11institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
22institutetext: Kungliga Tekniska Högskolan, 114 28 Stockholm, Sverige
33institutetext: Ecole Centrale de Lyon, 69 130 Ecully, France
44institutetext: Institut d’Astronomie et d’Astrophysique, Université Libre de
Bruxelles (ULB), Belgium 55institutetext: Royal Observatory of Belgium, Avenue
Circulaire 3, 1180 Brussels, Belgium 66institutetext: GEPI, Observatoire de
Paris, Université PSL, CNRS, 92190 Meudon, France

# Binaries masses and luminosities with Gaia DR3

S. Chevalier 112233 C. Babusiaux 11 T. Merle 4455 F. Arenou 66

(Received / Accepted )

###### Abstract

Context. The recent Gaia third data release (DR3) has brought some new
exciting data about stellar binaries. It provides new opportunities to fully
characterize more stellar systems and contribute to enforce our global
knowledge of stars behaviour.

Aims. By combining the new Gaia non-single stars catalog with double-lined
spectroscopic binaries (SB2), one can determine the individual masses and
luminosities of the components. To fit an empirical mass-luminosity relation
in the Gaia $G$ band, lower mass stars need to be added. Those can be derived
using Gaia resolved wide binaries combined with literature data.

Methods.  Using the BINARYS tool, we combine the astrometric non-single star
solutions in the Gaia DR3 with SB2 data from two other catalogs : the 9th
Catalogue of Spectroscopic Binary orbits (SB9) and APOGEE. We also look for
low mass stars resolved in Gaia with direct imaging and Hipparcos data or
literature mass fraction.

Results. The combination of Gaia astrometric non-single star solutions with
double-lined spectroscopic data enabled to characterize 43 binary systems with
SB9 and 13 with APOGEE. We further derive the masses of 6 low mass binaries
resolved with Gaia. We then derive an empirical mass-luminosity relation in
the Gaia $G$ band down to 0.12$\,{\cal M}_{\odot}$.

###### Key Words.:

binaries: general – binaries: spectroscopic – binaries: visual – Astrometry –
stars: fundamental parameters

##  1 Introduction

The $3^{\rm rd}$ data release (DR3) from the Gaia mission (Gaia Collaboration
et al. 2023b) provides for the first time non-single star solutions for
hundreds of thousands sources (Gaia Collaboration et al. 2023a). This brings a
new exciting dataset to fully characterize new binary systems in particular
their dynamical masses and luminosities.

The estimation of stellar masses is a fundamental process to improve the
understanding of their behaviour (luminosity, evolution, etc.). This can be
mainly achieved by characterizing binary systems, which is the aim of this
paper. The stars that can be fully characterized like the ones studied in this
paper are not that many, but they are crucial since they enable to calibrate
fundamental physical relations. These ones will then make possible to estimate
the parameters of single stars or less reachable objects. The main example is
the mass-luminosity relation. It is set on fully characterized star systems,
for which masses and luminosities of both components are known. Knowing the
dynamical masses also enables to constrain other characteristics of the stars,
such as their age through isochrone fitting.

One of the main purpose 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 due to its lower dependency on the metallicity than the visible (e.g.
Delfosse et al. 2000; Benedict et al. 2016; Mann et al. 2019). In the visible
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, the latter having the advantage of providing also 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 in general too small 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 since
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 already done
between the Gaia SB2 and astrometric orbital solutions in Gaia Collaboration
et al. (2023a). However the astrometric motion impacted the epoch radial
velocity measures of Gaia (Babusiaux et al. 2023), leading to bad goodness of
fit of the solutions. Those will therefore not be considered here. Here we
combine Gaia astrometric data with double-lined spectroscopic data from APOGEE
(Apache Point Observatory Galactic Evolution Experiment, 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 their flux in
the $G$ band.

Section 2 presents the data used: the double-lined spectroscopic data (Sect.
2.1) and the astrometric solutions from Gaia (Sect. 2.2). Then the method used
to determine the binary masses is explained in Sect. 3. The results obtained
are discussed in Sect. 4. Section 5 is dedicated to the mass-luminosity
relation, presenting first 6 low-mass stars resolved by Gaia with direct
imaging data (Sect. 5.1) and then the fit of mass-luminosity relation (Sect.
5.2).

##  2 Data

We determine binary system features using astrometric data combined with
double-lined spectroscopy.

###  2.1 The double-lined spectroscopic data

Spectroscopic data has been obtained by measuring the Doppler effect of the
star 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 star system. The estimation of the mass
requires the knowledge of the inclination which cannot be obtained from the
spectroscopic orbit.

We used double-lined spectroscopic data originating from two catalogs: the
Apache Point Observatory Galactic Evolution Experiment (APOGEE) and the 9th
Catalog of Spectroscopic Orbits (SB9).

SB9 (Pourbaix et al. 2004) is a huge compilation of spectroscopic orbits from
the literature over the past decades. It counts about 5000 orbits together
with the input radial velocities used to derive the orbit. From those, 55 have
epoch radial velocities available and a Gaia astrometric orbit counterpart.
The compatibility between the orbital solutions provided by Gaia astrometry
and SB9 spectroscopy has been checked first to remove triple systems. We
required a consistency at $10\sigma$ for the periods $P$ :

$\left|P_{Gaia}-P_{SB9}\right|<10\sqrt{\sigma_{P_{Gaia}}^{2}+\sigma_{P_{SB9}}^{2}}.$
(1)

We note that a consistency at $5\sigma$ would have removed the well behaved
solution Gaia DR3 1528045017687961856 (HIP 62935). The compatibility of the
eccentricity within $10\leavevmode\nobreak\ \sigma$ was also checked (like in
Eq. 1) but does not remove any star system, leading to 43 SB9 binaries
selected.

APOGEE (Majewski et al. 2017) is a survey conducted with two high resolution
spectrographs, covering the spectral band between $1.51$ and $1.7$ $\upmu$m.
Data used in this paper originates from Kounkel et al. (2021), in which 7273
double-lined spectroscopic systems have been detected. 183 star systems have
been found to have both double-lined SB2 spectroscopy data in APOGEE and an
astrometric orbital solution in Gaia non-single star (NSS) catalog, but only
126 that had more than one APOGEE epoch have been kept.

Among them, only one star system orbit could be solved through spectroscopy
only by Kounkel et al. (2021), Gaia DR3 $702393458327135360$ (HD 80234). The
direct combination of those spectroscopic parameters with Gaia astrometric
ones has already been achieved by Gaia Collaboration et al. (2023a) to derive
the masses of this system. For the other stars with a Gaia NSS solution
counterpart, the constrains from the astrometric orbit can be used to extract
the mass ratio from the raw radial velocity curves.

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

###  2.2 The Gaia DR3 astrometric orbits

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

Here the astrometric data is given by the orbital solutions from the Gaia
$3^{\rm rd}$ data release non-single star solutions catalog. This new catalog
enabled to increase significantly the number and the precision of binary
system solutions (Gaia Collaboration et al. 2023a). The catalog 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 combination of those. In this paper, we only use the
astrometric solutions. Among our final sample, 21 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, 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 (Annex A), we go back here to the raw
spectroscopic data to improve how the correlations between the parameters are
considered.

The combination of spectroscopy and astrometry is achieved using BINARYS
(orBIt determiNAtion with Absolute and Relative astrometRY and Spectroscopy,
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 heart which computes the
likelihoods is available online111https://gricad-gitlab.univ-grenoble-
alpes.fr/ipag-public/gaia/binarys. It needs initial values and uses the
automatic differentiation code TMB (Template Model Builder, Kristensen et al.
2016) to find the maximum likelihood. It gives in output the estimated orbital
parameters with the associated covariance matrix together with a convergence
flag. Due to the fact that Monte-Carlo techniques cannot be used with the
astrometric Thiele Innes coefficients of Gaia DR3 (see Section 6.1 of
Babusiaux et al. 2023), the MCMC (Markov Chain Monte Carlo) option of BINARYS
cannot be used in this study while the TMB automatic differentiation is
consistent with the local linear approximation result.

BINARYS provides among all the orbital parameters the primary semi-major axis
$a_{1}$, the mass ratio $q=\mathcal{M}_{2}/\mathcal{M}_{1}$ and the period $P$
with their associated covariance matrix. This enables to deduce the primary
and secondary masses (see Eq. 2 and 3) with the associated errors (Annex B).

$\mathcal{M}_{1}=\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{3}}$ (2)
$\mathcal{M}_{2}=\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{2}}$ (3)

with the period $P$ in years, $a_{1}$ in au and the masses $\mathcal{M}$ in
solar masses $\,{\cal M}_{\odot}$. It also gives the flux fraction of the
secondary $\beta$ in the G spectral band:

$\beta=\frac{F_{2}}{F_{1}+F_{2}}=\frac{q}{1+q}\left(1-\frac{a_{0}}{a_{1}}\right)$
(4)

with $a_{0}$ the semi-major axis of the photocentre in the same unit as
$a_{1}$ (see Annex A).

###  3.1 The 9th catalog of spectroscopic orbit - SB9

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

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

We first apply the variance weight factors $w$ provided in the SB9 database.
Those have been provided by some studies combining different observations and
gives their relative weights. We therefore start from the weighted
uncertainties $\sigma=\frac{\sigma_{0}}{\sqrt{w}}$.

Then those uncertainties are adjusted using the goodness of fit estimator
$\mathrm{F2}$ (Wilson & Hilferty 1931):

$\mathrm{F2}=\left(\frac{9\nu}{2}\right)^{1/2}\left[\left(\frac{\chi^{2}}{\nu}\right)^{1/3}+\frac{2}{9\nu}-1\right]$
(5)

with $\nu$ the number of degrees of freedom and $\chi^{2}$ the weighted sum of
the squares of the differences between the predicted and the observed values.
The radial velocity uncertainties are scaled to obtain $\mathrm{F2}=0$ i.e.
$\chi^{2}=\chi_{0}^{2}$:

$\chi_{0}^{2}=\nu\left(1-\frac{2}{9\nu}\right)^{3}$ (6)

The corrected uncertainties are then

$\sigma_{\rm corr}=\sqrt{\frac{\chi^{2}}{\chi_{0}^{2}}}\times\sigma.$ (7)

This correction factor is applied 3 times: the uncertainties are adjusted once
independently for each component with a SB1 correction, and then adjusted
again together with a SB2 correction over the whole system.

The process requires the number of degree of freedom to be positive i.e. to
have more epochs than parameters to fit. This is always the case except for
Gaia DR3 $1480959875337657088$ (HIP 69885), which has only 2 epochs for the
primary and the secondary. No uncertainty-correction at all is applied for
this one. 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 a SB1 solution necessary to apply the correction process:
Gaia DR3 $1441993625629660800$, $1517927895803742080$, $4145362250759997952$
and $4354357901908595456$, (respectively HIP 66511, HIP 61436, HD 163336 B and
HIP 81170). For them, the two other error correction factors, SB1 primary and
SB2, are still applied and the $\chi^{2}$ of the final solution on the
secondary has been checked to be small.

Around 10 star systems have had a significant SB1 correction factor over the
primary and/or the secondary, with $\sqrt{\frac{\chi^{2}}{\chi_{0}^{2}}}>1.3$.

###  3.2 The Apache Point Observatory Galactic Evolution Experiment - APOGEE

Similarly, BINARYS is provided the APOGEE radial velocity epoch data for each
component, the orbital astrometric solution from Gaia NSS. But since for
APOGEE the spectroscopic orbit is not known, the BINARYS code must be
initialized with various initial parameters. We used sampled initial values of
$\mathcal{M}_{1}$ from $0.6$ to $1.4$ solar mass with a $0.2$ step,
$\mathcal{M}_{2}$ from $0.6$ to $1.4$ solar mass with a $0.2$ step (keeping
$\mathcal{M}_{2}\leq\mathcal{M}_{1}$), $\beta$ from $0$ to $0.5$ with a $0.1$
step. Since the direction of motion is set by the spectroscopy, we also try
different configurations for the node angle, adding or not a $\pi$ angle to
both the node angle $\Omega$ and the argument of periastron $\omega$ to the
Gaia astrometric orbit values. For each system, there is then $15\times
6\times 2=180$ initial configurations tested for each star system.

The convergence of TMB towards a good solution is not expected for every
system: many will be triple systems with the short period binary seen by
APOGEE and the longer period one 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
$\mathrm{F2}<5$ and the flux fraction of the secondary should be within the
interval $[0;\leavevmode\nobreak\ 0.5]$ at $3\leavevmode\nobreak\ \sigma$. The
star system is kept only if those conditions are 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 where TMB converged is
kept. If no such solution exists, the star system is rejected.

From the 126 star systems studied, 35 remain at this point. Due to the small
number of radial velocity epochs, the solution may have a too low precision on
the masses to be interesting despite a good convergence. For the final
selection, we keep only stars with $\frac{\sigma_{q}}{q}<0.5$,
$\frac{\sigma_{\mathcal{M}_{1}}}{\mathcal{M}_{1}}<0.5$ and
$\sigma_{\mathcal{M}_{1}}<1\,{\cal M}_{\odot}$ leading to 13 systems. This
selection step is only applied for APOGEE, since the selection over SB9 star
systems has been performed through the compatibility of periods.

##  4 Results

We have obtained the dynamical masses and flux fraction of the individual
stars of 56 binary systems through the combination of 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 have characterized in the HR
diagram.

Figure 1: H-R diagram of the characterized binaries. The absolute magnitude of
the unresolved binary in the $G$-band $M_{G}$ is plotted as a function of the
colour $G_{BP}-G_{RP}$. 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.

Table 1: Solutions from the combination of Gaia NSS astrometric solutions with
SB9 double-lined spectroscopy. $q$ is the mass ratio, $\mathcal{M}_{1}$ is the
mass of the primary, $\mathcal{M}_{2}$ is the mass of the secondary (in
$\,{\cal M}_{\odot}$) and $\beta$ is the flux fraction of the secondary.
$\sigma_{q}$, $\sigma_{\mathcal{M}_{1}}$, $\sigma_{\mathcal{M}_{2}}$ and
$\sigma_{\beta}$ are their associated uncertainties. $\mathrm{F2}$ is the
goodness-of-fit estimator. Ref gives the reference of the raw radial velocity
data used in the fit, as provided by SB9.

Gaia DR3 ID | $q$ | $\sigma_{q}$ | $\mathcal{M}_{1}$ | $\sigma_{\mathcal{M}_{1}}$ | $\mathcal{M}_{2}$ | $\sigma_{\mathcal{M}_{2}}$ | $\beta$ | $\sigma_{\beta}$ | $\mathrm{F2}$ | Ref
---|---|---|---|---|---|---|---|---|---|---
48197783694869760 | 0.6771 | 0.0124 | 0.9707 | 0.0423 | 0.6573 | 0.0190 | 0.0936 | 0.0030 | 1.32 | 1
69883417170175488 | 0.9793 | 0.0224 | 0.8271 | 0.0289 | 0.8100 | 0.0273 | 0.3976 | 0.0084 | $-0.03$ | 2
308256610357824640 | 0.6970 | 0.0099 | 0.4283 | 0.2000 | 0.2985 | 0.1393 | 0.2770 | 0.0232 | $-0.15$ | 3
478996438146017280 | 0.9212 | 0.0064 | 0.8112 | 0.0249 | 0.7473 | 0.0228 | 0.3879 | 0.0030 | 1.25 | 4
544027809281308544 | 0.8336 | 0.0178 | 0.9465 | 0.0510 | 0.7890 | 0.0317 | 0.1363 | 0.0072 | 0.01 | 5
595390807776621824 | 0.8822 | 0.0061 | 0.7877 | 0.0105 | 0.6948 | 0.0066 | 0.3123 | 0.0017 | $-0.03$ | 6
660622010858088320 | 0.8707 | 0.0367 | 0.7304 | 0.0607 | 0.6360 | 0.0341 | 0.2252 | 0.0081 | 1.01 | 5
827608625636174720 | 0.9403 | 0.0129 | 0.7357 | 0.0392 | 0.6917 | 0.0367 | 0.4076 | 0.0045 | 0.29 | 7
882872210352301568 | 0.5027 | 0.0026 | 1.0044 | 0.0123 | 0.5049 | 0.0036 | 0.0257 | 0.0038 | $-0.07$ | 8
1067685718250692352 | 0.8884 | 0.0036 | 1.0279 | 0.0125 | 0.9132 | 0.0101 | 0.3035 | 0.0048 | 0.10 | 9
1074883087005896320 | 0.9388 | 0.0261 | 0.6640 | 0.0393 | 0.6233 | 0.0326 | 0.3977 | 0.0068 | 1.11 | 10
1324699172583973248 | 0.8839 | 0.0087 | 1.3136 | 0.0476 | 1.1611 | 0.0376 | 0.3167 | 0.0039 | 0.40 | 11
1441993625629660800 | 0.8344 | 0.0128 | 1.1033 | 0.0455 | 0.9205 | 0.0293 | 0.0706 | 0.0132 | $-0.10$ | 12
1480959875337657088 | 0.9351 | 0.0771 | 0.8679 | 0.1330 | 0.8116 | 0.0925 | 0.2884 | 0.0180 | 0.77 | 13
1517219363639758976 | 0.8198 | 0.0062 | 1.0650 | 0.0405 | 0.8731 | 0.0313 | 0.1576 | 0.0075 | 1.55 | 14
1517927895803742080 | 0.9076 | 0.0148 | 0.6354 | 0.0251 | 0.5767 | 0.0183 | 0.3546 | 0.0037 | 1.75 | 7
1528045017687961856 | 0.6793 | 0.0029 | 1.1179 | 0.0285 | 0.7593 | 0.0185 | 0.0623 | 0.0045 | 2.30 | 15
1615450866336763904 | 0.7864 | 0.0106 | 1.0138 | 0.0396 | 0.7973 | 0.0261 | 0.1317 | 0.0044 | 0.14 | 16
1918953867019478144 | 0.9308 | 0.0007 | 1.0465 | 0.0050 | 0.9741 | 0.0047 | 0.3866 | 0.0014 | 0.63 | 17
2012218158438964224 | 0.9486 | 0.0150 | 2.3368 | 0.1644 | 2.2168 | 0.1509 | 0.3589 | 0.0064 | 0.92 | 18
2035577729682322176 | 0.9024 | 0.0089 | 0.7027 | 0.0417 | 0.6341 | 0.0374 | 0.3227 | 0.0056 | $-0.16$ | 19
2067948245320365184 | 0.7863 | 0.0006 | 0.8374 | 0.0023 | 0.6584 | 0.0017 | 0.1521 | 0.0014 | 0.48 | 17
2129771310248902016 | 0.9071 | 0.0183 | 0.7408 | 0.0399 | 0.6720 | 0.0302 | 0.3184 | 0.0048 | $-0.16$ | 10
2185171578009765632 | 0.9629 | 0.0082 | 1.2178 | 0.0299 | 1.1727 | 0.0333 | 0.4436 | 0.0033 | 1.28 | 17
2198442167969655296 | 0.7508 | 0.0152 | 0.9844 | 0.0416 | 0.7391 | 0.0189 | 0.1963 | 0.0033 | 0.65 | 10
3283823387685219328 | 0.7419 | 0.0004 | 0.9864 | 0.0112 | 0.7319 | 0.0083 | 0.1478 | 0.0031 | 0.37 | 20
3312631623125272448 | 0.7062 | 0.0024 | 0.9920 | 0.0304 | 0.7005 | 0.0215 | 0.1249 | 0.0051 | 1.54 | 21
3366718833479009408 | 0.6407 | 0.0022 | 1.0494 | 0.0102 | 0.6724 | 0.0051 | 0.0155 | 0.0028 | 0.18 | 8
3409686270424363008 | 0.7076 | 0.0031 | 0.7927 | 0.0139 | 0.5610 | 0.0087 | 0.1086 | 0.0028 | 2.07 | 17
3427930123268526720  | 0.3311 | 0.0047 | 0.8066 | 0.0760 | 0.2671 | 0.0248 | 0.0616 | 0.0066 | 5.23 | 3
3536759371865789568 | 0.8392 | 0.0212 | 1.6490 | 0.0899 | 1.3838 | 0.0492 | 0.2000 | 0.0061 | 0.62 | 22
3549833939509628672 | 0.9406 | 0.0195 | 0.9450 | 0.0384 | 0.8889 | 0.0249 | 0.3567 | 0.0051 | 0.19 | 22
3931519127529822208 | 0.7750 | 0.0260 | 1.0791 | 0.0762 | 0.8363 | 0.0342 | 0.1403 | 0.0045 | 1.19 | 23
3935131126305835648 | 0.7039 | 0.0022 | 1.4860 | 0.0744 | 1.0460 | 0.0523 | 0.1374 | 0.0052 | 0.08 | 20
3954536956780305792 | 0.9053 | 0.0094 | 0.7713 | 0.7626 | 0.6982 | 0.6904 | 0.3262 | 0.0600 | 0.49 | 5
3964895043508685312 | 0.8650 | 0.0084 | 0.7531 | 0.1654 | 0.6514 | 0.1426 | 0.2201 | 0.0196 | $-0.09$ | 17
4145362250759997952 | 0.6960 | 0.0216 | 0.9414 | 0.0611 | 0.6552 | 0.0261 | 0.0757 | 0.0046 | 0.16 | 24
4228891667990334976 | 0.8629 | 0.0057 | 1.0907 | 0.0588 | 0.9412 | 0.0502 | 0.2872 | 0.0038 | 4.10 | 25
4354357901908595456 | 0.5832 | 0.0194 | 0.7005 | 0.0509 | 0.4085 | 0.0177 | 0.0606 | 0.0041 | 0.18 | 26,27
4589258562501677312 | 0.8201 | 0.0317 | 0.6679 | 0.0628 | 0.5478 | 0.0395 | 0.2924 | 0.0085 | $-0.13$ | 5
5762455439477309440 | 0.8041 | 0.0327 | 0.8058 | 0.0674 | 0.6480 | 0.0333 | 0.1768 | 0.0076 | 0.20 | 5
6244076338858859776 | 0.8820 | 0.0076 | 0.7730 | 0.0135 | 0.6818 | 0.0101 | 0.2943 | 0.0023 | 0.86 | 10
6799537965261994752 | 0.9000 | 0.0017 | 0.9036 | 0.0092 | 0.8132 | 0.0083 | 0.3267 | 0.0024 | 1.32 | 24

222
Full table including the correlations and the orbital parameters will be made
available on VizieR.

(1) Tomkin (2005); (2) Mermilliod et al. (2009); (3) Baroch et al. (2018); (4)
Fekel et al. (1994); (5) Goldberg et al. (2002); (6) Griffin (2009); (7)
Sperauskas et al. (2019); (8) Fekel et al. (2015); (9) Ginestet & Carquillat
(1995); (10) Halbwachs et al. (2018); (11) Griffin (2011); (12) Griffin
(1990); (13) Halbwachs et al. (2012); (14) Griffin (2008); (15) Kiefer et al.
(2016); (16) Griffin (2013); (17) Kiefer et al. (2018); (18) Griffin (1993);
(19) Imbert (2006); (20) Halbwachs et al. (2020); (21) Tomkin (2007); (22)
Griffin (2006); (23) Griffin (2012); (24) Tokovinin (2019); (25) Griffin
(2014); (26) Mazeh et al. (1997); (27) Mayor & Turon (1982)

The uncertainties on the masses of the binary system Gaia DR3
$3954536956780305792$ (HIP 61816) are extremely large, with
$\frac{\sigma_{\mathcal{M}_{1}}}{\mathcal{M}_{1}}\approx 1$. These results are
therefore unusable. We expect this result to be a consequence of the lack of
constrains over the inclination for this system which is $i=27.5\pm
12.7^{\circ}$. Being compatible with $0$ at less than $3\sigma$, the masses
are much less constrained too, 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 $\mathcal{M}_{1}=1.49\pm 0.44\,{\cal
M}_{\odot}$, $\mathcal{M}_{2}=1.34\pm 0.37\,{\cal M}_{\odot}$ and
$\beta=0.40\pm 0.03$, with $\mathrm{F2}=2.5$. These parameters make the system
an outlier in the mass-luminosity relation. Although a MCMC is not adapted to
the Thiele-Innes handling, we tested a short MCMC on this star that went to
lower mass values, indicating the presence of another solution. This binary
system is the only one of our sample with a low eccentricity consistent with
zero at $1\leavevmode\nobreak\ \sigma$. The system has then been additionally
tested with an eccentricity fixed to $0$, corresponding to a circular orbit.
The result obtained fits much better the mass-luminosity relation despite its
slightly larger $\mathrm{F2}$, and is the solution kept in Table 2.

Table 2: Solutions from the combination of Gaia NSS astrometric solutions with
APOGEE double-lined spectroscopy. $q$ is the mass ratio, $\mathcal{M}_{1}$ is
the mass of the primary, $\mathcal{M}_{2}$ is the mass of the secondary (in
$\,{\cal M}_{\odot}$) and $\beta$ is the flux fraction of the secondary.
$\sigma_{q}$, $\sigma_{\mathcal{M}_{1}}$, $\sigma_{\mathcal{M}_{2}}$ and
$\sigma_{\beta}$ are their associated uncertainties. $\mathrm{F2}$ is the
goodness-of-fit estimator.

Gaia DR3 ID | $q$ | $\sigma_{q}$ | $\mathcal{M}_{1}$ | $\sigma_{\mathcal{M}_{1}}$ | $\mathcal{M}_{2}$ | $\sigma_{\mathcal{M}_{2}}$ | $\beta$ | $\sigma_{\beta}$ | $\mathrm{F2}$
---|---|---|---|---|---|---|---|---|---
839401128363085568ee$e$particular case for which the eccentricity has been fixed to 0. | 0.9143 | 0.0993 | 0.6225 | 0.1162 | 0.5692 | 0.0918 | 0.3865 | 0.0266 | 4.03
683525873153063680 | 0.9376 | 0.0496 | 0.5124 | 0.1037 | 0.4804 | 0.0951 | 0.3985 | 0.0167 | 1.72
702393458327135360 | 0.9398 | 0.0252 | 1.4875 | 0.5001 | 1.3980 | 0.4663 | 0.3871 | 0.0186 | 4.53
790545256897189760 | 0.8549 | 0.0846 | 0.4236 | 0.0870 | 0.3621 | 0.0618 | 0.2850 | 0.0263 | 1.20
794359875050204544 | 0.8824 | 0.1294 | 1.1746 | 0.3294 | 1.0365 | 0.2566 | 0.2676 | 0.0360 | 0.99
824315485231630592 | 0.9035 | 0.3744 | 0.9467 | 0.2152 | 0.8554 | 0.2087 | 0.3104 | 0.1036 | 3.92
901170214141646592 | 0.8396 | 0.1134 | 0.9309 | 0.2506 | 0.7815 | 0.1951 | 0.2330 | 0.0404 | 0.46
1267970076306377344 | 0.6633 | 0.0586 | 1.1378 | 0.2014 | 0.7547 | 0.0912 | 0.0703 | 0.0157 | 0.61
1636132061580415488 | 0.7176 | 0.1342 | 1.1046 | 0.4127 | 0.7927 | 0.2574 | 0.1488 | 0.0446 | 3.24
2134829544776833280 | 0.7721 | 0.0680 | 0.8751 | 0.1245 | 0.6757 | 0.0738 | 0.1505 | 0.0251 | 0.73
2705239237909520128 | 0.5293 | 0.1035 | 0.2336 | 0.0683 | 0.1236 | 0.0326 | 0.1993 | 0.0427 | 0.53
3847995791877023104 | 0.8387 | 0.0788 | 1.1446 | 0.2029 | 0.9600 | 0.1550 | 0.3167 | 0.0244 | 1.05
5285071954833306368oo$o$High $\mathrm{F2}$, removed from the Section 5 study. | 0.7905 | 0.1019 | 0.4609 | 0.1058 | 0.3643 | 0.0853 | 0.1499 | 0.0416 | 1.58

333 $o$$o$footnotetext: Mass-luminosity outlier, removed from the Section 5
study.
Full table including the correlations and the orbital parameters will be made
available on VizieR.

Since 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
law444https://www.cosmos.esa.int/web/Gaia/edr3-extinction-law:

$M_{G_{1}}=G+2.5\log_{10}\left(\frac{1}{1-\beta}\right)+5+5\log_{10}\left(\frac{\varpi}{1000}\right)-A_{G}$
(8)
$M_{G_{2}}=G+2.5\log_{10}\left(\frac{1}{\beta}\right)+5+5\log_{10}\left(\frac{\varpi}{1000}\right)-A_{G}$
(9)

where $\varpi$ is the parallax in mas.

To estimate the uncertainties of those absolute magnitude (see Appendix B), a
10% relative error on the extinction with a minimum error of 0.01 mag is
assumed and a 0.01 mag error is quadratically added to the $G$ formal
magnitude errors. The extinction term $A_{G}$ remain 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 Figure 3 for APOGEE give the position of all the
individual stars we have characterized in the mass-luminosity diagram. They
are overplotted on top of the PARSEC solar-metallicity isochrones (PAdova and
TRieste Stellar Evolution Code, Bressan et al. 2012).

Figure 2: Mass-luminosity diagram of the characterized stars from the
combination of Gaia with SB9. The error bars at $1\sigma$ are given in grey.
The absolute magnitude of the individual stars in the $G$-band $M_{G}$ is
plotted as a function of the star mass (in $\,{\cal M}_{\odot}$). The black
dots represent the stars, with the associated error bars at $1\sigma$ 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 $\mathrm{F2}>5$. They are
overplotted on the solar-metallicity PARSEC isochrones (in green).  Figure 3:
Mass-luminosity diagram of the characterized stars from the combination of
Gaia with APOGEE. The error bars at $1\leavevmode\nobreak\ \sigma$ are given
in grey. The absolute magnitude of the individual stars in the $G$-band
$M_{G}$ is plotted as a function of the star mass (in $\,{\cal M}_{\odot}$).
The black dots represent the stars, with the associated error bars at
$1\sigma$ 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 $3\sigma$. 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 $\mathrm{F2}>5$. For
APOGEE, one main outlier exists, Gaia DR3 $5285071954833306368$ (HD 50199),
still compatible with the isochrones at less than $5\sigma$. Nothing specific
has been found about this system to justify its surprising position in the
diagram.

11 systems have a goodness-of-fit of their Gaia DR3 astrometric solution
higher than 10, but none are outliers in the mass-luminosity relation nor in
the $\mathrm{F2}$ of the combined fit nor in the relation between the flux
fraction and the mass ratio. We therefore decided to keep those systems in our
sample for the Section 5 mass-luminosity relation study.

###  4.1 Comparison with the direct calculation method for SB9

As a sanity check, we compare the masses obtained with the orbit fitting
process (Table 1) to those obtained through direct calculation, that is using
directly the orbital parameters provided by SB9 to derive the mass functions
without going back to the raw data, as detailed in Annex A (Table 5).

Going back to the raw spectroscopic data allows to take into account the
correlations between the spectroscopic parameters and then have a better
estimation of the orbital parameters and their uncertainties. Moreover, a
correction process is applied to the uncertainties of the radial velocity
epochs making them more realistic.

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

${\rm
compatibility}=\frac{\mathcal{M}_{OF}-\mathcal{M}_{DC}}{\max(\sigma_{OF},\sigma_{DC})}$
(10)

where $\mathcal{M}_{OF}$ is the mass obtained through orbit fitting and
$\mathcal{M}_{DC}$ the mass obtained through direct calculation.

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

Figure 4: Compatibility density of the masses obtained by direct calculation
and by orbit fitting for SB9 binaries. The compatibility is in $\sigma$. It is
given in orange for primary masses and in purple for secondary masses. Figure
5: Comparison of the uncertainties on the masses obtained by direct
calculation ($\sigma_{DC}$) or by orbit fitting ($\sigma_{OF}$) for binaries
from SB9. The primaries are given in orange, and secondaries 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 $y=x$.

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

A few points have slightly bigger uncertainties through the orbit fitting
process. This is the result of the correction process 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 contains only 3 epochs for the secondary. Thus, we can expect a strong
correlation between the SB9 orbital parameters. We performed a MCMC over the
raw spectroscopic data only and confirm that strong correlations appear and
that the distribution is strongly asymmetric, explaining the strong
improvement 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 et al. (2023a).
While our mass estimates are consistent with Gaia DR3 1067685718250692352 (HIP
45794) and Gaia DR3 2035577729682322176 (HIP 97640), Gaia DR3
595390807776621824 HIP 42418) is a 5$\sigma$ outlier. It has a photocentre
semi-major axis $a_{0}$ of 3.4 mas while the other two stars have a smaller
$a_{0}\sim 0.8$ mas, suggesting that the astrometric motion impacted the
spectroscopic measure, as suggested by Babusiaux et al. (2023). This is due to
the fact that the expected position of the spectra on the Gaia Radial Velocity
Spectrometer (RVS) detectors is predicted by the standard 5-parameter
astrometric motion instead of the epoch astrometric one 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)
($\mathcal{M}_{1}=0.670\pm 0.069$, $\mathcal{M}_{2}=0.602\pm 0.061$,
compatible with our results within $1\sigma$), for Gaia DR3
$2067948245320365184$ (HIP 101382) by Kiefer et al. (2018)
($\mathcal{M}_{1}=0.8420\pm 0.0014$, $\mathcal{M}_{2}=0.66201\pm 0.00076$,
compatible with our results within $2\sigma$) and for Gaia DR3
$3283823387685219328$ (HIP 20601) by Halbwachs et al. (2020)
($\mathcal{M}_{1}=0.9798\pm 0.0019\,{\cal M}_{\odot}$,
$\mathcal{M}_{2}=0.72697\pm 0.00094\,{\cal M}_{\odot}$, compatible with our
results within $1\sigma$). Halbwachs et al. (2020) combined the raw relative
astrometry and spectroscopic data. They obtained a parallax at 4.4$\sigma$
from the Gaia NSS one. 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 $\mathrm{F2}=1.5$, a parallax of $\varpi=16.573\pm 0.017$
and masses $\mathcal{M}_{1}=0.9816\pm 0.0014\,{\cal M}_{\odot}$ and
$\mathcal{M}_{2}=0.72808\pm 0.00076\,{\cal M}_{\odot}$. This new parallax is
at 3.4$\sigma$ from the Halbwachs et al. (2020) one and reduced to 2.6$\sigma$
if we take into account the Gaia DR3 parallax zero point (Lindegren et al.
2021), highlighting that SB2 stars with direct imaging will be excellent test
cases for Gaia DR4 epoch data validation. The orbit fit for this binary is
given on the Figure 6.

Figure 6: Radial velocity fit for the binary system Gaia DR3
$3283823387685219328$ (HIP 20601) from SB9. The radial velocity (km/s) is
plotted as a function of the phase. The orange dots correspond to the radial
velocity epochs of the primary, and the purple dots give the radial velocity
epochs for the secondary. The black curves represent the corresponding fits
obtained 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 Section 2.1. This star has been solved through spectroscopy only
(Kounkel et al. 2021) and then combined with Gaia astrometry through a direct
calculation process by Gaia Collaboration et al. (2023a) to obtain
$\mathcal{M}_{1}=1.14\pm 0.38\,{\cal M}_{\odot}$, $\mathcal{M}_{2}=1.06\pm
0.35\,{\cal M}_{\odot}$ and $F_{2}/F_{1}=0.567\pm 0.071$, fully compatible
with our results. The uncertainties reported here are larger than the one of
Gaia Collaboration et al. (2023a). This may be due to a slight discrepancy
between the orbital parameters of APOGEE and Gaia, specifically on the
eccentricities which are at 4$\sigma$ from each other. This discrepancy leads
to a high $\mathrm{F2}$ 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 on the Figure 7. The
difference of observation time clearly appears on the Figures 6 and 7, where
the lack of radial velocity epochs for APOGEE is rather obvious.

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

##  5 The Mass-Luminosity relation

The mass calculations presented above could enable to perform an empirical fit
of the mass-luminosity relation in the G-band using the Gaia photometry.
However, the masses calculated do not provide satisfying constrains in the
interesting region of the relation for low-mass stars ($\mathcal{M}<0.7\,{\cal
M}_{\odot}$). To fill in this part of the H-R diagram, we looked for low mass
stars resolved by Gaia and 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 star systems spatially resolved studied with direct imaging in
Mann et al. (2019) which 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 Mann et al. (2019) sample but
has direct imaging data from McAlister et al. (1989) and Mason et al. (2018)
and has been added to our sample. Those four stars are analysed by BINARYS by
combining the direct imaging data with Hipparcos TD and Gaia astrometric
parameters of both components following the methodology detailed in Leclerc et
al. (2023). All Gaia solutions have a 5-parameter solution except Gl 330 for
which the secondary component is a 2-parameter solution and Gl 860 for which
both components are 2-parameter solutions only. One Hipparcos TD outlier at
5$\upsigma$ had to be removed for both Gl 568 and Gl 227.

Two stars from the Mann et al. (2019) sample, Gl~65 and Gl~473, are resolved
by Gaia with separations consistent with the visual orbit, but no Hipparcos
data exist for them. However those two stars have literature mass fraction
$B=\frac{\mathcal{M}_{2}}{\mathcal{M}_{1}+\mathcal{M}_{2}}$: for Gl 65
$B=0.494\pm 0.04$ from Geyer et al. (1988) and for Gl 473 $B=0.477\pm 0.008$
from Torres et al. (1999). We incorporated this information within BINARYS for
those 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 compute the radial proper motion, that
is the relative change in distance per year, $\mu_{r}=V_{r}\varpi/A_{Z}$ in
yr-1 with $A_{Z}=9.7779222\times 10^{8}$ mas yr km s-1. The perspective
acceleration changes the along-scan abscissa $\nu$ (in mas) by adding :

$\Delta\nu=-\mu_{r}\Delta
T\left(\frac{\partial\nu}{\partial\varpi}\varpi+\frac{\partial\nu}{\partial\mu_{\alpha^{*}}}\mu_{\alpha^{*}}+\frac{\partial\nu}{\partial\mu_{\delta}}\mu_{\delta}\right)$
(11)

with $\Delta T$ the epoch in years relative to the reference epoch for the
astrometric parameters (that is 1991.25 for Hipparcos and 2016.0 for Gaia
DR3). This $\Delta\nu$ is subtracted to the Hipparcos abscissa residuals and
added to the Gaia simulated abscissa. However one has to take into account
that the perspective acceleration has been taken into account for DR3 for
stars with a Gaia DR2 radial velocity or in the table of nearby Hipparcos
stars with radial velocity used for DR2555
https://gea.esac.esa.int/archive/documentation/GDR2/Data\\_processing/chap\\_cu3ast/sec\\_cu3ast\\_cali/ssec\\_cu3ast\\_cali\\_source.html\\#Ch3.T3.
Here only the radial velocity for the A component of Gl 860 was applied for
the DR3 processing, using the same $V_{r}=-33.94$ $\rm\,km\,s^{-1}$ as we
used. For Gl 65 we used $V_{r}=39.04$ $\rm\,km\,s^{-1}$ (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 only
change marginally the $\chi^{2}$.

The results obtained for these six stars are given in Table 3. Our mass
estimates are consistent with Kervella et al. (2016), Delfosse et al. (2000)
and Benedict et al. (2016) for Gl 65, but we all use 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 are at 2.5$\sigma$ from
the values of Benedict et al. (2016) which is driven by the difference with
the RECONS parallax they used. Our mass estimate for Gl 860 is consistent with
Delfosse et al. (2000). We derive the dynamical masses for the first time, to
our knowledge, of the components of Gl 277, Gl 330 and Gl 568.

Table 3: Solutions for star systems resolved by Gaia with direct imaging data
and either Hipparcos Transit Data or literature mass fraction. The masses of
the primary $\mathcal{M}_{1}$ and secondary $\mathcal{M}_{2}$ are in $\,{\cal
M}_{\odot}$, the parallax $\varpi$ in mas,
$\rho_{\mathcal{M}_{1},\mathcal{M}_{2}}$ is the correlation between the mass
estimates and $\mathrm{F2}$ is the goodness of fit.

Name | $\mathcal{M}_{1}$ | $\sigma_{\mathcal{M}_{1}}$ | $\mathcal{M}_{2}$ | $\sigma_{\mathcal{M}_{2}}$ | $\varpi$ | $\sigma_{\varpi}$ | $\rho_{\mathcal{M}_{1},\mathcal{M}_{2}}$ | $\mathrm{F2}$
---|---|---|---|---|---|---|---|---
Gl277 | 0.5276 | 0.0046 | 0.2069 | 0.0031 | 83.465 | 0.052 | 0.77 | 3.93
Gl330 | 0.4969 | 0.0247 | 0.3184 | 0.0159 | 61.513 | 0.107 | 0.98 | 3.37
Gl568 | 0.3106 | 0.0067 | 0.2129 | 0.0047 | 87.439 | 0.069 | 0.95 | 1.53
Gl860 | 0.2915 | 0.0062 | 0.1789 | 0.0046 | 248.354 | 1.560 | 0.39 | 3.49
Gl65**$*$Solution using a literature mass fraction. | 0.1224 | 0.0012 | 0.1195 | 0.0012 | 371.396 | 0.453 | $-0.28$ | 1.19
Gl473**$*$Solution using a literature mass fraction. | 0.1379 | 0.0023 | 0.1258 | 0.0022 | 227.041 | 0.389 | $-0.76$ | 1.36
666 Full table including Gaia and Hipparcos IDs and full orbital solution will
be made available on VizieR.

###  5.2 Fitting the Mass-Luminosity relation

To fit the mass-luminosity relation, we implemented a TMB function that allows
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 Annex B). The true magnitudes of the stars are used as a random parameter:
they are marginalized, that is integrated out of the likelihood. The initial
value of TMB is 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 as for
fainter magnitudes the age dependency is well known to be too large.

We tested several degree 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 fitting
the log of the mass. The coefficients are given in Table 4. The fit
uncertainties have been estimated through a bootstrap, leading to
uncertainties smaller than 0.015$\,{\cal M}_{\odot}$ for magnitudes higher
than $M_{G}>6$ corresponding to masses $\mathcal{M}<0.77$ $\,{\cal
M}_{\odot}$.

The fit is displayed on Fig. 8 together with the PARSEC, the Baraffe et al.
(2015)777http://perso.ens-lyon.fr/isabelle.baraffe/BHAC15dir/ and the BASTI (A
BAg of Stellar Tracks and Isochrones, Hidalgo et al. 2018)888http://basti-
iac.oa-teramo.inaf.it/ isochrones. The age dependency of the mass-luminosity
relation starts to be significant for all isochrones at $\gtrsim 0.6$ $\,{\cal
M}_{\odot}$ and our fit follows the oldest isochrones. For masses $<0.5\,{\cal
M}_{\odot}$ our empirical relation indicate lower masses for a given
luminosity than the PARSEC isochrones. Our results are consistent with the
Baraffe et al. (2015) isochrones except in the low mass region where we find
slightly higher masses for a given luminosity.

Figure 8: Mass-luminosity relation fitted in this work (in blue), with areas
outside the $6<M_{G}<13.5$ range as dotted line. The grey area corresponds to
the 1$\sigma$ bootstrap interval. The stars from Table 3 (resolved Gaia stars
with direct imaging data) are in blue and stars from Tables 1 and 2 (SB2 data
combined with Gaia NSS astrometric solution) are in grey. Dashed green lines
are the solar metallicity PARSEC isochrones for main sequence stars
(isochrones with label=1). The dotdashed red line and the twodashed pink line
are the solar metallicity isochrone of Baraffe et al. (2015) and BASTI
respectively; they show a deviation only at masses lower than
$\sim$0.12$\,{\cal M}_{\odot}$.

.

Table 4: Coefficient values of the polynomial fit $\log_{10}(\mathcal{M})=C_{0}+C_{1}M_{G}+C_{2}M_{G}^{2}+C_{3}M_{G}^{3}+C_{4}M_{G}^{4}$, to be used within $6<M_{G}<13.5$ which corresponds to the $0.12<\mathcal{M}<0.77$ mass range. The coefficients are provided for the best fit as well as the upper and lower curves at 1$\sigma$ obtained through a bootstrap.  | best fit | upper | lower
---|---|---|---
C0 | $3.129$ | $3.798$ | $2.534$
C1 | $-1.5406$ | $-1.8377$ | $-1.2672$
C2 | $0.275\,13$ | $0.323\,73$ | $0.229\,54$
C3 | $-0.021\,661$ | $-0.025\,067$ | $-0.018\,413$
C4 | $0.000\,599\,1$ | $0.000\,685\,5$ | $0.000\,514\,8$

##  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, it
was the case for only one APOGEE star. We tested on SB9 the difference between
a direct calculation of the masses using the orbital parameters and a combined
fit using the raw radial velocity measures, the later estimating better the
parameters, their uncertainties and their correlations. We also estimated the
masses of 6 stars resolved by Gaia DR3 with literature direct imaging and
either Hipparcos data or literature mass fraction. Three of those stars have
dynamical masses derived for the first time.

The BINARYS tool have been 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 taking into account all the
correlations between the component masses and magnitudes. This empirical
relation is found to be in better agreement with the Baraffe et al. (2015)
isochrones than with the PARSEC ones.

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. It will enable to make a full combined fit on the raw data of both
spectroscopic or direct imaging data with Gaia astrometry. It will also enable
to dig into systems with a too low astrometric signal to have a full Gaia DR4
orbital solution but with a good spectroscopic or direct imaging one. We can
therefore expect a much more in depth study of the mass-luminosity relation
with Gaia DR4, in particular a study of the metallicity dependency should be
conducted with a larger sample.

###### Acknowledgements.

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.

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##  Appendix A The direct calculation process for SB9

Since the orbital parameters are both given by SB9 and Gaia, the combination
of astrometry with spectroscopy can be done easily. The features of the star
systems are directly computed from the astrometric and spectroscopic orbital
solutions, using commonly used formulae. This direct-calculation method is
approximately the same as applied for mass calculations by the Gaia
collaboration (Gaia Collaboration et al. 2023a).

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

$q=\frac{\mathcal{M}_{2}}{\mathcal{M}_{1}}=\frac{K_{1}}{K_{2}}.$ (12)

The dynamical mass of the secondary is

$\mathcal{M}_{2}=1.0385\times
10^{-7}K_{1}^{3}P(1-e^{2})^{3/2}\left(1+\frac{1}{q}\right)^{2}\frac{1}{\sin^{3}{i}},$
(13)

with $P$ the period in days, $K_{1}$ in km.s-1 and the inclination $i$ given
by the Gaia astrometry. The period and the eccentricity $e$ used are the
weighted mean of the Gaia and SB9 values.

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

The flux fraction of the secondary $\beta=\frac{F_{2}}{F_{1}+F_{2}}$ is given
through the relation between the individual stars orbits and the photocentre
orbit provided by astrometry:

$a_{0}=(B-\beta)\leavevmode\nobreak\ a_{21}$ (14)

with $B$ the mass fraction of the secondary

$B=\frac{\mathcal{M}_{2}}{\mathcal{M}_{1}+\mathcal{M}_{2}}=\frac{q}{1+q},$
(15)

$a_{21}$ the semi-major axis of the relative orbit between the primary and the
secondary

$a_{21}=a_{1}+a_{2}=a_{1}\left(1+\frac{1}{q}\right),$ (16)

and $a_{0}$ the semi-major axis of the the photocentre orbit.

Here, $a_{1/2}$ are the individual semi-major axis of the primary/secondary
respectively. They can be deduced from the semi-amplitudes:

$a_{1,2}=\frac{1}{10879}K_{1,2}P\frac{\sqrt{1-e^{2}}}{\sin{i}}$ (17)

with the semi-major axis in au and the period in days.

The masses $\mathcal{M}_{1}$, $\mathcal{M}_{2}$, and the flux fraction $\beta$
are then obtained through Eq. 2, 3, 4 respectively.

Astrometry is needed here to determine the photocentre orbit $a_{0}$ but also
to convert it from mas to au using the parallax $\varpi$.

The errors are estimated through a Monte-Carlo (with $n_{MC}=10^{3}$) assuming
all the parameter distributions to be Gaussian-like. The Gaia parameters are
generated simultaneously according to the covariance matrix, except the
photocentre semi-major axis $a_{0}$. 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) have been used for it. In order to
take into account potential asymmetry in final parameters distributions, the
lower and upper confidence levels at respectively 16% and 84% are determined.

The results of this direct calculation are given in Table 5.

Table 5: Results from direct calculation : SB9 Gaia DR3 ID | $\mathcal{M}_{1}$ | $\sigma_{\mathcal{M}_{1}}$ | $\mathcal{M}_{1_{low}}$ | $\mathcal{M}_{1_{up}}$ | $\mathcal{M}_{2}$ | $\sigma_{\mathcal{M}_{2}}$ | $\mathcal{M}_{2_{low}}$ | $\mathcal{M}_{2_{up}}$ | $\beta$ | $\sigma_{\beta}$ | $\beta_{low}$ | $\beta_{up}$
---|---|---|---|---|---|---|---|---|---|---|---|---
48197783694869760 | 0.994 | 0.038 | 0.954 | 1.031 | 0.679 | 0.022 | 0.656 | 0.700 | 0.101 | 0.001 | 0.097 | 0.105
69883417170175488 | 0.895 | 0.024 | 0.873 | 0.920 | 0.843 | 0.021 | 0.822 | 0.863 | 0.399 | 0.003 | 0.394 | 0.404
308256610357824640 | 0.440 | 0.341 | 0.168 | 0.478 | 0.312 | 0.241 | 0.119 | 0.338 | 0.283 | 0.010 | 0.233 | 0.288
478996438146017280 | 0.778 | 0.022 | 0.753 | 0.797 | 0.717 | 0.020 | 0.694 | 0.734 | 0.385 | 0.002 | 0.382 | 0.388
544027809281308544 | 0.954 | 0.059 | 0.897 | 1.018 | 0.796 | 0.036 | 0.761 | 0.833 | 0.137 | 0.001 | 0.131 | 0.144
595390807776621824 | 0.788 | 0.011 | 0.777 | 0.799 | 0.695 | 0.007 | 0.687 | 0.702 | 0.309 | 0.001 | 0.307 | 0.312
660622010858088320 | 0.715 | 0.065 | 0.656 | 0.783 | 0.626 | 0.038 | 0.588 | 0.665 | 0.227 | 0.003 | 0.217 | 0.237
827608625636174720 | 0.776 | 0.051 | 0.726 | 0.823 | 0.730 | 0.049 | 0.682 | 0.775 | 0.408 | 0.003 | 0.404 | 0.412
882872210352301568 | 1.007 | 0.011 | 0.996 | 1.018 | 0.506 | 0.003 | 0.503 | 0.509 | 0.025 | 0.001 | 0.021 | 0.029
1067685718250692352 | 1.035 | 0.013 | 1.022 | 1.048 | 0.919 | 0.012 | 0.908 | 0.931 | 0.300 | 0.002 | 0.294 | 0.305
1074883087005896320 | 0.641 | 0.037 | 0.608 | 0.680 | 0.602 | 0.034 | 0.572 | 0.638 | 0.396 | 0.005 | 0.389 | 0.403
1324699172583973248 | 1.309 | 0.056 | 1.254 | 1.365 | 1.154 | 0.044 | 1.109 | 1.198 | 0.315 | 0.002 | 0.311 | 0.319
1441993625629660800 | 1.116 | 0.046 | 1.062 | 1.151 | 0.932 | 0.028 | 0.897 | 0.951 | 0.092 | 0.001 | 0.084 | 0.098
1480959875337657088 | 0.864 | 0.257 | 0.655 | 1.122 | 0.811 | 0.169 | 0.664 | 0.978 | 0.291 | 0.014 | 0.258 | 0.324
1517219363639758976 | 1.114 | 0.053 | 1.066 | 1.170 | 0.913 | 0.042 | 0.875 | 0.958 | 0.163 | 0.001 | 0.157 | 0.170
1517927895803742080 | 0.584 | 0.013 | 0.571 | 0.597 | 0.549 | 0.017 | 0.533 | 0.566 | 0.361 | 0.002 | 0.357 | 0.365
1528045017687961856 | 1.125 | 0.030 | 1.095 | 1.154 | 0.764 | 0.019 | 0.744 | 0.783 | 0.068 | 0.001 | 0.064 | 0.071
1615450866336763904 | 1.000 | 0.045 | 0.956 | 1.044 | 0.787 | 0.031 | 0.757 | 0.816 | 0.131 | 0.001 | 0.125 | 0.136
1918953867019478144 | 1.040 | 0.006 | 1.036 | 1.047 | 0.969 | 0.005 | 0.964 | 0.974 | 0.390 | 0.001 | 0.388 | 0.392
2012218158438964224 | 2.072 | 0.129 | 1.957 | 2.207 | 1.952 | 0.119 | 1.848 | 2.078 | 0.338 | 0.004 | 0.331 | 0.345
2035577729682322176 | 0.708 | 0.048 | 0.664 | 0.759 | 0.641 | 0.044 | 0.602 | 0.687 | 0.326 | 0.003 | 0.321 | 0.332
2067948245320365184 | 0.839 | 0.002 | 0.837 | 0.842 | 0.660 | 0.002 | 0.658 | 0.662 | 0.153 | 0.001 | 0.151 | 0.154
2129771310248902016 | 0.748 | 0.038 | 0.712 | 0.785 | 0.672 | 0.032 | 0.640 | 0.702 | 0.316 | 0.002 | 0.311 | 0.321
2185171578009765632 | 1.190 | 0.030 | 1.160 | 1.218 | 1.093 | 0.032 | 1.059 | 1.124 | 0.412 | 0.005 | 0.405 | 0.419
2198442167969655296 | 1.021 | 0.046 | 0.976 | 1.067 | 0.772 | 0.027 | 0.745 | 0.800 | 0.203 | 0.001 | 0.198 | 0.208
3283823387685219328 | 1.017 | 0.019 | 0.995 | 1.032 | 0.755 | 0.014 | 0.739 | 0.766 | 0.124 | 0.004 | 0.093 | 0.151
3312631623125272448 | 0.990 | 0.077 | 0.925 | 1.076 | 0.700 | 0.054 | 0.655 | 0.761 | 0.139 | 0.002 | 0.128 | 0.150
3366718833479009408 | 1.050 | 0.011 | 1.038 | 1.061 | 0.673 | 0.006 | 0.667 | 0.679 | 0.017 | 0.001 | 0.012 | 0.022
3409686270424363008 | 0.794 | 0.016 | 0.778 | 0.811 | 0.561 | 0.010 | 0.551 | 0.571 | 0.105 | 0.001 | 0.102 | 0.109
3427930123268526720 | 0.933 | 0.087 | 0.849 | 1.018 | 0.308 | 0.029 | 0.282 | 0.336 | 0.074 | 0.001 | 0.067 | 0.079
3536759371865789568 | 1.588 | 0.075 | 1.514 | 1.662 | 1.380 | 0.051 | 1.331 | 1.428 | 0.208 | 0.002 | 0.201 | 0.214
3549833939509628672 | 0.957 | 0.042 | 0.916 | 1.001 | 0.900 | 0.032 | 0.868 | 0.931 | 0.359 | 0.004 | 0.352 | 0.366
3931519127529822208 | 1.085 | 0.076 | 1.015 | 1.164 | 0.838 | 0.034 | 0.806 | 0.872 | 0.144 | 0.001 | 0.139 | 0.148
3935131126305835648 | 1.418 | 0.086 | 1.338 | 1.507 | 0.998 | 0.060 | 0.942 | 1.061 | 0.135 | 0.001 | 0.129 | 0.141
3954536956780305792 | 0.694 | 3.100 | 0.262 | 0.933 | 0.629 | 2.756 | 0.238 | 0.853 | 0.333 | 0.019 | 0.271 | 0.352
3964895043508685312 | 0.674 | 0.183 | 0.531 | 0.835 | 0.599 | 0.163 | 0.473 | 0.743 | 0.218 | 0.006 | 0.198 | 0.236
4145362250759997952 | 1.492 | 0.400 | 1.145 | 1.929 | 0.956 | 0.191 | 0.782 | 1.154 | 0.105 | 0.003 | 0.079 | 0.133
4228891667990334976 | 1.294 | 0.095 | 1.192 | 1.376 | 1.116 | 0.081 | 1.029 | 1.187 | 0.304 | 0.002 | 0.300 | 0.308
4354357901908595456 | 0.756 | 0.067 | 0.693 | 0.825 | 0.428 | 0.023 | 0.408 | 0.452 | 0.060 | 0.001 | 0.056 | 0.064
4589258562501677312 | 0.681 | 0.066 | 0.617 | 0.750 | 0.559 | 0.041 | 0.520 | 0.600 | 0.292 | 0.003 | 0.284 | 0.300
5762455439477309440 | 0.802 | 0.069 | 0.739 | 0.874 | 0.646 | 0.034 | 0.613 | 0.680 | 0.177 | 0.002 | 0.169 | 0.185
6244076338858859776 | 0.776 | 0.015 | 0.760 | 0.790 | 0.684 | 0.011 | 0.673 | 0.694 | 0.294 | 0.001 | 0.291 | 0.296
6799537965261994752 | 0.892 | 0.006 | 0.890 | 0.900 | 0.802 | 0.005 | 0.801 | 0.809 | 0.326 | 0.002 | 0.322 | 0.331

##  Appendix B Jacobian calculations

The TMB algorithm provides in output the orbital parameters with their
associated uncertainties, in particular the period $P$ with $\sigma_{P}$ and
the parallax $\varpi$ with $\sigma_{\varpi}$. It additionally gives the semi-
major axis of the primary orbit $a_{1}$ with its error $\sigma_{a1}$, the mass
ratio $q$ with $\sigma_{q}$ and the flux fraction $\beta$ with
$\sigma_{\beta}$. The global covariance matrix is also calculated.

Using $a_{2}=\frac{a_{1}}{q}$, the masses are derived with:

$\mathcal{M}_{1}=\frac{(a_{1}+a_{2})^{3}}{P^{2}\leavevmode\nobreak\
(1+q)}=\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{3}}$ (18)
$\mathcal{M}_{2}=\frac{(a_{1}+a_{2})^{3}q}{P^{2}\leavevmode\nobreak\
(1+q)}=\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{2}}$ (19)

and the magnitudes are obtained through Eq. 8 and Eq. 9 from the parameters
($\varpi,\beta$) as well as the apparent magnitude $G$ and the absorption
$A_{G}$. The latest two are independent and independent of the other
parameters. We therefore for simplicity consider here only the extinction
corrected magnitude $G_{0}=G-A_{G}$ which has a variance
$\sigma_{G_{0}}^{2}=\sigma_{G}^{2}+\sigma_{A_{G}}^{2}$.

The covariance of the parameters ($a_{1},q,P,\varpi,\beta$) $\Sigma_{TMB}$ are
provided by TMB. The full covariance of our input parameters
($a_{1},q,P,\varpi,\beta,G_{0}$) is then

$\Sigma_{params}=\left(\begin{array}[]{ccc}\Sigma_{TMB}&0\\\
0&\sigma_{G_{0}}^{2}\\\ \end{array}\right)$ (20)

The covariance of our output parameters
($\mathcal{M}_{1},\mathcal{M}_{2},M_{G_{1}},M_{G_{2}}$), $\Sigma$, is then

$\Sigma=J^{T}\leavevmode\nobreak\ \Sigma_{params}\leavevmode\nobreak\ J$

with $J$ the Jacobian of the transformation from
($a_{1},q,P,\varpi,\beta,G_{0}$) to
($\mathcal{M}_{1},\mathcal{M}_{2},M_{G_{1}},M_{G_{2}}$):

$J=\left(\begin{array}[]{cccc}\partial_{a_{1}}\mathcal{M}_{1}&\partial_{a_{1}}\mathcal{M}_{2}&0&0\\\
\partial_{q}\mathcal{M}_{1}&\partial_{q}\mathcal{M}_{2}&0&0\\\
\partial_{P}\mathcal{M}_{1}&\partial_{P}\mathcal{M}_{2}&0&0\\\
0&0&\partial_{\varpi}M_{G_{1}}&\partial_{\varpi}M_{G_{2}}\\\
0&0&\partial_{\beta}M_{G_{1}}&\partial_{\beta}M_{G_{2}}\\\
0&0&\partial_{G_{0}}M_{G_{1}}&\partial_{G_{0}}M_{G_{2}}\\\ \end{array}\right)$
(21)

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

$\partial_{a_{1}}\mathcal{M}_{1}=\frac{3a_{1}^{2}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{3}}$ (22)
$\partial_{q}\mathcal{M}_{1}=\frac{a_{1}^{3}}{P^{2}}\left(\frac{2(1+q)}{q^{3}}-\frac{3(1+q)^{2}}{q^{4}}\right)$
(23) $\partial_{P}\mathcal{M}_{1}=-2\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{3}\leavevmode\nobreak\ q^{3}}$ (24)

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

$\partial_{a_{1}}\mathcal{M}_{2}=\frac{3a_{1}^{2}\leavevmode\nobreak\
(1+q)^{2}}{P^{2}\leavevmode\nobreak\ q^{2}}$ (25)
$\partial_{q}\mathcal{M}_{2}=\frac{a_{1}^{3}}{P^{2}}\left(\frac{2(1+q)}{q^{2}}-\frac{2(1+q)^{2}}{q^{3}}\right)$
(26) $\partial_{P}\mathcal{M}_{2}=-2\frac{a_{1}^{3}\leavevmode\nobreak\
(1+q)^{2}}{P^{3}\leavevmode\nobreak\ q^{2}}$ (27)

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

The derivatives of the primary magnitude are given below :

$\partial_{\varpi}M_{G_{1}}=\frac{5}{\log(10)}\frac{1}{\varpi}$ (28)
$\partial_{\beta}M_{G_{1}}=\frac{2.5}{\log(10)}\frac{1}{1-\beta}$ (29)
$\partial_{G_{0}}M_{G_{1}}=1$ (30)

The derivatives of the secondary magnitude are given below :

$\partial_{\varpi}M_{G_{2}}=\frac{5}{\log(10)}\frac{1}{\varpi}$ (31)
$\partial_{\beta}M_{G_{2}}=-\frac{2.5}{\log(10)}\frac{1}{\beta}$ (32)
$\partial_{G_{0}}M_{G_{2}}=1$ (33)