zotero-db/storage/V2T2AHJF/.zotero-ft-cache

A&A 649, A4 (2021) https://doi.org/10.1051/0004-6361/202039653 © ESO 2021
Gaia Early Data Release 3

AA&ssttrroonpohymsyics
Special issue

Gaia Early Data Release 3
Parallax bias versus magnitude, colour, and position
L. Lindegren1, U. Bastian2, M. Biermann2, A. Bombrun3, A. de Torres3, E. Gerlach4, R. Geyer4, J. Hernández5, T. Hilger4, D. Hobbs1, S. A. Klioner4, U. Lammers5, P. J. McMillan1, M. Ramos-Lerate6, H. Steidelmüller4, C. A. Stephenson7, and F. van Leeuwen8
1 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Sweden e-mail: lennart@astro.lu.se
2 Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany
3 HE Space Operations BV for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
4 Lohrmann Observatory, Technische Universität Dresden, Mommsenstraße 13, 01062 Dresden, Germany 5 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino bajo del Castillo, s/n, Urbanizacion Villafranca
del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain 6 Vitrociset Belgium for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo,
Villanueva de la Cañada, 28692 Madrid, Spain 7 Telespazio Vega UK Ltd for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo,
Villanueva de la Cañada, 28692 Madrid, Spain 8 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Received 12 October 2020 / Accepted 9 December 2020
ABSTRACT Context. Gaia Early Data Release 3 (Gaia EDR3) gives trigonometric parallaxes for nearly 1.5 billion sources. Inspection of the EDR3 data for sources identiﬁed as quasars reveals that their parallaxes are biased, that is, they are systematically offset from the expected distribution around zero, by a few tens of microarcseconds. Aims. We attempt to map the main dependences of the parallax bias in EDR3. In principle, this could provide a recipe for correcting the EDR3 parallaxes. Methods. Quasars provide the most direct way for estimating the parallax bias for faint sources. In order to extend this to brighter sources and a broader range of colours, we used differential methods based on physical pairs (binaries) and sources in the Large Magellanic Cloud. The functional forms of the dependences were explored by mapping the systematic differences between EDR3 and DR2 parallaxes. Results. The parallax bias is found to depend in a non-trivial way on (at least) the magnitude, colour, and ecliptic latitude of the source. Different dependences apply to the ﬁve- and six-parameter solutions in EDR3. While it is not possible to derive a deﬁnitive recipe for the parallax correction, we give tentative expressions to be used at the researcher’s discretion and point out some possible paths towards future improvements. Key words. astrometry – parallaxes – methods: data analysis – space vehicles: instruments – stars: distances

1. Introduction The (early) Third Gaia Data Release (hereafter EDR3; Gaia Collaboration 2021b) provides trigonometrically determined parallaxes for approximately 1.478 billion sources in the magnitude range G 6–21. The sources include stars, unresolved binaries, compact extragalactic objects such as active galactic nuclei (AGNs), and other objects that appear roughly pointlike at the angular resolution of Gaia (∼0.1 arcsec). Although Gaia in principle determines absolute parallaxes (e.g. Lindegren & Bastian 2011), imperfections in the instrument and data processing methods inevitably result in systematic errors in the published parallax values. For example, it is well known that the parallax solution is degenerate with respect to certain variations of the ‘basic angle’ between the viewing directions of the two

Gaia telescopes (Butkevich et al. 2017). This means that based on the Gaia astrometric observations alone, we cannot simultaneously determine absolute parallaxes and calibrate this particular perturbation of the instrument. Conversely, when the actual instrument perturbations contain a component of this form, it will produce biased parallax values in the astrometric solution.
Since the second release of Gaia data in April 2018 (DR2; Gaia Collaboration 2018b), a number of investigations have been published that have resulted in estimates of the parallax systematics in DR2 from a variety of astrophysical objects (see e.g. Chan & Bovy 2020, and references therein). Reported values of the Gaia DR2 ‘parallax zero-point’ (i.e. the quantity to be subtracted from the DR2 parallaxes) range from about −30 µas to −80 µas. While quasars in DR2 yield a median parallax of −29 µas (Lindegren et al. 2018), the cited studies, which typically

Article published by EDP Sciences

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A&A 649, A4 (2021)

use much brighter objects, tend to give more negative values. There is consequently a strong suspicion that the parallax offset in DR2 depends on the magnitude, and possibly also on the colour of the sources (e.g. Zinn et al. 2019). Furthermore, it is known that Gaia DR2 has position-dependent offsets of the parallaxes on angular scales down to ∼1 deg (Arenou et al. 2018, Lindegren et al. 2018). The systematic offset of parallaxes in DR2 could therefore be a complicated function of several variables x, including at least the magnitude, colour, and position of the object; in the following we write this ZDR2(x).
In Gaia EDR3 quasars have a median parallax of about −17 µas, and already a simple plot of the parallaxes versus the G magnitude or colour index reveals systematic variations at a level of ∼10 µas (see Sect. 4.1). Position-dependent variations are also seen on all angular scales, although with smaller amplitudes than in DR2 (Lindegren et al. 2021). Thus it is possible to deﬁne an offset function ZEDR3(x) that is in general different from ZDR2(x), although it may depend on the same variables x.
The letter Z adopted for these functions is a mnemonic for ‘zero-point’. What is meant is an estimate of the bias (or systematics) of the parallax estimate as a function of certain known variables x. The practical determination of this function is fraught with difﬁculties and uncertainties, and even if we knew the true parallax of every object in the catalogue, it would not be possible to deﬁne a unique function Z(x) short of tabulating the bias for every source. The function will necessarily depend on, for example, the choice of arguments in x and on their resolution and numerical representation. At best, we can achieve a prescription such that the use of i − Z(xi) for the parallax of source i, instead of the catalogue value i, will in most cases give more accurate and consistent results in astrophysical applications.
The aim of this paper is to map some of the main dependences of the Gaia EDR3 parallax bias (zero-point), as found in the course of the internal validations carried out by the Gaia astrometry team prior to the publication of the data. Because the parallax determinations in Gaia EDR3 are of two kinds, known as ﬁve- and six-parameter solutions, with distinctly different properties, the results are given in the form of two functions Z5(x) and Z6(x) describing the bias as a function of magnitude, colour, and position for each kind of solution. In the following, the functions Z5 and Z6 always refer to EDR3, while for DR2 there is only one kind of solution, with the bias function ZDR2.
Section 2 is a brief overview of some aspects of the Gaia instrument and data processing that are particularly relevant for the bias functions, such as the different observing modes depending on the magnitude of the source, and the use of colour information in the ﬁve- and six-parameter solutions. In Sect. 3 we discuss the systematic differences between DR2 and EDR3 parallaxes. While the DR2 parallaxes are completely superseded by the later release, the differences could give important clues to the systematic dependences. In Sect. 4 we estimate Z5, the bias function for the ﬁve-parameter solutions in EDR3. In Sect. 5 we use a differential procedure to estimate Z6, the bias function for the six-parameter solutions. A limited test of the derived bias functions is made in Sect. 6. Some possible future improvements are discussed in Sect. 7 before the ﬁndings are summarised in Sect. 8. Certain technical details are provided in the appendices.
2. Observing and processing modes for the EDR3 astrometry
The determination of biases in Gaia astrometry must rely on a comparison with external data. In principle, this process requires

Fraction of AF observations

1:0

0:9

0:8

0:7

0:6

0:5

0:4

gated

WC0a

0:3

WC0b

0:2

WC1

WC2

0:1

0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Magnitude G [mag]

Fig. 1. Fraction of observations in different modes. The shaded area represents gated observations; the curves show the fraction of AF observations made in window classes WC0a, WC0b, WC1, and WC2.

no prior knowledge on how the astrometry was generated, but the mapping of possible dependences is surely facilitated by some acquaintance with certain aspects of the instrument and data processing. This section provides a brief overview of relevant topics.

2.1. Window classes and gates The Gaia instrument and its routine operations are described in Sects. 3 and 5 of Gaia Collaboration (2016). Further details of the initial treatment of the data and the subsequent astrometric processing can be found in Rowell et al. (2021) and Lindegren et al. (2021). The astrometric results are derived almost exclusively from observations made with the 62 CCDs occupying the central part of the focal plane assembly, the so-called astrometric ﬁeld (AF; see Fig. 4 in Gaia Collaboration 2016). Of relevance here is that only small patches (‘windows’) of the CCD images around detected point sources are transmitted to the ground, and that different sampling schemes (window size, pixel binning, etc.) are used depending on the on-board estimate of the brightness of the source. Furthermore, to avoid pixel saturation for sources brighter than G 12 (where G is the magnitude in the Gaia passband; Evans et al. 2018), the CCD integration time may be reduced by means of gates (Crowley et al. 2016).
In principle, the different sampling schemes (known as window classes), and their different uses together with the gates, may require separate calibrations both in the initial treatment (image parameter determination, IPD) and in the astrometric solution. This need arises both because of subtle differences in the way the CCDs and associated electronics function depend on their mode of operation (Hambly et al. 2018), and because different data processing models are used, for example depending on whether the window contains a one- or two-dimensional image. Further complications are caused by unavoidable conﬂicts in the on-board resource management, for example overlapping and truncated windows.
In the EDR3 astrometric solution (Lindegren et al. 2021), distinct calibration models were used for observations in the four window classes designated WC0a, WC0b, WC1, and WC2. The ﬁrst two provide two-dimensional images with different gate usages, while WC1 and WC2 provide one-dimensional images (i.e. the pixels have been binned in the other dimension) of different sizes. For the IPD a similar division was made, but without the distinction between WC1 and WC2. Figure 1 shows the fraction of AF observations in each window class as a function

A4, page 2 of 31

L. Lindegren et al.: Gaia Early Data Release 3

of G. Clearly, the window classes map out distinct intervals in G, but the transitions around G = 11, 13, and 16 are slightly fuzzy because the decision on the sampling scheme is based on the on-board real-time estimate of G, which differs from the on-ground estimated mean magnitude used in the plot. The transition between WC0a and WC0b occurs gradually over a whole magnitude because it involves gating thresholds that are set individually for the CCDs.
In EDR3, the astrometric parameters for a given source are typically obtained by combining some 200–500 individual AF observations. Depending on the mean magnitude of the source, the observations will be distributed among the window classes and gates in proportion to the fractions shown in Fig. 1. This explains some of the magnitude-dependent effects described in later sections. As it turns out, it is also relevant whether the observations are gated or not. This creates another transition around G = 12, indicated by the shaded area in the diagram.

2.2. Five- and six-parameter solutions, and the use of colour information

For a source in EDR3 that has a parallax, the astrometric param-

eters were determined either in a ﬁve-parameter solution or in a

six-parameter solution. The choice of solution depends on the

availability of colour information, in the form of an effective

wavenumber νeff, at the time when the IPD is performed. The

effective wavenumber comes from the processing of BP and RP

spectra in the photometric pipeline (Riello et al. 2021; De Angeli

et al., in prep.) In IPD, a calibrated point-spread function (PSF)

or line-spread function (LSF) is ﬁtted to the CCD samples in a

window, yielding the accurate location and ﬂux (in instrumental

units) of the image in the pixel stream. In a previous step, the

PSF and LSF were calibrated as functions of several parameters,

including window class and effective wavenumber.

A ﬁve-parameter solution is computed if the source has a

reliable value of νeff that can be used to select the appropriate

PSF or LSF for the IPD. This means that most of the colour-

dependent image shifts, caused by diffraction phenomena in

the telescopes and electronic effects in the CCDs, have been

eliminated already in the data input to the astrometric solution

(AGIS). In AGIS, the image locations for a given source are ﬁt-

ted by the standard astrometric model with ﬁve unknowns, α,

δ, , µα∗, and µδ (Lindegren et al. 2012). These parameters (for some nearby stars, also the spectroscopic radial velocity) are suf-

ﬁcient to describe the astrometric observations of well-behaved

sources such as single stars and quasars.

When a source does not have a reliable νeff at the time of

the IPD, it will instead obtain a six-parameter solution in AGIS,

where the additional (sixth) parameter is an astrometric estimate

of νeff known as the pseudo-colour, denoted νˆeff. At the IPD

stage, image locations and ﬂuxes for the source are determined

by ﬁtting the PSF or LSF at the default effective wavenumber

fνaedifefnft=s1o.u4r3ceµsmth−1a.t

This value is make up the

close to majority

the mean νeff for the of sources without a

piwnhhoiemtroeamgνeeetfrfliocicsaνttehifofe.naTschttheuaautlsa(ebreuotfbuitahnsekenddoewifnanup)lrtcoocplooolruotriu.ornTihntoeIPνceoDfef f−rﬁecsνiueedflenftfst,

of proportionality is a property of the instrument, known as the

chromaticity; it varies with time and position in the ﬁeld, but

can be calibrated in the astrometric solution by means of stars

of known colours (for details, see Lindegren et al. 2021). Using

the calibrated chromaticity, corrections to the default colour

can be estimated for the individual sources, which gives their

pseudo-colours. The determination of pseudo-colour thus takes

advantage of the (extremely small) along-scan shifts of the image

centre versus wavelength, caused by optical wavefront errors and

other imperfections in the astrometric instrument.

The instrument chromaticity is calibrated in AGIS by means

of a special solution, using a subset of about eight million pri-

mary sources for which IPD was executed twice: once using

Fttrhooerphtahocettousemalseoterufifrceccpetrsioviceteiswssiapnvoges,nsaiubnmldeboetnorcceνoeufmfs,ipnkugntetohwbeondtheffraﬁouvmlet-vptahaleruaemsνpeeedtfeceff-r.

solutions (which are the published ones) and six-parameter solu-

tions; the latter are not published but used in Sect. 5 to derive

the systematic differences between the ﬁve- and six-parameter

solutions.

Because the parallax bias is colour dependent, the functions

Z5(x) and Z6(x) need to include a colour parameter among their arguments in x. Rather than using, for example, the colour index

GBP − GRP, which is not available for all sources, the colour parameter used here is the (photometric) effective wavenum-

ber νeff (nu_eff_used_in_astrometry) in Z5, and the (astrometric) pseudo-colour νˆeff (pseudocolour) in Z6. By deﬁnition, this colour

information is available for all sources with an astrometrically

determined parallax.

The use of colour information in IPD and AGIS has one

additional feature that signiﬁcantly affects the ﬁve-parameter

s1o.2lu4tiµomns−1w(iGthBPν−effG>RP

1.72 µm−1 (GBP−GRP 3.0). The calibration

0.14) of the

or νeff < PSF and

LSF versus colour is only made for the well-populated inter-

val 1.24 < νeff < 1.72 µm−1, where a quadratic variation of the displacement with νeff is assumed; if the IPD requests a

PSF or LSF for a wavenumber outside of this range, then the

calibration at the nearest boundary is used. This ‘clamping’

of the wavenumbers guarantees that the LSF or PSF model

used by the IPD is always sensible, but on the downside, it

introduces some biases for sources of extreme colours. In the

astrometric calibration model, the chromaticity is assumed to

be linear over the entire range of wavenumbers. The combina-

tion of the two models may produce astrometric biases that are

strongly non-linear in wavenumber, and there could in particu-

lar be discontinuities in ∂Z5/∂νeff at νeff = 1.24 and 1.72 µm−1.

As no clamping is used for sources with six-parameter solu-

tions, such abrupt changes in slope are not expected for Z6 versus

pseudo-colour, but the relation can still be non-linear from other

effects.

Because νeff is computed from the detailed BP and RP spec-

tra, while GBP − GRP depends on the ratio of the integrated ﬂuxes in the two bands, there is no strict one-to-one relation between

the two quantities. Nevertheless, for −0.5 ≤ GBP − GRP ≤ 7 the following simple formulae

νeff

1.76 −

1.61 π

atan

0.531(GBP − GRP)

µm−1 ,

(1)

GBP − GRP

1 0.531

tan

π 1.61

(1.76

−

νeff)

mag

(2)

represent the mean relation for stellar objects to within ±0.007 µm−1 in the effective wavenumber (Lindegren et al. 2021). We note that the values of νeff used in EDR3 actually come from the spectrophotometric processing of the previous cycle, corresponding to the DR2 photometry. This is a consequence of the overall cyclic processing scheme of DPAC (Fabricius et al. 2016), in which the IPD is one of the ﬁrst processes in a cycle and the photometric processing is further downstream.

A4, page 3 of 31

A&A 649, A4 (2021)

30
a 25

Parallax uncertainty ¾5($) [¹as]

20

15

10

5

0

0:3
b 0:2

Correlation coe±cient ½6($; º^e®)

0:1

0

¡0:1

¡0:2

¡0:3

Smoothed centred quasar parallax; $ + 17 ¹as

80

c

60

40

20

0

¡20

¡40

¡60

¡80

Fig. 2. Celestial maps in ecliptic coordinates of some selected statistics in Gaia EDR3. Panel a: mean parallax uncertainty for sources with ﬁve-parameter solutions and G < 14 mag. Panel b: mean correlation coefﬁcient between parallax and pseudo-colour for sources with six-parameter solutions. Panel c: smoothed median parallax of quasars, corrected for the global median offset of −17 µas. Owing to the small number of quasars identiﬁed at small Galactic latitudes, no data are shown for | sin b | < 0.1. The maps use a Hammer–Aitoff projection in ecliptic coordinates with λ = β = 0 at the centre, ecliptic north up, and ecliptic longitude λ increasing from right to left. All three statistics exhibit some degree of systematic dependence on ecliptic latitude, which may be symmetric (as in a) or antisymmetric (as in b and c), but also even larger variations as functions of both coordinates.

2.3. Scanning law The two telescopes in Gaia are continuously scanning the celestial sphere according to a pre-deﬁned schedule known as the scanning law. Details are given in Sect. 5.2. of Gaia Collaboration (2016). For thermal stability, it is necessary that the spin axis is kept at a ﬁxed angle (45◦) to the Sun at all times, and as a consequence, the scanning pattern is roughly symmetric about the ecliptic, at least in a statistical sense and after several years of observations. Thus, although equatorial (ICRS) coordinates are used throughout the processing and for the astrometric end products, many characteristics of the data are (approximately) aligned with ecliptic coordinates rather than equatorial.
Three examples of the ecliptic alignment are shown in Fig. 2. In the top panel a, the precision of the parallaxes is shown to depend systematically on the ecliptic latitude (β), with 50– 60% higher uncertainties along the ecliptic than near the ecliptic A4, page 4 of 31

poles. The middle panel b shows the mean correlation coefﬁcient between parallax and pseudo-colour in the six-parameter solutions. This correlation is systematically positive for β 45◦ and negative for β −45◦, which is relevant for the parallax bias because an offset in the assumed colours of the sources translates into a parallax bias that is proportional to the correlation coefﬁcient. Although this correlation coefﬁcient is only available for the six-parameter solutions, the correlation between the errors in colour and parallax exists also for sources with ﬁveparameter solutions. For example, if νeff is systematically too high in the ﬁve-parameter solutions, the correlations will produce a more positive parallax bias in the (ecliptic) northern sky than in the south. At intermediate latitudes the correlation coefﬁcient exhibits more complex (and larger) variations related to the scanning law. The bottom panel c is a smoothed map of quasar parallaxes, increased by a constant 17 µas to compensate for the global offset. Large regional variations are seen at middle latitudes, but a systematic difference between north and south is clearly visible for | β | 45◦. Although such a systematic could be produced by the correlation mechanism just described, several other explanations are possible.
The maps in panels a and b of Fig. 2 show substantial regional and local variations, especially for | β | 45◦. These features are related to the scanning law and the (still) relatively poor coverage of the ecliptic zone obtained in the 33 months of data used for the EDR3 astrometry (global coverage is optimised for a mission length of 60 months). It can be expected that the parallax bias has regional and local variations of a similar character, and the map of quasar parallaxes (panel c) seems to conﬁrm this, although the ﬁner details are made invisible by the smoothing, and small-number statistics contribute to the variations most clearly along the Galactic zone of avoidance. In practice, it is very difﬁcult to determine local or even regional variations in Z5 and Z6 with any degree of conﬁdence. Even if we trust (some of) the variations seen in the quasar map, they are probably only representative for the faint sources with colours similar to those of the quasars, that is, bluer than the typical faint Galactic stars.
Thus, while a detailed mapping of Z5 and Z6 versus position is not possible, we expect on theoretical grounds a global variation with ecliptic latitude, which is also seen empirically. The only positional argument in Z5 and Z6 is therefore taken to be the ecliptic latitude, β. In the Gaia archive this coordinate is given (in degrees) as ecl_lat. Alternatively, it can be computed to within ±43 mas using the formula

sin β = 0.9174820621 sin δ − 0.3977771559 cos δ sin α .

(3)

In this paper, all expressions involving the ecliptic latitude are written in terms of sin β.

3. Systematic differences between EDR3 and DR2 parallaxes
Although it is not our goal to study the parallax bias for Gaia DR2, that is, the function ZDR2(x), it is instructive to start this investigation by examining the systematic parallax difference between EDR3 and DR2, that is, the differential bias function

∆Z(G, νeff, β) = Z5(G, νeff, β) − ZDR2(G, νeff, β) .

(4)

In contrast to either Z5 or ZDR2, this difference can easily be mapped in considerable detail simply by computing mean differences in parallax for sources having the same identiﬁer (source_id) in the two releases. (We ignore here the evolution

L. Lindegren et al.: Gaia Early Data Release 3

of the source list between releases, which implies that a small

fraction of the sources in EDR3 may be found in DR2 under a

different source_id; see Gaia Collaboration 2021b for details.)

Obviously, ∆Z contains no direct quantitative information on

Z5, but it is reasonable to expect that it gives a good qualitative indication of the relevant dependences on G, νeff, and β. The deﬁnition of the general parametrised function Z(G, νeff, β) in Appendix A is largely guided by the shape of ∆Z.

In Eq. (4) we use Z5 rather than Z6 because there are too few sources with six-parameter solutions for G < 13 that

also appear in DR2. The colour argument is νeff given by nu_eff_used_in_astrometry in EDR3, which is available for all

ﬁve-parameter solutions.

Figure 3 shows the mean parallax difference EDR3 − DR2 for sources with ﬁve-parameter solutions in EDR3, subdivided

by colour. The top panel shows the dependence on magnitude,

while the middle and bottom panels show examples of the depen-

dence on β in two different magnitude intervals. In Fig. 4 we

similarly show the mean difference as a function of colour, but

subdivided by magnitude and ecliptic latitude. (In this and all

other diagrams with effective wavenumber or pseudo-colour on

the horizontal axis, the direction of the axis is reversed to follow

the usual convention for colour-magnitude diagrams, that is, blue

objects to the left and red to the right.) These ﬁgures and all sub-

sequent results on ∆Z were computed using all common sources

for G < 14 and a geometrically decreasing random fraction of

the fainter sources. In total about 26.7 million sources were used.

Mean values of the parallax differences were computed using

weights The

proportional to plots in Figs. 3

1/(σ2 and 4

,sEhDoRw3

+coσm2p, lDeRx2

). dependences

on

G,

νeff, and β, with interactions among all three arguments (i.e. the colour dependence is different depending on both G and β, etc.).

We use these results to design a continuous, multi-dimensional

function, relevant parts of which are used in Sects. 4 and 5 to rep-

resent the functions Z5 and Z6 ﬁtted to the EDR3 data. Owing to the scarcity of data available for these ﬁts, the general function

has to be quite schematic and cannot take many of the details

seen in ∆Z into account, especially for the brightest sources.

With this purpose in mind, the following features are noteworthy.

When ∆Z is considered as a function of G, there are jumps

at G 11.0, 12.0, 13.0, and 16.0, corresponding to the bound-

aries shown in Fig. 1. The transitions are not abrupt, but occur

over an interval of 0.2–0.4 mag. Features at G 9 are ignored

in the following because the number of sources is too small for a

reliable mapping of the parallax bias. Considering ∆Z as a func-

tion of νeff, the effect of the clamping at 1.24 and 1.72 µm−1 (Sect. 2.2) is visible in some plots as an abrupt change in slope;

within these limits, the relation is approximately linear for 1.48 <

νeff < 1.72 µm−1 and with no visible break

curved at 1.48

(cubic) for µm−1. The

1.24 hook

< νeff < 1.48 µm−1, for the reddest stars

(νeff < 1.18 µm−1) seen for G < 16 mag is not related to any known feature of the instrument or data processing, and because

it concerns very few objects, it is ignored in the following.

Finally, as a function of β, ∆Z typically shows an approximately

linear or quadratic dependence on sin β.

The parametrised function described in Appendix A is

designed to take these features into account. Using this form,

the differential bias is approximated by

∆Z(G, νeff, β) =

q jk(G) c j(νeff) bk(β) ,

(5)

jk

where c j and bk are basis functions in νeff and β, speciﬁed by Eqs. (A.3) and (A.4), respectively, and q jk(G) are piecewise

70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60

50

40

30

20

10

0

¡10 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Magnitude G [mag]
70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60

50

40

30

20

10

0
G = 6 − 11

¡10 ¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0

0:2 0:4 0:6 0:8 1:0

sin ¯

70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60

50

40

30

20

10

0
G = 12 − 13

¡10 ¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0

0:2 0:4 0:6 0:8 1:0

sin ¯

Fig. 3. Mean difference in parallax between EDR3 and DR2. Top: parallax difference vs. magnitude. Middle and bottom: parallax difference vs. ecliptic latitude in two magnitude intervals. Circles connected by solid lines are weighted mean values computed in bins of variable size, with at least 1000 sources per bin; dashed lines are mean values of the ﬁtted parametrised function ∆Z (Eq. (5)), binned as for the sources. Filled red circles are for νeff < 1.48 µm−1, and open blue circles for νeff > 1.48 µm−1.

linear functions of G given by Eqs. (A.2) and (A.6) in terms of the ﬁtted coefﬁcients zijk in Eq. (A.1).
The approximation in Eq. (5) has 13 × 5 × 3 = 195 free parameters, namely all possible combinations of i = 0 . . . 12, j = 0 . . . 4, and k = 0 . . . 2 in the ﬁtted coefﬁcients zijk. A simultaneous least-squares estimation of all 195 parameters shows that many of them are poorly determined and contribute little to the overall ﬁt. To avoid overﬁtting, the following procedure is used.
A4, page 5 of 31

A&A 649, A4 (2021)

100

70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60 50
50
0 40

¡50

30

¡100

¡150
G = 6 − 11

¡200

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

100

20

10

0

G = 13 − 16

¡10

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60 50
50
0 40

¡50

30

¡100

¡150
G = 11 − 12

¡200

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

100

20

10

0

G = 16 − 18

¡10

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

70

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

Mean parallax di®erence $EDR3 ¡ $DR2 [¹as]

60 50
50
0 40

¡50

30

¡100

¡150
G = 12 − 13

¡200

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

20

10

0

G = 18 − 20

¡10

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

Fig. 4. Mean difference in parallax between EDR3 and DR2 as a function of effective wavenumber for several ranges of the G magnitude. Circles connected by solid lines are weighted mean values computed in bins of variable size, with at least 1000 sources per bin, and dashed lines are mean values of the ﬁtted parametrised function ∆Z (Eq. (5)), binned as for the sources. Filled red circles are for sources with β > 0, and open blue circles for β < 0. The vertical dashed lines mark the breakpoints for the basis functions c j(νeff) in Eq. (A.3), i.e. the clamping limits at 1.24 and 1.72 µm−1 and the midpoint at 1.48 µm−1.

First, all 195 parameters are estimated (zˆijk), along with their formal uncertainties (σijk). The parameter with the smallest standard score S ijk = |zˆijk|/σijk is then removed (set to zero), and a new ﬁt calculated with updated uncertainties. This procedure is repeated until S ijk > 2 for all the retained parameters. However, the parameters zijk with j = k = 0 are always retained at their estimated values, regardless of their scores, to avoid that the sum in Eq. (5) defaults to zero at some G independently of νeff and β.
This procedure applied to the EDR3–DR2 parallax differences yields the 137 non-zero coefﬁcients shown in Table 1.

For compactness, no uncertainties are given; all values are signiﬁcant at least on the 2σ level, although some, indicated by a colon in the table, are below 3σ. The precise values of the coefﬁcients, as well as the subset of signiﬁcant coefﬁcients, will depend on the selection of sources used in the ﬁt, which was increasingly non-exhaustive for G > 14. If we had used more of the faint sources in EDR3 and DR2, we would undoubtedly have obtained a better determination of the coefﬁcients towards the faint end, and would have increased the number of signiﬁcant coefﬁcients.

A4, page 6 of 31

L. Lindegren et al.: Gaia Early Data Release 3 Table 1. Coefﬁcients for the function ∆Z(G, νeff, β) ﬁtted to the difference in parallax between EDR3 and DR2.

G

q00

q01

q02

q10

q11

q12

q20

q21

q22

q30

q31

q32

q40

q41

q42

6.0 +25.61 +3.66 −4.55 +173.9 −37.1 −40.8 −3662 −1451 +3070 +1783.0

− −1299.0 +408.8 +121.7: −609.6

10.8 +33.31 −14.26 +9.32 +101.8 −41.8 −52.5 −5556

− +2780 +2477.6 −99.4 −1314.0 +82.7: −512.0 −312.1

11.2 +19.91 −18.28 +6.92 −73.5 −35.8 −43.4 −4844 −955 +1363 +1827.7 −219.8 −1146.8 −110.5:

−

−

11.8 +12.45 −11.50 +2.53 −138.5 −25.9 −14.0 −2706 −549 +1712 +1078.5 −192.5 −469.3

−

− −322.9

12.2 +37.20 −1.49 −2.63 −151.2 +8.0 +10.8 −3548 −904 +1642 +489.6 −142.6 −386.1 −258.7 −96.6: −229.1:

12.9 +31.36 −9.14

− −99.1 +12.2

− −2730 −492 +922 +705.6 −265.8 −290.1 −63.6:

−

−

13.1 +15.20 −2.45 +3.99 +17.1 −9.5 −16.6 −159 −1087 +949 −51.9 −67.6

−

−

− +281.3

15.9 +10.94 −0.60 +2.80 −11.0 +13.3 −21.6 +655 −116: +737

−

− −301.4

− −176.6

−

16.1 +10.86 −3.26 +2.11 −32.6 −19.5 −22.3 +688 −345 +1107

−

− −321.2

− −188.5

−

17.5 +10.27

− +3.20 −79.0

− −21.2: +1000

− +1441

− +177.5: −287.8:

−

− +518.3

19.0 +7.00 +4.40 +8.64 −45.6

−

− +1851

−

− −271.1:

−

− +356.4

−

−

20.0 +0.43: +5.55: +14.53: +115.3

−

− +3827

− −8769:

−

−

−

−

−

−

21.0 +12.34:

−

−

−

−

−

−

−

−

−

−

−

−

−

−

Notes. The table gives q jk(G) in Eq. (5) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A dash (–) indicates that the coefﬁcient is not signiﬁcant at the 2σ level and should be taken to be zero. A colon (:) after the coefﬁcient indicates that it is not signiﬁcant at the 3σ level. Units are µas (for q0k), µas µm (q1k, q3k, and q4k), and µas µm3 (q2k).

As a veriﬁcation of the ﬁt, the function ∆Z deﬁned by the

coefﬁcients in Table 1 was evaluated for each of the 26.7 million

sources used in the ﬁt, and mean values in bins of magnitude,

colour, and ecliptic latitude were computed in exactly the same

way as was done for the parallax differences. The results, shown

by the dashed lines in Figs. 3 and 4, thus represent our multidi-

mensional ﬁt to the mean parallax differences shown as circles in

the same diagrams. The overall ﬁt is reasonable, but we comment

on a few systematic deviations below.

First, we note that the mean parallax differences around

G 7.0 and 8.0 show large (10–15 µas) positive and neg-

ative excursions that are not modelled by the ﬁt (Fig. 3, top).

This discrepancy is a direct consequence of our choice to use

a linear dependence on G in the interval 6.0–10.8, which in

turn is motivated by the scarcity of data for estimating Z5 in

this interval (see Sect. 4.4.2). Furthermore, as a function of β

(Fig. 3, middle and bottom) there are local excursions from the

general quadratic trend on a level of ±5 µas. These are related

to localised features on the sky (cf. Fig. 2), whereas the ﬁtted

model is only meant to represent the position dependences on

the largest scales. Finally, as ation between 1.48 and 1.72

a function of µm−1 is not

νeff (Fig. linear, as

4), the variassumed in

Eq. (A.3), which sometimes gives deviations of 5–10 µas for

moderately blue sources (νeff 1.65 µm−1). In the red end the

curvature of the cubic segment is not always sufﬁcient to give a

good ﬁt around νeff 1.25 µm−1. These discrepancies could be caused by speciﬁc features

in DR2, EDR3, or both. It is of course also possible that the

two data sets have similar systematics that cancel in the par-

allax difference, and which therefore have not been identiﬁed

and included in the adopted model (Appendix A). Nevertheless,

as shown in the next sections, the model seems to be general

and ﬂexible enough to describe Z5 and Z6 to the level of detail permitted by the data.

4. Five-parameter solutions The main methods for estimating Z5(G, νeff, β) in this paper are (i) quasars (Sect. 4.1), which provide a direct estimate of the parallax bias for G 14, although only in a rather narrow interval of νeff; (ii) stars in the Large Magellanic Cloud (LMC; Sect. 4.2),

which map differential variations in the bias over a range of magnitudes and colours, although only in a speciﬁc location near the south ecliptic pole; and (iii) physical pairs or binaries (Sect. 4.4), which can be used to map the differential variations for the bright stars. Additionally, we use red clump (RC) stars for a differential study of the bias in the red, where the number of quasars is too small (Appendix C). In subsequent sections, these methods are developed in full detail.
In this process no assumption is made concerning the distance to the LMC, or the absolute magnitude of the RC stars. These objects are used purely differentially, and our estimates of the parallax biases are completely anchored in the quasars, that is, placed on an absolute scale by means of their parallaxes.

4.1. Quasars

As part of EDR3, the Gaia Archive1 contains the table

agn_cross_id, listing a total of 1 614 173 sources constituting a

very clean sample of quasar-like objects, whose positions and

proper motions in EDR3 formally deﬁne the reference frame of

the catalogue, Gaia-CRF3. The list was constructed by cross-

matching the full EDR3 catalogue with 17 external catalogues of

AGNs followed by a ﬁltering based on the quality of the solutions

and the astrometric parameters: the proper motions are consistent

with zero, and the parallaxes with a constant offset of −17 µas,

to within ﬁve times their respective formal uncertainties. Full

details of the selection procedure, and further characterisation

of the sample, are given in Gaia Collaboration (2021c). In spite

of the strict selection criteria, it is likely that the list contains a

small number of stellar contaminants. As described below, we

take this into account in a statistical way.

The quasar sample used here is a subset of agn_cross_id, con-

sisting of 1 107 770 sources with ﬁve-parameter solutions and

effective wavenumbers (νeff) in the range 1.24 to 1.72 µm−1. (For this investigation we used a preliminary version of agn_cross_id,

resulting in a slightly smaller sample than the correspond-

ing 1.59

selection from µm−1, with 1st

the and

published table.) 99th percentiles at

The 1.44

median and 1.69

νµemff −i1s,

which makes this sample signiﬁcantly bluer than typical stars of

1 https://archives.esac.esa.int/gaia

A4, page 7 of 31

A&A 649, A4 (2021)

similar magnitudes, and covering a smaller range of colours. The magnitudes range from G 13.4 to 21.0; only 97 are brighter than G = 15 and 541 are brighter than G = 16.
Figure 5 shows the weighted mean parallax plotted against G, νeff, and β. On the whole, there is a negative parallax bias: the weighted mean parallax of the sample used here is approximately −21 µas and the median is about −17 µas. For reference, these values are indicated by the red lines in Fig. 5. Apart from this offset, the main trends are as follows. (i) As a function of G, there is a clearly non-linear variation with an approximately linear increase from G 17 to 20, with plateaus on either side of this interval or perhaps a decreasing trend for G > 20. (ii) As a function of νeff, the variation is approximately linear in the well-populated range of colours. If there is a curvature similar to what is seen in the EDR3–DR2 differences (Fig. 4) or the LMC data (Fig. 10), the interval in νeff covered by the quasars is too narrow to reveal it. (iii) As a function of β, the variation may be described by a quadratic polynomial in sin β.
Similarly to what was observed in the EDR3–DR2 differences in Sect. 3, interactions among the three main variables are also visible in the quasar parallaxes. For example, if the quasars are binned by effective wavenumber and a quadratic polynomial a0 + a1 sin β + a2 sin2 β is ﬁtted to the parallaxes in each bin, it is found that a1 has a strong dependence on νeff, whereas no clear trend is seen for a2 (Fig. 6).
The trends described above, including interactions, are well approximated by the parametrised function Z(G, νeff, β) deﬁned in Appendix A. However, the limited supports in G and νeff make it necessary to constrain the general expression in Eq. (A.1) in several ways. Speciﬁcally, for the basis functions in magnitude, gi(G) in Eq. (A.2), we only use i = 6 . . . 12 (i.e. zijk is not ﬁtted for i < 6), and for the basis function in colour, c j(νeff) in Eq. (A.3), we only use j = 0 and 1. For G < 17.5 there are not enough quasars to reliably determine a linear variation with G, as permitted by the basis functions; instead we assume that the bias is independent of G in the intervals 13.1 < G < 15.9 and 16.1 < G < 17.5, but allow a step around G = 16 representing the transition from window class WC1 to WC2 (Sect. 2.1)2. Additionally, the coefﬁcients q12 were all found to be insigniﬁcant and were therefore constrained to zero. The resulting ﬁt is given in Table 2.
At this point it is necessary to consider to what extent the ﬁt is affected by a possible contamination of the quasar sample by Galactic stars. The contaminating stars will on average have higher measured parallaxes than the quasars of similar magnitude, thus biasing the ﬁtted function towards more positive values. The effect is only expected to be important at the faint end and where the star density is high. In order to explore this, we introduce the confusion factor

X = log10 D21 + 0.3(G − 21) ,

(6)

where in the

D21 is the vicinity of

mean density of sources the quasar, expressed in

brighter than G = 21 deg−2. Densities are

calculated by counting the total number of sources in EDR3 in

pixels of solid angle 0.8393 deg2 (healpix level 6), divided by

the solid angle. Because the density of faint sources in EDR3

rsoeucognhdlytderomubilnesEwq.it(h6)eaischthme aagpnpirtouxdiem(adtelocgh1a0nDgGe/idnGdens0it.y3)w, tihthe

G. Thus X is simply a convenient proxy for log10 DG, the density

2 Algorithmically, this is achieved by using the combined basis functions g6(G) + g7(G) and g8(G) + g9(G) instead of the original four functions, or equivalently, by adding the constraints z6, jk = z7, jk and z8, jk = z9, jk for all combinations of indices j and k.

A4, page 8 of 31

20
a
10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60

14

15

16

17

18

19

20

21

Magnitude G [mag]

20

b
10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber ºe® [¹m¡1]

20

c
10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60 ¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0

0:2 0:4 0:6 0:8 1:0

Sine of ecliptic latitude sin ¯

Fig. 5. Mean parallax of quasars binned by magnitude (panel a), effective wavenumber (panel b), and sine of ecliptic latitude (panel c). In each bin the dot is the mean parallax in EDR3 weighted by σ−2, with error bars indicating the estimated standard deviation of the weighted mean. The two red lines indicate the weighted mean value (−21 µas) and median (−17 µas) of the full sample.

of sources brighter than the quasar. For our quasar sample, X ranges from about 1.25 to 5.5, with a median at 3.4. Less than 1% of the quasars have X > 4.5.
When the mean quasar parallax is plotted versus X, there is a strong increase over practically the whole range of X values. This is not primarily caused by contamination, but rather by the positive trend of parallaxes versus G shown in Fig. 5a, transferred to X by the second term in Eq. (6). In Fig. 7 we plot instead

L. Lindegren et al.: Gaia Early Data Release 3

Table 2. Coefﬁcients for the function Z5(G, νeff, β) ﬁtted to the quasar parallaxes.

G 13.1–15.9 16.1–17.5
19.0 20.0 21.0

q00 −30.90 ± 4.51 −27.04 ± 1.73 −16.39 ± 1.54 −10.21 ± 1.91 −10.58 ± 5.16

q01 +8.50 ± 6.54 −0.76 ± 2.66 +3.82 ± 2.38 −4.05 ± 2.92 −15.57 ± 7.64

q02 −2.44 ± 4.35 −1.63 ± 1.77 −3.93 ± 1.79 −9.91 ± 2.64 −26.32 ± 8.83

q10 −15.4 ± 31.3 −17.2 ± 12.0 −15.1 ± 11.7 −12.9 ± 16.2 −37.0 ± 47.6

q11 +31.1 ± 45.3 +40.4 ± 18.4 +14.6 ± 17.9 +97.0 ± 25.0 +124.3 ± 71.8

Notes. The table gives q jk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. Units are µas (for q0k) and µas µm (q1k).

30

20

Mean contamination bias [¹as]

20

15

Regression coe±cient [¹as]

10

10

0

5

¡10

0

¡20

¡30

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

Fig. 6. Example of interactions among the dependences of quasar parallaxes on colour and ecliptic latitude. In a quadratic regression of parallax vs. sin β, the linear coefﬁcient (ﬁlled blue squares) exhibits a strong variation with effective wavenumber, while no such trend is shown by the quadratic coefﬁcient (open red circles). The points have been slightly displaced sideways to avoid overlapping error bars.

80

70

60

Weighted mean residual [¹as]

50

40

30

20

10

0

¡10

¡20

1:5

2:0

2:5

3:0

3:5

4:0

4:5

5:0

5:5

Confusion factor X

Fig. 7. Mean residual in quasar parallax after a regression on G, νeff, and β (see text), plotted against the confusion factor from Eq. (6). In each bin the dot is the mean residual weighted by σ−2, with error bars indicating the estimated standard deviation of the weighted mean. The broken dashed line is the dependence modelled by Eq. (7).

the residual in parallax from the regression in Table 2 versus the confusion factor. Here, the mean residual is close to zero for X 3.7, after which it increases roughly linearly with X as suggested by the dashed line. Based on this plot, we assume that the

Fig. 8. Celestial map of the mean contamination bias of the quasars, as given by Eq. (8) and Table 3. The map uses the same projection in ecliptic coordinates as the maps in Fig. 2.

contamination bias at a particular position is proportional to

f (X) = max(0, X − 3.7) ,

(7)

with X given by Eq. (6). A dependence on position is expected not only from the varying star density, as encoded in X, but also from variations in the quality of Gaia astrometry created by the scanning law (see e.g. several plots in Lindegren et al. 2021 and Fabricius et al. 2021). Globally, the precision and number of observations improve towards the ecliptic poles, which to ﬁrst order can be described as a linear dependence on sin2 β. We consequently model the contamination bias in the mean quasar parallax by adding the nuisance terms

r0(G) cos2 β + r1(G) sin2 β f (X)

(8)

to the ﬁtted model. Here, r0(G) and r1(G) represent the contamination bias at β = 0 and β = ±90◦, respectively; both functions are piecewise linear functions of G using the basis functions gi(G) in Eq. (A.2). Only the last two basis functions (i = 11 and 12) are used, as the r-coefﬁcients are quite insigniﬁcant for G 20.
The resulting ﬁt, including the contamination terms r0 and r1, is given in Table 3. Compared with Table 2, where the ﬁt did not include these terms, the global contamination bias (as shown by the difference in q00) is about +4 µas at G = 21.0 and below 1 µas at G = 20.0. Figure 8 is a map of the bias, calculated from Eq. (8) and averaged over the quasars at each location. The expected increase in bias towards the Galactic plane is very evident, but several features related to the different surveys contributing to the sample appear as well. These features probably reﬂect variations in the magnitude completeness, made visible through the steep increase in estimated bias towards G = 21.0.
Our best estimate of Z5(G, νeff, β) from the quasar sample is therefore given by the coefﬁcients q jk in Table 3. The coefﬁcients
A4, page 9 of 31

A&A 649, A4 (2021)

Table 3. Coefﬁcients for the extended function Z5(G, νeff, β, X) ﬁtted to the quasar parallaxes.

G 13.1–15.9 16.1–17.5
19.0 20.0 21.0

q00 −30.90 ± 4.51 −27.04 ± 1.73 −16.39 ± 1.54 −10.57 ± 1.97 −14.62 ± 5.33

q01 +8.50 ± 6.54 −0.76 ± 2.66 +3.82 ± 2.38 −4.04 ± 2.92 −15.58 ± 7.64

q02 −2.44 ± 4.35 −1.63 ± 1.77 −3.94 ± 1.79 −10.78 ± 2.82 −18.43 ± 9.43

q10 −15.4 ± 31.3 −17.2 ± 12.0 −15.1 ± 11.7 −11.2 ± 16.3 −29.0 ± 47.8

q11 +31.1 ± 45.3 +40.4 ± 18.4 +14.6 ± 17.9 +97.4 ± 25.0 +125.4 ± 71.8

r0 − − −
−1.66 ± 8.61 +111.02 ± 31.48

r1 − − −
+10.82 ± 9.89 −31.38 ± 35.50

Notes. The function Z5 corrected for contamination bias is obtained by setting r0 = r1 = 0. The table gives q jk(G) and rk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A dash (–) indicates that the coefﬁcient should be ignored (taken as zero). Units are µas (for q0k), µas µm (for q1k), and µas dex−1 (for rk).

r0 and r1 must be ignored, as they represent the contamination bias, which should not be included in Z5.
Several interesting observations can be made concerning

the results in Table 3. The coefﬁcient q00 represents the mean quasar parallax at the given magnitudes, averaged over the

celestial sphere and reduced to the reference wavenumber,

νeff = 1.48 µm−1. As described in Appendix A, linear interpolation between the tabulated values should be used for other

magnitudes. Its values agree rather well with the mean rela-

tion displayed in Fig. 5a. Of the remaining coefﬁcients, the

most important ones (in terms of how much they decrease the

chi-square of the ﬁt) are q02 and q11. q02 describes a quadratic dependence on sin β; the consistently negative sign means that

the parallax bias is more negative towards the ecliptic poles, and

this effect is strongest at the faint end. The interaction coefﬁ-

cient q11 is consistently positive and increases with magnitude; in terms of the chi-square, it is much more important than the

corresponding simple coefﬁcients quasar colour, νeff = 1.59 µm−1, q11

q01 and q10. At describes a clear

the median north–south

asymmetry of the parallaxes, which is what is seen in Fig. 5c. At

the reference wavenumber 1.48 µm−1 the asymmetry, given by

q01, is smaller and less consistent. The coefﬁcients q10 represent the colour gradient averaged over the celestial sphere. They

are all consistent with a mean value of −15 µas µm, which is

much smaller than the slope −55 µas µm indicated by Fig. 5b.

This apparent contradiction is explained by a correlation between

magnitude and colour in the quasar sample: the faint quasars are

on average redder than the brighter ones; together with the over-

all trend in Fig. 5a, this creates a stronger variation with colour

in the sample as a whole than is present at a ﬁxed magnitude.

Together with the strong variation of q02 and q11 with magnitude, this illustrates the many complex dependences in the data

and the difﬁculty of determining a unique function Z5 based on a limited sample with intrinsic correlations. It also shows the dan-

ger of using simple plots of the quasar parallaxes versus a single

quantity such as colour for inferences on the parallax bias.

While Table 3 (ignoring r0 and r1) in principle deﬁnes Z5 for any combination of arguments G, νeff, and β, it is in practice only valid in the subspace of the arguments that is well populated by

the quasars. Most importantly, this does not include sources that

are brighter than G 14, redder than νeff 1.48, or bluer than νeff 1.72. In order to extend Z5 in these directions, we resort to differential methods using physical pairs and sources in the

LMC.

4.2. Large Magellanic Cloud The distance modulus of the LMC, (m − M)0 = 18.49 ± 0.09 mag (de Grijs et al. 2014), corresponds to a parallax in the range 19.2

to 20.9 µas. The depth and inclination of the system mean that

the parallaxes of individual stars have a true dispersion (and gra-

dient) of about one µas. For our analysis we did not assume any

speciﬁc mean distance to the LMC, only that the selected mem-

ber sources have the same (but unknown) parallax, regardless of

their colours and magnitudes. The LMC data can therefore be

used to map the bias function Z5(G, νeff, β) at the position of the LMC, β −85◦, up to an unknown additive constant.

The selection of sources in the LMC area for the analysis in

this section is described in Appendix B. The sample consists of

more than two million sources from EDR3, brighter than G = 19

and located within a 5◦ radius of the LMC centre. As discussed

in the appendix, it is thought to be reasonably clean at least down

to G 18. A colour-magnitude diagram (CMD) of the sample is

shown in the left panel of Fig. 9.

In the right panel of Fig. 9 the same CMD is colour-coded by

the median parallax at each point of the diagram. The predomi-

nantly greenish colour shows that the overall median parallax is

close to zero, roughly consistent with the expected true parallax

of 20 µas and a global parallax bias around −20 µas. The dark red

patch at G > 18, νeff < 1.4 is the only part of the diagram dominated by foreground stars. Several systematic deviations from

the overall median are clearly visible, and cannot be attributed

to foreground stars. These include the rather sharp divisions at

G 13.0 and G 16.0, coinciding with the magnitude bound-

aries of window classes WC0b, WC1, and WC2 (Sect. 2.1 and

Fig. 1); a strong difference (or gradient) in the bias for G < 13.0

between the main-sequence (at νeff 1.7) and the red supergiants (at νeff 1.4); an up-turn of the bias for the bluest stars, which is stronger for G > 16.0 than for the brighter stars; and a

down-turn of the bias for the reddest stars, at least for G 14

to 16. For the faintest stars there appears to be a depression of

the bias at intermediate colours (νeff 1.55). The cause of this depression is not known.

The systematic variation of parallax with colour and mag-

nitude is further explored in Fig. 10, where each panel displays

a different magnitude interval. The black dots show the median

parallax binned by νeff. The three panels for G ≥ 13.1 show that the up-turn of parallax values for the bluest stars sets in abruptly

around tion in

νeff = 1.72 µm−1, which is clearly related the range of wavenumbers to 1.24 ≤ νeff ≤

to the restric1.72 µm−1 for

the LSF and PSF calibrations, as discussed in Sect. 2.2. At the

other (red) end of the interval no clear break is seen, although

only a few stars in the LMC are redder than 1.24 µm−1. Between

1.24 and 1.72 µm−1 the relation is approximately linear in the left

(bluer) half of the interval, but it is clearly non-linear in the right

(redder) half, at least for G > 13.

These variations can be described by the general model in

Appendix A, although several simpliﬁcations are needed for a

A4, page 10 of 31

L. Lindegren et al.: Gaia Early Data Release 3

Magnitude G [mag] Count
Magnitude G [mag] Median parallax [¹as]

10

5000

11

2000

1000 12
500

13 200

14

100

15

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16

20

10 17
5

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¡40 18
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Fig. 9. Colour-magnitude diagrams of the LMC sample. Left: diagram colour-coded by the number of sources at a given position in the diagram. Right: diagram colour-coded by the median parallax in EDR3.

Table 4. Function EDR3(G, νeff) ﬁtted to the LMC sample.

G 6.0–10.8 11.2–11.8
12.2 12.9 13.1 15.9 16.1 17.5 19.0

p0 +9.76 ± 2.83 −4.35 ± 1.45 +9.42 ± 1.23 +18.14 ± 1.03 −5.08 ± 0.47 −11.74 ± 0.22 −6.18 ± 0.23 −5.31 ± 0.20 −7.61 ± 1.24

p1 −5.3 ± 17.3 −147.7 ± 8.1 −147.6 ± 6.8 −85.1 ± 5.9 −24.7 ± 3.0 −19.1 ± 1.9 −15.6 ± 2.5 −22.5 ± 2.5 −20.3 ± 15.3

p2 − − − − −1450 ± 192 −1197 ± 70 −1409 ± 136 −1122 ± 325 −

p3 − − − − +103.2 ± 141.7 +244.2 ± 48.0 +103.8 ± 82.7 +196.6 ± 231.7 −

p4 − − − − +101.3 ± 10.8 +156.2 ± 7.5 +171.9 ± 10.3 +315.9 ± 11.3 +354.3 ± 76.6

Notes. The table gives p j(G) in Eq. (9) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A dash (–) indicates that the coefﬁcient should be ignored (taken as zero). Units are µas (for p0), µas µm (p1, p3, and p4), and µas µm3 (p2).

well-determined ﬁt. Most importantly, because the LMC sample only covers a small area near the south ecliptic pole, the dependence of parallax on β cannot be determined; the resulting ﬁt is valid at the mean position of the LMC, β −85◦ or sin β −0.996, but not necessarily at other locations. Modelled on Eqs. (A.5) and (A.6), but using only the basis functions in G (Eq. (A.2)) and νeff (Eq. (A.3)), the function ﬁtted to the EDR3 parallaxes of the LMC sample is

4

EDR3(G, νeff) = p j(G) c j(νeff) ,

(9)

j=0

where

10

p j(G) = yijgi(G)

(10)

i=0

are piecewise linear functions in G, and yij are free parameters of the model, estimated by a weighted least-squares procedure. Here and elsewhere in this paper, the least-squares estimation is made robust against outliers by iteratively removing points that deviate by more than four times a robust estimate of the (weighted) RMS residual among all the data points. Owing to the small number of sources at the bright end and their limited coverage in νeff, the functions p j(G) are forced to be constant in

each of the magnitude intervals 6.0–10.8 and 11.2–11.8 (using the device described in footnote 2), and the parameters with j ≥ 2 are set to zero for i ≤ 6. At the faint end, the ﬁt is limited to sources brighter than G = 18 in order to minimise contamination effects, and the basis functions gi(G) for i = 11 and 12, which lack support for G < 18 (see Fig. A.1), are omitted in Eq. (10). The resulting ﬁt is given in Table 4, where coefﬁcients assumed to be zero are indicated by a dash (–). The ﬁtted function, evaluated at representative magnitudes, is shown by the blue curves in Fig. 10.
In Table 4 the coefﬁcients p0 at the different magnitudes give the mean parallax of the LMC sample reduced to the reference wavenumber νeff = 1.48 µm−1. As they refer to the location of the LMC, it is not useful to compare them with the coefﬁcients q00 from the quasars (Table 3), which are averages over the celestial sphere. Because we wish to avoid that our analysis depends in any way on an assumed distance modulus to the LMC, the ﬁtted coefﬁcients p0 were not further used in the determination of Z5. The remaining coefﬁcients p1 through p4 map the variation in parallax bias at the LMC location as a function of νeff. For WC0 (G ≤ 12.9) only the mean gradient in wavenumber (p2) is determined and exhibits very signiﬁcant variations with magnitude; the major breaks at G 11 and 13 are clearly visible in Fig. 9 (right). For WC1 and WC2 (G ≥ 13.1), the most striking feature is the relative constancy of p1, p2, and p3 with magnitude: the tabulated values all agree to within ±2σ with

A4, page 11 of 31

A&A 649, A4 (2021)

60

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40

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Fig. 10. Median parallaxes for the LMC sample in Fig. 9 plotted against νeff in six magnitude intervals, as indicated in the diagrams. The orange points show the parallaxes of individual sources and give an impression of the coverage in νeff and scatter in parallax. The black dots, with error bars, show the median parallax and its uncertainty in bins of νeff with at least 20 sources per bin. The solid blue curves show the ﬁtted combination of basis functions in Eq. (9) evaluated for a representative magnitude (G = 10.0, 11.5, 12.5, 15.0, 17.0, and 18.0) in each interval.

their weighted mean values, which are p1 = − 20.0 ± 1.2 µas µm,

p2 = −1257 ± 58 µas µm3, and p3 = +200 ± 39 µas µm. On the

tohtheebrlhuaenedn,dth(eνecfof e>fﬁ1c.i7e2ntµsmp−4,1

describing the ), show a very

colour gradient at clear progression

with magnitude.

4.3. Combined ﬁt using quasars and LMC sources (G > 13) The quasar sample contains only a few sources redder than νeff 1.44 µm−1 or bluer than 1.69 µm−1, and therefore cannot be used to estimate q jk for j = 2, 3, and 4. The LMC sample, on the other hand, gives a good determination of p j for j = 1 . . . 4 in A4, page 12 of 31

the magnitude range 13–18, but only for the speciﬁc location of the LMC. In this section we attempt to combine the two datasets in order to extend the model to the full range of colours for G > 13. This is not possible without certain additional assumptions, detailed below; however, the distance to the LMC is still left as a free parameter.
From Eq. (A.4), using sin βLMC −0.996, it can be seen that the coefﬁcients in a combined model must satisfy

p j = q j,0 − 0.996 q j,1 + 0.659 q j,2 , j = 1 . . . 4 .

(11)

Only for j = 1 is a direct comparison possible; this is shown in Table 5 for G ≥ 13.1. Although the agreement between pL1MC

L. Lindegren et al.: Gaia Early Data Release 3

Table 5. Comparison of the parallax gradient in colour, p1, at the position of the LMC, as estimated from LMC data, quasars (QSO), and physical pairs (PP).

G

pL1 MC

pQ1 SO

pP1P

10.8 −5.3 ± 17.3

+23.3

11.2 −147.7 ± 8.1

−101.0

11.8 −147.7 ± 8.1

−163.1

12.2 −147.6 ± 6.8

−143.8

12.9 −85.1 ± 5.9

−93.4

13.1 −24.7 ± 3.0 −46.5 ± 63.5 −19.6

15.9 −19.1 ± 1.9 −46.5 ± 63.5

16.1 −15.6 ± 2.5 −57.5 ± 22.7

17.5 −22.5 ± 2.5 −57.5 ± 22.7

19.0 −20.3 ± 15.3 −29.7 ± 22.1

NfwrooitmthescT.oaepbfL1lﬁeMcCi7e.insVtstaalfkureoenms farTroeambelxeTpa3rbe.lsepseP14Pd. =pinQ1qS1µO0a−=s

q10 − 0.996 q11 + 0.659 q12 0.996 q11 with coefﬁcients µm and refer to the mean

gradient in the interval νeff = 1.48–1.72 µm−1. Where applicable, uncer-

tainties are ±1σ from the covariance matrix of the respective solution.

and with

pthQ1 eSOuniscemrtoadinetriaetse.,

the The

differences are roughly compatible main conclusion from the compari-

son is that the LMC data are vastly superior to the quasars for

estimating the gradient of the bias with colour at the location of

the LMC.

If a combined solution of the quasar and LMC data were

attempted, in which all the parameters q jk for G > 13 are freely

adjusted, the result would be a model that ﬁts both datasets

very well, and is very well determined in the LMC area, but

has very large uncertainties in most other locations. This would

obviously not be very useful. To proceed, we need to con-

strain the model. We boldly make the following assumptions for

WC1 and WC2 (G ≥ 13.1): (i) that the gradient with colour

in the moderately blue region (νeff = 1.48–1.72 µm−1) is fully

described by the interaction term q11, that is, q10 = q12 = 0; (ii) that the curvature with colour in the moderately red region

(νeff = 1.24–1.48 µm−1) is fully described by the non-interaction

ttehremreqd20e,ndth(aνtefifs,<q121.2=4qµ2m2 =−10);is(iifiu)lltyhadtesthcreibceodlobuyr

gradient q30, that

at is,

(qν3e1ff=>q312.7=20µ; man−d1)

(iv) that the colour is fully described by

gradient q40, that

at is,

the q41

blue = q42

end = 0.

Assumption (i) is consistent with the observation in Sect. 4.1

that q12 is generally insigniﬁcant and that q11 is far more impor-

tant than q10 for the overall chi-square. Assumption (ii) cannot be tested by means of the quasars, but is partially supported

by the test in Appendix C, in which the red clump stars show

similar curvatures at sin β = ± 0.86. Indirect support for (iii) is

provided by the EDR3–DR2 differences ∆Z analysed in Sect. 3,

for example the panels in Fig. 4 for G = 13–16 and 16–18, where

the changes in gradient at νeff = 1.24 µm−1 are not distinctly dif-

ferent in the two hemispheres. As DR2 did not use the clamping

of νeff at 1.24 and 1.72 µm−1 described in Sect. 2.2, it is likely

that any abrupt changes in the gradient with colour seen in ∆Z

at these wavenumbers are caused by the EDR3 data. The same

argument may be advanced in support of (iv), although the evi-

dence in Fig. 4 for breaks at νeff > 1.72 µm−1 in the northern hemisphere is very weak.

With these assumptions the coefﬁcients q jk are effectively

determined by the LMC data through the conditions in Eq. (11),

and it is possible to ﬁt both datasets with a single model. To

account for the offset between p0 and the parallax bias at the LMC location, one additional unknown must be introduced, representing the true mean parallax of the LMC; this is constrained to be independent of G. Results are given in Table 6, except for the ﬁtted LMC parallax, which is +22.11 ± 1.10 µas.

4.4. Physical pairs (G < 13) After mapping Z5(G, νeff, β) for G > 13 by means of the quasars and the LMC, we now turn to the brighter sources. In the LMC area the gross variation in parallax bias with colour was mapped in Sect. 4.2 (Table 4), but this local result must now be extended to the whole sphere. To this end, we used binaries (here called ‘physical pairs’), in which it can be assumed that the components have similar true parallaxes, although their magnitudes and colours may be very different. Using the results from previous sections to anchor the parallax bias of the fainter component among the quasars, it is then possible to estimate the bias of the brighter component. Details of the method are given in Appendix D, which also describes the selection of data used in the analysis below.

4.4.1. Results for G > 10

The method outlined in Appendix D was applied to pairs where

the magnitude of the bright component (G1) is in the range 10–

14 mag, and that of the faint component (G2) is in the range max(G1, 13.1)–G1 + 4 mag. The bias-corrected parallax of the

faint component was computed as 2 − Z5(G2, νeff2, β), where

2 and νeff2 are the published EDR3 parallax and effective wavenumber of the faint component, and Z5 is deﬁned by the

coefﬁcients in Table 6. The investigated magnitude interval over-

laps with Table 6 for G = 13.1–14, which provides a consistency

check and possibly improved estimates of the parallax bias in an

interval that is poorly covered by the quasars.

The LMC data show that signiﬁcant variations in bias as a

function of (at least) G and νeff exist for sources brighter than G 13. Owing to the relatively small number of available phys-

ical pairs, the analysis cannot be very complex but should still

include the main known or expected dependences. A severe lim-

itation is that the >1.6 µm−1 makes and q40. From the

scarcity of bright sources with νeff < 1.3 or

it practically impossible LMC data we have q20

to determine −1257 µas

qµ2m0,3qf3o0r,

G > 13.1, so that even though it cannot be usefully estimated

from the physical pairs in the magnitude interval 13.1–14, consis-

tency requires that the corresponding term is subtracted from the

parallax of the brighter component before the remaining parame-

ters are ﬁtted. This procedure indeed reduces the sum minimised

in Eq. (D.4), but not by a signiﬁcant amount. A similar improve-

ment is obtained by applying the same a priori correction for

G < 13.1. We therefore assumed that q20 −1257 µas µm3 throughout the range G < 14. Concerning q30 and q40 we

assumed that they are zero for G < 13.1. Noting that a ﬁt includ-

ing the six coefﬁcients q jk ( j ≤ 1, k ≤ 2) gives insigniﬁcant

results for q12 at all magnitudes (lower right panel of Fig. 11), we also assumed q12 = 0.

Figure 11 shows results for the remaining ﬁve coefﬁcients

q00, q01, q02, q10, and q11. The interaction of these terms with magnitude is fully mapped by independent ﬁts in 35 bins of G1

covering the interval 10 < G1 < 14. All ﬁve coefﬁcients show signiﬁcant variations with G, which in most cases can be related

to the boundaries discussed in Sect. 2.1.

The different symbols in Fig. 11 show the results from

using different intervals in ρ∆µ. The ﬁlled black circles are for

A4, page 13 of 31

A&A 649, A4 (2021) Table 6. Coefﬁcients for the extended function Z5(G, νeff, β, X) as obtained in a combined ﬁt to the quasar and LMC samples.

G

q00

q01

q02

q11

q20

q30

q40

r0

r1

13.1 −23.69 ± 5.84 +7.27 ± 5.16 +5.54 ± 12.11 +26.5 ± 2.8 −1475 ± 179 +107.9 ± 132.2 +104.3 ± 10.2

−

−

15.9 −38.33 ± 2.47 +5.61 ± 2.81 +15.42 ± 5.35 +18.7 ± 1.8 −1189 ± 65 +243.8 ± 44.5 +155.2 ± 7.0

−

−

16.1 −31.05 ± 1.78 +2.83 ± 2.09 +8.59 ± 3.95 +15.5 ± 2.3 −1404 ± 125 +105.5 ± 76.4 +170.7 ± 9.4

−

−

17.5 −29.18 ± 0.80 −0.09 ± 1.01 +2.41 ± 1.98 +24.5 ± 2.1 −1165 ± 299 +189.7 ± 214.2 +325.0 ± 9.5

−

−

19.0 −18.40 ± 0.64 +5.98 ± 1.33 −6.46 ± 1.73 +5.5 ± 10.0

−

−

+276.6 ± 55.3

−

−

20.0 −12.65 ± 0.99 −4.57 ± 3.02 −7.46 ± 2.97 +97.9 ± 25.8

−

−

−

+41.22 ± 10.57 −3.13 ± 10.48

21.0 −18.22 ± 3.38 −15.24 ± 8.01 −18.54 ± 10.50 +128.2 ± 74.0

−

−

−

+73.60 ± 30.84 −16.71 ± 31.30

Notes. The function Z5 corrected for contamination bias is obtained by setting r0 = r1 = 0. The table gives q jk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. r0 and r1 are the ﬁtted coefﬁcients of the contamination terms. A dash (–) indicates that the coefﬁcient should be ignored (taken as zero). Units are µas (for q0k), µas µm (q1k, q30, q40), µas µm3 (q20), and µas dex−1 (r0, r1).

Table 7. Coefﬁcients of Z5(G, νeff, β) as estimated from physical pairs for G > 10.

G
10.0–10.8 11.2 11.8 12.2 12.9
13.1–14.0

q00
−27.75 −31.24 −33.87 −13.37 −19.61 −37.99

q01
−1.43 −8.84 −7.33 +1.68 −0.68 +2.63

q02
+27.56 +8.40 +12.98 +7.82 +15.98 +16.14

q10
+33.3 −88.9 −135.6 −105.6 −68.1 −5.7

q11
+10.0 +12.1 +27.6 +38.4 +25.4 +14.0

q20
−1257 −1257 −1257 −1257 −1257 −1257

Notes. The functions q jk(G), obtained by linear interpolation in the table, are expressed in µas (q0k), µas µm (q1k), and µas µm3 (q20). Values for q20 are assumed as described in the text.

ρ∆µ < 2 arcsec mas yr−1, which gives the most precise estimates; the error bars are ±1σ uncertainties obtained by bootstrap resampling. The coloured symbols (open circles, triangles, and squares) are for non-overlapping intervals in ρ∆µ. The results for the different non-overlapping intervals in ρ∆µ are to a high degree statistically independent, which gives an additional indication of the uncertainty. (They are not completely independent, as systems with more than two components may appear in more than one interval of ρ∆µ.) The absence of any obvious trend in q00 with ρ∆µ suggests that contamination bias is negligible.
More reliable estimates of the coefﬁcients were obtained in a simultaneous ﬁt of all the parameters, using Eq. (D.4). Here, q jk were constrained to be piecewise linear functions of G with breakpoints at G = 10.8, 11.2, 11.8, 12.2, 12.9, and 13.1 (see Appendix A). For numerical stability, we required that q jk are constant for G < 10.8 and >13.1. The resulting ﬁt is given in Table 7 and is shown by the dashed blue lines in Fig. 11.
The coefﬁcients in the last row of Table 7 are in excellent agreement with the joint quasar and LMC results (Table 6) at G = 15.9, while the agreement is poorer at G = 13.1, where the coefﬁcients in Table 6 are considerably more uncertain. In Table 8 we joined the two datasets by adopting the quasar/LMC results for G ≥ 15.9 and the results from the physical pairs for G ≤ 13.1, but taking q30 and q40 at G = 13.1 from Table 6 as they could not be determined from the pairs.
4.4.2. Extending the analysis to G = 6 The previous analysis of physical pairs did not reach brighter than G 10 owing to the restrictions G2 − G1 < 4 mag and G2 > 13.1, where the latter condition came from the necessity A4, page 14 of 31

of using the bias function from Sect. 4.3 (valid for G > 13.1) to correct the parallax of the faint component. As a consequence, the number of available pairs rapidly decreased towards the bright end, not only because of the general scarcity of bright stars, but also because a decreasing fraction of them have faint components in the required range.
Using the coefﬁcients in Table 7 to deﬁne a provisional parallax bias for G > 10, we can now extend the analysis to pairs as bright as G1 6 by removing the second constraint, that is, by including all pairs with G1 < G2 < G1 + 4 mag. Not only does this extend the analysis to G > 6, but the results for G > 10 are also improved by the many more pairs included. For example, between G1 = 10.0 and 10.8, twice as many pairs become available for the analysis. This also allows us to remove the constraint that the coefﬁcients are constant for G < 10. The resulting estimates are different from those in Table 7, with the most important differences seen towards the bright end. A repetition of the analysis using the improved coefﬁcients for the biases of the faint components results in yet another slightly different set of coefﬁcients. However, after a few more iterations, the coefﬁcients were found to be completely stable, and hence internally consistent with the data for all useful pairs. The end result is shown in Table 9, which is our ﬁnal estimate of the bias function Z5(G, νeff, β) for sources with ﬁve-parameter solutions in EDR3. Figure 21 is a visualisation of this function. The top panel of Fig. 20 shows values of the bias function for a representative selection of sources, taking into account their actual distribution in effective wavenumber at a given magnitude.
5. Six-parameter solutions Because of their different treatments in the image parameter determination and astrometric solution, the ﬁve- and sixparameter solutions have different systematics and it is necessary to consider their parallax biases separately. In principle, the same method as was used for the ﬁve-parameter solutions could be applied to the six-parameter solutions, but in practice, this in not possible owing to the much smaller number of six-parameter solutions at all magnitudes, except for G 19 and at the very bright end (Fig. 12). Recalling (Sect. 2.2) that six-parameter solutions are used for sources that did not have reliable colour information from the BP and RP photometers in Gaia DR2, we may also note that their observations are more often disturbed by other sources than the ﬁve-parameter solutions, and therefore they are generally more problematic.
To circumvent the scarcity of suitable six-parameter solutions, we bootstrapped the estimation of Z6 in the following

L. Lindegren et al.: Gaia Early Data Release 3

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[q12]
¡400 10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0 Magnitude G [mag]

Fig. 11. Coefﬁcients q jk estimated from physical pairs as functions of G. Results for q12 are considered insigniﬁcant and set to zero when ﬁtting the other ﬁve coefﬁcients. The different symbols represent different selections on ρ∆µ: 0–0.5 arcsec mas yr−1 (red circles), 0.5–1 arcsec mas yr−1 (green triangles), 1–2 arcsec mas yr−1 (blue squares), and 0–2 arcsec mas yr−1 (ﬁlled black circles with lines and error bars). The dashed blue line is the global ﬁt in Table 7. The vertical grey lines show the breakpoints for the basis functions in G deﬁned by Eq. (A.2).

way on the already determined Z5. For 8.16 million of the primary sources we have both ﬁve- and six-parameter solutions (see Sect 2.2), and this sample was used here to map the systematic differences in parallax between the two kinds of solution. Figure 13 shows the median of 6 − 5 versus G and νeff, where
5 and 6 are the parallaxes of a given source as obtained in the ﬁve- and six-parameter solutions. The most prominent features in the plots are the positive gradient in νeff for 16 G 19.5 and 11 G 12, and an opposite gradient for the faintest stars. There are clear differences between the southern and northern

ecliptic hemispheres. The systematics of 6 − 5 thus depend in a complex way on at least G, νeff, and β. Owing to the restrictions in the selection of primary sources for the astrometric solution, the colour region outside of the interval 1.24–1.72 µm−1 in νeff cannot be mapped with this sample.
In the Gaia archive, no effective wavenumber derived from photometry is provided for sources with six-parameter solutions, and many of them also lack a colour index such as GBP − GRP. The parallax bias function Z6 must therefore be expressed in terms of the pseudo-colour νˆeff instead of νeff. Figure 14 shows

A4, page 15 of 31

A&A 649, A4 (2021)

Table 8. Coefﬁcients of Z5(G, νeff, β) obtained by joining the results in Tables 6 and 7.

G 10.0–10.8
11.2 11.8 12.2 12.9 13.1 15.9 16.1 17.5 19.0 20.0 21.0

q00 −27.75 −31.24 −33.87 −13.37 −19.61 −37.99 −38.33 −31.05 −29.18 −18.40 −12.65 −18.22

q01 −1.43 −8.84 −7.33 +1.68 −0.68 +2.63 +5.61 +2.83 −0.09 +5.98 −4.57 −15.24

q02 +27.56 +8.40 +12.98 +7.82 +15.98 +16.14 +15.42 +8.59 +2.41 −6.46 −7.46 −18.54

q10 +33.3 −88.9 −135.6 −105.6 −68.1
−5.7 − − − − − −

q11 +10.0 +12.1 +27.6 +38.4 +25.4 +14.0 +18.7 +15.5 +24.5 +5.5 +97.9 +128.2

q20 −1257 −1257 −1257 −1257 −1257 −1257 −1189 −1404 −1165
− − −

q30 − − − − −
+107.9 +243.8 +105.5 +189.7
− − −

q40 − − − − −
+104.3 +155.2 +170.7 +325.0 +276.6
− −

Notes. The table gives q jk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A dash (–) indicates that the coefﬁcient should be ignored (taken as zero). Units are µas (for q0k), µas µm (q1k, q30, q40), and µas µm3 (q20).

Table 9. Final coefﬁcients of Z5(G, νeff, β) obtained by joining the results in Table 6 with the analysis in Sect. 4.4.2.

G

q00

q01

q02

q10

q11

q20

q30

q40

6.0 −26.98 −9.62 +27.40 −25.1 −0.0 −1257

−

−

10.8 −27.23 −3.07 +23.04 +35.3 +15.7 −1257

−

−

11.2 −30.33 −9.23 +9.08 −88.4 −11.8 −1257

−

−

11.8 −33.54 −10.08 +13.28 −126.7 +11.6 −1257

−

−

12.2 −13.65 −0.07 +9.35 −111.4 +40.6 −1257

−

−

12.9 −19.53 −1.64 +15.86 −66.8 +20.6 −1257

−

−

13.1 −37.99 +2.63 +16.14 −5.7 +14.0 −1257 +107.9 +104.3

15.9 −38.33 +5.61 +15.42

− +18.7 −1189 +243.8 +155.2

16.1 −31.05 +2.83 +8.59

− +15.5 −1404 +105.5 +170.7

17.5 −29.18 −0.09 +2.41

− +24.5 −1165 +189.7 +325.0

19.0 −18.40 +5.98 −6.46

− +5.5

−

− +276.6

20.0 −12.65 −4.57 −7.46

− +97.9

−

−

−

21.0 −18.22 −15.24 −18.54

− +128.2

−

−

−

Notes. The table gives q jk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A dash (–) indicates that the coefﬁcient should be ignored (taken as zero). Results are uncertain for νeff > 1.72 µm−1 (GBP − GRP 0.15) and νeff < 1.24 µm−1 (GBP − GRP 3.0). Units are µas (for q0k), µas µm (q1k, q30, q40), and µas µm3 (q20).

1:0

P5

0:9

P6

P2

0:8

0:7

0:6

Fraction

0:5

0:4

0:3

0:2

0:1

0 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Magnitude G [mag]

Fig. 12. Fraction of sources in Gaia EDR3 with different kinds of solutions: ﬁve-parameter solutions (P5 = solid red curve), sixparameter solutions (P6 = dashed blue curve), and two-parameter solutions (P2 = thin grey curve). The two-parameter solutions are ignored in this paper as they do not have parallaxes.

A4, page 16 of 31

the same differences 6 − 5 as in Fig. 13, but plotted versus

νˆeff. The main trends are the same, only ampliﬁed for the faintest

sources. interval

Uncertainties in 1.24–1.72 µm−1

νˆeff scatter some especially at the

points outside of faint end, where

the the

pseudo-colour becomes rather uncertain. In the southern hemi-

sphere, the strong gradient versus νˆeff at the faint end is produced by the predominantly negative correlation between parallax and

pseudo-colour seen in Fig. 2 (panel b).

To estimate the parallax bias function for six-parameter solu-

tions, Z6(G, νˆeff, β), the following procedure was used. First, a corrected ﬁve-parameter parallax was calculated as

corr 5

=

5 − Z5(G, νeff, β)

(12)

for each of the ∼8 million primary sources that have both kinds of solution, and for which both νeff and νˆeff are available. This calculation used the coefﬁcients in Table 9. Next, an estimate of the parallax bias in the six-parameter solutions was obtained as

zˆ6 =

6−

corr 5

.

(13)

Finally, a robust weighted least-squares ﬁt of the general model in Eqs. (A.5) and (A.6) to zˆ6 was made, with νˆeff replacing νeff

L. Lindegren et al.: Gaia Early Data Release 3

Magnitude G [mag]

Magnitude G [mag]

6

7a

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

6

7b

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

6

7c

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

Magnitude G [mag]

Fig. 13. Median parallax difference between six- and ﬁve-parameter solutions as a function of νeff and G for the 8 million sources with both kinds of solutions. The panels show selections depending on ecliptic latitude β: southern hemisphere (panel a), all latitudes (panel b), and northern hemisphere (panel c). In this sample no sources lie outside of the interval 1.24 < νeff < 1.72 µm−1.

in Eq. (A.3). In the ﬁt, the data were weighted by the inverse

variance, calculated as the sum of the formal variances of 5 and

6. This overestimates the random errors (because the ﬁve- and six-parameter solutions are positively correlated), but neglects

the uncertainty of the correction Z5. Figure 15 shows the median values of zˆ6 in Eq. (13), plotted

versus G and νˆeff using the same divisions by ecliptic latitude

as be

in Figs. usefully

13 and 14. determined

From Fig. for νˆeff >

11.752itµmis −c1lefaror mthatht eZa6vacialnanbolet

Median $6¡$5 [¹as] Magnitude G [mag]

Median $6¡$5 [¹as] Magnitude G [mag]

Median $6¡$5 [¹as] Magnitude G [mag]

Median $6¡$5 [¹as]

Median $6¡$5 [¹as]

6

7a

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

7b

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

7c

50

8

40

9 30
10

11

20

12

10

13 0
14

15

¡10

16

¡20

17

18

¡30

19

¡40

20

¡50

21 1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

Median $6¡$5 [¹as]

Fig. 14. Same data as in Fig. 13, but plotted against the astrometrically determined pseudo-colour (νˆeff). Uncertainties in the pseudo-colour scatter some points outside of the interval 1.24–1.72 µm−1.

sample of common ﬁve- and six-parameter solutions, at least not

for G 18 (and even for fainter sources it would be very dubious,

given the correlation mentioned above). For a similar reason, the

results at the Because the

red end νˆeff boundaries

< at

1.24 1.24

µm−1 are and 1.72

also highly µm−1 have

uncertain. no special

signiﬁcance in the six-parameter solutions (except that the chro-

maticity calibration does not extend beyond these limits), it could

well be that an extrapolation of the ﬁt gives reasonable results at

the more extreme colours. However, rather than relying on this,

we chose to assume tively means that Z6

q3k = q4k = is clamped

0 in the ﬁtted to its value at

model; 1.24 or

this 1.72

effecµm−1

for more extreme pseudo-colours. This has the added beneﬁt of

A4, page 17 of 31

A&A 649, A4 (2021)

Magnitude G [mag]

Magnitude G [mag]

6

30

7a

20

8

9

10

10

0

11 ¡10
12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7b

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7c

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

Magnitude G [mag]

Fig. 15. Median zˆ6 as a function of νeff and G for the sample in Figs. 13 and 14. The panels show selections depending on ecliptic latitude: southern hemisphere (panel a), all latitudes (panel b), and northern hemisphere (panel c).

restricting Z6 to a ﬁnite interval even when the pseudo-colour is grossly incorrect. In line with the analysis of Z5, we also assumed q21 = q22 = 0.
The resulting coefﬁcients for Z6 are given in Table 10. The function Z6(G, νˆeff, β) is visualised in Figs. 16 and 22. To facilitate comparison with the sample data in Fig. 15, the data shown in Fig. 16 were averaged over all latitudes in the middle panel, and over each hemisphere in the top and bottom panels. (To calculate the average over the whole celestial sphere, only the coefﬁcients q j0 should be included; similarly, the averages in the two hemispheres are obtained from q j0 ± q j1/2.) A4, page 18 of 31

Median z^6 [¹as] Magnitude G [mag]

Median z^6 [¹as] Magnitude G [mag]

Median z^6 [¹as] Magnitude G [mag]

Mean Z6 [¹as]

Mean Z6 [¹as]

6

30

7a

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7b

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7c

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

Mean Z6 [¹as]

Fig. 16. Parallax bias Z6 according to Table 10. The panels show mean values for the southern hemisphere (panel a), all latitudes (panel b), and the northern hemisphere (panel c).

6. Validation of the bias corrections In this section the bias functions Z5 and Z6 deﬁned in Tables 9 and 10 are applied to EDR3 parallaxes in order to conﬁrm the validity of the corrections. To some extent, the same data are used as for deriving the corrections in Sects. 4.1 and 4.2, in which case the tests are not a true validation of the functions but rather a consistency check of the procedures used to derive them. Exceptions are the tests of the quasar sample with six-parameter solutions and the bright (G < 13) LMC sample, neither of which were used in the derivation, and the mean parallax of the LMC, which was a free parameter in the ﬁt. Additional tests are described in Fabricius et al. (2021) and Gaia Collaboration (2021a).

L. Lindegren et al.: Gaia Early Data Release 3

Table 10. Coefﬁcients for the function Z6(G, νˆeff, β).

G

q00

q01

q02

q10

q11

q12

q20

6.0 −27.85 −7.78 +27.47 −32.1 +14.4 +9.5 −67: 10.8 −28.91 −3.57 +22.92 +7.7 +12.6 +1.6: −572 11.2 −26.72 −8.74 +9.36 −30.3 +5.6 +17.2 −1104 11.8 −29.04 −9.69 +13.63 −49.4 +36.3 +17.7 −1129 12.2 −12.39 −2.16 +10.23 −92.6 +19.8 +27.6 −365 12.9 −18.99 −1.93 +15.90 −57.2 −8.0 +19.9 −554 13.1 −38.29 +2.59 +16.20 −10.5 +1.4 +0.4: −960 15.9 −36.83 +4.20 +15.76 +22.3 +11.1 +10.0 −1367 16.1 −28.37 +1.99 +9.28 +50.4 +17.2 +13.7 −1351 17.5 −24.68 −1.37 +3.52 +86.8 +19.8 +21.3 −1380 19.0 −15.32 +4.01 −6.03 +29.2 +14.1 +0.4: −563 20.0 −13.73 −10.92 −8.30 −74.4 +196.4 −42.0: +536: 21.0 −29.53 −20.34 −18.74: −39.5: +326.8 −262.3 +1598:

Notes. The table gives q jk(G) at the values of G in the ﬁrst column. For other values of G, linear interpolation should be used. A colon (:) after the coefﬁcient indicates that it is not signiﬁcant at the 3σ level. Results are very uncertain for νˆeff > 1.72 µm−1 (nominally corresponding to GBP − GRP 0.15) and νˆeff < 1.24 µm−1 (nominally GBP − GRP 3.0). Units are µas (for q0k), µas µm (q1k), and µas µm3 (q20).

6.1. Using quasars We applied the bias corrections to the quasar sample in EDR3, expecting the mean corrected parallax to be close to zero regardless of magnitude, colour, etc. The EDR3 table agn_cross_id contains 1 215 942 sources with ﬁve-parameter solutions and 398 231 with six-parameter solutions. In the ﬁve-parameter case, this sample is nearly the same as was used in Sect. 4 to derive the faint part of Z5. By contrast, the six-parameter sample was not previously used: as detailed in Sect. 5, Z6 was estimated differentially with respect to Z5, using a sample of sources for which both kinds of solution were available.
Figure 17 shows the results for the ﬁve-parameters solutions, divided according to magnitude, effective wavenumber, and ecliptic latitude. Mean values of the uncorrected parallaxes ( ) are shown as open black circles, and those of the corrected values ( − Z5) as ﬁlled blue circles. The yellow dots, showing individual uncorrected values, are mainly intended to give an impression of the distributions in G, νˆeff, and sin β of the sources. Although the corrected parallaxes are not perfectly centred on zero, especially for the brightest and reddest sources, the overall improvement is fairly satisfactory.
Figure 18 shows the corresponding results for the sixparameters solutions. Although the corrected values ( − Z6) are clearly better than the uncorrected values, it appears that a slightly larger correction than Z6 might often be required. A peculiar effect noted in this sample is that the mean corrected parallax is a strong function of various goodness-of-ﬁt measures such as the renormalised unit weight error (RUWE) and excess source noise. For example, the mean corrected parallax is generally closer to zero for the subset of quasars with insigniﬁcant excess source noise (astrometric_excess_noise_sig < 2), as shown by the red squares in Fig. 18. This trend is not present in the ﬁve-parameter sample, and we currently have no explanation for it.

6.2. Using the LMC sample The LMC data in Table 4 for G < 13 were not used anywhere in the analysis that led to the bias estimates Z5 in Table 9.

For example, the faint components of the physical pairs were anchored in the combined quasar–LMC solution, which was derived based only on sources with G > 13. We can therefore use the bright LMC data for a partial validation of Z5. Of particular interest is the conspicuous difference in parallax bias between the blue upper main sequence and the red giants, shown in the right panel of Fig. 9 and in Fig. 10 for G = 11.2–11.8 and 12.2– 12.9. Figure 19 is a CMD of the LMC sample, similar to Fig. 9, but colour-coded by the median value of − Z5 at each point, where Z5 is deﬁned by the coefﬁcients in Table 9. The diagram clearly shows that Z5 provides a reasonable correction in most parts of the CMD, including the bright part, although there is a suggestion that the bias is more negative (by 10 µas) than as given by Z5 for the blue branch at G < 13. This could indicate that the sizes of q10 and/or q11, as obtained from the physical pairs, are underestimated in Table 9. As remarked in Sect. 4.2, the dark red patch in the lower right part of the diagram (roughly G > 18, νeff < 1.4) is caused by foreground stars that dominate this part of the CMD. The unmodelled depression of the bias for the faintest stars at intermediate colours, mentioned in Sect. 4.2, is visible as a greenish patch in Fig. 19.
In Sect. 4.3 the distance the LMC was a free parameter in a ﬁt of Z5 to the combined quasar and LMC data, yielding a mean parallax of +22.11 ± 1.10 µas. This is about two standard deviations higher than commonly accepted values, for example, +20.17 ± 0.25 µas (Pietrzyn´ski et al. 2019; including systematic uncertainty). In absolute measure, however, the difference of about 2 µas is small compared with the regional variations of the quasar parallaxes shown for example in Fig. 2c. The angular power spectrum of quasar parallaxes in Gaia EDR3 is discussed in Lindegren et al. (2021), who estimate that the RMS variation of the parallax systematics (excluding the global offset) is about 10 µas on angular scales 10◦. The uncertainty of ±1.1 µas from the ﬁt in Sect. 4.3 does not take these variations into account because the LMC only probes a smaller area. We can therefore conclude that the mean corrected parallax of the LMC agrees remarkably well with the accepted value.
7. Way forward The bias functions Z5 and Z6 provide a recipe for the systematic correction of the EDR3 parallaxes based on the particular choices of data, method, and bias models described in the preceding sections. These choices contain considerable elements of uncertainty and arbitrariness, and are no doubt also coloured by our preconceived notions. The results should therefore not be regarded as deﬁnitive. On the contrary, it is vitally important that alternative routes are explored towards better compensating for the systematics in Gaia data. Gaia Data Release 3, which is expected to be released in 2022, will be a superset of EDR3, so that investigations made based on EDR3 parallaxes will not be superseded until Data Release 4 (DR4) much later. Although DR4 is expected to be signiﬁcantly better in terms of systematics, it will not be unbiased. Methodological developments made using the current data will therefore remain applicable for a long time. We point out a few possible directions for this work.
Global analysis: the approach taken here is highly heuristic, where models are built up gradually in interaction with the data analysis, using a range of different analysis tools. This is often a good way to arrive at a reasonable approximation quickly, and it naturally incorporates the already existing knowledge of the structures and difﬁculties inherent in the data. It probably does not give the optimal result, however. When a general model
A4, page 19 of 31

A&A 649, A4 (2021)

30
20 a

10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60

¡70

14

15

16

17

18

19

20

21

Magnitude G [mag]

30
20 b

10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60

¡70

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber [¹m¡1]

30
20 c

10

Weighted mean parallax [¹as]

0

¡10

¡20

¡30

¡40

¡50

¡60

¡70 ¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0

0:2 0:4 0:6 0:8 1:0

Sine of ecliptic latitude sin ¯

Fig. 17. Parallaxes for 1.2 million quasars with ﬁve-parameter solutions in Gaia EDR3. Yellow dots show the individual values plotted vs. magnitude (panel a), effective wavenumber (panel b), and sine of ecliptic latitude (panel c). Open black circles show mean values of the uncorrected parallaxes ( ) in bins of magnitude etc., and ﬁlled blue circles show mean values of the corrected parallaxes ( − Z5) in the same bins. Mean values are calculated using weights σ−2. Error bars indicate the estimated standard deviation of the weighted mean in each bin.

has been established, it would be advantageous to make a singe global ﬁt to all the data. Using standard statistical techniques (e.g. Burnham & Anderson 2002), we can select the appropriate submodel by objective criteria and can evaluate the conﬁdence intervals. This should give more precise estimates by combining datasets optimally and avoiding the accumulation of errors in the current multi-step ﬁts. It might also reveal new dependences and interactions.

30
20 a

10

Weighted mean parallax [¹as]

0 ¡10 ¡20 ¡30

¡40 ¡50 ¡60

¡70 13 14 15 16 17 18 19 20 21 22 Magnitude G [mag]
30
20 b

10

Weighted mean parallax [¹as]

0 ¡10 ¡20 ¡30

¡40 ¡50

¡60

¡70

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

Pseudocolour [¹m¡1]

30
20 c

10

Weighted mean parallax [¹as]

0 ¡10 ¡20 ¡30

¡40 ¡50

¡60

¡70 ¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0

0:2 0:4 0:6 0:8 1:0

Sine of ecliptic latitude sin ¯

Fig. 18. Parallaxes for 0.4 million quasars with six-parameter solutions in Gaia EDR3. Yellow dots show the individual values plotted vs. magnitude (panel a), pseudo-colour (panel b), and sine of ecliptic latitude (panel c). Open black circles show mean values of the uncorrected parallaxes ( ) in bins of magnitude etc., and ﬁlled blue circles show mean values of the corrected parallaxes ( − Z6) in the same bins. Red open squares show mean values of the corrected parallaxes for the 78% of the quasars that have insigniﬁcant excess source noise, using a slightly different binning. Mean values are calculated using weights σ−2. Error bars indicate the estimated standard deviation of the weighted mean in each bin.

More and different data: increasing the amount of data included in a global analysis could improve the precision of the bias estimation and perhaps extend its validity in magnitude– colour space. Examples of datasets that should be explored are the stars in open and globular clusters, which have the potential to cover a wide range in magnitudes and different environments,

A4, page 20 of 31

L. Lindegren et al.: Gaia Early Data Release 3

Magnitude G [mag]

10

60

11

50

40

12 30

13

20

14

10

0

15 ¡10

16

¡20

17

¡30

¡40

18

¡50

19

¡60

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

Fig. 19. Colour-magnitude diagram of the LMC sample colour-coded by the median of the EDR3 parallax after subtracting Z5 as given by the cofﬁcients in Table 9.

for instance crowded areas. They could also provide a more direct link between Z5 (for the brighter stars) and Z6 (for the faint members) in each cluster.
Data-driven modelling: the access to high-quality astrometric, photometric, and radial-velocity data for very large stellar samples makes it possible to construct and ﬁt statistical models that do not depend on the physical models of speciﬁc types of stars (e.g. Schönrich et al. 2019; Hogg et al. 2019). These methods depend on having a very large number of objects to beat down statistical uncertainties, and will therefore not provide a high resolution in several variables, but rather an independent overall validity check.
Bias modelling for speciﬁc applications: general bias models of the kind developed here are not necessarily the best way to handle parallax bias in speciﬁc applications. It might for example be better to include it as a free parameter directly in the physical model, for instance for luminosity calibrations. This approach was taken by many researchers using Gaia DR2 data (see Sect. 1 for some examples), and it remains a valid alternative to correcting the data.
Systematics in proper motions: A priori there is no reason to expect that the proper motions in EDR3 are less affected by systematics than the parallaxes. After all, they are jointly determined from the same observations. Fabricius et al. (2021) showed an example (their Fig. 25) in which a discontinuity of about 25 µas yr−1 is detected at G 13 in the EDR3 proper motions µα∗ of cluster stars. It is therefore clearly interesting to extend our analysis to the components of proper motion. Although less crucial than the parallax bias in most astrophysical and galactic astronomy applications of Gaia data, proper motion biases are relevant for a number of the most exacting applications, such as the search for exoplanets. Mapping these biases in clusters, however, may be non-trivial in view of internal motions, which may include systematic patterns from expansion, rotation, etc.
Feedback to Gaia calibration models: Several distinct features in Z5 and Z6, such as the abrupt changes and reversal of gradients in colour at G 11 and 13 (Fig. 20), can be traced back to speciﬁc elements of the instrument calibration in the data-processing chain for EDR3. This can be used in a kind of reversed engineering to understand how the calibration models

Median parallax [¹as] Z5 [¹as]

40

1:9

E®ective wavenumber ºe® [¹m¡1]

20

1:8

1:7 0
1:6 ¡20
1:5 ¡40
1:4
¡60 1:3

¡80

1:2

¡100

1:1

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Magnitude G [mag]

40

1:9

Pseudocolour º^e® [¹m¡1]

Z6 [¹as]

20

1:8

1:7 0
1:6 ¡20
1:5 ¡40
1:4
¡60 1:3

¡80

1:2

¡100

1:1

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Magnitude G [mag]

Fig. 20. Parallax bias Z5 (top) and Z6 (bottom) computed according to Tables 9 and 10 for a representative sample of sources with ﬁve- and six-parameter solutions in EDR3. The dots show values for the complete sample of sources (for G < 11.5) or for a random selection (G > 11.5). The colour scale indicates the mean νeff or νˆeff at a given point, thus giving an impression of the mean colour dependence of the bias. The black curves show the 1st, 10th, 50th, 90th, and 99th percentiles. The thick curve is the 50th percentile or median.

need to be improved in order to decrease systematics in future releases. This is part of the normal cyclic development work in the Gaia data-processing consortium, and is much helped by having tools at our disposal to evaluate systematics in the astrometric solution.

8. Summary and conclusions We have investigated the parallax bias in Gaia EDR3 and the variation of this bias with magnitude, colour, and ecliptic latitude. The direct estimation of the bias using quasars was complemented by indirect methods using physical binaries and stars in the LMC. The indirect methods were strictly differential with respect to the quasars; in particular, no speciﬁc value was assumed for the distance to the LMC. The expected functional form of the dependences on magnitude, colour, and ecliptic latitude was derived partly from a consideration of the known properties of the instrument and data processing, and partly from a mapping of the systematic difference between parallaxes in Gaia EDR3 and DR2 (Table 1).

A4, page 21 of 31

A&A 649, A4 (2021)

Magnitude G [mag]

Magnitude G [mag]

6

30

7a

20

8

9

10

10

0

11 ¡10
12

13

¡20

14

¡30

15 ¡40
16

17

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18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

6

30

7b

20

8

9

10

10

0

11

¡10

12

13

¡20

14

¡30

15 ¡40
16

17

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18

¡60

19

20

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21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

6

30

7c

20

8

9

10

10

0

11

¡10

12

13

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14

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15 ¡40
16

17

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18

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19

20

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21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

E®ective wavenumber [¹m¡1]

Magnitude G [mag]

Fig. 21. Parallax bias Z5(G, νeff, β) according to Table 9. The panels show cuts at (a) ecliptic latitude β = −90◦, (b) β = 0◦, and (c) β = +90◦.

Z5(G; ºe®; +90±) [¹as] Magnitude G [mag]

Z5(G; ºe®; 0±) [¹as] Magnitude G [mag]

Z5(G; ºe®; ¡90±) [¹as] Magnitude G [mag]

Z6(G; º^e®; ¡90±) [¹as]

Z6(G; º^e®; ¡90±) [¹as]

6

30

7a

20

8

9

10

10

0

11 ¡10
12

13

¡20

14

¡30

15 ¡40
16

17

¡50

18

¡60

19

20

¡70

21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7b

20

8

9

10

10

0

11

¡10

12

13

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14

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15 ¡40
16

17

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18

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19

20

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21

¡80

1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

6

30

7c

20

8

9

10

10

0

11

¡10

12

13

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14

¡30

15 ¡40
16

17

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18

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19

20

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21

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1:9 1:8 1:7 1:6 1:5 1:4 1:3 1:2 1:1

Pseudocolour [¹m¡1]

Z6(G; º^e®; ¡90±) [¹as]

Fig. 22. Parallax bias Z6(G, νˆeff, β) according to Table 10. The panels show cuts at (a) ecliptic latitude β = −90◦, (b) β = 0◦, and (c) β = +90◦.

Complex dependences on all three variables are evident in all the data, and the dependences are not the same for sources that have ﬁve- and six-parameter solutions. For sources with ﬁve-parameter solutions in Gaia EDR3 (astrometric_params_solved = 31) the ﬁtted bias function, Z5(G, νeff, β), is given by Eqs. (A.3)–(A.5) using the functions q jk(G) obtained by linear interpolation in Table 9. Here, G is the mean magnitude of the source in the broad-band Gaia photometric system (phot_g_mean_mag), νeff is the effective wavenumber (nu_eff_used_in_astrometry), and β is the ecliptic latitude (ecl_lat). A dash (−) in the table should be interpreted as zero. The function Z5 is illustrated in Fig. 21 and in the top panel of Fig. 20. Nominally, the bias ranges from −94 to +36 µas, with an RMS scatter of 18 µas when the full range of colours and magnitudes is considered; when it is weighted by the actual distribution of colours per magnitude in EDR3, as in Fig. 20 (top), the RMS scatter is about 13 µas. A4, page 22 of 31

For sources with six-parameter solutions in Gaia EDR3

(astrometric_params_solved = 95), the parallax bias function,

Z6(G, νˆeff, β), is similarly given by the functions q jk(G) obtained

by interpolation in Table 10. Here, νˆeff is the astrometrically estimated effective wavenumber, known as pseudo-colour

(pseudocolour). The function Z6 is illustrated in Fig. 22 and in

the bottom panel of Fig. 20. <1.24 µm−1 or >1.72 µm−1.

It is The

very uncertain when νˆeff is bias ranges from −151 µas

to +130 µas, with an RMS scatter of 21 µas for the full range

of colours, or 15 µas if weighted by the actual distribution of

colours.

The derived relations are only applicable to the parallaxes in

Gaia EDR3. Regarded as a systematic correction to the parallax,

the bias function Z5 or Z6 should be subtracted from the value (parallax) given in the archive. Python implementations of both

functions are available in the Gaia web pages3.

3 https://www.cosmos.esa.int/web/gaia/edr3-code

L. Lindegren et al.: Gaia Early Data Release 3

As recipes for the systematic correction of the EDR3 parallaxes, these functions should be regarded as provisional and indicative. While we have reason to believe that their application will in general reduce systematics in the parallaxes, this may not always be the case. Users are urged to make their own judgement concerning the relevance of the indicated bias correction for their speciﬁc applications. Whenever possible, depending on the type and number of sources under consideration, users of EDR3 should try to derive more targeted bias estimates for their speciﬁc use cases.
It is difﬁcult to quantify uncertainties in Z5 and Z6. In the region of the parameter space that is well populated by the quasars (essentially G 16 and 1.4 νeff 1.7 µm−1), they may be as small as a few µas, but beyond that region, uncertainties are bound to be greater because of the indirect methods we used. For redder sources (νeff 1.4 µm−1, corresponding to GBP−GRP 1.6), both Z5 and Z6 depend critically on assumption (ii) in Sect. 4.3 that the curvature in colour, seen in the LMC, is the same over the whole sky. If this turns out to be incorrect, it could mean that the bias function for very red sources in the northern hemisphere is off by more than 40 µas. As discussed in Appendix C, we do not think this is likely, but the possibility should be kept in mind. Beyond the non-clamped interval 1.72 > νeff > 1.24 µm−1 (corresponding to 0.15 GBP − GRP 3.0) all results are in any case quite uncertain owing to the way colour information is handled in the LSF and PSF calibrations and astrometric solutions for EDR3 (Sect. 2.2).
Two unmodelled features mentioned in the paper could merit further investigation. One is the depression of the bias for G 18, νeff 1.55 µm−1 that is visible as a greenish patch in Fig. 19. As this region of colour–magnitude space is well covered by the quasars, the feature is probably not generally present but might be particularly strong in the LMC area. The other unmodelled feature is the dependence of the parallax bias for six-parameter solutions on excess source noise illustrated in Fig. 18. It is possible that both features are caused by to a hitherto unexplored tendency at faint magnitudes for the bias to become more negative in crowded areas, where the source excess noise or RUWE also tends to be higher.
As discussed in connection with Fig. 2 (panel c) and more extensively elsewhere (Lindegren et al. 2021), additional systematics in the EDR3 parallaxes have been identiﬁed, for example depending on position on small and intermediate angular scales. These cannot easily be mapped to any useful precision, and should rather be modelled as correlated random errors. The angular power spectrum of quasar parallaxes presented in Sect. 5.6 of Lindegren et al. (2021) may be used to that end. Depending on the angular scale considered, the estimated RMS variation with position ranges from 5 to 26 µas, that is, of the same order of magnitude as the RMS variations in Z5 and Z6 as functions of colour or magnitude.
While it is easy enough to demonstrate that the EDR3 parallaxes contain signiﬁcant systematics, it is extremely difﬁcult to obtain accurate estimates of the bias beyond the limited region of parameter space that is well populated by the quasars. This paper does not claim to give a deﬁnitive answer but merely a rough characterisation of what we have found to be the main

dependences. It is likely that better and possibly quite different estimates can be obtained in the future by means of more reﬁned and comprehensive analysis methods. Continued exploration of the systematics is important not least in order to gain a better understanding of their causes. In the end, this will hopefully lead to much lower levels of systematics in future Gaia data releases.
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 work was ﬁnancially supported by the Swedish National Space Agency (SNSA/Rymdstyrelsen), the European Space Agency (ESA) in the framework of the Gaia project, and the German Aerospace Agency (Deutsches Zentrum für Luft- und Raumfahrt e.V., DLR) through grants 50QG0501, 50QG0601, 50QG0901, 50QG1401 and 50QG1402. We thank the Centre for Information Services and High Performance Computing (ZIH) at the Technische Universität (TU) Dresden for generous allocations of computer time; C. Babusiaux, A.G.A. Brown, J. Maíz Apellániz, A. Vallenari, and the anonymous referee for many helpful suggestions and corrections to the manuscript; and P. Ramos for implementing the Python version of the bias functions. Diagrams were produced using the astronomy-oriented data handling and visualisation software TOPCAT (Taylor 2005).
References
Arenou, F., Luri, X., Babusiaux, C., et al. 2018, A&A, 616, A17 Burnham, K., & Anderson, D. 2002, Model Selection and Multimodel Inference:
A Practical Information-Theoretic Approach (New York: Springer) Butkevich, A. G., Klioner, S. A., Lindegren, L., Hobbs, D., & van Leeuwen, F.
2017, A&A, 603, A45 Chan, V. C., & Bovy, J. 2020, MNRAS, 493, 4367 Crowley, C., Kohley, R., Hambly, N. C., et al. 2016, A&A, 595, A6 de Grijs, R., Wicker, J. E., & Bono, G. 2014, AJ, 147, 122 Evans, D. W., Riello, M., De Angeli, F., et al. 2018, A&A, 616, A4 Fabricius, C., Bastian, U., Portell, J., et al. 2016, A&A, 595, A3 Fabricius, C., Luri, X., Arenou, F., et al. 2021, A&A, 649, A5 (Gaia EDR3 SI) Gaia Collaboration (Prusti, T., et al.) 2016, A&A, 595, A1 Gaia Collaboration (Babusiaux, C., et al.) 2018a, A&A, 616, A10 Gaia Collaboration (Brown, A. G. A., et al.) 2018b, A&A, 616, A1 Gaia Collaboration (Helmi, A., et al.) 2018c, A&A, 616, A12 Gaia Collaboration (Antoja, T., et al.) 2021a, A&A, 649, A8 (Gaia EDR3 SI) Gaia Collaboration (Brown, A. G. A., et al.) 2021b, A&A, 649, A1 (Gaia
EDR3 SI) Gaia Collaboration (Klioner, S. A., et al.) 2021c, A&A, 649, A9 (Gaia EDR3 SI) Girardi, L. 2016, ARA&A, 54, 95 Hambly, N. C., Cropper, M., Boudreault, S., et al. 2018, A&A, 616, A15 Hogg, D. W., Eilers, A.-C., & Rix, H.-W. 2019, AJ, 158, 147 Lindegren, L., & Bastian, U. 2011, EAS Pub. Ser., 45, 109 Lindegren, L., Lammers, U., Hobbs, D., et al. 2012, A&A, 538, A78 Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2 Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2 (Gaia
EDR3 SI) Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G., & Worley, C. E.
2001, AJ, 122, 3466 Murphy, D. C., & May, J. 1991, A&A, 247, 202 Pietrzyn´ski, G., Graczyk, D., Gallenne, A., et al. 2019, Nature, 567, 200 Riello, M., De Angeli, F., Evans, D. W., et al. 2021, A&A, 649, A3 (Gaia
EDR3 SI) Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14 Rowell, N., Davidson, M., Lindegren, L., et al. 2021, A&A, 649, A11 (Gaia
EDR3 SI) Schönrich, R., McMillan, P., & Eyer, L. 2019, MNRAS, 487, 3568 Taylor, M. B. 2005, ASP Conf. Ser., 347, 29 Zinn, J. C., Pinsonneault, M. H., Huber, D., & Stello, D. 2019, ApJ, 878,
136

A4, page 23 of 31

A&A 649, A4 (2021)

Appendix A: Parametrised functions

1:0 i = 0

1 2 3 4 56

78

9

10 11 12

gi(G)

0:5

0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Magnitude G [mag]
Fig. A.1. Basis functions gi(G), i = 0 . . . 12 according to Eq. (A.2).

The various absolute or differential bias functions discussed in the paper are written as linear combinations of a ﬁnite set of three-dimensional basis functions, with G, νeff (or νˆeff), and β as independent variables. To allow interactions among the variables, the set of three-dimensional basis functions should in the most general case be the outer product of the one-dimensional basis functions on each axis. The generic function is therefore

Z(G, νeff, β) =

zijk gi(G) c j(νeff) bk(β) ,

i jk

(A.1)

where gi(G) are the basis functions in magnitude, c j(νeff) the basis functions in colour, and bk(β) the basis functions in ecliptic latitude. The coefﬁcients zijk are the free parameters used to ﬁt Z to the given data.
The magnitude dependence is modelled as a continuous piecewise linear function with breakpoints (knots) at γ0...12 = 6.0, 10.8, 11.2, 11.8, 12.2, 12.9, 13.1, 15.9, 16.1, 17.5, 19.0, 20.0, and 21.0 mag. The corresponding basis functions are

g0(G) = 1(0γ1 − G)/(γ1 − γ0)

if G ≤ γ0, if γ0 < G ≤ γ1, if γ1 < G,

gi(G)

=

0(G − (0γi+1

γi−1)/(γi − − G)/(γi+1

γi−1) − γi)

for i = 1 . . . 11,

if G ≤ γi−1, if γi−1 < G ≤ γi, if γi < G ≤ γi+1, if γi+1 < G,

g12(G) = 0(1G − γ11)/(γ12 − γ11)

if G ≤ γ11, if γ11 < G ≤ γ12, if γ12 < G



(A.2)

(Fig. A.1). A linear combination of these functions may provide a reasonable approximation of variations such as those shown in the top panel of Fig. 3. In particular, the knots at G = 11.0 ± 0.2, 12.0 ± 0.2, 13.0 ± 0.1, and 16.0 ± 0.1 correspond to the transitions suggested in Fig. 1. An important property of this basis set is that at every breakpoint, exactly one basis function is = 1 and the rest are = 0. This means that for arbitrary coefﬁcients qi, the function q(G) = j qigi(G) can be evaluated by linear interpolation among the coefﬁcients, since q(γi) = qi. This property is useful in connection with the alternative form in Eq. (A.5).
The dependence on colour is also modelled as a continuous piecewise polynomial, but of a more speciﬁc form inspired by plots like Figs. 4 and 10. Interior breakpoints are placed at νeff = 1.24, 1.48, and 1.72 µm−1. The segments below 1.24, between 1.48 and 1.72, and above 1.72 are linear, while between

A4, page 24 of 31

1.24 and 1.48 it is cubic with continuous ﬁrst and second derivatives at 1.48 µm−1. The basis functions are

(c0(νeff) = 1,

c1(νeff) = ν+−e00ff..22−441.48

if νeff ≤ 1.24, if 1.24 < νeff ≤ 1.72, if 1.72 < νeff,

c2(νeff) = 0((1+.04.824−)3νeff)3

if νeff ≤ 1.24, if 1.24 < νeff ≤ 1.48, if 1.48 < νeff,

c3(νeff) =

νeff − 1.24 0

if νeff ≤ 1.24, if 1.24 < νeff,

c4(νeff) =

0 νeff − 1.72

if νeff ≤ 1.72, if 1.72 < νeff,



(A.3)

(Fig. A.2). Their coefﬁcients thus represent the value at νeff = 1.48 µm−1, the linear gradient between 1.24 and 1.72 µm−1, the added cubic trend in the moderately red interval (from 1.24 to 1.48 µm−1), and the linear gradients at the extreme colours (below 1.24 and above 1.72 µm−1).
Finally, the basis functions in ecliptic latitude,

b0(β) b1(β) b2(β)

= = =

1, sin β, sin2 β

−

1 3



(A.4)

(Fig. A.3) describe an arbitrary quadratic dependence on sin β.

Tunhiefotremrmdi−st13ribinutibo2nmoaf kseosurtchees

three functions on the celestial

orthogonal for sphere (which

a is

also uniform in sin β).

An equivalent form of Eq. (A.1) is

Z(G, νeff, β) =

q jk(G) c j(νeff) bk(β) ,

jk

(A.5)

where the functions

q jk(G) = zijk gi(G) , j = 0 . . . 4 , k = 0 . . . 2
i

(A.6)

are piecewise linear in G. Equation (A.5) is useful because the functions q jk(G) can be evaluated by linear interpolation among the coefﬁcients zijk, which allows Z(G, νeff, β) to be given in the compact tabular form that we extensively use in this paper.
The coefﬁcients zijk may be determined by standard curveﬁtting techniques. The problem is simple in the sense that it is linear in all the coefﬁcients, but in practice, it is complicated by the presence of outliers and the often very incomplete coverage of the three-dimensional space (G, νeff, β). Robust techniques such as L1-norm minimisation can be used to cope with outliers. Data coverage is more problematic and may require some judicious modiﬁcation of the set of basis functions. A simple remedy could be to remove basis functions without support, for example c3 and c4 if there are too few sources of extreme colours; this is equivalent to setting the corresponding coefﬁcients (zi3k and zi4k) equal to 0. The dependence on G can be simpliﬁed by adding constraints to the ﬁt; for example, the constraint zi, jk = zi+1, jk will force the function q jk(G) to be constant for γi ≤ G ≤ γi+1.

L. Lindegren et al.: Gaia Early Data Release 3

cj(ºe®) + j=2

2:5
j=4 2:0
j=3 1:5
j = 2 (×10) 1:0
j=1 0:5

0

1:9

1:8

1:7

1:6

1:5

1:4

1:3

1:2

1:1

E®ective wavenumber ºe® [¹m¡1]

Fig. A.2. Basis functions c j(νeff), j = 1 . . . 4 according to Eq. (A.3). The (constant) function c0 = 1 is not shown. For better visibility the functions are vertically displaced by j/2, and the amplitude of c2(νeff) is increased by a factor 10.

k=0 1

The positions and proper motions (including uncertainties and correlations of the latter) were converted into Galactic coordinates and then into rectangular orthographic components (x, y, µx, µy) using Eq. (2) in Gaia Collaboration (2018c), but replacing α, δ, µα∗, and µδ by l, b, µl∗, µb, and (αC, δC) everywhere by (lC, bC) (279.77◦, −33.77◦). Using a robust (L1-norm minimisation) algorithm, we obtained the following linear relation in a ﬁt including only stars brighter than G = 18:

µx µy

−0.602 − 0.409 x − 4.755 y +1.760 + 4.180 x − 1.464 y

[mas yr−1] .

(B.1)

For each star in the sample, deviations in (µx, µy) from this model were transformed back into proper-motion residuals (∆µl∗, ∆µb), from which the normalised deviations

 ∆µN = 

∆µl∗ σµl∗

2
+

∆µb σµb

2
− 2ρ(µl∗, µb)

1 − ρ(µl∗, µb)2

∆µl∗ σµl∗

∆µb σµb

1/2

(B.2)

bk(¯)

k=2 0
k=1 ¡1
¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0 0:2 0:4 0:6 0:8 1:0 sin ¯
Fig. A.3. Basis functions bk(β), k = 0 . . . 2 according to Eq. (A.4).
Appendix B: Construction of the LMC sample This appendix describes the selection of sources in the LMC area that we used for the analysis in Sect. 4.2. We also present some tests of the purity of the sample.
Adopting the LMC centre (αC, δC) = (78.77◦, −69.01◦) as in Gaia Collaboration (2018c), we extracted all sources in Gaia EDR3 within a radius of 5◦ with ﬁve-parameter solutions, G < 19, and ruwe < 1.4. A colour-magnitude (Hess) diagram of the resulting sample is shown in Fig. B.1a. The effective wavenumber νeff was used instead of a colour index on the horizontal axis, and the direction was reversed so that bluer stars are to the left and redder stars to the right. Several prominent features in this diagram are produced by Galactic foreground stars, and some additional ﬁltering is clearly required. Because the purpose here is to study biases in parallax, it is essential that no ﬁltering uses the actual parallax values, while the ﬁltering already done based on position and ruwe cannot introduce a selection bias on the parallaxes. For the further selection we used the residuals in proper motion relative to a ﬁtted, very simple kinematic model.

were computed. Figure B.1b shows the joint distribution of parallax and ∆µN for the sample in Fig. B.1a. The selection ∆µN < 10 is likely to produce a relatively clean sample of LMC stars. The colour-magnitude diagram of the ﬁltered sample is shown in Fig. B.1c. This sample, which we used for the analysis in Sect. 4.2, contains 1457 sources with G < 13.0, 88 285 with G < 16.0, 519 203 with G < 17.5, and 2 371 761 with G < 19.0.
To validate and further quantify the cleanliness of the resulting sample, we performed the analogous selection and ﬁltering of sources in two adjacent areas of the same size, but centred on (lC ± 10◦, bC). Their positions and proper motions were transformed into x, y, µx, µy relative to the centre of the respective offset area, while residuals and ∆µN were computed relative to the ﬁxed values in Eq. (B.1). This should give approximately the same number and kinematic selection of Galactic stars in the three areas because their latitudes are equal and their spread in longitude is small. (Here it is essential that Galactic coordinates are used, so that the orientations of the x- and y-axes relative to the Galactic plane are the same in the different areas.) The right panels d–f of Fig. B.1 show the results for one of the offset areas; plots for the other offset area are not shown but are qualitatively similar. The –∆µN plots in the middle panels conﬁrm that the number and distribution of Galactic stars (the structure extending towards positive and high ∆µN) are quite similar in the LMC and offset areas. Further statistical analysis shows that the sample in Fig. B.1c has negligible contamination by foreground stars down to G 18 at all colours, and down to G = 19 for the bluer sources (νeff 1.6).

A4, page 25 of 31

A&A 649, A4 (2021)

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Fig. B.1. Illustrating the selection and validation of the LMC sample. (a) Colour-magnitude diagram for the original sample centred on (lC, bC). (b) Joint distribution for the original sample of parallax and ∆µN, the normalised proper-motion difference to the ﬁtted model. The vertical line marks the cut-off used for the selection based on proper motions. (c) Colour-magnitude diagram for the sub-sample with ∆µN < 10. (d–f ) Same as (a–c), but for a sample centred on (lC − 10◦, bC), containing far fewer LMC stars, but roughly the same number of Galactic foreground stars as in (a–c).

Appendix C: Test using red clump stars A crucial assumption for the joint quasar and LMC model in Sect. 4.3 is that the curvature in colour, represented by the basis function c2(νeff) in Eq. (A.3), is the same over the whole celestial sphere; in other words, that interaction terms between νeff and β are negligible. This assumption is needed because too few quasars are red enough (νeff 1.35) to reliably determine the curvature at any point; instead it is derived entirely from the LMC sample, as illustrated by the curved segments in the three panels of Fig. 10 for G > 13. Such interactions, if they existed,

would be represented by non-zero coefﬁcients q21 and q22 in Table 6. The assumed invariance of the colour curvature with position rests mainly on the following analysis of the parallaxes of red clump (RC) stars in EDR3.
In the observational Hertzsprung–Russell diagram (HRD), the RC is a prominent feature made up of low-mass stars in the core helium-burning stage of their evolution. Red clump stars are widely used as standard candles because their scatter in absolute magnitudes is relatively small, especially at near-infrared and infrared wavelengths (for a general overview, see Girardi 2016). Differences in the absolute magnitudes of RC stars are

A4, page 26 of 31

L. Lindegren et al.: Gaia Early Data Release 3

¡10

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Fig. C.1. Hertzsprung–Russell diagrams for the union of samples N and S. Top: absolute magnitudes in the G band. Bottom: absolute magnitudes from W in Eq. (C.1).

nevertheless expected, depending on factors such as their ages,

effective temperatures, metallicities ([Fe/H]), and alpha-element

abundances ([α/Fe]).

In the HRD of nearby giants (e.g. Fig. 10 in Gaia

Collaboration 2018a), for which extinction is negligible, the RC

stars occupy a narrow range of GBP = 1.2 ± 0.1, corresponding

colours, approximately to νeff = (1.48 ± 0.02)

GµmRP−1−.

Intrinsically, therefore, the RC stars are not nearly red enough

to determine the curvature of the parallax bias versus colour,

which only becomes signiﬁcant for νeff 1.3 µm−1. In the Galac-

tic plane and bulge, however, it is possible to ﬁnd many RC stars

that are sufﬁciently reddened by interstellar extinction.

Because the absolute magnitudes of the RC stars are uncer-

tain and may depend on many unknown factors, as mentioned

above, our analysis of the RC parallaxes must be strictly dif-

ferential and only compare samples with similar population

characteristics. At the same time, we need locations at widely

different ecliptic latitudes in order to see a possible variation in

curvature with β. These conditions rule out the use of the inner

part of the Galaxy, which only covers a limited range of ecliptic

latitudes. A better strategy is to compare two areas in the Galactic

plane that are symmetrically placed on either side of the Galactic

centre and therefore probe similar ranges of galactocentric dis-

tances. The difference in sin β between the areas is maximised

for Galactic longitudes l = ±90◦, and we therefore chose areas

Count Parallax di®erence $EDR3¡$RC(G; GBP¡GRP) [¹as]

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Fig. C.2. Differential parallax bias estimated by means of RC stars. The plots show differences between the measured parallaxes EDR3 and the photometric parallaxes RC from Eq. (C.2) in area N (a) and S (b). Contours of constant density are shown as thin grey lines. The thick black curve traces the ridge of the feature created by the red clump stars.

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Fig. C.3. Differential parallax bias estimated by means of RC stars. The points show the locations of the ridges in Fig. C.2 for area N (ﬁlled red circles) and S (open blue squares). The curves show the expected locations at G = 17 according to Table 6 for area N (solid red) and S (long-dashed blue), assuming that the curvature with colour ( j = 2) is the same in both areas. If the curvature had ﬂipped sign with β, the ridge location for area N should instead follow the up-bending, short-dashed black curve.

A4, page 27 of 31

A&A 649, A4 (2021)

around (l, b) = (90◦, 0) or (α, δ) = (318.00◦, +48.33◦) (hereafter

called ‘N’), and (270◦, 0) or (α, δ) = (138.00◦, −48.33◦) (‘S’).

Within a 5◦ radius of these points, we extracted all sources

with ﬁve-parameter solutions satisfying 13 < G < 17.5 and

RUWE < 1.4 from EDR3. This gave 1.022 million sources in

N and 0.686 million in S. The reason for this strong asymme-

try seems to be the presence of a nearby (<1 kpc) complex of

dust clouds in the middle of the S area, possibly part of the

Vela Molecular Ridge (Murphy & May 1991). While the high

extinction produced by the dust clouds is desirable for our pur-

pose, their proximity reduces the number of stars in the sample

and increases their mean parallax, which is unfavourable for a

precise determination of the bias. We therefore added two areas

of 5◦ radius, centred on (l, b) = (85◦, 0) and (275◦, 0) to the pre-

vious selection. The resulting union sets (1.361 million sources

in N; 1.561 million in S) are more similar in terms of the over-

all distribution of colours and distances, and still approximately

symmetric in Galactic longitudes. The mean value of sin β is

+0.852 for the sources in N, and −0.867 in S.

Figure C.1 is an HRD for the union of the two sets N and S.

The absolute magnitude is (simplistically) computed using 1/

for the distance and ignoring extinction. In the top panel a the

RC stars are seen as the concentration of points along a diagonal

line about ﬁve magnitudes above the main sequence. In effective

wavenumber, the feature starts at least to 1.24 µm−1 because

at νeff of the

1.43 µm−1, and it extends extinction reddening. In the

bottom panel b the reddening-free Wesenheit magnitude

W = G − λ (GBP − GRP)

(C.1)

is used instead of G to compute the values on the vertical axis; here λ 1.9 is the slope of the reddening line for the photometric bands of Gaia (Ripepi et al. 2019). With this transformation the RC stars have an absolute magnitude MW −1.7 that is nearly independent of the colour. By lucky coincidence, the RC for unreddened nearby giants has a similar slope in the Gaia HRD, so the use of W instead of G not only eliminates the effect of the reddening, but reduces the intrinsic scatter of the absolute magnitudes. Assuming that the RC stars have a ﬁxed and known MW, their photometric parallaxes can be computed as

RC(G, GBP −GRP) = (100 mas) × 100.2[MW−G+λ(GBP−GRP)] . (C.2)

In Fig. C.2 we have plotted the differences EDR3 − RC(G, GBP −GRP) for the sources in areas N and S versus νeff,

using λ = 1.9 and MW = −1.68 mag. If all the sources had absolute magnitude MW, the points in these diagrams would outline

the parallax bias as a function of νeff. Most of the sources are nearby main-sequence stars with more positive parallaxes; they

are shown in the upper left corners of the diagrams. The RC stars

form a concentration of points between 0 and −100 µas on the

vertical axis. The ridge locations, shown as the black curves in

Fig. C.2, were estimated by binning the points in colour, using

a bin width of 0.01 µm−1, and ﬁnding the maximum of the dis-

tribution for a Gaussian kernel of 10 µas standard deviation. In

Fig. C.3 the ridge locations for the two areas are plotted in the

same diagram for easier comparison.

The ridge locations in Figs. C.2 and C.3 are very sensitive

to the assumed values of λ and MW. The value λ = 1.90 used

here was adopted from Ripepi et al. (2019), while MW = −1.68

was selected to give approximately the expected bias 1.4 µm−1 according to the analysis in Sect. 4.3. The red

at νeff and blue

curves in Fig. C.3 show the expected variation of the bias accord-

ing to Table 6, evaluated at G = 17, where most of the sources in

A4, page 28 of 31

the RC area are found. By adjusting MW, it is possible to obtain agreement at a speciﬁc νeff for any reasonable choice of λ, but the slope and curvature of the relations deﬁned by the points will be different. If λ is a function of the total extinction, this could also change the curvature of the relation. Little weight should therefore be given to the circumstance that the points in Fig. C.3 roughly follow the curves computed using the values in Table 6. However, it is signiﬁcant and supports the assumption made in Sect. 4.3 that the empirical relation is similar in the two areas for any reasonable MW and reddening law. If, for example, the curvature instead of being independent of β were proportional to sin β, the expected relation in the N area would rather follow the up-bending black curve in the diagram, which is clearly contradicted by the RC data.
For νeff 1.26 µm−1 the ridges and points in Fig. C.2 reach much more negative values on the vertical axis than expected from Table 6. Although deviations from the simplistic reddening law (constant λ) could contribute to this trend, it is probably mainly a selection effect, similar to the Malmquist bias. The strong down-turn sets in for sources redder than approximately 1.26 µm−1, at which point the total extinction in V is at least 4 mag and most of the sources deﬁning the ridge are close to the faint magnitude limit of our sample at G = 17.5. A preferential selection of RC stars that are intrinsically 0.2–0.3 mag brighter than the mean population is enough to explain the discrepancy at νeff = 1.24 µm−1. For νeff > 1.30 µm−1 this selection bias should be much smaller because the sources are on average at least one magnitude brighter. The conclusion from the analysis is that we see no evidence in the RC data for a difference in the curvature of parallax bias versus νeff between the northern and southern hemispheres.

Appendix D: Data and method for physical pairs

This appendix describes the selection of data used for the analysis of physical pairs in Sect. 4.4. We also present the method applied to these data to estimate the parallax bias.
In this method only pairs with apparent separation ρ < 10 arcsec (5 × 10−5 rad) are considered. When their parallax is 20 mas, as is the case for 99.9% of the bright stars, it is not likely that the true parallax difference in the pair exceeds ± 1 µas. In the following we use subscripts 1 and 2 for the bright and faint components in a pair. When G2 > 13.1, the recipe derived from the quasars can be used to correct the EDR3 parallax of the faint component, 2, thus providing an (approximately) unbiased estimate of the true parallax of the pair. Considering a sample of physical pairs with similar G1, νeff1, and β, we can then estimate the parallax bias as

Z5(G1, νeff1, β) = med Z5(G2, νeff2, β) + 1 − 2 ,

(D.1)

where med[x] is the sample median, which is used for robustness. A limitation of the method is that the components in a pair always have practically the same β (which is why β is not subscripted in Eq. (D.1)), so that the mapping of Z5 with respect to this parameter for the bright components must rely on the presumed known dependence on β for the faint components.
A main concern with the method is contamination by optical pairs, that is, chance alignments of physically unconnected sources with different true parallaxes. The use of the median in Eq. (D.1) provides good protection against outliers but is still biased if the outliers tend to fall mainly on one side of the median. This is the case for optical pairs, where the fainter component is likely to be more distant than the brighter, that

L. Lindegren et al.: Gaia Early Data Release 3

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Fig. D.1. Illustration of the selection of physical pairs and the contamination by optical pairs. Top panels: proper motion difference (∆µ) vs. separation (ρ) for source pairs in sample A (a) and B1 (b) before making the cut ρ > 2.2 arcsec indicated by the dashed vertical line. The three solid lines correspond to ρ∆µ = 0.5, 1, and 2 arcsec mas yr−1. The points are colour-coded by the median parallax difference between the components. In panel b the separation of the faint source is measured from the position of the bright source displaced by +0.1◦ in declination. Bottom panels: normalised parallax difference vs. ρ∆µ for the samples in the top panels, but now after the selection ρ > 2.2 arcsec. The three vertical lines correspond to the values of ρ∆µ indicated in the top panels. The thin green curves show the 16th, 50th, and 84th percentiles of the distribution in normalised parallax difference.

is, 1 − 2 > 0, leading to a positive contamination bias in Eq. (D.1). Rather than eliminating the contamination bias by using a very clean sample, which may then be too small for our purpose, we adopt a heuristic approach, where a small amount of contamination is accepted and the calculated median is corrected by a statistical procedure. This requires accurate knowledge of the selection function of the sample, which in practice precludes using pre-deﬁned catalogues such as the Washington Double Star Catalog (Mason et al. 2001).
Instead, pairs need to be selected entirely based on EDR3 data. It is imperative that the parallax values themselves are not used anywhere in the selection process. Without risk of introducing selection biases, it is possible to deﬁne samples in terms of angular separation ρ, proper motion difference ∆µ = [(µα∗1 − µα∗2)2 + (µδ1 − µδ2)2]1/2, parallax uncertainty and other quality indicators, and various photometric parameters. The main sample (denoted A) consists of pairs with G1 < 14, G1 < G2 < 19, and ρ < 10 arcsec. To correct for the contamination bias, we need in addition two comparison samples (B1 and B2), where faint components in the range G1 < G2 < 19 are selected within 10 arcsec of positions that are offset in declination by ± 0.1◦

from the bright components in sample A. We assume that B1 and B2 contain similar numbers and distributions of potential contaminants as A. The use of two comparison ﬁelds, rather than one, reduces the statistical noise introduced by the correction procedure, and their symmetrical placement around the bright component reduces the effect of local variations in star density, etc.
For the actual samples used, we started by extracting sources in Gaia EDR3 with ﬁve-parameter solutions that satisﬁed

G < 14 & σ < 0.05 mas & RUWE < 2 ,

(D.2)

which resulted in 14 561 255 sources. These are the bright components in sample A. Next, within a radius of 10 arcsec around each such source, we extracted faint components (physical and optical) with ﬁve-parameter solutions and magnitudes G2 in the range [G1, 19]. This selection is very incomplete for ρ 2 arcsec, mainly because such sources usually do not have a reliable νeff from the BP and RP photometry, and therefore lack a ﬁve-parameter solution. For G2 − G1 5 mag the selection is incomplete at much larger separations as well. For a well-deﬁned selection we therefore in addition required ρ > 2.2 arcsec and

A4, page 29 of 31

A&A 649, A4 (2021)

G2 − G1 < 4 mag, which gave 3 336 571 faint components in sample A. No condition was imposed on σ or RUWE for the faint components.
To obtain the faint components in B1 and B2, the same criteria were used as in A (that is G1 < G2 < 19, G2 − G1 < 4, and 2.2 < ρ < 10 arcsec), only with ρ measured from the offset positions. This gave 3 309 468 and 3 311 657 sources, respectively. Because B1 and B2 probably contain some faraway (ρ 0.1◦) physical companions or members of clusters that include a brighter component in A, it can be inferred that sample A contains at least 26 000 physical pairs.
We note that these selections were made on a pre-release version of the EDR3 catalogue, which did not yet include the EDR3 photometric data but inherited the G magnitudes from DR2. Making the corresponding selections on the published EDR3 may therefore result in slightly different numbers.
The further selection of physical pairs was based on separation (ρ) and proper-motion difference (∆µ). Low values of ρ and/or ∆µ are clearly much more likely in physical pairs than in optical. However, orbital motion may give a signiﬁcant proper motion difference in a physical pair, especially when the separation is small. On the other hand, the risk of selecting an optical pair is proportional to ρ2, so that we can afford to increase the tolerance on ∆µ for small separations. The product ρ∆µ was found to be convenient for separating optical from physical pairs. This is illustrated in the top panels of Fig. D.1. Panel a shows ∆µ versus ρ for sample A, but for illustration purposes, the diagram includes also pairs with ρ < 2.2 arcsec. Panel b is the corresponding plot for sample B1; the plot for B2 (not shown) is extremely similar. Both panels are colour-coded by the parallax difference in the pair, and it is obvious that A contains mainly physical pairs for ρ∆µ 2 arcsec mas yr−1, that is, below the topmost diagonal line. From a comparison of panels a and b it should be clear why a cut like ρ > 2.2 arcsec is needed: without it, the number of optical pairs in the top left corner of the diagram is grossly overestimated. Similar plots of G2 − G1 versus ρ motivate the selection G2 − G1 < 4 mag (cf. Fig. 6 of Lindegren et al. 2021).
The bottom panels of Fig. D.1 show the normalised parallax difference ( 1 − 2 divided by the combined uncertainty) versus ρ∆µ for sample A (panel c) and B1 (panel d), after applying the selection ρ > 2.2 arcsec. For ρ∆µ 1 arcsec mas yr−1, the normalised parallax differences roughly follow the expected normal distribution in sample A, whereas the distribution is much wider and displaced towards positive values in B1, and for ρ∆µ 1 in both samples. Subsamples of A with a varying degree of contamination can be obtained by selecting on ρ∆µ; speciﬁcally, we used the limits indicated by the three solid lines in Fig. D.1, corresponding to ρ∆µ = 0.5, 1, and 2. For example, ρ∆µ < 0.5 is the cleanest (and smallest) subsample, while ρ∆µ < 2 gives a larger but more contaminated subsample. Increasing the upper limit on ρ∆µ reduces the statistical uncertainty of the parallax bias, but could instead increase contamination bias. However, with a good procedure to correct for the contamination, the result should not depend systematically on the upper limit used.
The procedure to correct for contamination bias is illustrated in Fig. D.2. With the selection ρ∆µ < 1 arcsec mas yr−1, sample A contains 69 993 pairs and the median of 1 − 2 is +4.1 ± 0.2 µas (uncertainty estimated by bootstrapping). The distribution of the parallax differences in sample A is shown by the red histogram (hA) in the top panel of the ﬁgure. With the same selection on ρ∆µ, samples B1 and B2 contain 7226 and

7097 pairs, respectively. The mean of the distributions in B1 and B2, (hB1 + hB2)/2, shown by the blue histogram, is clearly skewed towards positive values, as expected from the contaminants; its median parallax difference is +29.5 ± 1.4 µas. On the assumption that the contaminants in sample A are similar, in number and distributions, as in B1 or B2, we obtain an estimate of the distribution for the true pairs in A by taking the difference between the observed distributions, that is, ∆h ≡ hA − (hB1 + hB2)/2. This is shown by the shaded histogram in Fig. D.2, although only the part with positive count differences is visible owing to the logarithmic scale. Calculating the median of the difference histogram ∆h, including negative counts, gives the corrected estimate +3.4 ± 0.2 µas. The contamination correction is smaller than a µas in this case, but it can be much larger for other subsamples.
The bottom panel of Fig. D.2 shows the uncorrected and corrected medians as functions of the upper limit on ρ∆µ. With increasing limit the sample sizes increase and the statistical uncertainties decrease (as shown by the error bars), but the increasing contamination bias is also apparent in the uncorrected medians. The corrected medians, on the other hand, remain virtually constant up to a limit of about 2 arcsec mas yr−1 on ρ∆µ. This result is only meant to illustrate the principle of the contamination correction and ignores the complex dependences of both the parallax bias and the contamination bias on G, νeff, and β. The important point is that the correction enables us to beneﬁt from the larger samples obtained with a less strict limit on ρ∆µ, or alternatively, to use a ﬁner division in magnitude or colour without sacriﬁcing the accuracy of the estimated parallax bias.
Because we are only interested in the median of the distribution difference, it is not necessary to compute and subtract histograms; a simpler and more exact way is to use a weighted median. In an array of values {xi} with (relative) weights {wi}, the weighted median is the value xˆ, such that the sum of weights is the same on either side of xˆ. In our case x = 1 − 2, and the weighted median is computed for the union of A, B1, and B2, setting w = 1 for pairs in sample A, and w = −0.5 for pairs in B1 and B2.
It is well known that the median minimises the sum of absolute deviations, that is, the L1 norm of the residuals. An alternative deﬁnition of the weighted median therefore is

xˆ = arg min |xi − x| wi .

x

i

(D.3)

This expression can immediately be generalised to the multidimensional case, where an arbitrary function of the parameter vector z (such as the general function described in Appendix A) is ﬁtted to the data by weighted L1-norm minimisation,

ˆz = arg min |xi − f (z)| wi .

z

i

(D.4)

In all our analyses of the pairs (and for the results shown as black

dots in the bottom panel of Fig. D.2), we used Eq. (D.4) with

two additional reﬁnements. First, the uncertainty of the resulting

median or ﬁt was estimated by bootstrap resampling of the union

set. Secondly, the weights in Eq. (D.4) were adjusted to take the

uacnccoeurtnati.nTtiheissowf athsedpoanrealblayxsdetitfifnegretnhceesw,eσig∆hts=t(oσ121/+σσ∆22)1i/n2

, into sam-

ple σ−∆1

A and to −0.5/σ∆ are consistent with

in B1 the L1

and B2. Weights formalism in Eq.

proportional (D.4).

to

A4, page 30 of 31

L. Lindegren et al.: Gaia Early Data Release 3

1e4 5000

Count

2000 1000
500
200 100 50
20 10
5

2

1

0:5

¡1:5

¡1:0

¡0:5

0

0:5

1:0

1:5

Parallax di®erence $1 ¡ $2 [mas]

10

9

8

Estimated $1 ¡ $2 [¹as]

7

6

5

4

3

2

1

0

0:1

0:2

0:5

1

2

5

Upper limit on ½¢¹ [arcsec mas yr¡1]

Fig. D.2. Illustrating the procedure for contamination bias correction. Top: distribution of parallax differences in sample A (thick red histogram) and for the mean of comparison samples B1 and B2 (thin blue histogram). The shaded grey histogram is the difference between the red and blue histograms. The dashed line is the corrected estimate of the parallax difference, equal to the median of the difference histogram. Bottom: uncorrected (open red circles) and corrected (ﬁlled black) estimates of the mean parallax difference vs. the cut in ρ∆µ. For better visibility, the points have been slightly displaced sideways.

A4, page 31 of 31