Orbital parallax of binary systems compared to GAIA DR3 and the parallax zero-point offset at bright magnitudes

(abridged)Multiple systems for which the astrometric and spectroscopic orbit are known offer the unique possibility of determining the distance to these systems directly without any assumptions. They are therefore ideal objects for a comparison of Gaia data release 3 (GDR3) parallax data, especially since GDR3 presents the results of the non-single star (NSS) analysis that potentially results in improved parallaxes. An sample of 192 orbital parallax determinations for 186 systems is compiled from the literature. The stars are also potentially in wide binary systems, and 37 candidates were found. Only for 21 objects does the NSS analysis provide information, including 8 from the astrometric binary pipeline, for which the parallaxes do improve significantly compared to those in the main catalogue. It appears that most of the objects in the sample are eliminated in the pre-filtering stage of the NSS analysis. The difference between the orbital parallax and the (best) \G\ parallax was finally obtained for 170 objects. When objects with large parallax errors or unrealistically large differences between the orbital and \G\ parallaxes are eliminated, and objects with a GOF<100 or<8 are selected, samples of 68 and 20 stars remain. Three recipes that calculate the PZPO are tested. After these corrections are applied the remaining parallax differences are formally consistent with zero within the error bar for all three recipes. The method of using orbital parallaxes is shown to work, but the full potential is not reached as an improved parallax from the NSS analysis is available for only few systems. In the final selection, the orbital parallax of 18 of 20 stars is known to better than 5%. In the full sample, 148 objects reach this precision and therefore the full potential of using orbital parallaxes may hopefully be reached with GDR4.


Introduction
With the 2nd data release (DR; Gaia Collaboration et al. 2018) of the Gaia mission (Gaia Collaboration et al. 2016) it was demonstrated that the mean parallax of quasi-stellar objects (QSOs) was not zero but slightly negative, −0.029 mas (Lindegren et al. 2018).This was confirmed in the third early DR (GEDR3; Gaia Collaboration et al. 2021), where the average and median parallax zero-point offset (PZPO) of QSOs are −21 and −17 µas, respectively (Lindegren et al. 2021;hereafter L21).L21 presented a python script to the community which returned the PZPO (without an error bar) as a function of input parameters, namely ecliptic latitude (β), G-band magnitude, the astrometric_params_solved parameter, and either the effective wavenumber of the source used in the astrometric solution (ν eff , nu_eff_used_in_astrometry for the five-parameter solution astrometric_params_solved = 31) or the astrometrically estimated pseudo colour of the source (pseudocolour) Send offprint requests to: Martin Groenewegen ⋆ Tables 1 and 2 are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.
for the six-parameter solution (astrometric_params_solved = 95).The module is defined in the range 6 < G < 21 mag, 1.72 > ν eff > 1.24 µm −1 , corresponding to about 0.15 < (G BP −G RP ) < 3.0 mag where G, G BP , and G RP are the magnitudes in the Gaia G-, Bp-, and Rp-band, respectively.The script is based on the parallax values of QSOs and wide-binary systems (WBS; see L21 for all details).
One disadvantage of using alternative classes of objects is the intrinsic assumption is that the distances to those objects are known exactly, and with an accuracy comparable to that of Gaia.However, with the exception of the QSOs that truly have a parallax of zero for all practical purpose, this is based on direct or indirect assumptions or model dependencies, like reddening, a (linear) period-luminosity relation, an absolute magnitude, a surface-brightness colour relation, etc.
Binaries where the astrometric and spectroscopic orbits are known offer the possibility to derive the orbital parallax free from any assumption, based on Kepler laws.The only limitation in the accuracy of the distance is ultimately in the quality of the data.GDR3 (Gaia Collaboration et al. 2022b) offers for the possibility to compare Gaia trigonometric parallaxes to orbital parallaxes.In previous releases the Gaia astrometric solution assumed single stars.Therefore the quality parameters of the astrometric solution, like the the goodness-of-fit (GOF, astrometric_gof_al) or the the renormalised unit weight error (RUWE) were (very) poor for (close) binary systems.In GDR3 non-single star (NSS) solutions are considered (Halbwachs et al. 2022;Gaia Collaboration et al. 2022a) which means that for a subset of stars (improved) parallaxes are determined that take into binary motion.Individual studies of astrometric and spectroscopic binary orbits also often derived the orbital parallax and compared it to Hipparcos or Gaia data but using mostly single objects (Gallenne et al. 2019 used four objects in the comparison of the orbital parallax and GDR2 data) no real systematic differences can be identified as the PZPO is a small quantity.
The paper is structured as follows.In Section 2 the sample of binaries is introduced and confronted with G(E)DR3 data.The impact and limitations of the DR3 NSS analysis is discussed in Sections 3.1 and 3.2, and the PZPO is is discussed in Section 3.3.A brief discussion and summary concludes the paper.

Sample
Recently Piccotti et al. (2020) compiled a list of 69 SB2s with both visual and spectroscopic orbits but the emphasis of that paper was on the masses and ages of the systems.The sample selection follows that of Piccotti et al. (2020) and is based on an extensive literature search using the Sixth Catalogue of Orbits of Visual Binary Stars (ORB6; Hartkopf et al. 2001, starting from orbits with grades 1, 2, and 3)1 and the 9th Catalogue of Spectroscopic Binary orbits (SB9; Pourbaix et al. 2004) 2 .This initial correlation pointed to other literature that was then searched.Articles by specific authors were searched through the ADS, and the ArXiv was monitored for relevant papers.The literature search ended May 10th, 2022.Table 1 lists the adopted orbital elements and velocity amplitudes with references for a total of 186 systems.For six binaries several components have been resolved (WDS 03272+0944, 06024+0939, 06290+2013, 17247+3802, 20396+0458, and 22388+4419) and therefore the table has 192 entries.This almost triples the number of systems compared to Piccotti et al. (2020).Although it was attempted to make this list complete this can not be guaranteed.Obviously, GDR3 provides orbital elements and velocity amplitudes for selected systems and the impact of this will be discussed in Sect.3.2.
The orbital parallax follows from the orbital elements as where a is the semi-major axis in mas, K 1 and K 2 the velocity semi-amplitudes of the two components (in km s −1 ), P the orbital period in years, e the eccentricity of the orbit, i the inclination in degrees, and π o the orbital parallax in mas.The error in π o is calculated through standard error propagation assuming all errors are independent.The orbital parallaxes and errors are listed in Table 2.
No selection on the accuracy of period, major axis, inclination or velocity amplitude is made to be included in the sample, even if a large error implies a large error on the orbital parallax, and likely a value not competitive in accuracy with the Gaia value.For objects where no error bars on the orbital elements was published an error of (1.3, 6, 40%) on period, (2.5, 15, 40%) on a, (5, 12, 27 • ) on i, and (0.01, 0.02, 0.10) on e was adopted for orbits of grade (1, 2, 3) in ORB6, respectively.If no error on the velocity amplitude was reported an error of 5% was adopted.These cases are marked in Table 1 and are good targets for further observations to improve on the orbital parameters.
As a first step, and as preparation for the DR3 release, the objects were identified in GEDR3 and also in the Hipparcos catalogue (the new reduction version by van Leeuwen 2008).The later proved essential in many cases as the epoch 1991.25 Hipparcos coordinates could be transformed to the epoch 2016.0 of GEDR3/GDR3 to properly identify the correct object that was achieved by comparing coordinates, parallax and magnitude.It was then trivial to identify the correct object with the release of DR3, and Table 2 gives for the 192 objects the orbital parallax (based on the data in Table1 and Eq.1), and some information from Hipparcos and DR3.The latter includes some parameters that were used in the pre-filtering stage of the astrometric binary pipeline (Halbwachs et al. 2022) and that will be discussed in detail later.
Inspecting Table 1 reveals that quite a number of objects (59, or about 30%) are not listed in GDR3, or are listed without parallax.The former are six of the very brightest objects (all have V < ∼ 2.7 mag).The others only have 2-parameter solutions (astrometric_params_solved = 3).However these objects may potentially be in WBS.To investigate this further the objects were correlated with the catalogue of El-Badry et al. (2021) (based on GEDR3 data).Fourteen matches were found, up to 57 ′′ distance.However, objects with no parallax or proper motion in GEDR3 are obviously missing from El-Badry et al. (2021).In a second step all sources within 1 ′ were retrieved from GDR3 and a potential list of WBS was compiled based on the orbital parallax and criteria on the parallax difference so that the search would retrieve all fourteen matches in El-Badry et al. (2021).In a final step the GDR3 and Hipparcos parallaxes and proper motions were inspected to make a final list of likely WBS, keeping only stars with Bp and Rp photometry.The information of these 37 sources is listed in Table A.1.Only thirty objects obey all pre-selection criteria.Additional objects are likely to have been removed in the processing and the post-processing steps (see details in Halbwachs et al. 2022).Similar criteria must have been adopted in the spectroscopic binary processing but the paper describing this has not been published at the time of submission.As the aim of the paper is to use improved parallaxes from the astrometric binary pipeline the details the processing in the spectroscopic binary pipeline are less relevant for this paper.

Comparing the orbital elements of the NSS processing with the literature
Comparing the the orbital elements in  1, as period, eccentricity or velocity amplitude do not match.In all cases except one the period in Table A.2 is the shorter one suggesting that the elements refer to some inner orbit of the multiple system.Consulting the ORB6 suggests that none of these orbits was known.
In the other cases the elements found by the NSS analysis are not more precise than known in the literature, which are therefore kept.The median and 1.4826 • median-absolute-deviation (MAD; equivalent to 1σ in a Gaussian distribution) are calculated of (x 1 − x 2 )/ σ 2 x 1 + σ 2 x 2 , where x represents period, eccentricity, or the velocity amplitudes, from the literature and the NSS analysis, and which are expected to be zero and unity, respectively.
Although the sample is small the errors on the parameters in the SB2 solution appear to be underestimated, as indeed suggested at the end of section 6 in Babusiaux et al. (2022).Taking the formal errors the width of the distribution is about 4-9σ.Scaling the errors with √ GOF this is reduced to 0.8-1.5σ.For the non-SB2 solutions the width is 0.6-1.6σsuggesting the error estimates are realistic.

PZPO
The main aim of the paper is to the investigate the PZPO. Figure 1 shows the difference of DR3 parallax minus orbital parallax for the objects plotted against G-mag.The range in ordinate is ±12 mas to display all points.The Gaia parallax is the improved parallax from the NSS analysis (with its associated GOF parameter) for the eight stars in Table A.2 and from Table 1 otherwise.Added are the 37 WBS from Table A.1 for a total of 170 determinations plotted.In cases that there are two Gaia parallaxes available (from the counterpart of the orbital parallax source and from a WBS) these are not averaged as the PZPO de-pends on magnitude and possibly colour.The figure shows that some of parallax errors (dominated by the error in orbital parallax) are large and do not constrain any difference with the DR3 parallaxes.Unless specified otherwise a standard selection (SS) is applied from now on choosing objects where: the error in the orbital parallax should be less than five times the error in the DR3 parallax, the error in the orbital and DR3 parallaxes should be less than 2 mas, the ratio of the absolute difference between orbital and DR3 parallax to the combined error bar should be less than five, and the GOF (either from the astrometric solution or the NSS solution) should be less than 100, reducing the number of points to 68.In Figure 1 these points are plotted in blue and Figure 2 shows a zoom with ordinates of ±0.15 mas using this selection.The data is also binned in G-mag, using five bins that start at G= 0, 5.0, 6.0, 7.4, and 9.2 mag (this last bin includes all fainter objects) where the weighted mean (and error on the mean) is calculated, and plotted at the mean G-mag of the objects in that bin.The first bin collects all of the brightest objects.Bright limits of 5.0 and 6.0 mag were used in G21, and L21/MA22, respectively while 7.4 and 9.2 mag are cardinal magnitudes used in MA22.The weighted average over all objects results in an offset of −41.7 ± 10.7 µas (see Model 1 in Table 3 that also includes the values per magnitude bin).
Several corrections to the G(E)DR3 parallaxes have been proposed and it is interesting to compare the corrected parallaxes to the orbital ones.To guide the eye the solid line in Figure 2 shows the magnitude correction from G21 with a constant spatial offset of −0.013 mas added (the average spatial correction of the sample following G21 at HEALPix level 0).It is recalled that the HEALPix formalism (Górski et al. 2005) is a convenient way to divide the sky in equal-area pixels.At HEALPix level 0 there are 12 pixels, and this increases by a factor of four for every next higher level.The HEALPix formalism is used by the Gaia team and is encoded in the source_id3 .
Below we apply different corrections on a star by star basis to the GDR3 parallaxes.If the corrections work the weighted mean difference should be consistent with zero.The first correction is the one by L21 (Model 2).The number of sources is reduced to 30 as the correction is only defined for stars fainter than magnitude 6.After correction the difference with the orbital parallax is −8.7 ± 11.5 µas.Model 3 shows the results for the MA22 correction, which is an extension of L21.The resulting difference is marginally closer to zero.One may note that the errors on the correction are identical to applying no correction.This is related to the fact that the L21 and MA22 provide the offset without error bar.Finally, the G21 formalism is tested.The correction depends on the chosen HEALPix level and consists of a spatial correction defined at G= 20 mag (depending on the source coordinates (that is, a certain pixel) at a chosen HEALPix level) and an additive magnitude correction.Following G21 only pixels with more than 40 QSOs have been considered.Models 4-8 show the results for HEALPix levels 0-4, respectively.As the G21 correction is defined for G > 5 there are more objects at the lower HEALPix levels, but then the number goes down with increasing HEALPix level as the number of objects in pixels with insufficient QSOs increases.HEALPix level 2 seems a good compromise between the sampling of the spatial correction and the loss of objects and the result is comparable to the ones for the other correction methods.If the G21 correction is limited to G > 6 mag as it is for the other two methods the resulting difference is very close to zero (Model 6a).The SS is useful to get insight into the overall behaviour of the PZPO but now a stricter final selection (FS) is introduced.As there are few objects fainter than 10th magnitude that nevertheless cover a wide range, and very few extremely bright objects the sample is restricted in magnitude and two bins from 4-6.162 and 6.162-10.591mag (two breakpoints in Eq. 7 in Groenewegen 2021) will be considered now.As there is an general offset between orbital and Gaia parallaxes the condition on the difference of the two is modified to |(π Gaia − (−0.04 mas)) − π orb |/σ c < 5.0, where σ c is the combined error of the Gaia and the orbital parallax (k • σ π Gaia ) 2 + σ 2 π orb .The factor k is the error inflation factor, which was set to unity in the SS.Several papers have found that the error bars in the astrometric solution are underestimated (Fabricius et al. 2021;Maíz Apellániz et al. 2021;Maíz Apellániz 2022).Here the formalism by El-Badry et al. ( 2021) is used.There correction was derived for G < 7 mag but it will be used for brighter magnitudes as well.The effect is in any case small, k = 1.10 − 1.15 for G < 10 mag.Finally a more stringent cut on the GOF is imposed.The distribution of the GOF of QSOs was discussed in Groenewegen (2021) and in that paper an interval from −4 to +5 was chosen for acceptable solutions.Here the upper limit is slightly relaxed to +8, resulting in a sample of 20 objects.This is consistent with the other often used selection criteria of RUWE < 1.4 that would result in 23 objects.
Model 9 in Table 3 gives the results and they are displayed in Figure 3.This figure also shows the offset as a function of Bp-Rp colour with the range in ordinate chosen to show all objects, indicating that there is no evident dependence on colour.Models 10-16 show the results after applying the various correction methods.The results are very similar as before.All three proposed models bring the differences closer to zero, and consistent with zero within the error bars.When restricted to G magnitudes fainter than 6, the G21 again provides the correction closest to zero for HEALPix level 2. Nevertheless most of the results suggest that the corrections may be slightly underestimated (that is, should be more negative), at least on average over the 6-10 magnitude range.

Conclusions
The results from the NSS analysis specific to GDR3 show the potential but also the limitations of the current release.Of the sample of 186 known binaries compiled from the literature to have an orbital parallax only 8 have an astrometric and 13 an Fig. 2. A zoom of Figure 1 with the SS applied as described in the text.There are points outside the plot range.The blue points are the weighted averages plotted at the mean magnitude of the objects in the magnitude bins.The solid lines represents the magnitude dependence of the PZPO from G21 with an average spatial correction of −0.013 mas added.spectroscopic orbit determined from the NSS analysis.Most objects are eliminated at the pre-filtering stage of the NSS analysis.The analysis of the parallax difference between Gaia and orbital parallax is therefor strongly influenced by the large GOF parameter in the main astrometric catalogue, limiting the number of useful objects.The PZPO corrections proposed by L21, G21, and M22 give similar residuals.After applying these corrections the remaining parallax differences are formally consistent with zero within the error bar for all three recipes.The current data and analysis does therefor not prefer a particular PZPO correction scheme over the other two.
Several improvements may be anticipated in the near future.The number of systems for which an orbital parallax will become available will likely grow, or orbital elements of existing systems will improve.For about one-third of the systems there exist separate astrometric and spectroscopic orbits that were used to obtain the orbital parallax.For consistency the existing data could be combined to obtain a single solution.
Potentially the situation may improve by DR4.For 18 out of 20 stars in the final selection sample the orbital parallax has an accuracy of better then 5%, and in six cases it is even better than the Gaia value.In the full sample 148 stars have an orbital parallax determination better than 5%.Having NSS solutions available for a eight times larger sample would lead to a significantly more precise determination of the PZPO at bright magnitudes.Whether this is realistic would depend on how much the criteria in the pre-filtering and post-processing stages stages could be relaxed.The impact of the former has been discussed, but some of the stars in the sample may also have been eliminated at the latter stage.Halbwachs et al. (2022) mentions three criteria that have been applied as well.The one on parallax accuracy4 would eliminate about 45% of the sample.The selection on eccentricity accuracy5 would eliminate 1% of the sample, and the one on the significance of the photocenter major-axis6 about 4%.Alternatively, with the planned release of the complete astrometric and spectroscopic time series data with DR4 the community could combine Gaia data with literature data in order to obtain the best-determined orbit.Notes.Results are given for the SS (Cols.1-4) and the FS (Cols.5-8), and give the model number, the weighted mean parallax difference with error, the average G-mag of the objects in the bin, and the the number of objects considered.Remarks and further details are given in Column 9.

Fig. 1 .
Fig. 1.The parallax difference in the sense DR3 minus orbital parallax for the entire sample.Note the range in ordinate of ±12 mas.The stars from the standard selection (SS) are plotted in blue (see text).

Fig. 3 .
Fig.3.Top panel: As Figure2for the final selection as described in the text.There are points outside the plot range.Middle panel: The offset plotted against Bp-Rp colour.The ordinate is chosen to be ± 1 mas to show all data points in the sample, while the bottom panel has the same range in ordinate as in the top panel.

Table 1 .
Sample of stars (selected entries)

Table 2 .
Halbwachs et al. (2022) entries) by the NSS pipeline(s).Although this is the reality of the current release it might be instructive to investigate how these binaries went "missing".Halbwachs et al. (2022)describes the astrometric binary star processing.Stars were selected to be brighter than G = 19 mag and have 12 or more visibility periods.The former criteria has no impact, and the latter removes 23 objects.There is a selection on keeping sources with RUWE < 1.4, removing 93 objects (of the 130 which have a RUWE listed), on ipd_frac_multi_peak ≤ 2, removing 64 objects, on ipd_gof_harmonic_amplitude < 0.1, removing 57 objects, and on the significance of the modified Bp-Rp excess, |C ⋆ |/σ C ⋆ < 1.645, removing 113 objects.

Table 3 .
Parallax differences for different assumptions.