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Astronomy & Astrophysics manuscript no. GAIA_ZPOﬀset_nCom June 16, 2021

©ESO 2021

arXiv:2106.08128v1 [astro-ph.GA] 15 Jun 2021

The parallax zero point offset from Gaia EDR3 data ⋆
M. A. T. Groenewegen
Koninklijke Sterrenwacht van België, Ringlaan 3, B–1180 Brussels, Belgium e-mail: martin.groenewegen@oma.be
received: ** 2021, accepted: * 2021
ABSTRACT
The second data release of Gaia revealed a parallax zero point oﬀset of −0.029 mas based on quasars. The value depended on the position on the sky, and also likely on magnitude and colour. The oﬀset and its dependence on other parameters inhibited an improvement in the local distance scale using e.g. the Cepheid and RR Lyrae period-luminosity relations. Analysis of the recent Gaia Early Data Release 3 (EDR3) reveals a mean parallax zero point oﬀset of −0.021 mas based on quasars. The Gaia team addresses the parallax zero point oﬀset in detail and proposes a recipe to correct for it, based on ecliptic latitude, G-band magnitude, and colour information. This paper is a completely independent investigation into this issue focussing on the spatial dependence of the correction based on quasars and the magnitude dependence based on wide binaries. The spatial and magnitude corrections are connected to each other in the overlap region between 17 < G < 19. The spatial correction is presented at several spatial resolutions based on the HEALPix formalism. The colour dependence of the parallax oﬀset is unclear and in any case secondary to the spatial and magnitude dependence. The spatial and magnitude corrections are applied to two samples of brighter sources, namely a sample of ∼100 stars with independent trigonometric parallax measurements from HST data, and a sample of 75 classical cepheids using photometric parallaxes. The mean oﬀset between the observed GEDR3 parallax and the independent trigonometric parallax (excluding outliers) is about −39 µas, and after applying the correction it is consistent with being zero. For the classical cepheid sample it is suggested that the photometric parallaxes may be underestimated by about 5%.
Key words. Stars: distances - parallaxes

1. Introduction
Data from the Gaia mission (Gaia Collaboration et al. 2016) has impacted most areas in astronomy. One the ﬁelds were the Gaia 2nd data release (GDR2, Gaia Collaboration et al. 2018) was eagerly awaited was in reliably establishing the local distance scale through calibration of the period-luminosity (PL) relation of classical cepheids (CCs) and RR Lyrae (RRL) variables.
Riess et al. (2018b) analysed a sample of 50 CCs. They derived a parallax zero point oﬀset (hereafter PZPO) of −0.046 ± 0.013 mas, compared to the −0.029 mas derived for quasars by Lindegren et al. (2018) and concluded that the need to independently determine the PZPO largely countered the higher accuracy of the parallaxes in determining an improved zero point of the PL-relation. Independently, Groenewegen (2018) (hereafter G18) derived a PZPO of −0.049 ± 0.018 mas based on a comparison of nine CCs with the best non-Gaia parallaxes (mostly from HST data). Ripepi et al. (2019) re-classiﬁed all 2116 stars reported by Clementini et al. (2019) to be Cepheids in the Milky Way (MW). Period-Wesenheit relations in the Gaia bands were presented. Assuming a canonical distance modulus to the LMC of 18.50, a Gaia PZPO of ∼ −0.07 to −0.1 mas was found. PZPOs based on GDR2 data were also reported for RRL stars (∼ −0.056 mas, Muraveva et al. 2018; −0.042 ±0.013 mas, Layden et al. 2019), and many other classes of objects (Stassun & Torres 2018, Graczyk et al. 2019, Xu et al. 2019, Schönrich et al. 2019). These values were mostly all-sky
Send oﬀprint requests to: Martin Groenewegen ⋆ Table 1 is 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/.

averages, but when suﬃcient data was available it was clear that the PZPO depended on position in the sky, magnitude, and colour (Zinn et al. 2019; Khan et al. 2019; Leung & Bovy 2019; Chan & Bovy 2020).
The Gaia Early Data Release 3 (GEDR3) presents the most recent information on parallax, proper motions, position and colour information for about 1.8 billion objects (Gaia Collaboration et al. 2021a). Lindegren et al. (2021b) presents the general properties of the astrometric solution, while Lindegren et al. (2021a) (hereafter L20) specifically addresses the PZPO and make a python script available to the community in order to calculate the PZPO. This module gives PZPO (without an error bar) as a function of input parameters ecliptic latitude (β), Gband magnitude, the astrometric_params_solved parameter (Lindegren et al. 2021b), and either the eﬀective wavenumber of the source used in the astrometric solution (νeﬀ, nu_eff_used_in_astrometry for the 5-parameter solution astrometric_params_solved = 31) or the astrometrically estimated pseudo colour of the source (pseudocolour) for the 6parameter solution (astrometric_params_solved= 95). The module is deﬁned in the range 6 < G < 21 mag, 1.72 > νeﬀ > 1.24 µm−1, corresponding to about 0.15 < (GBP−GRP) < 3.0 mag where G, GBP, and GRP are the magnitudes in the Gaia G-, Bp-, and Rp-band, respectively.
Several papers have already applied the L20 correction to the raw GEDR3 parallaxes. Riess et al. (2021) applied it to a sample of 75 CCs in the Milky Way (6.1 ∼< G ∼< 11.2 mag, 0.9 ∼< (GBP − GRP) ∼< 2.5 mag). For this sample, there is a strong dependence of the correction on β (ranging from −4 to −38 µas) with only a small dependence (of order 1.8 µas) on magnitude and colour,

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with a median correction of −24 µas. Allowing for a remaining PZPO after application of the L20 correction and ﬁtting the data to the independently calibrated PL relation of CCs in the Large Magellanic Cloud (LMC) reveals an oﬀset of 14 ± 6 µas, in the sense that the L20 corrections are too much negative, that is the values are over corrected.
A similar conclusion is reached by Zinn (2021) who analysed a sample of 2000 ﬁrst-ascent red-giant branch stars with asteroseismic parallaxes in the Kepler ﬁeld (9.0 ∼< G ∼< 13.0 mag, 1.0 ∼< (GBP − GRP) ∼< 2.3 mag) and concludes that the L20 corrections are too much negative by 15 ± 3 µas for G ∼<10.8 mag.
Bhardwaj et al. (2021) apply a theoretical period-luminositymetallicity relation in the K−band to a sample of about 350 Milky Way RR Lyrae stars (8.9 ∼< G ∼< 17.8 mag, 0.4 ∼< (GBP − GRP) ∼< 1.3 mag) to ﬁnd a ZP of (−7 ± 3) µas when compared to the raw GEDR3 parallaxes. The mode of the L20 correction for this sample is −32 ± 4 µas so, again, the L20 formula over corrects the parallaxes, in this case by 25 ± 5 µas.
Stassun & Torres (2021) continue their previous analysis using eclipsing binaries as reference objects. Stassun & Torres (2018) found an oﬀset of (−82 ± 33) µas based on 89 stars from GDR2 while, while their latest analysis using GEDR3 indicates an oﬀset of (−37 ± 20) µas based on 76 objects (5 ∼< G ∼<12 mag, −0.1 ∼< (GBP − GRP) ∼< 2.2 mag). After applying the L20 correction the PZPO becomes (−15 ± 18) µas, indicating no over- or under-correction.
Huang et al. (2021) use a sample of over 69 000 primary red clump (PRC; 9.5 ∼< G ∼< 15 mag, 1.32 ∼< νeﬀ ∼< 1.5, or about 1.0 ∼< (GBP −GRP) ∼< 2.2 mag) stars based on LAMOST data from Huang et al. (2020). The distances come from Schönrich et al. (2019) that are based on a Bayesian analysis of DR2 data and include an PZPO of −0.054 mas. The reference distance is compared to the raw GEDR3 parallax, and the GEDR3 parallax after applying the L20 correction. The diﬀerence (Gaia - PRC) is −26 in the former and +3.7 µas in the latter case (no errors are reported). They also show the trends of the parallax diﬀerence against G, νeﬀ, and ecliptic latitude. They show that applying the L20 correction removes some of these trends (in particular against G magnitude), but not all, and show that there is a trend with ecliptic longitude, especially for ecliptic latitudes <30◦.
The aim of the present paper is to have an independent (and alternative) investigation into the PZPO, and in particular into the spatial dependence. This will be achieved by using a large sample of reliable QSOs (selected diﬀerently from the sample used in various GEDR3 papers). In addition physical binaries will be considered to derive the dependence of the PZPO on G magnitude. At the bright end the PZPO derived in the present paper will be applied to a sample of stars that has not been systematically considered in previous works, namely stars that have an independent parallax measurement from the Hubble Space Telescope (HST). In addition, the PZPO will be applied to the sample of CCs by Riess et al. (2021), and results will be compared to using the correction in L20.
The paper is structured as follows. In Section 2 the main methodology is introduced. In Section 3 the sample of stars is described. Section 4 presents the results of the calculations and the derivation of the PZPO, while Sect. 5 applies the PZPO to the QSO sample itself and the two samples of stars. A brief discussion and summary concludes the paper.

2. Methodology The parallax zero point oﬀset is deﬁned through

πt = πo − ZP

(1)

where πt is the true parallax and πo the observed parallax (as listed in the GEDR3 catalogue). The PZPO is parameterised in the present paper as a sum of linear functions that are assumed to hold over a range in magnitudes:

ZP = C0(α, δ) + C1 (G − Gref) + C2 (G − Gref)2

+ C3 ((GBP − GRP) − BRref)

(2)

= πo − πt

where Gref= 20.0 mag and BRref= 0.8 mag are reference values. They are chosen to represent the typical colours of QSOs.
In the present analysis also binaries are considered. In that case the true parallaxes for the two components are essentially the same and as they are essentially at the same position on the sky the C0 term may be assumed to cancel out. The diﬀerence of Eq. 2 for the two components (labelled as subscripts as primary, p, and secondary, s) becomes:

∆ZP = C1 (Gp − Gs) +

C2 ((Gp − Gref)2 − (Gs − Gref)2) +

C3 ((GBP − GRP)p − (GBP − GRP)s)) + ǫ

(3)

= (πo)p − (πo)s

where ǫ is a term that can be thought oﬀ as the spatial correlation on the extent of the binary separation.
The diﬀerence between the formalism outlined in Appendix A in L20 and that is used here is two-fold. The main diﬀerence is that the spatial dependence is made explicit here rather than using a second-order polynomial in sin β. The other diﬀerence is that the spatial, magnitude and colour dependence are assumed to be separable while the L20 correction allows for cross-terms.
The term C0 is allowed to vary over the sky and the HEALPix formalism (Górski et al. 2005, the NESTED variant) is used to transform (α, δ) to a sky pixel. This is done using a implementation in python, HEALPy (Zonca et al. 2019). The number of sky pixels depends on the chosen resolution; resolution levels 0, 1, 2, 3, and 4 are considered in the present paper which correspond to 12, 48, 192, 768 and 3072 pixels, respectively. The highest resolution corresponds to a mean spacing of 3.7 degrees between sky pixels.
The ﬁtting of Eqs. 2 or 3 to the data is done with the singular value decomposition algorithm (routine svdﬁt) as implemented in Fortran77 in Press et al. (1992). This algorithm minimises the χ2 taking into account the errors in the ordinate (the parallax (difference)) and gives the best-ﬁt parameter values with error bars. In order to also consider the errors in magnitudes and colours Monte Carlo simulations are performed where new datasets are generated taking into account Gaussian errors in the parallaxes, magnitudes, and colours. The parameter values quoted below (in Table 2) are the median values for the parameters among these simulations with as error the dispersion among the parameter values, calculated as 1.4826 times the median-absolute-deviation (MAD), equivalent to 1σ in a Gaussian distribution.

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3. The sample

In order to apply Eq. 2 or 3 and determine ZP as a function of sky position, magnitude and colour a large sample of sources with known true distances is required. In this paper QSOs, physical binaries, and stars with an independent trigonometric parallax determination will be considered.
For the samples discussed below the following parameters were retrieved from the GEDR3 main catalogue1: parallax and parallax error (parallax, parallax_error), proper motion in Ra and Dec with errors (pmra, pmra_error, pmdec, pmdec_error), the source identiﬁer (source_id)2, which parameters have been solved for (astrometric_params_solved; ﬁve- and six-parameter solutions are relevant for the present paper, see Lindegren et al. 2021b), the renormalised unit weight error (RUWE), the goodness-of-ﬁt (GOF, astrometric_gof_al), the effective wavenumber of the source used in the astrometric solution (νeﬀ, nu_eff_used_in_astrometry), the astrometrically estimated pseudocolour of the source (pseudocolour), the G, GBP, and GRP magnitudes with errors (phot_g_mean_mag, phot_g_mean_mag_error, phot_bp_mean_mag, phot_bp_mean_mag_error, phot_rp_mean_mag, phot_rp_mean_mag_error).
3.1. Quasars
Several of the GEDR3-team papers use a QSO sample (Gaia Collaboration et al. 2021b; Lindegren et al. 2021b,a) but the paper describing the selection of this sample has not been published at the time of submission. What is available is the list of 1.61 million source_ids which contains a reference to a catalogue but without any quality ﬂag. For this reason, and because this project started before the release of GEDR3, a diﬀerent QSO sample was created.
The Million Quasars (Milliquas) catalog (version 7.0b, Flesch 2019) is used which contains of order 830 000 type-I QSOs and AGN plus about 500 000 quasar candidates. From the full catalogue the 1.37 million objects with a conﬁrmed redshift > 0.1 or a probability of being a quasar of >98% are selected. The cross-match facility (xMatch) at the Centre des Données (CDS) in Strasbourg was used to match this list with GEDR3 using a search radius of 0.15′′, and this returned 998 220 matches, of which 855 518 QSOs have a parallax, G, GBP, and GRP magnitude available. The true parallax for these sources is assumed to be zero.
First the general properties of the QSO sample are discussed in particular the distribution of the GOF and the RUWE. It is recalled that the GOF parameter should follow a Gaussian distribution with zero mean and unit dispersion (Wilson & Hilferty 1931). In GDR2 this was not the case due to the "degree-offreedom" bug (see Appendix A in Lindegren et al. 2018 and the discussion in Groenewegen 2018). The RUWE was introduced after GDR2 (Lindegren 2018) as an other quality indicator of the astrometric solution. It compared the unit-weight-error (UWE, the square-root of the reduced χ2) to that of a sample of unproblematic stars as a function of magnitude and colour. Although the χ2 values were actually numerically incorrect the ratio of the UWEs was probably deemed representative of the relative quality of the astrometric solution. In Gaia EDR3, the "degree-
1 The data is downloaded from the copy available via ViZier at the CDS. 2 The source_id can also be used to determine the pixel in the HEALPix scheme, as source_id/(235 · 4(12−level)).

Fig. 1. Distribution of the GOF with a Gaussian ﬁt. Top panel, the about 841 000 QSO with parallaxes and PMs consistent with zero. A significant tail toward large GOF is visible. Lower panel, the ﬁt with GOF restricted to < +2.0.
of-freedom" bug has been corrected and the GOF is the main parameter to describe the quality of the astrometric solution.
Although the QSO sample was selected to be pure it still contains non-QSOs. As QSOs should have zero parallax and proper motion (within the error bars) the following conditions are applied:
| (π + 0.0202) | /σπ < 5 (PMRA/σPMRA)2 + (PMDE/σPMDE)2 < 25
where −0.0202 mas is the average oﬀset of QSOs (see below). Similar cuts were applied to the QSO sample in Lindegren et al. (2021a), but no selection on astrometric_params_solved and νeﬀ is applied. The 5σ limit on the parallax and proper motions (PMs) in Ra and Dec corresponds to the expected level of outliers following Chauvenet’s criterion with ∼ 800 000 objects.
The upper panel in Figure 1 shows the distribution of the GOF for the 841 000 QSOs that remain after applying the selection on parallax and PM, together with a Gaussian ﬁt. The mean and dispersion are 0.483 and 1.187 with negligible formal errors. The mean is slightly larger than expected but the tail to negative GOF (the extreme value is −5.4) is not inconsistent with the distribution. On the other hand the tail to larger GOF (the extreme value is +200) is obvious. To obtain a better estimate of the mean and width of the distribution undisturbed by outliers, the lower panel shows the distribution when the GOF and the Gaussian ﬁt to the distribution is restricted to < +2.0. The mean is 0.388 with σ = 1.109. Based on this analysis a condition −4 < GOF < 5 is imposed on all selections described in this paper, allowing for a small excess of sources towards larger GOF. Figure 2 shows the distribution in G, (GBP − GRP) colour, and RUWE after imposing these conditions, as well as RUWE <1.4 to eliminate a few extreme outliers in that parameter. The distribution is RUWE has a peak slightly above one, consistent with the fact that the distribution in GOF peaks slightly above zero. It indicates that the error bars in the parallax are likely underestimated by a few percent (at least in this range of magnitude and colour), consistent with the ﬁndings in Fabricius et al. (2021) and El-Badry et al. (2021). A ﬁnal sample of 824 819 QSOs is retained. The median parallax in that sample is −0.0202 mas with a dispersion (calculated as 1.4826 times the MAD) of 0.393 mas.

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ology to use the WFC3 to obtain parallaxes was described in Riess et al. (2014) and Casertano et al. (2016) and has been used to obtain distances to eight CCs (Riess et al. 2014, 2018a). Although parallaxes from Hipparcos are in general no longer competitive when compared to Gaia, Table 1 includes one exception, Polaris A, the most nearby CC, that had no parallax listed in GDR1 and GDR2.
The independent trigonometric parallax data and basic GEDR3 data on the 111 objects are listed in Table 1. Only one object is not listed in GEDR3 (Polaris A), and 8 objects have no parallax listed (all these have exceptionally large GOF values). Applying the selection on GOF and RUWE further eliminates 37 objects for a useful sample of 57 objects.
4. Results
In this section the results are being discussed related to the derivation of the PZPO correction.

4.1. QSOs

Fig. 2. Distribution of G, (GBP − GRP) colour and RUWE for the QSO sample after applying selections on parallax, proper motion, GOF, and RUWE.
3.2. Wide binaries
In this paper the catalogue of wide binaries (WBs) of El-Badry et al. (2021) is used which is based on GEDR3 data. In its raw form it contains 1.8 million candidate physical binaries. For wide and close-by binaries the hypothesis that both components are essentially at the same physical distance may no longer be correct. The procedure outlined in Sect. 5 of that paper is used to eliminate objects where the true parallax diﬀerence between the two stars (Eq. 13 of that paper) is estimated to contribute more than 5% to the error in the observed parallax diﬀerences (eliminating about 14 000 of the 1.8 million pairs). The sample is also restricted to the subset of about 784 000 pairs with a <1% probability of chance alignment, which is stricter than the high-conﬁdence sample (chance alignment <10%) of 1.2 million objects considered in El-Badry et al. (2021).
Fabricius et al. (2021) and Lindegren et al. (2021a) also consider samples of binaries to validate the GEDR3 results and analyse the PZPO, respectively. Both samples are directly constructed from GEDR3 data (but they are not identical) and careful selection is needed to reach a pure sample.
3.3. Sources with trigonometric parallax determinations
The Fine Guidance Sensor (FGS) and the Wide Field Camera 3 (WFC3) on board the Hubble Space Telescope (HST) have been used to determine parallaxes and proper motions. The review of Benedict et al. (2017) describes the methodology and provides a list of 105 targets for which the parallax has been determined by using the FGS. Some close binaries are included in the list with both components. One component was removed from the list as the binary can not be resolved by Gaia. Some recent works by van Belle et al. (2020) and Bond et al. (2018) (on Polaris B) are added to give a list of 102 targets. The method-

Figure 3 gives another representation of the QSO sample, similar to Fig. 5 in L20. It shows a binned version of the PZPO (weighted mean and error) as a function of G magnitude, (GBP − GRP) colour (L20 shows it as a function of νeﬀ), and β. The lines in the top and middle panels are not ﬁts to the data but represent the ﬁnally applied corrections based on a full analysis (see later in this section, Eqs. 4, 5). In the middle panel, the line does not seem to ﬁt the points very well. This is related to the fact that in the QSO sample the brighter QSOs are bluer than fainter ones. In addition, the error bar on this slope is quite large.
Although the QSO sample is diﬀerent from that used in GEDR3-team papers the behaviour is very similar to that shown in L20, as expected. There is a quite noticeable correlation with G for G ∼>17 mag, a small (if any) correlation with colour (or νeﬀ), and a relatively modest correlation with β. In particular the latter correlation is interesting. An identical binning is used as in L20 (40 bins) and the distribution of the black open squares is quasi identical to that shown in L20. The blue points show another representation, and the main reason why a diﬀerent spatial dependence of the PZPO is proposed here. The binning is now done based on HEALPix level 1, which has 48 pixels, very similar to the binning in L20. The point and the vertical error bar have the same meaning, while the horizontal line represents the range in β for that HEALPix pixel. One can clearly observe a signiﬁcantly larger spread even for pixels with very similar ecliptic latitudes. This indicates that the PZPO is a more complicated function than of β alone. The result is qualitatively similar to Huang et al. (2021) who demonstrate that there is a trend of the PZPO with ecliptic longitude, especially for β <30◦.
To investigate this further Eq. 2 is ﬁt to this sample, solving for C0 (only a spatial component), C0+C1+C2, and all parameters. This is done for several HEALPix levels, and the results are summarised in Table 2. As the ﬁtting routine is based on minimising the χ2 one expects the value for C0 to be equal to the weighted mean of the parallaxes of all QSOs in that pixel when only the spatial component is solved for. As a sanity check to the implementation of the numerical code it was veriﬁed that this is indeed the case.
As the distribution of known quasars is not uniform over the sky (typically underrepresented in the Galactic plane) the number of objects per sky pixel varies strongly. At HEALPix level 3 there are 10 pixels with no QSOs, and 63 with 40 or less objects. On the other hand 50% of pixels have 810 objects or more,

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with a maximum of 3762. Inspecting the error in the parallax oﬀset and the signal-to-noise suggests that forty objects or more are required for the results to be robust. The median oﬀset over these 706 pixels is −21.0 µas with a dispersion over the pixels (calculated via the MAD) of 12.4 µas when ﬁtting only the spatial component. Averaging only over the pixels with 100 objects or more changes the parameter and the error by ∼ 0.3 µas.
The ﬁrst entries in Table 2 (models 1-9) include all QSOs but based on the trends seen in Fig. 3 models restricted in G magnitude are more realistic, and several ranges have been explored. Models 10-28 show the main the results, and based on these the following linear correction of the parallax is proposed (in µas/mag) in the range G >17 mag (∆π to be subtracted from the catalogued GEDR3 parallax):

 ∆π = 

+6.0 (G − 19.9) +0.0

17.0 ≤ G < 19.9 19.9 ≤ G < 20.0

(4)

−16.0 (G−Gref) 20.0 ≤ G < 22.5

The presence of a colour dependence is less clear. As can be seen from the results in Table 2 the term is not very signiﬁcant (at the 2σ level at best). Nevertheless, the colour correction (in µas/mag) that will be tested is:

∆π = −3.5 ((GBP − GRP) − BRref)

(5)

Figure 3 shows no real trend with magnitude for brighter magnitudes. Fitting a constant at HEALPix level 0, as there are only ∼3300 QSOs brighter than 17 mag, gives a value of about −31 µas (model 29).
Models 30-34 and 35-39 give the results when the parallaxes are corrected according to Eq. 4, respectively, Eq. 4 and 5. The results are listed for several HEALPix levels. The average spatial correction of the PZPO (at G = 20) is essentially independent of the chosen HEALPix level and suggests systematic errors of ∼< 0.5 µas. Tests using a slope of −6.1, respectively, 17.0, at the bright and faint magnitude end, and shifting the nodes by 0.1 mag indicate global diﬀerences of ∼< 0.2 µas and changes in the spatial PZPO in individual pixels of ∼< 0.2σ. Adding the colour correction increases the dispersion over the pixels, suggesting again that the colour term is not a signiﬁcant factor.
The detailed results of models 30-39 are available through the CDS, and an example of the content is given in Tab. 3. These ﬁles list the PZPO and error for each individual HEALPix pixel for levels 0, 1, 2, 3, and 4, and the number of QSOs in each pixel.
Some properties are summarised in Tab. 4. It lists the HEALPix level, the number of pixels, the number of pixels with only 0 or 1 object, and in column 4 the range in the PZPO errors for the pixels with 40 or more objects, which typically increases with increasing spatial resolution.
Figure 4 illustrates how this spatial and magnitude correction works for the QSO sample with G >17 mag. The black open circles give the observed parallaxes of the 821 000 QSOs averaged and binned over sin β. The blue open circles give the L20 correction for the individual QSOs again averaged and binned over sin β, while the black ﬁlled circles give the corrected parallaxes according to Eq. 4 and the red ﬁlled circles the spatial correction at HEALPix level 2. Weighted averages are used except for the blue circles of the L20 correction. The unweighted mean is shown as the L20 correction carries no error. The shape of the L20 correction is due to the fact that L20 uses a second order polynominal in sin β (Eq. A4 in L20 and the discussion in their Sect. 4.1). The spatial correction at HEALPix level 2 gives a good description of the parallaxes corrected according to Eq. 4

Fig. 3. PZPO for the QSO sample as a function of G, (GBP −GRP) colour and ecliptic latitude (open circles). Only bins with ≥5 objects are plotted. Bins with 100 objects or less are plotted in blue, with 30 objects or less in red. The lines in the top panel and the lines in the middle panel are not a ﬁt to the data, but are based on Eq. 4. The blue points in the lower panel indicate the PZPO for the 48 HEALPix level 1 pixels. The horizontal bar gives the range in sin β for each HEALPix pixel.
even though the ﬁtting was done according to the HEALPix level and not speciﬁcally to ecliptic latitude. Section 5.1 discusses the results when the correction in L20 and the current one are applied to the QSO sample.
4.2. Wide binaries
The top panel in Fig. 5 shows a binned version of the parallax diﬀerence between the primary and secondary component as a function of primary G in the top panel. A similar diagrams was shown in Fabricius et al. (2021) (their Fig. 22) What is striking is the sharp decrease of the parallax diﬀerence for faint magnitudes. This is due to selection eﬀects in the El-Badry et al. (2021) sample. Readily visible in the bottom panel are the conditions πp > 1 and πs > 1 mas that were imposed (but there are others on the (relative) parallax accuracy, see their section 2),

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Table 2. Result of the ﬁtting to the QSO sample.

Model HEALPix level

1

2

2

2

3

2

4

2

C0 (µas)
−20.6 ± 9.0 −14.5 ± 9.2 −13.8 ± 8.9 −14.1 ± 9.0

N

C1

(µas/mag)

all G magnitudes

190/192

-

190/192 4.00 ± 0.25

190/192 4.91 ± 0.63

190/192 5.16 ± 0.60

C2 (µas/mag2)

C3 (µas/mag)

0.24 ± 0.13 0.29 ± 0.14

−3.08 ± 1.16

5

3

−21.0 ± 12.4 706/768

-

-

-

6

3

−14.9 ± 12.2 706/768 3.97 ± 0.23

-

-

7

3

−14.2 ± 12.7 706/768 5.07 ± 0.60 0.29 ± 0.13

-

8

3

−14.9 ± 12.5 706/768 5.18 ± 0.65 0.28 ± 0.13 −3.61 ± 1.23

9

4

−15.1 ± 19.0 2600/3072 4.85 ± 0.45 0.22 ± 0.08 −3.96 ± 1.67

17 < G < 20

10

1

−12.1 ± 7.4

48/48

5.97 ± 0.41

-

-

11

2

−20.7 ± 9.0 190/192

-

-

-

12

2

−12.4 ± 9.1 190/192 6.03 ± 0.32

-

-

13

2

−12.5 ± 9.2 190/192 6.11 ± 0.44

-

−3.55 ± 1.49

14

2

−13.4 ± 9.2 190/192 4.13 ± 1.51 −0.71 ± 0.52

-

15

2

−13.2 ± 9.3 190/192 4.54 ± 1.71 −0.53 ± 0.54 −3.17 ± 1.35

16

3

−12.5 ± 12.6 698/768 6.05 ± 0.45

-

-

17

3

−12.9 ± 12.5 698/768 4.70 ± 1.32 −0.53 ± 0.50 −4.06 ± 1.55

19.875 < G < 20.125

18

0

−13.4 ± 6.5

12/12

-

-

-

19

1

−14.3 ± 11.3 47/48

-

-

-

20

2

−12.8 ± 22.0 180/192

-

-

-

21

3

−13.6 ± 44.2 595/768

-

-

-

22

4

−13.5 ± 62.8 921/3072

-

-

-

20.125 < G < 22.5

23

1

−22.5 ± 23.8 47/48

-

-

-

24

1

−16.8 ± 22.1 47/48 −16.35 ± 7.90

-

-

25

2

−13.86 ± 33.3 182/192 −15.64 ± 9.59

-

-

26

2

−13.39 ± 33.1 182/192 −16.30 ± 7.30

-

−3.71 ± 4.17

27

2

−13.99 ± 34.2 182/192 −15.36 ± 30.71 −1.96 ± 33.7 −3.21 ± 5.51

28

3

−15.4 ± 51.9 640/768 −15.43 ± 7.66

-

-

0 < G < 17

29

0

−30.6 ± 5.5

12/12

-

-

-

17 < G < 22.5, magnitude corrected according to Eq. 4

30

0

−11.8 ± 3.0

12/12

-

-

-

31

1

−13.1 ± 7.0

48/48

-

-

-

32

2

−12.8 ± 8.9 190/192

-

-

-

33

3

−12.4 ± 12.3 705/768

-

-

-

34

4

−13.1 ± 20.1 2599/3072

-

-

-

17 < G < 22.5, G and (GBP − GRP) corrected according to Eqs. 4-5

35

0

−12.4 ± 3.4

12/12

-

-

-

36

1

−13.4 ± 7.3

48/48

-

-

-

37

2

−13.0 ± 9.1 190/192

-

-

-

38

3

−13.1 ± 12.2 705/768

-

-

-

39

4

−13.5 ± 19.7 2599/3072

-

-

-

Notes. The result of the ﬁtting Eq. 3 to the QSO sample. The value for C0 is the median and dispersion over all HEALPix pixels with 40 objects per pixel or more based on Monte Carlo simulations. The number of these pixels is listed in column 4.

Article number, page 6 of 18

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data

Table 3. Example of PZPO and error over the HEALPix pixels.

HEALPix pixel value error Number

0 -6.55 9.00

1 -20.79 10.05

2 -18.46 6.86

3 -16.25 12.93

4 -28.44 15.57

5 -8.94 16.20

6 -21.67 12.68

7 -22.48 13.13

8 -27.29 8.09

9 -34.50 14.58

10 -40.48 9.50

...

... ...

760 -8.36 9.88

761 3.38 8.43

762 -8.08 12.00

763 0.46 10.69

764 5.99 11.31

765 -9.45 8.54

766 11.18 9.30

767 -8.84 8.87

1308 488 838 635 592 453 647 515 956 656 2146 ... 550 562 404 542 486 776 591 775

Fig. 4. PZPO for the QSO sample with G >17 mag as a function of sin β. Black open circles represent the observed data (the weighted mean) to be compared to the blue open circles that represents the L20 correction (the unweighted mean as the L20 correction carries no error). The black ﬁlled circles represent the corrected parallax data (according to Eq. 4) to be compared to the red ﬁlled circles that represent the spatial correction at G = 20 mag at HEALPix level 2 (both are weighted means, but the error in the red ﬁlled circles is too small to be visible).

Notes. Example of PZPO and error over the HEALPix pixels for level 3. The ﬁles (corresponding to models 30-39 from Table 2) are available through the CDS. The ﬁle contains the HEALPix pixel number, the PZPO with error (in µas), and column 4 is the number of objects in that pixel.

Table 4. Properties of the solutions.

HEALPix level 0 1 2 3 4

pixels
12 48 192 768 3072

number of ill deﬁned pixels
0 0 0 12 157

range in error (µas) 0.63 - 1.74 1.16 - 10.8 1.99 - 41 3.23 - 82 1.93 - 72

Notes. Column 1: HEALPix level, Column 2: total number of pixels on the Sky, Column 3: number of pixels with 0 or 1 object, Column 4: range of the error in the PZPO for the pixels with 40 or more objects,

and that for a given πp there are many more objects with πs < πp than the inverse.
Restricting the magnitudes to less than 19 mag seems to largely remove this asymmetric behaviour (top panel Fig. 6) and also removes the unexpected tendency of the parallax diﬀerence as a function of magnitude (bottom panel).
It is now possible to iteratively study the PZPO based on WBs as a function of G magnitude. The ﬁrst step is to correct the parallaxes according to Eq. 4. As the G magnitude of the binary sample is limited to G = 19 mag this implies correcting the parallaxes of all objects with G >17 by +6.0 µas/mag. One can then plot the parallax diﬀerence against magnitude, only considering secondaries fainter than 17 mag. The top panel of Fig. 7 shows the result. The PZPO is essentially independent of G at the faint end, the weighted mean of the 11 bins fainter than 17.1 mag is 1.6±0.6 µas. In the range between ∼13.3 and ∼17 mag the PZPO can be well approximated by a linear behaviour as indicated by the black line. In a second step this oﬀset can be applied as well in this magnitude range, and the PZPO can be determined us-

Fig. 5. Top panel. Parallax diﬀerence between primary and secondary component in wide binaries, as a function of primary G magnitude. Bins with more than 1000 objects are plotted in blue. Bottom panel. Parallax of the secondary binary component plotted against that of the primary. Objects with primary G magnitude larger than 5, 15, 17, 19, and 20 mag are plotted as black, red, green, dark blue, and light blue dots, respectively. Plotted are about 590 000 binaries where both components pass the criteria on GOF and RUWE.

Article number, page 7 of 18

A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom
is the inverse. In addition, the dependence on β increases with brighter magnitudes.

∆π

=

−++410.67..280 +−++4115.2772...6323

(G − (G − (G −
(G − (G − (G− (G−

19.900) 13.265) 12.755)
11.735) 10.545)
6.295) 5.275)

+0.000 − 26.372
−7.823 −8.064 − 29.531 − 10.366 − 16.579 − 67.589

16.450 ≤ G < 19.900 13.218 ≤ G < 16.450 12.761 ≤ G < 13.218 12.243 ≤ G < 12.761 11.713 ≤ G < 12.243 10.591 ≤ G < 11.713
6.162 ≤ G < 10.591 5.275 ≤ G < 6.162
(6)

As error in this parallax correction a 1 µas systematic error is added in quadrature to a random error of 2.7 µas for G > 6 and 13 µas for G ≤ 6.

5. Application of the PZPO correction
In this section the results are being discussed related to the application of the PZPO correction.

5.1. QSOs

Fig. 6. As Fig. 5 for Gs restricted to < 19.0 mag. After eliminating ∼700 extreme outliers (those outside the two plotted lines in the top panel), a sample of about 480 000 objects remain. The bottom panel shows the parallax diﬀerence between primary and secondary component for that sample as a function of Gp. Note the diﬀerent range in the ordinate compared to Fig. 5.
ing secondaries fainter than 13.3 mag. The consecutive panels in Fig. 7 show how this procedure can be applied to brighter and brighter magnitudes. The bottom panel shows the ﬁnal result. The weighted mean of the residuals is 0.05 µas with an rms of 2.7 µas (for G > 6), 4.3 µas (for G > 5), and 13 µas (for 5 < G < 6 mag). The corrections that were applied with the range of G magnitudes determined so that the correction is continuous in G is given by Eq. 6. It is show as the black line in the bottom panel.
The bottom panel of Fig. 7 also shows for comparison the correction by L20 for νeﬀ = 1.55 (corresponding to (GBP − GRP) = 0.8 mag) and β = 0◦ (small black circles), +60◦ (green), and −60◦ (red circles). The behaviour for β = 0◦ of the L20 correction is very similar in shape and amplitude to the correction derived here. There is an oﬀset due to the fact that the L20 corrections are absolute while the corrections in Eq. 6 are relative to the correction at G = 20 which is ≈ −12.6 µas (see Table 2) which is indeed about the diﬀerence at the faint end. What is remarkable is that the L20 correction also depends in a particular way on ecliptic latitude. For bright magnitudes the L20 correction for −60◦ lies above that for +60◦, while for G ∼>13 mag it

The ﬁrst application of the PZPO correction is to the QSO sample itself and a comparison to using the L20 correction. Table 5 provides the L20 correction, and the spatial, magnitude and total correction in the present work, as well as the oﬀset after applying the L20 correction and the correction in the present work for the diﬀerent HEALPix levels. The weighted mean and the error therein are quoted in all cases. As the L20 correction comes without an error one has been assigned. It has been chosen to be a constant such that the error in the weighted mean of the L20 and the correction in the present work (Cols 2 and 5) are the same for HEALPix level 0 and equals 3.0 µas. This choice has no practical impact. The error in the weighted mean after applying the corrections (Cols 6 and 7) is independent of this choice, and is actually virtually the same for both type of corrections (∼0.3 µas), as this error is dominated by the error in the observed parallaxes.
The results in Table 5 give the overall comparison for ∼821 000 QSOs in the sample, but as the main diﬀerence between the approach in L20 and in the present work is in the dependence of the correction on sky position this dependence is of interest. Figure 8 shows the corrected parallax after applying the correction in the present work (in black) and that in L20 (in blue) using 60 bins in sin β. The black points tend to be closer to the line of zero oﬀset and Table 6 contains the details for the different HEALPix levels. The table lists the median over the bins and the scatter around the median (calculated as 1.4826·MAD), and shows that the scatter decreases with increasing spatial resolution when using the present correction. As shown in the next two subsections, this, will not be the case in general however. As the sample to deﬁne the spatial correction is the same as to which it is applied there are no undeﬁned spatial pixels being used. In general, increasing the spatial resolution (increasing the HEALPix level) will lead to an increasing number of stars to be in spatial pixels that are undeﬁned (insuﬃcient number of QSOs), so that there will be an optimal HEALPix level to be used.

Article number, page 8 of 18

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data

Table 5. Parallax corrections for the QSO sample

HEALPix level
0 1 2 3 4

L20 correction
(µas)
−17.11 ± 0.0034 −17.11 ± 0.0034 −17.11 ± 0.0034 −17.11 ± 0.0034 −17.11 ± 0.0034

PW spatial correction
(µas)
−13.49 ± 0.0010 −13.25 ± 0.0019 −13.31 ± 0.0036 −13.38 ± 0.0070 −14.08 ± 0.0104

PW magnitude correction (µas)
−5.04 ± 0.0032 −5.04 ± 0.0032 −5.04 ± 0.0032 −5.04 ± 0.0032 −5.04 ± 0.0032

PW total correction
(µas)
−18.66 ± 0.0034 −18.73 ± 0.0038 −18.70 ± 0.0050 −18.68 ± 0.0079 −19.16 ± 0.0114

∆(PZPO corrected parallax, L20) (µas)
−0.502 ± 0.295 −0.502 ± 0.295 −0.502 ± 0.295 −0.502 ± 0.295 −0.502 ± 0.295

∆(PZPO corrected parallax, PW) (µas)
+0.008 ± 0.295 +0.016 ± 0.295 +0.002 ± 0.295 −0.007 ± 0.295 +0.005 ± 0.296

Notes. Column 1 gives the HEALPix level, column 2 gives the weighted mean and error of the L20 correction, columns 3-5 give the weighted mean and error for the spatial correction, the magnitude correction, and the total correction of the present work (PW), Column 6-7 give the weighted mean and error after applying the correction in L20 and of the present work, respectively.

Table 6. Parallax corrections for the QSO sample when binned against ecliptic latitude

tude (7 ∼< G ∼< 10) and colour (0.5 ∼< (GBP − GRP) ∼< 2.3 mag). Appendix A gives some more details on this subsample.

HEALPix level
0 1 2 3 4

∆(PZPO corrected parallax, L20) (µas) −0.82 ± 2.75 −0.82 ± 2.75 −0.82 ± 2.75 −0.82 ± 2.75 −0.82 ± 2.75

∆(PZPO corrected parallax, present work)
(µas) −0.086 ± 3.06 +0.078 ± 2.50 −0.080 ± 2.09 −0.166 ± 1.80 −0.090 ± 1.59

Notes. Column 1 gives the HEALPix level, column 2-3 give the median and 1.4826·MAD using the L20 correction and the present work, respectively.

5.2. Independent trigonometric parallaxes
In Sect. 3.3 a sample of 111 stars with independent trigonometric paralax data was introduced (Table 1) of which 57 pass the selection on GOF and RUWE. Figure 9 compares these parallaxes to the GEDR3 ones in the top panel, while the residuals are shown in the bottom panel.
Two stars are excluded in the further analysis, VY Pyx and HD 285876. Benedict et al. (2017) mention that VY Pyx is an outlier, lying 1.19 mag of the PL-relation they derived. Adopting the Gaia parallax would shift this object 1.06 mag closer, and hence in agreement, with the PL-relation. Although Benedict et al. (2017) carefully analysed all steps in their procedure, it is likely that the FGS parallax is in error. For vA 645 (HD 285876) the diﬀerence between Gaia and FGS parallax is 20σ, much larger than one can reasonably attribute to a statistical outlier.
Figure 10 plots the residuals against G magnitude, (GBP − GRP) colour and sin β. Although these are the objects with the best independent trigonometric parallaxes the error bars on the diﬀerential parallax are dominated by the error in the external parallax and the range in the ordinate (∼4 mas) is much larger than when intercomparing GEDR3 parallaxes where diﬀerences in parallax due to sky position, magnitude and colour are of order 100 times smaller (∼0.04 mas, e.g. Figs. 3 or 7). This is probably the reason that no trends are obvious.
Table 7 gives the median and weighted mean with error of the diﬀerence between observed and the independent trigonometric parallax. The ﬁrst ﬁve entries are for the entire sample applying increasingly stricter selection criteria. The last two entries are speciﬁcally for the CCs, Type-ii cepheids (T2C) and RRL stars in the sample. These objects are of special interest to the distance scale, and they are all radially pulsating stars of similar magni-

Table 8 provides the spatial, magnitude and total correction, as well as the oﬀset between the corrected GEDR3 and independent trigonometric parallaxes. This is done for three representative samples, and for the diﬀerent HEALPix levels. Ideally, the weighted mean of the diﬀerence between the corrected GEDR3 parallax and the independent trigonometric parallax (Col. 5) should be zero within the error bars, and this is indeed the case. However, some trends are observed. For the larger HEALPix levels an increasing number of stars will be missing and this results in a marked increase in the scatter. On the other hand one should favour the best mapping of the spatial variations. For the samples discussed here this would imply using the results for HEALPix level 2 as the most appropriate. However this choice will depend on the properties of the external sample (number of stars, distribution on the sky, accuracy of the external parallaxes).
5.3. Classical cepheids
As a second application of the spatial and magnitude corrections derived in the present paper the sample of Galactic CCs of Riess et al. (2021) is studied. They discuss a sample of 75 CCs with HST photometry which is used to calibrate the extragalactic distance scale along the lines outlined in earlier works (Riess et al. 2016, 2018b, 2019). They correct the GEDR3 parallaxes using the L20 formalism and ﬁt the slope, zero point and metallicity dependence of the PL relation as well as a constant oﬀset between the photometric parallaxes and the corrected GEDR3 values. Fits where some of these parameters are ﬁxed are also presented. In their analysis they increased the GEDR3 parallax uncertainties by 10%, which is not done here. One important conclusion in the present context is that Riess et al. (2021) ﬁnd that the L20 procedure over corrects the PZPO by 14 ± 6 µas.
Table 9 contains the result of the calculations for two samples. The ﬁrst is the sample of 66 stars retained by Riess et al. (2021). This is the full sample of 75 stars minus 9 stars excluded in their best ﬁt analysis. Six were excluded there because their GOF > 12.5 (SV Per, RW Cam, U Aql, DL Cas, SY Nor, RX Cam), one, CY Aur, because it is an outlier in the L20 correction, and S Vul and SV Vul that are marginal outliers of the PL-

Article number, page 9 of 18

A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom

Table 7. PZPOs for the sample with independent trigonometric parallaxes

median weighted mean and error N description

(µas)

(µas)

−39

−35 ± 14

57 all that pass the GOF/RUWE selection

−39

−45 ± 14

55 excluding VY Pyx and vA 645 (10σ outliers)

−45

−71 ± 14

53 excluding 8σ outliers

−39

−36 ± 14

46 excluding 6σ outliers

−39

−39 ± 14

39 excluding 6σ outliers, σπ < 0.4 mas

−33

−31 ± 16

15 all CCs, T2C, RRL that pass the GOF/RUWE selection (and excluding VY Pyx as well)

−33

−29 ± 16

13 as above, excluding Polaris B and FF Aql as well

Notes. Columns 1 and 2 list the median and weighted mean oﬀset between the GEDR3 parallax and the independent trigonometric parallax for the sample discussed in Sects. 3.3 and 5.2. Column 3 lists the number of stars, and column 4 gives a description of the selection criteria.

Table 8. Parallax corrections for the sample with independent trigonometric parallaxes

HEALPix spatial correction magnitude correction total correction ∆ (PZPO corrected parallax) N

level

(µas)

(µas)

(µas)

(µas)

Sample: 55 stars that pass GOF/RUWE selection

0

−13.18 ± 0.14

−17.47 ± 0.40

−30.10 ± 0.44

−11.17 ± 13.88

54

1

−14.67 ± 0.29

−17.47 ± 0.40

−32.11 ± 0.53

−13.03 ± 13.93

54

2

−13.60 ± 0.58

−17.47 ± 0.40

−32.07 ± 0.78

− 7.69 ± 14.16

54

3

−10.72 ± 1.12

−17.58 ± 0.41

−30.35 ± 1.26

+ 2.46 ± 18.63

53

4

−16.11 ± 1.46

−18.07 ± 0.44

−35.74 ± 1.65

+ 2.40 ± 23.03

45

Sample: 39 stars that pass GOF/RUWE selection, excluding 6σ outliers, σπ < 0.4 mas

0

−13.67 ± 0.19

−16.68 ± 0.48

−29.50 ± 0.53

− 4.86 ± 14.50

38

1

−14.94 ± 0.37

−16.68 ± 0.48

−31.55 ± 0.65

− 6.85 ± 14.55

38

2

−13.98 ± 0.75

−16.68 ± 0.48

−31.95 ± 0.99

− 2.16 ± 14.81

38

3

−12.58 ± 1.53

−16.81 ± 0.49

−31.56 ± 1.70

+13.81 ± 20.16

37

4

−25.05 ± 1.94

−17.58 ± 0.54

−43.10 ± 2.24

− 4.42 ± 25.71

30

Sample: 13 CCs, T2C, RRL

0

−13.91 ± 0.35

−13.47 ± 0.80

−26.72 ± 0.88

− 6.75 ± 15.91

13

1

−14.49 ± 0.70

−13.47 ± 0.80

−27.49 ± 1.12

− 2.87 ± 15.97

13

2

−14.31 ± 1.41

−13.47 ± 0.80

−28.13 ± 1.85

+ 3.92 ± 16.30

13

3

−10.48 ± 3.19

−13.47 ± 0.80

−24.47 ± 3.56

+ 2.70 ± 23.70

13

4

−7.38 ± 5.54

−12.95 ± 1.01

−20.13 ± 5.79

+16.91 ± 32.86

8

Notes. Column 1 gives the HEALPix level considered (deﬁning the spatial correction term), columns 2-5 give the weighted mean and error for the spatial correction, the magnitude correction, the total correction, and the oﬀset between the corrected GEDR3 parallax and the external parallax. Column 6 gives the number of stars. The samples refer to those deﬁned in Tab. 7.

relation3. The second sample are the 54 stars that remain after the applying the criteria on GOF and RUWE used in this paper.
The reference parallax is the photometric parallax (with error) from Table 1 in Riess et al. (2021) that is derived from the HST photometry, the pulsation period, and the PL relation from Riess et al. (2016, 2019).
Column 2 of Tab. 9 gives the weighted mean oﬀset between the observed GEDR3 parallax and the photometric parallax. It is unusually large (−6.4 to −7.7 µas, see below). Column 3 lists the weighted mean of the L20 correction, and the upper panel of Fig. 11 show the dependence on β. A similar diagram was shown for the 75 CCs in Riess et al. (2021). What is striking is the close to parabolic shape of the correction which is build-in in
3 It was conﬁrmed (Riess 2021, private communication) that two corrections are necessary in Table 1 of Riess et al. (2021) to match their analysis; Z Sct is missing there but is available in Table 1 of Riess et al. 2018b, and the high GOF ﬂag on AD Pup (GOF = 12.48) should instead appear on RX Cam (GOF = 28.7).

the L20 approach. The other columns show the weighted mean spatial, magnitude, and total correction, the oﬀset between the corrected GEDR3 parallax and the photometric parallax, and the number of objects. If the GEDR3 parallaxes are corrected by the L20 formalism (on a star-by-star basis) the weighted mean oﬀset with the photometric parallax becomes +14.3 ± 2.9 µas, consistent with Riess et al. (2021), and indicating an overcorrection by the L20 formalism. By increasing the photometric parallaxes by ∼3.3% one can obtain a weighted mean oﬀset between the L20 corrected and photometric parallax consistent with zero (0.00 ± 2.93 µas). Such an increase is consistent with the result reported in the last entry of table 2 in Riess et al. (2021) where they forced a ﬁt without additional PZPO to determine the zero point of the PL relation. The value reported there (−5.865 ± 0.013) is consistent with ﬁnding here that implies a zero point of −5.93 + 5 log 1.0325 = −5.861.
The second block in Table 9 shows similar results for the smaller sample that fulﬁls the criteria on GOF and RUWE im-

Article number, page 10 of 18

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data
Fig. 8. Residual in the observed parallax after applying the correction in the present work (in black open circles, oﬀset by −0.004 units horizontally) and that in L20 (in blue ﬁlled circles, oﬀset by +0.004 units) versus sin β for the QSO sample. Sixty bins have been used, and HEALPix level 2 has been used in the calculations.

Fig. 9. Independent trigonometric parallax plotted against GEDR3 parallax. The bottom panel displays the residual, where the error bar in the ordinate combines the error in the Gaia and the independent parallax in quadrature. Two stars where the residual is more than ten times the combined error bar are plotted as open triangles.

Fig. 7. As Fig. 5 with parallaxes corrected according to Eq. 6 in consecutive steps (see main text). The bottom panel shows the applied correction (Eq. 6) as the black line. The small black, red, and green circles represent the L20 correction for β = 0, −60, and +60◦. There is an oﬀset as the L20 corrections are absolute, while the corrections applied to the WB sample are relative to the correction at G = 20 mag.

posed here. The bottom panel of Fig. 11 shows that there is no the dependence of the total correction (with error bar) proposed in the present paper on β. It is remarked that the error in the average total correction (Col. 6) is similar or smaller than the average L20 correction (Col. 3) for HEALPix levels 0, 1, and 2.

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A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom

Fig. 10. Diﬀerence between the independent trigonometric parallax and the GEDR3 parallax plotted against G, (GBP − GRP) colour, and sin β. The two outliers mentioned in Fig. 9 have been removed.

Adopting the standard photometric parallax leads to overcorrection of ∼20 µas. Increasing the photometric parallax by a factor 1.0505 ± 0.0080 (implying a PL zero point of −5.823 ± 0.016, and H0 = 76.2 ± 1.3 km/s/Mpc) will lead to weighted mean oﬀset between the corrected GEDR3 and the photometric parallax consistent with zero. It also implies a weighted mean oﬀset of the observed GEDR3 and the photometric parallax of −29 ± 3 µas, which is very similar to other bright (G ∼< 10-11 mag) samples, for example the stars with external trigonometric parallaxes and the subsample of pulsating stars (−39 ± 14, respectively, −29 ± 16 µas from Tab. 7) or the sample of EBs (−37 ± 20 µas, Stassun & Torres 2021) or WUMa-type EBs (−28.6 ±0.6 µas for the 5-parameter solution, Ren et al. 2021).
6. Discussion and summary
The presence of a parallax zero point oﬀset that was identiﬁed in GDR2 received a lot of attention. The L20 paper analysing the new GEDR3 data oﬀers a lot of insight into the issue and they presented a python script to calculate the correction

Fig. 11. Top panel: PZPO correction by L20 for the sample of 66 CCs analysed by Riess et al. (2021) (cf. their Fig. 2). The colours represent diﬀerent ranges in G: black (G ≤ 7), red (7 < G ≤ 8.5), green (8.5 < G ≤ 9.0), and blue (9 < G ≤ 11.5). Bottom panel: Correction proposed here for the stricter selected sample of 54 stars at HEALPix level 2.
based on G, β, and the pseudocolour or νeﬀ (depending on the astrometric_params_solved parameter).
On the other hand, L20 remark that ‘the results should . . . in no way be regarded as deﬁnitive’, and that ‘alternative routes are explored towards getting a better handle on the systematics in Gaia data’. The present paper should be viewed in this light. An alternative procedure to the one in L20 is proposed which is oﬀered to the community for further scrutiny.
The two approaches are similar in that both use samples of QSOs and wide binaries (albeit selected in diﬀerent ways). The main diﬀerences to the L20 approach are that (1) there is no separation between 5- and 6-parameter solutions, (2) the colour dependence uses the (GBP − GRP) colour rather than pseudocolour

Article number, page 12 of 18

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data

Table 9. Parallax corrections for samples of Galactic CCs

HEALPix ∆(uncorrected L20

PW spatial PW magnitude PW total ∆(PZPO corrected N Remarks

level

parallax) correction correction correction correction parallax, PW)

(µas)

(µas)

(µas)

(µas)

(µas)

(µas)

66 CCs following Riess et al. (2021)

0

−6.36 ± 2.83 −22.1 ± 1.23 −14.10 ± 0.17 −14.55 ± 0.35 −27.82 ± 0.39 +22.0 ± 2.87 66

1

−14.13 ± 0.38 −14.55 ± 0.35 −27.57 ± 0.55 +20.3 ± 2.90 66

2

−14.43 ± 1.25 −14.55 ± 0.35 −28.94 ± 1.31 +25.1 ± 3.26 66

3

−10.43 ± 4.12 −14.57 ± 0.36 −25.34 ± 4.17 +17.4 ± 6.73 63

4

− 8.58 ± 8.57 −14.67 ± 0.47 −23.90 ± 8.64 +24.7 ± 11.5 38

2

−20.5 ± 2.83

−14.43 ± 1.25 −14.55 ± 0.35 −28.94 ± 1.31 +10.4 ± 3.26 66 πphot · 1.0325a

54 CCs following the GOF/RUWE selection in the present work

0

−7.66 ± 3.04 −22.0 ± 1.36 −13.66 ± 0.19 −14.54 ± 0.39 −27.39 ± 0.44 +20.4 ± 3.08 54

1

−13.51 ± 0.42 −14.54 ± 0.39 −27.15 ± 0.60 +18.5 ± 3.12 54

2

−13.57 ± 1.41 −14.54 ± 0.39 −28.04 ± 1.47 +22.2 ± 3.53 54

3

− 8.05 ± 4.66 −14.57 ± 0.40 −22.70 ± 4.73 +13.4 ± 7.62 51

4

− 6.94 ± 9.24 −14.82 ± 0.55 −22.35 ± 9.32 +22.7 ± 12.7 27

2

−29.2 ± 3.05

−13.57 ± 1.41 −14.54 ± 0.39 −29.22 ± 3.05 +0.00 ± 3.53 54 πphot · 1.0505

Notes. Column 1 gives the HEALPix level considered (deﬁning the spatial correction term), column 2 gives the weighted mean and error of the
oﬀset between the observed GEDR3 parallax and the photometric parallax, column 3 gives the weighted mean and error of the L20 correction, columns 4-7 give the weighted mean and error for the spatial correction, the magnitude correction, the total correction, and the oﬀset between the corrected GEDR3 parallax in the present work (PW) and the photometric parallax. Column 8 gives the number of stars. The sample sizes of 66 and 54 stars are explained in Sect. 5.3. (a) This model results in a ∆(PZPO corrected parallax) of 0.00 ± 2.93 µas when using the L20 correction.

or νeﬀ, (3) the dependence on sky position and magnitude are separated, and are treated as additive terms, and that (4) the present approach gives a correction including an error estimate.
It is shown that the PZPO shows a more complicated behaviour than only on the ecliptic latitude (Fig. 3, also see Huang et al. 2021). L20 argue that such a dependence is theoretically expected and related to the scanning law but this would not explain the diﬀerent behaviour at bright (Fig. 11; a range of ∼35 µas with the largest correction around β ∼ 5◦) and faint magnitudes (Fig. 4; a range of ∼20 µas with a slow increase with β) in the L20 recipe.
Here, the practical approach is taken to calculate the PZPO over the sky using the HEALPix formalism. Using the dependence of the PZPO as a function of G, a spatial PZPO at G = 20 mag is determined for several HEALPix levels, based on the QSO sample for G > 17 mag. A large sample of WBs with very low chance alignments is used to derive the magnitude dependence of the PZPO for magnitudes <19 mag. The range of 17 − 19 mag is used to connect the QSO to the WB sample.
The L20 recipe does not provide an error on the correction. It is shown here that error on the PZPO is dominated by the error on the spatial correction, and that it can be substantial (up to several tens of µas depending on sky position). Increasing the sample of QSOs, especially in the direction of the Galac√tic plane, will help in reducing the statistical error but only as 1/ N.
The recipes provided here can not be easily transformed into a simple script. This may be seen as a disadvantage, on the other hand is requires the user to make informed choices. The procedure to be followed is as follows:
– Obtain the source_id, G magnitude, parallax and error, and (GBP − GRP) colour from GEDR3 for your source(s).
– Get the pixel number in the HEALPix scheme from the source_id following footnote 2 for levels 0, 1, 2, 3, and 4.

– Use the results from models 30-34 available from the CDS to obtain the spatial correction and error at G = 20 for the various HEALPix levels.
– For G >19.9 mag apply Eq. 4, otherwise apply Eq. 6. The error in this correction is a 1 µas systematic error to be added in quadrature to a random error of 2.7 µas for G > 6 and 13 µas for G ≤ 6.
– If a colour term is to be included use the results from models 35-39, and additionally apply Eq. 5. This colour term is derived for the QSOs sample (G ∼>17 mag, 0.2 ∼< (GBP − GRP) ∼< 1.6 mag) and is untested outside this range.
– Add the spatial and magnitude (+colour) correction, and add the errors in quadrature. Subtract the total from the observed parallax to obtain the corrected parallax, that is, an estimate of the true parallax (Eq. 1). Also in this last step the errors should be added in quadrature.
Following the examples described in Section 5, it is recommended do this for all available HEALPix levels and then choose the highest level that does not compromise the S/N.
Acknowledgements. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the SIMBAD database, the VizieR catalogue access tool and the cross-match service provided by CDS, Strasbourg. Thanks to Francois-Xavier Pineau for explaining the best use of the cross-match service. Some of the results in this paper have been derived using the healpy and HEALPix package. I would like to thank Drs. Kareen El-Badry and Valentin Ivanov for discussions on wide binaries and quasar catalogues, respectively, and the referee for a careful reading of the manuscript and helpful suggestions.
References
Anderson, R. I. 2018, A&A, 611, L7

Article number, page 13 of 18

A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom
Benedict, G. F., McArthur, B. E., Nelan, E. P., & Harrison, T. E. 2017, PASP, 129, 012001
Bhardwaj, A., Rejkuba, M., de Grijs, R., et al. 2021, ApJ, 909, 200 Bond, H. E., Nelan, E. P., Remage Evans, N., Schaefer, G. H., & Harmer, D.
2018, ApJ, 853, 55 Casertano, S., Riess, A. G., Anderson, J., et al. 2016, ApJ, 825, 11 Chan, V. C. & Bovy, J. 2020, MNRAS, 493, 4367 Clementini, G., Ripepi, V., Molinaro, R., et al. 2019, A&A, 622, A60 El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, MNRAS[arXiv:2101.05282] Fabricius, C., Luri, X., Arenou, F., et al. 2021, A&A, 649, A5 Flesch, E. W. 2019, arXiv e-prints, arXiv:1912.05614 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021a, A&A, 649, A1 Gaia Collaboration, Klioner, S. A., Mignard, F., et al. 2021b, A&A, 649, A9 Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1 Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759 Graczyk, D., Pietrzyn´ski, G., Gieren, W., et al. 2019, ApJ, 872, 85 Groenewegen, M. A. T. 2018, A&A, 619, A8 Huang, Y., Schönrich, R., Zhang, H., et al. 2020, ApJS, 249, 29 Huang, Y., Yuan, H., Beers, T. C., & Zhang, H. 2021, ApJ, 910, L5 Khan, S., Miglio, A., Mosser, B., et al. 2019, in The Gaia Universe, 13 Layden, A. C., Tiede, G. P., Chaboyer, B., Bunner, C., & Smitka, M. T. 2019,
AJ, 158, 105 Leung, H. W. & Bovy, J. 2019, MNRAS, 489, 2079 Lindegren, L. 2018, GAIA-C3-TN-LU-LL-124 Lindegren, L., Bastian, U., Biermann, M., et al. 2021a, A&A, 649, A4 Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2 Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021b, A&A, 649, A2 Muraveva, T., Delgado, H. E., Clementini, G., Sarro, L. M., & Garofalo, A. 2018,
MNRAS, 481, 1195 Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. 1992, Numerical Recipes
in C (Cambridge: Cambridge University Press) Ren, F., Chen, X., Zhang, H., et al. 2021, ApJ, 911, L20 Riess, A. G., Casertano, S., Anderson, J., MacKenty, J., & Filippenko, A. V.
2014, ApJ, 785, 161 Riess, A. G., Casertano, S., Yuan, W., et al. 2021, ApJ, 908, L6 Riess, A. G., Casertano, S., Yuan, W., et al. 2018a, ApJ, 855, 136 Riess, A. G., Casertano, S., Yuan, W., et al. 2018b, ApJ, 861, 126 Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ,
876, 85 Riess, A. G., Macri, L. M., Hoﬀmann, S. L., et al. 2016, ApJ, 826, 56 Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14 Schönrich, R., McMillan, P., & Eyer, L. 2019, MNRAS, 487, 3568 Stassun, K. G. & Torres, G. 2018, ApJ, 862, 61 Stassun, K. G. & Torres, G. 2021, ApJ, 907, L33 van Belle, G. T., Schaefer, G. H., von Braun, K., et al. 2020, PASP, 132, 054201 Wilson, E. B. & Hilferty, M. M. 1931, Proceedings of the National Academy of
Science, 17, 684 Xu, S., Zhang, B., Reid, M. J., Zheng, X., & Wang, G. 2019, ApJ, 875, 114 Zinn, J. C. 2021, AJ, 161, 214 Zinn, J. C., Pinsonneault, M. H., Huber, D., & Stello, D. 2019, ApJ, 878, 136 Zonca, A., Singer, L., Lenz, D., et al. 2019, Journal of Open Source Software, 4,
1298
Article number, page 14 of 18

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data
Appendix A: Parallax difference for classical pulsators
Figure A.1 is as Figs. 9 and 10 for the sample of 15 CCs, T2C, and RRL stars. Two outliers are plotted as open triangles. They are FF Aql and Polaris B. FF Aql is very bright (G = 5.1 mag) and this may be the reason for the oﬀset. For Polaris B the reason is less clear. The GOF (3.55) and RUWE (1.15) easily fall within the applied selection criteria. The diﬀerence between the FGS parallax for Polaris B and the Hipparcos parallax for Polaris A have been discussed in the literature without reaching a conclusion on its implications (Bond et al. 2018; Anderson 2018; appendix B in Groenewegen 2018).
Fig. A.1. Diﬀerence between the independent trigonometric parallax and the GEDR3 parallax plotted against G, (GBP − GRP) colour, and sin β for the 15 CCs, T2C, and RRL stars. Two outliers are plotted as open triangles.
Article number, page 15 of 18

A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom Article number, page 16 of 18

Table 1. Stars with independent trigonometric parallaxes, ordered by Right Ascension.

Identiﬁer

parallax Ref.

Ra

Dec

Source ID

parallax

(mas)

(deg)

(deg)

(mas)

GJ1005A

166.6 ± 0.3 1

3.869880 -16.136572 2368293487261055488

GJ22C

99.2 ± 0.6 1

8.142840 +67.233278 527956488339113600 100.397 ± 0.037

GJ22A

99.2 ± 0.6 1

8.143031 +67.234328 527956488340229632 101.086 ± 0.461

υ Aan

73.71 ± 0.1 1 24.198321 +41.403762 348020482735930112 74.194 ± 0.208

VX Per

0.42 ± 0.0744 5 31.952002 +58.443534 506779550797525760 0.364 ± 0.017

RW Tri

2.93 ± 0.33 1 36.400769 +28.097391 130692247044752768 3.267 ± 0.022

Polaris B

6.26 ± 0.24 2 37.664816 +89.260830 576402619921510144 7.287 ± 0.018

Polaris A

7.62 ± 0.08 3 (37.967198) (+89.264051)

Feige 24

14.6 ± 0.4 1 38.782009 +3.732482 2503828498910129664 12.868 ± 0.036

REJ 0317-853

34.38 ± 0.26 1 49.311809 -85.540486 4613612951211823104 34.035 ± 0.029

LB 9802

33.28 ± 0.24 1 49.326169 -85.542124 4613612951211823616 34.021 ± 0.019

GK Per

2.1 ± 0.12 1 52.799999 +43.904220 238540495056450048 2.306 ± 0.042

ǫ Eri

311.37 ± 0.11 1 53.228293 -9.458168 5164707970261890560 310.577 ± 0.136

Cl* Melotte 22 HII 3030 7.41 ± 0.18 1 57.855681 +23.889172 66481734354737792 7.371 ± 0.018

Cl* Melotte 22 HII 3063 7.43 ± 0.16 1 57.874842 +23.899036 66481837433947392 7.498 ± 0.025

Cl* Melotte 22 HII 3179 7.45 ± 0.16 1 57.987027 +23.901753 66471220274934272 7.565 ± 0.017

XO-3

5.67 ± 0.14 1 65.469581 +57.817210 470650560779277952 4.869 ± 0.026

vA 310

20.13 ± 0.17 1 66.071101 +18.002784 3314079508140198528 21.497 ± 0.015

vA 383

21.53 ± 0.2 1 66.520077 +15.041285 3311179340063437952 20.839 ± 0.028

vA 472

21.7 ± 0.15 1 67.018941 +13.867854 3307844864893938304 20.876 ± 0.021

vA 548

20.69 ± 0.17 1 67.379585 +16.244692 3312899491645515776 21.818 ± 0.021

vA 622

24.11 ± 0.3 1 67.871614 +17.718519 3314150048683152896 22.236 ± 0.022

vA 627

21.74 ± 0.25 1 67.905103 +17.709637 3314151251273992832 28.508 ± 0.581

vA 645

17.46 ± 0.21 1 67.969120 +15.499373 3312564037520033792 21.872 ± 0.016

HD 33636

35.6 ± 0.2 1 77.944347 +4.402925 3238810137558836352 33.798 ± 0.053

SY Aur

0.428 ± 0.054 4 78.163452 +42.831780 201509768065410944 0.427 ± 0.020

TV Col

2.7 ± 0.11 1 82.356418 -32.817651 2901783160488793728 1.951 ± 0.014

GJ1081A

65.2 ± 0.37 1 83.329996 +44.814683 207910437566174592

β Dor

3.14 ± 0.16 1 83.406311 -62.489769 4757601523650165120 2.931 ± 0.139

HD 38529

25.11 ± 0.19 1 86.645126 +1.167567 3219847066672970368 23.571 ± 0.042

SS Aur

5.99 ± 0.33 1 93.343503 +47.740270 968824328534823936 3.978 ± 0.028

RT Aur

2.4 ± 0.19 1 97.142032 +30.492976 3435571660360952704 1.815 ± 0.122

GJ234A

240.98 ± 0.4 1 97.350795 -2.817131 3117120863523946368

RR Pic

1.92 ± 0.18 1 98.900296 -62.640097 5477422099543150592 1.996 ± 0.021

HD 47536

8.71 ± 0.16 1 99.448976 -32.339444 5583831735369515008 7.990 ± 0.053

G250-029A

95.59 ± 0.28 1 103.522633 +60.867338 1003752587430126464 91.611 ± 0.503

G193-027A

110.2 ± 1.1 1 105.987163 +52.697727 981548637301374336 110.826 ± 0.695

ζ Gem

2.78 ± 0.18 1 106.027177 +20.570294 3366754155291545344 3.073 ± 0.218

SS CMa

0.389 ± 0.0287 5 111.529960 -25.257297 5616601820448126336 0.286 ± 0.013

X Pup

0.277 ± 0.0469 5 113.195965 -20.909682 5620098679741674496 0.376 ± 0.020

YY Gem

67.22 ± 0.4 6 113.654977 +31.869063 892348454394856064 66.311 ± 0.024

U Gem

9.96 ± 0.37 1 118.771670 +22.001222 674214551557961984 10.705 ± 0.034

ρ1 Cnc

79.78 ± 0.3 1 133.146761 +28.329783 704967037090946688 79.448 ± 0.043

PN A66 31

1.61 ± 0.21 1 133.554806 +8.898001 597324024095840512 1.842 ± 0.055

VY Pyx

6.44 ± 0.23 1 133.623514 -23.521695 5653136461526964224 3.950 ± 0.019

HIP 46120

15.01 ± 0.12 1 141.092846 -80.517086 5195968563310843008 14.776 ± 0.014

ℓ Car

2.01 ± 0.2 1 146.311593 -62.507867 5250032958818831360 1.984 ± 0.110

HD 84937

12.24 ± 0.2 1 147.235464 +13.740818 615943806835727872 13.498 ± 0.044

GoF
809.35 12.87 91.24 77.72
3.49 6.10 3.55

RUWE

G

(mag)

10.268 ± 0.005

1.54 11.107 ± 0.003

5.89 9.567 ± 0.003

7.25 3.966 ± 0.003

1.12 8.894 ± 0.009

1.34 13.246 ± 0.033

1.15 8.630 ± 0.003

(GBP − GRP) (mag)
2.819 ± 0.005 2.620 ± 0.012 2.304 ± 0.005 0.723 ± 0.007 1.551 ± 0.048 0.631 ± 0.153 0.492 ± 0.005

11.01 0.69 2.49
48.50 31.41
0.47 8.04 2.34 9.03 0.69 25.01 0.68 6.31 5.07 488.26 2.77 20.84 2.50 2.96 647.99 47.30 1.31 15.02 66.45 761.25 2.50 -3.12 130.21 234.89 21.96 3.97 1.19 0.16 31.00 -3.81 0.95 -3.57 -1.60 24.35 1.94

1.58 1.02 1.09 3.35 2.72 1.02 1.30 1.08 1.25 1.02 1.89 1.02 1.24 1.18 43.27 1.10 1.88 1.08 1.10
4.54 1.05 1.69 6.43
1.10 0.88 9.05 18.90 2.78 1.11 1.04 1.01 2.41 0.86 1.03 0.89 0.94 2.39 1.10

12.210 ± 0.003 14.754 ± 0.004 14.079 ± 0.003 12.557 ± 0.007 3.466 ± 0.003 13.465 ± 0.003 13.028 ± 0.003 9.927 ± 0.003 9.743 ± 0.003 9.650 ± 0.003 11.470 ± 0.003 8.840 ± 0.003 9.932 ± 0.003 11.184 ± 0.003 9.248 ± 0.003 10.498 ± 0.003 6.865 ± 0.003 8.798 ± 0.008 13.981 ± 0.011 11.019 ± 0.004 3.593 ± 0.014 5.748 ± 0.003 14.318 ± 0.024 5.336 ± 0.021 9.630 ± 0.005 12.425 ± 0.006 4.874 ± 0.003 10.003 ± 0.006 11.699 ± 0.003 3.540 ± 0.006 9.563 ± 0.007 8.348 ± 0.012 8.296 ± 0.003 13.903 ± 0.011 5.733 ± 0.003 15.475 ± 0.003 7.107 ± 0.005 9.938 ± 0.003 3.471 ± 0.014 8.207 ± 0.003

0.193 ± 0.006 -0.373 ± 0.014 -0.182 ± 0.005 1.393 ± 0.029 1.140 ± 0.011 1.739 ± 0.007 1.571 ± 0.007 0.752 ± 0.005 0.565 ± 0.005 1.248 ± 0.005 1.912 ± 0.005 0.990 ± 0.005 1.373 ± 0.005 1.879 ± 0.005 1.180 ± 0.005 1.573 ± 0.005 0.750 ± 0.005 1.376 ± 0.038 0.456 ± 0.050 2.678 ± 0.005 1.050 ± 0.053 0.913 ± 0.005 0.978 ± 0.098 0.828 ± 0.086 3.080 ± 0.012 0.013 ± 0.021 1.399 ± 0.006 2.534 ± 0.005 3.210 ± 0.005 0.987 ± 0.030 1.673 ± 0.032 1.697 ± 0.052 1.932 ± 0.008 1.280 ± 0.049 1.009 ± 0.005 -0.475 ± 0.006 0.811 ± 0.017 0.814 ± 0.005 1.503 ± 0.047 0.606 ± 0.005

M. A. T. Groenewegen: The parallax zero point oﬀset from Gaia EDR3 data Article number, page 17 of 18

Table 1. continued.
Identiﬁer
VY Car XY Car HIP 54639 SU Dra GJ469A AM CVn EX Hya GP Com V803 Cen NSVS 0103 NSVS 0103-REF68 CR Boo Proxima Cen G166-037 HD 128311 HD 132475 HD 136118 GU Boo HP Lib HD 140283 G16-025 GJ623A UV Oct CM Dra CM Dra-REF47 TRES HER0 HIP 87062 X Sgr HIP 87788 Barnard Star W Sgr DQ Her WZ Sgr Y Sgr V603 Aql V1223 Sgr κ Pav FF Aql GJ748A RR Lyr XZ Cyg S Vul GJ1245A GJ1245C NGC6853 HIP 98492 WZ Sge GJ791.2A

parallax Ref. (mas)
0.586 ± 0.0438 5 0.438 ± 0.0469 5
11.12 ± 0.11 1 1.42 ± 0.16 1 76.41 ± 0.46 1
1.65 ± 0.3 1 15.5 ± 0.29 1 13.34 ± 0.33 1 2.88 ± 0.24 1 14.92 ± 0.53 6 14.84 ± 0.66 6 2.97 ± 0.34 1 768.7 ± 0.3 1
5.2 ± 0.7 1 60.53 ± 0.15 1 10.18 ± 0.21 1 19.12 ± 0.22 1 3.15 ± 0.56 6 5.07 ± 0.33 1 17.15 ± 0.14 1
3.8 ± 1 1 125 ± 0.3 1 1.71 ± 0.1 1 68.23 ± 0.38 6 65.1 ± 1.4 6 5.58 ± 0.53 6 8.21 ± 0.11 1
3 ± 0.18 1 10.83 ± 0.13 1 545.4 ± 0.3 1
2.28 ± 0.2 1 2.59 ± 0.21 1 0.512 ± 0.0373 5 2.13 ± 0.29 1 4.01 ± 0.14 1 1.96 ± 0.18 1 5.57 ± 0.28 1 2.81 ± 0.18 1
98.4 ± 0.3 1 3.77 ± 0.13 1 1.67 ± 0.17 1 0.322 ± 0.0396 5 219.9 ± 0.5 1 219.9 ± 0.5 1 2.47 ± 0.16 1 3.49 ± 0.14 1 22.97 ± 0.15 1 113.4 ± 0.2 1

Ra (deg)
161.136160 165.566881 167.747457 174.485330 187.237102 188.727770 193.100317 196.425059 200.935556 206.392715 206.476952 207.229937 217.392321 218.712738 219.003258 224.954678 229.730587 230.478556 233.971000 235.757857 240.339103 246.046510 248.103953 248.575581 248.580692 252.586282 266.867655 266.890075 268.993631 269.448503 271.255132 271.876040 274.248822 275.345761 282.227705 283.759644 284.237527 284.561446 288.068738 291.365640 293.122787 297.099176 298.479759 298.481926 299.901564 300.139629 301.902433 307.454338

Dec (deg)
-57.565357 -64.262893 +6.417567 +67.329393 +8.424238 +37.628978 -29.248754 +18.017867 -41.741460 +79.397010 +79.387440 +7.959982 -62.676075 +25.166043 +9.745402 -22.014952
-1.592288 +33.935715 -14.220117 -10.934848 +5.393949 +48.350869 -83.903451 +57.167574 +57.174461 +46.650513
-8.781545 -27.830835 -16.411888 +4.739420 -29.580110 +45.859101 -19.075831 -18.860034 +0.584085 -31.163883 -67.233423 +17.360872 +2.884048 +42.783489 +56.388085 +27.286481 +44.412330 +44.412906 +22.721214 +9.352701 +17.703984 +9.689504

Source ID
5351161399793209984 5240441472232302848 3817965105665685504 1058066262817534336 3902745286187581312 1519860699806445184 6185040879503491584 3938156295111047680 6137049739573759872 1715299716278321408 1715287999607537408 3721961488404743040 5853498713190525696 1255095276181144320 1176209886733406592 6232043867720079616 4415515934099120768 1278589709364139520 6265476408553544320 6268770373590148224 4425854676297423104 1411178510887026048 5768557209320424320 1431176943768690816 1431176943768691328 1407718450873494784 4165370682239910144 4057701830728920064 4144902306908889600 4472832130942575872 4050309195613114624 2116226254706461568 4094784475310672128 4096107909387492992 4266547566124966912 6760253239457454592 6434564460631076864 4514145288240593408 4268226078065241600 2125982599343482624 2142052889490819328 2027971514401523456 2079074130463898624 2079073928612821760 1827256624493300096 4299974407538484096 1809844934461976832 1752805741531173632

parallax (mas)
0.553 ± 0.017 0.378 ± 0.014 12.141 ± 0.020 1.332 ± 0.014 72.266 ± 0.696 3.311 ± 0.030 17.572 ± 0.017 13.731 ± 0.045 3.489 ± 0.060 16.572 ± 0.018 16.576 ± 0.018 2.844 ± 0.037 768.067 ± 0.050 5.155 ± 0.014 61.279 ± 0.043 10.671 ± 0.025 19.812 ± 0.034 6.187 ± 0.011 3.567 ± 0.031 16.267 ± 0.026 3.433 ± 0.014
1.838 ± 0.012 67.288 ± 0.034 67.354 ± 0.021 7.158 ± 0.017 8.718 ± 0.019 2.806 ± 0.140 10.760 ± 0.016 546.976 ± 0.040 2.365 ± 0.176 2.016 ± 0.017 0.574 ± 0.028 1.975 ± 0.058 3.106 ± 0.035 1.745 ± 0.024 5.245 ± 0.122 1.906 ± 0.071
3.985 ± 0.026 1.586 ± 0.015 0.205 ± 0.020
214.575 ± 0.048 2.570 ± 0.037 2.660 ± 0.018 22.104 ± 0.030

GoF
-2.35 2.08 2.60 0.35 201.12 1.86 2.65 1.21 12.95 11.67 4.20 -1.98 -1.20 1.59 5.14 3.22 8.15 1.39 2.31 1.31 0.75 775.27 0.00 9.09 0.36 2.42 -3.96 5.11 -7.45 1.89 35.76 1.50 -1.14 13.06 0.31 1.66 37.63 1.51 1152.54 0.93 4.09 1.13 1646.33 15.35 5.53 -4.13 1.29 1078.85

RUWE
0.92 1.07 1.12 1.01 21.31 1.07 1.08 1.04 1.66 1.51 1.17 0.93 0.97 1.06 1.31 1.16 1.43 1.04 1.11 1.06 1.03
1.00 1.42 1.01 1.09 0.88 1.22 0.70 1.08 3.95 1.06 0.94 1.76 1.01 1.08 2.29 1.05
1.04 1.17 1.03
1.61 1.16 0.85 1.04

G (mag)
7.338 ± 0.009 8.941 ± 0.009 11.131 ± 0.003 9.784 ± 0.012 10.866 ± 0.006 14.059 ± 0.003 13.246 ± 0.008 15.929 ± 0.004 15.731 ± 0.106 12.363 ± 0.004 14.486 ± 0.003 15.467 ± 0.052 8.985 ± 0.003 12.453 ± 0.003 7.181 ± 0.003 8.391 ± 0.003 6.813 ± 0.003 13.044 ± 0.003 13.603 ± 0.003 7.036 ± 0.003 13.146 ± 0.003 9.254 ± 0.003 9.536 ± 0.010 11.491 ± 0.003 14.849 ± 0.003 14.547 ± 0.003 10.352 ± 0.003 4.327 ± 0.008 11.095 ± 0.003 8.194 ± 0.003 4.585 ± 0.019 14.589 ± 0.011 7.682 ± 0.016 5.475 ± 0.012 11.867 ± 0.013 13.029 ± 0.006 4.263 ± 0.010 5.171 ± 0.005 9.886 ± 0.010 7.619 ± 0.015 9.914 ± 0.006 8.152 ± 0.007 11.535 ± 0.003 11.908 ± 0.003 14.037 ± 0.003 11.373 ± 0.003 15.181 ± 0.004 11.485 ± 0.003

(GBP − GRP) (mag)
1.511 ± 0.038 1.549 ± 0.043 0.990 ± 0.005 0.622 ± 0.050 2.781 ± 0.006 -0.283 ± 0.009 0.416 ± 0.034 0.021 ± 0.014 0.232 ± 0.475 2.451 ± 0.012 2.741 ± 0.007 0.066 ± 0.208 3.805 ± 0.006 0.911 ± 0.005 1.164 ± 0.005 0.794 ± 0.005 0.690 ± 0.005 1.663 ± 0.009 -0.153 ± 0.006 0.759 ± 0.005 0.849 ± 0.005 2.401 ± 0.007 0.706 ± 0.043 2.924 ± 0.005 0.685 ± 0.005 2.585 ± 0.008 0.889 ± 0.005 1.098 ± 0.029 0.915 ± 0.005 2.834 ± 0.005 1.108 ± 0.078 0.462 ± 0.055 1.779 ± 0.065 1.166 ± 0.050 0.172 ± 0.060 0.135 ± 0.026 0.915 ± 0.041 1.056 ± 0.018 2.624 ± 0.014 0.574 ± 0.062 0.639 ± 0.031 2.231 ± 0.033 3.701 ± 0.007 3.823 ± 0.005 -0.541 ± 0.005 0.898 ± 0.005 0.170 ± 0.017 3.168 ± 0.006

A&A proofs: manuscript no. GAIA_ZPOﬀset_nCom Article number, page 18 of 18

Table 1. continued.

Identiﬁer
T Vul HIP 103269 HD 202206 GJ831A HIP 106924 SS Cyg HIP 108200 RU Peg PN DeHt5 HD 213307 δ Cep NGC7293 RZ Cep GJ 876 γ Cep

parallax Ref. (mas)
1.9 ± 0.23 1 14.12 ± 0.1 1 22.98 ± 0.13 1 125.3 ± 0.3 1 14.47 ± 0.1 1
8.3 ± 0.41 1 12.4 ± 0.09 1 3.55 ± 0.26 1
2.9 ± 0.15 1 3.65 ± 0.15 1 3.66 ± 0.15 1 4.64 ± 0.27 1 2.12 ± 0.16 1 214.6 ± 0.2 1 74.27 ± 0.12 1

Ra (deg) 312.867677 313.820146 318.740516 322.832981 324.813858 325.679037 328.821293 333.510604 334.890158 337.288645 337.292885 337.410790 339.805873 343.324111 354.835781

Dec (deg) +28.250482 +42.298456 -20.789745 -9.790965 +60.284865 +43.586222 +32.645318 +12.703148 +70.934134 +58.404113 +58.415208 -20.837167 +64.859351 -14.266689 +77.633125

Source ID
1857884212378132096 2065901676227318272 6832155218215202944 6894054664842632448 2203746967971153024 1972957892448494592 1946297900868982016 2727974767550030080 2229624931896924160 2200153214212849024 2200153454733285248 6628874205642084224 2211629018927324288 2603090003484152064 2281778105594488192

parallax (mas)
1.688 ± 0.058 13.960 ± 0.012 21.939 ± 0.028
15.019 ± 0.012 8.854 ± 0.030 12.373 ± 0.014 3.662 ± 0.021 2.982 ± 0.036 3.454 ± 0.051 3.555 ± 0.147 5.012 ± 0.044 2.401 ± 0.012 214.038 ± 0.036 72.517 ± 0.147

GoF
5.28 -1.26 0.82 1186.79 -5.10 33.77 3.43 5.72 5.46 9.58 31.03 -1.02 -0.29 6.49 37.24

RUWE
1.20 0.95 1.03
0.79 2.33 1.09 1.29 1.22 1.42 2.71 0.95 0.99 1.34 3.21

G (mag)
5.500 ± 0.010 10.091 ± 0.003 7.922 ± 0.003 10.471 ± 0.003 10.155 ± 0.003 11.671 ± 0.014 10.783 ± 0.003 12.334 ± 0.012 15.462 ± 0.003 6.300 ± 0.003 3.851 ± 0.014 13.459 ± 0.003 9.294 ± 0.009 8.875 ± 0.003 2.943 ± 0.003

(GBP − GRP) (mag)
0.824 ± 0.041 0.844 ± 0.005 0.848 ± 0.005 3.083 ± 0.008 0.871 ± 0.005 1.177 ± 0.078 0.932 ± 0.005 0.964 ± 0.058 -0.304 ± 0.006 -0.025 ± 0.005 0.971 ± 0.057 -0.588 ± 0.006 0.740 ± 0.039 2.809 ± 0.005 1.257 ± 0.022

Notes. Column 1. Identiﬁer. Column 2. Trigonometric parallax with error. Column 3. References for the parallax. 1=Benedict et al. (2017), 2=Bond et al. (2018), 3=Groenewegen (2018), 4=Riess et al. (2014), 5=Riess et al. (2018a), and 6=van Belle et al. (2020), Column 4,5. Ra and Declination from GEDR3. Stars not in GEDR3 have coordinates listed between parentheses. Column 6. source identiﬁer from GEDR3. Column 7. parallax with error from GEDR3. Column 8,9. Goodness-of-ﬁt and Renormalised Unit Weight Error. Column 10,11. G magnitude and (GBP − GRP) colour.