Determining the true mass of radial-velocity exoplanets with Gaia: 9 planet candidates in the brown-dwarf/stellar regime and 27 confirmed planets

Mass is one of the most important parameters for determining the true nature of an astronomical object. Yet, many published exoplanets lack a measurement of their true mass, in particular those detected thanks to radial velocity (RV) variations of their host star. For those, only the minimum mass, or $m\sin i$, is known, owing to the insensitivity of RVs to the inclination of the detected orbit compared to the plane-of-the-sky. The mass that is given in database is generally that of an assumed edge-on system ($\sim$90$^\circ$), but many other inclinations are possible, even extreme values closer to 0$^\circ$ (face-on). In such case, the mass of the published object could be strongly underestimated by up to two orders of magnitude. In the present study, we use GASTON, a tool recently developed in Kiefer et al. (2019)&Kiefer (2019) to take advantage of the voluminous Gaia astrometric database, in order to constrain the inclination and true mass of several hundreds of published exoplanet candidates. We find 9 exoplanet candidates in the stellar or brown dwarf (BD) domain, among which 6 were never characterized. We show that 30 Ari B b, HD 141937 b, HD 148427 b, HD 6718 b, HIP 65891 b, and HD 16760 b have masses larger than 13.5 M$_\text{J}$ at 3-$\sigma$. We also confirm the planetary nature of 27 exoplanets among which HD 10180 c, d and g. Studying the orbital periods, eccentricities and host-star metallicities in the BD domain, we found distributions with respect to true masses consistent with other publications. The distribution of orbital periods shows of a void of BD detections below $\sim$100 days, while eccentricity and metallicity distributions agree with a transition between BDs similar to planets and BDs similar to stars about 40-50 M$_\text{J}$.


Introduction
A large fraction of exoplanets published in all up-to-date catalogs, such as www.exoplanet.eu (Schneider et al. 2011) or the NASA exoplanet archive (Akeson et al. 2013) were detected thanks to radial velocity variations of their host star. If the minimum mass m sin i, with i being a common symbolic notation for 'orbital inclination', is located below the planet/brown-dwarf (BD) critical mass of 13.5 M J such detection has to be considered as a new "candidate" planet. If any observed system is inclined according to an isotropic distribution, there is indeed a non-zero probability 1 − cos I c , with I c the inclination of the candidate orbit, that the m sin i underestimates the true mass of the companion by a factor larger than 1/ sin I c . With I c =10 • , this leads already to a factor ∼6, with a probability of 1.5%. Such small rate, considering the ∼500-1000 planets detected through RVs, implies that only few tens of planets have a mass strongly underestimated. However, exoplanets catalogs usually neglect an important number of companions which m sin i is larger than 13.5 M J . The RV-detected samples of exoplanets in catalogs are partly biased to small m sin i candidates and thus to small inclinations (see e.g. Han 2001 for a related discussion). They likely contains more than few tens of objects which actual mass is larger than 13.5 M J .
Please send any request to flavien.kiefer@obspm.fr Recovering the exact mass ratio distribution of binary companions from their mass function, therefore bypassing the issue of unknown inclination by using inversion techniques, such as the Lucy-Richardson algorithm (Richardson 1972, Lucy 1974, is a famous well-studied problem (Mazeh & Goldberg 1992, Heacox 1995, Sahaf et al. 2017. Such inversion algorithms were applied to RV exoplanets mass distribution (Zucker & Mazeh 2001b, Jorissen, Mayor & Udry 2001, Tabachnik & Tremaine 2002. However, it is a statistical problem that cannot determine individual masses. It also lacks strong validation by comparing to exact mass distributions in the stellar or planetary regime. Moreover, the distribution of binary companions in the BD/Mdwarf, with orbital periods of 1-10 4 days, from which exoplanet candidates could be originating, is not well described. Detections are still lacking in the BD regime -the so-called BD desert (Marcy et al. 2000) -although this region is constantly being populated (Halbwachs et al. 2000, Sahlmann et al. 2011a, Diáz et al. 2013, Ranc et al. 2015, Wilson et al. 2016. Our sparse knowledge of the low-mass tail of the population of stellar binary companions does not allow disentangling low sin i BD/M-dwarf from real exoplanets. It also motivates to extensively characterize the orbital inclination and true mass of companions in the exoplanet to M-dwarf regime. The true mass of an individual RV exoplanet candidate can be determined by directly measuring the inclination angle of its orbit compared to the plane of the sky. If the companion is on Article number, page 1 of 32 arXiv:2009.14164v1 [astro-ph.EP] 29 Sep 2020 A&A proofs: manuscript no. exoplanet_mass_gaia an edge-on orbit (I c ∼90 • ), then it is likely transiting and could be detected using photometric monitoring. Commonly the transiting exoplanets are detected first with photometry -with e.g. Kepler, WASP, TESS -and then characterized in mass with RV. About half of the exoplanets observed in RV were detected by transit photometry. The other half are not known to transit and the main options to measure the inclination of an exoplanet orbit are mutual interactions in the case of multiple planets systems, and astrometry.
Astrometry has been used to determine the mass of exoplanet candidates in many studies. Observations with the Hubble Space Telescope Fine-Guidance-Sensor (FGS) led to confirm few planets, in particular GJ 876 b (Benedict et al. 2002), and -Eri b (Benedict et al. 2006). It led also to corrected mass of planet candidates beyond the Deuterium-burning limit, such as HD 38529 b with a mass in the BD regime of 17 M J , and HD 33636 b, with an M-dwarf mass of 140 M J (Bean et al. 2007). Hipparcos data were also extensively used to that purpose (Perryman et al. 1996, Mazeh et al. 1999, Zucker & Mazeh 2001a, Sozzetti & Desidera 2010, Sahlmann et al. 2011a, Reffert & Quirrenbach 2011, Diáz et al. 2012, Wilson et al. 2016) but only yielded masses in the BD/M-dwarf regime. More recently, Gaia astrometric data were used for the first time to determine the mass of RV exoplanet candidates with different methods: either based on astrometric excess noise for HD 114762 b, showing it is stellar in nature (Kiefer 2019), either by comparing Gaia proper motion to Hipparcos proper motion on the case of Proxima b, confirming its planetary nature (Kervella et al. 2020). It is expected that Gaia will provide by the end of its mission the most precise and voluminous astrometry able to characterize exoplanet companions and even to detect new exoplanets (Perryman et al. 2014).
In the present study, we aim at assessing the nature of numerous RV-detected exoplanet candidates publicaly available in exoplanets catalogs using the astrometric excess noise from the first data release, or DR1, of the Gaia mission (Gaia Collaboration et al. 2016). We use the recently developed method GAS-TON , Kiefer 2019, to constrain, from the astrometric excess noise and RV-derived orbital parameters, the orbital inclination and true mass of these companions.
In Section 2, we define the sample of companions and hoststars selected from this study. In Section 3, we explore the Gaia archive for the selected systems and reduce the sample of companions to those with exploitable Gaia DR1 data and astrometric excess noise. We summarize the GASTON method in Section 4. The GASTON results are presented in Section 5. They are then discussed in Section 6. We conclude in Section 7.

Initial exoplanet candidates selection
In order to measure the inclination and true mass of orbiting exoplanet candidates, complete information on orbital parameters are required. We thus need first to select a database in which the largest number of published exoplanets fullfill several criteria. They are the followings: (1) A measurement for period P, eccentricity e, RV semiamplitude K and star mass M must exist; (2) K, P and M should be >0; (3) If e > 0.1, a measurement for T p and ω, respectively the time of periastron passage and the angle of periastron, must exist. If e<0.1, the orbit is about circular, so the phase is not taken into account in the GASTON method and thus T p and ω are spurious parameters; (4) A given value for m sin i (otherwise calculated from other orbital parameters); (5) Recently updated.
We compared the 3 main exoplanets databases available on-line, which are the www.exoplanet.eu (Schneider et al. 2011), www.exoplanets.org (Han et al. 2014) and NASA exoplanet archive, applying these above criteria. A complete review on the current state of on-line catalogs has been achieved in Christiansen (2018). The result of this comparison is shown in Table 1.
The www.Exoplanets.org although not updated since June 2018 is the most complete database available, with respect to planetary, stellar and orbital parameters, with a complete set of orbital data for 911 companions. For comparison, in the NASA exoplanet archive (NEA), there are only 580 exoplanets for which a complete set of parameters is given. In the Exoplanet.eu database, a reference in terms of up-to-date data (4302 against 4197 in the NEA on 12th of August 2020), suffers from inhomogeneities in the reported data, with e.g. some masses expressed in Earth mass while most are given in Jupiter mass, or radial velocities semi-amplitudes that are only sparsely reported. We found best to rely on the www.Exoplanets.org database, the most homogeneous, although counting only 3262 objects. It constitutes a robust yet not too old reference sample of objects that will remain unchanged in the future, since updates have ceased.
In this database, applying the above criteria, the sample of companions reduces down to 924 companions. A measurement of m sin i is provided with uncertainties for 911 of them, following Wright et al. (2011). There thus remains 13 objects for which the m sin i was not provided. Those planets are all transiting, but for 12 of them no RV signal is detected (Marcy et al. 2014) and K is only an MCMC estimation with large errorbars. We will exclude those 12 objects from our analysis. The remaining planet with no m sin i given in the database is Kepler-76 b. However, a solid RV-variation detection is reported in Faigler et al. (2013), leading to an m sin i of 2±0.3 M J . We thus keep Kepler-76 b in our list of targets and insert its m sin i measurement.
The selected sample also includes 358 exoplanets detected with transit photometry and Doppler velocimetry. These companions with known inclination of their orbit -edge-on in virtually all cases -will be useful to assess the quality of the inclinations obtained independently with GASTON. The full list of 912 selected companions orbiting 782 host-stars are shown in Table 2.

Gaia DR1 data for the target list
The GASTON algorithm determine the inclination of RV companion orbits using the Gaia DR1 astrometric excess noise (Gaia Collaboration et al. 2016, Lindegren et al. 2016. The most recent Gaia DR2 release cannot be used similarly because it is based on a different definition of the astrometric excess noise and moreover cursed by the so-called 'DOF-bug' directly affecting the measurement of residual scatter (Lindegren et al. 2018). For that reason, from Kiefer et al. (2019) it was decided to rely the GASTON analysis on the more reliable, although preliminary, Gaia DR1 data.
The list of host stars constituted in Section 2 is uploaded in the Gaia archive of the DR1 to retrieve astrometric data around each star, with a search radius of 5". Among the 782 host stars of Article number, page 2 of 32 F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia  Table 2. List of selected exoplanet companions (see Section 2). Only the 10 first companions of the sample are shown here. The whole sample of 912 companions will be made available online. our initial sample defined in the previous Section, we found 679 entries in the DR1 catalog. Most stars are reported singles, but among the 679 DR1 sources, 44 (with 50 reported exoplanets) have a close background star, a visual companion, or a duplicated (but non-identified) source, with a separation to the main source smaller than 5". In particular, 7 stars (with 12 reported exoplanets) have a "visual companion" with a different ID, at less than 5" distance but with an equal magnitude ±0.01. This is strongly suspicious, and must be due to duplication in the catalogue. Duplication is only reported in the Gaia DR1 database for one of those sources, YZ Cet. We consider safer to exclude these 7 sources from our analysis. However, in general we want to keep those that are marked as duplicate. Duplication separates the dataset of a single source into two different IDs. In the worst case scenario, duplication lead to ignore outlying measurements, and thus to underestimate the astrometric scatter. This can only be problematic if GAS-TON leads to characterize a mass in the regime of planets, since underestimating the astrometric excess noise implies underestimating the mass. More generally, duplication is not an issue because GASTON characterizes masses in the regime of BD or stars, allowing to exclude a planetary nature.
Finally, we identified three supplementary problematic hosts with a magnitude difference with commonly adopted values, as in e.g. SIMBAD, larger than 3. These are Proxima Cen, HD 142 and HD 28254 (see e.g. Lindegren et al. (2016) for Proxima Cen) . We also exclude them from our studied sample. We also note the presence of 11 sources with a null parallax, which are also taken off the sample.
The Gaia DR1 sources are divided into two different datasets : the 'primary' and the 'secondary' (Lindegren et al. 2016). The primary dataset contains two million of targets also observed with Tycho/Hipparcos for which there is a robust measurement of parallax and proper motion out of a 24-year baseline astrometry. It is also sometimes referred to as the TGAS (for the joint Tycho-Gaia Astrometric Solution) dataset. The secondary dataset contains 1.141 billion sources that do not have a supplementary constraint on position from Tycho/Hipparcos, some of those being also newly discovered objects. In the secondary dataset, the proper motion and parallax are fitted to the Gaia data, leaving from a prior based on magnitude (Michalik et al. 2015b), but they are discarded in the DR1. In Lindegren et al. (2016), it is reported that the astrometric residuals scatter is generally larger in the secondary dataset that in the primary dataset (see also Section 3.3 below). We will thus separate those secondary dataset objects from those in the primary dataset in the rest of the study and treat them specifically.
In total, we constituted a sample of 755 exoplanets with both RV and Gaia DR1 data, orbiting 658 stars of which Table 3 gives the full list. Among those, 508 exoplanets orbit 436 stars in the primary dataset, for 247 exoplanets around 222 stars in the secondary dataset. We list among all DR1 parameters the G-band magnitude, the parallax, the belonging to primary or secondary dataset, the source duplication (see e.g. Lindegren et al. 2016), the number of field-of-view transits N FoV (matched_observations in the DR1 catalog), the total number of recorded along-scan angle (AL) measurements N tot , the astrometric excess noise ε DR1 and its significance parameter D ε (Lindegren et al. 2012).

Magnitude, color and parallax correlations with astrometric excess noise
The astrometric excess noise is the main measured quantity that will be used in this study to derive a constraint on the RV companion masses listed in Table 2. The fundamental hypothesis assumed in GASTON relates the astrometric excess noise to astrometric orbital motion. It is thus crucial to identify possible systematic correlations of this quantity with respect to other intrinsic data such as magnitude, color or DR1 dataset that would reveal instrumental or modelisation effects. As can be seen in Fig 1, stars brigher than magnitude 6.4 show a significant drift of increasing excess noise with decreasing magnitude. This is a sign of instrumental systematics (PSF, jitter, CCD sensibility etc.), that are recognised to occur in Gaia data (Lindegren et al. 2018). With G-mag<6.4, the astrometric excess noise are all larger than 0.4 mas, with a median value Article number, page 3 of 32 A&A proofs: manuscript no. exoplanet_mass_gaia Table 3. List of selected stellar hosts from the initial sample and selected from the Gaia DR1 archive (see Section 3.1). Only the 10 first sources of the sample are shown here. The whole list of 658 stars hosting 755 exoplanet candidates will be made available online.

Notes.
(a) The Gaia recorded flux magnitude in the G-band. (b) The parallax. For the sources from the secondary dataset, the values are given without errorbars since missing from the DR1. They are taken from the DR2. For those, we will assume 10% errorbars in the rest of the study. (c) DR1 primary (1) or secondary (2) dataset. (d) Duplicate source (true) or not (false), as explained in Lindegren et al. (2016).
(e) Number of field-of-view transits of the sources (matched_observations in the DR1 database). (f) Total number of astrometric AL observations reported (astrometric_n_good_obs_al in the DR1 database). (g) Astrometric excess noise in mas.
(i) G b -G r color index as presented in Lindegren et al. (2018).  Table 3. The red dashed line shows the G=6.4 limit discussed in the text. We separate targets from the primary dataset (in blue) and from the secondary dataset (orange). The gray line is a moving median filter of the data. about 0.7 mas. In the rest of the paper, we will thus exclude any source with a G-mag<6.4, reducing the sample to 614 sources. Moreover, the astrometric excess noise also shows a correlation with color indices, i.e. the B-V as found in Simbad for 498 sources with G-mag>6.4, and the Gaia DR2 b−r available in the DR2 database for all the 614 sources with G-mag>6.4, as plotted in Figs 2 and 3. A moving median filter of the data indeed shows a correlation of ε DR1 with B-V beyond 1, and with DR2 b − r beyond 1.4. The B-V index is not available for all the 614 sources, we thus prefer using the DR2 b − r index as a limiting parameter. As for the magnitude, the astrometric excess noise is larger than 0.4 mas whatever b−r larger than 1.8. A correlation of the alongscan (AL) angle residuals with V-I color was already reported in Section D.2 of Lindegren et al. (2016). These two correlations likely have a common optical origin due to the chromaticity of the star centroid location on the CCD. In the rest of the paper, we will thus also exclude any source with a b − r>1.8, reducing the sample to 580 sources. Figure 4 shows that the parallax and magnitude are correlated in both primary and secondary datasets, which is not surprising as we expect distant sources to be on average less luminous than sources close-by. Sources of the secondary dataset are located much farther away from the Sun than sources of the primary dataset, which is expected from the absence of Tycho data for the secondary dataset. The astrometric excess noise is not correlated with parallax, but we observe astrometric excess noise measurements on the same order -and even larger -than the parallax for sources of the secondary dataset. This strongly suggests issues with parallax and proper motion modeling, reminding that those parameters are poorly fitted from rough priors in the secondary dataset. We will thus discard from the rest of the study secondary sources for which log π∼log ε DR1 ±0.5. We think wiser to postpone their thorough analysis to the future Gaia DR3 release. Moreover, the largest ε DR1 in the secondary dataset are generally obtained for small parallax (<10 mas). This behavior is different from what is observed in the primary dataset where the astrometric excess noise is not correlated with parallax.
The final sample contains 597 planet candidates orbiting around 524 host stars with G>6.4, b − r<1.8, and for sources in the secondary dataset with log π-log ε DR1 >0.5.

Distribution of astrometric excess noise
In order to get a sense of how ε DR1 is a relevant quantity to characterize a binary or planetary system, it is crucial to understand how the astrometric excess noise generally varies with respect Article number, page 4 of 32   Article number, page 5 of 32 A&A proofs: manuscript no. exoplanet_mass_gaia to the known or unknown inclination of the gravitational systems observed -transiting or not -the presence of a long-period outer companion in the system -presence of RV drift -and the quality of their observations with Gaia -primary or secondary dataset. We perform here an analysis of the distribution of astrometric excess noise of our selected sample of companions and sources as defined in Section 3.2, with respect to following subsets selection criteria: -Dataset (primary/secondary); -All planets around the host star are transiting; -At least one planet is not transiting; -Detection or hint of an RV drift; -No hint of an RV drift.
In principle, with orbital inclination fixed to ∼90 • , the semimajor axis of transiting planets host stars should not reach more than a few µas, and remain undetectable in the DR1 astrometric excess noise. The astrometric scatter is dominated by the instrumental and measurement noises on the order of ∼0.6 mas (Lindegren et al. 2016). The distribution of ε DR1 for transiting planet hosts should be close to the distribution of astrometric scatter due to pure instrumental and measurement noises. On the other hand, system with non-transiting planets allow inclinations down to 0 • , and host star semi-major axis beyond a few 0.1 mas. We expect their astrometric excess noise to be generally larger than for systems with transiting-only planets. Finally, the detection of a drift in the RV suggest the presence of a hidden outer companion in the system. The astrometric excess noise might be systematically larger for those systems, implying that the astrometric signal is not only due to the companion with a well-defined orbit. This is however certainly not a rule, as shown e.g. in the case of HD 114762 (Kiefer 2019) for which the astrometric excess noise is dominated by the effect of the short period companion HD 114762 Ab.
In Table 4, we present the 10th, 50th and 90th percentiles of the astrometric excess noise distribution according to the different sub-samples defined above. We confront them to the Lindegren et al. (2016) percentiles derived for the whole primary, secondary and Hipparcos DR1 datasets (Tables 1 and 2 in Lindegren et al. 2016) based on more than 1 billion sources observed with Gaia. In Figure 5, we compare in a first panel the distributions of astrometric excess noise for the sources from the primary and secondary datasets with transiting-only planets, and in a second panel, sources from the primary dataset with transiting-only planets to those with at least one non-transiting planet.
The median and 90th percentile value of the ε DR1 distribution for all subset in the primary dataset are generally compatible with the Lindegren et al. (2016) values. Although Lindegren et al. (2016) study shows that the Hipparcos subset is associated to larger excess noise, Table 4 shows that excluding G-mag>6.4 and b − r>1.8 objects as proposed in Section 3.1 above leads to decreasing the extent of the astrometric excess noise distribution with values agreeing with the primary dataset. The Hipparcos subset excluding bright and late-type sources is thus likely not different from the full primary dataset.
We observe a clear distinction in the distributions of ε DR1 between the primary and secondary datasets, with significantly higher astrometric excess noise in the secondary dataset. This could be well explained by the absence, for the secondary dataset, of the Tycho/Hipparcos supplementary positions 24years in the past that allows deriving robust proper motion and parallax for the sources in the primary dataset. The derivation of proper motion and parallax from Gaia data only with Galactic priors based on magnitude (Michalik et al. 2015b, Lindegren et al. 2016) certainly leads to larger scatter in the residuals of the 5-parameters solution.
For transiting sources of the primary dataset, the 90th percentile of the astrometric excess noise distribution is 0.70 mas. This is compatible with, and even lower than, Lindegren et al. (2016) values of the global DR1 solution. For this subset, the 95-th percentile is 0.81 mas, still lower than the 90th-percentile of Lindegren et al. (2016). This generally small astrometric excess noises of the sources with transiting planets is compatible with statistical noise and the non-detection by Gaia of any orbital motion of a star orbited by a planet at short separation (<0.1 a.u. or P<50 days) and with an edge-on inclination of its orbit.
The systems in the primary dataset with a non-transiting planet have the highest median among all other subsets (0.47 mas) and the highest 90th percentile (0.78 mas). More importantly, the astrometric excess noise of sources with nontransiting companions is significantly larger than for sources with transiting-only planets. This can also be seen in the lower panel of Figure 5 with a net shift between the two ε DR1 distributions. This confirms that ε DR1 contains a non-negligible fraction of astrometric motion for systems with a companion which orbital inclination is not known.
The ε DR1 in the secondary dataset generally reaches larger values than in the primary dataset, with a 90th percentile for the subset of systems with transiting-only planets ∼0.8 mas. This was expected by the less accurate fit of the proper motion and parallax in the secondary dataset compared to the primary. However, this is also much smaller than the 2.3 mas 90th-percentile derived for the whole secondary dataset in Lindegren et al. (2016). Therefore once cleaned from problematic systems, in particular those with log π/ε DR1 >0.5 (Section 3.2), the astrometric excess noise of remaining objects in the secondary dataset seems robust, with parallax and proper motion most likely well determined (although not published in the DR1).
Interestingly, we find no correlation of the astrometric excess noise distribution with the presence of any drift in the RV data, and even smaller values than in the other subsets. This could be due to the smaller number of sources in this category, which if following a inclination probability density function ∼ sin I c would preferentially have inclinations close to 90 • , and thus smaller astometric motion. It also suggests that the presence of an outer companion does not have a strong effect on the astrometric excess noise compared to the enhanced astrometric motion due to a small inclination of a non-transiting companion.

Testing the noise model
From the 133 and 113 stars with transiting-only planets from the primary and secondary datasets we can test the model of noise used in the simulations of GASTON. In previous studies , Kiefer 2019, we chose to use values based on published estimations of the measurement uncertainty and of the typical external noise (including modeling noise and instrument jitter), respectively σ AL =0.4 mas (Michalik et al. 2015a) and σ syst =0.5 mas (Lindegren et al. 2016). As we showed in the preceding section, the sources with transiting companion must be generally more similar to sources with no astrometric motion. Therefore, the astrometric excess noise measured by Gaia for these sources should be close to purely instrumental and photonic stochastic scatter.
The distribution of ε DR1 for these 300 sources from primary and secondary datasets is plotted in Fig. 6. It is compared to simulations of astrometric excess noise of sources with no orbital motion, in the framework of different noise models. We Article number, page 6 of 32 F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia assumed for each simulation random numbers of FoV transits (N FoV ) and numbers of measurements per FoV transit (N AL ) in the same ranges as those of the sample presented here, e.g. N FoV =15±8 and N AL =7±2 with Gaussian distribution, and imposing that 48>N FoV >5 and 9>N AL >2. We singled out 5 different noise models of (σ syst ,σ AL ), either based on literature values, or based on the best fit of a bi-uniform distribution of σ syst with a fixed median to the ε DR1 cumulative density function ( The bi-uniform distributions models were found to lead to the best least square fit of the observed ε DR1 cdf. All 5 models are compared to the data in Figure 6. A model with a wide range of systematic noise better explain the observed distributions for values of astrometric excess noise assumed to be compatible with pure stochastic and systematic noise, e.g. below 0.85 mas and 2.3 mas for sources in the primary and secondary datasets respectively. The constant noise model tends to overestimate the astrometric excess noise, and the simulated distribution decreases too steeply at values closer to 1 mas. The noise σ syst taken from a bi-uniform distribution of values within bounds (e.g. 0.36-0.7 mas in the primary dataset) explains better the full distribution of observed astrometric excess noises. The distribution is damped beyond about 0.85-1 mas for the primary dataset and ∼1.2-1.4 mas in the secondary dataset, with less than 1% of the simulations beyond.
Article number, page 7 of 32 A&A proofs: manuscript no. exoplanet_mass_gaia We can exclude that the σ syst is much larger than 0.7 mas in the primary dataset, and 0.8 mas in the secondary dataset, since that would extend the core of the distribution towards larger excess noise, therefore leading to a poorer agreement. Finally, we observe that the AL angle measurement uncertainty σ AL does not have a strong impact on the astrometric excess noise. We tested two values, a small uncertainty 0.1 mas, as given in Lindegren et al. (2018), and a more conservative value as assumed by  of 0.4 mas that corresponds well to the typical AL residuals reported in Lindegren et al. (2016) of 0.65 mas ( (σ AL = 0.4) 2 + (σ syst = 0.5) 2 =0.64 mas). We thus fix σ AL to 0.4 mas.
In conclusion, the systematic noise is typically about 0.5 and 0.6 mas respectively in the primary and secondary dataset. But it is likely that for an individual observed source, σ syst can be somewhat larger or smaller by a few fraction of mas. We thus adopt a random systematic noise for each simulation in GASTON uniformly distributed on both sides of the median at 0.4 mas down to 0.36 mas and up to 0.7 mas for sources in the primary dataset, and about the median at 0.45 mas down to 0.4 mas and up to 0.8 mas for sources in the secondary dataset.

Detection threshold and source selection
GASTON cannot be used to characterize the mass of as much as 597 planet candidates. This would require several weeks of calculation, while many of them cannot be truly characterized, because the astrometric excess noise is compatible with an edgeon inclination. We thus need to define a robust threshold above which ε DR1 can be considered significantly non-stochastic -and thus astrophysical -and below which it could be explained by pure stochastic noise.
Given that close to 40-50% of the sources in the Milky Way are part of binary systems (Duquennoy & Mayor 1991, Raghavan et al. 2010), a consequent fraction of Gaia sources -possibly more than 10% -could show a detectable astrometric motion. We thus consider using the 90th-percentiles of Lindegren et al. (2016) based on a sample of more than 1 billion stars, as a detection threshold, above which a significant fraction of ε DR1 could be imputed to astrometric motion. This was assumed in previous work (Kiefer 2019), but the astrometric excess noise distribution of the present sample might differ from the sample upon which these percentiles were calculated.
In Sections 3.3 and 3.4, we showed that the 90th-percentile of 0.85 mas in the primary sample, as derived in Lindegren et al. (2016) is robust, but the 2.3 mas threshold for secondary dataset sources is excessive and could be more reasonably lowered to ∼1.2 mas. This overestimation of the 90th-percentile in Lindegren et al. (2016) must be due to the inclusion of small magnitude, large b − r color sources and badly modelled parallax, which we did clean out from our sample. We will thus use the detections thresholds thresh,prim =0.85 mas, and thresh,sec =1.2 mas, above which the astrometric excess noise would be mainly due to supplementary astrometric motion. This reduces our sample to 28 sources (29 planet candidates) with an astrometric detection by Gaia in the DR1. They are the best candidates for orbit inclination and true mass measurement of the companion. They constitutes the 'detection sample'.
We counted 312 non-transiting planet candidates (around 254 sources) for which the inclination is not constrained from photometry and for which the astrometric motion of their host star leads to an astrometric excess noise smaller than the threshold. Even though compatible with pure noise, the astrometric excess noise allow constraining the true astrometric extent of the star's orbit. This leads to deriving a minimum inclination and a maximum true mass of the exoplanet candidates beyond which they are not compatible with a non-detection. In this sample, we exclude the so-called 'duplicate sources' in the Gaia DR1, because, for such target, possibly large sets of astrometric measurements are attributed to another source with another ID, and thus lead to underestimate its AL-angle residuals and its astrometric excess noise. This will be of crucial importance if the mass of a companion is found to be smaller than 13.5 M J . A larger astrometric excess noise leads to a larger mass range. We thus focus on the 227 non-duplicate companions orbiting 187 sources which will constitute the 'non-detection sample'.
The complete list of 29 detected exoplanet candidates is presented in Table 5, and the list of 312 non-detected non-transiting planet candidates is given in Appendix B. Table 6 summarizes all source selection steps applied from Section 2 up to the present section. Table 5.
List of the 29 selected planets which astrometric excess noise overpass the detection threshold (see text). Where uncertainties are missing we will assume 10% errors on the corresponding parameter. The parallax with uncertainties are all taken from the DR1, while those without uncertainties are taken from SIMBAD. We expand this table on-line to also include the 227 exoplanet candidates for which the astrometric excess noise is smaller than the threshold (see Table 6).
RV data

General principle
In the present study, our goal is to constrain the inclination and true mass of RV planet candidates using the released Gaia astrometric data. To do so, we are applying the GASTON method described in Kiefer et al. 2019& Kiefer 2019. This algorithm simulates the residuals of Gaia's 5-parameters fit of a source accounting for a supplementary astrometric motion due to a perturbing RV-detected companion. It leads to simulated astrometric excess noise ε simu depending on the actual inclination of the RV-detected orbital motion. It also accounts for measurement noise and modeling errors in the reduction of the DR1 through the noise model adopted in Section 3.4. These simulations are then compared to the astrometric excess noise actually measured by Gaia and reported in the DR1 database (Table 3) to derive a matching orbital inclination. The GASTON algorithm is embedded into an MCMC process, with emcee (Foreman-Mackey et al. 2013), that allows deriving the posterior distributions of orbital inclination and true mass of the RV companion among other parameters. To sum up, the varied physical parameters in the MCMC run are the orbital period P, the eccentricity e, the longitude of periastron ω, the periastron time of passage T p , the inclination I c , the minimum mass m sin i, the star mass M , the parallax π, an hyper-parameter f ε to scale error bars on ε 2 DR1 , and a jitter term σ K, jitter . Some of these parameters have strong gaussian priors from RV (P, e, T p , ω, m sin i), or from other analysis (M , π). The hyper-parameter f ε follows a Gaussian prior about 0 with a standard deviation of 0.1. The jitter term follows a flat prior between 0 and √ 3 assuming that the published uncertainty on K could be underestimated by as much as a factor 1 + σ 2 K, jitter =2. Generally, we adopt a dp(I c )=sin I c dI c prior probability distribution for the inclination, assuming the inclination of orbits among RV-candidates is isotropic. If the MCMC converges to an inclination strongly different from 90 • despite the low prior probability, that implies the data inputs have a significant weight in the likelihood.

Dealing with proper motion and parallax in the simulations
For sources in the primary dataset, we assume that the proper motion fit as performed by Gaia in the DR1 is disentangled from the hidden astrometric orbit. We thus assume that the astrometric excess noise is purely composed of noise and orbital motion, and that it is not needed to fit out excess parallax and proper motion to the simulated astrometric orbit. This is justified by the addition of past Hipparcos or Tycho-2 positions in the Gaia's reduction for fitting proper motion of primary dataset sources, thus based on astrometric measurements spanning more than 24 years. Given that the orbital periods of all studied companions are smaller than 14 years, the fit of proper motion to the simulated orbits reduces the amplitude of the simulated residuals -and thus of the astrometric excess noise -only by a small amount. Numerical simulations show that in the worst case scenario with a Tycho-2 position uncertainty of ∼100 mas, the average simulated astrometric excess noise ε simu is lowered by less than 0.2 mas. This offset reduces to less than 0.05 mas if a Hipparcos position (σ RA,DEC ∼1 mas) is used instead or if P<10 days. Hipparcos positions are available for 171 over the 190 primary sources in our sample, while only 6 sources have a Tycho-2 position with more than 20 mas of uncertainty and a companion with P>10 days. These 6 sources, HD 95872, NGC 2423 3, BD+20 2457, HD 233604, BD+15 2375, and M67 SAND 364, all belongs to the non-detection sample. For those, the astrometric excess noise that we simulate with GASTON for a given companion orbit and at a given orbital inclination could be overestimated by up to ∼0.2 mas. Thus GASTON possibly underestimates the upper-limit on the companion true mass for those stars.
For sources in the secondary dataset, the proper motion given in the DR1 is derived from the Gaia data only, without a supplementary data point from Tycho-2 or Hipparcos. An important part of the orbital motion could thus be mistaken for proper motion during the Gaia data reduction of the DR1, especially for orbital periods at which the Gaia measurements along the 416days time baseline of the DR1 campaign could appear almost linear. For sources from the secondary dataset, we thus perform a fit of linear motion to the simulated astrometric orbit, from which residuals we derive ε simu .
For sources of both datasets, fitting the parallax to the astrometric orbit does not have a significant effect on ε simu even if P∼365 days and if the orbital and parallax motions are aligned. Numerical tests of parallax fit to simulated data along an astrometric orbit with P∼365 days and randomizing along the unknown longitude of ascending node, Ω from 0 to 2π, leads to a typical reduction of the average ε simu smaller than 0.05 mas. Fitting parallax to the simulated astrometric orbit is thus unnecessary, leading to negligible deviations on the simulated astrometric excess noise.

Recent improvements
Since Kiefer (2019), we have done few improvements, with the list below: -The number of walkers was reduced from 200 to 20, as it improved the speed of convergence of the MCMC while leading to equivalent results; -The maximum number of iterations is increased to 1,000,000. The MCMC stops whenever the autocorrelation length of every parameters stops progressing by more than 1% and is at least 50 times smaller than the actual number of iterations; -The host star and companion magnitude are calculated using a continuous series of model from planetary mass up to stellar mass of 30 M . We also implemented the reflection of stellar light on the surface of the companion. These issues are discussed in Appendix A.
We highlight here an important effect of the modeling of the light reflected from the companion surface on the motion of the photocenter, and developed in more details in the Appendix. For mass ratios M c /M ∼10 −5 − 10 −3 and companion orbit semi-major axis a c <0.5 a.u, the companion reflected light can become more important than the star's emission in the calculation of the photocenter semi-major axis, with a ph =L c a c + L a . The astrometric motion of the system observed from Earth can even follow the motion of the companion itself rather than the motion of the stellar host. This could lead to wrongly determine the orientation (retrograde/prograde) of the primary star orbit, and strongly underestimate the primary star semi-major axis and thus the mass of the companion. This effect cannot be seen in the present study because it is smaller than the adopted detection thresholds (Section 3.5), but should be taken into account in future analysis of Gaia's time series of systems with planets.
Concerning the definition of the parameters explored in the MCMC corresponding to inclination, eccentricity and longitude Article number, page 10 of 32 F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia  (2019), solving singularity issues at the border of the domain expected for these parameters: -Adopting λ I c =tan(2I c − π/2) as was used in Kiefer (2019) led the StretchMove algorithm of emcee (Foreman-Mackey et al. 2013) to get stuck in low probability regions with large or small inclinations, much wider in terms of λ I c . We thus considered instead simply varying I c imposing rigid boundaries at I c =0 and π/2. -The exploration of the (e,ω) space was not optimal, especially around the singularity e=0. We thus varied instead λ c =tan( π 2 e cos ω) and λ s =tan( π 2 e sin ω), with e and ω being then obtained from the simple transformations and combinations of λ c and λ s .

Application of GASTON to the defined samples
We apply GASTON on the 29 candidate exoplanets of the detection sample, orbiting the 28 sources which astrometric excess noise exceed the detection thresholds fixed in Section 3.5 and listed in Table 5. For those, with the star's orbit a priori detected in the astrometric data, an inclination and true mass could technically be measured.
For the 227 non-transiting companions of the non-detection sample listed in Table B.1, we also used GASTON to derive the lowest inclination and largest mass possible for the companion, beyond which the astrometric excess noise would become too large to be compatible with ε DR1 . In order to limit the computation time, and since these calculations only leads to parameter ranges and not strict measurements, we reduced the maximum number of MCMC steps in GASTON to 50,000 for these 227 companions. Moreover, conversely to what adopted for the detection sample, for those 227 companions we adopt a flat prior for the inclination. The shape of the prior distribution of the inclination tends to dictate the shape of the posterior distributions if the simulated astrometric excess noise is compatible with ε DR1 for inclinations of 90 c irc down to ∼0 • . This prior artificially increases the conventional lower limit, such as 3-σ, for inclination -and thus decreases the upper-limit on mass. This is typically the case for companions in the non-detection sample with ε DR1 compatible with noise and a sin i ε DR1 . Adopting a flat prior for the inclination favours instead the likelihood -and thus the data -to dictate the shape of the posterior distributions down to small inclination. This better reveals the variations of the inclination and mass posteriors only due to incompatibilities between the ε simu and ε DR1 at inclinations close to 0 • .
In the following, we will only report for the resulting posteriors of the inclination I c and its deriving parameters: the true mass of the companion, the photocenter semi-major axis, and the magnitude difference between the companion and its host star.

General results
Out of the results produced by GASTON, we identified 3 possible situations: 1. Orbits leading to a firm measurement of the RV orbit inclination and the true mass of the companions. This concerns 9 exoplanet candidates out of 29 in the detection sample. This is summarized in Section 5.2.1; 2. Orbits for which the astrometry cannot constrain the inclination. Because of the noise, producing a measured astrometric excess noise compatible with the RV orbital motion is possible for a large range of inclinations. The derived solution follows mainly the sin i prior distribution of inclination, with a median about 60 • , 1-σ confidence interval within 30-80 • and a 3-σ (99.85%) percentile larger than 89.5 • . Only the upper-limit on the mass and the lower-limit on the inclination is informative. This concerns 18 exoplanet candidates from the detection sample and the 227 exoplanet candidates from the non-detection sample. This is summarized in Sections 5.2.2 and 5.3; 3. Companions for which the astrometric excess noise could never be reached in the simulations testing any inclinations from 0.001 to 90 • . The Gaia astrometric excess noise is incompatible with the published RV orbit. Two companions from the detection sample enter this situation, WASP-43 b and WASP-156 b (see Section 5.2.3).
For the 29 companions of the detection sample, the results of GASTON according to different situations introduced above are presented in Tables 8 & 7. Moreover, Table 9 summarizes the parameter limits derived for the 227 companions of the nondetection sample. In both tables, we list the resulting corrected mass, astrometric orbit semi-major axis, estimated magnitude Article number, page 11 of 32 F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia difference between the host and the companion, MCMC acceptance rate and convergence indicator N steps /max(τ λ ) (see below).
The acceptance rate delivered by emcee allows to quantify the probability of reaching ε DR1 through all simulations performed during the MCMC process. Typically, if an MCMC performs well, the acceptance rate must reach 0.2-0.4. This is the case for all 9 companions entering situation #1, except one, HD 96127 b for which it is 0.06. Low values of the acceptance rate usually imply too large steps in the Monte-Carlo process (Foreman-Mackey et al. 2013). We can firmly exclude any "steps issue", since the geometry of the parameter space is the same for all systems, and the steps for the different parameters have been tuned such that well behaved cases fullfill the 0.2-0.4 criterion. Rather we explain this low acceptance rate by the presence of noise in our simulations. A fortuitous pile-up of noise can allow some simulations to be compatible with ε DR1 =1.124 mas even with an inclination close to 90 • and a negligible photocenter orbit. With a sin i-prior on inclination favouring the edge-on configuration, this is sufficient to drag the MCMC towards exploring regions where producing such astrometric excess noise is not frequent. The low acceptance rate is a reflection of this low frequency. This leads, in the case of HD 96127 b, to a 3-σ upper-limit on the inclination of 89.54 • . This is the same mechanism that explains the small acceptance rates associated to mass upper-limits for all companions entering situation #2.
The autocorrelation length τ λ probes the quality of a λparameter exploration by the MCMC during a run. With emcee and its Goodman-Weare algorithm (Goodman & Weare 2010) it can be considered that convergence is reached if at least N steps /τ λ >50 (Foreman-Mackey et al. 2013), and at best if δτ λ /τ λ <1% for all parameters λ. The errors on the estimations of the posteriors are then reduced by a factor smaller than 1/ √ 50∼0.14. Longer chains obviously produce more accurate results, but are also more time consuming. This paper is not aiming perfect accuracy, since only based on a preliminary estimation of one quantity, the astrometric excess noise, by Gaia. We thus decided to stop the MCMC whenever N steps,max is reached or N steps /max(τ λ )>50 and δτ λ /τ λ <1% for all parameters λ. With up to 1,000,000 steps and 20 walkers for 10 parameters to explore, the MCMC should have enough time to converge. This allows to identify problematic systems, such as e.g. HD 96127 b, for which the exploration of the parameter space is inefficient. In Table 8, we identify 3 companions -including HD 96127 b -for which GASTON did not converge after N step =1,000,000 iterations, with a maximum autocorrelation length larger than N step /50 and a small acceptance rate. The posteriors for those companions cannot be reliable, and the width of the confidence intervals on their mass is most likely underestimated.

Detection sample
5.2.1. SItuation #1: mass measurement for 2 possible massive exoplanets, 2 BDs and 5 M-dwarfs We illustrates this first case scenario in Figure 7 with the example of 30 Ari B b for which with a period of 335 days, the astrometric excess noise of 1.78 mas leads to an inclination of 4.14 •+0.96 −0.90 and a corrected mass of 148 +42 −27 M J instead of an m sin i=10±1 M J . The top panel of Fig. 7 shows the simulated astrometric excess noise obtained for 10,000 different values of inclinations from 0.001 to 90 • . For any inclinations below 1 • the true mass of 30 Ari B b is too large and the magnitude difference with the primary star is smaller than 2.5; these simulations are ignored since they would imply the presence of a detectable secondary component in the spectrum of this system, conversely to what observed.
The bottom panel of Fig. 7 compares the I c posterior distribution -probability density function or PDF -to the PDF of an ensemble of same size drawn from the assumed prior density function, dp=sin I c dI c . This posterior PDF is well distinct from the prior PDF which thus have a minor impact on the posterior distributions output from the MCMC. The corner plot of all posterior distributions for 30 Ari B b is shown in Figure 8. In this category, all other GASTON runs work similarly as well as 30 Ari B b, with the exception of HD 96127 b which MCMC run could not converge after 1,000,000 iterations. In total, the true masses for 9 exoplanet candidates could be determined using GASTON, with 8 orbiting sources from the primary dataset and one, HD 16760 b, from the secondary dataset. We determined that 7 of the companions are not planets, and two, could be likely brown-dwarf or M-dwarf, but the planetary nature cannot be excluded at 3-σ.
Among the primary sources, we find that HD 5388  The two possible planets are HD 141937 b and HD 96127 b. The true mass of HD 141937 b is located just beyond the boundary between massive planets and low-mass brown dwarfs with M=27.5 +6.9 −10.8 M J at 1-σ but a mass possibly as low as 9 M J at 3-σ.
The true mass of HD 96127 b is most likely well within the stellar domain with M=190 +284 −184 M J and an inclination I c =1.364 +38.527 −0.763 • at 1-σ. Within the 1-σ confidence interval, a true mass of HD 96127 b as low as 6 M J could also be compatible with ε DR1 . However, we already noted that GASTON did not converge for this precise case, due to a marginal but possible compatibility of the Gaia DR1 astrometry with an edge-on configuration, as revealed by the low 0.05 acceptance ratio of the MCMC run. The 1-σ bounds of HD 96127 b's mass are thus questionable and its true nature is still uncertain.
The results for the single source from the Gaia DR1's secondary dataset within situation #1, HD 16760 b, are given in table 8, and illustrated in Fig. 9. HD 16760 b (Bouchy et al. 2009), is the first companion with a possible planetary mass discovered with the SOPHIE spectrograph (Perruchot et al. 2013) actually is not a planet. With a parallax of 14 mas and an astrometric excess noise of 2.99 mas, we found its astrometry to be rather compatible with an M-dwarf which true mass is larger than 13.5 M J at 3-σ.

SItuation #2: upper-limit constraint on companion mass
The orbit inclination of 18 companions from 17 different systems in the detection sample cannot be fully determined using GAS-Article number, page 13 of 32 F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia TON. For those orbits, the simulated astrometric excess noise is often compatible with ε DR1 from I c =90 • down to ∼0 • . Accounting for the sin I c prior probability distribution on the inclination, the MCMC leads to a posterior distribution for which the 3-σ upper-bound on inclination is located beyond 89.5 • . More accurately, the posterior distribution on their orbit inclination and mass are mainly fixed by the sin I c prior on inclination.
As presented in Table 8, all these candidates are possible planets at the 1-σ limit. Excluding the transiting planets which are known to be bona-fide planets on edge-on orbits, only two of them have a true mass below, but close to, the Deuterium burning limit of 13.5 M J . They are HD 164595 b and HD 185269 b with a mass smaller than respectively 12.9 and 12.6 M J at 3-σ. Those two seem thus likely to be actual planets with a mass in the Neptunian (0.06 M J for HD 164595 b) and Jupiterian (1.12 M J for HD 185269 b) domain.
We note however that HD 164595 is a duplicate source in the Gaia DR1. Its astrometric excess noise, and thus the mass of HD 164595 b, might be underestimated (see the discussion on this specific issue in Section 3.1). Moreover, for both companions, the simulated astrometric excess noise is indeed compatible with ε DR1 on a large range of inclinations (Figs. 10 and 11). The posterior distributions of I c and M c are essentially due to the prior distribution on I c . If the actual prior distribution of I c is biased towards 0 • (see the related discussion in Section 6.2), it cannot be excluded that the masses of HD 164595 b and HD 185269 b are actually larger than 13.5 M J . The MCMC acceptance rates are smaller than 0.01 with a star semi-major axis smaller than 1 µas. It can be excluded that Gaia will truly detect the reflex motion of these stars due to their transiting exoplanets.
Two exoplanet candidates are part of a common multiple system, HD 154857 b and c. The Gaia observations are compatible with an edge-on inclination and masses of 2.2 and 2.5 M J . At 1-σ the posterior distributions, conformally to the sin I c prior distribution on I c , allow inclinations as low as 20 • with masses as large as 6 M J , but at 3-σ their mass could be as large as 135 and 175 M J . Both companions are thus possible Jupiter-mass planets with masses within 2-6 M J , but their true nature could not be confirmed.

Situation #3: incompatible RV orbit and Gaia astrometry
The GASTON results for the two companions within this situation are presented in Table 7. They are WASP-43 b and WASP-156 b, both transiting planets on compact orbit (P=0.8 days and 3.8 days). In these two systems, none of the published companions are adequate for explaining Gaia's observations. The maxi-  mum astrometric excess noise that could be simulated from RV orbital parameters were respectively 1.25 and 1.32 mas, well below the ε DR1 of these two sources, respectively 2 mas and 1.5 mas. These two sources from the secondary dataset are not mentioned as duplicated sources in the DR1 database. There are three possible scenarios for explaining this RV-Gaia discrepancy: -The value of the astrometric excess noise could depend on the presence of fortuitous outliers. With a number of astrometric measurements ∼50 per source, outliers of several mas could slightly inflate ε DR1 with a discrepancy of a few 0.1 mas. Outliers larger than 4.8×ε DR1 ∼10 mas (see note 7 in Lindegren et al. 2016) are flagged as "bad" during the AGIS reduction and discarded. Therefore, the discrepancy observed in Table 7 for the 2 companions between the highest ε simu and ε DR1 of 0.2-0.7 mas could be explained by numerous or large outliers. We cannot exclude this possibility without analysing the time series, which will not be available until the final Gaia release in a few years. -Instrumental and modeling noises larger than those adopted in Section 3.4 could allow reaching the astrometric excess noise. Indeed, for the astrometric data of secondary dataset targets the parallax and proper motion fit is not of good quality, and could individually reach high astrometric excess noise, as indicated by the 90th-percentile ε DR1 =2.3 mas measured by Lindegren et al. (2016) in the full secondary dataset. Although plausible, as already discussed in Section 3.4, the good match between the distribution of simulated and observed ε DR1 implies that the instrumental and modeling noise cannot be much larger than the adopted range of 0.4-0.9 mas in the present sample. -A hidden outer companion to the system, unseen in the RV variations, could be responsible for the astrometric signal. This issue is discussed in Section 6.4.
Although the presence of outliers cannot be excluded, this RV-Gaia discrepancy motivates the search for supplementary yet hidden companions in these systems.

Non-detection sample: 27 confirmed planets
For a given RV orbit with given m sin i of the companion, an increasing true mass and thus decreasing orbital inclination imply increasing astrometric motion of the star. The non-detection of an astrometric excess noise larger than the defined threshold thus allow deriving an upper-limit on the true mass of the companion and a lower-limit on its orbital inclination with GASTON.
Among the 227 non-transiting companions of the nondetection sample, we constrained true masses lower than 13.5 M J within 3-σ confidence interval for a total of 27 companions. They are summarized in Table 9. Nine planets have a true mass lower than 5 M J , and 19 have a true mass lower than 10 M J .
We confirm that 6 multiple system contains several true planets. They are HD 10180, HD 176986, HD 181433, HD 215152, HD 7924, and HD 40307. In the 6-planets system, HD 10180, we can confirm that the a priori less massive companions c (P=5.8 days, m sin i=0.041 M J ), d (P=16.4 days, m sin i=0.037 M J ), and g (P=602 days, m sin i=0.067 M J ) are planets with a mass strictly lower than 12 M J at 3-σ. Fig. 12 shows the ε simu -I c relationship and I c posterior distribution for HD 10180 c. A study of the effect of mutual inclinations on the stability of this system led to constrain the masses of the planets within a factor of 3, with I c >10 • for all planets (Lovis et al. 2010). While our result is not as much as restrictive, it excludes a full face-on inclination with I c >0.2 • at 3-σ and confirms planetary mass for planets c, d, and g. Notes.
( †) After 1,000,000 iterations MCMC did not reach convergence, with a final maximum autocorrelation length larger than N step /50.
Among the 200 other candidate planets, as summarized in Table B.2, 103 companions can be confirmed substellar but may be as massive as brown dwarfs with a mass strictly smaller than 85 M J at 3-σ, and 59 others have a mass upper-limit within the M-dwarf domain. For the remaining 48 companions, GASTON could not converge within the 50,000 steps, with an autocorrelation length larger than 1000. At the end of the GASTON run, the posterior distributions for all of them led to an upper-limit on the mass larger than 13.5 M J . This non-convergence is due to a large astrometric excess noise but smaller than the detection limit. Simulations are less often compatible with ε DR1 , GASTON thus needs more time to converge. Their nature is undetermined between planet, BD and M-dwarf. We do not publish GASTON results for those 48 candidates.
While most of companions with a mass possibly greater than 13.5 M J have large orbital periods, 30 of them have an orbital period smaller than 100 days. Those are possible BD located within the driest region of the brown-dwarfs detection desert (Kiefer et al. 2019). They are particularly interesting objects that need to be further characterized in order to better constrain the shores of the BD mass-period phase space.  Figure 13 summarizes the corrected mass derived with GAS-TON compared to the initial m sin i as given in the Exoplanets.org database. The firm measurements for the 9 companions identified in Section 5.2.1 lead to true masses significantly different from the m sin i with an non edge-on inclination. Their revised mass is generally comprised between 10 and 500 M J , as are the 3-σ upperlimits reported for companions from situation #2 and in the nondetection sample.

A revised mass for 9 companions
This shows that Gaia will be best at detecting astrometric motions due to companions beyond ∼10 M J . But with improved precision in the future releases and the use of time series, it will certainly allow the detection of Jupiter mass planets.

Small inclinations <4 •
To our knowledge, no exoplanet RV candidate from the exoplanets.org database were yet found with an inclination strictly lower than 4 • . The exoplanet with the smallest known orbital inclination is Kepler-419 c with I c =2.5±3 • , thanks to transit timing variations (Dawson et al. 2014). In Table 8, among the 9 non-transiting systems with a firmly detected inclination, and accounting for their 3-σ bounds, we find zero companion with I c strictly smaller than 1 • , one companion, HD 148427 b, with I c strictly smaller than 2 • , and four others with I c strictly smaller than 4 • . Many other companions from the detection and non-detection samples could have such small inclinations, but also posibly larger than 1, 2 or 4 • . Assuming isotropy of orbits within the ∼600 known non-transiting RV exoplanets in the exoplanets.org database leads to less than 0.4 orbits with I c ≤2 • and less than 1.5 orbits with I c ≤4 • . Finding more than 1 system with an orbit less inclined than 2 • and more than 4 with I c <4 • suggests that the distribution of inclinations within exoplanet candidates deviates from a uniform distribution at least below 4 • . This questions the isotropy of orbits within the population of discovered RV exoplanets advocated in e.g. Zucker & Mazeh (2001b), Jorissen et al. (2001) and Tabachnik & Tremaine (2002). It was indeed already noticed in Halbwachs et al. (2000) and Han et al. (2001) that RV-planet systems are possibly biased towards small sin i.
The uniform distribution of inclinations certainly applies on a larger sample of systems than just RV exoplanets hosting systems. SB1 binary companions (M c <0.6 M ) on orbits with inclinations <2 • would most likely fall within exoplanetary domain. A sin i<0.035 would indeed lead to M sin i<20(M /M ) M J . Several thousands of binary systems and stars were, and are still being, followed-up for RV variations for many years. About 1300 SB1 binaries are collected in the SB9 database (Pourbaix et al. 2004) and about 600 RV systems (excluding all transiting planets that are F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia Table 9. GASTON results for 27 exoplanet candidates from the non-detection sample. Their 3-σ upper-limit on mass is smaller than 13.5 M J and the convergence criterion N step /max(τ λ ≥50. We expand this table in the Appendix, Table B.2, with the rest of the exoplanet candidates among the non-detection sample and for which M max,3σ >13.5 M J .  1927 biased to 90 • ). Probably twice as many are still being followedup, neither characterized nor published yet. We thus estimate the full population of RV-monitored systems possibly, or actually, harboring planets or hidden binary component to reach at least 4000 individuals, among which FGK stars are the dominating class of stellar primaries. Assuming isotropy of orbits in this larger sample, we expect to observe at least ∼0.6 systems with an inclination ≤1 • , ∼2.4 systems with I c ≤2 • and ∼9.7 systems with I c ≤4 • . This is in better agreement with our findings, validating the GASTON determination of orbital inclinations for RV companions with the Gaia DR1.

Systems with edge-on transiting orbits
In Section 3.3, we found 9 over 246 systems (i.e. 3.7% of them) with transiting-only planets to have an astrometric excess noise larger than the defined thresholds for a significant astrometric motion. Six of them are systems around sources from the primary dataset. Given their edge-on orbit, with an expected astrometric semi-major axis of the photocenter on the order of a few µas, it is technically impossible to detect their host star's reflex motion with Gaia. But the measurement of ε DR1 beyond the threshold suggests on the contrary that a significant astrometric motion was actually detected.
Running GASTON brings a solution to this inconsistency. For 7 among 9 companions, an edge-on astrometric orbit is compatible with the value of ε DR1 , because instrumental and measurement noises can pile up to produce an astrometric excess noise above the threshold. However, this compatibility is far from being frequent in the MCMC runs, as shown by the low acceptance ratio in both cases (<1.5%; see Table 8). Could it be that ε DR1 is actually not due to instrumental and measurement noise for some of them, and leads to truly incompatible astrometric motion?
To test this possibility, we simulated many Gaia observations of non-accelerating stars, with N FoV and N AL drawn from the sam-ple of 133 systems with transiting-only planets from the primary dataset, and the 113 systems with transiting-only planets from the secondary dataset. In the primary dataset, this results into 6 over 133 astrometric excess noise values beyond 0.85 mas, produced only from noise, with a probability of 0.0014, i.e. below 3 σ. Producing 5 over 133 astrometric excess noise values beyond 0.85 mas has a probability of 0.0072, i.e. above 3 σ. In the secondary dataset, the probability to produce 2 or 3 over 113 astrometric excess noise values beyond 1.2 mas is smaller than 0.0001, while producing 1 over 113 is significantly more likely with a probability of 1.2%.
Thus for at least one system from the primary dataset, possibly K2-34 with the lowest MCMC acceptance rate of 0.002, Gaia detected a signal that cannot be fully explained by the combination of the published orbit and noise. This could be the sign of an unseen and unknown outer companion in the system of K2-34, or a measurable effect of outliers (see the discussion in Section 5.2.3 for WASP-43 b and WASP-156 b).
In the secondary dataset, no more than one system among 113 could be simulated with an astrometric excess noise as large as 1.2 mas. Consistently, we recall obtaining a marginal compatibility of GASTON simulations with ε DR1 only for K2-110 b (Section 5.2.2), and no compatibility at all for two other planets, WASP-43 b and WASP-156 b (Section 5.2.3). We conclude that noise is the likely explanation for the astrometric excess noise of K2-110 b. As already discussed in Section 5.2.3, the most reasonable explanations for the large inconsistent ε DR1 of WASP-43 and WASP-156 are either unmodeled outliers, or the presence of an outer companion in both systems.
We conclude that follow-ups of K2-34, WASP-43 and WASP-156 should be conducted to search for outer companions in these edge-on systems.

Outer companions
In the cases with the lowest acceptance rates in Table 8, it could be that the astrometric signal is better explained by the influence of another outer companion to the system, especially if as for HD 4203, a long-period RV-drift is detected (Kane et al. 2014). With a minimum mass of at least 2 M J , this outer companion with an orbital period of several thousands of days (tens of year) might also be at the origin of the astrometric signal.
As mentioned in Section 6.3, the astrometric excess noise of few compact edge-on systems (K2-34, WASP-43 and WASP-156) is difficult, and even impossible, to produce with our simulations. Thus, while outliers could explain the discrepancy of the RV orbit and Gaia astrometry, a more simple explanation could be the presence of an outer companion. There are no clues of long RV drifts in neither of these systems, but this does not invalidate the hypothesis of an outer companion because the orbital inclination can be adjusted to make the RV signal vanish.
More generally, for any system in situation #1 leading to a mass measurement of an RV companion, the presence of an unknown outer companion cannot be totally excluded. Nevertheless, as far as we know these systems, the solution with the less complexity is preferred, i.e. the known RV-orbit with a realistic inclination is responsible for the Gaia measurement of the star motion. Fine-tuned mass, semi-major axis and inclination of an unknown unseen companion's orbit would be necessary to explain the astrometric excess noise, while using the existing known companion on a known orbit only require to fit a single parameter, the inclination.
We have shown in Section 3.3 that the astrometric excess noise is not correlated to the presence of an RV-drift, and thus to the presence of an outer companion on an undetermined orbit. Thus, a large astrometric excess noise does not imply the presence of an outer companion and conversely the presence of an outer companion does not imply a large astrometric excess noise. It was already shown for HD 114762 b (Kiefer 2019) that the binary companion HD 114762 B, a low-mass star at several hundred au separation and with orbital period much larger than the Gaia DR1 campaign duration of 416 days, only has a minor impact on the motion of HD 114762 A. It was better explained by a small orbital inclination and larger mass of HD 114762 A b.
Solving this issue is not the scope of the present paper, but learning from the specific case of HD 114762, we expect that outer companions, moreover not observed in the RV variations, with period much greater than 416 days could be neglected.

Comparison with already published mass
The true mass for 86 exoplanet candidates of our samples were also constrained in Reffert & Quirrenbach (2011) using Hipparcos-2 data. The results of the two studies are compared together in Fig. 14. This comparison shows that GASTON leads to better contraints with generally lower upper-limits on the mass for 69 over the 86 companions. The Gaia DR1 astrometric excess noise is thus compatible the Hipparcos-2 astrometry even revealing smaller scatter and better astrometric precision. Among these 86 exoplanet candidates, Table 10 lists 5 companions with a well-constrained mass in the present study -30 Ari B b, HD 114762 b, HD 141937 b, HD 148427 b, and HD 5388 b -for which Reffert & Quirrenbach (2011) analysis only led to upper-limits on mass. The true mass and inclination that we obtain with GASTON all stand within the bounds they derived. We also list 5 companions for which GASTON could only derive limits -HD 190228 b, HD 87883 b, HD 142022 b, HD 181720 b, and HD 131664 b -but Reffert & Quirrenbach (2011) published upper and lower bounds for both inclination and mass. We reduce the interval of possible mass for HD 142022 b and HD 181720 b, now respectively within 4.6-39 M J and 6-32 M J , i.e. in the giant planet or BD domain. We also confirm the upper-bound on HD 87883 b mass, which ranges within 3-21 M J at 3-σ. HD 87883 b is thus most likely a giant planet on a long-period 7.5-years orbit.
Besides Reffert & Quirrenbach (2011), we found several other publications of true mass and inclinations for 12 companions. They are summarized on Table 10 and discussed individually below.
HD 5388 b The mass of this exoplanet candidate was already measured to be 62.2±19.9 M J with Hipparcos measurements (Sahlmann et al. 2011b HD 33636 b Bean et al. (2007) rejected the planetary nature of this candidate, with a mass determined in the M-dwarf domain M=140±11 M J with an orbital inclination of ∼4 • . Interestingly, the small astrometric excess noise measured by Gaia ε DR1 =0.53 mas leads to a probability for the mass of this companion to be higher than 93.5 M J is 0.27%. The mass measurements from Gaia and FGS astrometry are thus incompatible at 3σ. However, with an inclination of 4 • , our simulations could produce ε DR1 smaller than 0.53 mas with a probability of 0.7%. Thus, the Gaia DR1 astrometric excess noise is compatible with Bean et al. (2007) results at 3σ. The disagreement between the parallax of this measured by FGS and Hipparcos (∼35-36 mas) with the parallax measured by Gaia (∼34) may also explain the small ε DR1 if part of the orbital motion was wrongly fitted as parallax motion.
HD 92788 b Han et al. (2001) proposed a true mass of 45 M J , with an inclination of 6.3 • for this Jupiter-mass candidate (m sin i=3.6 M J ) on an Earth-like orbit (P=325 days). Simpson et al. (2010) later proposed a derivation of the orbit inclination based on the assumption of coplanarity of the stellar equator and the companion orbit and by measuring the rotation speed of the star compared to its v sin i. This led to a lower mass of 9-28 M J . This method is however not fully reliable as coplanarity of the stellar equator and the companion orbit is never a robust assumption. Both results are compatible with the 3-σ limit that we derived here (I c >3.9 • , M b <54 M J ) with ε DR1 =0.32 mas. It confirms that this companion is most likely a brown-dwarf if not a massive planet. Table 10. Comparison of results in the present studies to 1 or 3-σ upper-limits published in other articles. Inclinations should be compared modulo 180 • since in the present study the prograde or retrograde orientation of the orbital motion cannot be determined. We present the 1-σ confidence interval for the inclinations and masses when they are well-constrained in the present study and obtained with a sin I c prior distribution on the inclination, and the 3-σ limits obtained with a flat prior distribution (  There are no constraint on the inclination or true mass of companion b, but an assumed coplanarity with planet c orbit would imply a planetary nature as well with a mass close to 2 M J . Coplanarity is not generic, and it remains thus possible that planet c is actually circumbinary, possibly leading to an interesting configuration in 2:1 resonance. The non-detection of the astrometric motion of the host star by the Gaia DR1 astrometric excess noise with ε DR1 =0.6 mas puts a 3-σ upper-limit on the mass at 46 and 48 M J for companions b and c. We can therefore exclude a stellar nature for planet c in agreement with McArthur et al. (2014), as well as for object b, but it could still be a brown dwarf.
HD 130322 b As for HD 154345 b, assuming the coplanarity of HD 130322's equator and companion b orbit, Simpson et al. (2010) proposes a mass of 1.1 M J for this companion. The low astrometric excess noise of 0.3 mas for this source allows deriving with GAS-TON a 3-σ upper-limit on the mass of HD 130322 b of 136 M J . The planetary nature of this object cannot be confirmed here.
HD 131664 b This 5-years period candidate brown dwarf (m sin i=18 M J ) was characterized using Hipparcos astrometry by Sozzetti & Desidera (2010) and Hipparcos-2 data by Reffert & Quirrenbach (2011). Both found a possible small orbital inclinations for this companion down to ∼10-20 • . Sozzetti & Desidera (2010) could not reject an edge-on inclination while Reffert & Quirrenbach (2011) obtain inclinations smaller than 30 • at 3σ with a mass of at least 42 M J . GASTON cannot help settling the true mass of HD 131664 b but constrains its mass to less than 170 M J at 3σ.
HD 136118 b Using the Hubble Space Telescope Fine Guidance Sensor (HST/FGS), Martioli et al. (2010) could measure the astrometric motion of the F9V star HD 136118. They obtained an inclination of 163±3 • and a true mass for the exoplanet candidate of 42 11 −18 M J instead of the 12 M J deduced from RV assuming an orbit seen edge-on. The 3-σ upper-limit that we derived with GASTON for the true mass of HD 136118 b is close to the 3-σ upper-limit derived from Hipparcos-2 measurements by Reffert & Quirrenbach (2011) about 95-97 M J . The Gaia DR1 astrometric excess noise of HD 136118 (0.51 mas) is thus compatible with the true astrometric motion ∼1.45 mas of the host star due to companion 'b'.
HD 154345 b This long-period (3325 days) companion is a planet confirmed by Simpson et al. (2010) by measuring the rotation axis angle of the host star. But this conclusion relies on the hypothesis of coplanar orbital and stellar equator planes, which is never guaranteed. Our 3-limit based on ε DR1 =0.35 mas shows that HD 154345 b is indeed a planet with a mass smaller than 11.6 M J .
HD 177830 b Its true mass was tentatively determined at 55 M J , with an inclination of 1.3 • by Han et al. (2001) using Hipparcos data. This result is within the 3-σ limit that we derived here (I c >1.0 • , M b <79 M J ). It confirms that the ε DR1 =0.87 mas for this source incorporates a consequent fraction of real astrometric orbital motion.
HD 190228 b Using Hipparcos astrometry, this companion was previously identified as a brown dwarf with a mass of 67±29 M J (Zucker & Mazeh 2001a). Its mass was then reduced to 49±18 M J and an inclination of 4.3 •+1.8 −1.0 (Sahlmann et al. 2011a). The orbit significance they obtained for this star was 2-3σ. Using GASTON and a sin i-prior on the inclination, we measure a 1-σ upper-limit for the same mass of <24 M J and a 3-σ upper-limit of 111 M J . The inclination is >14 • at 1-σ and >3.2 • at 3-σ. Our result agrees with the most precise measurements of the mass of HD 190228 b, but cannot bring significant improvements. Given the astrometric orbit semi-major axis as large as 2 mas, Gaia will certainly provide the best measurement for this brown-dwarf once the astrometric series will be available.
The global compatibility of the true masses derived with GAS-TON with the true masses already published for these systems validates the GASTON method and confirms it can lead to better characterize candidate planetary systems.

An updated mass-period diagram
The masses derived with GASTON allow us to update the massperiod diagram of planet and brown dwarf companions. It is represented in Fig. 15, compared to companions with true mass from the Exoplanet.eu database and massive companions reported in Wilson et al. (2016) and Kiefer et al. (2019). We selected only systems within 60 pc from the Sun, surrounding FGK host stars with masses within 0.52-1.7 M , with a published inclination measurement. Such systems are objects of extensive surveys (e.g.  Sahlmann et al. 2011a with better observational completion and detection of planets and BD with mass larger than 1 M J . We include any mass compatible with as much as 150 M J in order to encompass the surroundings of the substellar domain. We exclude GASTON masses of transiting planets, which are better determined in the Exoplanet.eu database. We also exclude the GASTON mass of candidates which host star RVs have a long-term drift, in order to remove possible bias due to an outer companion. Upper-limits are not represented. The mass measurements of the present study add new points to the M-P diagram in the BD-to-stellar domain at orbital periods larger than 100 days. There still remains blank regions: the BD domain below 100-days period (Kiefer et al. 2019a), the short-period Neptunian desert (Mazeh et al. 2016), and the observationally biased triangular area from short-period Earth-mass planets to longperiod Jupiter mass planets.
In the BD domain, the M-P distribution presents a strong cut in the region of brown dwarf companions at ∼100 days. But below 100 days, several tens of other companions which mass cannot be well-constrained may reside in the BD mass regime. In Fig. 16, we included the 3-σ upper-limits derived with GASTON around and in the BD domain. Even then, the region bounded by masses 20-85 M J and periods 0-100 days remains significantly emptier than its surrounding. This tends to confirm the most recent estimation of the brown dwarfs desert boundaries (Ma & Ge 2014, Ranc et al. 2015).

Star-host metallicity and orbital eccentricity distributions in the BD domain
Brown dwarfs companion stand at the boundary between stellar binaries and giant planets. It remains unknown if some BD companions belong to one or the other population, or if BDs have a main formation channel. Core-accretion scenarios (Pollack et al. 1996) predict that giant planets are more difficult to form around metal-deficient stars (Ida & Lin 2005)   expected that binary companions have the same metallicity distribution whatever their mass (Maldonado et al. 2017).
The study of eccentricity distributions in the BD domain by Ma & Ge (2014) revealed the existence of a sharp transition at ∼42.5 M J . Below 42.5 M J the eccentricity distribution is consistent with mass-limited eccentricity pumping by planet-planet scattering (Rasio & Ford 1996), and beyond 42.5 M J the eccentricity distribution of BD companions is similar to that of binary stars (Halbwachs et al. 2003). A consistent transition at similar mass was found by Maldonado et al. (2017) in the distribution of host star metallicities. The host-star metallicity of BD with a mass >42.5 M J spans a large range of values from sub-solar to super-solar, while those with mass <42.5 M J have host-stars metallicities more similar to those of giant exoplanets with a prevalence for metal-rich hosts.
This limiting mass of ∼42.5 M J could thus be separating lowmass BDs formed like planets by core-accretion from high-mass BDs formed like stars by gravitational instability in molecular clouds.
Here, we add the new GASTON measurements to exoplanet.eu companions, and the published companions in Wilson et al. (2016) and Kiefer et al. (2019a) to obtain metallicity and eccentricity distributions with respect to true masses in Figs. 17 and 18. We select, as in Section 6.6, systems within 60 pc from the Sun, surrounding FGK host stars with masses within 0.52-1.7 M , and with a published inclination measurement. Metallicity, or [Fe/H], measurements are taken from the exoplanets.org database for the sample studied in the present paper, from exoplanet.eu for the corresponding sample, and from the Wilson et al. (2016) and Kiefer et al. (2019a) for the rest of the considered companions.
Beyond the brown dwarf domain, metallicity reaches sub-solar values, while giant planets are indeed found preferably around stars with super-solar metallicity. No clear boundary can be derived from the still sparse distribution of measured companion mass, although a transition could be occurring about 50 M J in agreement with the 42.5 M J found by Maldonado et al. (2017). The distribution of eccentricity with companion mass in Fig. 18 does not exhibit a well-defined transition at 42.5 M J , as reported in Ma & Ge (2014). Nevertheless, four BD with M>45 M J stand above e=0.7 while all BD companions have e<0.7 below 45 M J . The eccentricity distribution within our sample thus seems to match with that of Ma & Ge (2014).
Our current sample of exoplanets, BD and low-mass M-dwarf companions around FGK stars at less than 60 pc of the Sun with a true mass measured still need to be populated, but it agrees with previous non-volume limited studies on a transition in the brown  dwarf domain at a mass of ∼42.5 M J . This critical mass possibly separates two populations of BD, those formed like stars from those formed like planet and mainly following predictions of the core-accretion scenario.

Conclusions
We use the GASTON method developed in Kiefer et al. (2019) & Kiefer (2019 with Gaia DR1 data to determine the true mass of the 911 RV-detected exoplanet candidates published in the exoplanets.org database. Reliable DR1 data were found for the host stars of 755 companions. Among those, a total of 29 companions induce an orbital motion of their host star significant enough to be detected as large astrometric excess noise, constituting a 'detection sample'. With GASTON, an inclination, and thus a true mass could be determined for 8 of them. For the remaining 21 companions we could only constrain an upper-limit on the true mass, with an astrometric motion compatible with the edge-on inclination within measurement noise. For other 227 candidates, the astrometric excess noise is not large enough to imply a firm detection of the astrometric motion of their host star, but it allows deriving an upper-limit on their true mass. They constitute a 'non-detection sample'.
We found that among the detection sample, 30 Ari B b, HD 114762 b, HD 148427 b, HD 5388 b, HD 6718 b, HD 16760 b, and HIP 65891 b are not planets, but brown dwarfs or M-dwarfs. Moreover, we measured a true mass of HD 141937 b, within 9-50 M J compatible at 3-σ with a planetary nature, altthough more likely a brown-dwarf.
Among the 227 candidates of the non-detection sample, GAS-TON applied on the small astrometric excess noise measured by Gaia DR1 confirms that 27 exoplanet candidates are indeed planets. The lower-limit on their inclination deduced from the small value of their astrometric excess noise led to an upper-limit on mass below 13.5 M J at 3-σ.
These new measurements populate the mass-period diagram in the BD-to-M-dwarf domain constraining the driest region of the desert of brown-dwarf companions detection, also-known-as the brown-dwarf desert, to orbital periods smaller than 100-days and mass larger than 20 M J . We thus confirm previous estimates of the period threshold of the brown-dwarf desert, ∼100 days, obtained by Ma & Ge (2014) accounting for the true mass of whole set of detected BD companions as of 2014, and Kiefer et al. (2019) analysing the m sin i distribution of companions to FGK-stars at less than 60 pc from Earth. Moreover, the distributions of eccentricities and metallicities among brown dwarf companions are consistent with a transition from planet-like formation to star-like formation at about 40-50 M J (Ma & Ge 2014, Maldonado et al. 2017. Since GASTON allowed determining companion masses of few tens of M J using only preliminary Gaia data products at a precision of ∼1 mas, we can rejoice that future orbital solutions from Gaia astrometric time series at a precision of a few 10-100 µas, will allow measuring the orbital inclination and masses of many RV Jupiter-mass exoplanets and brown dwarfs, as well as new detections among the several billions of sources monitored with Gaia. F. Kiefer et al.: Determining the true mass of radial-velocity exoplanets with Gaia Appendix A: The magnitude of the companion To calculate the impact of the light emitted from the companion on the apparent primary semi-major axis, as observed by Gaia, also known as the photocenter semi-major axis, we have to take into account two effects: the emission from the companion (c) itself if not planetary and the star ( ) light reflected by the companion towards the observer. We recall the equation of the photocenter semi-major axis given in Kiefer et al. (2019) (see also van de Kamp 1975 for the original calculation): a phot = (B − β)a tot with a tot = a c + a (A.1) We introduced the luminosity fraction β=L c /(L c + L ) and the mass fraction B=q/(1 + q) with q=M c /M the mass ratio. The keyparameter here is the luminosity fraction.
In order to calculate the emitted V-magnitude of the components, we use different empirical models. We use the given V magnitude as an approximation of the G magnitude. Since only the δG is important for our study, we do not expect strong deviations due to this approximation. They are listed below and presented on Only visual magnitudes for objects with mass larger than 0.1 M could be found in the literature, due to the difficulty for observing faint objects such as brown dwarfs and exoplanets in the optical band. Therefore, we can only assess the validity of the Vmag models in the stellar domain and we will assume their validity in the G-band down to the planetary domain owing moreover to the relative compatibility of the AMES-cond models with observations in the infrared (Chabrier et al. 2000, Baraffe et al. 2003, Allard et al. 2012. The BT-settl model seems more accurate than AMEScond in the IR for massive brown-dwarfs (Allard et al. 2012), but it does not give magnitudes for objects with mass below 0.75 M . In order to insure continuity of the mass-magnitude relation we prefer using the AMES-cond model from masses of 0.1 M down to 10 M J .
For the reflected light, we calculate the mean reflection along an entire orbit. For the whole domain from planets to stars, we assume the body is a Lambertian sphere with a typical Bond albedo of 0.3. The radius of the sphere is related to the mass of the body. There is no continuous exact law R= f (M) on the whole planetto-star domain, with a wide diversity of densities for planets and also for stars if we account for (sub-)giants stars. However, for the sake of continuousness and simplicity, we will assume a common continuous law relating the radius of a body to its mass. This will avoid issues with gaps in the MCMC and allow us to derive well-behaved posterior distributions. This law is established using several segments and is presented in Fig. A.2: -Low-mass planets, up to 0.5 M J with the empirical relations of Bashi et al. (2017), -Jovian-mass planets and brown-dwarfs up to 0.1 M stars along an AMES-cond isochrone at 5 Gyr (Allard et al. 2012), -Low-mass stars from 0.1 to 1.1 M with the BT-settl model (Allard et al. 2012, and references therein) along an isochrone at 5 Gyr, -More massive stars up to 10 M with the empirical R ∝ M 0.57 relationship (e.g. Demircan & Kahraman 1991, Torres et al. 2010).
Taking into account emission and reflection from the companion, we can calculate magnitude differences between the primary  Malkov et al. (2007). Solid lines are models, as presented in the legend. and the companion for different values of their mass and for different companion's orbit semi-major axis. This is plotted on Fig. A.3. The net impact on the apparent primary semi-major axis can be measured by comparing the photocenter semi-major axis to the primary semi-major axis. This is presented on Fig. A.4. In general the semi-major axis of the photocenter orbit decreases with increasing contribution of the companion in the total luminosity of the system.
In the stellar regime, the impact of the companion starts to be significant on the apparent primary semi-major axis for companion masses larger than about 20% of the primary star mass. For a primary of mass 0.5 M the magnitude of the secondary has a measurable effect for a mass larger than about 0.1 M . If the secondary is too luminous compared to the primary, the semi-major axis of the photocenter can even reach 0. However this effect remains hidden in the present study, because we impose that the companion be dark with (∆V>2.5).
In the planetary mass regime on the other hand, the impact of the companion on the photocenter can be strong if the semi-major axis of the companion orbit is smaller than 0.5 au and the mass ratio q=M c /M ∼10 −5 − 10 −3 . The impact is stronger for earlier primaries. This is due to the shortening of the primary star orbit with decreasing companion mass, while the companion radius is relatively constant about 1 R J down to a few fractions of Jupiter mass (Bashi et al. 2017). In this regime, the astrometric motion of the system observed from Earth, although of small extent, can actually follow the motion of the companion rather than that of the stellar host. This happens precisely when the luminosity ratio L c /L >q.
Nevertheless, this effect will not have a strong role in the present study, because q∼10 −5 − 10 −3 with a c <0.5 a.u. implies a a c and a ph <10 −3 a.u. In the worst case scenario, it could only lead at most to an astrometric motion of ∼0.5 mas if the parallax is ∼500 mas. This is well below the detection thresholds (ε thresh,prim =0.85 mas and ε thresh,second =1.2 mas) defined in Section 3.5 and this situation is thus undetectable within the diverse noises accounted for in Gaia measurements. However, it will be an important effect to account for in future analysis of Gaia time series when this precision will be reached. Neglecting it might lead to strongly underestimate the star semi-major axis, and thus the mass of the companion.   . The adopted continuous model of ∆V with respect to the mass ratio. On this plot, the primary is assumed with a mass ranging from 0.1 to 1.5 M , and the secondary is a Lambertian sphere with average Bond albedo of 0.3 on an orbit with semi-major axis between 0.1 and 1 au.