<figure><img src="image_49.png" /><figcaption>**Fig. 47.** HRD of sources with low astrometric mass functions ($<0.001 M_\odot$; green dots); the grey background is the DR3 low-extinction HRD. A very large fraction are not stars with low-mass companions, but rather binaries with a mass ratio similar to their flux ratio. The two blue sources are HD 12800 and HD 3221, described in Sect. 8.7 while the four WDs are discussed in Sect. 8.8.</figcaption></figure>

programs in the solar neighbourhood (e.g. Cumming et al. 2008; Howard et al. 2010; Mayor et al. 2011; Bonfils et al. 2013; Butler et al. 2017; Rosenthal et al. 2021; Pinamonti et al. 2022). The Gaia DR3 astrometric performance levels reach the sensitivity to detect substellar companions around a statistically significant number of stars, enabling first-time measurements of their three-dimensional orbital architectures and true masses.

## 8.1. Astrometry: substellar companions

A naive search for substellar companions detected by Gaia astrometry might simply select solutions with low values of the astrometric mass function, say $f(M) < 0.001 M_\odot$. However, inspection of Fig. 47 shows that a sizeable fraction of the companions of sources with low $f(M)$ do not have small secondary mass with a negligible flux ratio but rather have a flux ratio close to the mass ratio, leaving them clearly above the main sequence.

Keeping this in mind, we browsed the catalogue of masses presented in Sect. 5.1 to investigate the regime of astrometrically detected companions with lower mass bound $M_c$ (assuming they contribute no flux) in the substellar regime, operationally defined as having $20 \le M_c \le 80 M_{Jup}$ and $M_c \le 20 M_{Jup}$ for brown dwarfs (BDs) and exoplanets (EPs), respectively. For a subset of sources with orbital solutions in the Gaia DR3 archive but without a companion mass estimate in our catalogue of masses, the information was derived based on primary mass estimates from the Starhorse catalogue (Anders et al. 2022).

A total of 1843 BDs and 72 EPs were identified in the catalogue of companion masses. This includes 20 sources with AstroSpectroSB1 solution type that have upper bounds to the companion mass $<80 M_{Jup}$, that is, for which the assumption of negligible flux ratio is confirmed. A small subset of 10 BDs were already known, identified in ground-based RV surveys for planets (Ma & Ge 2014; Wilson et al. 2016; Kiefer et al. 2019; Dalal et al. 2021). Table 11 reports the basic comparison between period and eccentricity from Gaia and the literature, as well as minimum and true mass estimates. A total of 9 known EPs were also validated against literature sources, and we report the same information in Table 11. As an illustrative example, the astrometric orbital solution for HD 81040 b is shown online<fnref n="18" />. Additional plots of Gaia DR3 orbits of substellar companions can also be found in Holl et al. (2023b).

## 8.2. Astrometric masses: transition regimes

The results of long-term Doppler surveys have allowed detailed studies of the shape of the mass distribution of relatively close-in ($a \le 5$ au or so) companions to solar-type (F-G-K-type) stars, particularly in the two transition regimes between EPs and BDs and between BDs and stars. The most notable feature is the so-called ‘brown dwarf desert’: the (minimum) mass distribution has a clear decline moving from the planetary-mass to the BD-mass regime, reaches an apparent plateau with a minimum at $\sim 40–50 M_{Jup}$ (0.04–0.05 $M_\odot$), and then rises again reaching the stellar-mass regime (e.g. Grether & Lineweaver 2006; Sahlmann et al. 2011; Ma & Ge 2014; Grieves et al. 2017).

Figure 48 shows the distribution of primary masses for sources with astrometrically detected substellar-mass companions for the cases of NSS solution types (Orbital and OrbitalTargetedSearch*). The medians of the two distributions are $0.42 M_\odot$ and $0.91 M_\odot$, respectively. The striking difference stems from the different ways in which the input lists for the two analysis channels were constructed (see Sects. 2.2.1 and 2.2.2, and references therein). In particular, the bulk of the sources with known solutions input to the alternative orbit determination algorithms are solar-type stars, and this is reflected in Fig. 48. The calibration levels in the bright-star regime are still suboptimal for Gaia DR3; consequently, it is expected that nearby, relatively faint low-mass stars might be the sample of primaries around which the chances of detecting substellar companions are maximised, and this is also reflected in Fig. 48.

The three panels of Fig. 49 show the distribution of substellar companion masses for three samples. The distribution for the OrbitalTargetedSearch* sample (top panel) corroborates the notion – already provided by Doppler surveys – of a minimum in the occurrence of $\sim 40 M_{Jup}$ close-in BDs around solar-type stars. For the first time, Gaia DR3 offers the opportunity to see the feature in the distribution based on true companion mass estimates, without the ambiguities inherent to studies of the population of substellar companions based on minimum mass values and/or simulation-driven upper mass limits (e.g. Kiefer et al. 2021, and references therein).

The centre and bottom panels of Fig. 49 show the equivalent distribution for the main NSS sample, split into two regimes of companions with substellar masses around M dwarfs ($M_\star < 0.6 M_\odot$) and higher-mass primaries, respectively. The first notable feature is the difference in slope between the two samples rising towards the highest masses: a simple power-law fit returns $N \propto M_c^{1.56}$ and $N \propto M_c^{3.03}$ for companions around

<footnote n="18">¹⁸ https://www.cosmos.esa.int/web/gaia/iow_20220131</footnote>
<table><caption>**Table 11.** Known substellar companions with a confirmed mass in the planetary and brown dwarf regime, respectively.</caption><thead><tr><th>Gaia DR3</th><th>Name</th><th>M<sub>c</sub> sin i<br>(M<sub>Jup</sub>)</th><th>M<sub>c</sub><br>(M<sub>Jup</sub>)</th><th>i<br>(deg)</th><th>P<sub>lit</sub><br>(days)</th><th>P<sub>Gaia</sub><br>(days)</th><th>e<sub>lit</sub></th><th>e<sub>Gaia</sub></th><th>a<sub>0</sub><br>(mas)</th><th>Refs.</th></tr></thead><tbody><tr><td>6421118739093252224</td><td>HD 175167 b</td><td>7.8 ± 3.5</td><td>9.5 ± 0.9</td><td>28 ± 19</td><td>1290 ± 22</td><td>898 ± 198</td><td>0.54 ± 0.09</td><td>0.19 ± 0.12</td><td>0.22 ± 0.02</td><td>1</td></tr><tr><td>4062446910648807168</td><td>HD 164604 b</td><td>2.7 ± 1.3</td><td>14.3 ± 5.5</td><td>29 ± 19</td><td>606 ± 9</td><td>615 ± 12</td><td>0.24 ± 0.14</td><td>0.61 ± 0.34</td><td>0.56 ± 0.22</td><td>1, 2</td></tr><tr><td>1594127865540229888</td><td>HD 132406 b</td><td>5.6</td><td>6.7 ± 2.1</td><td>122 ± 14</td><td>974 ± 39</td><td>893 ± 251</td><td>0.34 ± 0.09</td><td>0.31 ± 0.29</td><td>0.16 ± 0.04</td><td>3</td></tr><tr><td>4745373133284418816</td><td>HR 810 b</td><td>2.26 ± 0.18</td><td>6.2 ± 0.5</td><td>87 ± 6</td><td>312 ± 5</td><td>332 ± 6</td><td>0.15 ± 0.05</td><td>0.04 ± 0.2</td><td>0.30 ± 0.02</td><td>4, 5</td></tr><tr><td>2367734656180397952</td><td>BD −17 0063 b</td><td>5.1 ± 0.12</td><td>4.3 ± 0.5</td><td>80 ± 6</td><td>656 ± 0.6</td><td>649 ± 36</td><td>0.54 ± 0.005</td><td>0.28 ± 0.22</td><td>0.22 ± 0.02</td><td>6</td></tr><tr><td>5855730584310531200</td><td>HD 111232 b</td><td>6.8</td><td>8.3 ± 0.6</td><td>97 ± 4</td><td>1143 ± 14</td><td>882 ± 34</td><td>0.20 ± 0.01</td><td>0.50 ± 0.10</td><td>0.51 ± 0.03</td><td>7</td></tr><tr><td>637329067477530368</td><td>HD 81040 b</td><td>6.8 ± 0.7</td><td>7.9 ± 0.9</td><td>108 ± 6</td><td>1002 ± 7</td><td>851 ± 113</td><td>0.53 ± 0.04</td><td>0.37 ± 0.15</td><td>0.39 ± 0.03</td><td>8</td></tr><tr><td>4976894960284258048</td><td>HD 142 b</td><td>1.3 ± 0.2</td><td>7.1 ± 1.0</td><td>59 ± 7</td><td>350 ± 4</td><td>319 ± 7</td><td>0.26 ± 0.18</td><td>0.26 ± 0.23</td><td>0.21 ± 0.03</td><td>5, 9, 10</td></tr><tr><td>2603090003484152064</td><td>GJ 876 b</td><td>2.1 ± 0.2</td><td>3.6 ± 0.4</td><td>101 ± 8</td><td>61.08 ± 0.01</td><td>61.4 ± 0.2</td><td>0.027 ± 0.002</td><td>0.16 ± 0.15</td><td>0.43 ± 0.05</td><td>11–18</td></tr><tr><td>2651390587219807744</td><td>BD −00 4475 b</td><td>25 ± 2</td><td>48.4 ± 7.6</td><td>129 ± 7</td><td>723.2 ± 0.7</td><td>780 ± 84</td><td>0.39 ± 0.01</td><td>0.48 ± 0.11</td><td>1.91 ± 0.28</td><td>19</td></tr><tr><td>2778298280881817984</td><td>HD 5433 b</td><td>49 ± 3</td><td>53.8 ± 1.7</td><td>12 ± 39</td><td>576.6 ± 1.6</td><td>576.7 ± 10.6</td><td>0.81 ± 0.02</td><td>0.46 ± 0.12</td><td>1.04 ± 0.03</td><td>19</td></tr><tr><td>3309006602007842048</td><td>HD 30246 b</td><td>55<sup>+20</sup><sub>-8</sub></td><td>40.6 ± 8.3</td><td>78 ± 2</td><td>990 ± 6</td><td>814 ± 141</td><td>0.84 ± 0.08</td><td>0.59 ± 0.10</td><td>1.34 ± 0.24</td><td>19</td></tr><tr><td>3750881083756656128</td><td>HD 91669 b</td><td>30.6 ± 2.1</td><td>43.2 ± 2.2</td><td>58 ± 3</td><td>497.5 ± 0.6</td><td>500.4 ± 6.9</td><td>0.448 ± 0.002</td><td>0.32 ± 0.06</td><td>0.73 ± 0.04</td><td>19</td></tr><tr><td>3751763647996317056</td><td>HD 89707 b</td><td>54<sup>+8</sup><sub>-7</sub></td><td>82.5 ± 12.7</td><td>54 ± 10</td><td>298.5 ± 0.1</td><td>297 ± 2</td><td>0.90 ± 0.04</td><td>0.68 ± 0.20</td><td>1.82 ± 0.30</td><td>19</td></tr><tr><td>685029558383335168</td><td>HD 77065 b</td><td>41 ± 2</td><td>64.2 ± 5.1</td><td>42 ± 3</td><td>119.113 ± 0.003</td><td>119.1 ± 0.2</td><td>0.694 ± 0.0004</td><td>0.70 ± 0.04</td><td>1.04 ± 0.07</td><td>20</td></tr><tr><td>855523714036230016</td><td>HD 92320 b</td><td>59.4 ± 4.0</td><td>70 ± 3.1</td><td>111 ± 2</td><td>145.4 ± 0.01</td><td>145.1 ± 0.3</td><td>0.323 ± 0.001</td><td>0.26 ± 0.05</td><td>0.82 ± 0.01</td><td>20</td></tr><tr><td>824461960796102528</td><td>HD 82460 b</td><td>73.2 ± 3.0</td><td>62.5 ± 6.4</td><td>66 ± 1</td><td>590.9 ± 0.2</td><td>579 ± 6</td><td>0.84 ± 0.01</td><td>0.73 ± 0.04</td><td>1.63 ± 0.09</td><td>21</td></tr><tr><td>873616860770228352</td><td>BD +29 1539 b</td><td>59.7 ± 2.0</td><td>60.7 ± 23.5</td><td>120 ± 9</td><td>175.87 ± 0.01</td><td>173 ± 3</td><td>0.275 ± 0.001</td><td>0.43 ± 0.12</td><td>0.61 ± 0.14</td><td>21</td></tr><tr><td>5563001178343925376</td><td>HD 52756 b</td><td>59.3 ± 2.0</td><td>61.2 ± 8.6</td><td>73 ± 4</td><td>52.8657 ± 0.0001</td><td>52.9 ± 0.1</td><td>0.678 ± 0.0003</td><td>0.54 ± 0.16</td><td>0.54 ± 0.08</td><td>22</td></tr></tbody></table>

**References.** (1) Arriagada et al. (2010); (2) Feng et al. (2019); (3) da Silva et al. (2007); (4) Kürster et al. (2000); (5) Butler et al. (2006); (6) Moutou et al. (2009); (7) Mayor et al. (2004); (8) Sozzetti et al. (2006); (9) Tinney et al. (2002); (10) Wittenmyer et al. (2012); (11) Trifonov et al. (2018); (12) Rivera et al. (2005); (13) Rivera et al. (2010); (14) Benedict et al. (2002); (15) Marcy et al. (1998); (16) Marcy et al. (2001); (17) Correia et al. (2010); (18) Nelson et al. (2016); (19) Dalal et al. (2021); (20) Wilson et al. (2016); (21) Kiefer et al. (2019); (22) Sahlmann et al. (2011).

<figure><img src="image_50.png" /><figcaption>**Fig. 48.** Primary mass distributions for sources with astrometrically detected substellar companions with Orbital (long-dashed histogram) and OrbitalTargetedSearch* (solid-line histogram) solution types.</figcaption></figure>

M dwarfs and higher mass stars, respectively. A Kolmogorov-Smirnov (K-S) test indicates that the two distributions of substellar companion masses around M dwarfs and higher mass stars in the centre and bottom panels of Fig. 49 have a *p*-value of ~1 × 10⁻⁶, allowing us to reject the hypothesis that they are drawn from the same distributions. Secondly, the occurrence of detected companions around higher mass dwarfs appears roughly flat in the approximate range 0.04 ≤ *M*<sub>c</sub> ≤ 0.06 *M*<sub>☉</sub>. For M dwarf primaries, the distribution continues to show a declining trend towards the low-mass end (corresponding to super-Jupiter-mass objects with *M*<sub>c</sub> ≤ 0.01 *M*<sub>☉</sub>).

The above differences could in principle be mostly due to the not-yet-well-characterised sensitivity to substellar companions in different mass and orbital separation regimes of Gaia DR3 astrometry. However, a few considerations help to reinforce the idea that we are seeing, at least in part, intrinsic features in the

distributions rather than effects due to the selection function. Most notably, (1) the flat shape of the distribution in the bottom panel of Fig. 49 is consistent with that of the substellar companion distribution derived from RV surveys (e.g. Kiefer et al. 2019), and (2) the declining trend in the distribution towards the lowest-mass end in the central panel of Fig. 49 is consistent with the well-established notion of a much lower frequency of giant planets around M dwarfs with respect to solar-type primaries (e.g. Endl et al. 2006; Cumming et al. 2008; Bonfils et al. 2013; Pinamonti et al. 2022). With Gaia DR3, we therefore achieve the first-ever characterisation of a conspicuous population of substellar companions with true mass estimates within typically 1–2 au of nearby M dwarfs.

### 8.3. Substellar companion frequency in the 100 pc sample

As low-mass stars provide the primary sample around which substellar companions have been detected with Gaia DR3 astrometry, we can attempt to derive a first-order estimate of their occurrence rate. Clearly, a detailed assessment of the Gaia sensitivity in terms of completeness (estimation of the number of missed companions) and reliability (estimate of the number of false detections) is warranted, but goes beyond the scope of this work, and will be presented elsewhere (Giacobbe et al., in prep.).

Gaia Collaboration (2021b) show that Gaia DR3 is complete down to the M7 spectral sub-type within 100 pc from the Sun, with an M dwarf sample amounting to 218 366 sources. The NSS sample encompasses 790 astrometrically detected companions with likely substellar masses around M dwarfs within 100 pc. Of these, the vast majority (~94%) are BDs. Under the optimistic assumptions that (i) Gaia has homogeneous sensitivity to and is 100% complete for BD companions across the M dwarf 100-pc sample, (ii) none of the orbital solutions corresponding to BD companions around this sample are spurious, and (iii) the companion does not contribute light, we propose an initial value for the frequency of BDs around M dwarfs with *P* ≤ 1000 days of ~0.3%.
<figure><img src="image_51.png" /><figcaption>**Fig. 49.** Top: substellar companion mass distribution for the OrbitalTargetedSearch* solution type. Centre: same as above, but for the Orbital solution type, with a cut-off in the primary mass $M_{\star} < 0.6 M_{\odot}$. Bottom: same again, but for the Orbital solution type, with a cut-off in the primary mass $M_{\star} > 0.6 M_{\odot}$.</figcaption></figure>

Dieterich et al. (2012) report a BD companion frequency around M dwarfs of $2.3_{-0.7}^{+5.0\%}$ for separation in the range 10–70 au. Bowler et al. (2015) find that such companions in the 10–100 au orbital radius range have an occurrence rate of $2.8_{-1.5}^{+2.4\%}$ around M dwarfs in young moving groups. Susemehl & Meyer (2022) find a similar frequency ($2.7_{-0.7}^{+1.0\%}$) for field M dwarfs in the same

separation interval. Winters et al. (2019) report a formally lower value of $1.3 \pm 0.3\%$ for the frequency for BDs around M dwarfs within 25 pc in the separation range out to ~300", but this is still compatible within the uncertainties.

Our occurrence rate estimate is likely an underestimate (pending detailed assessment of the numbers of missed companions versus those of spurious solutions and incorrectly classified objects), but nevertheless it is a clear example of the fact that Gaia DR3 provides critical constraints on the M dwarf binary fraction at close separations and very low mass ratios. Even with corrections for completeness and reliability still to be accounted for, the Gaia M dwarf sample in the solar neighbourhood with sensitivity to substellar companions within a few astronomical units is orders of magnitude larger than those of all other spectroscopic surveys combined.

## 8.4. Astrometric masses: trends with stellar metallicity

The number of new Gaia detections of substellar companions in the EP regime is still too small to provide an independent assessment of well-known trends of EP frequency with stellar properties, such as the strong dependence of giant planet occurrence with metallicity (e.g. Fischer & Valenti 2005; Sozzetti et al. 2009; Santos et al. 2011; Mortier et al. 2012; Adibekyan 2019, and references therein). The population of likely BD companions is instead conspicuous, and can be used to verify the outcome of recent statistical investigations.

As an example, Ma & Ge (2014) showed that, using the iron abundance relative to the Sun [Fe/H] as a proxy, the metallicity distribution of BD solar-type hosts has a median and standard deviation [Fe/H] = -0.04 ± 0.28. Using the most recent compilation of BD companions (based on Wilson et al. 2016; Kiefer et al. 2019; Dalal et al. 2021), the corresponding values are [Fe/H] = +0.01±0.25. The stellar sample is therefore not particularly metal rich, as is the case of giant-planet hosts (median [Fe/H]~ +0.12, e.g. Adibekyan 2019), but is also not as metal deficient as the typical field stars in the solar neighbourhood, depending on stellar sample, with median [Fe/H] in the range [-0.10, -0.15] (see e.g. Nordström et al. 2004; Raghavan et al. 2010; Sousa et al. 2011; Adibekyan 2019).

Gaia Collaboration (2023a) outline recipes for selection of sources with global metallicity ([M/H]) of good and intermediate quality determined based on Gaia data. A total of 17 129 sources with an astrometric orbital solution were selected to have intermediate-quality [M/H] from the archive. Unsurprisingly, the overwhelming majority of [M/H] determinations is for the brighter solar-type stars, and therefore the typical M-dwarf primary with a substellar companion does not have a metallicity determination. However, we find [M/H] = -0.02 ± 0.29 for a sample of 143 F-G-K-type BD hosts. For reference, applying a more strict recipe for good-quality [M/H] returns a sample of 74 sources with [M/H] = +0.01 ± 0.27. Both estimates are in excellent agreement, and indeed indistinguishable, with literature results. If no quality constraints are used, 327 sources with a substellar companion and a Gaia-derived metallicity have [M/H] = -0.25 ± 0.49, which is indicative of the need to restrict ourselves to the regime of primaries with better-calibrated metallicities.

## 8.5. Known substellar objects: statistics and notable examples

The comparison between Gaia orbital solutions and companion mass estimates for known substellar companions and literature
results, despite the small sample sizes, is interesting for a number of reasons.

First of all, a quick look at Table 11 allows us to underline how the angular orbit size for known BDs is always $\ge 0.5$ mas, while the opposite, with a few exceptions, holds for the known EPs, HD 132406 b being the record holder with the smallest measured angular semi-major axis: $a_1 = 136 \pm 40$ µas. Overall, for both EPs and BDs there is a tendency to underestimate orbital periods longer than the time-span of Gaia observations. Not unexpectedly, orbital eccentricities are typically more loosely constrained by Gaia astrometry than by Doppler spectroscopy, and very high-eccentricity orbits ($e \ge 0.8$) are typically underestimated. The loss of accuracy in the long-period and high-eccentricity regimes are known effects, already quantified via detailed simulations by Casertano et al. (2008), and further discussed in by Holl et al. (2023b), for example.

Inspection of Table 11 also shows that there is no simple mapping of $M_c \sin i$ from literature into $M_c$ given the derived $i$ value. For some objects (e.g. HD 132406 b, HD 81040 b, and HD 52756 b), the minimum mass estimate translates into a larger true mass, in agreement with the determined value of inclination, but in other cases (e.g. HD 164604 b, HR 810 b, and HD 142 b), $M_c$ is estimated to be much larger than if it were inferred from $M_c \sin i$ together with the $\sin i$ value, or even lower (e.g. HD 30246 b and HD 82460 b) than $M_c \sin i$. When the parameter uncertainties are taken into account, the discrepancies are typically not very statistically significant, but the effect will nevertheless require elucidation. As both minimum and true mass estimates depend on the assumptions made for the mass of the primary, part of the reason for the discrepancy might be due to the heterogeneity of methods used to derive the latter. However, the more fundamental explanation is likely to be found in the overall limitations of Gaia DR3 detection sensitivity, including the selection effects and biases introduced by astrometric NSS processing, as discussed in Sect. 4.1 and Appendix C, particularly in the limit of relatively low astrometric S/Ns and suboptimal redundancy in the number of visibility periods with respect to the number of model parameters<fnref n="19" />.

Among the companions with a derived mass in the planetary regime, the case of GJ 876 b stands out. The planet constitutes one of the earliest radial-velocity discoveries in the field, first announced by Marcy et al. (1998) as a gas giant with $M_c \sin i \sim 2 M_{Jup}$ orbiting a mid-M dwarf in the backyard of the Sun ($d = 4.67$ pc). Benedict et al. (2002), using HST/FGS data, published the astrometric orbit of GJ 876 b (constrained by the RV solution), determining an orbit size of $0.25 \pm 0.06$ mas, an inclination of $84^\circ$, and a true mass very close to the minimum mass limit. The GJ 876 planetary system was subsequently found to host four planets: the hot super Earth GJ 876 d with period of ~2 d, the two Jupiter-type planets GJ 876 c,b with periods of ~30 and ~61 d, respectively, and the Neptune-mass companion GJ 876 e with a period of ~125 d. The three outermost companions are dynamically interacting, locked in a 1:2:4 Laplace mean-motion resonance, which has been the subject of many studies (e.g. Rivera et al. 2005, 2010; Correia et al. 2010; Nelson et al. 2016; Trifonov et al. 2018). The more recent investigations, based on dynamical considerations, infer a close-to-coplanar configuration for the three interacting planets, and a likely inclination of GJ 876 b ~50–60°.

This implies a true mass of $M_c \sim 2.3–2.7 M_{Jup}$. The amplitude of the astrometric perturbation determined with Gaia, $0.43 \pm 0.05$ mas, is larger than that measured by HST/FGS, and discrepant at the $2.3\sigma$ level. The inferred mass, $3.6 M_{Jup}$, is correspondingly larger, and also in this case the derived inclination $i = 101^\circ$ does not allow for a simple mapping from the $M_c \sin i$ value.

A number of known companions, which in the literature have minimum mass estimates in the planetary regime, appear in Table 3 with much higher true mass estimates from Gaia. These appear in the Gaia DR3 archive as validated orbital solutions of type OrbitalTargetedSearchValidated. A few cases of particular interest are discussed below.

HD 114762 (Gaia DR3 3937211745905473024): The first substellar companion candidate around a solar-type star with minimum mass $M_c \sin i = 0.011 \pm 0.001 M_\odot$ was inferred from radial velocity variations by Latham et al. (1989). Gaia DR1 noise modelling resulted in a considerably higher companion mass estimate of $0.103_{-0.025}^{+0.030} M_\odot$ (Kiefer 2019). The Gaia DR3 orbital solution has a period of $83.73 \pm 0.12$ d in agreement with the radial-velocity orbit and an orbit size of $a_0 = 1.80 \pm 0.07$ mas. Using the primary mass estimate from Table 3 and standard linear propagation of the uncertainties, the inferred companion mass is $M_c = 0.21 \pm 0.01 M_\odot$. Gaia DR3 therefore establishes that the companion is a low-mass M dwarf and not a substellar object.

HD 164604 (Gaia DR3 4062446910648807168): Arriagada et al. (2010) announced a low-confidence detection of a $M_c \sin i \sim 2.7 M_{Jup}$ companion on a $606 \pm 9$ d orbit, whose parameters were then refined by Feng et al. (2019) who reported $M_c \sin i = 1.99 \pm 0.26 M_{Jup}$ and $P = 641 \pm 9$ d. The Gaia DR3 orbital solution has $P = 615 \pm 12$ d (in agreement at the $1.7\sigma$ and $0.6\sigma$ level with Feng et al. 2019 and Arriagada et al. 2010, respectively) and $a_0 = 0.56 \pm 0.22$ mas. The inferred companion mass is $M_c = 14.3 \pm 5.5 M_{Jup}$.

HD 162020 (Gaia DR3 5957920668132624256): Udry et al. (2002) published the discovery of slightly eccentric ($e = 0.28$) $M_c \sin i \sim 14 M_{Jup}$ companion with $P = 8.428$ d, whose minimum mass was recently updated to $9.8 \pm 2.7 M_{Jup}$ by Stassun et al. (2017). The Gaia DR3 orbital solution has $P = 8.429 \pm 0.001$ d, $e = 0.23 \pm 0.05$, and $a_0 = 0.91 \pm 0.03$ mas. The detected companion is a low-mass star with $M_c = 0.39 \pm 0.02 M_\odot$.

KIC 7917485 (Gaia DR3 2075978592919858432): Murphy et al. 2016 published the detection of a $M_c \sin i = 11.8_{-0.6}^{+0.8} M_{Jup}$ companion on a $P = 840_{-20}^{+22}$ d orbit around the Delta Scuti, A-type star Kepler-1648, based on a pulsation timing variation technique. The Gaia DR3 orbital solution has $P = 810 \pm 28$ d and $a_0 = 0.42 \pm 0.02$ mas. At a distance of 1.38 kpc, the companion turns out to be an M dwarf with $M_c = 0.55 \pm 0.03 M_\odot$.

## 8.6. Validated orbital solutions that can imply new exoplanet discoveries

Two sources have validated astrometric orbital solutions (see Holl et al. 2023b, for details) that imply the presence of previously unpublished planetary-mass companions if a ‘binary scenario’ can be excluded. In such a scenario, the small apparent orbit size would be caused by a binary star with components of similar mass and brightness ratios. These two sources are discussed below.

HIP 66074 (Gaia DR3 1712614124767394816): The Gaia DR3 orbital solution has $P = 297 \pm 2.8$ d, $e = 0.46 \pm 0.17$, $a_0 = 0.21 \pm 0.03$ mas. Given the primary mass estimate corre-

<footnote n="19">¹⁹ The typical number of visibility periods is only twice the number of fitted parameters in an orbital model (see Appendix C, and also Holl et al. 2023b).</footnote>
sponding to an M0 dwarf, the inferred companion mass in the exoplanet scenario is $M_c = 7.3 \pm 1.1 M_{\text{Jup}}$.

HIP 28193 (Gaia DR3 2884087104955208064): the orbital period, semi-major axis, and eccentricity of this new exoplanet are $827 \pm 50$ d, $e = 0.07 \pm 0.10$, and $a_0 = 0.25 \pm 0.02$ mas, respectively. Using the K-dwarf primary mass from Table 3, the inferred companion in the exoplanet scenario is a super-Jupiter with a mass of $5.3 \pm 0.6 M_{\text{Jup}}$.

We note that, had we used Monte Carlo resampling, the semi-major axis distribution would have been asymmetric with larger uncertainties, and this would have been the case for the companion-mass distribution as well. This is an example of a solution with a poorly constrained eccentricity ($e = 0.07 \pm 0.10$) for which the uncertainties in the Thiele-Innes coefficients are likely overestimated and Monte Carlo resampling is not advisable (cf. Babusiaux et al. 2023; Holl et al. 2023b).

A third source (Gaia DR3 1035000055055287680, HIP 40497) that initially also fell into this category has been identified as SB2 in the literature (Busà et al. 2007); see also the discussion in Holl et al. (2023b). This illustrates that the risk of confusing the binary and exoplanet scenarios is real when only considering the Gaia astrometric orbit.

For HIP 66074 and HIP 28193 on the other hand, we can make use of auxiliary radial-velocity information. To evaluate the binary scenario where the companion is a MS star, we used the mass–luminosity relationships of Henry & McCarthy (1993) to estimate that the two components must have masses that agree within a few percent to be compatible with both the orbital parameters and the photocentre orbit size. Consequently, the RV semi-amplitudes of the primary components of HIP 66074 and HIP 28193 would be $K_1 \sim 20 \text{ km s}^{-1}$ and $K_1 \sim 9 \text{ km s}^{-1}$, respectively, clearly incompatible with the high-precision RVs used for validating the orbital solution that have dispersions that are three orders of magnitude smaller (Butler et al. 2017; Holl et al. 2023b). Similarly, the Gaia DR3 radialVelocity_error computed from the dispersion of individual Gaia RV measurements is $0.15 \text{ km s}^{-1}$ and $0.16 \text{ km s}^{-1}$, respectively, which lies in the first percentile for sources with $G < 12$, and therefore also appears incompatible with the binary scenario. The blending of the SB2 spectra could possibly lead to such suppressed RV variability. In the case of HIP 28193, the small uncertainties in the ground-based RVs coupled with the estimated FWHM of the underlying cross-correlation functions of $\le 9 \text{ km s}^{-1}$ speak against this possibility<fnref n="20" />. In terms of absolute magnitudes of these systems, the difference between the exoplanet and binary scenario amounts to ~0.8 mag, which, because of the width of the observed HRD, can also not be used to definitely rule out the binary scenario.

In the HIPPARCOS-Gaia catalogues of accelerations produced by Brandt (2021) and Kervella et al. (2022), no statistically significant PMa is reported for HIP 66074 ($S/N \sim 1$), while a moderately high PMa ($S/N \sim 10$) at the Gaia mean epoch is found for HIP 28193. For the relatively short-period orbit of the companion around HIP 66074, the inferred companion mass is approximately in line with a $S/N \sim 1$ in the proper motion difference. In the case of the longer-period orbit of the companion around HIP 28193, the PMa value might point to a companion with a larger mass than the one inferred nominally. An alternative possibility would be

that the Gaia orbit has significantly underestimated the true period.

In summary, the most likely scenario for both HIP 66074 and HIP 28193 is the presence of a newly discovered giant exoplanet. A more detailed analysis and probably more auxiliary data are needed to definitely rule out the binary scenario. When that is achieved, these are to be considered as the first Gaia astrometric planet detections and the first examples of confirmed exoplanet discoveries with the astrometry technique.

## 8.7. Candidates with substellar masses: statistics and notable examples

Among the substellar mass candidates with solution type OrbitalTargetedSearch, two are worth particular mention.

HD 12800 (Gaia DR3 522135261462534528): the Gaia DR3 orbital solution for this bright source (54 Cas) has $P = 401 \pm 12$ d, $a_0 = 0.25 \pm 0.05$ mas, and $M_c = 5.6 \pm 1.4 M_{\text{Jup}}$. This is the only candidate companion around a MS solar-type star with a mass well within the planetary regime.

HD 3221 (Gaia DR3 4901802507993393664): the Gaia DR3 orbital solution for this bright solar-type star has $P = 476 \pm 5$ d, $a_0 = 0.36 \pm 0.01$ mas, and $M_c = 14.2 \pm 0.6 M_{\text{Jup}}$. The primary is a fast-rotating ($v \sin i \sim 70 \text{ km s}^{-1}$), very young star with an estimated age of ~10–30 Myr in the Tucana/Horologium association. The candidate companion, at the planet–brown dwarf mass boundary according to a classical definition (Burrows et al. 2001), if confirmed, would be the first of this type in an orbital separation regime virtually inaccessible to Doppler and direct imaging surveys.

As mentioned above, the population of substellar mass candidates with solution type OrbitalTargetedSearch* is typically found around solar-type primaries. It is interesting to note how the eccentricity distributions for EPs and BDs in this sample are marginally different based on a K-S test (p-value of 0.04), with the more massive BDs ($M_c \ge 50 M_{\text{Jup}}$) having a median of $e \sim 0.5$, while EPs and lower-mass BDs ($M_c \le 40 M_{\text{Jup}}$) have typically $e \sim 0.3$. This is in agreement with Ma & Ge (2014), and with the notion that the former might correspond to the low-mass tail of objects formed in the same manner as stars while the latter sample would map the high-mass tail of objects formed in the same manner as planets.

The population of substellar mass candidates with solution type Orbital is instead predominantly found around low-mass M primaries. In this case, the eccentricity distributions of candidate EPs and BDs detected around this sample are indistinguishable, with typically $e \sim 0.4$ in both cases. This might indicate that most of the detected companions might have formed in the same way, and that some of them are actually of intrinsically larger mass compared to the lower mass bounds adopted in this work as discussed in Sect. 8.1.

## 8.8. White dwarfs with a substellar companion candidate

The nss_two_body_orbit table contains 38 orbital solutions for sources on the WD sequence, and all of them correspond to astrometric orbits. Assuming a fixed mass of $0.6 M_\odot$ for the WD host and that the companion is dark, there are four sources with substellar companion candidates, namely Gaia DR3 2813020961166816512 (LP 522–46) with $M_c \sim 34 M_{\text{Jup}}$, Gaia DR3 2098419251579450880 with $M_c \sim 34 M_{\text{Jup}}$, Gaia DR3 6471102606408911360 (L 279–25) with $M_c \sim 22 M_{\text{Jup}}$, and Gaia DR3 4698424845771339520 (WD 0141–675), with this last one standing out as being part of the 10 pc

<footnote n="20">²⁰ At the times of minimum/maximum RV, the separation between the primary and companion RV should be ~18 km s⁻¹, i.e. wider than the spectroscopic cross-correlation function.</footnote>
sample and with a companion candidate in the planetary-mass range.

In Fig. 39 these four sources are marked with red symbols. Two sources (LP 522–46 and L 279–25) are located above the hydrogen sequence, which suggests that the companion could itself be degenerate and luminous enough to cause an excess in luminosity and at the same time dilute the astrometric orbit signal. The companions of these sources are therefore unlikely to be either dark or substellar. On the other hand, Gaia DR3 2098419251579450880 and WD 0141–675 lie within the hydrogen sequence and therefore Fig. 39 appears consistent with the interpretation that these sources have substellar companions.

WD 0141–675 in particular was included in the OrbitalTargetedSearch sample (Holl et al. 2023b) because it is nearby and metal-polluted (Debes & Kilic 2010), and therefore represents a promising target to search for the presence of an orbiting giant planet that could act as the much sought-after perturber of the circumstellar material (Debes et al. 2012). The Gaia DR3 orbital solution has a period of 33.65 ± 0.05 d and with a WD mass of 0.57 ± 0.03 $M_\odot$ (Subasavage et al. 2017), the estimated planet mass is $M_c = 9.26_{-1.15}^{+2.64} M_{\text{Jup}}$, where we accounted for all parameter covariances using Monte Carlo resampling. The resulting $M_c$ distribution is asymmetric with its mode at ~8.3 $M_{\text{Jup}}$.

With only a handful of known giant planets orbiting white dwarfs (Veras 2021; Blackman et al. 2021), the Gaia discovery of a super-Jupiter candidate planet orbiting WD 0141–675 is remarkable. If the Gaia DR3 orbit is confirmed and other possible scenarios can be excluded – such as some kind of WD binary system with nearly equal-brightness components (WD 0141–675 is marked with an orange square in Fig. 39) –, this would represent the discovery of the most nearby planet-hosting WD and the first giant planet around a metal-enriched WD, which would make it an important test for our understanding of the fate of stars and their planetary systems.

## 8.9. Radial velocity: Substellar companions

Out of $6 \times 10^4$ minimum mass estimates for SB1 solutions (see Table 3), about 10% (5723) have $M_c \sin i < 0.08 M_\odot$, and about 10% of these (437) have $M_c \sin i < 0.02 M_\odot$. Not unexpectedly, the mass distribution for the primaries (see Fig. 50, top panel) has a median of ~1.0 $M_\odot$, similar to that of the primary mass distribution of OrbitalTargetedSearch* sources rather, but with a significantly larger contribution from bright, earlier type stars. The distribution of $M_c \sin i$ shown in the lower panel of Fig. 50 contains an expected feature, namely a decline in numbers of companions at the low-mass end due to the intrinsic lack of sensitivity of Gaia RVS to RV amplitudes of signals typically induced by planetary-mass companions (significantly <1 km s⁻¹). On the other hand, the number of higher mass substellar companions appears constant all the way into the low-mass star regime, and independently of primary mass. This is unexpected, as an intrinsically lower frequency of intermediate-mass BDs is observed in RV surveys (Ma & Ge 2014), particularly in the short-period regime to which Gaia RVs are sensitive. The eccentricity distributions of SB1 companions with mass estimates in the EP and BD regimes appear entirely indistinguishable, independently of primary mass, and this is also not in agreement with the Ma & Ge (2014) analysis.

For more than 80% of the SB1 sample of close-in companions with $M_c \sin i < 0.08 M_\odot$, the orbital solutions have $P < 10$ days, and this fraction grows to >90% for companions

<figure><img src="image_52.png" /><figcaption>**Fig. 50.** Top: primary mass distribution for sources with SB1 solution type and inferred minimum companion masses in the substellar regime. Bottom: substellar companion mass distribution for the SB1 solution type.</figcaption></figure>

with $M_c \sin i < 0.02 M_\odot$. Such companions, which are detected with $K_1$-values with significance of below 10, typically correspond to primaries with $\text{ruwe} > 1.4$, but they are not expected to be those responsible for high $\text{ruwe}$ values, as the typical sizes of the induced astrometric perturbations would escape detection by Gaia (e.g. Belokurov et al. 2020). As discussed in Sect. 2.3, a sizeable fraction of these short-period orbits are rather a kind of alias of longer period orbits. For example, the companions reported around HIP 24329, HD 35956, and HD 8691 have $M_c \sin i \sim 3 M_{\text{Jup}}$, ~11 $M_{\text{Jup}}$, and ~14 $M_{\text{Jup}}$, respectively, and $P = 0.63$ d, 3.02, and 3.77 d, respectively. However, these sources also have Orbital solutions with $P = 1499$ d, 1203 d, and 581 d. The latter $P$ values very closely match the published periods of the spectroscopic orbits for the three stars reported by Wilson et al. (2016), Katoh et al. (2013), and Sperauskas et al. (2019), respectively.

The sample of short-period SB1 orbits with minimum masses corresponding to substellar companions should therefore be considered with caution, although not all solutions can be incorrect or spurious. For example, a cross-check with the NASA Exoplanet Archive shows that the spectroscopic orbit of WASP-18b (Gaia DR3 4955371367334610048), a transiting super-Jupiter with $M_c \sim 10 M_{\text{Jup}}$ and $P = 0.94$ d discovered by Hellier et al. (2009) around the Hyades-age F6 dwarf HD 10069, is recovered as an SB1 with the correct period and RV semi-amplitude ($K_1 \sim 1.8$ km s⁻¹). This source has also been detected
<figure><img src="image_53.png" /></figure>

**Fig. 51.** Distribution of outer vs. inner periods for triple systems found by matching *Orbital* with SB1 (circles), SB2 (squares), and SB2C solutions (triangles), coloured by the ratio of spectroscopic to astrometric semi-amplitudes. The solid line shows the limit $P_{\text{Orbital}} = 5 P_{\text{SB}}$. Top panel: integrated distribution of inner periods.

in the Gaia photometry (Eyer et al. 2023) and is present in the
vari_planetary_transit table.

The list of most exoplanet candidates detected by Gaia using either astrometry, transit, or radial velocities is published online<fnref n="21" />.

**9. Multiple stars**

Although the NSS pipeline in Gaia DR3 produces solutions of
binary stars, the results can also be used to uncover higher mul-
tiplicity stars. Triple (and higher multiplicity) stars are of par-
ticular interest. The study of the architecture and dynamics of
hierarchical stellar systems provides precious information on the
mechanisms at work during star formation. For example, the fact
that the orbits of a hierarchical system are coplanar would indi-
cate that the stars are formed in a viscous accretion disc, while
their mass ratios shed light on the disc fragmentation mechanism
(see e.g. Tokovinin 2017).

9.1. Multiplicity from spectroscopic and astrometric solutions

A first way to find multiple stars is to look at the sources for
which the pipeline produced an astrometric Orbital solution
and an SB1/SB2 solution, which were not combined. Given that
the astrometry is sensitive to long periods and larger orbits while
the spectroscopy is sensitive to shorter periods, in the case of a
triple star the astrometry would detect the outer period while the
spectroscopy would unveil the inner period.

However, many of the sources for which the Orbital and
SB1/SB1C solution was found but not combined by the pipeline
are not triple stars. In many cases, they have similar periods but

<figure><img src="image_54.png" /></figure>

**Fig. 52.** Distribution of inner periods for triple systems found by comparing astrometric acceleration and SB solutions.

were not combined because of the inconsistency of the other
orbital parameters. There are also many cases where the SB solu-
tion is actually some kind of alias of the astrometric period; these
cases can be spotted noting that the semi-amplitude $K_0$ of the
astrometric motion (defined by Eq. (B.10), substituting $a_0/\bar{\omega}$
for $a_1$) is similar to the semi-amplitude $K_1$ of the radial veloc-
ity curve. These confusing cases typically have a significance of
lower than 10. We then identify genuine triple stars by select-
ing those for which the significance of both the astrometric and
the spectroscopic solution is larger than 10, $K_1 > 3 \cdot K_0$ and
$P_{\text{Orbital}} > 5 \cdot P_{\text{SB}}$. With this selection, we obtain 81 triple systems
from matching Orbital with SB1 solutions, 55 from matching
with SB2, 16 from matching with SB2C, and none from matching
with SB1C. The distribution of the outer versus inner periods of
these sources is reported in Fig. 51.

One can also find multiple stars by looking at the sources for
which the pipeline produced an astrometric acceleration solution
and an SB1/SB2 solution. In this case, the astrometric accelera-
tion would detect outer periods which are similar to or longer
than the length of Gaia observations (~1000 days), while the SB
solutions detect the inner periods.

In order to avoid the situation whereby the astrometric accel-
eration and the SB solutions are in reality of the same orbit, we
selected only SB solutions with a period of <300 days. We also
restrain our search to SB solutions with significance > 20 to
avoid pollution from some kind of aliasing.

The distribution of inner periods for triple systems found by
matching astrometric acceleration and SB solutions is shown in
Fig. 52. We can note that the mode of the distribution is at around
3 days.

9.2. Multiplicity in wide visual binaries

Another method to discover triple or higher multiplicity stars
consists in using catalogues of wide visual binaries and check-
ing whether or not one of the two components is detected as
a binary. We started from the El-Badry et al. (2021) catalogue,
and selected sources with $R < 0.01$ (0.08% contamination from
chance-alignment), cross-matched with nss_two_body_orbit
solutions and spectroscopic trends, and we found 10 063 systems
for which one of the two components is a non-single star and

<footnote n="21">²¹ https://www.cosmos.esa.int/web/gaia/exoplanets</footnote>
<figure><img src="image_55.png" /><figcaption>**Fig. 53.** Distribution of outer vs. inner periods for triple systems found in the El-Badry catalogue. *Left scale:* period. *Right scale:* separation. The line in the bottom right corner shows the limit $P_{\text{out}} = 5 P_{\text{in}}$.</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>**Fig. 54.** Distribution of inner periods for triple systems found in the El-Badry catalogue.</figcaption></figure>

52 systems where both components are non-single stars. For 10 of the first group, the non-single component is actually a triple star (Orbital+SB1/SB2, selected as in Sect. 9.1), making these quadruple systems.

Figure 53 shows the distribution of the outer versus inner periods for the triple systems found in the El-Badry catalogue, where the outer period $P_{\text{out}}$ is computed from the separation $s$ (provided in the El-Badry catalogue) as $P_{\text{out}} = \sqrt{s^3/7.496 \times 10^{-6}(M_1 + M_2)}$ and assuming $M_1 + M_2 = 1.5 M_{\odot}$, while the inner period $P_{\text{in}}$ is the period in the NSS solution.

The drop in the distribution for separations below 200 au is certainly due to selection effects introduced by the blending of the two components in a large range of scanning angles. The cut at longer inner periods is dictated by the limited baseline of Gaia observations. The shortening of the maximum inner period with the increase in separation is due to the fact that sources at large separations are also at larger distances, where the astrometric signal of the internal orbit drops.

We note from Fig. 53 that the distribution of the inner period is multimodal. As shown in Fig. 54, this is in part due to the different methods of orbital detection (highlighted with different colours). However, if we compare the distribution of spectroscopic solutions in this sample with respect to the whole NSS solutions (Fig. 2), we can see that the overabundance of solutions with $2 \text{ days} < P_{\text{in}} < 30 \text{ days}$ is real. This overabundance is also visible in the Multiple Star catalogue by Tokovinin (2018); see their Fig. 7. Tokovinin (2004) attributed the overdensity he observed at $P_{\text{in}} < 7 \text{ days}$ to dynamical interaction between the orbits and consequent tidal interaction within the inner couple, as suggested by Kiseleva et al. (1998). This short analysis shows that the study of multiple systems will greatly benefit from the results of the NSS catalogue.

## 10. Conclusions

On 24 April 1610, Galileo brought his telescope to demonstrate its performances to his opponents and other scholars in Bologna. Martin Horký, a student of Kepler noted: “I tested the instrument of Galileo’s in a thousands ways”, . . . , “below it works wonderfully; in the heavens, it deceives one as some fixed stars are seen double” (Feyerabend 1970). This is how the discovery of some of the first telescopic double stars failed and how easy it is to consider that facts can be contradicted by prejudice.

Over four centuries later, the validations and the results obtained so far suggest that NSS entries in Gaia DR3 are binaries and not Gaia telescope artefacts. Nevertheless, in fairness, this catalogue is expected to contain some spurious solutions. We therefore need to start this conclusion by stressing that the counterpart of such a large catalogue is some unavoidable contamination, which should be kept in mind when analysing the results.

In particular, the selection of distribution tails (e.g. small periods and small or large mass functions) is likely to preferentially select incorrect solutions, as we experienced in the course of this verification paper. The abundant partially resolved sources and their impact on the astrometry and epoch radial velocities conspire with the scanning law periodic motions to produce solutions that look like bona fide unresolved binaries. To cope with this, increasing the threshold on the significance of the solutions appears satisfactory but may well represent a Pyrrhic victory, as this decreases the sample sizes drastically.

The NSS catalogue is already the result of a drastic selection of sources. The impact of the total selection effects – due successively to the input list, the data processing, and the posterior source filtering – need to be taken into account, and statistical studies of the NSS sample will require dedicated efforts. This has not been attempted here, although the main points have probably been mentioned. There are subjects that are already known to represent pitfalls, such as the acceleration solutions, which should not be used for physical interpretations.

Although the main impact of the NSS catalogue originates from the simultaneous presence of the main kind of unresolved binaries, and consequently an impressive coverage of binary periods, the novelty is mainly brought by the exquisite astrometric precision, allowing the detection of many astrometric binaries. However, there is one strength and one weakness concerning astrometric orbits: if the astrometric and the photometric effects are not decoupled, the actual size of the orbit of the primary cannot be found. The decoupling here is difficult, but when it can be achieved, then both masses and luminosities
of the components become available. We have tried to take this into account for the masses estimated here, but the combination with external data, as some examples show, will prove very useful if not sometimes mandatory.

The whole HRD can be studied with the abundant number of astrometric orbits, spectroscopic orbits, or both. Some analyses may be more interesting than others. Here, new ultracool dwarf binaries are found, and the small mass ratios can also be studied down to the substellar domain. True masses are found for substellar companions and two new super-Jupiter candidates may have been found. Twin degenerates and one WD hosting a super-Jupiter companion are also proposed. Concerning compact companions with larger masses, potential companions are present in the mass range of neutron stars or black holes, but these may also be Algol or triple systems: indeed, at this step, we stress that most findings here should be considered as tentative; although verifications have been done, further analyses are warranted. The eccentricity-period relation will also undoubtedly be under scrutiny although interesting substructures for giants are already shown here. The detection of ellipsoidal variables mistaken as long-period variables, or the detection of sources found in a rare evolutionary stage as in EL CVn, underlines the potential of acquiring both photometric and spectroscopic data. This is further testimony that *Gaia* is an impressively complete observatory in orbit.

Without any detailed content study or modelling, which will be tasks of the scientific exploitation to come, the multiple topics tackled in the sections above, while only a very preliminary and tentative exploration of the *Gaia* DR3 binaries, clearly demonstrate the great scientific potential of this catalogue. Paraphrasing Aaron Levenstein on the topic of statistics, what these analyses reveal is suggestive; what they conceal may be essential.

*Acknowledgements.* This work is dedicated to the memory of our late colleague Dimitri Pourbaix who suggested the title of this article. He had been for many years leader of the *Gaia* CU4 Coordination Unit and passed away before seeing the *Gaia* DR3 NSS outcome. This catalogue will remain in a large part the testimony of his involvement. We would like to thank the referee Andrei Tokovinin, and the DPAC reporter, Lennart Lindegren, for their useful suggestions. 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. Further acknowledgements are given in Appendix A. This publication has also made use of observations collected with the SOPHIE spectrograph on the 1.93-m telescope at Observatoire de Haute-Provence (CNRS), France (program 21B.PNPS.AREN) using support by the French Programme National de Physique Stellaire (PNPS). We thank all the staff of Haute-Provence Observatory for their support at the 1.93 m telescope and on SOPHIE. Observations have also been obtained with the Mercator Telescope and the HERMES spectrograph, which is supported by the Research Foundation - Flanders (FWO), Belgium, the Research Council of KULeuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. [951549]). Ts.M. research was supported by Grant No. 2016069 of the United States-Israel Binalional Science Foundation (BSF) and by Grant No. I-1498-303.7/2019 of the German-Israeli Foundation for Scientific Research and Development (GIF). The full acknowledgements are available in Appendix A.

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5 Institut d'Astronomie et d'Astrophysique, Université Libre de Bruxelles CP 226, Boulevard du Triomphe, 1050 Brussels, Belgium

6 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 Place Jules Janssen, 92190 Meudon, France

7 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland

8 RHEA for European Space Agency (ESA), Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

9 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel

10 INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, TO, Italy

11 Institut d'Astrophysique et de Géophysique, Université de Liège, 19c Allée du 6 Août, 4000 Liège, Belgium

12 CRAAG – Centre de Recherche en Astronomie, Astrophysique et Géophysique, Route de l'Observatoire Bp 63 Bouzareah, 16340 Algiers, Algeria

13 F.R.S.-FNRS, Rue d'Egmont 5, 1000 Brussels, Belgium

14 Université de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR 7550, 11 rue de l'Université, 67000 Strasbourg, France

15 Department of Astronomy, University of Geneva, Chemin d'Ecogia 16, 1290 Versoix, Switzerland

16 University of Turin, Department of Physics, Via Pietro Giuria 1, 10125 Torino, Italy

17 Telespazio UK S.L. for European Space Agency (ESA), Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

18 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, 1000 Ljubljana, Slovenia

19 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands

20 INAF – Osservatorio astronomico di Padova, Vicolo Osservatorio 5, 35122 Padova, Italy

21 European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

22 Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany

23 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice Cedex 4, France

24 Laboratoire d'Astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, Allée Geoffroy Saint-Hilaire, 33615 Pessac, France

25 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

26 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

27 Aurora Technology for European Space Agency (ESA), Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

28 Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain

29 Lohrmann Observatory, Technische Universität Dresden, Mommsenstraße 13, 01062 Dresden, Germany

30 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Sweden

31 CNES Centre Spatial de Toulouse, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France

32 INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy

33 Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany

34 European Space Agency, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

<footnote n="1">¹ GEPI, Observatoire de Paris, Université PSL, CNRS, 5 place Jules Janssen, 92190 Meudon, France
e-mail: Frederic.arenou@obspm.fr</footnote>

<footnote n="2">² Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France</footnote>

<footnote n="3">³ School of Physics and Astronomy/Space Park Leicester, University of Leicester, University Road, Leicester LE1 7RH, UK</footnote>

<footnote n="4">⁴ School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel</footnote>
35 INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy

36 DAPCOM for Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain

37 Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium

38 Observational Astrophysics, Division of Astronomy and Space Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden

39 ALTEC S.p.a., Corso Marche, 79, 10146 Torino, Italy

40 Sednai Sârl, Geneva, Switzerland

41 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

42 Gaia DPAC Project Office, ESAC, Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

43 SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, LNE, 61 Avenue de l'Observatoire, 75014 Paris, France

44 National Observatory of Athens, I. Metaxa and Vas. Pavlou, Palaia Penteli 15236 Athens, Greece

45 IMCCE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Lille, 77 Av. Denfert-Rochereau, 75014 Paris, France

46 Serco Gestión de Negocios for European Space Agency (ESA), Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

47 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

48 ATG Europe for European Space Agency (ESA), Camino Bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada 28692, Madrid, Spain

49 CIGUS CITIC – Department of Computer Science and Information Technologies, University of A Coruña, Campus de Elviña s/n, A Coruña 15071, Spain

50 Kavli Institute for Cosmology Cambridge, Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK

51 Leibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany

52 CENTRA, Faculdade de Ciências, Universidade de Lisboa, Edif. C8, Campo Grande, 1749-016 Lisboa, Portugal

53 Department of Informatics, Donald Bren School of Information and Computer Sciences, University of California, Irvine, 5226 Donald Bren Hall, 92697-3440, CA Irvine, USA

54 INAF – Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy

55 Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, Via S. Sofia 64, 95123 Catania, Italy

56 INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone, Roma, Italy

57 Space Science Data Center – ASI, Via del Politecnico SNC, 00133 Roma, Italy

58 Department of Physics, University of Helsinki, PO Box 64, 00014 Helsinki, Finland

59 Finnish Geospatial Research Institute FGI, Geodeetinrinne 2, 02430 Masala, Finland

60 Institut UTINAM CNRS UMR6213, Université Bourgogne Franche-Comté, OSU THETA Franche-Comté Bourgogne, Observatoire de Besançon, BP1615, 25010 Besançon Cedex, France

61 HE Space Operations BV for European Space Agency (ESA), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

62 Dpto. de Inteligencia Artificial, UNED, c/ Juan del Rosal 16, 28040 Madrid, Spain

63 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH), MTA Centre of Excellence, Konkoly Thege Miklós út 15-17, 1121 Budapest, Hungary

64 ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1A, 1117 Budapest, Hungary

65 Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium

66 Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands

67 University of Vienna, Department of Astrophysics, Türkenschanzstraße 17, 1180 Vienna, Austria

68 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

69 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands

70 Thales Services for CNES Centre Spatial de Toulouse, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France

71 Depto. Estadística e Investigación Operativa. Universidad de Cádiz, Avda. República Saharaui s/n, 11510 Puerto Real, Cádiz, Spain

72 Center for Research and Exploration in Space Science and Technology, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD, USA

73 GSFC – Goddard Space Flight Center, Code 698, 8800 Greenbelt Rd, 20771, MD Greenbelt, USA

74 EURIX S.r.l., Corso Vittorio Emanuele II 61, 10128 Torino, Italy

75 Porter School of the Environment and Earth Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

76 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 15, Cambridge, MA 02138, USA

77 ATOS for CNES Centre Spatial de Toulouse, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France

78 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

79 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

80 LFCA/DAS, Universidad de Chile, CNRS, Casilla 36-D, Santiago, Chile

81 SISSA – Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy

82 Telespazio for CNES Centre Spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

83 University of Turin, Department of Computer Sciences, Corso Svizzera 185, 10149 Torino, Italy

84 Dpto. de Matemática Aplicada y Ciencias de la Computación, Univ. de Cantabria, ETS Ingenieros de Caminos, Canales y Puertos, Avda. de los Castros s/n, 39005 Santander, Spain

85 Centro de Astronomía – CITEVA, Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile

86 DLR Gesellschaft für Raumfahrtanwendungen (GfR), mbH Münchener Straße 20, 82234 Weßling, Germany

87 Centre for Astrophysics Research, University of Hertfordshire, College Lane AL10 9AB, Hatfield, UK

88 University of Turin, Mathematical Department “G. Peano”, Via Carlo Alberto 10, 10123 Torino, Italy

89 INAF – Osservatorio Astronomico d’Abruzzo, Via Mentore Maggini, 64100, Teramo, Italy

90 Instituto de Astronomia, Geofisica e Ciências Atmosféricas, Universidade de São Paulo, Rua do Matão, 1226, Cidade Universitaria, 05508-900 São Paulo, SP, Brazil

91 APAVE SUDEUROPE SAS for CNES Centre Spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

92 Mésocentre de calcul de Franche-Comté, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

93 Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK

94 Centre de Données Astronomique de Strasbourg, Strasbourg, France
95 Institute for Computational Cosmology, Department of Physics, Durham University, Durham DH1 3LE, UK

96 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany

97 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany

98 Data Science and Big Data Lab., Pablo de Olavide University, 41013 Seville, Spain

99 Barcelona Supercomputing Center (BSC), Plaça Eusebi Güell 1-3, 08034 Barcelona, Spain

100 ETSE Telecomunicación, Universidade de Vigo, Campus Lagoas-Marcosende, 36310 Vigo, Galicia, Spain

101 Asteroid Engineering Laboratory, Space Systems, Luleå University of Technology, Box 848, 981 28 Kiruna, Sweden

102 Vera C Rubin Observatory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA

103 Department of Astrophysics, Astronomy and Mechanics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos 15783, Athens, Greece

104 TRUMPF Photonic Components GmbH, Lise-Meitner-Straße 13, 89081 Ulm, Germany

105 IAC – Instituto de Astrofísica de Canarias, Via Láctea s/n, 38200 La Laguna S.C., Tenerife, Spain

106 Department of Astrophysics, University of La Laguna, Via Láctea s/n, 38200 La Laguna S.C., Tenerife, Spain

107 Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands

108 Radagast Solutions, Simon Vestdijkpad 24, 2321 WD Leiden, The Netherlands

109 Laboratoire Univers et Particules de Montpellier, CNRS Université Montpellier, Place Eugène Bataillon, CC72, 34095 Montpellier Cedex 05, France

110 Université de Caen Normandie, Côte de Nacre Boulevard Maréchal Juin, 14032 Caen, France

111 SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands

112 Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland

113 Scalian for CNES Centre Spatial de Toulouse, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France

114 Université Rennes, CNRS, IPR (Institut de Physique de Rennes), UMR 6251, 35000 Rennes, France

115 INAF – Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy

116 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, PR China

117 University of Chinese Academy of Sciences, No. 19(A) Yuquan Road, Shijingshan District, Beijing 100049, PR China

118 Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark

119 DXC Technology, Retortvej 8, 2500 Valby, Denmark

120 Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117, USA

121 CIGUS CITIC, Department of Nautical Sciences and Marine Engineering, University of A Coruña, Paseo de Ronda 51, 15071 A Coruña, Spain

122 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK

123 IPAC, Mail Code 100-22, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA

124 IRAP, Université de Toulouse, CNRS, UPS, CNES, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France

125 MTA CSFK Lendület Near-Field Cosmology Research Group, Konkoly Observatory, MTA Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, 1121 Budapest, Hungary

126 Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, 28040 Madrid, Spain

127 Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia

128 Villanova University, Department of Astrophysics and Planetary Science, 800 E Lancaster Avenue, Villanova, PA 19085, USA

129 INAF – Osservatorio Astronomico di Brera, Via E. Bianchi, 46, 23807 Merate, LC, Italy

130 STFC, Rutherford Appleton Laboratory, Harwell, Didcot OX11 0QX, UK

131 Charles University, Faculty of Mathematics and Physics, Astronomical Institute of Charles University, V Holesovickach 2, 18000 Prague, Czech Republic

132 Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544, USA

133 Departamento de Astrofísica, Centro de Astrobiología (CSIC-INTA), ESA-ESAC, Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain

134 naXys, University of Namur, Rempart de la Vierge, 5000 Namur, Belgium

135 CGI Deutschland B.V. & Co. KG, Mornewegstr. 30, 64293 Darmstadt, Germany

136 Institute of Global Health, University of Geneva, Geneva, Switzerland

137 Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, Poznań, Poland

138 H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK

139 Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell'Osservatorio 3, 35122 Padova, Italy

140 CERN, Esplanade des Particules 1, PO Box 1211, Geneva, Switzerland

141 Applied Physics Department, Universidade de Vigo, 36310 Vigo, Spain

142 Association of Universities for Research in Astronomy, 1331 Pennsylvania Ave. NW, Washington, DC 20004, USA

143 European Southern Observatory, Alonso de Córdova, 3107 Casilla 19, Santiago, Chile

144 Sorbonne Université, CNRS, UMR7095, Institut d'Astrophysique de Paris, 98bis Bd. Arago, 75014 Paris, France
# Appendix A: Acknowledgements

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

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

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

- the BELgian federal Science Policy Office (BELSPO) through various PROgramme de Développement d'Expériences scientifiques (PRODEX) grants, the Fonds Wetenschappelijk Onderzoek through grant VS.091.16N, 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 Aperfeiçoamento 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 PIOFGA-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 galaxies' (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 HAR-MONIA 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 Gaia project and data processing have made use of:

- the Set of Identifications, Measurements, and Bibliography for Astronomical Data (SIMBAD, Wenger et al. 2000), the 'Aladin sky atlas' (Bonnarel et al. 2000; Boch & Fernique 2014), and the VizieR catalogue access tool (Ochsenbein et al. 2000), all operated at the Centre de Données astronomiques de Strasbourg (CDS);

- the National Aeronautics and Space Administration (NASA) Astrophysics Data System (ADS);

- the Space ENVironment Information System (SPENVIS), initiated by the Space Environment and Effects Section (TEC-EES) of ESA and developed by the Belgian Institute for Space Aeronomy (BIRA-IASB) under ESA contract through ESA's General Support Technologies Programme (GSTP), administered by the BELgian federal Science Policy Office (BELSPO);

- the software products TOPCAT, STIL, and STILTS (Taylor 2005, 2006);

- Matplotlib (Hunter 2007);

- IPython (Pérez & Granger 2007);

- Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration 2018);

- SCIPY (Jones et al. 2001),

- NUMPY (Oliphant 2007),
- PANDAS (Reback et al. 2022),

- R (R Core Team 2013);

- Vaex (Breddels & Veljanoski 2018);

- the HIPPARCOS-2 catalogue (van Leeuwen 2007b). The HIPPARCOS and *Tycho* catalogues were constructed under the responsibility of large scientific teams collaborating with ESA. The Consortia Leaders were Lennart Lindegren (Lund, Sweden: NDAC) and Jean Kovalevsky (Grasse, France: FAST), together responsible for the HIPPARCOS Catalogue; Erik Høg (Copenhagen, Denmark: TDAC) responsible for the *Tycho* Catalogue; and Catherine Turon (Meudon, France: INCA) responsible for the HIPPARCOS Input Catalogue (HIC);

- the *Tycho*-2 catalogue (Høg et al. 2000), the construction of which was supported by the Velux Foundation of 1981 and the Danish Space Board;

- The *Tycho* double star catalogue (TDSC, Fabricius et al. 2002), based on observations made with the ESA HIPPARCOS astrometry satellite, as supported by the Danish Space Board and the United States Naval Observatory through their double-star programme;

- data products from the Two Micron All Sky Survey (2MASS, Skrutskie et al. 2006), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center (IPAC)/California Institute of Technology, funded by the National Aeronautics and Space Administration (NASA) and the National Science Foundation (NSF) of the USA;

- the ninth data release of the AAVSO Photometric All-Sky Survey (APASS, Henden et al. 2016), funded by the Robert Martin Ayers Sciences Fund;

- the first data release of the Pan-STARRS survey (Chambers et al. 2016; Magnier et al. 2020a; Waters et al. 2020; Magnier et al. 2020b,c; Flewelling et al. 2020). The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration (NASA) through grant NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation through grant AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation;

- the second release of the Guide Star Catalogue (GSC2.3, Lasker et al. 2008). The Guide Star Catalogue II is a joint project of the Space Telescope Science Institute (STScI) and the Osservatorio Astrofisico di Torino (OATo). STScI is operated by the Association of Universities for Research in Astronomy (AURA), for the National Aeronautics and Space Administration (NASA) under contract NAS5-26555. OATo is operated by the Italian National Institute for Astrophysics (INAF). Additional support was provided by the European Southern Observatory (ESO), the Space Telescope European Coordinating Facility (STECF), the International

GEMINI project, and the European Space Agency (ESA) Astrophysics Division (nowadays SCI-S);

- the eXtended, Large (XL) version of the catalogue of Positions and Proper Motions (PPM-XL, Roeser et al. 2010);

- data products from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEO-WISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEO-WISE are funded by the National Aeronautics and Space Administration (NASA);

- the first data release of the United States Naval Observatory (USNO) Robotic Astrometric Telescope (URAT-1, Zacharias et al. 2015);

- the fourth data release of the United States Naval Observatory (USNO) CCD Astrograph Catalogue (UCAC-4, Zacharias et al. 2013);

- the sixth and final data release of the Radial Velocity Experiment (RAVE DR6, Steinmetz et al. 2020a,b). Funding for RAVE has been provided by the Leibniz Institute for Astrophysics Potsdam (AIP), the Australian Astronomical Observatory, the Australian National University, the Australian Research Council, the French National Research Agency, the German Research Foundation (SPP 1177 and SFB 881), the European Research Council (ERC-StG 240271 Galactica), the Istituto Nazionale di Astrofisica at Padova, the Johns Hopkins University, the National Science Foundation of the USA (AST-0908326), the W.M. Keck foundation, the Macquarie University, the Netherlands Research School for Astronomy, the Natural Sciences and Engineering Research Council of Canada, the Slovenian Research Agency, the Swiss National Science Foundation, the Science & Technology Facilities Council of the UK, Opticon, Strasbourg Observatory, and the Universities of Basel, Groningen, Heidelberg, and Sydney. The RAVE website is at https://www.rave-survey.org/;

- the first data release of the Large sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST DR1, Luo et al. 2015);

- the K2 Ecliptic Plane Input Catalogue (EPIC, Huber et al. 2016);

- the ninth data release of the Sloan Digital Sky Survey (SDSS DR9, Ahn et al. 2012). Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the United States Department of Energy Office of Science. The SDSS-III website is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofísica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University;
- the thirteenth release of the Sloan Digital Sky Survey (SDSS DR13, Albareti et al. 2017). Funding for SDSS-IV has been provided by the Alfred P. Sloan Foundation, the United States Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is https://www.sdss.org/. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University;

- the second release of the SkyMapper catalogue (SkyMapper DR2, Onken et al. 2019, Digital Object Identifier 10.25914/5ce60d31ce759). The national facility capability for SkyMapper has been funded through grant LE130100104 from the Australian Research Council (ARC) Linkage Infrastructure, Equipment, and Facilities (LIEF) programme, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University, and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University's Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at the Australian National University. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth's Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS);

- the Gaia-ESO Public Spectroscopic Survey (GES, Gilmore et al. 2022; Randich et al. 2022). The Gaia-ESO Survey is based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 188.B-3002. Public data releases are available through the ESO Science Portal. The project has received funding from the Leverhulme Trust (project RPG-2012-541), the European Research Council (project

ERC-2012-AdG 320360-Gaia-ESO-MW), and the Istituto Nazionale di Astrofisica, INAF (2012: CRA 1.05.01.09.16; 2013: CRA 1.05.06.02.07).

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.

## Appendix B: Parameters describing the orbital motion

The parameters relative to the orbital motion as presented in the `nss_two_body_orbit` are introduced in what follows. More details can be found in the online documentation and articles accompanying the data release Halbwachs et al. (2023), Holl et al. (2023b), Gosset et al. (in prep.), and Siopis (in prep.) for astrometric, spectroscopic, and eclipsing binaries respectively.

### B.1. Astrometry

The Gaia AL astrometric measurement $w$ (abscissa) for a binary system can be modelled by the combination of a single-source model $w_{ss}$, describing the standard astrometric motion of the system's barycentre, and a Keplerian model $w_{kl}$.

The single-source model can be written as

$$w_{ss} = (\Delta\alpha^* + \mu_{\alpha^*} t) \sin\psi + (\Delta\delta + \mu_{\delta} t) \cos\psi + \varpi f_{\varpi}, \quad (B.1)$$

where $\Delta\alpha^* = \Delta\alpha \cos\delta$ and $\Delta\delta$ are small offsets in equatorial coordinates from some fixed reference point, $\mu_{\alpha^*}$, $\mu_{\delta}$ are proper motions in these coordinates, $t$ is time, $\varpi$ is the parallax, $f_{\varpi}$ is the parallax factor, and $\psi$ is the scan angle. The astrometric motion corresponding to a Keplerian orbit of a binary system has generally seven independent parameters, namely the Campbell elements. These are the period $P$, the epoch of periastron passage $T_0$, the eccentricity $e$, the inclination $i$, the ascending node $\Omega$, the argument of periastron $\omega$, and the semi-major axis of the photocentre $a_0$. The Thiele-Innes coefficients $A, B, F$, and $G$ are defined as

$$ \begin{aligned} A &= a_0 (\cos \omega \cos \Omega - \sin \omega \sin \Omega \cos i) \\ B &= a_0 (\cos \omega \sin \Omega + \sin \omega \cos \Omega \cos i) \\ F &= -a_0 (\sin \omega \cos \Omega + \cos \omega \sin \Omega \cos i) \\ G &= -a_0 (\sin \omega \sin \Omega - \cos \omega \cos \Omega \cos i). \end{aligned} \quad (B.2) $$

The elliptical rectangular coordinates $X$ and $Y$ are functions of eccentric anomaly $E$ and eccentricity:

$$E - e \sin E = \frac{2\pi}{P} (t - T_0) \quad (B.3)$$

$$X = \cos E - e \quad (B.4)$$

$$Y = \sqrt{1 - e^2} \sin E. \quad (B.5)$$
The single Keplerian model can then be written as

$$w_{k1} = (BX + GY) \sin \psi + (AX + FY) \cos \psi. \qquad (B.6)$$

The combined model $w^{(\text{model})}$ for the Gaia AL astrometry is

$$ \begin{aligned} w^{(\text{model})} &= w_{\text{ss}} + w_{k1} \\ &= (\Delta\alpha^* + \mu_{\alpha^*} t) \sin \psi + (\Delta\delta + \mu_{\delta} t) \cos \psi + \bar{\omega} f_{\bar{\omega}} \\ &\quad + (BX + GY) \sin \psi + (AX + FY) \cos \psi. \end{aligned} \qquad (B.7) $$

This model has been extensively used for modelling the HIPPARCOS epoch data of non-single stars (e.g. Sahlmann et al. 2011).

## B.2. Spectroscopy

The radial motion of the primary is given by

$$RV_1(t) = \gamma + K_1 [\cos(v(t) + \omega) + e \cos(\omega)], \qquad (B.8)$$

with $v(t)$ the true anomaly deriving from the eccentric anomaly $E$ by

$$\tan \frac{v}{2} = \sqrt{\frac{1+e}{1-e}} \tan \frac{E}{2}, \qquad (B.9)$$

and with the semi-amplitude of the primary (resp. secondary)

$$K_{1,2} = \frac{\kappa a_{1,2} \sin i}{P \sqrt{1-e^2}}, \qquad (B.10)$$

where $\kappa \sim 10879$ when $K_i$ is expressed in km s$^{-1}$, $P$ in days and $a_i$ the semi-major axis of the primary (resp. secondary) in astronomical units (au).

This model has been used to produce the NSS solutions for SB1 and SB2. When astrometry and spectroscopy were combined (AstroSpectroSB1 solutions) in order to avoid correlations with the Campbell elements, it was easier to complete the Thiele-Innes elements with

$$ \begin{aligned} C &= +a_1 \sin \omega \sin i \\ H &= +a_1 \cos \omega \sin i, \end{aligned} \qquad (B.11) $$

and these parameters have been published instead.

# Appendix C: Analytic orbit detection sensitivity as a function of orbit inclination

Our toy orbit model is shown in Fig. C.1. To keep the expressions simple, we aligned the ellipse major-axis $a$ with $\alpha^*$ and introduced the inclination as rotation around the $\alpha^*$-axis. The orbital-phase ($\phi$)-dependent position $(x(t), y(t))$ is projected to the on-sky position $\alpha^*, \delta$ via:

$$\alpha^*(\phi) = x(\phi) \qquad = a \sin \phi \qquad (C.1)$$

$$\delta(\phi) = y(\phi) \cos i \qquad = b \cos \phi \cos i \qquad (C.2)$$

The orbit signal in the AL (i.e. along $\psi$) and across-scan (AC, rotated 90 degrees counter-clockwise) directions is

$$\mathbf{d}_{\text{AL}} = \begin{pmatrix} \alpha^*(\phi) \\ \delta(\phi) \end{pmatrix} \begin{pmatrix} \sin \psi \\ \cos \psi \end{pmatrix}^T \qquad (C.3)$$

$$\mathbf{d}_{\text{AC}} = \begin{pmatrix} \alpha^*(\phi) \\ \delta(\phi) \end{pmatrix} \begin{pmatrix} -\cos \psi \\ \sin \psi \end{pmatrix}^T. \qquad (C.4)$$

To derive the RMS of $\mathbf{d}_{\text{AL}}$ and $\mathbf{d}_{\text{AC}}$, we make the following simplifying assumptions: (a) the orbit is circular, i.e. $a = b$: The angular velocity $\phi$ is constant and integration over $\phi$ is straightforward. (b) At each observation the orbital phase $\phi$ is random: the orbital-average value can be obtained by simply integrating over $\phi$. (c) At each observation the scan-angle $\psi$ is random. The mean square amplitude of $\mathbf{d}_{\text{AL}}$ for a given scan angle $\psi$ under assumptions (a) and (b) reduces to

$$ \begin{aligned} \hat{d}_{\text{AL},\psi}^2 &= \frac{1}{2\pi} \int_0^{2\pi} \|\mathbf{d}_{\text{AL}}\|^2 d\phi \\ &= \frac{1}{2\pi} \int_0^{2\pi} (a \sin \phi \sin \psi + a \cos \phi \cos i \cos \psi)^2 d\phi \\ &= \frac{a^2}{2} (\sin^2 \psi + \cos^2 \psi \cos^2 i), \end{aligned} \qquad (C.5) $$

where we used the identity

$$\int_0^{2\pi} (u \sin x + v \cos x)^2 dx = \pi (u^2 + v^2). \qquad (C.6)$$

Similarly, we obtain in the AC direction

$$\hat{d}_{\text{AC},\psi}^2 = \frac{a^2}{2} (\cos^2 \psi + \sin^2 \psi \cos^2 i). \qquad (C.7)$$

Applying assumption (c) we get

$$ \begin{aligned} \hat{d}_{\text{AL}}^2 &= \frac{1}{2\pi} \int_0^{2\pi} \hat{d}_{\text{AL},\psi}^2 d\psi \\ &= \frac{a^2}{4\pi} \int_0^{2\pi} \sin^2 \psi + \cos^2 \psi \cos^2 i \, d\psi \\ &= \frac{a^2}{4} (1 + \cos^2 i), \end{aligned} \qquad (C.8) $$

and in AC

$$\hat{d}_{\text{AC}}^2 = \frac{a^2}{4} (1 + \cos^2 i). \qquad (C.9)$$

Assuming constant weights $W_{\text{AL}}$ and $W_{\text{AC}}$ for the AL and AC observations, respectively, the resulting RMS level is

$$\text{RMS}(i) = \sqrt{\hat{d}_{\text{AL}}^2 + \hat{d}_{\text{AC}}^2} = \frac{a}{2} \sqrt{(W_{\text{AL}} + W_{\text{AC}}) (1 + \cos^2 i)}. \qquad (C.10)$$

It is instructive to examine the two extreme scenarios:

1. One-dimensional astrometry, i.e. $W_{\text{AL}} = 1$ and $W_{\text{AC}} = 0$, which approximates the Gaia DR3 astrometric solution: we obtain

$$\text{RMS}_{1\text{D}}(i) = \frac{a}{2} \sqrt{1 + \cos^2 i}, \qquad (C.11)$$

which results in a $\sqrt{2}$ reduction for edge-on systems compared to the face-on configuration.

2. Two-dimensional astrometry with equal weights, i.e. $W_{\text{AL}} = W_{\text{AC}} = 1$, which approximates conventional CCD imaging:

$$\text{RMS}_{2\text{D}}(i) = \frac{a}{2} \sqrt{2(1 + \cos^2 i)}, \qquad (C.12)$$
<figure><img src="image_57.png" /><figcaption>**Fig. C.1.** Left: Orbital plane ellipse with width *a* and height *b* (the circular case *a* = *b* = 1 is shown). Middle: Sky-projected plane with scan angle *ψ* and projected orbit due to inclination *i* shown in the right panel.</figcaption></figure>

i.e. same inclination-dependency as one-dimensional observations but with a $\sqrt{2}$ boost in RMS because the number of observations is doubled<fnref n="22" />.

# Appendix D: Biases and uncertainties of Thiele-Innes parameters

## D.1. Biases introduced by fitting Thiele-Innes parameters to noisy data

All Gaia processing pipelines for fitting astrometric orbits<fnref n="23" /> follow a similar parametrisation scheme (Sect. B.1; DR3 documentation; Halbwachs et al. 2023; Holl et al. 2023b): the three non-linear parameters *P*, *e*, and *T*₀ are fitted using different algorithms but the four remaining parameters are represented by the Thiele-Innes coefficients *A*, *B*, *F*, *G*, which linearise part of the equations.

As the features in the distributions of DR3 orbit parameters hinted at signal-dependent effects, we simulated the *A*, *B*, *F*, *G* recovery performance as a function of noise level. We applied the same simulation approach as in Sect. 3.2 to compute the AL signal *w*<sub>k1</sub> of the Keplerian orbit, Eq. (B.6), for an ensemble of binary systems with the same distance, period, and semi-major axis. We computed the rectangular coordinates *X* and *Y* according to the simulated parameters and then solved the linear equation

$$w_{k1} + \epsilon_w = (BX + GY) \sin\psi + (AX + FY) \cos\psi \quad (D.1)$$

for the unknown *A*, *B*, *F*, *G*, where $\epsilon_w$ is a random noise term that we added to the noiseless simulated abscissa. For simplicity, we chose uniform abscissa uncertainties $\sigma_w$ and zero covariances. We used a standard matrix-inversion solver<fnref n="24" /> to determine the solution.

For the noise term $\epsilon_w$, we chose its amplitude relative to the simulated semi-major axis, which was the same for all simulated systems ($a_0 = 0.059$ mas, see Sect. 3.2) but independent of the number of simulated observation epochs for a given source. For S/N=5, for example, a random noise term was added to *w* according to a normal distribution with dispersion $\epsilon_{w,5} = a_0/5$. This setup does intentionally not account for the inclination-dependent sensitivity (Sect. 3.2).

Figure D.1 shows the simulated and recovered distributions of $\cos i$ for various levels of noise. At high S/N=100, the simulated distribution is recovered, but for progressively smaller S/N, a lack of face-on configurations starts to appear and the inclinations extracted from the fitted *A*, *B*, *F*, *G* are biased towards edge-on configurations. At very small S/N=0.01, where the *A*, *B*, *F*, *G* are essentially unconstrained except for their amplitude term dependent on $a_0$, the $\cos i$ distribution is peaked at $\cos i = 0$, i.e. edge-on configurations.

The apparent dearth of face-on configurations for Orbital solutions (Fig. 11) is therefore the consequence of extracting the Thiele-Innes parameters *A*, *B*, *F*, *G* from noisy data. The dependence on S/N expected from the simulations is observed in the actual data as well (Fig. 12). As the DR3 solutions contain a continuum of S/N levels, the corresponding $\cos i$ distribution is a superposition of the distributions for individual S/N levels.

Figure D.2 shows the recovered $\Omega$ distribution in the lowest S/N case we simulated. The suppression of orbits with $\Omega = 90^\circ$ observed for Orbital solutions is reproduced by the simulation. This modulation becomes very weak for higher levels of S/N. We conclude that this feature can therefore also be attributed to a bias introduced by fitting the *A*, *B*, *F*, *G* coefficients.

To simulate the effect on $\omega$ we simulated non-circular orbits with eccentricity $e = 0.5$, all other parameters remained the same. Figure D.3 shows a bimodal modulation of the recovered $\omega$ distribution. However, the minima at $\omega = 0^\circ$ or $180^\circ$ do not match the ones observed for Orbital solutions at $\omega = 90^\circ$ or $270^\circ$. Instead, they reproduce the minima seen in the orb6 solutions shown in Fig. 11.

## D.2. Selection effect causing the modulation in the $\omega$ distribution

The accepted Orbital solutions have to pass a period-dependent threshold on their significance, defined as the ratio between the semi-major axis and its uncertainty $a_0/\sigma_{a_0}$ (Halbwachs et al. 2023). Figure D.4 shows that there is a bimodal correlation between the published significance and $\omega$ with minima at $\omega = 90^\circ$ or $270^\circ$. The application of an $\omega$-independent significance threshold therefore leads to a

<footnote n="22">²² This two-dimensional case was formally derived with a random rotation of the frame per observation, which does not correspond to fixed-axis orientation CCD observation. In our toy model, the scan-angle integral takes care of considering the random ($\Omega$-)orientation of the orbit since we assumed the rotation axis is aligned with $\alpha^*$. Alternatively the sky rotation angle reminiscent of $\Omega$ could be introduced and integrated over to achieve this averaging, and then there is no need to integrate over scan angle for a fixed orientation telescope. As long as $\psi$ and/or $\Omega$ are assumed to be randomly oriented without range restriction, the simpler model used here is accurate.</footnote>

<footnote n="23">²³ There are three flavours: (1) the ‘binary pipeline’ (Halbwachs et al. 2023) (2) the Genetic Algorithm channel of the ‘exoplanet pipeline’ (Holl et al. 2023b) (3) the Markov Chain Monte Carlo channel of the ‘exoplanet pipeline’.</footnote>

<footnote n="24">²⁴ https://github.com/Johannes-Sahlmann/linearfit</footnote>
<figure><img src="image_58.png" /><figcaption>**Fig. D.1.** Simulated (filled grey) and recovered (solid line) distributions of $cos i$ for six levels of signal-to-noise as explained in the text. The vertical dashed line at $i = 90^\circ$ indicates the edge-on configurations.</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>**Fig. D.2.** Simulated (filled grey) and recovered (solid line) distributions of $\Omega$ at $S/N = 0.01$.</figcaption></figure>

suppression of solutions around these values. We argue that this is the explanation for the observed modulation in the $\omega$ distribution of Gaia Orbital solutions.

### D.3. Geometric element conversion from Thiele-Innes parameters in the presence of uncertainties

The archive tables list the fitted astrometric-orbit parameters $P$, $e$, $T_0$, $A, B, F, G$ with their formal uncertainties and the `corr_vec` field contains the correlation strengths between those parameters. When converting $A, B, F, G$ to geometric orbit elements and determining their uncertainties we therefore have to account for their covariances. This can be done with classical error propagation using some form of linearisation of the parameter dependencies or by Monte Carlo simulations. As also pointed out in Babusiaux et al. (2023), the distributions of the

<figure><img src="image_60.png" /><figcaption>**Fig. D.3.** Simulated (filled grey) and recovered (solid line) distributions of $\omega$ for non-circular orbits with $e = 0.5$ at $S/N = 0.01$.</figcaption></figure>

<figure><img src="image_61.png" /><figcaption>**Fig. D.4.** Density histogram of published significance as a function of $\omega$ for DR3 Orbital solutions. The solid curve shows the running median value. The dashed curve shows the median of the Monte Carlo-resampled significance.</figcaption></figure>

geometric parameters are often non-Gaussian, which favours the latter approach.

We used Python implementations of both methods<fnref n="25" /> to investigate the value and uncertainty estimates of $i, \Omega$, and $\omega$ for the non-circular Orbital solution in DR3. One finding is that the inclination uncertainty generally increases towards face-on configurations and that the linearised uncertainty estimate is usually comparable to the Monte Carlo estimate, except for large uncertainties $\ge 10^\circ$ where the former is larger. There are only few cases where the discrepancy between the two different inclination estimates is larger than the linearised uncertainty.

The significance estimates using linearised error propagation can be different from alternative estimates that use Monte Carlo resampling for obtaining $\sigma'_{a_0}$. These differences typically become important for solutions with poorly constrained eccentricities (i.e. $e/\sigma_e < 1$) for which Monte Carlo resampling is not recommended because of overestimated variances of the Thiele-Innes coefficients (Babusiaux et al. 2023; Holl et al. 2023b). In Fig. D.4 we see that the correlation between the Monte Carlo-resampled semi-major axis significance and $\omega$ is slightly weaker, i.e. has a smaller amplitude variation, than for the published significance.

## Appendix E: Primary mass computation

We use the PARSEC isochrones<fnref n="26" /> (Bressan et al. 2012) with ages $\tau$ by steps of 0.01 in $log(\tau)$ and metallicity [M/H] from -2 to 0.4 dex by steps of 0.05. Considering that the mass distribution of the isochrones is not uniform and that the age

<footnote n="25">²⁵ https://github.com/Johannes-Sahlmann/pystrometry (Sahlmann 2019).</footnote>

<footnote n="26">²⁶ http://stev.oapd.inaf.it/cmdV3.6</footnote>
<figure><img src="image_62.png" /><figcaption>Fig. E.1. Density plot of the comparison between the primary masses derived here with the FLAME ones for systems with fluxratio < 0.01.</figcaption></figure>

sampling is in log, weights need to be applied to each isochrone point i: $P(i) = P(M, \tau, [M/H]) = P(M)P(\tau)P([M/H])$. For $P(M)$ we use the Chabrier (2001) IMF, for $P(\tau)$ a flat star formation rate and for $P([M/H])$ a Gaussian centred on zero with a dispersion of 0.05 so that by default the solar isochrones will drive the mass determination. The default mass sampling being too sparse at the bottom of the main sequence, we interpolated the isochrones with finer mass steps. We removed the lowest mass stars which present a step feature in the isochrone HRD which is not observed in the Gaia DR3 HRD, so that the masses can only be derived for $M_G < 14.4$ which corresponds to $M > 0.1 M_\odot$. We also removed points outside a very crude age-metallicity relation that is $[M/H] < -0.4-0.05\tau$ for $\tau > 10$ or $[M/H] > 0.5-0.05\tau$.

Our observables are the absolute magnitude $M_G$ and the colour ($BP - RP)_0$, noted below $\tilde{O}$. Through Bayes's theorem and considering that the isochrone point $i$ contains the $M$ information, we have:

$$P(M|\tilde{O}) \propto \sum_i P(\tilde{O}|i)P(i). \qquad (E.1)$$

To allow for small systematic errors in the Gaia photometry, we quadratically add a 0.01 mag error to the $G$, $G_{BP}$, $G_{RP}$ formal magnitude errors.

To avoid the presence of outliers in azero_gspphot, especially at the bottom of the MS (Babusiaux et al. 2023), we use the 3D extinction map of Lallement et al. (2019) to provide an estimate of the extinction $A_0$. Most of the sources in the MS are within the completeness limit of this map. A 10% relative error on the extinction with a minimum error of 0.01 is assumed. We derive the extinction coefficients $k_G$, $k_{BP}$, $k_{RP}$ from the EDR3 extinction law<fnref n="27" />.

We know that the position on the HRD provides a direct estimate on the mass only for MS stars. We therefore remove all isochrone points with PARSEC label=0 (pre-MS stage) before applying Eq. E.1 and only provide a mass estimate for stars with more than half of the isochrone points at $3\sigma$ from the observables with PARSEC label=1 (main sequence stage). When several flux ratios are tested, the label of the smallest valid one is used for the MS star selection. Therefore no mass estimate is provided for giant stars and mass estimates for pre-MS stars will not be valid.

<figure><img src="image_63.png" /><figcaption>Fig. E.2. HRD of the NSS astrometric solutions without a primary mass estimate. The grey background is the HRD of the full DR3 low-extinction stars ($A_0 < 0.05$ according to Lallement et al. 2019), equivalent to Fig. 5 of Gaia Collaboration (2018a). The orange points are identified as giants. The red points do not have isochrone match. Violet + do not have an extinction estimate and uncorrected magnitude and colour are therefore used for those. Green points are the white dwarfs with a mass of $0.65 \pm 0.16 M_\odot$ assumed for Table 3.</figcaption></figure>

For each flux ratio $F_2/F_1$ tested, we obtain the absolute magnitude to be compared with the isochrones with:

$$M_G = G - k_G A_0 + 5 + 5 \log(\sigma/1000) + 2.5 \log(1 + \frac{F_2}{F_1}), \quad (E.2)$$

and the colour simply with $G_{BP} - G_{RP} - (k_{BP} - k_{RP})A_0$. We note that we do not consider the change in colour induced by the presence of the secondary. The mass is then fully driven by $M_G$. We consider that the resulting mass distribution obtained through Eq. E.1 is valid if we have more than five isochrone points within $3\sigma$ in both magnitude and colour to be compared with our star and if the closest isochrone point has a $\chi^2$ p-value larger than 0.01.

Our derived masses are compared with the FLAME ones for systems with a small flux ratio in Fig. E.1. As expected those are fully consistent within the errors quoted, with less than 1% of $5\sigma$ outliers. For the small masses, we see the overestimation trend of the FLAME masses due to the overestimation of the GSP-Phot extinction for stars with $M_G > 7$ (Babusiaux et al. 2023), corresponding to $M < 0.7 M_\odot$.

Figure E.2 shows the location in the HRD of the sources with an astrometric solution but no primary mass estimate obtained from our procedure. They have all been corrected by the extinction except those without an extinction estimate. Apart from giants, young stars above the main sequence are missed by construction as well as a few subdwarfs and sources too faint to have a reliable $G_{BP}$.

<footnote n="27">²⁷ https://www.cosmos.esa.int/web/gaia/edr3-extinction-law</footnote>
Appendix F: Acronyms

<table><thead><tr><td>Acronym</td><td>Description</td></tr></thead><tbody><tr><td>2MASS</td><td>Two-Micron All Sky Survey</td></tr><tr><td>AC</td><td>Across scan (direction)</td></tr><tr><td>ADQL</td><td>Astronomical Data Query Language</td></tr><tr><td>AGB</td><td>Asymptotic Giant Branch (star)</td></tr><tr><td>AL</td><td>ALong scan (direction)</td></tr><tr><td>AP</td><td>Astrophysical Parameters</td></tr><tr><td>BH</td><td>Black Hole</td></tr><tr><td>BP</td><td>Gaia Blue Photometer</td></tr><tr><td>CMD</td><td>Colour Magnitude Diagram</td></tr><tr><td>DoF</td><td>Degree(s) of Freedom</td></tr><tr><td>DPAC</td><td>Data Processing and Analysis Consortium</td></tr><tr><td>DR1</td><td>Gaia Data Release 1</td></tr><tr><td>DR2</td><td>Gaia Data Release 2</td></tr><tr><td>DR3</td><td>Gaia Data Release 3</td></tr><tr><td>EB</td><td>Eclipsing Binary</td></tr><tr><td>EDR3</td><td>Gaia Early Data Release 3</td></tr><tr><td>EUV</td><td>Extreme Ultra Violet</td></tr><tr><td>FLAME</td><td>Final Luminosity Age Mass Estimator</td></tr><tr><td>FUV</td><td>far-ultraviolet</td></tr><tr><td>FWHM</td><td>Full Width at Half-Maximum</td></tr><tr><td>GALEX</td><td>GALaxy Evolution eXplorer</td></tr><tr><td>GoF</td><td>Goodness of Fit</td></tr><tr><td>GSPPhot</td><td>Generalised Stellar Parametriser PHOTometry</td></tr><tr><td>GSPSpec</td><td>Generalised Stellar Parametriser SPECTroscopy</td></tr><tr><td>GUCD</td><td>Gaia Ultra-cool Dwarf</td></tr><tr><td>HealPix</td><td>Hierarchical Equal Area isoLatitude Pixelisation</td></tr><tr><td>HPM</td><td>High Proper Motion</td></tr><tr><td>HRD</td><td>Hertzsprung-Russell Diagram</td></tr><tr><td>IMF</td><td>Initial Mass Function</td></tr><tr><td>LMC</td><td>Large Magellanic Cloud</td></tr><tr><td>LPV</td><td>Long Period Variables</td></tr><tr><td>MAD</td><td>Median Absolute Deviation</td></tr><tr><td>MS</td><td>Main Sequence (star)</td></tr><tr><td>NLS</td><td>Non-linear Spectro</td></tr><tr><td>NS</td><td>Neutron Star</td></tr><tr><td>NSS</td><td>Non-Single Star</td></tr><tr><td>NUV</td><td>near-ultraviolet</td></tr><tr><td>PMa</td><td>Proper Motion anomaly</td></tr><tr><td>RGB</td><td>Red Giant Branch (star)</td></tr><tr><td>RMS</td><td>Root-Mean-Square</td></tr><tr><td>RP</td><td>Gaia Red Photometer</td></tr><tr><td>RUWE</td><td>Re-normalised Unit-Weight Error</td></tr><tr><td>RV</td><td>Radial Velocity</td></tr><tr><td>RVS</td><td>Radial Velocity Spectrometer</td></tr><tr><td>SB</td><td>Spectroscopic Binary</td></tr><tr><td>SB1</td><td>Single-line Spectroscopic Binary</td></tr><tr><td>SB2</td><td>Double-line Spectroscopic Binary</td></tr><tr><td>SB*C</td><td>Circular orbit</td></tr><tr><td>SNR</td><td>Signal-to-Noise ratio (also denoted SN and S/N)</td></tr><tr><td>TBO</td><td>Two body orbits</td></tr><tr><td>UCD</td><td>Ultra cool dwarf (star)</td></tr><tr><td>UV</td><td>Ultra Violet</td></tr><tr><td>VIMF</td><td>variable-induced movers fixed</td></tr><tr><td>WD</td><td>White dwarf</td></tr></tbody></table>