the median ρ is 0.93 mas, and the X ρ values are larger than3.3. Among them, 54 sources have the directions of the position di ff erences along the jet directions within 25 degrees and their ρ are in the range0.2–28.0 mas; 17 sources have the directions of the position di ff erences opposite to the jet directions and their ρ are in the range 0.2–39.1 mas. Fig. 6.
MOJAVE images of source 0743 −
006 (CARMS = the optical jet structure tends to be also strong for the sourceswith extended structure at cm-wavelengths.
5. Conclusion
We made the conclusion based on the position di ff erences be-tween the Gaia
EDR3 and the ICRF3 as follows:1. The arc lengths ρ of the Gaia and VLBI position di ff erencesincrease with the CARMS values. Article number, page 11 of 13 & A proofs: manuscript no. gaia_crf_edr3_R1
Fig. 7.
Explanation of the large
Gaia -VLBI position di ff erences for two sources, 0923 +
392 with ρ = = +
415 with ρ = = +
392 isthe western, weak component, and the core of the source 0429 +
415 is the north-east component. Their
Gaia positions are located close to thecores, given that the VLBI positions are located at the peaks of flux. These two sources strongly demonstrate the e ff ects of source structure onthe position di ff erences between VLBI and Gaia — the source positions from geodetic VLBI are dominated by the positions of the peak fluxes,whereas the optical positions from
Gaia are located close to the cores.
2. The majority of the sources with statistically significant arclengths, X ρ >
4, are associated with the extended sources. Forinstance, the median CARMS of the 432 sources with X ρ > ρ > X ρ >
4, the majority, 70percent, have extended structure. The source 0429 +
415 hasbeen used as an example to demonstrate that based on theMOJVAE image shown in Fig. 7.4. Distinct relations between the optical magnitudes and theredshifts are found for the sources with and without statisti-cally significant position di ff erences. The sources with X ρ > ∼ ff er-ences if the source has extended structure.6. The Gaia and VLBI position di ff erences can be well ex-plained through the radio images for several sources as ex-amples. The vectors of the Gaia and VLBI position di ff er-ences are parallel to the radio-jet directions, which is con-firmed with stronger evidence. Acknowledgements.
We would like to thank the reviewer François Mignard forhis helpful comments. This research has made use of data from the MOJAVEdatabase that is maintained by the MOJAVE team (Lister et al. 2018). All compo-nents of the International VLBI Service for Geodesy and Astrometry are deeplyappreciated for providing the VLBI observations. This research was supportedby the Academy of Finland project No. 315721 and the National Natural Sci-ence Foundation of China No. 11973023. SL is supported by the DFG grant No.HE59372-2.
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