represents periods derived from Gaia DR3 photometry that are exactly twice
those obtained from the ground.

It is important to note that the adopted algorithm for photometric inversion
produces for each object a set of 15 different inversion solutions for each
object. The reason is that an intrinsic property of the adopted genetic
algorithm is that it does not always converge to a unique final solution. This
happens because the evolution of the population of possible solution
parameters can often evolve along a dead branch, leading to a poor-quality
solution. Fifteen different genetic solutions have been found to be a
reasonable compromise between the need to obtain a good-quality solution and
the need of minimising the CPU execution time.

We stress that at this stage, only the best obtained solution for each object,
namely the solution that produces the smallest residuals with respect to the
measured magnitudes at the available transits, was considered and compared
with ground-based results. This choice has some consequences, because in
several cases, more than one inversion solution gave equivalent residuals. We
defined as equivalent residuals those that differ by no more than $0.0015$
mag. As a consequence, the inversion solution is not unique in some cases. In
these cases, the nominally best solution may not correspond to the rotation
period and pole listed in ground-based catalogues, whereas another equivalent
solution that is not considered for the moment would correspond to these
values. For this reason, the results presented here are conservative.

We recall that for the purposes of inversion of a set of sparse photometric
measurements, it is of paramount importance to have a good sampling of the
possible observing circumstances for any given object. This means that the
data should include measurements that adequately sample the whole interval of
360 deg in ecliptic longitude. This is especially important for the
determination of the spin axis orientation (the asteroid pole). However, this
is not yet the case for the data that are available in Gaia DR3. For each
object, large gaps exist in the interval of the covered ecliptic longitudes.
The publication of photometric inversion of Gaia data is scheduled as an end-
of-mission task. The results of the preliminary inversion attempts presented
here must be considered as no more than a useful tool for the scientific
validation of Gaia DR3 data. We expect to obtain much better inversion
solutions in future data releases as new measurements will become available.

We considered as a successful determination of the rotation period $P$ an
inversion solution for which the absolute value of the difference between the
resulting $P$ solution and the $P$ value determined by ground-based
observations, expressed in units of the absolute value of the ground-based $P$
value, expressed in hours, is not higher than $0.001$. This criterion takes
into account the fact that for fast rotations of just a few hours, small
errors in $P$ lead to strong differences in the rotational phase of the
object, which is computed at epochs differing by few years. On the other hand,
in the case of very long rotation periods, longer than several dozen or
hundreds of hours, it is not reasonable to impose a required accuracy of about
a few seconds, for example, on the determination of the period.

Based on our adopted criterion, we obtained the correct $P$ solution for $229$
out of $430$ asteroids of our sample. These results are shown in Fig. 24. This
figure shows an interesting feature: in addition to the $229$ cases of correct
$P$ determinations, a significant number of cases exist,for which the $P$
value determined by photometric inversion is nearly exactly twice the $P$
value determined from ground-based photometry. This feature is not entirely
unexpected considering the assumptions of the adopted model. In particular,
the algorithm assumes that over a full rotation of the object, the brightness
reaches two maxima and two minima, which are expected to be (nearly) equal. In
the real world, however, and in particular when an object is not strongly
elongated and the maxima and minima tend to be shallow and/or strongly
asymmetric, the inversion algorithm might derive a rotation period that can be
twice the correct one. Moreover, at least in some cases, a derived double
period might be indicative of a photometric behaviour that is dominated not by
the shape, as assumed by the inversion model, but by variation in surface
albedo. This is the case, for instance, of the large asteroid (4) Vesta
(Cellino et al., 1987), which is known to have a light curve producing only
one maximum and minimum per cycle.

It seems therefore that the results shown in Fig. 24 are very encouraging when
the limits of the adopted shape model, the small minimum number of accepted
transits per object, the still limited variety of observing circumstances, and
the conservative criteria of definition of the inversion solution are taken
into account. Ambiguous cases are not considered, together with the the non-
negligible number of solutions corresponding to a $P$ twice as large as
ground-based determination.

To determine the pole, the situation is intrinsically more complicated. In the
vast majority of cases, two or more different pole solutions per object are
listed in the literature. Moreover, in our assumption of a triaxial ellipsoid
shape model, an ambiguity of 180 degrees in the determination of the ecliptic
longitude of the pole may be present in some cases due to the symmetry of this
shape model. In particular, a $180^{\circ}$ ambiguity on the longitude of the
pole can be triggered by a low orbital inclination of the object or by an
unfavourable distribution of the observations in ecliptic longitude. Moreover,
simulations have shown that a low ecliptic latitude of the pole of the object
tends to make the photometric inversion more challenging with the assumed
shape model (Santana-Ros et al., 2015). We limited our analysis to a
comparison between the obtained pole solutions and the DAMIT database of
asteroid poles, which is considered to be the most accurate list of ground-
based asteroid period and pole determinations. Based on the results of our
analysis, we decided to subjectively define a small number of pole solution
quality classes (QC). In particular, we assigned pole QC = 2 to pole solutions
that differed by no more than about 10 deg (separately) in ecliptic longitude
and latitude with respect to one existing DAMIT pole solution, and we also
imposed that the rotation period determination was correct. We assigned pole
QC = 1 to objects for which the best pole solution was not so close to a DAMIT
pole solution, independently of the obtained period. We assigned pole QC = -1
in situations in which the obtained pole solution had little to do with any
existing DAMIT solution. Finally, we assigned pole QC = 0 to objects for which
no DAMIT pole solution exists.

Figure 25 shows plots of the period determined by inversion of Gaia DR3 data
versus the $P$ value known from ground-based determinations for objects
belonging to different pole solution QCs. The figure shows that except for the
case of pole solution QC = 2, for which a (nearly) perfect agreement with
ground-based period solutions is imposed by definition, when objects belong to
different pole solution QCs, the relation between the QC of the pole solution
and the success in the period determination by inversion of Gaia DR3 data is
not always obvious. In particular, poor pole determinations (red points in the
bottom right panel of Fig. 25) do not correspond to a larger fraction of
erroneous period determinations. We note, however, that the fraction of cases
in which the period from Gaia DR3 photometric data inversion tends to disagree
with ground-based period determinations tends to increase when we consider
objects for which no pole solution from ground-based data is available (see
the bottom left panel in Fig. 25). This might be considered as an indication
that asteroids for which no ground-based pole determination is available may
well be challenging cases, for which even the ground-based period solutions
could be more uncertain and possibly incorrect. The relatively high number of
objects with long rotation periods among those determined by ground-based data
should be noted in particular, and for which ground-based data are so far
insufficient to derive a pole solution. Long rotation periods correspond in
many cases to relatively large uncertainties in the determination of the
period.

Based on the results shown in this and the previous section, we conclude that
Gaia DR3 photometric data are of a good if not excellent quality because we
successfully produced a correct inversion of a large number of objects using a
clearly simplistic shape model and relatively small numbers of measurements
covering a still partial fraction of the possible observing circumstances. We
limited our analysis for the moment to measured Gaia DR3 magnitudes with
nominal errors not exceeding $0.02$ mag. The results of our photometric
inversion attempts seem to be very encouraging. In principle, we cannot rule
out the possibility that in some cases, the error bar of some Gaia DR3
magnitude might be higher than the nominal value. In some cases, this might
affect the results of photometric inversion negatively, and might explain some
incorrect results.

More convincing tests should be based on the analysis of data of asteroids for
which our knowledge of the rotation period and spin axis direction are of the
best possible reliability. This is the case of a very small number of objects
that were visited in situ by space probes. The results for some of them, for
which we have a reasonable number of Gaia DR3 observations, are shown in the
next section.

####  6.2.1 Photometry of (21) Lutetia and (2867) Šteins

Table 4: Rotation period ($P$, in hours), the ecliptic longitude and latitude of the pole ($\lambda$ and $\beta$, in degrees), and the Tholen taxonomical classes for asteroids (21) Lutetia and (2867) Šteins. For the latter, there are two possible pole solutions I and II. Numbers in parentheses depict the published uncertainty in units of the last digit shown. | (21) Lutetia | (2867) Šteins
---|---|---
$P$ | 8.168270(1) | 6.04681(2)
$\lambda$, $\beta$ | 52.2(4), -7.8(4) | I: 96(5), -85(5)
|  | II: 142(5), -83(5)
Class | M | E

Figure 26: Shape model of asteroid (21) Lutetia (left). Shape model of
asteroid (2867) Šteins (right). The models are based on in situ images
obtained by the Rosetta space mission. In the panels, the top and bottom
plates on the left correspond to polar views along the $z$-axis (axis of
rotation). The top (bottom) plates in the middle and to the right correspond
to viewing along the $x$-axis ($y$-axis).

Figure 27: Observed $G$ magnitudes of asteroid (21) Lutetia included in Gaia
DR3, relative to the mean of the magnitudes ($G_{0}$), as a function of days
after the first observation ($t_{0}$) (top left). The top right panel shows
the same as the left panel for (2867) Šteins. The bottom panel shows the same
as the top panel, but the magnitudes are depicted against the phase angle.

Our validation consists of comparing the observed Gaia photometry to
photometry computed for known asteroids that have accurately determined
rotation periods, pole orientations, and high-resolution shape models. Gaia
DR3 contains photometric data for more than a dozen asteroids studied by space
missions. By using the shape models, rotational parameters, and taxonomical
classifications of (21) Lutetia and (2867) Šteins, which are asteroids that
were visited by the ESA Rosetta space mission, we studied whether it is
possible to validate the Gaia DR3 SSO photometry. We note that (21) Lutetia
and (2867) Šteins were assessed earlier in the documentation of Gaia DR2.

For (21) Lutetia and (2867) Šteins, the Planetary Data System provides the
shape models illustrated in Fig. 26 as well as the rotation periods and pole
orientations described in Table 4 (Farnham & Jorda, 2013; Jorda et al., 2012;
Farnham, 2013; Sierks et al., 2011). The table also includes the Tholen
taxonomical classes of the two asteroids.

The Gaia DR3 photometric measurements for these two asteroids are plotted as a
function of the observation epochs, expressed in days after the first
observation, and against the phase angle in degrees, in Fig. 27. Both phase-
magnitude relations show a decreasing trend in disk-integrated brightness with
increasing phase angle. Furthermore, the apparent slope of decreasing
brightness is steeper for Lutetia, in agreement with Lutetia and Šteins being
lower-albedo M-type and higher-albedo E-type objects in the Tholen taxonomy,
respectively.

Figure 28: Observed $G$ magnitudes of asteroid (21) Lutetia included in Gaia
DR3 as a function of days after the first observation ($t_{0}$, top left) and
against the phase angle ($\alpha$, top right) together with the modelled
magnitudes. The bottom left panel shows the same as in top right panel, but
only for the observations concerning the Lutetia hemisphere observed by
Rosetta. The bottom right panel shows the same as in the top right panel, but
only for the observations of the Lutetia hemisphere that was not observed by
Rosetta.

Figure 29: Observed $G$ magnitudes of asteroid (2867) Šteins included in Gaia
DR3 against the phase angle $\alpha$ together with the modelled magnitudes.
Two models are shown, corresponding to the pole orientations I (left) and II
(right) given in Table 4.

We studied (21) Lutetia with the 23 transit magnitudes obtained by Gaia (case
I) using a shape model derived from combined ground-based relative photometry
of 50 light curves with 4012 photometric points in total (Ďurech et al., 2010)
and Gaia photometry with convex inversion methods (Muinonen et al., 2020;
Kaasalainen et al., 2001; Kaasalainen & Torppa, 2001). The results showed that
the Gaia measurements contributed to the shape modelling with reasonable
residuals, with a low RMS value of 0.016 mag. The RMS value for the entire
data set of 51 light curves was 0.015 mag. Next (case II), it was seen that
using the best available shape model (Farnham, 2013), constructed by combining
a high-resolution shape model based on disk-resolved imaging by Rosetta and a
lower-resolution model based on ground-based relative photometry and
silhouette observations with adaptive optics, it was not possible to produce a
straightforward fit to the Gaia observations (see Fig. 28). By optimising the
rotational phase of Lutetia and the slope of the phase curve implied by the
Lommel-Seeliger scattering model, the high-resolution shape model resampled at
a 5-degree resolution reproduced the Gaia photometry with a high RMS value of
0.047 mag, whereas for the entire data set, the RMS value was 0.017 mag. We
interpreted the RMS values for cases I and II as giving a strong indication
that the Gaia data offer new information on the shape or surface properties of
Lutetia.

A closer inspection of the time distribution of the 23 Gaia DR3 transit
magnitudes shows that for 12 of them, the Lutetia hemisphere that
predominantly visible to Gaia was the hemisphere that was not mapped by
Rosetta. It is therefore highly probable that the Gaia data provide additional
information about this portion of the asteroid surface. In particular, the
Gaia photometry, with its absolute phase angle dependence, relates the size
and geometric albedo characteristics of the two hemispheres.

We used the high-resolution shape resampled in 5-degree resolution to model
the photometry of the remaining 11 observations that were obtained with the
hemisphere mapped by Rosetta in view directly (case III). Allowing for
optimisation in only the rotational phase and the slope of the phase curve
implied by the Lommel-Seeliger scattering model, the high-resolution shape
model reproduced the Gaia photometry with an excellent RMS value of 0.019 mag
(Fig. 28). We then applied an analogous approach to the 12 observations
obtained with the hemisphere that was not mapped by Rosetta (case IV). With
constant geometric albedo characteristics, the RMS value was high at 0.057
mag.

The photometric slopes implied by the Lommel-Seeliger scattering model were
then compared in detail. In the cases of convex optimisation using all Gaia
measurements (case I), optimisation of the rotational phase and slope using
the high-resolution shape model and all Gaia measurements (II), measurements
corresponding to the hemisphere observed by Rosetta (III), and measurements
corresponding to the hemisphere that was not observed by Rosetta (IV), the
slopes were 1.89 mag rad-1, 1.56 mag rad-1, 1.65 mag rad-1, and 1.47 mag
rad-1, respectively. First, case I stands out as showing the steepest
photometric slope for a convex shape solution. A plausible explanation is that
global self-shadowing effects due to a non-convex shape steepen the
photometric slope. Second, case IV stands out as showing the shallowest
photometric slope for the high-resolution non-convex shape model. Third, case
III shows the most realistic photometric slope for the high-resolution shape
model: the least-squares fit to the Gaia data is successful with a small RMS-
value. Finally, the case II photometric slope is a compromise between cases
III and IV. Whereas it is beyond the scope of the present study to improve the
Lutetia model, the present analysis indicates varying scattering properties
for the hemisphere that was not imaged by Rosetta. This is also supported by
the fact that all the ground-based light curves, consisting of extensive
observations of both hemispheres and treated in relative sense in the
validation, were well fitted by the high-resolution shape model.

The final photometric slope analysis for Lutetia was carried out in case III
using Markov chain Monte Carlo (MCMC) sampling. The photometric slope for the
hemisphere observed by Rosetta was estimated to be (1.677$\pm$0.075) mag rad-1
using the high-resolution shape model resampled at a 5-degree resolution.
Based on Penttilä et al. (2016), Muinonen et al. (2020), and Martikainen et
al. (2021), these photometric slopes agree with those for Tholen S- and
M-class asteroids.

For Lutetia, the apparent $G$ magnitudes range from 11.1 to 13.1 mag, with
minute nominal observational errors of $0.00033-0.00070$ mag. Our RMS values
above are significantly higher. We conclude that Gaia photometry has an
accuracy better than 0.01-0.02 mag with the limitations on first, the shape
model accuracy, which does not allow us to push the analysis further. Second,
we assume that the same Lommel-Seeliger scattering model is valid everywhere
on the surface of Lutetia. In particular, we assume that a single combination
of geometric albedo and scattering phase function is representative of the
entire surface of Lutetia. Third, it is not possible to exclude that some
anomalous brightness values are present.

The results obtained for (2867) Šteins, whose high-resolution shape model is
not limited by hemisphere scales similar to those of (21) Lutetia, support the
idea that Gaia photometry is indeed accurate. In the case of this object,
there are two pole solutions (Šteins cases I-II, see Table 4) that essentially
only differ in terms of the zero-point of the rotational phase. By directly
using the shape model to reproduce the Gaia data, resampled at a 5-degree
resolution, the RMS values of the observed-computed magnitudes are 0.022 mag
and 0.023 mag for the two pole solutions. These fits are based on 30
photometric points, 3 of which had to be omitted as outliers. Again, the fits
were obtained by optimising only the rotational phase and the photometric
slope implied by the Lommel-Seeliger scattering model.

We continued the analysis of Šteins photometry by resampling the shape model
at a higher 3-degree resolution and simultaneously fitting five parameters,
that is, the rotation period, pole orientation, rotational phase, and
photometric slope. The RMS values for the two pole solutions were lower at
0.018 mag and 0.019 mag, respectively. To compute these fits, the rotational
parameters were regularised to lie within their domains of uncertainty (Table
4). The photometric slope obtained values of 1.515 mag rad-1 (case I) and
1.522 mag rad-1 (II), that is, the values were essentially equal.

Finally, MCMC sampling was carried out for the rotational phase and
photometric slope in cases I-II using the high-resolution shape model re-
sampled at 5-degree resolution. For 5000 samples, the means and standard
deviations of the photometric slope obtained the values of (1.497$\pm$0.064)
mag rad-1 (case I) and (1.458$\pm$0.062) mag rad-1 (II). The values agree
mutually, and they are realistic for high-albedo E-class asteroids (Penttilä
et al., 2016; Muinonen et al., 2020; Martikainen et al., 2021). The mean
values agreed within the given uncertainties with the least-squares values for
the resampled 5-degree and 3-degree shape models.

For Šteins, the apparent $G$ magnitudes ranged over $16.3-18.4$ mag, with
nominal observational errors within $0.0028-0.014$ mag. Considering the
simplicity of the surface-scattering modelling we used, our results can be
considered completely satisfactory (see Fig. 29). The remaining limitations in
the case of (2867) Šteins are related to details of the shape model and
surface-scattering characteristics. In particular, certain assumptions were
made on the scattering properties when the high-resolution shape model was
derived from the Rosetta images. It remains possible that the three
observations that we omitted as outliers were omitted due to modelling issues
rather than low observational accuracy.

In conclusion, we cannot rule out the possibility that the sample published in
Gaia DR3 could still include a non-negligible fraction of anomalous data. We
recommend detailed analyses and careful verifications when the Gaia DR3
photometry of asteroids is applied. However, especially for objects as bright
as Lutetia, our current approach to magnitude prediction cannot investigate
Gaia photometry of SSOs at the level that would be required by the very high
accuracy.

##  7 Conclusions

The Solar System data processing, developed and trained over several years,
reaches maturity with Gaia DR3 by providing the expected large survey for
asteroids and planetary satellites. The different data types (astrometry,
photometry, and low-resolution reflectance spectra) and their accuracy and
homogeneity are an impressive achievement of the Gaia mission. We have
summarized the approach that we followed for the processing pipeline and in
particular explained the procedures that clean the data set and provide a
reliable outcome.

The highly accurate epoch astrometry obtained by Gaia is confirmed by the
orbit adjustment, resulting in residuals at submilliarcsecond level for
$G<18$. The astrometric performance is clearly improved with respect to Gaia
DR2.

This accuracy for a large number of asteroids brings new capabilities of
investigation, revealing effects related to the partially resolved shape of
asteroids. The amplitude of the photocentre wobble that we measured on (21)
Lutetia shows that this effect probably affects the orbit computation even for
asteroids that are three to four times smaller. The capability of detecting
the wobbling, associated with the orbital motion of a satellite, discloses an
impressive domain of investigation for the search of asteroid binaries with
the astrometric method. This approach was simply not possible before. A
comprehensive exploration of Gaia DR3 should reveal a variety of binary
systems.

A search for the best orbital modelling for near-Earth objects that can
exhibit the Yarkovsky drift shows that in some circumstances, Gaia can detect
this effect, even in the absence of the radar ranging data that have been
essential in the recent past. A careful coupling to ground-based data will
fully disclose this potential.

The G-band photometric data are the other valuable source of information that
will be exploited to obtain new constraints on the rotation and the shape of a
very large number of objects. For them, it remains difficult to assess the
quality at the level of their expected uncertainty, simply because there is no
direct comparison to other data sets of comparable accuracy. The simulated
photometry from accurate shape models brings, however, positive results,
despite the remaining uncertainties. It also shows that some outliers remain
probably present among the released measurements.

In conclusion, our review shows that the data processing of Solar System data
in Gaia DR3 has produced an extremely rich data set standing out for its
unique properties with respect to other existing surveys in many aspects. Gaia
DR3 will certainly be exploited in many ways by the community of planetary
scientists. New properties and features beyond those illustrated in this
article will probably be found, and will be a major driver for improvements of
the data quality in future data releases.

###### Acknowledgements.

We dedicate this work to the memory of Dimitri Pourbaix, who managed the
Coordination Unit 4 of DPAC with immense dedication, enthusiasm and
intellectual honesty. His legacy is also present in this work. This work
presents results from the European Space Agency (ESA) space mission Gaia. Gaia
data are being processed by the Gaia Data Processing and Analysis Consortium
(DPAC). Funding for the DPAC is provided by national institutions, in
particular the institutions participating in the Gaia MultiLateral Agreement
(MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia
archive website is https://archives.esac.esa.int/gaia. Acknowledgements are
given in Appendix A

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## Appendix A

The Gaia mission and data processing have financially been supported by, in
alphabetical order by country:

  * •

the Algerian Centre de Recherche en Astronomie, Astrophysique et Géophysique
of Bouzareah Observatory;

  * •

the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Hertha
Firnberg Programme through grants T359, P20046, and P23737;

  * •

the BELgian federal Science Policy Office (BELSPO) through various PROgramme
de Développement d’Expériences scientifiques (PRODEX) grants and the Polish
Academy of Sciences - Fonds Wetenschappelijk Onderzoek through grant
VS.091.16N, and the Fonds de la Recherche Scientifique (FNRS), and the
Research Council of Katholieke Universiteit (KU) Leuven through grant
C16/18/005 (Pushing AsteRoseismology to the next level with TESS, GaiA, and
the Sloan DIgital Sky SurvEy – PARADISE);

  * •

the Brazil-France exchange programmes Fundação de Amparo à Pesquisa do Estado
de São Paulo (FAPESP) and Coordenação de Aperfeicoamento de Pessoal de Nível
Superior (CAPES) - Comité Français d’Evaluation de la Coopération
Universitaire et Scientifique avec le Brésil (COFECUB);

  * •

the Chilean Agencia Nacional de Investigación y Desarrollo (ANID) through
Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Regular
Project 1210992 (L. Chemin);

  * •

the National Natural Science Foundation of China (NSFC) through grants
11573054, 11703065, and 12173069, the China Scholarship Council through grant
201806040200, and the Natural Science Foundation of Shanghai through grant
21ZR1474100;

  * •

the Tenure Track Pilot Programme of the Croatian Science Foundation and the
École Polytechnique Fédérale de Lausanne and the project TTP-2018-07-1171
‘Mining the Variable Sky’, with the funds of the Croatian-Swiss Research
Programme;

  * •

the Czech-Republic Ministry of Education, Youth, and Sports through grant LG
15010 and INTER-EXCELLENCE grant LTAUSA18093, and the Czech Space Office
through ESA PECS contract 98058;

  * •

the Danish Ministry of Science;

  * •

the Estonian Ministry of Education and Research through grant IUT40-1;

  * •

the European Commission’s Sixth Framework Programme through the European
Leadership in Space Astrometry (ELSA) Marie Curie Research Training Network
(MRTN-CT-2006-033481), through Marie Curie project PIOF-GA-2009-255267 (Space
AsteroSeismology & RR Lyrae stars, SAS-RRL), and through a Marie Curie
Transfer-of-Knowledge (ToK) fellowship (MTKD-CT-2004-014188); the European
Commission’s Seventh Framework Programme through grant FP7-606740
(FP7-SPACE-2013-1) for the Gaia European Network for Improved data User
Services (GENIUS) and through grant 264895 for the Gaia Research for European
Astronomy Training (GREAT-ITN) network;

  * •

the European Cooperation in Science and Technology (COST) through COST Action
CA18104 ‘Revealing the Milky Way with Gaia (MW-Gaia)’;

  * •

the European Research Council (ERC) through grants 320360, 647208, and 834148
and through the European Union’s Horizon 2020 research and innovation and
excellent science programmes through Marie Skłodowska-Curie grant 745617 (Our
Galaxy at full HD – Gal-HD) and 895174 (The build-up and fate of self-
gravitating systems in the Universe) as well as grants 687378 (Small Bodies:
Near and Far), 682115 (Using the Magellanic Clouds to Understand the
Interaction of Galaxies), 695099 (A sub-percent distance scale from binaries
and Cepheids – CepBin), 716155 (Structured ACCREtion Disks – SACCRED), 951549
(Sub-percent calibration of the extragalactic distance scale in the era of big
surveys – UniverScale), and 101004214 (Innovative Scientific Data Exploration
and Exploitation Applications for Space Sciences – EXPLORE);

  * •

the European Science Foundation (ESF), in the framework of the Gaia Research
for European Astronomy Training Research Network Programme (GREAT-ESF);

  * •

the European Space Agency (ESA) in the framework of the Gaia project, through
the Plan for European Cooperating States (PECS) programme through contracts
C98090 and 4000106398/12/NL/KML for Hungary, through contract
4000115263/15/NL/IB for Germany, and through PROgramme de Développement
d’Expériences scientifiques (PRODEX) grant 4000127986 for Slovenia;

  * •

the Academy of Finland through grants 299543, 307157, 325805, 328654, 336546,
and 345115 and the Magnus Ehrnrooth Foundation;

  * •

the French Centre National d’Études Spatiales (CNES), the Agence Nationale de
la Recherche (ANR) through grant ANR-10-IDEX-0001-02 for the ‘Investissements
d’avenir’ programme, through grant ANR-15-CE31-0007 for project ‘Modelling the
Milky Way in the Gaia era’ (MOD4Gaia), through grant ANR-14-CE33-0014-01 for
project ‘The Milky Way disc formation in the Gaia era’ (ARCHEOGAL), through
grant ANR-15-CE31-0012-01 for project ‘Unlocking the potential of Cepheids as
primary distance calibrators’ (UnlockCepheids), through grant ANR-19-CE31-0017
for project ‘Secular evolution of galxies’ (SEGAL), and through grant
ANR-18-CE31-0006 for project ‘Galactic Dark Matter’ (GaDaMa), the Centre
National de la Recherche Scientifique (CNRS) and its SNO Gaia of the Institut
des Sciences de l’Univers (INSU), its Programmes Nationaux: Cosmologie et
Galaxies (PNCG), Gravitation Références Astronomie Métrologie (PNGRAM),
Planétologie (PNP), Physique et Chimie du Milieu Interstellaire (PCMI), and
Physique Stellaire (PNPS), the ‘Action Fédératrice Gaia’ of the Observatoire
de Paris, the Région de Franche-Comté, the Institut National Polytechnique
(INP) and the Institut National de Physique nucléaire et de Physique des
Particules (IN2P3) co-funded by CNES;

  * •

the German Aerospace Agency (Deutsches Zentrum für Luft- und Raumfahrt e.V.,
DLR) through grants 50QG0501, 50QG0601, 50QG0602, 50QG0701, 50QG0901,
50QG1001, 50QG1101, 50QG1401, 50QG1402, 50QG1403, 50QG1404, 50QG1904,
50QG2101, 50QG2102, and 50QG2202, and the Centre for Information Services and
High Performance Computing (ZIH) at the Technische Universität Dresden for
generous allocations of computer time;

  * •

the Hungarian Academy of Sciences through the Lendület Programme grants
LP2014-17 and LP2018-7 and the Hungarian National Research, Development, and
Innovation Office (NKFIH) through grant KKP-137523 (‘SeismoLab’);

  * •

the Science Foundation Ireland (SFI) through a Royal Society - SFI University
Research Fellowship (M. Fraser);

  * •

the Israel Ministry of Science and Technology through grant 3-18143 and the
Tel Aviv University Center for Artificial Intelligence and Data Science (TAD)
through a grant;

  * •

the Agenzia Spaziale Italiana (ASI) through contracts I/037/08/0, I/058/10/0,
2014-025-R.0, 2014-025-R.1.2015, and 2018-24-HH.0 to the Italian Istituto
Nazionale di Astrofisica (INAF), contract 2014-049-R.0/1/2 to INAF for the
Space Science Data Centre (SSDC, formerly known as the ASI Science Data
Center, ASDC), contracts I/008/10/0, 2013/030/I.0, 2013-030-I.0.1-2015, and
2016-17-I.0 to the Aerospace Logistics Technology Engineering Company (ALTEC
S.p.A.), INAF, and the Italian Ministry of Education, University, and Research
(Ministero dell’Istruzione, dell’Università e della Ricerca) through the
Premiale project ‘MIning The Cosmos Big Data and Innovative Italian Technology
for Frontier Astrophysics and Cosmology’ (MITiC);

  * •

the Netherlands Organisation for Scientific Research (NWO) through grant
NWO-M-614.061.414, through a VICI grant (A. Helmi), and through a Spinoza
prize (A. Helmi), and the Netherlands Research School for Astronomy (NOVA);

  * •

the Polish National Science Centre through HARMONIA grant 2018/30/M/ST9/00311
and DAINA grant 2017/27/L/ST9/03221 and the Ministry of Science and Higher
Education (MNiSW) through grant DIR/WK/2018/12;

  * •

the Portuguese Fundação para a Ciência e a Tecnologia (FCT) through national
funds, grants SFRH/BD/128840/2017 and PTDC/FIS-AST/30389/2017, and work
contract DL 57/2016/CP1364/CT0006, the Fundo Europeu de Desenvolvimento
Regional (FEDER) through grant POCI-01-0145-FEDER-030389 and its Programa
Operacional Competitividade e Internacionalização (COMPETE2020) through grants
UIDB/04434/2020 and UIDP/04434/2020, and the Strategic Programme
UIDB/00099/2020 for the Centro de Astrofísica e Gravitação (CENTRA);

  * •

the Slovenian Research Agency through grant P1-0188;

  * •

the Spanish Ministry of Economy (MINECO/FEDER, UE), the Spanish Ministry of
Science and Innovation (MICIN), the Spanish Ministry of Education, Culture,
and Sports, and the Spanish Government through grants BES-2016-078499,
BES-2017-083126, BES-C-2017-0085, ESP2016-80079-C2-1-R, ESP2016-80079-C2-2-R,
FPU16/03827, PDC2021-121059-C22, RTI2018-095076-B-C22, and TIN2015-65316-P
(‘Computación de Altas Prestaciones VII’), the Juan de la Cierva Incorporación
Programme (FJCI-2015-2671 and IJC2019-04862-I for F. Anders), the Severo Ochoa
Centre of Excellence Programme (SEV2015-0493), and
MICIN/AEI/10.13039/501100011033 (and the European Union through European
Regional Development Fund ‘A way of making Europe’) through grant
RTI2018-095076-B-C21, the Institute of Cosmos Sciences University of Barcelona
(ICCUB, Unidad de Excelencia ‘María de Maeztu’) through grant
CEX2019-000918-M, the University of Barcelona’s official doctoral programme
for the development of an R+D+i project through an Ajuts de Personal
Investigador en Formació (APIF) grant, the Spanish Virtual Observatory through
project AyA2017-84089, the Galician Regional Government, Xunta de Galicia,
through grants ED431B-2021/36, ED481A-2019/155, and ED481A-2021/296, the
Centro de Investigación en Tecnologías de la Información y las Comunicaciones
(CITIC), funded by the Xunta de Galicia and the European Union (European
Regional Development Fund – Galicia 2014-2020 Programme), through grant
ED431G-2019/01, the Red Española de Supercomputación (RES) computer resources
at MareNostrum, the Barcelona Supercomputing Centre - Centro Nacional de
Supercomputación (BSC-CNS) through activities AECT-2017-2-0002,
AECT-2017-3-0006, AECT-2018-1-0017, AECT-2018-2-0013, AECT-2018-3-0011,
AECT-2019-1-0010, AECT-2019-2-0014, AECT-2019-3-0003, AECT-2020-1-0004, and
DATA-2020-1-0010, the Departament d’Innovació, Universitats i Empresa de la
Generalitat de Catalunya through grant 2014-SGR-1051 for project ‘Models de
Programació i Entorns d’Execució Parallels’ (MPEXPAR), and Ramon y Cajal
Fellowship RYC2018-025968-I funded by MICIN/AEI/10.13039/501100011033 and the
European Science Foundation (‘Investing in your future’);

  * •

the Swedish National Space Agency (SNSA/Rymdstyrelsen);

  * •

the Swiss State Secretariat for Education, Research, and Innovation through
the Swiss Activités Nationales Complémentaires and the Swiss National Science
Foundation through an Eccellenza Professorial Fellowship (award PCEFP2_194638
for R. Anderson);

  * •

the United Kingdom Particle Physics and Astronomy Research Council (PPARC),
the United Kingdom Science and Technology Facilities Council (STFC), and the
United Kingdom Space Agency (UKSA) through the following grants to the
University of Bristol, the University of Cambridge, the University of
Edinburgh, the University of Leicester, the Mullard Space Sciences Laboratory
of University College London, and the United Kingdom Rutherford Appleton
Laboratory (RAL): PP/D006511/1, PP/D006546/1, PP/D006570/1, ST/I000852/1,
ST/J005045/1, ST/K00056X/1, ST/K000209/1, ST/K000756/1, ST/L006561/1,
ST/N000595/1, ST/N000641/1, ST/N000978/1, ST/N001117/1, ST/S000089/1,
ST/S000976/1, ST/S000984/1, ST/S001123/1, ST/S001948/1, ST/S001980/1,
ST/S002103/1, ST/V000969/1, ST/W002469/1, ST/W002493/1, ST/W002671/1,
ST/W002809/1, and EP/V520342/1.

The GBOT programme uses observations collected at (i) the European
Organisation for Astronomical Research in the Southern Hemisphere (ESO) with
the VLT Survey Telescope (VST), under ESO programmes 092.B-0165, 093.B-0236,
094.B-0181, 095.B-0046, 096.B-0162, 097.B-0304, 098.B-0030, 099.B-0034,
0100.B-0131, 0101.B-0156, 0102.B-0174, and 0103.B-0165; and (ii) the Liverpool
Telescope, which is operated on the island of La Palma by Liverpool John
Moores University in the Spanish Observatorio del Roque de los Muchachos of
the Instituto de Astrofísica de Canarias with financial support from the
United Kingdom Science and Technology Facilities Council, and (iii) telescopes
of the Las Cumbres Observatory Global Telescope Network.

L. Liberato acknowledges support by the Coordena cão de Aperfei coamento de
Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001, also by CAPES-
PRINT Process 88887.570251/2020-00.

The authors want to acknowledge Valéry Lainey (IMCCE, Paris obseervatory) for
providing the ephemerides of planetary satellites, and Josselin Desmars
(IMCCE, Paris observatory) for providing extensive external and quality checks
on the orbit computations from the NIMA software.
We made use of the software products TOPCAT, (Taylor 2005); Matplotlib (Hunter
2007); Astropy, a community-developed core Python package for Astronomy
(Astropy Collaboration et al. 2013, 2018).