© The Authors 2023

1. Introduction

The third data release (DR3) from the Gaia mission (Gaia Collaboration 2023b) provides non-single star solutions for hundreds of thousands sources for the first time (Gaia Collaboration 2023a). This is a new exciting dataset based on which new binary systems can be fully characterized, their dynamical masses and luminosities in particular.

Estimating stellar masses is a fundamental process with which the understanding of the stellar behaviour (luminosity, evolution, etc.) can be improved. This can mainly be achieved by characterizing binary systems, and this is the aim of this paper. The stars that can be fully characterized, such as those studied in this paper, are few, but they are crucial because they enable calibration of fundamental physical relations. These relations will then enable estimating the parameters of single stars or objects that are hard to reach, such as mass using a mass-luminosity relation for main-sequence stars. The mass-luminosity relation is derived from fully characterized star systems, the masses and luminosities of which are known for both components. Knowing the dynamical masses also enables constraining other characteristics of the stars, such as their age, through isochrone fitting.

One of the main purposes of this paper is to use these new Gaia DR3 data to provide new dynamical masses and a first mass-luminosity relation in the G band. Empirical mass-luminosity relations are mostly provided in the near-infrared range because it does not depend on the metallicity as much as the visible range (e.g. Delfosse et al. 2000; Benedict et al. 2016; Mann et al. 2019). In the visible range, empirical mass-luminosity relations are provided in the V band (e.g. Delfosse et al. 2000; Benedict et al. 2016).

Masses of double-lined spectroscopic binaries (SB2) have been obtained so far mainly through eclipsing binaries and a smaller sample of visually resolved binaries. Visually resolved binaries have the advantage of also providing a measure of the parallax of the system (e.g. Pourbaix 2000). Masses can also be estimated together with the luminosity of the stars through the astrometric motion of the photocentre. However, this motion was too small in general to be detected by HIPPARCOS (ESA 1997; see e.g. Jancart et al. 2005), except in a few cases (Arenou et al. 2000). An observing program of SB2 has been initiated in 2010 to allow the determination of masses at the 1% level using future Gaia astrometric orbits (Halbwachs et al. 2020). The new Gaia DR3 astrometric orbits allow the determination of new masses of SB2 systems, as was done between the Gaia SB2 and astrometric orbital solutions in Gaia Collaboration (2023a). However, the astrometric motion affected the epoch radial velocity measures of Gaia (Babusiaux et al. 2023), leading to a poor goodness of fit of the solutions. They are therefore not considered here. Here, we combine Gaia astrometric data with double-lined spectroscopic data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE; Kounkel et al. 2021) and the 9th Catalogue of Spectroscopic Binary orbits (SB9; Pourbaix et al. 2004) to derive the dynamical mass of each component as well as the flux of the components in the G band.

Section 2 presents the data we used: the double-lined spectroscopic data (Sect. 2.1) and the astrometric solutions from Gaia (Sect. 2.2). Then the method we used to determine the binary masses is explained in Sect. 3. The results we obtained are discussed in Sect. 4. Section 5 is dedicated to the mass-luminosity relation. We first present six low-mass stars resolved by Gaia with direct-imaging data (Sect. 5.1) and then the fit of the mass-luminosity relation (Sect. 5.2).

2. Data

2.1. Double-lined spectroscopic data

Spectroscopic data have been obtained by measuring the Doppler effect of the stellar system. The stellar motion induces a periodic translation of their spectrum depending on their motion in the line of sight from the Earth. For double-lined spectroscopy, the motion of the two sources of the binary are well identified, and the ratio of the amplitude of the radial velocity motion provides the mass ratio of the stellar system. Estimating the mass requires knowing the inclination, which cannot be obtained from the spectroscopic orbit. We used double-lined spectroscopic data originating from two catalogues: APOGEE and SB9.

The SB9 catalogue (Pourbaix et al. 2004) is a huge compilation of spectroscopic orbits from the literature over the past decades. It lists about 5000 orbits together with the input radial velocities used to derive the orbit. Fifty-five of these have available epoch radial velocities and a Gaia astrometric orbit counterpart. The compatibility of the orbital solutions provided by Gaia astrometry and SB9 spectroscopy was first checked to remove triple systems. We required a consistency at 10σ for the periods P,

(1)

We note that a consistency at 5σ would have removed the well-behaved solution Gaia DR3 1528045017687961856 (HIP 62935). The compatibility of the eccentricity within 10σ was also checked (as in Eq. (1)), but did not remove any stellar system. In this way, 43 SB9 binaries were selected.

The APOGEE survey (Majewski et al. 2017) is conducted with two high-resolution spectrographs, covering the spectral band between 1.51 and 1.7 μm. The data used in this paper originated from Kounkel et al. (2021), who detected 7273 double-lined spectroscopic systems. One hundred eighty-three star systems have been found to have both double-lined SB2 spectroscopy data in APOGEE and an astrometric orbital solution in the Gaia non-single star (NSS) catalogue, but only 126 with more than one APOGEE epoch have been kept.

Only one orbit of these star systems could be solved through spectroscopy alone by Kounkel et al. (2021): Gaia DR3 702393458327135360 (HD 80234). The direct combination of these spectroscopic parameters with Gaia astrometry was achieved by Gaia Collaboration (2023a) to derive the masses of this system. For the other stars with a Gaia NSS solution counterpart, the constraints from the astrometric orbit can be used to extract the mass ratio from the raw radial velocity curves.

The SB9 radial velocity curves have a much wider observation time range than APOGEE. There are enough data from various epochs for an independent fit of the orbit with spectroscopy alone. The orbital parameters are directly given with their associated errors in the catalogue. This is not the case for our APOGEE sample, except for Gaia DR3 702393458327135360 (HD 80234).

2.2. Gaia DR3 astrometric orbits

Astrometric data are obtained by observing the corkscrew-like motion of the photocentre, that is, the apparent light source of the binary system. This provides the required inclination, but also the parallax, and, combined with an SB2 solution, the flux ratio.

Here the astrometric data are given by the orbital solutions from the Gaia DR3 non-single star solutions catalogue. This new catalogue has a significantly increased number and precision for binary system solutions (Gaia Collaboration 2023a). The catalogue provides several types of solutions depending on the collected data and the detection method or instrument used, that is, eclipsing, spectroscopic and astrometric solutions, and potential combinations of these. In this paper, we only use the astrometric solutions. Twenty-one objects in our final sample have a combined AstroSpectroSB1 solution for which only the astrometric part is taken into account. In Gaia DR3, the orbit is not described by the Campbell elements (semi-major axis, inclination, node angle, and periastron angle), but by the Thiele-Innes coefficients (Halbwachs et al. 2023).

3. Data processing

While for SB9 the orbital parameters are known and the computation of the masses can be derived directly (Appendix A), we return to the raw spectroscopic data here to improve the correlations between the parameters.

We combined pectroscopy and astrometry with the code BINARYS (Leclerc et al. 2023). BINARYS can combine HIPPARCOS and/or Gaia absolute astrometric data with relative astrometry and/or radial velocity data. It has been updated to handle Gaia NSS solutions, and its core, which computes the likelihoods, is available online¹. It needs initial values and uses the automatic differentiation code Template Model Builder (TMB; Kristensen et al. 2016) to find the maximum likelihood. The output is the estimated orbital parameters with the associated covariance matrix, together with a convergence flag. Because Monte Carlo techniques cannot be used with the astrometric Thiele-Innes coefficients of Gaia DR3 (see Sect. 6.1 of Babusiaux et al. 2023), the Markov chain Monte Carlo (MCMC) option of BINARYS cannot be used in this study, while the TMB automatic differentiation is consistent with the local linear approximation result.

BINARYS provides the primary semi-major axis a₁, the mass ratio q = ℳ₂/ℳ₁, and the period P with their associated covariance matrix for all the orbital parameters. This enables us to derive the primary and secondary masses (see Eqs. (2) and (3)) with the associated errors (Appendix B),

(2)

(3)

where the period P is given in years, a₁ is given in au, and the masses ℳ are given in solar masses ℳ_⊙. It also gives the flux fraction of the secondary β in the G spectral band,

(4)

where a₀ is the semi-major axis of the photocentre in the same unit as a₁ (see Appendix A).

3.1. SB9

The radial velocity epoch data for each component, the orbital astrometric solution from Gaia NSS, and initial parameters were used as input for BINARYS. They were chosen for SB9 to be the result of the direct calculation process (Appendix A).

Inflation of the raw radial velocity uncertainties is quite often needed, either due to an under-estimation of the formal errors, a template mismatch, or stellar variability effects. We therefore applied a procedure similar to Halbwachs et al. (2020) to correct for the uncertainties.

We first applied the variance weight factors w provided in the SB9 database. They were provided by some studies that combined different observations and give their relative weights. We therefore started from the weighted uncertainties .

Then these uncertainties were adjusted using the goodness-of-fit estimator F2 (Wilson & Hilferty 1931),

(5)

where ν is the number of degrees of freedom, and χ² is the weighted sum of the squares of the differences between the predicted and observed values. The radial velocity uncertainties were scaled to obtain F2 = 0 i.e. ,

(6)

The corrected uncertainties are then

(7)

This correction factor was applied three times: the uncertainties were adjusted once independently for each component with an SB1 correction, and they were then adjusted again together with a SB2 correction over the whole system.

The process requires the number of the degrees of freedom to be positive, that is, to have more epochs than parameters to fit. This was always the case, except for Gaia DR3 1480959875337657088 (HIP 69885), which has only two epochs for the primary and the secondary. No uncertainty correction was applied here. In the literature, an orbit fit could have been achieved using additional blended radial velocity epochs, which could not be used here. For this star, the orbital parameters are mainly driven by the Gaia NSS solution. Four other star systems do not have enough radial velocity epochs for the secondary to have the SB1 solution that is necessary to apply the correction process: Gaia DR3 1441993625629660800, 1517927895803742080, 4145362250759997952, and 4354357901908595456 (HIP 66511, HIP 61436, HD 163336 B, and HIP 81170, respectively). For these, the two other error correction factors, SB1 primary and SB2, were still applied, and the χ² of the final solution on the secondary was checked to be small.

About ten star systems have had a significant SB1 correction factor over the primary and/or the secondary, with .

3.2. APOGEE

Similarly, BINARYS was provided the APOGEE radial velocity epoch data for each component, and the orbital astrometric solution from Gaia NSS. Because the spectroscopic orbit is not known for APOGEE, the BINARYS code had to be initialized with various initial parameters. We used sampled initial values of ℳ₁ from 0.6 to 1.4 solar mass with a 0.2 step, ℳ₂ from 0.6 to 1.4 solar mass with a 0.2 step (keeping ℳ₂ ≤ ℳ₁), and β from 0 to 0.5 with a 0.1 step. Because the direction of motion is set by the spectroscopy, we also tried different configurations for the node angle, adding or not a π angle to both the node angle Ω and the argument of periastron ω to the Gaia astrometric orbit values. For each system, 15 × 6 × 2 = 180 initial configurations were tested for each star system.

The convergence of TMB towards a good solution is not expected for every system: many will be triple systems for which the short-period binary is seen by APOGEE and the longer-period binary is seen by Gaia. Each TMB output corresponding to an initial configuration of a given star system must then fit the following criteria to be kept: it must converge, with a goodness-of-fit estimator F2 < 5, and the flux fraction of the secondary should be within the interval [0;  0.5] at 3σ. The star system was kept only when these conditions were met by at least 10% of the 180 initial configurations tested. Then, for each star system, only the solution obtained for more than 80% of the cases for which TMB converged was kept. When no such solution existed, the star system was rejected.

Thirty-five of the 126 star systems we studied remained at this point. Due to the small number of radial velocity epochs, the precision of the solution may be too low on the masses to be interesting even though the convergence is good. For the final selection, we kept only stars with , and σ_ℳ1 < 1 ℳ_⊙. This led to 13 systems. This selection step was only applied for APOGEE because the selection over SB9 star systems was performed through the compatibility of periods.

4. Results

We obtained the dynamical masses and flux fraction of the individual stars of 56 binary systems by combining Gaia DR3 NSS astrometric solutions with SB2 solutions from SB9 (43 systems) and APOGEE (13 systems). Figure 1 provides the position of the binaries we characterized in the HR diagram.

Fig. 1. Hertzsprung-Russel diagram of the characterized binaries. The absolute magnitude of the unresolved binary in the G band MG is plotted as a function of the colour GBP − GRP. The blue dots correspond to the APOGEE binaries, and the red dots correspond to the SB9 binaries. They are overplotted on the low-extinction Gaia DR3 HR diagram (in grey).

The results obtained through the orbit fitting process are given in Table 1 for SB9.

The uncertainties on the masses of the binary system Gaia DR3 3954536956780305792 (HIP 61816) are extremely large, with . These results therefore cannot be used. We expect this result to be a consequence of the lack of constraints over the inclination for this system, which is i = 27.5 ± 12.7°. Being compatible with 0 at less than 3σ, the masses are much less constrained as well, leading to the large uncertainties.

The characterization of the binaries from APOGEE is detailed in Table 2. A particular case to be considered is Gaia DR3 839401128363085568 (LP 129-155). The results lead to ℳ₁ = 1.49 ± 0.44ℳ_⊙, ℳ₂ = 1.34 ± 0.37 ℳ_⊙, and β = 0.40 ± 0.03, with F2 = 2.5. These parameters make the system an outlier in the mass-luminosity relation. Although MCMC is not adapted to the Thiele-Innes handling, we tested a short MCMC on this star that indicated the presence of another solution with lower mass values. This binary system is the only one in our sample with a low eccentricity consistent with zero at 1σ. The system was additionally tested with an eccentricity fixed to 0, corresponding to a circular orbit. The result we obtained fits the mass-luminosity relation much better, despite its slightly larger F2, and this solution is kept in Table 2.

Because Gaia provides the G magnitude of the binary system, the individual absolute magnitude in the G band can be deduced for each star using A_G, the extinction in the G band derived from the 3D extinction map of Lallement et al. (2022), and using the Gaia DR3 extinction law²,

(8)

(9)

where ϖ is the parallax in mas.

To estimate the uncertainties of these absolute magnitudes (see Appendix B), a 10% relative error on the extinction with a minimum error of 0.01 mag was assumed, and a 0.01 mag error was added in quadrature to the G formal magnitude errors. The extinction term A_G remained negligible for 90% of our sample, with a maximum A_G of 0.03 for APOGEE and 0.15 for SB9.

Figure 2 for SB9 and Fig. 3 for APOGEE show the positions of all the individual stars we characterized in the mass-luminosity diagram. They are overplotted on the PARSEC solar-metallicity isochrones (Bressan et al. 2012).

Fig. 2. Mass-luminosity diagram of the characterized stars from the combination of Gaia with SB9. The error bars at 1σ are given in grey. The absolute magnitude of the individual stars in the G band MG is plotted as a function of the stellar mass (in ℳ⊙). The black dots represent the stars, and the associated error bars at 1σ are shown in grey. The red triangles represent Gaia DR3 3954536956780305792, for which the uncertainties are really large and not represented here. The blue diamonds represent Gaia DR3 3427930123268526720, for which F2 > 5. They are overplotted on the solar-metallicity PARSEC isochrones (in green).

Fig. 3. Mass-luminosity diagram of the characterized stars from the combination of Gaia with APOGEE. The error bars at 1σ are given in grey. The absolute magnitude of the individual stars in the G band MG is plotted as a function of the stellar mass (in ℳ⊙). The black dots represent the stars, and the associated error bars at 1σ are shown in grey. The red triangles represent the outlier Gaia DR3 ID 5285071954833306368. They are overplotted on the isochrones (in green).

Almost all the stars are compatible with the isochrones at 3σ. For SB9, one star can be considered as an outlier, Gaia DR3 3427930123268526720 (GJ 220), which is the only system of the SB9 sample with F2 > 5. For APOGEE, one main outlier exists, Gaia DR3 5285071954833306368 (HD 50199). This is still compatible with the isochrones at less than 5σ. Nothing specific has been found about this system to justify its surprising position in the diagram.

The goodness of fit of the Gaia DR3 astrometric solution is higher than 10 for 11 systems, but none are outliers in the mass-luminosity relation, in the F2 of the combined fit, or in the relation between the flux fraction and the mass ratio. We therefore decided to keep these systems in our sample for the mass-luminosity relation study in Sect. 5.

4.1. Comparison with the direct calculation method for SB9

As a sanity check, we compared the masses obtained with the orbit-fitting process (Table 1) to those obtained through direct calculation, that is, using the orbital parameters provided by SB9 directly to derive the mass functions without returning to the raw data, as detailed in Table A.1.

Returning to the raw spectroscopic data allowed us to take the correlations between the spectroscopic parameters into account and then gain a better estimation of the orbital parameters and their uncertainties. Moreover, a correction process was applied to the uncertainties of the radial velocity epochs, which made them more realistic.

Figure 4 presents the distribution of the compatibility (in σ) between the masses obtained for the orbit-fitting process and the direct calculation. The compatibility is defined as

(10)

Fig. 4. Compatibility density of the masses obtained by direct calculation and by orbit fitting for SB9 binaries. The compatibility is given in σ. It is given in orange for primary masses and in purple for secondary masses.

where ℳ_OF is the mass obtained through orbit fitting, and ℳ_DC is the mass obtained through direct calculation.

As expected, this shows that the results are nicely compatible, but with a difference that is not fully negligible in some cases.

Figure 5 provides the uncertainties over the masses obtained with one method with respect to the other. The majority of the mass uncertainties lie below the identity line, meaning that the uncertainties coming from the direct calculation process are generally higher than those from orbit fitting. This confirms that the uncertainties over the orbital parameters are often reduced when accounting for the correlations between them.

Fig. 5. Comparison of the uncertainties on the masses obtained by direct calculation (σDC) or by orbit fitting (σOF) for binaries from SB9. The primaries are given in orange, and secondaries are plotted in purple. The outlier star system Gaia DR3 4145362250759997952 is represented in green for its primary and secondary. The grey line is the identity line y = x.

A few points have slightly larger uncertainties through the orbit-fitting process. This is the result of the correction process we applied to the uncertainties (see Eq. (7)).

One noteworthy case is Gaia DR3 4145362250759997952 (HD 163336 B), for which the difference between the uncertainties is quite large. The radial velocity data contain only three epochs for the secondary. Thus, we can expect a strong correlation between the SB9 orbital parameters. We performed an MCMC over the raw spectroscopic data alone and confirm that strong correlations appear and that the distribution is strongly asymmetric. This explains the strong improvement we obtained by including the knowledge of the Gaia orbit in the spectroscopic fit.

4.2. Reference comparison

Three star systems from our SB9 sample are identified as SB2 in the Gaia DR3 catalogue with direct masses derived by Gaia Collaboration (2023a). While our mass estimates are consistent with those of Gaia DR3 1067685718250692352 (HIP 45794) and Gaia DR3 2035577729682322176 (HIP 97640), Gaia DR3 595390807776621824 HIP 42418) is a 5σ outlier. It has a photocentre semi-major axis a₀ of 3.4 mas, while the other two stars have a smaller a₀ ∼ 0.8 mas, suggesting that the astrometric motion impacted the spectroscopic measure, as suggested by Babusiaux et al. (2023). The reason is that the expected position of the spectra on the Gaia Radial Velocity Spectrometer (RVS) detectors is predicted by the standard five-parameter astrometric motion instead of the epoch astrometric motion, which would not be precise enough.

Masses were obtained combining a visual orbit with an SB2 orbit for Gaia DR3 2129771310248902016 (HIP 95575) by Piccotti et al. (2020) (ℳ₁ = 0.670 ± 0.069, ℳ₂ = 0.602 ± 0.061, compatible with our results within 1σ), for Gaia DR3 2067948245320365184 (HIP 101382) by Kiefer et al. (2018) (ℳ₁ = 0.8420 ± 0.0014, ℳ₂ = 0.66201 ± 0.00076, compatible with our results within 2σ), and for Gaia DR3 3283823387685219328 (HIP 20601) by Halbwachs et al. (2020) (ℳ₁ = 0.9798 ± 0.0019 ℳ_⊙, ℳ₂ = 0.72697 ± 0.00094 ℳ_⊙, compatible with our results within 1σ). Halbwachs et al. (2020) combined the raw relative astrometry and spectroscopic data. They obtained a parallax at 4.4σ from the Gaia NSS parallax. We tested a combined fit with the radial velocity, interferometry from Halbwachs et al. (2020), and Gaia NSS solution and obtained a goodness of fit of F2 = 1.5, a parallax of ϖ = 16.573  ±  0.017, and masses ℳ₁ = 0.9816 ± 0.0014 ℳ_⊙ and ℳ₂ = 0.72808 ± 0.00076 ℳ_⊙. This new parallax is at 3.4σ from that of Halbwachs et al. (2020) and reduces to 2.6σ when the Gaia DR3 parallax zero-point is taken into account (Lindegren et al. 2021). This highlights that SB2 stars with direct imaging will be excellent test cases for a validation of Gaia DR4 epoch data. The orbit fit for this binary is given in Fig. 6.

Fig. 6. Radial velocity fit for the binary system Gaia DR3 3283823387685219328 (HIP 20601) from SB9. The radial velocity (km s−1) is plotted as a function of the phase. The orange dots correspond to the radial velocity epochs of the primary, and the purple dots show the radial velocity epochs for the secondary. The black curves represent the corresponding fits obtained by combining the SB9 epoch radial velocity with the Gaia DR3 NSS astrometric orbital solution with BINARYS.

For APOGEE, the only star system for which masses have been obtained in the literature is the binary Gaia DR3 702393458327135360, which has been discussed in Sect. 2.1. This star was solved through spectroscopy alone (Kounkel et al. 2021) and was then combined with Gaia astrometry through a direct calculation process by Gaia Collaboration (2023a) to obtain ℳ₁ = 1.14 ± 0.38 ℳ_⊙, ℳ₂ = 1.06 ± 0.35 ℳ_⊙, and F₂/F₁ = 0.567 ± 0.071. This is fully compatible with our results. The uncertainties reported here are larger than those of Gaia Collaboration (2023a). This may be due to a slight discrepancy between the orbital parameters of APOGEE and Gaia, specifically on the eccentricities, which are at 4σ from each other. This discrepancy leads to a high F2 of 4.6 in our solution and to higher uncertainties than what is obtained by a direct calculation. The orbit fit for the binary Gaia DR3 702393458327135360 (HD 80234) is given in Fig. 7. The difference in the observation time clearly appears in Figs. 6 and 7, where the lack of radial velocity epochs for APOGEE is rather obvious.

Fig. 7. Radial velocity fit for the binary system Gaia DR3 702393458327135360 (HD 80234) from APOGEE. The radial velocity (km s−1) is plotted as a function of the phase. The orange dots correspond to the radial velocity epochs of the primary, and the purple dots show the radial velocity epochs for the secondary. The black curves represent the corresponding fits obtained by combining the APOGEE epoch radial velocity with the Gaia DR3 NSS astrometric orbital solution with BINARYS.

5. Mass-luminosity relation

The mass calculations presented above might enable an empirical fit of the mass-luminosity relation in the G band using the Gaia photometry. However, the masses we calculated do not provide satisfying constraints in the interesting region of the relation for low-mass stars (ℳ < 0.7 ℳ_⊙). To fill in this part of the H-R diagram, we searched for low-mass stars that were resolved by Gaia and with direct-imaging data from the literature, following the work of Leclerc et al. (2023) on HIP 88745.

5.1. Low-mass systems resolved by Gaia with direct-imaging data

We found three spatially resolved stellar systems that were studied with direct imaging in Mann et al. (2019) that also have HIPPARCOS (van Leeuwen 2007) transit data (TD) and Gaia-resolved observations consistent with the direct-imaging data: Gl 330, Gl 860, and Gl 277. Gl 568 is not in the sample of Mann et al. (2019), but has direct-imaging data from McAlister et al. (1989) and Mason et al. (2018), and it was added to our sample. These four stars were analysed with BINARYS by combining the direct-imaging data with HIPPARCOS TD and Gaia astrometric parameters of both components following the method detailed in Leclerc et al. (2023). All Gaia solutions have a five-parameter solution except for Gl 330, for which the secondary component has a two-parameter solution, and Gl 860, for which both components only have a two-parameter solution. One HIPPARCOS TD outlier at 5σ had to be removed for both Gl 568 and Gl 227.

Two stars from the sample of Mann et al. (2019), Gl 65 and Gl 473, are resolved by Gaia with separations consistent with the visual orbit, but no HIPPARCOS data exist for them. However, these two stars have a literature mass fraction : for Gl 65, B = 0.494  ±  0.04 from Geyer et al. (1988), and for Gl 473, B = 0.477  ±  0.008 from Torres et al. (1999). We incorporated this information within BINARYS for these stars.

As our sample contains very nearby stars, we added the perspective acceleration terms in BINARYS following the description detailed in ESA (1997) and Halbwachs et al. (2023). We first computed the radial proper motion, that is, the relative change in distance per year, μ_r = V_rϖ/A_Z in yr⁻¹ with A_Z = 9.7779222 × 10⁸ mas yr km s⁻¹. The perspective acceleration changes the along-scan abscissa ν (in mas) by adding

(11)

where ΔT is the epoch in years relative to the reference epoch for the astrometric parameters (i.e. 1991.25 for HIPPARCOS and 2016.0 for Gaia DR3). This Δν was subtracted from the HIPPARCOS abscissa residuals and was added to the Gaia-simulated abscissa. However, the perspective acceleration was taken into account for DR3 for stars with a Gaia DR2 radial velocity or in the table of nearby HIPPARCOS stars with the radial velocity used for DR2³. Here, only the radial velocity for the A component of Gl 860 was applied for the DR3 processing, using the same V_r = −33.94 km s⁻¹ as we used. For Gl 65, we used V_r = 39.04 km s⁻¹ (Kervella et al. 2016), while for Gl 473, all literature V_r are consistent with zero. We checked that taking into account the perspective acceleration for our stars changed χ² only marginally.

The results obtained for these six stars are listed in Table 3. Our mass estimates are consistent with those from Kervella et al. (2016), Delfosse et al. (2000), and Benedict et al. (2016) for Gl 65, but we all used the same literature mass fraction. For Gl 473, using again the same literature mass fraction, our masses are consistent with Delfosse et al. (2000), but differ at 2.5σ from the values of Benedict et al. (2016). This is caused by the difference with the RECONS parallax they used. Our mass estimate for Gl 860 is consistent with that of Delfosse et al. (2000). To our knowledge, we derived the dynamical masses of the components of Gl 277, Gl 330, and Gl 568 for the first time.

5.2. Fitting the mass-luminosity relation

To fit the mass-luminosity relation, we implemented a TMB function that allowed us to take into account the uncertainties on both the mass and the magnitude, and more importantly, the correlations between the parameters of the two components of the same system (the calculation of the covariance is detailed in Appendix B). The true magnitudes of the stars were used as a random parameters: they were marginalized, that is, integrated out of the likelihood. The initial value of TMB was provided by a classical polynomial fit (R lm function). We selected only components with M_G > 5 to be used as input for the fit because the dependence on age for fainter magnitudes is well known to be too strong.

We tested several degrees for the polynomial and fitting the logarithm of the mass instead of the mass itself and used the Bayesian information criterion (BIC) to compare the models. The BIC favoured a polynomial of degree 4 to fit the log of the mass. The coefficients are listed in Table 4. The fit uncertainties were estimated through a bootstrap method, leading to uncertainties smaller than 0.015 ℳ_⊙ for magnitudes higher than M_G > 6, corresponding to masses ℳ < 0.77 ℳ_⊙.

The fit is displayed in Fig. 8 together with the PARSEC, the Baraffe et al. (2015)⁴, and the BASTI isochrones (Hidalgo et al. 2018)⁵. The age dependence of the mass-luminosity relation starts to be significant for all isochrones at ≳0.6 ℳ_⊙, and our fit follows the oldest isochrones. For masses < 0.5 ℳ_⊙, our empirical relation indicates lower masses for a given luminosity than the PARSEC isochrones. Our results are consistent with the isochrones of Baraffe et al. (2015), except in the low-mass region, where we find slightly higher masses for a given luminosity.

Fig. 8. Mass-luminosity relation fitted in this work (in blue). The areas outside the 6 < MG < 13.5 range are shown by the dotted line. The grey area corresponds to the 1σ bootstrap interval. The stars from Table 3 (resolved Gaia stars with direct-imaging data) are plotted in blue, and stars from Tables 1 and 2 (SB2 data combined with Gaia NSS astrometric solution) are shown in grey. The dashed green lines show the solar metallicity PARSEC isochrones for main-sequence stars (isochrones with the label 1). The dot-dashed red line and the two-dash pink line are the solar metallicity isochrone of Baraffe et al. (2015) and BASTI, respectively; they only deviate at masses lower than ∼ 0.12 ℳ⊙.

6. Conclusion

We have estimated the masses of binary systems by combining the astrometric orbits from Gaia DR3 with spectroscopy, 43 star systems from SB9, and 13 from APOGEE. While the spectroscopic orbit was already known for the SB9 stars, this was the case for only one APOGEE star. We tested the difference between a direct calculation of the masses using the orbital parameters and a combined fit using the raw radial velocity measures on SB9. A combined fit estimates the parameters, their uncertainties, and their correlations better. We also estimated the masses of 6 stars resolved by Gaia DR3 with literature direct-imaging and either HIPPARCOS data or literature mass fraction. The dynamical masses of three of these stars were derived for the first time.

The BINARYS tool was used to perform the combined fits. BINARYS was extended for this study to handle Gaia DR3 NSS solutions and perspective acceleration within the HIPPARCOS and Gaia observation time.

Using the derived masses and the G magnitudes, we derived a first empirical mass-luminosity relation in the G band that took all the correlations between the component masses and magnitudes into account. This empirical relation agrees better with the isochrones of Baraffe et al. (2015) than with the PARSEC isochrones.

We expect Gaia DR4 to significantly increase the sample of stars that could be used in such a study. Moreover, Gaia DR4 will provide access to the epoch astrometry. This will enable a full combined fit on the raw data of spectroscopic or direct-imaging data with Gaia astrometry. It will also enable studying systems whose astrometric signal is too low for a full Gaia DR4 orbital solution but whose spectroscopic or direct-imaging signal is good. We can therefore expect a much deeper study of the mass-luminosity relation with Gaia DR4. In particular, a study of the metallicity dependence should be conducted with a larger sample.

Acknowledgments

We thank J.-B. Le Bouquin for helping debugging our test on HIP 20601 and X. Delfosse for providing lists of close-by M-dwarfs to dig into. We thank S. Cassisi for his prompt feedback on the BASTI isochrones. T.M. is granted by the BELSPO Belgian federal research program FED-tWIN under the research profile Prf-2020-033_BISTRO. This work has made use of data from the European Space Agency (ESA) space mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.

References

Appendix A: Direct calculation process for SB9

Because the orbital parameters are both given by SB9 and Gaia, it is easy to combine astrometry with spectroscopy. The features of the star systems were directly computed from the astrometric and spectroscopic orbital solutions, using commonly used formulae. This direct-calculation method is approximately the same as the method that was applied to calculate the mass by the Gaia collaboration (Gaia Collaboration 2023a).

From the semi-amplitudes given by spectroscopy, the mass ratio is

(A.1)

The dynamical mass of the secondary is

(A.2)

where P is the period in days, K₁ is given in km.s⁻¹, and the inclination i is given by the Gaia astrometry. The period and the eccentricity e were the weighted mean of the Gaia and SB9 values.

The primary mass was then deduced from the mass ratio and the secondary mass.

The flux fraction of the secondary is given through the relation between the individual stars orbits and the photocentre orbit provided by astrometry,

(A.3)

where B is the mass fraction of the secondary,

(A.4)

a₂₁ is the semi-major axis of the relative orbit between the primary and the secondary,

(A.5)

and a₀ is the semi-major axis of the photocentre orbit.

Here, a_1/2 are the individual semi-major axis of the primary and secondary, respectively. They can be deduced from the semi-amplitudes,

(A.6)

where the semi-major axis is given in au and the period in days.

The masses ℳ₁, ℳ₂, and the flux fraction β are then obtained through Eq. 2, 3, 4 respectively.

Astrometry is needed here to determine the photocentre orbit a₀, but also to convert it from mas into au using the parallax ϖ.

The errors were estimated through a Monte Carlo method (with n_MC = 10³) assuming all the parameter distributions to be Gaussian-like. The Gaia parameters were generated simultaneously according to the covariance matrix, except for the photocentre semi-major axis a₀. Its error has been shown by Babusiaux et al. (2023) to be over-estimated by a Monte Carlo approach, and the local linear approximations of Halbwachs et al. (2023) was used for it. In order to take potential asymmetry in the final parameter distributions into account, the lower and upper confidence levels were determined at 16% and 84%. The results of this direct calculation are listed in Table A.1.

Appendix B: Jacobian calculations

The TMB algorithm provides as output the orbital parameters with their associated uncertainties, in particular, the period P with σ_P and the parallax ϖ with σ_ϖ. It additionally gives the semi-major axis of the primary orbit a₁ with its error σ_a1, the mass ratio q with σ_q, and the flux fraction β with σ_β. The global covariance matrix is also calculated.

Using , we derived the masses with

(B.1)

(B.2)

and the magnitudes were obtained through Eq. 8 and Eq. 9 from the parameters (ϖ, β) as well as the apparent magnitude G and the absorption A_G. The last two are independent and independent of the other parameters. For simplicity, we therefore considered here only the extinction-corrected magnitude G₀ = G − A_G, which has a variance of .

The covariance of the parameters (a₁, q, P, ϖ, β) Σ_TMB are provided by TMB. The full covariance of our input parameters (a₁, q, P, ϖ, β, G₀) is then

(B.3)

The covariance of our output parameters (ℳ₁, ℳ₂, M_G1, M_G2), Σ, is then

with J the Jacobian of the transformation from (a₁, q, P, ϖ, β, G₀) to (ℳ₁, ℳ₂, M_G1, M_G2),

(B.4)

The derivatives of the primary mass are given in the equations below,

(B.5)

(B.6)

(B.7)

The derivatives of the secondary mass are given in the equations below,

(B.8)

(B.9)

(B.10)

As a reminder, the absolute magnitudes are given by the Eq. 8 and 9.

The derivatives of the primary magnitude are given below,

(B.11)

(B.12)

(B.13)

The derivatives of the secondary magnitude are given below,

(B.14)

(B.15)

(B.16)

All Tables

All Figures

	Fig. 1. Hertzsprung-Russel diagram of the characterized binaries. The absolute magnitude of the unresolved binary in the G band MG is plotted as a function of the colour GBP − GRP. The blue dots correspond to the APOGEE binaries, and the red dots correspond to the SB9 binaries. They are overplotted on the low-extinction Gaia DR3 HR diagram (in grey).
In the text

Fig. 2. Mass-luminosity diagram of the characterized stars from the combination of Gaia with SB9. The error bars at 1σ are given in grey. The absolute magnitude of the individual stars in the G band MG is plotted as a function of the stellar mass (in ℳ⊙). The black dots represent the stars, and the associated error bars at 1σ are shown in grey. The red triangles represent Gaia DR3 3954536956780305792, for which the uncertainties are really large and not represented here. The blue diamonds represent Gaia DR3 3427930123268526720, for which F2 > 5. They are overplotted on the solar-metallicity PARSEC isochrones (in green).

In the text

Fig. 3. Mass-luminosity diagram of the characterized stars from the combination of Gaia with APOGEE. The error bars at 1σ are given in grey. The absolute magnitude of the individual stars in the G band MG is plotted as a function of the stellar mass (in ℳ⊙). The black dots represent the stars, and the associated error bars at 1σ are shown in grey. The red triangles represent the outlier Gaia DR3 ID 5285071954833306368. They are overplotted on the isochrones (in green).

In the text

	Fig. 4. Compatibility density of the masses obtained by direct calculation and by orbit fitting for SB9 binaries. The compatibility is given in σ. It is given in orange for primary masses and in purple for secondary masses.
In the text

	Fig. 5. Comparison of the uncertainties on the masses obtained by direct calculation (σDC) or by orbit fitting (σOF) for binaries from SB9. The primaries are given in orange, and secondaries are plotted in purple. The outlier star system Gaia DR3 4145362250759997952 is represented in green for its primary and secondary. The grey line is the identity line y = x.
In the text

Fig. 6. Radial velocity fit for the binary system Gaia DR3 3283823387685219328 (HIP 20601) from SB9. The radial velocity (km s−1) is plotted as a function of the phase. The orange dots correspond to the radial velocity epochs of the primary, and the purple dots show the radial velocity epochs for the secondary. The black curves represent the corresponding fits obtained by combining the SB9 epoch radial velocity with the Gaia DR3 NSS astrometric orbital solution with BINARYS.

In the text

Fig. 7. Radial velocity fit for the binary system Gaia DR3 702393458327135360 (HD 80234) from APOGEE. The radial velocity (km s−1) is plotted as a function of the phase. The orange dots correspond to the radial velocity epochs of the primary, and the purple dots show the radial velocity epochs for the secondary. The black curves represent the corresponding fits obtained by combining the APOGEE epoch radial velocity with the Gaia DR3 NSS astrometric orbital solution with BINARYS.

In the text

Fig. 8. Mass-luminosity relation fitted in this work (in blue). The areas outside the 6 < MG < 13.5 range are shown by the dotted line. The grey area corresponds to the 1σ bootstrap interval. The stars from Table 3 (resolved Gaia stars with direct-imaging data) are plotted in blue, and stars from Tables 1 and 2 (SB2 data combined with Gaia NSS astrometric solution) are shown in grey. The dashed green lines show the solar metallicity PARSEC isochrones for main-sequence stars (isochrones with the label 1). The dot-dashed red line and the two-dash pink line are the solar metallicity isochrone of Baraffe et al. (2015) and BASTI, respectively; they only deviate at masses lower than ∼ 0.12 ℳ⊙.

In the text