Astronomy & Astrophysics manuscript no. DR3-Parallaxes ©ESO 2020
December 4, 2020
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
2
Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Ger-
many
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, United Kingdom
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 systematically offset from the expected
distribution around zero, by a few tens of microarcsec.
Aims. We attempt to map the main dependencies of the parallax bias in EDR3. In principle this could provide a recipe for correcting
the EDR3 parallaxes.
Methods. For faint sources the quasars provide the most direct way to estimate parallax bias. In order to extend this to brighter
sources and a broader range of colours, we use differential methods based on physical pairs (binaries) and sources in the Large
Magellanic Cloud. The functional forms of the dependencies are 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 dependencies 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
et al. 2020b) provides trigonometrically determined
parallaxes for nearly 1478 million sources in the magnitude
range G 6 to 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), without relying on distant background objects, imperfections
in the instrument and data processing methods inevitably
result in systematic errors in the published astrometric data. For
example, it is well known that the parallax solution is degenerate
with respect to certain variations of the ‘basic angle’ between
Corresponding author: L. Lindegren
e-mail: lennart@astro.lu.se
the viewing directions of Gaia’s two telescopes (Butkevich et al.
2017). This means that one cannot, purely from Gaia’s own astrometric
observations, simultaneously determine absolute parallaxes
and calibrate this particular perturbation of the instrument.
Conversely, if 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 et al. 2018b), a number of investigations have
been published, using a variety of astrophysical objects, that
have resulted in estimates of the parallax systematics in DR2
(see, for example, 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 use much brighter objects, tend to give more
Article number, page 1 of 32
arXiv:2012.01742v1[astro-ph.IM]3Dec2020
A&A proofs: manuscript no. DR3-Parallaxes
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. 2020). 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, their resolution and numerical
representation. At best, what can be achieved is 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 dependencies
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 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 dependencies. 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 put in 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
no prior knowledge on how the astrometry was generated, but the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Magnitude G [mag]
0
0:1
0:2
0:3
0:4
0:5
0:6
0:7
0:8
0:9
1:0
FractionofAFobservations
gated
WC0a
WC0b
WC1
WC2
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.
mapping of possible dependencies 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 et al. (2016). Further details
on the initial treatment of the data and the subsequent astrometric
processing can be found in Rowell et al. (2020) and Lindegren
et al. (2020). 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 et al. 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, WC), 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 depending
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. 2020),
distinct calibration models were used for observations in the four
window classes designated WC0a, WC0b, WC1, and WC2. The
first two window classes have two-dimensional images, but different
gate usages, while WC1 and WC2 have one-dimensional
images (i.e., the pixels have been binned in the other dimension),
but 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 of G. Clearly, the window classes map out distinct
Article number, page 2 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
0
5
10
15
20
25
30
Parallaxuncertainty¾5($)[¹as]
a
¡0:3
¡0:2
¡0:1
0
0:1
0:2
0:3
Correlationcoe±cient½6($;^ºe®)
b
¡80
¡60
¡40
¡20
0
20
40
60
80
Smoothedcentredquasarparallax;$+17¹as
c
fig02 (A20)
Fig. 2. Celestial maps in ecliptic coordinates of some selected statistics
in Gaia EDR3: (a) mean parallax uncertainty for sources with fiveparameter
solutions and G < 14 mag. (b) mean correlation coefficient
between parallax and pseudocolour for sources with six-parameter solutions.
(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.
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 parameters
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 image parameter determination
(IPD) is performed. The effective wavenumber comes
from the processing of BP and RP spectra in the photometric
pipeline (Riello et al. 2020; De Angeli et al. 2021).1
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 have been
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 colourdependent
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 fitted
by the standard astrometric model with five unknowns α, δ,
, µα∗, and µδ (Lindegren et al. 2012). These parameters (plus,
for some nearby stars, the spectroscopic radial velocity) are sufficient
to describe the astrometric observations of well-behaved
sources such as single stars and quasars.
If 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
extra (sixth) parameter is an astrometric estimate of νeff known as
the pseudocolour, denoted ˆνeff. At the IPD stage, image locations
and fluxes for the source are determined by fitting the PSF or
LSF at the default effective wavenumber νdef
eff = 1.43 µm−1
. This
value is close to the mean νeff for the faint sources that make
up the majority of sources without a photometric νeff. The use
of the default colour in IPD results in image locations that are
biased in proportion to νeff − νdef
eff , where νeff is the actual (but
unknown) colour. The coefficient 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. 2020). Using the calibrated chromaticity, corrections
to the default colour can be estimated for the individual
sources, which gives their pseudocolours. The determination
of pseudocolour 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 primary
sources for which IPD was executed twice: once using the
actual effective wavenumber νeff, known from the spectrophotometric
processing, and once using the default value νdef
eff . For
1
Because the IPD is one of the first processes in the overall cyclic processing
scheme of DPAC (Fabricius et al. 2016), while the photometric
processing is further downstream, the values of νeff used in EDR3 actually
come from the spectrophotometric processing of the previous cycle,
corresponding to the DR2 photometry.
Article number, page 3 of 32
A&A proofs: manuscript no. DR3-Parallaxes
these sources it is possible to compute both five-parameter solutions
(which are the published ones) and six-parameter solutions;
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 its
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 wavenumber νeff
(nu_eff_used_in_astrometry) in Z5, and the (astrometric) pseudocolour
ˆνeff (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 solutions
with νeff > 1.72 µm−1
(GBP−GRP 0.14) or νeff <
1.24 µm−1
(GBP−GRP 3.0). The calibration of the PSF and
LSF versus colour is only done for the well-populated interval
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/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 combination of the two models may produce
astrometric biases that are strongly non-linear in wavenumber,
and there could in particular be discontinuities in ∂Z5/∂νeff at
νeff = 1.24 and 1.72 µm−1
. As no clamping is used for sources
with six-parameter solutions, such abrupt changes in slope are
not expected for Z6 versus pseudocolour, but the relation can still
be non-linear from other effects.
Because νeff is computed from the detailed BP and RP spectra,
while GBP −GRP depends on the ratio of the integrated fluxes
in the two band, 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.
2020).
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
et al. (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 pattern of scanning 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
poles. The middle panel (b) shows the mean correlation coefficient
between parallax and pseudocolour 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 five-parameter
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 for | β | 45◦
there is clearly a systematic difference
between north and south. Although such a systematic could be
produced by the correlation mechanism just described, several
other explanations can be envisaged.
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 however
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.
Recognising that a detailed mapping of Z5 and Z6 versus position
is not possible, but that a global variation with ecliptic
latitude is expected on theoretical grounds and also seen empirically,
the only positional argument in Z5 and Z6 is taken to be
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 looking at 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
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L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
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 qjk(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 µm2
(q20).
(source_id) in the two releases.2
Obviously, the difference 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. In fact, 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
rather few sources with six-parameter solutions for G < 13
that appear also 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 dependence
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.3
These figures,
and all subsequent 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 proportional to 1/(σ2
, EDR3 + σ2
, DR2).
The plots in Figs. 3 and 4 show complex dependencies on
G, νeff, and β, with interactions among all three arguments (that
is, the colour-dependence is different depending on both G and
β, etc.). We use these results to design a continuous, multidimensional
function, relevant parts of which are used in Sects. 4
and 5 to represent the functions Z5 and Z6 fitted to the EDR3
data. Owing to the scarcity of data available for those fits, the
general function has to be quite schematic and cannot take into
account many of the details seen in ∆Z, especially for the brightest
sources. With that purpose in mind, the following features in
∆Z are noteworthy:
2
For a small fraction of the sources in EDR3, no source with the same
source_id is found in DR2 (and vice versa), owing to the evolution
of the source list between releases (Gaia Collaboration et al. 2020b).
That complication is presently ignored, as the EDR3–DR2 differences
are not a main topic of the paper.
3
In this figure, and in all other diagrams with effective wavenumber or
pseudocolour on the horizontal axis, the direction of the axis has been
reversed to follow the usual convention for colour-magnitude diagrams,
that is, blue objects to the left and red to the right.
– As a function of G, there are jumps at G 11.0, 12.0, 13.0,
and 16.0, corresponding to the boundaries shown in Fig. 1.
The transitions are not abrupt, but occur over an interval of
0.2–0.4 mag. Features at G 9 will in the following be ignored
because the number of sources is too small for a reliable
mapping of the parallax bias.
– As a function 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 of slope; within these limits the relation is
approximately linear for 1.48 < νeff < 1.72 µm−1
and
curved (cubic) for 1.24 < νeff < 1.48 µm−1
, with no visible
break at 1.48 µm−1
. The ‘hook’ 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
since it concerns very few objects it will be ignored in the
following.
– As a function of β, the differences typically show an approximately
linear or quadratic dependence on sin β.
The parametrised function described in Appendix A is designed
to take into account these features. Using that form, the differential
bias is approximated by
∆Z(G, νeff, β) =
j k
qjk(G) cj(νeff) bk(β) , (5)
where cj and bk are basis functions in νeff and β, specified by
Eqs. (A.3) and (A.4), respectively, and qjk(G) are piecewise 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. First, all 195 parameters are estimated (ˆzijk), along with
their formal uncertainties (σijk). The parameter with the smallest
standard score S ijk = |ˆzijk|/σijk is then removed (set to zero), and
a new fit calculated with updated uncertainties. This procedure
is repeated until Sijk > 2 for all the retained parameters. However,
the parameters zijk with j = k = 0 are always retained at
their estimated values independent of their scores. This is done
Article number, page 5 of 32
A&A proofs: manuscript no. DR3-Parallaxes
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Magnitude G [mag]
¡10
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10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0 0:2 0:4 0:6 0:8 1:0
sin¯
¡10
0
10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 6 − 11
¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0 0:2 0:4 0:6 0:8 1:0
sin¯
¡10
0
10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 12 − 13
fig
Fig. 3. Mean difference in parallax between EDR3 and DR2 versus
magnitude (top) and, in two magnitude intervals, versus ecliptic
latitude (middle and bottom). 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
parametrized function ∆Z (Eq. 5), binned as for the sources. Red filled
circles are for νeff < 1.48 µm−1
, blue open circles for νeff > 1.48 µm−1
.
in order to avoid that the sum in Eq. (5) defaults to zero at some
G independently of νeff and β.
Applying the above procedure 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. Using more of the faint
sources in EDR3 and DR2 would undoubtedly give a better determination
of the coefficients towards the faint end, and increase
the number of significant coefficients.
As a check 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 multidimensional
fit to the mean parallax differences shown as circles
in the same diagrams. The overall fit is reasonable, but there are
a few systematic deviations that should be commented on:
– Around G 7.0 and 8.0 there are large (10–15 µas) positive
and negative excursions in the mean parallax differences
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 to 10.8, which in turn
is motivated by the scarcity of data for estimating Z5 in this
interval (see Sect. 4.4.2).
– 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.
– As a function of νeff (Fig. 4), the variation between 1.48
and 1.72 µm−1
is not linear, as assumed 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 parallax 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 of 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 will
be 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, put on an absolute scale by means of their parallaxes.
Article number, page 6 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
1:11:21:31:41:51:61:71:81:9
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]
¡200
¡150
¡100
¡50
0
50
100Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 6 − 11
1:11:21:31:41:51:61:71:81:9
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]
¡200
¡150
¡100
¡50
0
50
100
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 11 − 12
1:11:21:31:41:51:61:71:81:9
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]
¡200
¡150
¡100
¡50
0
50
100
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 12 − 13
1:11:21:31:41:51:61:71:81:9
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]
¡10
0
10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 13 − 16
1:11:21:31:41:51:61:71:81:9
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]
¡10
0
10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 16 − 18
1:11:21:31:41:51:61:71:81:9
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]
¡10
0
10
20
30
40
50
60
70
Meanparallaxdi®erence$EDR3¡$DR2[¹as]
G = 18 − 20
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; dashed lines are mean
values of the fitted parametrized function ∆Z (Eq. 5), binned as for the sources. Red filled circles are for sources with β > 0, blue open circles for
β < 0. The vertical dashed lines mark the breakpoints for the basis functions cj(νeff) in Eq. (A.3), namely the clamping limits at 1.24 and 1.72 µm−1
and the midpoint at 1.48 µm−1
.
4.1. Quasars
As part of EDR3, the Gaia Archive4
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
4
https://gea.esac.esa.int/archive/
proper motions in EDR3 formally define the reference frame of
the catalogue, Gaia-CRF3. The list was constructed by crossmatching
the full EDR3 catalogue with 17 external catalogues
of active galactic nuclei (AGN) 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
Article number, page 7 of 32
A&A proofs: manuscript no. DR3-Parallaxes
14 15 16 17 18 19 20 21
Magnitude G [mag]
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
Weightedmeanparallax[¹as]
a
1:11:21:31:41:51:61:71:81:9
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]
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
Weightedmeanparallax[¹as]
b
¡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¯
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
Weightedmeanparallax[¹as]
c
fi
Fig. 5. Mean parallax of quasars binned by magnitude (a), effective
wavenumber (b), and sine of ecliptic latitude (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.
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 (Klioner et
al. 2020). In spite of the strict selection criteria it is likely that
the list contains a small number of stellar contaminants. As described
further down, we take this into account in a statistical
way.
The quasar sample used here is a subset of
agn_cross_id, consisting of 1 107 770 sources5
with
five-parameter solutions and effective wavenumbers
(νeff = nu_eff_used_in_astrometry) in the range 1.24 to
1.72 µm−1
. The median νeff is 1.59 µm−1
, with 1st and 99th
percentiles at 1.44 and 1.69 µm−1
, which makes this sample
significantly bluer than typical stars of 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. The main trends are as follows.
– 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.
– 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.
– 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.
– 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 seen
also 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 (that is, zijk is not
fitted for i < 6), and for the basis function in colour, cj(νeff) in
Eq. (A.3), we only use j = 0 and 1. For G < 17.5 there are not
enough quasars to determine reliably 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).6
Additionally,
the terms with j = 1 and k = 2 were all found to be
insignificant, and 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
5
This investigation used a preliminary version of the table, resulting
in a slightly smaller sample than the corresponding selection from the
published table. The final version was used for the check in Sect. 6.
6
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.
Article number, page 8 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
fig06a
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡30
¡20
¡10
0
10
20
30
Regressioncoe±cient[¹as]
Fig. 6. An example of interactions among the dependences of quasar
parallaxes on colour and ecliptic latitude. In a quadratic regression of
parallax versus 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 that error bars overlap.
1:5 2:0 2:5 3:0 3:5 4:0 4:5 5:0 5:5
Confusion factor X
¡20
¡10
0
10
20
30
40
50
60
70
80
Weightedmeanresidual[¹as]
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).
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 D21 is the mean density of sources brighter than G = 21
in the vicinity of the quasar, expressed in 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. Since the density of faint sources in EDR3 roughly
doubles with each magnitude (d log10 DG/dG 0.3), the second
term in Eq. (6) is the approximate change in density with G.
Thus X is simply a convenient proxy for log10 DG, the density of
sources brighter than the quasar. For the present quasar sample,
0
5
10
15
20
Meancontaminationbias[¹as]
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 in Fig. 2.
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.
If 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 via the second term in Eq. (6). In Fig. 7 we plot instead
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
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, for example, several plots in Lindegren et al.
2020 and Fabricius et al. 2020). 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. Comparing with Table 2, where the fit
did not include these terms, we conclude that the global 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 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 also several features related to the different surveys
contributing to the sample. 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 qjk in Table 3. The coefficients
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
Article number, page 9 of 32
A&A proofs: manuscript no. DR3-Parallaxes
Table 2. Coefficients for the function Z5(G, νeff, β) fitted to the quasar parallaxes.
G q00 q01 q02 q10 q11
13.1–15.9 −30.90 ± 4.51 +8.50 ± 6.54 −2.44 ± 4.35 −15.4 ± 31.3 +31.1 ± 45.3
16.1–17.5 −27.04 ± 1.73 −0.76 ± 2.66 −1.63 ± 1.77 −17.2 ± 12.0 +40.4 ± 18.4
19.0 −16.39 ± 1.54 +3.82 ± 2.38 −3.93 ± 1.79 −15.1 ± 11.7 +14.6 ± 17.9
20.0 −10.21 ± 1.91 −4.05 ± 2.92 −9.91 ± 2.64 −12.9 ± 16.2 +97.0 ± 25.0
21.0 −10.58 ± 5.16 −15.57 ± 7.64 −26.32 ± 8.83 −37.0 ± 47.6 +124.3 ± 71.8
Notes. The table gives qjk(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).
Table 3. Coefficients for the extended function Z5(G, νeff, β, X) fitted to the quasar parallaxes. The function Z5 corrected for contamination bias is
obtained by setting r0 = r1 = 0.
G q00 q01 q02 q10 q11 r0 r1
13.1–15.9 −30.90 ± 4.51 +8.50 ± 6.54 −2.44 ± 4.35 −15.4 ± 31.3 +31.1 ± 45.3 − −
16.1–17.5 −27.04 ± 1.73 −0.76 ± 2.66 −1.63 ± 1.77 −17.2 ± 12.0 +40.4 ± 18.4 − −
19.0 −16.39 ± 1.54 +3.82 ± 2.38 −3.94 ± 1.79 −15.1 ± 11.7 +14.6 ± 17.9 − −
20.0 −10.57 ± 1.97 −4.04 ± 2.92 −10.78 ± 2.82 −11.2 ± 16.3 +97.4 ± 25.0 −1.66 ± 8.61 +10.82 ± 9.89
21.0 −14.62 ± 5.33 −15.58 ± 7.64 −18.43 ± 9.43 −29.0 ± 47.8 +125.4 ± 71.8 +111.02 ± 31.48 −31.38 ± 35.50
Notes. The table gives qjk(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). .
1:11:21:31:41:51:61:71:81:9
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]
10
11
12
13
14
15
16
17
18
19
MagnitudeG[mag]
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
1:11:21:31:41:51:61:71:81:9
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]
10
11
12
13
14
15
16
17
18
19
MagnitudeG[mag]
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
40
50
60
Medianparallax[¹as]
fi
Fig. 9. Colour-magnitude diagrams of the LMC sample. Left: colour-coded by the number of sources at a given point in the diagram. Right:
colour-coded by the median parallax in EDR3.
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 relation displayed in
Fig. 5a. Of the remaining coefficients, the most important ones
(in terms of how much they lessen 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 coefficient q11 is consistently positive,
and increasing with magnitude; in terms of the chi-square
it is much more important than the corresponding simple coefficients
q01 and q10. At the median quasar colour, νeff = 1.59 µm−1
,
q11 describes a clear 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 overall trend in
Fig. 5a this creates a stronger variation with colour in the sample
as a whole than is present at a fixed magnitude. This, as well
as the strong variation of q02 and q11 with magnitude, illustrates
the many complex dependencies in the data and the difficulty to
determine a unique function Z5 based on a limited sample with
intrinsic correlations. It also shows the danger in 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
Article number, page 10 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
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 (LMC)
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 means that
the parallaxes of individual stars have a true dispersion (and gradient)
of the order of one µas. For the present analysis we do
not assume any specific mean distance to the LMC, only that
the selected member sources have the same (but unknown) parallax,
independent 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 2 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 believed 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 predominantly
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 boundaries
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 magnitude
is further explored in Fig. 10, where each panel displays
a different magnitude interval. The black dots show the median
parallax binned by νeff. As seen in the three bottom panels, the
up-turn of parallax values for the bluest stars sets in abruptly
around νeff = 1.72 µm−1
, which is clearly related to the restriction
in the range of wavenumbers to 1.24 ≤ νeff ≤ 1.72 µm−1
for the LSF/PSF calibration, as discussed in Sect. 2.2. At the
other (red) end of the interval no clear break is seen, although
rather 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 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
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
EDR3(G, νeff) =
4
j=0
pj(G) cj(νeff) , (9)
where
pj(G) =
10
i=0
yijgi(G) (10)
are piecewise linear functions in G, and yij are free parameters
of the model, estimated by a robust weighted least-squares pro-
cedure.7
Owing to the small number of sources at the bright end
and their limited coverage in νeff, the functions pj(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 6), 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 give, at the different magnitudes,
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. Indeed, as we do not want the present analysis to
depend in any way on an assumed distance modulus to the LMC,
the fitted coefficients p0 are not further used in the determination
of Z5. The remaining coefficients p1 through p4 map the variation
of the 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
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
other hand, the coefficients p4, describing the colour gradient in
the blue end (νeff > 1.72 µm−1
), show a very clear progression
with magnitude.
4.3. Combined fit using quasars and LMC sources (G > 13)
The quasar sample contains few sources redder than νeff
1.44 µm−1
or bluer than 1.69 µm−1
, and therefore cannot be used
to estimate qjk for j = 2, 3, and 4. The LMC sample, on the other
hand, gives a good determination of pj for j = 1 . . . 4 in the magnitude
range 13 to 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.
7
Here, and elsewhere in the paper, whenever (weighted) least-squares
estimation is used, it 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.
Article number, page 11 of 32
A&A proofs: manuscript no. DR3-Parallaxes
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 6.0 − 10.8
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 11.2 − 11.8
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 12.2 − 12.9
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 13.1 − 15.9
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 16.1 − 17.5
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡80
¡60
¡40
¡20
0
20
40
60
Parallax[¹as]
G = 17.5 − 18.0
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 in the interval (respectively G = 10.0, 11.5, 12.5, 15.0, 17.0, and 18.0.)
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
pj = qj,0 − 0.996 qj,1 + 0.659 qj,2 , j = 1 . . . 4 . (11)
The case j = 1 is the only one for which all the coefficients pj
and qjk are available in Tables 3 and 4, and for which a direct
comparison is therefore possible; this is shown in Table 5 for
G ≥ 13.1. Although the agreement between pLMC
1 and pQSO
1 is
not impressive, the differences are roughly compatible with the
uncertainties. The main conclusion from the comparison 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 using both the quasar and LMC data
is attempted, retaining all the free parameters qjk for G > 13,
the result would be a model that fits both datasets very well,
Article number, page 12 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
Table 4. Function EDR3(G, νeff) fitted to the LMC sample.
G p0 p1 p2 p3 p4
6.0–10.8 +9.76 ± 2.83 −5.3 ± 17.3 − − −
11.2–11.8 −4.35 ± 1.45 −147.7 ± 8.1 − − −
12.2 +9.42 ± 1.23 −147.6 ± 6.8 − − −
12.9 +18.14 ± 1.03 −85.1 ± 5.9 − − −
13.1 −5.08 ± 0.47 −24.7 ± 3.0 −1450 ± 192 +103.2 ± 141.7 +101.3 ± 10.8
15.9 −11.74 ± 0.22 −19.1 ± 1.9 −1197 ± 70 +244.2 ± 48.0 +156.2 ± 7.5
16.1 −6.18 ± 0.23 −15.6 ± 2.5 −1409 ± 136 +103.8 ± 82.7 +171.9 ± 10.3
17.5 −5.31 ± 0.20 −22.5 ± 2.5 −1122 ± 325 +196.6 ± 231.7 +315.9 ± 11.3
19.0 −7.61 ± 1.24 −20.3 ± 15.3 − − +354.3 ± 76.6
Notes. The table gives pj(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 µm2
(p2).
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 pLMC
1 pQSO
1 pPP
1
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
Notes. pLMC
1 is taken from Table 4. pQSO
1 = q10 − 0.996 q11 + 0.659 q12
with coefficients from Table 3. pPP
1 = q10 − 0.996 q11 with coefficients
from Table 7. Values are expressed in µas µm and refer to the mean gradient
in the interval νeff = 1.48 to 1.72 µm−1
. Where applicable, uncertainties
are ±1σ from the covariance matrix of the respective solution.
and is very well-determined in the LMC area, but has very large
uncertainties in most other locations. This is obviously not very
useful. To proceed, we need to constrain the model. We boldly
make the following assumptions for WC1 and WC2 (G ≥ 13.1):
1. The gradient with colour in the mid-blue region (νeff = 1.48
to 1.72 µm−1
) is fully described by the interaction term q11,
that is q10 = q12 = 0. This is consistent with the observation
in Sect. 4.1 that q12 is generally insignificant and that q11 is
far more important than q10 for the overall chi-square.
2. The curvature with colour in the mid-red region (νeff = 1.24
to 1.48 µm−1
) is fully described by the non-interaction term
q20, that is q21 = q22 = 0. This assumption cannot be tested
by means of the quasars, but is partially supported by the
test in Appendix C, using red clump stars, showing similar
curvatures at sin β = ±0.86.
3. The colour gradient in the red end (νeff < 1.24 µm−1
) is fully
described by q30, that is q31 = q32 = 0.
4. The colour gradient in the blue end (νeff > 1.72 µm−1
) is fully
described by q40, that is q41 = q42 = 0.
Indirect support for the third assumption is provided by the
EDR3–DR2 differences ∆Z analysed in Sect. 3, for example the
bottom 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 different
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 the fourth assumption,
although the evidence in Fig. 4 for breaks at νeff > 1.72 µm−1
in
the northern hemisphere is very weak.
With these assumptions the coefficients qjk are effectively determined
by the LMC data via 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)
Having mapped Z5(G, νeff, β) for G > 13 by mean of the quasars
and the LMC, we now turn to the brighter sources. In the LMC
area the gross variation of the 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 use 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 to 14 mag, and that of the faint component (G2) is in the
range max(G1, 13.1) to 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
overlaps with Table 6 for G = 13.1 to 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 of the bias
as a function of (at least) G and νeff exist for sources brighter
than G 13. Owing to the relatively small number of physical
Article number, page 13 of 32
A&A proofs: manuscript no. DR3-Parallaxes
Table 6. Coefficients for the extended function Z5(G, νeff, β, X) as obtained in a combined fit to the quasar and LMC samples. The function Z5
corrected for contamination bias is obtained by setting r0 = r1 = 0.
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 table gives qjk(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 q00 q01 q02 q10 q11 q20
10.0–10.8 −27.75 −1.43 +27.56 +33.3 +10.0 −1257
11.2 −31.24 −8.84 +8.40 −88.9 +12.1 −1257
11.8 −33.87 −7.33 +12.98 −135.6 +27.6 −1257
12.2 −13.37 +1.68 +7.82 −105.6 +38.4 −1257
12.9 −19.61 −0.68 +15.98 −68.1 +25.4 −1257
13.1–14.0 −37.99 +2.63 +16.14 −5.7 +14.0 −1257
Notes. The functions qjk(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.
pairs available, the analysis cannot be very complex but should
still include the main known or expected dependencies. A severe
limitation is that the scarcity of bright sources with νeff < 1.3
or > 1.6 µm−1
makes it practically impossible to determine q20,
q30, and q40. From the LMC data we have q20 −1257 µas µm3
for G > 13.1, so even though it cannot be usefully estimated
from the physical pairs in the magnitude interval 13.1 to 14,
consistency requires that the corresponding term is subtracted
from the parallax of the brighter component before fitting the
remaining parameters. It turns out that this procedure indeed reduces
the sum minimised in Eq. (D.4), albeit not by a significant
amount. A similar improvement is obtained by applying the
same a priori correction for G < 13.1. We therefore assume that
q20 −1257 µas µm3
throughout the range G < 14. Concerning
q30 and q40 we assume that they are zero for G < 13.1. Noting
that a fit including the six coefficients qjk ( j ≤ 1, k ≤ 2) gives
insignificant results for q12 at all magnitudes (lower right panel
of Fig. 11), we also assume 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
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, and are thus
statistically independent,8
which gives an additional indication
of the uncertainty. The absence of any obvious trend in q00 with
ρ∆µ suggests that contamination bias is negligible.
More reliable estimates of the coefficients are obtained in a
simultaneous fit of all the parameters, using Eq. (D.4). Here, qjk
are 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 require that qjk are constant
for G < 10.8 and > 13.1. The resulting fit is given in Table 7
and shown by the blue dashed 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 less good at G = 13.1
where the coefficients in Table 6 are considerably more uncertain.
In Table 8 we have 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 could not go brighter than
G = 10 owing to the restrictions G2−G1 < 4 mag and G2 > 13.1,
where the latter condition came from the necessity to use 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 was rapidly decreasing 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, it is now possible to 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. Naturally, the
resulting estimates are different from those in Table 7, with the
most important differences seen towards the bright end. Repeating
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 coeffi-
8
They are not completely independent, though: systems with more
than two components may appear with the same bright component in
different intervals of ρ∆µ.
Article number, page 14 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡50
¡45
¡40
¡35
¡30
¡25
¡20
¡15
¡10
¡5
0q00[¹as]
q00
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡30
¡20
¡10
0
10
20
30
q01[¹as]
q01
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡30
¡20
¡10
0
10
20
30
40
50
60
70
q02[¹as]
q02
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡200
¡150
¡100
¡50
0
50
100
q10[¹as¹m]
q10
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡150
¡100
¡50
0
50
100
150
200
q11[¹as¹m]
q11
10:0 10:5 11:0 11:5 12:0 12:5 13:0 13:5 14:0
Magnitude G [mag]
¡400
¡300
¡200
¡100
0
100
200
300
q12[¹as¹m]
[q12]
fi
Fig. 11. Coefficients qjk 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); 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).
cients 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 the actual distribution of
effective wavenumbers at a given magnitude.
5. Six-parameter solutions
Because of their different treatments in the image parameter determination
and astrometroc solution, the five- and six-parameter
Article number, page 15 of 32
A&A proofs: manuscript no. DR3-Parallaxes
Table 8. Coefficients of Z5(G, νeff, β) obtained by joining the results in Tables 6 and 7.
G q00 q01 q02 q10 q11 q20 q30 q40
10.0–10.8 −27.75 −1.43 +27.56 +33.3 +10.0 −1257 − −
11.2 −31.24 −8.84 +8.40 −88.9 +12.1 −1257 − −
11.8 −33.87 −7.33 +12.98 −135.6 +27.6 −1257 − −
12.2 −13.37 +1.68 +7.82 −105.6 +38.4 −1257 − −
12.9 −19.61 −0.68 +15.98 −68.1 +25.4 −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 qjk(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 qjk(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).
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Magnitude G [mag]
0
0:1
0:2
0:3
0:4
0:5
0:6
0:7
0:8
0:9
1:0
Fraction
P5
P6
P2
Fig. 12. Fraction of sources in Gaia EDR3 with different kinds of solutions:
five-parameter solutions (P5 = red solid curve), six-parameter
solutions (P6 = blue dashed curve), and two-parameter solutions (P2 =
thin grey curve). The two-parameter solutions are ignored in this paper
as they do not have parallaxes.
solutions have different systematics and it is necessary to consider
their parallax biases separately. In principle the same
methodology 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 in 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
generally more problematic.
To circumvent the scarcity of suitable six-parameter solutions
we bootstrap the estimation of Z6 in the following 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 is 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
Article number, page 16 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
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 to 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 pseudocolour ˆνeff instead of νeff. Figure 14 shows
the same difference 6 − 5 as in Fig. 13, but plotted versus
ˆνeff. The main trends are the same, only amplified for the faintest
sources. Uncertainties in ˆνeff scatter some points outside of the
interval 1.24 to 1.72 µm−1
especially in the faint end, where the
pseudocolour becomes rather uncertain. In the southern hemisphere,
the strong gradient versus ˆνeff in the faint end is produced
by the predominantly negative correlation between parallax and
pseudocolour seen in Fig. 2 (panel b).
To estimate the parallax bias function for six-parameter solutions,
Z6(G, ˆνeff, β), the following procedure was used:
1. For each of the ∼8 million primary sources with both kinds
of solution, and for which both νeff and ˆνeff are available, a
corrected five-parameter parallax is calculated as
corr
5 = 5 − Z5(G, νeff, β) , (12)
using the coefficients in Table 9.
2. From this, an estimate of the parallax bias in the sixparameter
solutions is obtained as
ˆz6 = 6 − corr
5 . (13)
3. Finally, a robust weighted least-squares fit of the general
model in Eqs. (A.5) and (A.6) to ˆz6 is made, with ˆνeff replacing
νeff in Eq. (A.3). In the fit, the data are 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 ˆz6 from step 2, plotted versus
G and ˆνeff using the same divisions by ecliptic latitude as in
Figs. 13 and 14. From Fig. 15 it is clear that Z6 cannot be usefully
determined for ˆνeff > 1.72 µm−1
, from the available 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 in the red end ˆνeff < 1.24 µm−1
are also highly uncertain.
Since the boundaries at 1.24 and 1.72 µm−1
have no special
significance in the six-parameter solutions (except that the chromaticity
calibration does not go 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 have chosen to assume q3k = q4k = 0 in the fitted model;
effectively this means that Z6 is ‘clamped’ to its value at 1.24 or
1.72 µm−1
for more extreme pseudocolours. This has the added
benefit of restricting Z6 to a finite interval even when the pseudocolour
is completely wrong. In line with the analysis of Z5 we
also assume q21 = q22 = 0.
The resulting coefficients for Z6 are given in Table 10. The
function Z6(G, ˆνeff, β) is visualised in Figs. 16 and 22. To facilitate
comparison with the sample data in Fig. 15, the data shown
in Fig. 16 have been 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 one
only needs to include the coefficients qj0; similarly, the averages
in the two hemispheres are obtained from qj0 ± qj1/2.)
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 check 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 on 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. (2020) and Gaia Collaboration et al.
(2020a).
6.1. Using quasars
We apply the bias corrections to the quasar sample in EDR3, expecting
the mean corrected parallax to be close to zero independent
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 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, pseudocolour, and ecliptic latitude.
Mean values of the uncorrected parallaxes ( ) are shown
as open black circles, 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, ˆνeff, 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 ones, 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 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 have at the present time no explanation for it.
6.2. Using the LMC sample
The LMC data in Table 4 for G < 13 were not used in any of
the analysis leading 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 using
only 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, seen 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. It is clear from the diagram that
Article number, page 17 of 32
A&A proofs: manuscript no. DR3-Parallaxes
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
6
7
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21
MagnitudeG[mag]
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Median$6¡$5[¹as]
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MagnitudeG[mag]
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Median$6¡$5[¹as]
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1:11:21:31:41:51:61:71:81:9
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]
6
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21
MagnitudeG[mag]
¡50
¡40
¡30
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¡10
0
10
20
30
40
50
Median$6¡$5[¹as]
c
fig12 (fig
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 (a), all latitudes (b), and northern hemisphere
(c). In this sample there are no sources outside of the interval
1.24 < νeff < 1.72 µm−1
.
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
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
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13
14
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18
19
20
21
MagnitudeG[mag]
¡50
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0
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30
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50
Median$6¡$5[¹as]
a
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
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MagnitudeG[mag]
¡50
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0
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Median$6¡$5[¹as]
b
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡50
¡40
¡30
¡20
¡10
0
10
20
30
40
50
Median$6¡$5[¹as]
c
Fig. 14. Same data as in Fig. 13 but plotted against the astrometrically
determined pseudocolour (ˆνeff). Uncertainties in the pseudocolour scatter
some points outside of the interval 1.24 to 1.72 µm−1
.
the lower right part of the diagram (roughly G > 18, νeff < 1.4)
is caused by foreground stars dominating this part of the CMD.
The unmodelled depression of the bias for the faintest stars at intermediate
colours, mentioned in Sect. 4.2, is seen 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
Article number, page 18 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
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21
MagnitudeG[mag]
¡80
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0
10
20
30
Median^z6[¹as]
a
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
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12
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18
19
20
21
MagnitudeG[mag]
¡80
¡70
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¡20
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0
10
20
30
Median^z6[¹as]
b
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Median^z6[¹as]
c
fig14 (figA1
Fig. 15. Median ˆz6 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 (a), all latitudes (b), and northern hemisphere (c).
standard deviations higher than commonly accepted values, e.g.
+20.17 ± 0.25 µas (Pietrzy´nski 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. (2020), where it is estimated that the RMS
variation of the parallax systematics (excluding the global off-
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
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21
MagnitudeG[mag]
¡80
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MeanZ6[¹as]
a
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
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MagnitudeG[mag]
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MeanZ6[¹as]
b
1:11:21:31:41:51:61:71:81:9
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]
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MagnitudeG[mag]
¡80
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0
10
20
30
MeanZ6[¹as]
c
Fig. 16. Parallax bias Z6 according to Table 10. The panels show mean
values for the southern hemisphere (a), all latitudes (b), and northern
hemisphere (c).
set) 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, as the LMC only probes a smaller area. We can
therefore conclude that the mean corrected parallax of the LMC
is in remarkably good agreement with the accepted value.
Article number, page 19 of 32
A&A proofs: manuscript no. DR3-Parallaxes
14 15 16 17 18 19 20 21
Magnitude G [mag]
¡70
¡60
¡50
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0
10
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30Weightedmeanparallax[¹as]
a
1:11:21:31:41:51:61:71:81:9
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]
¡70
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30
Weightedmeanparallax[¹as]
b
¡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¯
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Weightedmeanparallax[¹as]
c
fig
Fig. 17. Parallaxes for 1.2 million quasars with five-parameter solutions
in Gaia EDR3. Yellow dots show the individual values plotted versus
magnitude (a), effective wavenumber (b), and sine of ecliptic latitude
(c). Open black circles show mean values of the uncorrected parallaxes
( ) in bins of magnitude etc.; 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.
7. The 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, methodology, 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 in no
13 14 15 16 17 18 19 20 21 22
Magnitude G [mag]
¡70
¡60
¡50
¡40
¡30
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0
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Weightedmeanparallax[¹as]
a
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
¡70
¡60
¡50
¡40
¡30
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10
20
30
Weightedmeanparallax[¹as]
b
¡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¯
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Weightedmeanparallax[¹as]
c
Fig. 18. Parallaxes for 0.4 million quasars with six-parameter solutions
in Gaia EDR3. Yellow dots show the individual values plotted versus
magnitude (a), effective wavenumber (b), and sine of ecliptic latitude
(c). Open black circles show mean values of the uncorrected parallaxes
( ) in bins of magnitude etc.; 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.
way be regarded as definitive. On the contrary, it is vitally important
that alternative routes are explored towards getting a better
handle on the systematics in Gaia data. It should be noted
that Gaia Data Release 3 expected in 2022 will be a superset
of EDR3, so investigations made on EDR3 parallaxes will not
be superseded until Data Release 4 (DR4) much later. Although
Article number, page 20 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
10
11
12
13
14
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16
17
18
19
MagnitudeG[mag]
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
40
50
60
Medianparallax[¹as]
Fig. 19. Colour-magnitude diagram of the LMC sample color-coded by
the median of the EDR3 parallax after subtracting Z5 as given by the
cofficients in Table 9.
DR4 is expected to be significantly better in terms of systematics,
it will not be unbiased. Thus methodological developments
made using the current data will remain applicable for a long
time. Here 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 on the
structures and difficulties inherent in the data. But it probably
does not give the optimal result. Once a general model 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) the appropriate submodel can be selected
by objective criteria, and confidence intervals evaluated.
This should give more precise estimates by combing datasets
optimally and avoiding the accumulation of errors in the current
multi-step fits, and could also reveal new dependencies and
interactions.
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,
e.g. crowded areas. They could for example 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
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Magnitude G [mag]
¡100
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0
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40
Z5[¹as]
1:1
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E®ectivewavenumberºe®[¹m¡1
]
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Magnitude G [mag]
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Z6[¹as]
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1:8
1:9
Pseudocolour^ºe®[¹m¡1
]
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 sixparameter
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 ˆνeff 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.
physical model, e.g. 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. (2020)
show an example (their Fig. 25) where a discontinuity of about
25 µas yr−1
is seen at G 13 in the EDR3 proper motions µα∗
of cluster stars. It is therefore clearly interesting to extend the
present analysis to the components of proper motion. Although
less crucial in most astrophysical and galactic astronomy applications
of Gaia data than the parallax bias, proper motion biases
are relevant for a number of the most exacting applications, such
as the search for exoplanets. Mapping these biases in clusters
may however 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
Article number, page 21 of 32
A&A proofs: manuscript no. DR3-Parallaxes
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z5(G;ºe®;¡90±
)[¹as]
a
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z5(G;ºe®;0±
)[¹as]
b
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z5(G;ºe®;+90±
)[¹as]
c
fig16 (figA14)
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◦
.
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 need to be
improved in order to bring down 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 at our
disposal tools 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 lati-
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z6(G;^ºe®;¡90±
)[¹as]
a
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z6(G;^ºe®;¡90±
)[¹as]
b
1:11:21:31:41:51:61:71:81:9
Pseudocolour [¹m¡1
]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
MagnitudeG[mag]
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
10
20
30
Z6(G;^ºe®;¡90±
)[¹as]
c
Fig. 22. Parallax bias Z6(G, ˆνeff, β) according to Table 10. The panels
show cuts at (a) ecliptic latitude β = −90◦
, (b) β = 0◦
, and (c) β = +90◦
.
tude. The direct estimation of the bias using quasars is complemented
by indirect methods using physical binaries and stars in
the LMC. The indirect methods are strictly differential with respect
to the quasars; in particular, no particular value is assumed
for the distance to the LMC.
The expected functional form of the dependencies on magnitude,
colour, and ecliptic latitude was derived partly from a
consideration of the known properties of the instrument and data
processing, partly from a mapping of the systematic difference
between parallaxes in Gaia EDR3 and DR2 (Table 1).
Complex dependencies on all three variables are evident
in all the data, and the dependencies are not the
same for sources that have five- and six-parameter soluArticle
number, page 22 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
Table 10. Coefficients for the function Z6(G, ˆνeff, 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 qjk(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 ˆνeff > 1.72 µm−1
(nominally corresponding to
GBP − GRP 0.15) and ˆνeff < 1.24 µm−1
(nominally GBP − GRP 3.0).
Units are: µas (for q0k), µas µm (q1k), and µas µm3
(q20).
tions. 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
qjk(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 the ecliptic latitude
(ecl_lat). A dash (−) in the table should be interpreted as zero.
The function Z5 is illustrated in Fig. 21 and 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; weighted by the actual distribution of
colours per magnitude in EDR3, as in Fig. 20 (top), the RMS
scatter is about 13 µas.
For sources with six-parameter solutions in Gaia EDR3
(astrometric_params_solved = 95) the parallax bias function,
Z6(G, ˆνeff, β), is similarly given by the functions qjk(G) obtained
by interpolation in Table 10. Here, ˆνeff is the astrometrically
estimated effective wavenumber, known as pseudocolour
(pseudocolour). The function Z6 is illustrated in Fig. 22 and in
the bottom panel of Fig. 20. It is very uncertain when ˆνeff is
< 1.24 µm−1
or > 1.72 µm−1
. The 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 via the Gaia web pages at
https://www.cosmos.esa.int/web/gaia/edr3-code.
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 parameter space 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 used. For redder sources
(νeff 1.4 µm−1
, corresponding to GBP−GRP 1.6), both Z5 and
Z6 depend critically on the assumption (item 2 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 wrong, it could mean that the
bias function for very red sources in the northern hemisphere is
wrong by several tens of 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/PSF calibration 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
seen 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 could
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 tend to be higher.
As discussed in connection with Fig. 2 (panel c) and more
extensively elsewhere (Lindegren et al. 2020), 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. (2020) 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 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 dependencies. 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.
Article number, page 23 of 32
A&A proofs: manuscript no. DR3-Parallaxes
We thank the Centre for Information Services and High Performance Computing
(ZIH) at the Technische Universität (TU) Dresden for generous allocations
of computer time. Diagrams were produced using the astronomy-oriented data
handling and visualisation software TOPCAT (Taylor 2005).
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L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
Appendix A: Parametrised functions
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 ˆνeff), 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) =
i j k
zijk gi(G) cj(νeff) bk(β) , (A.1)
where gi(G) are the basis functions in magnitude, cj(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 if G ≤ γ0,
(γ1 − G)/(γ1 − γ0) if γ0 < G ≤ γ1,
0 if γ1 < G,
gi(G) =



0 if G ≤ γi−1,
(G − γi−1)/(γi − γi−1) if γi−1 < G ≤ γi,
(γi+1 − G)/(γi+1 − γi) if γi < G ≤ γi+1,
0 if γi+1 < G,
for i = 1 . . . 11,
g12(G) =



0 if G ≤ γ11,
(G − γ11)/(γ12 − γ11) if γ11 < G ≤ γ12,
1 if γ12 < G



(A.2)
(Fig. A.1). A linear combination of these functions may provide
a reasonable approximation of variations such as those seen 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 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) =



−0.24 if νeff ≤ 1.24,
νeff − 1.48 if 1.24 < νeff ≤ 1.72,
+0.24 if 1.72 < νeff,
c2(νeff) =
+0.24 if νeff ≤ 1.24,
(1.48 − νeff)3
if 1.24 < νeff ≤ 1.47,
c3(νeff) =
νeff − 1.24 if νeff ≤ 1.24,
0 if 1.24 < νeff,
c4(νeff) =
0 if νeff ≤ 1.72,
νeff − 1.72 if 1.72 < νeff,



(A.3)
(Fig. A.2). Their coefficients thus represent, respectively, 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 middle 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) = 1,
b1(β) = sin β,
b2(β) = sin2
β − 1
3



(A.4)
(Fig. A.3) describe an arbitrary quadratic dependence on sin β.
The term −1
3 in b2 makes the three functions orthogonal for a
uniform distribution of sources on the celestial sphere (which is
also uniform in sin β).
An equivalent form of Eq. (A.1) is
Z(G, νeff, β) =
j k
qjk(G) cj(νeff) bk(β) , (A.5)
where the functions
qjk(G) =
i
zijk gi(G) , j = 0 . . . 4 , k = 0 . . . 2 (A.6)
are piecewise linear in G. Equation (A.5) is useful because the
functions qjk(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 extensively used in this paper.
The coefficients zijk may be determined by standard curve fitting
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 putting 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 qjk(G) to be constant for γi ≤ G ≤ γi+1.
Article number, page 25 of 32
A&A proofs: manuscript no. DR3-Parallaxes
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Magnitude G [mag]
0
0:5
1:0
gi(G)
i = 0 1 2 3 4 5 6 7 8 9 10 11 12
Fig. A.1. Basis functions gi(G), i = 0 . . . 12 according to Eq. (A.2).
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber ºe® [¹m¡1
]
0
0:5
1:0
1:5
2:0
2:5
cj(ºe®)+j=2
j = 1
j = 2 (×10)
j = 3
j = 4
Fig. A.2. Basis functions cj(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.
¡1:0 ¡0:8 ¡0:6 ¡0:4 ¡0:2 0 0:2 0:4 0:6 0:8 1:0
sin¯
¡1
0
1
bk(¯)
k = 2
k = 1
k = 0
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, used for the analysis in Sect. 4.2, and some tests on the
purity of the sample.
Adopting the LMC centre (αC, δC) = (78.77◦
, −69.01◦
) as
in Gaia Collaboration et al. (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 is used instead of a colour index on the horizontal
axis, and the direction has been reversed so that bluer stars
are to the left and redder to the right. Several prominent features
in this diagram are produced by Galactic foreground stars, and
some additional filtering is clearly required. Since 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 use the residuals in
proper motion relative to a fitted, very simple kinematic model.
The positions and proper motions (including uncertainties
and correlations of the latter) were converted to Galactic co-
ordinates,9
and then to rectangular orthographic components
(x, y, µx, µy) using Eq. (2) in Gaia Collaboration et al. (2018c),
but replacing everywhere α, δ, µα∗, and µδ by l, b, µl∗, µb, and
(αC, δC) by (lC, bC) (279.77◦
, −33.77◦
). Using a robust (L1norm
minimisation) algorithm, the following linear relation was
obtained in a fit including only stars brighter than G = 18:
µx −0.602 − 0.409 x − 4.755 y
µy +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 to proper-motion residuals (∆µl∗, ∆µb),
from which the normalised deviations
∆µN =


∆µl∗
σµl∗
2
+ ∆µb
σµb
2
− 2ρ(µl∗, µb) ∆µl∗
σµl∗
∆µb
σµb
1 − ρ(µl∗, µb)2


1/2
(B.2)
were computed. Figure B.1b shows the joint distribution of parallax
and ∆µN for the sample in Fig. B.1a. It is seen that 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 is the one 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
to 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, thanks to their equal latitude and limited spread in
longitude. 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 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).
Appendix C: Test using red clump (RC) 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 there
are too few quasars that 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. In Table 6 such
interactions, if they existed, would be represented by non-zero
coefficients q21 and q22. The assumed invariance of the colour
9
The selection described here could equally well have been made using
the original equatorial values, but for the corresponding selection
in the comparison areas, described in the next paragraph, it is essential
that Galactic coordinates are used in order to preserve the orientation of
the x and y axes relative to the Galactic plane.
Article number, page 26 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
10
11
12
13
14
15
16
17
18
19
Gmagnitude
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
a
0:1 1 10 100 1000 1e4
¢¹N + 0:1
¡2
¡1
0
1
2
3
Parallax[mas]
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
b
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
10
11
12
13
14
15
16
17
18
19
Gmagnitude
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
c
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
10
11
12
13
14
15
16
17
18
19
Gmagnitude
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
d
0:1 1 10 100 1000 1e4
¢¹N + 0:1
¡2
¡1
0
1
2
3
Parallax[mas]
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
e
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
10
11
12
13
14
15
16
17
18
19
Gmagnitude
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
f
fi
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 much fewer LMC stars but roughly the same number of Galactic foreground stars as in a–c.
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 red clump is a prominent feature made up of low-mass stars
in the core helium-burning stage of their evolution. RC stars are
widely used as standard candles thanks to their relatively small
scatter in absolute magnitudes, especially at near-infrared and
infrared wavelengths (for a general overview, see Girardi 2016).
Differences in the absolute magnitudes of RC stars are 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
et al. 2018a), for which extinction is negligible, the RC stars
occupy a narrow range of colours, approximately GRP − GBP =
1.2±0.1, corresponding to νeff = (1.48±0.02) µm−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 Galactic plane and bulge
it is however possible to find many RC stars that are sufficiently
reddened by interstellar extinction.
Article number, page 27 of 32
A&A proofs: manuscript no. DR3-Parallaxes
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡10
¡5
0
5
10
15
G+5log10($=100mas)
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
a
1:11:21:31:41:51:61:71:81:9
E®ective wavenumber [¹m¡1
]
¡10
¡5
0
5
10
15
W+5log10($=100mas)
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
b
figC1
Fig. C.1. Hertzsprung–Russell diagrams for the union of samples N and
S. Top (a): absolute magnitudes in the G band. Bottom (b): absolute
magnitudes from W in Eq. (C.1).
Recognising that the absolute magnitudes of the RC stars
are uncertain and may depend on many unknown factors, as
mentioned above, our analysis of the RC parallaxes must be
strictly differential, comparing only samples with similar population
characteristics. At the same time we need locations at
widely different ecliptic latitudes in order to see a possible variation
in curvature with β. These conditions rule out the use of the
inner part of the Galaxy, which only covers a limited range of
ecliptic latitudes. A better strategy is to compare two areas in the
Galactic plane, symmetrically placed on either side of the Galactic
centre, and therefore probing similar ranges of galactocentric
distances. The difference in sin β between the areas is maximised
for Galactic longitudes l = ±90◦
, so we should choose areas
around (l, b) = (90◦
, 0) or (α, δ) = (318.00◦
, +48.33◦
) (hereafter
called ‘N’), and (270◦
, 0) or (α, δ) = (138.00◦
, −48.33◦
) ( ‘S’).
Within 5◦
radius of these points, we extracted from EDR3 all
sources with five-parameter solutions satisfying 13 < G < 17.5
and RUWE < 1.4. This gave 1.022 million sources in N and
0.686 million in S. The reason for this strong asymmetry seems
to be the presence of a nearby (< 1 kpc) complex of dust clouds
in the middle of the S area, possibly part of the Vela Molecular
Ridge (Murphy & May 1991). While the high extinction produced
by the dust clouds is desirable for our purpose, their proximity
reduces the number of stars in the sample and increases
their mean parallax, which is unfavourable for a precise determi-
1:151:201:251:301:351:401:451:50
E®ective wavenumber [¹m¡1
]
¡300
¡200
¡100
0
100
200
300
Parallaxdi®erence$EDR3¡$RC(G;GBP¡GRP)[¹as]
0
10
20
30
40
50
60
70
80
90
100
Count
a
1:151:201:251:301:351:401:451:50
E®ective wavenumber [¹m¡1
]
¡300
¡200
¡100
0
100
200
300
Parallaxdi®erence$EDR3¡$RC(G;GBP¡GRP)[¹as]
0
20
40
60
80
100
120
140
Count
b
Fig. C.2. Differential parallax bias estimated by means of red clump
stars. The plots show differences between the measured parallaxes
EDR3 the photometric parallaxes RC from Eq. (C.2) in area N (a)
and S (b). Contours of constant density are shown in thin grey lines.
The thick black curve traces the ridge of the feature created by the red
clump stars.
nation of the bias. We therefore added to the previous selection
two areas of 5◦
radius, centred on (l, b) = (85◦
, 0) and (275◦
, 0).
The resulting union sets (1.361 million sources in N; 1.561 million
in S) are more similar in terms of the overall distribution
of colours and distances, and still approximately symmetric in
Galactic longitudes. The mean value of sin β is +0.852 for the
sources in N, and −0.867 in S.
Figure C.1 is an HRD for the union of the two sets N and S.
The absolute magnitude is (simplistically) computed using 1/
for the distance and ignoring extinction. In the top panel (a) the
RC stars are seen as the concentration of points along a diagonal
line about five magnitudes above the main sequence. In effective
wavenumber the feature starts at νeff 1.43 µm−1
, and extends
at least to 1.24 µm−1
thanks to the extinction reddening. In the
bottom panel (b) the ‘reddening-free’ Wesenheit magnitude
W = G − λ (GBP − GRP) (C.1)
is used instead of G to compute the values on the vertical axis;
here λ 1.9 is the slope of the reddening line for the photometric
bands of Gaia (Ripepi et al. 2019). With this transformation
the RC stars have an absolute magnitude MW −1.7 that is
Article number, page 28 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
1:151:201:25
m¡1
]
0
1:241:261:281:301:321:341:361:381:40
E®ective wavenumber [¹m¡1
]
¡100
¡90
¡80
¡70
¡60
¡50
¡40
¡30
¡20
¡10
0
Parallaxdi®erence$EDR3¡$RC(G;GBP¡GRP)[¹as]
N
S
N
Fig. C.3. Differential parallax bias estimated by means of red clump
stars. The points show the locations of the ridges in Fig. C.2 for area
N (red filled circles) and S (blue open 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. Had the curvature flipped sign with
β, the ridge location for area N should instead follow the up-bending,
short-dashed black curve.
nearly independent of the colour.10
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 seen
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 distribution 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 at νeff
1.4 µm−1
according to the analysis in Sect. 4.3. The red and
blue curves in Fig. C.3 show the expected variation of the bias
according to Table 6, evaluated at G = 17, where most of the
sources in 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
10
By a 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.
values in Table 6. What is significant, and supports the assumption
made in Sect. 4.3, is 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 go
to 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, we believe that it
is 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 the present 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 would be much smaller, as 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 and the method applied to these
data for estimating the parallax bias.
We only consider pairs with apparent separation ρ < 10 arcsec
(5 × 10−5
rad). If 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.
If 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, the parallax bias can then be estimated as
Z5(G1, νeff1, β) = med Z5(G2, νeff2, β) + 1 − 2 , (D.1)
where med[x] is the sample median, which we use 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 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 major 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 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).
Article number, page 29 of 32
A&A proofs: manuscript no. DR3-Parallaxes
1 2 3 4 5 10
Separation ½ [arcsec]
0:01
0:02
0:05
0:1
0:2
0:5
1
2
5
10
Propermotiondi®erence¢¹[masyr¡1
]
¡0:10
¡0:08
¡0:06
¡0:04
¡0:02
0
0:02
0:04
0:06
0:08
0:10
Median$1¡$2[mas]
a
1 2 3 4 5 10
Separation ½ [arcsec]
0:01
0:02
0:05
0:1
0:2
0:5
1
2
5
10
Propermotiondi®erence¢¹[masyr¡1
]
¡0:10
¡0:08
¡0:06
¡0:04
¡0:02
0
0:02
0:04
0:06
0:08
0:10
Median$1¡$2[mas]
b
0:01 0:1 1 10 100 1000
½¢¹ [arcsec mas yr¡1
]
¡10
¡8
¡6
¡4
¡2
0
2
4
6
8
10
Normalisedparallaxdi®erence($1¡$2)=
p
¾2
1+¾2
2
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
c
0:01 0:1 1 10 100 1000
½¢¹ [arcsec mas yr¡1
]
¡10
¡8
¡6
¡4
¡2
0
2
4
6
8
10
Normalisedparallaxdi®erence($1¡$2)=
p
¾2
1+¾2
2
1
2
5
10
20
50
100
200
500
1000
2000
5000
Count
d
Fig. D.1. Illustration of the selection of physical pairs and the contamination by optical pairs. Top panels: proper motion difference (∆µ) versus
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 versus 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.
For the present study, pairs are 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. For correcting 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. It is
assumed 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.
We started by extracting sources in Gaia EDR3 with fiveparameter
solutions that satisfy
G < 14 & σ < 0.05 mas & RUWE < 2 , (D.2)
which resulted in 14 561 255 sources.11
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 no
five-parameter solution. For G2 − G1 5 mag the selection is
incomplete also at much larger separations. For a well-defined
selection we therefore required, in addition, ρ > 2.2 arcsec and
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.
11
This selection was made on a pre-release version of the EDR3 catalogue,
which did not yet include the EDR3 photometric data; the G
magnitudes (and νeff) therefore come from DR2 and the resulting number
of sources may be slightly different in the published release.
Article number, page 30 of 32
L. Lindegren et al.: Gaia EDR3 – Parallax bias versus magnitude, colour, and position
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.
Since 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.
The further selection of physical pairs is based on separation
(ρ) and proper motion difference (∆µ). Small 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 if the separation
is small. On the other hand, the risk of selecting an optical pair
is proportional to ρ2
, so we can afford to increase the tolerance
on ∆µ for small separations. It turns out that the product ρ∆µ is
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. 2020).
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
will use 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 less than a
µas in this case, but for other subsamples it can be much larger.
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 samples get larger and the statistical uncertainties
smaller (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 dependencies of both the parallax
bias and the contamination bias on G, νeff, and β. The important
point is that the correction makes it possible 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.
Since we only want 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. Given an array
of values {xi} with (relative) weights {wi}, the weighted median
is the value ˆx, such that the sum of weights is the same on either
side of ˆx. In the present 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 is, therefore,
ˆx = arg min
x i
|xi − x| wi . (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
z i
|xi − f(z)| wi . (D.4)
In all our analysis of the pairs (and for the results shown as the
black dots in the bottom panel of Fig. D.2), we use Eq. (D.4)
with two additional refinements. First, the uncertainty of the resulting
median or fit is estimated by bootstrap resampling of
the union set. Secondly, the weights in Eq. (D.4) are adjusted
to take into account the uncertainties of the parallax differences,
σ∆ = (σ2
1 +σ2
2)1/2
. This is done by setting the weights to 1/σ∆
in sample A, and to −0.5/σ∆ in B1 and B2. Using weights proportional
to σ−1
∆ is consistent with the L1 formalism in Eq. (D.4).
Article number, page 31 of 32
A&A proofs: manuscript no. DR3-Parallaxes
¡1:5 ¡1:0 ¡0:5 0 0:5 1:0 1:5
Parallax di®erence $1 ¡ $2 [mas]
0:5
1
2
5
10
20
50
100
200
500
1000
2000
5000
1e4
Count
0:1 0:2 0:5 1 2 5
Upper limit on ½¢¹ [arcsec mas yr¡1
]
0
1
2
3
4
5
6
7
8
9
10
Estimated$1¡$2[¹as]
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.
Botton: uncorrected (open red circles) and corrected (filled black) estimates
of the mean parallax difference versus the cut in ρ∆µ. For better
visibility, the points have been slightly displaced sideways.
Article number, page 32 of 32