# Dynamical Masses and Ages of Sirius-like Systems

Hengyue Zhang,1,2 Timothy D. Brandt,1 Rocio Kiman,3 Alexander Venner,4 Qier
An,1 Minghan Chen,1 Yiting Li1
1Department of Physics, University of California, Santa Barbara, Santa
Barbara, CA 93106, USA
2Sub-department of Astrophysics, Department of Physics, University of Oxford,
Keble Road, Oxford OX1 3RH, UK
3Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA
4Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD
4350, Australia E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We measure precise orbits and dynamical masses and derive age constraints for
six confirmed and one candidate Sirius-like systems, including the Hyades
member HD 27483. Our orbital analysis incorporates radial velocities, relative
astrometry, and Hipparcos-Gaia astrometric accelerations. We constrain the
main-sequence lifetime of a white dwarf’s progenitor from the remnant’s
dynamical mass and semi-empirical initial-final mass relations and infer the
cooling age from mass and effective temperature. We present new relative
astrometry of HD 27483 B from Keck/NIRC2 observations and archival HST data,
and obtain the first dynamical mass of ${0.798}_{-0.041}^{+0.10}$ $M_{\sun}$,
and an age of ${450}_{-180}^{+570}$ Myr, consistent with previous age
estimates of Hyades. We also measure precise dynamical masses for HD 114174 B
($0.591\pm 0.011$ $M_{\sun}$) and HD 169889 B (${0.526}_{-0.037}^{+0.039}$
$M_{\sun}$), but their age precisions are limited by their uncertain
temperatures. For HD 27786 B, the unusually small mass of $0.443\pm 0.012$
$M_{\sun}$ suggests a history of rapid mass loss, possibly due to binary
interaction in its progenitor’s RGB phase. The orbits of HD 118475 and HD
136138 from our RV fitting are overall in good agreement with Gaia DR3
astrometric two-body solutions, despite moderate differences in the
eccentricity and period of HD 136138. The mass of ${0.580}_{-0.039}^{+0.052}$
$M_{\sun}$ for HD 118475 B and a speckle imaging non-detection confirms that
the companion is a white dwarf. Our analysis shows examples of a rich number
of precise WD dynamical mass measurements enabled by Gaia DR3 and later
releases, which will improve empirical calibrations of the white dwarf
initial-final mass relation.

###### keywords:

white dwarfs – stars: kinematics and dynamics – stars: fundamental parameters
– binaries: general – stars: imaging – astrometry

††pubyear: 2022††pagerange: Dynamical Masses and Ages of Sirius-like
Systems–10

##  1 Introduction

While isolated white dwarfs (WDs) simply cool and fade indefinitely, WDs in
binary systems can give rise to some of the most dramatic events in the
Universe. These include novae and Type Ia supernovae, the source of much of
the Universe’s iron-peak elements (Iwamoto et al., 1999) and standardizable
candles used to measure the expansion history of the Universe (e.g., Phillips,
1993; Riess et al., 1998; Perlmutter et al., 1999). The nearest known WD,
Sirius B, is itself one component in a binary system (Bond, 1862); it orbits
20 AU (Bond et al., 2017) from the early A-type star (Morgan et al., 1953)
Sirius A. The Sirius system is too widely separated for the stars to interact
either in the past or in the future (Bond et al., 2017). However, widely
separated Sirius-like systems enable mass measurements based only on Newtonian
dynamics and they form the widely-separated analogs of the progenitors of
novae and supernovae.

Stars below $\approx$8 $M_{\odot}$ will end their lives as WDs, while stars
below $\approx$0.8 $M_{\odot}$ live longer than a Hubble time on the main
sequence (MS), according to stellar evolutionary models (e.g., Choi et al.,
2016). At least half of the stars in this mass interval are born in binaries
(Raghavan et al., 2010). The Solar neighborhood is rich in the Sirius-like
systems left after the more massive star in the binary has evolved off the MS.
Before turning into a WD, the star loses mass by expelling its outer shell and
exposing its central core during the asymptotic giant phase (Jeans, 1924;
Hadjidemetriou, 1963; Höfner & Olofsson, 2018). The final mass of the remnant
WD is related to the initial mass of its progenitor by the initial-final mass
relation (IFMR). Constraining the relation is crucial to understanding the
complex physical processes underlying the final stages of stellar evolution.
Observations of orbits of Sirius-like systems enable precise measurements of
WD masses without requiring constraints on surface gravity and are independent
of WD evolutionary models. Combing the mass with a $T_{\rm eff}$ or luminosity
measurement constrains the WD’s cooling age, which informs us about the WD’s
initial mass when compared to the age of the MS companion. These provide
direct measurements of the IFMR. Conversely, one can derive the total age of
the WD by assuming a particular IFMR. Comparing the WD’s age to the MS star’s
age tests the accuracy of the assumed IFMR.

Fitting dynamical masses and orbits to Sirius-like systems requires either
long-term astrometric monitoring like that available for Sirius (e.g., Bond et
al., 2017), or the combination of different data types to compensate for
limited phase coverage. On the one hand, absolute astrometry from the
Hipparcos (ESA, 1997) and Gaia (Gaia Collaboration et al., 2016) missions
probe the transverse acceleration of the MS star due to the tug of a WD
companion. These powerful astrometric measurements have been complied and
calibrated to the same reference frame by the Hipparcos-Gaia Catalog of
Accelerations (HGCA, Brandt, 2018, 2021). On the other hand, spectrographs
like AAT-UCLES (Diego et al., 1990), Keck-HIRES (Vogt et al., 1994), APF-Levy
(Vogt et al., 2014), and Lick-Hamilton (Fischer et al., 2014) provide high-
precision radial velocities (RVs) that probe the MS star’s acceleration along
the line-of-sight. These measurements, together with relative astrometry of a
short orbital arc, enable a direct measurement of WD mass (e.g., Brandt et
al., 2019; Bowler et al., 2021; Zeng et al., 2022).

In this paper, we combine absolute astrometry from the HGCA with radial
velocities and relative astrometry from various sources to measure orbits and
masses of six confirmed and one candidate nearby Sirius-like systems,
including the Hyades member HD 27483. For HD 118475 and HD 136138, we compare
our results against Gaia DR3 two-body solutions (Gaia Collaboration et al.,
2022; Holl et al., 2022). We use the open-source code orvara (Brandt et al.,
2021b) to perform the fits and to derive constraints on the WDs’ masses and
orbits. Then, we infer age constraints on four WDs from their dynamical masses
and effective temperatures ($T_{\rm eff}$).

We structure the paper as follows. Section 2 introduces our target selection
method and summarizes the properties of the selected targets. In Section 3, we
outline the observational data available for each target and present our new
observation for HD 27483 and the corresponding data reduction. We introduce
our orbit-fitting method and discuss our results in Section 4. In Section 5,
we perform a Bayesian age analysis on the WDs and compare our results with
existing age estimates by other methods. We conclude in Section 6.

##  2 Target Selection

We selected our sample from the list of known Sirius-like systems in Holberg
et al. (2013) and the more recently detected WD + late MS star binaries in the
Montreal White Dwarf
Database111https://www.montrealwhitedwarfdatabase.org/home.html (Dufour et
al., 2017). Of more than 100 such systems, we first selected 63 targets that
have astrometric accelerations in the HGCA (Brandt, 2018, 2021). Then, we
excluded 49 targets with neither high-precision RVs nor at least two epochs of
relative astrometry. We choose not to redo the dynamical mass analysis for
seven systems that already have precise dynamical mass measurements: 40 Eri
(Mason et al., 2017), Procyon (Bond et al., 2015), Sirius (Bond et al., 2017),
Stein 2051 (Sahu et al., 2017), GJ 86 Brandt et al. (2019), 12 Psc and HD
159062 (Bowler et al., 2021). In addition to the selection criteria, we
removed HD 149499 as we found that its $\sim 0.45$ $M_{\rm\sun}$ inner
companion (Tokovinin et al., 2019) alone explains the host star’s astrometric
acceleration and RV trend, which leaves almost no information to constrain the
outer WD companion (first discovered by Jordan et al. 1997). Furthermore, we
added HD 118475 B, an unconfirmed WD detected by RV in Kane et al. (2019). We
ended up with seven targets, three of which have both high-precision RVs and
relative astrometry, two of which only have relative astrometry (HD 27483 and
HD 27786), and two of which (HD 118475 and HD 136138) only have RVs. The host
star properties are summarized in Table 1. We collected the spectral types
from Skiff (2014) and parallaxes from Gaia DR3 (Gaia Collaboration et al.,
2022). We adopted the most recent isochronological mass for each host star
except HD 27483 A, for which we used the total dynamical mass of the two F6V
components of the spectroscopic binary, measured by Konacki & Lane (2004).

Table 1: Summary of host star properties.

HD | SpT | $\varpi$ | $M$ | Ref.
---|---|---|---|---
(mas) | ($M_{\sun}$)
19019 | G0V | $31.979\pm 0.029$ | $1.06\pm 0.06$ | 1
27483 | F6V+F6V | $21.094\pm 0.032$ | $2.77\pm 0.26$ | 2
27786 | F4IV-V | $23.79\pm 0.11$ | $1.54\pm 0.08$ | 3
114174 | G3IV | $37.868\pm 0.024$ | $0.97\pm 0.04$ | 4
118475 | F9V | $29.537\pm 0.017$ | $1.16\pm 0.06$ | 3
136138 | G8IIIa | $9.011\pm 0.051$ | $1.84\pm 0.40$ | 5
169889 | G7V | $28.279\pm 0.026$ | $0.98\pm 0.05$ | 3

The spectral types are collected from (Skiff, 2014) and parallaxes from Gaia
DR3 (Gaia Collaboration et al., 2022). The references for stellar masses are:
1. Landstreet & Bagnulo (2020); 2. Konacki & Lane (2004); 3. Kervella et al.
(2019); 4. Rosenthal et al. (2021); 5. Stefanik et al. (2011)

##  3 Observations and Data

We collect high-precision RVs, relative astrometry, and the Hipparcos-Gaia
absolute astrometry to determine each system’s orbital parameters and masses.
Because a WD companion is fainter than a MS companion with the same mass, it
needs to be further from the primary star to be detectable by direct imaging.
However, such a companion is often too widely separated to impose significant
radial or astrometric acceleration on the primary star. Hence, high-precision
RVs and multiple epochs of relative astrometry are not simultaneously
available for more than half of our systems. Below we summarize the data that
we adopted for each system.

###  3.1 Absolute Astrometry

Absolute astrometry shows the proper motion anomaly of stars and is thus
powerful in constraining the orbits of massive, long-period companions. We use
the absolute astrometry in the EDR3 version of the HGCA (Brandt, 2021), which
calibrates Gaia EDR3 and Hipparcos astrometry to reveal the proper motion
difference between Hipparcos, Gaia, and the Hipparcos-Gaia mean motion. This
gives the astrometric acceleration of the host star, which is directly
proportional to the companion’s mass. Table 2 lists the difference between the
Gaia proper motion ($\mu_{\alpha\star,\delta}^{\rm Gaia}$) and the Hipparcos-
Gaia mean motion ($\mu_{\alpha\star,\delta}^{\rm H-G}$) for each target, with
the corresponding significance levels of astrometric accelerations. Five of
our targets have levels $\gg 5\sigma$, meaning that they have accelerated
significantly due to the gravitational pull of the companions.

Table 2: Proper motion anomaly ($\mu_{\alpha\star,\delta}^{\rm Gaia}-\mu_{\alpha\star,\delta}^{\rm H-G}$) of the targets as given by the HGCA. HD | $\Delta\mu_{\alpha\star}(\hbox{mas\,yr${}^{-1}$})$ |  $\Delta\mu_{\delta}$ (mas yr-1) | Sig. level
---|---|---|---
19019 | $0.045\pm 0.049$ | $-0.017\pm 0.040$ | $0.7\sigma$
27483 | $0.79\pm 0.06$ | $2.15\pm 0.04$ | $52\sigma$
27786 | $-0.91\pm 0.21$ | $-19.42\pm 0.13$ | $150\sigma$
114174 | $0.54\pm 0.04$ | $-3.55\pm 0.04$ | $88\sigma$
118475 | $28.99\pm 0.51$ | $18.37\pm 0.57$ | $64\sigma$
136138 | $-0.89\pm 0.30$ | $-0.97\pm 0.32$ | $4.1\sigma$
169889 | $0.51\pm 0.04$ | $-1.77\pm 0.04$ | $48\sigma$

We also checked Gaia DR3 for astrometric non-single stars among our targets.
We found two-body fits (Holl et al., 2022) for HD 118475 and HD 136138, which
significantly corrected the proper motion and parallax measurements from their
one-body fits. Therefore, we decided not to use the uncorrected proper motions
in the HGCA v.EDR3, fitting the two systems only from their RVs and the
corrected parallaxes. Section 4.2 compares our best-fitting orbital parameters
to the Gaia DR3 two-body solution. Checking the consistency between the two
types of solutions helps validate Gaia two-body astrometry. The companion of
HD 19019 is detected in Gaia at a separation of $11.7$ arcsec. We use the Gaia
proper motions of the companion, calibrated to the same reference frame as the
primary using the prescription in Cantat-Gaudin & Brandt (2021), to help
constrain the orbit.

###  3.2 Radial Velocity and Relative Astrometry

Our RVs and relative astrometry come from various literature sources, archival
data, and, in the case of HD 27483, new data. In this section, we summarize
the data available for each system.

####  3.2.1 HD 19019

The HD 19019 system contains a strongly magnetic WD, its spectroscopic
signature detected by Landstreet & Bagnulo (2020) with the ISIS spectrograph
on the William Herschel telescope (Boksenberg, 1985). It has 17 RV
observations between 2002 and 2014 (Butler et al., 2017) by the HIRES
instrument on Keck (Vogt et al., 1994). The system is in the Washington Double
Star Catalog as WDS J03038+0608, first resolved by Gaia DR2 and reported by
Knapp & Nanson (2019) as a common proper motion pair. The WD companion was
also detected in Gaia DR3, which gives a total of two epochs of relative
astrometry with precisions better than 0.1 $\rm{mas}$. The extreme precision
dominates over the precision of the rest of the data, causing the orbital fit
routine (described in Section 4) to ignore the RV and absolute astrometry and
struggle to converge. We inflated the uncertainties of Gaia relative
astrometry by a factor of 5 to account for potential systematics and to avoid
convergence issues.

####  3.2.2 HD 27483

Boehm-Vitense (1993) discovered a WD companion to the Hyades F6V binary from
the IUE spectrum of the system. The companion was then resolved twice by HST
in July 1999 (Barstow et al., 2001) and November 2011 (in WFC3 F218W,
unpublished, HST proposal 12606, PI Martin Barstow). We derive relative
astrometry and photometry from the unpublished data, using the host star’s PSF
as a template to fit the companion’s position and magnitude. We apply a least-
squares routine, assuming uniform $\sigma_{\rm data}$ on the intensity at each
pixel and demanding a reduced $\chi^{2}$ of unity. We fit for three
parameters: an offset in each of two directions, and a contrast. We estimate
the uncertainty of a parameter by fixing all other parameters at their best-
fitting values and finding the $\Delta\chi^{2}=1$ interval of the free
parameter. The results are listed in the first row of Table 3. The two
existing epochs of relative astrometry gave only loose constraints on the
orbital parameters when we applied our orbit fitting procedure in Section 4.

To obtain additional relative astrometry and photometry, we observed HD 27483
on 2021 November 25 UT with the NIRC2 instrument and the natural guide star
adaptive optics system at the Keck II telescope (Wizinowich et al., 2000). In
the $K_{s}$ filter, we took 21 deep exposures with an integration time of
0.18s per coadd, 150 coadds, and 1024 $\times$ 1024 pixels, obtaining high S/N
images for the companion, but with the host star saturated. We then took 11
shallow (0.006s integration time per coadd, 150 coadds, 128 $\times$ 120
pixels) exposures with unsaturated host star PSFs. We used the unsaturated
PSFs in shallow exposures as templates to fit both the host star and the
companion in deep exposures. For the host star, we masked saturated pixels,
fitting the template only to the outer, unsaturated speckles. By comparing the
results from different PSF templates, we confirm that the uncertainties of the
fit were smaller than 0.1 pixels in both directions.

After centering the host star in each frame, we detected the companion by
performing angular differential imaging (Marois et al., 2006, ADI) using the
vip-hci (Gomez Gonzalez et al., 2017) package. Finally, we applied the
negative fake companion method (Wertz et al., 2017, NEGFC) to fit the
companion, subtracting the PSF template from the ADI annulus at different
locations and magnitudes until minimizing the root-mean-square residual of the
subtraction. We used the MCMC routine in vip-hci to perform the fit and
estimate the uncertainties. We applied the distortion correction in Service et
al. (2016), rotating the images by $\theta_{\rm north}=0.262\degree\pm
0.020\degree$ clockwise to align them with the celestial north. The plate
scale we adopted is $9.971\pm 0.004\pm 0.001$ mas pixel-1 (Service et al.,
2016). We added these uncertainties to the uncertainties of the astrometry, in
pixels, from the MCMC posteriors. The left panel of Figure 1 shows the ADI
annulus, with the companion clearly revealed at a separation of $1.08$ arcsec,
and the right panel shows the residual of the NEGFC subtraction. The derived
relative astrometry and photometry are given in the second row of Table 3.

Figure 1: PSF fitting for the NIRC2 observations of HD 27483. The left panel shows the annulus obtained by performing ADI on the data cube. The companion (enlarged in the figure inset) is detected at a separation of $1.08$ arcsec. The right panel shows the residual of subtracting the best-fitting fake companion from the annulus. The host star, represented by a star symbol, is at the center of the annulus. Table 3: New relative astrometry and photometry from this work. Target | Date (UT) | Instrument | Sep (arcsec) | PA (deg) | $\Delta m$ | Filter
---|---|---|---|---|---|---
HD 27483 | 2011 Nov 26 | WFC3 | $1.1880\pm 0.0038$ | $25.32\pm 0.18$ | $3.11\pm 0.17$ | F218W
HD 27483 | 2021 Nov 25 | NIRC2 | $1.0832\pm 0.0016$ | $38.67\pm 0.07$ | $7.79\pm 0.06$ | $K_{s}$
HD 27786 Aa-Ab | 2003 Jan 08 | WFPC2 | $0.4120\pm 0.0070$ | $288.56\pm 0.97$ | $2.67\pm 0.22$ | F170W
Ba-Bb | 2003 Jan 08 | WFPC2 | $0.6200\pm 0.0096$ | $290.18\pm 0.65$ | $1.74\pm 0.22$ | F170W
Aa-B | 2003 Jan 08 | WFPC2 | $4.2924\pm 0.0072$ | $14.31\pm 0.10$ | … | F170W
HD 27786 Aa-Ab | 2003 Dec 14 | WFPC2 | $0.4024\pm 0.0096$ | $280.41\pm 1.00$ | $2.55\pm 0.24$ | F170W
Ba-Bb | 2003 Dec 14 | WFPC2 | $0.6173\pm 0.0059$ | $289.23\pm 0.54$ | $1.63\pm 0.23$ | F170W
Aa-B | 2003 Dec 14 | WFPC2 | $4.2681\pm 0.0071$ | $14.11\pm 0.10$ | … | F170W
HD 27786 Aa-Ab | 2012 Feb 28 | WFC3 | $0.2159\pm 0.0061$ | $191.49\pm 1.62$ | $4.64\pm 0.32$ | F218W
Ba-Bb | 2012 Feb 28 | WFC3 | $0.5272\pm 0.0041$ | $273.22\pm 0.45$ | $2.41\pm 0.17$ | F218W
Aa-B | 2012 Feb 28 | WFC3 | $4.2347\pm 0.0026$ | $12.95\pm 0.04$ | … | F218W

####  3.2.3 HD 27786

HD 27786 (56 Per) is a hierarchical four-body system with astrometric
observations (e.g., Dembowski, 1870; Rabe, 1953; Kallarakal et al., 1969)
dating back to 1847. Landsman et al. (1996) first discovered an inner WD
companion. Barstow et al. (2001) first resolved all four components, finding
that the outer companion is itself two M-dwarfs. From their astrometry and
contrast values, we modeled the mass ratio between the two outer M-dwarfs to
be $1.475\pm 0.042$ and derived that their barycenter is at a separation of
$4.325$ arcsec and a PA of $14.\\!\\!\degree 95$ relative to the host star.
This allows us to treat the outer binary as a single companion and perform a
three-body fit.

Unpublished observations by HST resolved the system twice in the WFPC2 F170W
filter in January 2003 (HST proposal 9334, PI Howard Bond) and December 2003
(HST proposal 9964, PI Howard Bond) and once in the WFC3 F218W filter in
February 2012 (HST proposal 12606, PI Martin Barstow). We derived relative
astrometry and photometry from these unpublished observations using methods
similar to that for the HD 27483 HST observation. The only difference is that
the WD companion is too close to the host star that we cannot use the host
star PSF as a template. Instead, we adopted the observation of HIP 66578 on
2000 January 18 (HST proposal 8496, PI Stefano Casertano) as the PSF template
for WFPC F170W observations and the previously analyzed image of HD 27483 A as
the PSF template for the WFC3 F218W observation. We find that adopting a
different image as the PSF template results in positional differences of up to
7 mas, comparable to our statistical uncertainties. We add this to our error
budget. The results are listed in Table 3.

Other epochs of relative astrometry did not resolve all four components and
only measured the photocenter of the outer binary, which has a filter-
dependent offset from the barycenter. We chose not to include them in the fit.

####  3.2.4 HD 114174

We have 66 high-precision RV measurements between 1997 and 2019, all of which
are from HIRES as part of the California Legacy Survey (Rosenthal et al.,
2021). Relative astrometry of the system consists of 4 observations by Crepp
et al. (2013) in 2011 (first discovery) and 2012, one from Matthews et al.
(2014) in 2013, one from Bacchus et al. (2017) in 2014, and 18 observations by
Gratton et al. (2021) between 2014 and 2019 using the Spectro-Polarimetric
High-contrast Exoplanet REsearch (SPHERE, Beuzit et al., 2019) instrument at
VLT.

####  3.2.5 HD 118475

Kane et al. (2019) acquired RV data of the system using the UCLES high-
resolution spectrograph (Diego et al., 1990) on the Anglo-Australian Telescope
(AAT). The data consist of 11 observations from 2002 to 2014 and reveal a
companion with a minimum mass of $0.445\mbox{$M_{\sun}$}$. Kane et al. (2019)
found that a $0.445\mbox{$M_{\sun}$}$ M-dwarf would be inconsistent by
$3.3\sigma$ with the non-detection in their 880 nm direct imaging
observations, showing evidence that the companion is a compact object.

####  3.2.6 HD 136138

de Medeiros & Mayor (1999) first identified the system as a spectroscopic
binary using RV observations, and the RV orbit was derived by Massarotti et
al. (2008) and Griffin (2009). The IUE spectrum of the system shows that the
companion is a hot ($T_{\rm eff}=30400\pm 780$) WD (Stefanik et al., 2011).
Stefanik et al. (2011) acquired 46 additional measurements of the RV between
2003 and 2009 using the CfA Digital Speedometers (Mayor, 1985; Latham, 1992).
We adopt all of them in our fit. Previous RVs dating back to 1924 are
available in Griffin (2009), but we did not include those data because their
measurement uncertainties are unavailable.

####  3.2.7 HD 169889

We have five epochs of relative astrometry published by Crepp et al. (2018),
who first discovered the WD companion. Four observations were conducted using
the NIRC2 instrument on Keck, and the observation on 2016 June 20 UT came from
the LMIRCam on the LBT (Skrutskie et al., 2010). Eight precise RV measurements
were obtained with HIRES in the California Planet Survey (Howard et al.,
2010), showing a significant RV trend.

##  4 Orbit Fitting

###  4.1 Method

We use the orbit-fitting code orvara (Brandt et al., 2021b) to infer the
orbital parameters of each system. orvara implements a parallel-tempered MCMC
with  ptemcee (Foreman-Mackey et al., 2013; Vousden et al., 2016, 2021). At
each chain step, it uses htof (Brandt et al., 2021a) to model the positions
and proper motions of the host star relative to the system’s barycenter as
they would be seen by the Hipparcos and Gaia intermediate data. orvara then
computes the likelihood of each sample orbit given the relative astrometry,
HGCA absolute astrometry, and RVs.

We run the MCMC with 20 temperatures, 100 walkers per temperature, and at
least 100,000 steps per walker. We discard at least the first 25% and up to
the first 75 % of the chain as the burn-in. Finally, we thin the chain by a
factor of 50. We confirm that the chains have converged by reading orvara’s
diagnostic plots that show the path of each walker in the parameter space,
checking that every walker has reached the same posterior and has fully
sampled the posterior.

We impose Gaussian mass priors on the host stars, adopting masses in Table 1,
but use $1/M$ mass priors for the companions unless otherwise stated in
Section 4.2. A dynamical mass prior is available for HD 27483 A because the
star is itself a spectroscopic binary with Hipparcos astrometry (Konacki &
Lane, 2004). For stars without previous dynamical mass measurements, we use
their most recent isochronological masses as priors. We adopt the Gaia DR3
parallaxes as our parallax priors. We assume the standard geometric prior for
the inclination and log-uniform priors for the semimajor axis and the RV
jitter. We use uniform priors for all other parameters.

###  4.2 Results

Our MCMC analysis improves the orbital parameters and gives unprecedented
dynamical mass constraints for all systems except HD 118475 and HD 136138, for
which we have imprecise minimum masses due to leaving out Hipparcos-Gaia
proper motions. Figure 2 shows the best-fitting relative astrometric orbits of
the systems. Figures 4, 7, 9, 10, 12, 15, 17, and 19 are the corner plots of
the key orbital parameters. Figures 5, 13, 16,18, and 20 gives the fitted RV
orbits and the corresponding residuals. Figures 6, 8, 11, 14, and 21 show the
fits to relative and absolute astrometry and the corresponding residuals.
Tables 4 to 10 list the priors and posteriors of all fitted and derived
parameters. We discuss the details of each system below.

(a)  (b)

(c)  (d)

(e)  (f)  (g)

Figure 2: Relative astrometric orbits of companions. The thick black line
indicates the maximum likelihood orbit. The thin lines, color-coded by the
companion mass, are 50 orbits drawn randomly from the posterior distribution.
The dotted line connects the primary star to the periastron. The dashed line
is the intersection between the orbital plane and the sky plane. The blue dots
show the relative astrometry measurements, with error bars typically smaller
than the size of the symbols. The black circles give the predicted positions
of the companion at several past and future times along the orbit.

####  4.2.1 HD 19019

The dynamical mass constraint on the WD is loose because of the system’s long
orbital period and insignificant astrometric acceleration. We place a uniform
prior instead of a $1/M$ prior on the companion because the $1/M$ prior would
dominate over the weak dynamical mass constraint and give a mass posterior
very close to zero. The companion mass posterior of ${0.32}_{-0.23}^{+0.37}$
$M_{\sun}$ is surprisingly in tension with (2-$\sigma$ away from) the
spectroscopic mass of $1.12\pm 0.15$ $M_{\sun}$ in Landstreet & Bagnulo (2020)
and does not rule out $M_{\rm comp}=0$. The left panel of Figure 5 shows that
the RV data cover only a tiny fraction of the orbit with an insignificant RV
trend. Our best-fitting RV jitter of ${12.2}_{-2.0}^{+2.6}$ m s-1is within the
error bars of the jitter of $13.36\pm 2.86$ m s-1derived by Luhn et al. (2020)
that assumed no companions, suggesting that the companion does not induce a
measurable RV signal. The proper motion of the $G=6.8$ star in Gaia DR4 will
likely be $\sim 4$ times more precise than that in Gaia DR3 (Gaia
Collaboration et al., 2022). The uncertainties will be better than $0.01$ mas
yr-1and comparable to the star’s proper motion change over the baseline of the
Gaia mission, allowing substantial improvement in the system’s orbital
parameters.

####  4.2.2 HD 27483

The host star, HD 27483 A, is itself a tight binary of roughly equal masses
and magnitudes, with a period of 3.06 d (Konacki & Lane, 2004). We treat the
host binary as a single star because the photocenter of its two components
almost coincides with the barycenter.

The WD companion, HD 27483 B, has a long history of spectroscopic
observations. Boehm-Vitense (1993) derived a mass of $\sim 0.6$ $M_{\sun}$
from the WD’s IUE spectrum. Burleigh et al. (1998) re-analyzed the spectrum
and got $M_{\rm WD}\approx 0.94$ $M_{\sun}$. Joyce et al. (2018) analyzed two
HST spectra of the WD and got $0.748\pm 0.072$ $M_{\sun}$ and $0.711\pm 0.137$
$M_{\sun}$, respectively.

With a $1/M$ prior, our dynamical mass posterior of ${0.798}_{-0.041}^{+0.10}$
$M_{\sun}$ agrees reasonably with and has comparable precision to the
spectroscopic masses in Joyce et al. (2018). Compared to Burleigh et al.
(1998), the smaller WD mass implies a longer progenitor lifetime and would
bring the total age of the WD to closer agreement with that of the Hyades
cluster. We perform a detailed analysis of the WD’s age in Section 5.3.2. The
WD’s orbital inclination of ${30\degree}_{-15\degree}^{+13\degree}$ is
reasonably aligned with the inclination of the host binary’s orbit,
$45.1\degree\pm 1.7\degree$ (Konacki & Lane, 2004). Using a uniform companion
mass prior instead of the log-flat prior does not change the MCMC posterior
noticeably.

####  4.2.3 HD 27786

While HD 27786 is a quadruple system, we treat the tight outer M-dwarf binary
as a single unit and perform a three-body fit with the primary and the inner
WD companion. We place a uniform mass prior on the outer binary. The three-
body fit to the system has 16 separate parameters and is therefore difficult
to converge. Hence, we adopt the orbital parameters for the outer companion in
Tokovinin & Kiyaeva (2016) as the starting condition for the MCMC to speed up
convergence.

Our dynamical mass of $M_{\rm WD}=0.443\pm 0.012$ $M_{\sun}$ places the
companion near the low-mass end in the known sample of WDs. Landsman et al.
(1996) fitted possible spectroscopic masses of the WD for a grid of assumed
distances. On their grid, the closest point to the Gaia DR3 distance of
$42.0\pm 0.2$ pc was at 43 pc, for which they derived a mass of $0.435$
$M_{\sun}$. The mass is within 1-$\sigma$ of our dynamical mass.

If we assume a standard evolutionary track, such a small dynamical mass
implies a $<1$ $M_{\sun}$ progenitor, which does not have enough time to
evolve off the MS within the primary star’s age of ${2.11}_{-0.79}^{+1.26}$
Gyr (David & Hillenbrand, 2015). One plausible explanation is that the WD had
a $\sim 2$ $M_{\sun}$ progenitor but underwent faster-than-usual mass loss in
its RGB phase due to interactions with the primary star, such as common
envelope evolution (Althaus et al., 2001). This scenario is not uncommon in
compact binary systems with periods of days (Kilic et al., 2007; Brown et al.,
2011) but is not known to happen for wider WD companions. Future spectroscopic
observations of the system may provide evidence for or against such
interaction.

Because the inner WD companion contributes $>99\%$ of the astrometric
acceleration, it is unrealistic to tightly constrain the mass of the outer
binary companion. The dynamical mass of ${1.36}_{-0.41}^{+0.51}$ $M_{\sun}$
agrees loosely with the total photometric mass of the two companions, $0.84$
$M_{\sun}$ (Tokovinin & Kiyaeva, 2016). Our result of $P=1400\pm 200$ yr and
$a=4.4\pm 0.4$ arcsec is marginally consistent with $P=1945\pm 535$ yr and
$a=7.5\pm 2.7$ arcsec from Izmailov (2019).

####  4.2.4 HD 114174

Literature spectroscopic masses of the WD companion are discrepant, depending
on assumptions on core composition and the specific WD-evolution model
applied. Matthews et al. (2014) reported a mass of $0.54\pm 0.01$ $M_{\sun}$
using the theoretical models in Tremblay et al. (2011) for a pure-H atmosphere
and a C/O core. Bacchus et al. (2017), however, added new photometry and got
$M=0.336\pm 0.014$ $M_{\sun}$ for low-mass He-core evolutionary models
(Althaus et al., 2001) and $M=1.198\pm 0.006$ $M_{\sun}$ for high-mass C/O
core evolutionary models (Fontaine et al., 2001). Recent observations with
SPHERE (Gratton et al., 2021) gave $M=0.75\pm 0.03$ $M_{\sun}$, derived from
pure-H C/O core models in Bergeron et al. (1995). The only previous
measurement of the companion’s dynamical mass was from Crepp et al. (2013),
who inferred a minimum mass of $0.260\pm 0.010$ $M_{\sun}$ from the host
star’s RV trend.

Our mass of $M=0.591\pm 0.011$ $M_{\sun}$ is the first precise dynamical mass
for the WD companion and agrees with the dynamical mass lower limit from Crepp
et al. (2013). Comparing to the WD’s spectroscopic masses, it is the closest
($\approx 3\sigma$) to the mass of $0.54\pm 0.01$ $M_{\sun}$ by Matthews et
al. (2014), suggesting that the WD has a C/O core. Our orbital period of
$108.2_{-4.8}^{+5.2}$ yr and inclination of $88.86\degree\pm 0.21\degree$
agree reasonably with $P\approx 124$ yr and $i=88.11\degree\pm 0.11\degree$
published by Gratton et al. (2021). Our semimajor axis of
$26.25_{-0.66}^{+0.71}$ AU and eccentricity of $0.690\pm 0.017$ are slightly
smaller than $a=30.11\pm 0.03$ AU and $e=0.89$ by Gratton et al. (2021)
because those authors assumed a companion mass higher than the dynamical mass
we found. Other orbital parameters of the system are summarized in Table 7.

####  4.2.5 HD 118475

Because we fitted only to the RV data, there are no informative constraints on
the orbital inclination. Instead of a precise dynamical mass, we obtain a
minimum mass of $M_{\rm sec}\sin{i}={0.461}_{-0.015}^{+0.017}$ $M_{\sun}$,
agreeing with $M_{\rm sec}\sin{i}=0.445\pm 0.025$ from Kane et al. (2019),
using the same RVs but a different fitting procedure. Other parameters
($a={3.76}_{-0.06}^{+0.07}$ AU, $P=2070.47\pm 0.16$ d, $e=0.128\pm 0.001$,
$\omega=237.8\degree\pm 0.2\degree$) are nearly identical to the values in
Kane et al. (2019) ($a=3.69\pm 0.11$ AU, $P={2070.47}_{-0.2}^{+0.19}$ d,
$e=0.128\pm 0.001$, $\omega=237.7\degree\pm 0.3\degree$). The complete set of
orbital parameters is listed in Table 8.

In Gaia DR3, HD 118475 has a two-body solution, given in terms of $P$, $e$,
time of periastron $T_{0}$, and the Thiele-Innes constants $A$, $B$, $F$, $G$
in units of mas (multiplied by the semi-major axis of the photocenter motion
$a_{0}$). We first convert the Thiele-Innes constants into physical orbit
parameters with the following equations:

$\displaystyle a_{0}$ $\displaystyle=\sqrt{p+\sqrt{p^{2}-q^{2}}}$ (1)
$\displaystyle i$ $\displaystyle=\cos^{-1}{\frac{q}{a_{0}^{2}}}$ (2)
$\displaystyle\omega$ $\displaystyle=\frac{r+s}{2}$ (3) $\displaystyle\Omega$
$\displaystyle=\frac{r-s}{2}$ (4)

with $p$, $q$, $r$, $s$ defined to be

$\displaystyle p$
$\displaystyle\equiv\frac{1}{2}\left(A^{2}+B^{2}+F^{2}+G^{2}\right)$ (5)
$\displaystyle q$ $\displaystyle\equiv AG-BF$ (6) $\displaystyle r$
$\displaystyle\equiv\tan^{-1}\left(\frac{F-B}{G+A}\right)$ (7) $\displaystyle
s$ $\displaystyle\equiv\tan^{-1}\left(\frac{F+B}{G-A}\right)$ (8)

Because the primary star is much brighter than the companion, the system’s
photocenter almost coincides with the primary. Therefore, the semi-major axis
of the photocenter motion is approximately that of the primary motion, which
is related to the total semi-major axis $a$ by:

$a=\frac{M_{\rm tot}}{M_{\rm sec}}a_{0}$ (9)

We also have Kepler’s Third Law:

$P^{2}=\frac{a^{3}}{M_{\rm tot}}$ (10)

where $P$ is in years, $a$ is in AU, and $M_{\rm tot}=M_{\rm pri}+M_{\rm sec}$
is in solar masses.

Combining Equations (9) and (10), we get

$M_{\rm sec}^{3}-(M_{\rm pri}+M_{\rm sec})^{2}\frac{a_{0}^{3}}{P^{2}}=0$ (11)

from which we solve for the companion mass $M_{\rm sec}$, given a primary mass
of $M_{\rm pri}=1.16\pm 0.06$ $M_{\sun}$ (Kervella et al., 2019).

The last two columns of Table 8 compare the orbital elements from our RV fit
to those from the Gaia DR3 astrometric solution. Both solutions tightly
constrain $M_{\rm sec}$, $a$, $P$, $e$, and the argument of periastron
$\omega$, and they agree remarkably well. Once the Gaia IAD are published, a
joint analysis of Hipparcos IAD, Gaia IAD, and RV will place tight constraints
on all orbital parameters. Our analysis is an example of verifying a Gaia
orbital solution against orbital parameters derived by other means. The same
procedure could be applied to a large sample of binary systems with known
orbits to validate the Gaia two-body solutions.

Our $M_{\rm sec}\sin{i}$ of ${0.461}_{-0.015}^{+0.017}$ $M_{\sun}$ and the
Gaia DR3 inclination of ${53.\\!\\!\degree 1}_{-5.\\!\\!\degree
5}^{+4.\\!\\!\degree 8}$ give a dynamical mass of ${0.580}_{-0.039}^{+0.052}$
$M_{\sun}$. The precise dynamical mass is much greater than the minimum mass
of $\approx 0.445$ $M_{\sun}$ in Kane et al. (2019), implying that if the
companion were a MS star, it would have a magnitude more inconsistent with the
non-detection in the direct imaging observations by Kane et al. (2019) than
previously derived. Using orvara, we confirm that the separation between the
primary star and the companion was $\approx 0.09$ arcsec at the observation
epoch, so we refer to Figure 3 of Kane et al. (2019) as the non-detection
significance curve. According to MIST evolutionary tracks (Choi et al., 2016),
a 0.58 $M_{\sun}$ MS star would be only $\approx 3.87$ mag fainter, near 880
nm, than the 1.16 $M_{\sun}$ primary at the current age ($\sim 4.1$ Gyr) and
metallicity ($\rm{[Fe/H]}=0.10$) of the system (Valenti & Fischer, 2005).
Hence, we can rule out a MS companion at a significance level of more than
$8\sigma$. The dynamical mass also eliminates the possibility that the
companion is a neutron star or a black hole. Therefore, we conclude that HD
118475 B is a WD.

####  4.2.6 HD 136138

Stefanik et al. (2011) inferred a WD mass of $M_{\rm WD}=0.79\pm 0.09$
$M_{\sun}$ from the IUE spectrum of the system (Landsman et al., 1996), using
C/O-core cooling models in Wood (1995). The authors also obtained a dynamical
mass of $0.59\pm 0.12$ $M_{\sun}$ from a joint fit of Hipparcos IAD and RV.
Analyzing only the RVs, we get a minimum dynamical mass of
$M\sin{i}={0.389}_{-0.080}^{+0.081}$ $M_{\sun}$, agreeing well with the
dynamical mass and $i=42.9\degree\pm 6.7\degree$ in Stefanik et al. (2011)
($M\sin{i}={0.395}^{+0.099}_{-0.091}$ $M_{\sun}$) but only marginally with the
spectroscopic mass. With a semimajor axis of only $a={1.65}_{-0.12}^{+0.10}$
AU, it is possible that the progenitor of the WD once transferred some of its
mass to the giant primary. The spectroscopic analysis did not consider the
effects of such binary interactions on the WD mass-radius relation and could
overestimate the mass. Our period of $509.6\pm 1.2$ d, eccentricity of
$e={0.336}_{-0.014}^{+0.015}$, and RV semi-amplitude of
${6.22}_{-0.11}^{+0.14}$ km s-1all agree reasonably with $P=506.45\pm 0.18$ d,
$e=0.3353\pm 0.0056$, and $K=6.340\pm 0.044$ km s-1in Stefanik et al. (2011).

HD 136138 also has a two-body solution in Gaia DR3. We derive the
corresponding orbital elements with Equations (1) to (11) and list the results
in the last column of Table 9. The Gaia solution gives a semi-major axis and
$M\sin{i}$ almost identical to our RV solution, but it has a slightly larger
($1.7\sigma$) eccentricity, shorter (by $1.7\sigma$) period, and larger (by
$3.2\sigma$) argument of periastron, all in moderate tension with our values.

####  4.2.7 HD 169889

Our dynamical mass of ${0.526}_{-0.037}^{+0.039}$ $M_{\sun}$ is the first
precise mass of the WD, consistent with the dynamical mass lower limit of
$0.369\pm 0.010$ $M_{\sun}$ from Crepp et al. (2018). The eccentricity of
$e={0.896}_{-0.088}^{+0.064}$ is the largest in our sample. A large
eccentricity is not surprising because the eccentricity distribution of wide
MS binaries with separations smaller than 100 AU is nearly uniform (Hwang et
al., 2022), and the isotropic and adiabatic mass loss of a star during the AGB
phase does not modify the eccentricity of its orbit (Dosopoulou & Kalogera,
2016). Our fit to relative separation has a $\chi^{2}$ of 32 for only five
observations, suggesting that the relative astrometry in Crepp et al. (2018)
may have underestimated uncertainties. Inflating the uncertainties to get a
reduced $\chi^{2}$ of 1 does not affect the results significantly. The WD only
has two broadband photometry measurements in the $H$-band and the
$L^{\prime}$-band, which are insufficient to precisely measure its $T_{\rm
eff}$, $\log(g)$, or photometric mass (Crepp et al., 2018). Additional
photometry or spectroscopy will be necessary to characterize the fundamental
properties of the WD.

##  5 WD Age Inference

A dynamical mass comes directly from the solution to the Kepler problem,
independent of stellar models and their theoretical uncertainties. Hence, it
is a reliable starting point to constrain the fundamental parameters of a WD
or test WD evolutionary models. A Sirius-like binary system is an especially
ideal testing ground because the age of the system is often known by measuring
the activity or the rotation of the MS star. The dynamical mass of the WD,
closely related to the WD’s cooling age and progenitor lifetime, allows for an
independent measurement of the system’s age. In this section, we outline our
methods to infer the MS lifetimes and the cooling ages of the WDs and discuss
our results for each system.

###  5.1 MS Lifetime

The mass of a WD implies the mass of its MS progenitor, and the progenitor
mass determines the progenitor’s lifetime before evolving into a WD. We adopt
the empirical IFMR in El-Badry et al. (2018) to compute the likelihood of a
WD’s progenitor mass from its dynamical mass posterior in the MCMC chain.
Then, we multiply the distribution with the initial-mass function (IMF) in
Chabrier (2003) (corrected for binaries) to obtain the posterior distribution
of the initial mass. Finally, we perform cubic spline interpolation on MIST
evolutionary tracks (Choi et al., 2016) to find the MS lifetime distribution
from the initial mass posterior, assuming that the WD companion has the same
metallicity as the MS primary. We remove any probability density beyond the
age of the universe (Planck Collaboration et al., 2020, 13.8 Gyr) and re-
normalize the distribution.

Our analysis is subject to potential systematics in our choice of IFMR, IMF,
and stellar evolutionary model. We repeat the above procedures with the
Cummings et al. (2018) IFMR, the Kroupa (2001) IMF, and PARSEC evolutionary
tracks (Bressan et al., 2012), confirming that our conclusions in Section 5.3
do not change significantly.

###  5.2 Cooling Age

The cooling age of a WD relates primarily to the starting condition of cooling
(decided by its mass) and by how much it has cooled (its present effective
temperature). A dynamical mass alone does not constrain the cooling age, but
when paired with a $T_{\rm eff}$ measurement or multi-band photometry, it
tightens existing constraints. We perform multi-dimensional linear
interpolation on the latest generation of Montréal cooling
sequences222http://www.astro.umontreal.ca/ bergeron/CoolingModels (Bédard et
al., 2020) to compute the cooling age from dynamical mass and other
parameters. The synthetic photometry in the Montréal cooling sequences is
computed using prescriptions in Holberg & Bergeron (2006) and relies on models
of Blouin et al. (2018), Tremblay et al. (2011), and Bédard et al. (2020) for
low, intermediate, and high-temperature DA WDs, respectively. Our
interpolation method is similar to that in Kiman et al. (2022).

###  5.3 Results

We place informative constraints on the ages of all WDs in this work except HD
27786 Ab, HD 118475 B, and HD 136138 B. The unusually low mass of HD 27786 Ab
is smaller than the applicable range of the El-Badry et al. (2018) IFMR and
the Cummings et al. (2018) IFMR and is not explainable by standard single-star
evolutionary tracks. One needs to consider binary interaction to infer the age
of the WD. HD 118475 B and HD 136138 B only have dynamical mass lower limits.
The corresponding upper limits on age are over the age of the universe. We
present our results for the rest of the systems below.

Figure 3: Comparison of the WD age posteriors (plotted as histograms) derived
from our dynamical masses to the $1\sigma$ intervals (represented by shaded
regions) of previous age measurements. In each panel, the vertical solid black
line marks the 50th percentile of the distribution, while the two dashed back
lines correspond to the 16th and the 84th percentiles. Left panel (HD 27483):
our result does not resolve the age tension between Brandt & Huang (2015)
(gold region, the age of the Hyades cluster derived from rotating isochrones)
and Martín et al. (2018) (pink region, the lithium depletion boundary age of
the Hyades cluster). Middle panel (HD 114174): two discrepant $T_{\rm eff}$
give different ages. The age derived from $\mbox{$T_{\rm eff}$}=5890\pm 270$ K
(solid blue histogram, Gratton et al., 2021) agrees nicely with the primary’s
gyrochronological age (pink region, Crepp et al., 2013), but the result from
$\mbox{$T_{\rm eff}$}=4260\pm 360$ K (dashed green histogram, Matthews et al.,
2014) agrees better with the primary’s isochronal age (orange region, Tucci
Maia et al., 2016). Right panel (HD 169889): our age constraint is loose but
matches well with the primary’s gyrochronological age (orange region, Mamajek
& Hillenbrand, 2008).

####  5.3.1 HD 19019

The precision of our dynamical mass is insufficient to place good constraints
on the MS lifetime of HD 19019 B. A considerable fraction of the mass
posterior is below the applicable range of any IFMRs. If we naively apply the
El-Badry et al. (2018) IFMR, we would obtain a progenitor mass of $M_{\rm
pro}={2.8}_{-1.3}^{+2.3}$ $M_{\sun}$ and a maximum likelihood MS lifetime of
$\sim 0.5$ Gyr (assuming $[\rm{Fe}/\rm{H}]=-0.15$ as in Arentsen et al.,
2019), but we cannot rule out any lifetime within the age of the universe. If
we instead adopt the spectroscopic mass of $1.12\pm 0.15$ $M_{\sun}$
(Landstreet & Bagnulo, 2020), we would get a progenitor mass of $M_{\rm
pro}={6.3}_{-1.7}^{+1.9}$ $M_{\sun}$ and a MS lifetime of ${110}_{-60}^{+320}$
Myr.

Because the spectroscopic analysis in Landstreet & Bagnulo (2020) gives a
relatively hot $T_{\rm eff}$ of $18200\pm 3000$ K, but our dynamical analysis
favors small masses, we get a young cooling age of ${51}_{-44}^{+157}$ Myr. If
adopting $\mbox{$\log(g)$}=8.85\pm 0.15$ in Landstreet & Bagnulo (2020)
instead of our dynamical mass, we obtain a much older cooling age of
${530}_{-220}^{+380}$ Myr. Given that the age of the MS primary is $\sim 3.0$
Gyr (Landstreet & Bagnulo, 2020), if the WD had a mass of $\sim 0.54$
$M_{\sun}$, it would have a cooling age of $\sim 0.1$ Myr and a MS lifetime of
$\sim 2.9$ Myr for a $\sim 1.4$ $M_{\sun}$ progenitor, giving a consistent
total age. This mass is within the 1-$\sigma$ interval of our dynamical mass.

####  5.3.2 HD 27483

HD 27483 is a member of the Hyades cluster. For the cluster, different age
determination methods give values in moderate tension.

Perryman et al. (1998) determine an age of $625\pm 50$ Myr from non-rotating
stellar models. Brandt & Huang (2015) infer an age of $750\pm 100$ Myr from
rotational stellar models from Ekström et al. (2012). Gossage et al. (2018)
infer a somewhat younger age of $\approx$680 Myr from models with a different
treatment of stellar rotation (Paxton et al., 2013). Martín et al. (2018)
report a younger age of $650\pm 70$ Myr by determining the lithium depletion
boundary of two members. De Gennaro et al. (2009) obtained an age of $648\pm
45$ Myr by fitting the WD portion of the color-magnitude diagram to stellar
evolution models. Here, we present an independent age estimate for the Hyades
by inferring the age of HD 27483 B.

The ${0.798}_{-0.041}^{+0.10}$ $M_{\sun}$ WD has a ${3.59}_{-0.78}^{+1.43}$
$M_{\sun}$ progenitor, which lived a lifetime of ${350}_{-180}^{+570}$ Myr,
assuming a metallicity of $[\rm{Fe}/\rm{H}]=0.04$ (Bochanski et al., 2018).
The imprecision originates mainly from the uncertainty in the empirical IFMR.
We neglect uncertainties in the MS lifetime from, e.g., the effects of stellar
rotation (Brandt & Huang, 2015); these would be $\lesssim$20% of the age.
Using a theoretical IFMR, like the MIST IFMR described in Choi et al. (2016),
would reduce statistical uncertainties but overlook the systematics in
theoretical modeling.

The photometry of a WD constrains its effective temperature, mass and hence
the cooling age. To find the magnitudes of HD 27483 B from the $\Delta m$
measurements in Table 3, we need precise magnitudes of the host star in the
two filters. We adopt $K_{\rm pri}=5.062\pm 0.018$ from Cutri et al. (2003) as
the $K$-band apparent magnitude of the primary. However, we are unaware of any
measurement of the primary’s F218W magnitude. Hence, we apply the species
package (Stolker et al., 2020) to compute the magnitude from synthetic
photometry. We modeled the spectrum of the primary with the BT-NextGen model
(Allard et al., 2012), adopting $\mbox{$T_{\rm eff}$}=6549\pm 80$ K,
$\mbox{$\log(g)$}=4.04$ from Casagrande et al. (2011) and
$R={2.02}^{+0.10}_{-0.11}$ $R_{\sun}$ and $\rm{[Fe/H]}={0.04}^{+0.15}_{-0.13}$
from Bochanski et al. (2018). Integrating the model spectrum with the F218W
filter, we obtain an F218W absolute magnitude of ${5.29}_{-0.24}^{+0.26}$ for
HD 27483 A. Finally, we add the host star magnitudes to the contrast values in
Table 3 and get $M_{K}=11.47\pm 0.06$ and $M_{\rm F218W}=8.40\pm 0.30$ for the
WD companion.

Comparing the WD absolute magnitudes to the Montréal cooling sequences, we get
a cooling age of ${61}_{-17}^{+21}$ Myr, an effective temperature of $21000\pm
3000$ K, and a photometric mass of ${0.669}_{-0.066}^{+0.079}$ $M_{\sun}$. Our
$T_{\rm eff}$ agrees well with the spectroscopic $T_{\rm eff}$ of $20790\pm
187$ K (Joyce et al., 2018) but has a much larger uncertainty. The photometric
mass is in mild tension with ($1.4\sigma$ below) our dynamical mass. The
observed magnitudes depend mainly on the WD’s $T_{\rm eff}$ and size, while
the size depends on the WD’s mass via the mass-radius relation (e.g., Joyce et
al., 2018; Romero et al., 2019; Chandra et al., 2020). The Montréal cooling
sequences match the photometry of HD 27483 B at a radius larger than that
expected for a 0.8 $M_{\odot}$ WD, suggesting a lower mass. The lower mass
would, in turn, imply a less massive progenitor and a longer MS lifetime (and
older Hyades age). We note that with $\mbox{$T_{\rm eff}$}\approx 21000$ K,
the WD has its emission peak at about 1400 Å, much bluer than both photometric
bands. Additional photometry at shorter wavelengths and a more precise
dynamical mass will hopefully resolve the discrepancy.

Combining all of our constraints, we perform a joint analysis of photometry
and dynamical mass, treating the dynamical mass as a prior when fitting the
cooling sequences to the photometry. We get a mass posterior of
${0.763}_{-0.026}^{+0.034}$ $M_{\sun}$ and a cooling age of ${45}_{-9}^{+12}$
Myr. The cooling time is slightly smaller than that from photometry alone due
to the mass tension.

Adding the cooling age from the joint analysis to the MS lifetime, we get a
total age of ${400}_{-180}^{+570}$ Myr. The left panel of Figure 3 compares
our age posterior (blue histogram) to the $1\sigma$ intervals of previous age
estimates (light orange region for Brandt & Huang (2015) and light pink region
for Martín et al. (2018)). Our current analysis does not have the precision to
resolve the age tension, mainly due to the uncertainties in the semi-empirical
IFMR. Reducing the dynamical mass uncertainty would improve the cooling age
but have negligible effects on the precision of the MS lifetime. On the
contrary, if one could reduce the uncertainty of the IFMR by a factor of 4,
they would constrain the total age to 10% ($\approx 70$ Myr), comparable to
the precision of other age estimates of Hyades.

####  5.3.3 HD 114174

The El-Badry et al. (2018) IFMR gives a progenitor mass of
${1.90}_{-0.27}^{+0.30}$ $M_{\sun}$ for the $0.591\pm 0.011$ $M_{\sun}$ WD.
The Cummings et al. (2018) IFMR suggests a smaller progenitor of $1.28\pm
0.47$ $M_{\sun}$. We adopt the former mass because El-Badry et al. (2018) has
more low-mass ($<0.6$ $M_{\sun}$) WDs in their calibration sample. Comparing
the progenitor mass to MIST tracks, we get a MS lifetime of
${1.46}_{-0.48}^{+0.84}$ Gyr (for a metallicity of $[\rm{Fe}/\rm{H}]=0.056$ as
in Casali et al., 2020).

Previous works on the WD reported disagreeing effective temperatures leading
to discrepant cooling ages. Our dynamical mass and $\mbox{$T_{\rm
eff}$}=4260\pm 360$ K from Matthews et al. (2014) combing spectroscopy and
photometry suggests a cooling age of $t_{\rm cool}={8.3}_{-2.1}^{+1.9}$ Gyr,
but assuming $\mbox{$T_{\rm eff}$}=5890\pm 270$ K from Gratton et al. (2021)
using the SPHERE IFS instead gives $t_{\rm cool}={2.40}_{-0.41}^{+0.52}$ Gyr.
Other $T_{\rm eff}$ measurements are too imprecise to constrain the cooling
age. Summing the MS lifetime and the cooling age, the former $T_{\rm eff}$
corresponds to a total age of ${8.3}_{-2.1}^{+1.9}$ Gyr, while the latter
gives a much younger age of ${3.93}_{-0.74}^{+0.98}$ Gyr. The middle panel of
Figure 3 compares the total age posterior of the WD, for different $T_{\rm
eff}$, to the $1\sigma$ intervals of existing age measurements for the
primary. The total age from our dynamical mass and the Gratton et al. (2021)
$T_{\rm eff}$ agrees remarkably with the primary’s gyrochronological age by
Crepp et al. (2013) but only marginally with the isochronal age from Tucci
Maia et al. (2016). On the contrary, the much older age from the Matthews et
al. (2014) $T_{\rm eff}$ is consistent with Tucci Maia et al. (2016) but in
moderate tension with Crepp et al. (2013). Additional spectroscopy will be
necessary to resolve the $T_{\rm eff}$ discrepancy and confirm the age of the
WD.

####  5.3.4 HD 169889

The relatively light (${0.526}_{-0.037}^{+0.039}$ $M_{\sun}$) WD has a
${1.33}_{-0.26}^{+0.38}$ $M_{\sun}$ progenitor, with a MS lifetime of
${4.50}_{-2.38}^{+3.90}$ Gyr (assuming $[\rm{Fe}/\rm{H}]=-0.14$ as in Brewer
et al., 2016). Due to the uncertainty of the IFMR, our progenitor mass is not
precise enough to rule out low-mass progenitors that lived up to the age of
the universe.

Crepp et al. (2018) could not determine the WD’s precise $T_{\rm eff}$ from
their $H$-band and $L^{\prime}$-band photometry, proposing two different
temperatures consistent with observation. The cooler temperature,
$\mbox{$T_{\rm eff}$}\approx 2150$ K, plus our dynamical mass, would bring the
cooling age to $\approx 12$ Gyr, much longer than the primary’s
gyrochronological age of ${5.2}_{-1.5}^{+1.3}$ Gyr (Mamajek & Hillenbrand,
2008). The maximum likelihood total age would be even beyond the universe’s
age. A WD this cool and old has never been detected. Hence, we adopt the
hotter temperature, $\mbox{$T_{\rm eff}$}\approx 10000$ K, assuming a
conservative uncertainty of 1000 K. We find a young cooling age of
${0.48}_{-0.15}^{+0.20}$ Gyr. The WD’s total age is, thus,
${5.0}_{-2.4}^{+3.9}$ Gyr, in good agreement with the primary’s
gyrochronological age. The right panel of Figure 3 displays our age posterior
on top of a shaded region representing the $1\sigma$ interval of the
gyrochronological age. We cut the tail of the distribution at the universe’s
age. The age constraint could be considerably improved if one obtains a more
precise IFMR in the low-mass range, ruling out light progenitors with
extremely long lifetimes.

##  6 Discussion and Conclusions

In this paper, we have derived the masses and orbits for six confirmed and one
candidate Sirius-like systems from a joint analysis of high-precision RVs,
relative astrometry, and Hipparcos-Gaia proper motion anomaly. From the
dynamical masses of the WDs, we have constrained the ages of four systems and
compared them to age estimates by other means. We summarize our main results
below:

  1. 1.

For HD 27483 B, our analysis of unpublished Hubble data and new observations
with NIRC2 introduce two additional epochs of relative astrometry and
photometry, allowing us to obtain the first-ever dynamical mass of the WD
(${0.798}_{-0.041}^{+0.10}$ $M_{\sun}$). The $\sim$10%-precision mass leads to
a total age of ${400}_{-180}^{+570}$ Myr, consistent with previous age
estimates of the Hyades cluster but not sufficiently precise to resolve the
age discrepancy between various methods. Additional relative astrometry would
improve the mass precision considerably, but the MS lifetime precision is
limited by the calibration uncertainty of semi-empirical IFMRs. Constraints on
the cooling age could be improved by additional photometry or adopting a
$T_{\rm eff}$ from high-resolution spectroscopy.

  2. 2.

Our three-body fit on HD 27786 results in an unusually low mass ($0.443\pm
0.012$ $M_{\sun}$) for the WD HD 27786 Ab, suggesting the possibility of
common envelope evolution with the primary star during its progenitor’s RGB
phase.

  3. 3.

We obtain the first precise dynamical masses of HD 114174 B ($0.591\pm 0.011$
$M_{\sun}$) and HD 169889 B (${0.526}_{-0.037}^{+0.039}$ $M_{\sun}$). The ages
of both WDs depend strongly on the assumed $T_{\rm eff}$, showing the need for
high-resolution spectroscopy to resolve the $T_{\rm eff}$ uncertainty and
improve the age constraints.

  4. 4.

For HD 118475, the orbital elements we derived from RVs agree remarkably with
the Gaia DR3 two-body solution from intermediate astrometry. Whereas for HD
136138, the best-fitting eccentricity and period from the RVs are in moderate
tension with that from Gaia. Such a comparison, applicable to a large sample
of systems, provides validations of Gaia’s two-body solutions.

  5. 5.

With $M\sin{i}={0.461}_{-0.015}^{+0.017}$ $M_{\sun}$ and a Gaia DR3
inclination of ${53.\\!\\!\degree 1}_{-5.\\!\\!\degree 5}^{+4.\\!\\!\degree
8}$, HD 118475 B has a dynamical mass of ${0.580}_{-0.039}^{+0.052}$
$M_{\sun}$. By comparing the magnitude of a $0.58$ $M_{\sun}$ MS star to the
non-detection significance curve in Kane et al. (2019), we rule out the
hypothesis that the companion is a MS star at $>8\sigma$ level, confirming
that it is a WD.

We note that for WDs with better than $\sim 15\%$ dynamical masses, such as HD
27483 B, HD 114174 B, and HD 169889 B, the dominant source of age uncertainty
(assuming a good $T_{\rm eff}$) is the scatter in semi-empirical IFMRs.
Theoretically modeled IFMRs, in comparison, are free from calibration
uncertainties but replace them with systematic uncertainties of specific mass-
loss models (e.g., Reimers, 1975; Bloecker, 1995). Hence, we choose not to use
them to disentangle our analysis from systematics. From another perspective,
our dynamical masses are themselves measurements of the IFMR. With a dynamical
final mass, one can infer an initial mass, independent of mass-loss models,
from the difference between the MS primary’s age and the WD’s cooling age.
With help from precise astrometric solutions in Gaia DR3 and later releases,
repeating our analysis on additional Sirius-like systems would increase the
sample size for a future re-calibration of the WD IFMR.

## ACKNOWLEDGEMENTS

This research has made use of the SIMBAD database (Wenger et al., 2000),
operated at CDS, Strasbourg, France. This research has made use of the SVO
Filter Profile Service (Rodrigo & Solano, 2020) supported by the Spanish
MINECO through grant AYA2017-84089. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data Availability

The observations used in this article are available in the Keck Observatory
Archive (https://koa.ipac.caltech.edu), the Hubble Legacy Archive
(https://hla.stsci.edu), and the references listed in Section 3.

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##  Appendix A Figures and tables of fit results

Figure 4: Best-fitting orbital parameters for the HD 19019 system and their
posterior distributions from orvara. The selected parameters include the
primary mass (in solar masses) $M_{\rm pri}$, the secondary mass (in Jupiter
masses) $M_{\rm sec}$, the semi-major axis (in A.U.) $a$, the eccentricity
$e$, and the inclination (in degrees) $i$. The contours on the 2-d joint
posterior distributions give the 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$
levels. The vertical dashed lines on the 1-d marginalized distributions
indicate the 16% and 84% quantiles.

(a)  (b)

Figure 5: Left panel: RV orbits of HD 19019 over the entire best-fitting
orbital period. Right panel: RV orbits of HD 19019 over the observational time
frame (top) and the residuals after subtracting the best-fitting orbit
(bottom). For both panels, an RV of zero corresponds to the barycentric
velocity of the system. The thick black line indicates the maximum likelihood
RV orbit. The thin lines, color-coded by the companion mass, are 50 other
orbits drawn randomly from the posterior distribution. The colored dots are
the RV measurements, with a different color indicating a separate telescope.
Most color bars are too small to be visible on the left panel.

(a)  (b)

(c)

Figure 6: Relative (top panel) and absolute (bottom panel) astrometry for the
HD 19019 system. The observations are shown as blue dots with error bars. The
thick black line is the maximum likelihood orbit, and the thin lines, color-
coded by the companion mass, are 50 other orbits drawn randomly from the
posterior distribution. The bottom part of each panel shows the residuals
after subtracting the maximum likelihood orbit from the measurements.

Parameter | Prior Distribution | Posteriors $\pm 1\sigma$
---|---|---
Fitted parameters
Primary mass ($M_{\odot}$) |  $1.06\pm 0.06$ | ${1.061}_{-0.062}^{+0.057}$
Companion mass ($M_{\odot}$) | Uniform | ${0.32}_{-0.23}^{+0.37}$
Parallax (mas) |  $31.979\pm 0.029$ | $31.979\pm 0.028$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) | ${276}_{-47}^{+64}$
Inclination $i$ ($\degree$) |  $\sin{i}$ | ${52.8}_{-15}^{+9.0}$
$\sqrt{e}\sin{\omega}$ | Uniform | ${0.18}_{-0.43}^{+0.42}$
$\sqrt{e}\cos{\omega}$ | Uniform | ${0.53}_{-0.26}^{+0.24}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform | ${207}_{-11}^{+22}$
PA of the ascending node $\Omega$ ($\degree$) | Uniform | ${23}_{-14}^{+326}$
RV jitter $\sigma$ (m s-1) | Log-flat over [0,1000 m s-1] | ${12.2}_{-2.0}^{+2.6}$
Derived parameters
Orbital period (yr) | … | ${3888}_{-956}^{+1304}$
Semimajor axis (mas) | … | ${8830}_{-1518}^{+2031}$
Eccentricity $e$ | … | ${0.54}_{-0.27}^{+0.18}$
Argument of periastron $\omega$ $(\degree)$ | … | ${59}_{-40}^{+281}$
Time of periastron $T_{0}$ (JD) | … | ${3033863}_{-164261}^{+373450}$
Mass ratio | … | ${0.30}_{-0.22}^{+0.35}$
Table 4: Posteriors of the HD 19019 system. Parameter | Prior Distribution | Posteriors $\pm 1\sigma$
---|---|---
Fitted parameters
Primary mass ($M_{\odot}$) |  $2.77\pm 0.26$ | ${2.86}_{-0.22}^{+0.25}$
Companion mass ($M_{\odot}$) |  $1/M$ (log-flat) | ${0.798}_{-0.041}^{+0.10}$
Parallax (mas) |  $21.094\pm 0.032$ | $21.094\pm 0.025$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) | ${49.8}_{-5.4}^{+12}$
Inclination $i$ ($\degree$) |  $\sin{i}$ | ${30}_{-15}^{+13}$
$\sqrt{e}\sin{\omega}$ | Uniform | ${0.07}_{-0.49}^{+0.41}$
$\sqrt{e}\cos{\omega}$ | Uniform | ${0.01}_{-0.47}^{+0.47}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform | ${156}_{-126}^{+80}$
PA of the ascending node $\Omega$ ($\degree$) | Uniform | ${174}_{-159}^{+121}$
Derived parameters
Orbital period (yr) | … | ${184}_{-30}^{+65}$
Semimajor axis (mas) | … | ${1050}_{-115}^{+250}$
Eccentricity $e$ | … | ${0.342}_{-0.19}^{+0.094}$
Argument of periastron $\omega$ $(\degree)$ | … | ${163}_{-111}^{+141}$
Time of periastron $T_{0}$ (JD) | … | ${2474412}_{-2469}^{+4137}$
Mass ratio | … | ${0.284}_{-0.024}^{+0.033}$
Table 5: Posteriors of the HD 27483 system.

Figure 7: Best-fitting orbital parameters for the HD 27483 system. Other
descriptions are identical to those for Figure 4.

(a)  (b)

(c)

Figure 8: Relative (top panel) and absolute (bottom panel) astrometry for the
HD 27483 system. Other descriptions are identical to those for Figure 6.

Figure 9: Best-fitting orbital parameters for HD 27786 Ab. Other descriptions
are identical to those for Figure 4.

Figure 10: Best-fitting orbital parameters for HD 27786 B. Other descriptions
are identical to those for Figure 4.

(a)  (b)

(c)  (d)

(e)

Figure 11: Relative (top and middle panels) and absolute (bottom panel)
astrometry for the HD 27786 system. Other descriptions are identical to those
for Figure 6.

Parameter | Prior Distribution | Posteriors $\pm 1\sigma$ | Posteriors $\pm 1\sigma$
---|---|---|---
(Both companions) | (HD 27786 Ab) | (HD 27786 B)
Fitted parameters
Primary mass ($M_{\odot}$) |  $1.540\pm 0.077$ | ${1.714}_{-0.060}^{+0.061}$
Companion mass ($M_{\sun}$) | Uniform |  $0.443\pm 0.012$ | ${1.36}_{-0.41}^{+0.51}$
Parallax (mas) |  $23.79\pm 0.11$ | $23.87\pm 0.11$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) |  ${12.04}_{-0.14}^{+0.13}$ | $183\pm 16$
Inclination $i$ ($\degree$) |  $\sin{i}$ |  ${172.8}_{-4.0}^{+3.2}$ | ${158.8}_{-9.2}^{+10}$
$\sqrt{e}\sin{\omega}$ | Uniform |  ${0.574}_{-0.16}^{+0.053}$ | ${0.07}_{-0.26}^{+0.21}$
$\sqrt{e}\cos{\omega}$ | Uniform |  ${0.15}_{-0.37}^{+0.31}$ | ${0.09}_{-0.21}^{+0.16}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform |  ${48}_{-34}^{+294}$ | ${211}_{-140}^{+72}$
PA of the ascending node $\Omega$ ($\degree$) | Uniform |  ${199}_{-33}^{+35}$ | ${221}_{-140}^{+67}$
Derived parameters
Orbital period (yr) | … |  ${28.42}_{-0.28}^{+0.31}$ | ${1436}_{-251}^{+203}$
Semimajor axis (mas) | … |  ${287.4}_{-3.4}^{+3.3}$ | ${4357}_{-367}^{+385}$
Eccentricity $e$ | … |  ${0.4016}_{-0.0084}^{+0.0083}$ | ${0.074}_{-0.052}^{+0.085}$
Argument of periastron $\omega$ $(\degree)$ | … |  ${77}_{-32}^{+34}$ | ${124}_{-82}^{+176}$
Time of periastron $T_{0}$ (JD) | … |  ${2456909}_{-28}^{+29}$ | ${2648314}_{-89023}^{+202895}$
Mass ratio | … |  ${0.2585}_{-0.0050}^{+0.0059}$ | ${0.80}_{-0.25}^{+0.31}$
Table 6: Posteriors of the HD 27786 system. Parameter | Prior Distribution | Posteriors $\pm 1\sigma$
---|---|---
Fitted parameters
Primary mass ($M_{\odot}$) |  $0.968\pm 0.044$ | $0.955\pm 0.043$
Companion mass ($M_{\odot}$) |  $1/M$ (log-flat) | $0.591\pm 0.011$
Parallax (mas) |  $37.8677\pm 0.0243$ | $37.867\pm 0.016$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) | ${26.25}_{-0.66}^{+0.71}$
Inclination $i$ ($\degree$) |  $\sin{i}$ | $88.86\pm 0.21$
$\sqrt{e}\sin{\omega}$ | Uniform | ${0.8088}_{-0.0048}^{+0.0044}$
$\sqrt{e}\cos{\omega}$ | Uniform | ${0.190}_{-0.029}^{+0.027}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform | $21.1\pm 2.1$
PA of the ascending node $\Omega$ ($\degree$) | Uniform | ${169.31}_{-0.52}^{+0.53}$
RV jitter $\sigma$ (m s-1) | Log-flat over [0,1000 m s-1] | ${3.55}_{-0.34}^{+0.39}$
Derived parameters
Orbital period (yr) | … | ${108.2}_{-4.8}^{+5.2}$
Semimajor axis (mas) | … | ${994}_{-25}^{+27}$
Eccentricity $e$ | … | $0.690\pm 0.017$
Argument of periastron $\omega$ $(\degree)$ | … | ${76.8}_{-1.7}^{+1.9}$
Time of periastron $T_{0}$ (JD) | … | ${2481070}_{-1524}^{+1682}$
Mass ratio | … | ${0.620}_{-0.030}^{+0.033}$
Table 7: Posteriors of the HD 114174 system.

Figure 12: Best-fitting orbital parameters for the HD 114174 system. Other
descriptions are identical to those for Figure 4.

(a)  (b)

Figure 13: RV orbits of HD 114174. Other descriptions are identical to those
for Figure 5.

(a)  (b)

(c)

Figure 14: Relative (top panel) and absolute (bottom panel) astrometry for the
HD 114174 system. Other descriptions are identical to those for Figure 6.

Figure 15: Best-fitting orbital parameters for the HD 118475 system. Because
fitting only the RVs does not break the degeneracy between secondary mass and
inclination, we plot the minimum mass $M_{\rm sec}\sin{i}$ instead of $M$ and
$i$ separately. Other descriptions are identical to those for Figure 4.

(a)  (b)

Figure 16: RV orbits of HD 118475. Other descriptions are identical to those
for Figure 5.

Parameter | Prior Distribution | Posteriors $\pm 1\sigma$ |  Gaia solution $\pm 1\sigma$
---|---|---|---
Fitted parameters | …
Primary mass ($M_{\odot}$) |  $1.158\pm 0.058$ |  ${1.154}_{-0.055}^{+0.056}$ | …
Companion mass ($M_{\odot}$) |  $1/M$ (log-flat) | … | ${0.557}^{+0.08}_{-0.072}$
Parallax (mas) |  $29.537\pm 0.017$ |  $29.537\pm 0.017$ | $29.537\pm 0.017$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) |  ${3.758}_{-0.060}^{+0.070}$ | $3.68\pm 0.18$
Inclination $i$ ($\degree$) |  $\sin{i}$ | … | ${53.1}^{+4.8}_{-5.5}$
$\sqrt{e}\sin{\omega}$ | Uniform |  ${-0.3024}_{-0.0006}^{+0.0007}$ | ${0.07}^{+0.06}_{-0.26}$
$\sqrt{e}\cos{\omega}$ | Uniform |  ${-0.1907}_{-0.0016}^{+0.0017}$ | ${0.34}^{+0.06}_{-0.65}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform |  $44.52\pm 0.05$ | …
PA of the ascending node $\Omega$ ($\degree$) | Uniform | … | ${212}^{+11}_{-191}$
RV jitter $\sigma$ (m s-1) | Log-flat over [0,10 m s-1] |  ${0.02}_{-0.02}^{+1.77}$ | …
Derived parameters | …
Companion minimum mass $M_{\rm sec}\sin{i}$ ($M_{\odot}$) | … |  ${0.461}_{-0.015}^{+0.017}$ | ${0.444}^{+0.081}_{-0.076}$
Orbital period (d) | … |  $2070.47\pm 0.16$ | $1968.6\pm 154.0$
Semimajor axis (mas) | … |  ${111.0}_{-1.8}^{+2.1}$ | $108.6\pm 5.3$
Eccentricity $e$ | … |  $0.1278\pm 0.0008$ | $0.143\pm 0.036$
RV semi-amplitude $K$ (m s-1) | … |  ${5560.3}_{-8.4}^{+8.1}$ | …
Argument of periastron $\omega$ $(\degree)$ | … |  $237.76\pm 0.22$ | ${19.5}^{+191.7}_{-9.7}$
Time of periastron $T_{0}$ (JD) | … |  ${2455344}_{-66}^{+77}$ | $2456380\pm 160$
Mass ratio | … |  $0.419_{-0.025}^{+0.066}$ | ${0.481}^{+0.070}_{-0.061}$
Table 8: Posteriors of the HD 118475 system. Parameter | Prior Distribution | Posteriors $\pm 1\sigma$ |  Gaia solution $\pm 1\sigma$
---|---|---|---
Fitted parameters | …
Primary mass ($M_{\odot}$) |  $1.84\pm 0.40$ |  ${1.80}^{+0.37}_{-0.39}$ | …
Companion mass ($M_{\odot}$) |  $1/M$ (log-flat) | … | ${0.543}^{+0.078}_{-0.080}$
Parallax (mas) |  $9.011\pm 0.051$ |  $9.011\pm 0.051$ | $9.011\pm 0.051$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) |  ${1.65}^{+0.10}_{-0.12}$ | ${1.66}^{+0.10}_{-0.12}$
Inclination $i$ ($\degree$) |  $\sin{i}$ | … | ${46.5}^{+5.6}_{-6.6}$
$\sqrt{e}\sin{\omega}$ | Uniform |  ${0.377}^{+0.025}_{-0.029}$ | ${0.557}^{+0.043}_{-0.044}$
$\sqrt{e}\cos{\omega}$ | Uniform |  ${0.438}^{+0.028}_{-0.024}$ | ${0.325}^{+0.052}_{-0.057}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform |  $341.0^{+2.4}_{-2.6}$ | …
PA of the ascending node $\Omega$ ($\degree$) | Uniform | … | ${52.4}^{+5.1}_{-5.4}$
RV jitter $\sigma$ (m s-1) | Log-flat over [0,1000 m s-1] |  ${0.07}^{+9.94}_{-0.07}$ | …
Derived parameters | …
Companion minimum mass $M_{\rm sec}\sin{i}$ ($M_{\odot}$) | … |  ${0.385}^{+0.046}_{-0.056}$ | ${0.389}^{+0.081}_{-0.080}$
Orbital period (d) | … |  $509.6\pm 1.2$ | $505.0\pm 2.5$
Semimajor axis (mas) | … |  ${14.88}^{+0.89}_{-1.11}$ | ${14.94}^{+0.93}_{-1.06}$
Eccentricity $e$ | … |  ${0.336}^{+0.015}_{-0.014}$ | $0.419\pm 0.046$
RV semi-amplitude $K$ (m s-1) | … |  ${6218}^{+135}_{-113}$ | …
Argument of periastron $\omega$ $(\degree)$ | … |  ${40.5}^{+3.4}_{-3.5}$ | ${59.6}^{+5.5}_{-5.0}$
Time of periastron $T_{0}$ (JD) | … |  ${2455451}^{+184}_{-157}$ | $2455281\pm 8$
Mass ratio | … |  ${0.24}^{+0.15}_{-0.04}$ | ${0.297}^{+0.035}_{-0.028}$
Table 9: Posteriors of the HD 136138 system.

Figure 17: Best-fitting orbital parameters for the HD 136138 system. Because
fitting only the RVs does not break the degeneracy between secondary mass and
inclination, we plot the minimum mass $M_{\rm sec}\sin{i}$ instead of $M$ and
$i$ separately. Other descriptions are identical to those for Figure 4.

(a)  (b)

Figure 18: RV orbits of HD 136138. Other descriptions are identical to those
for Figure 5.

Figure 19: Best-fitting orbital parameters for the HD 169889 system. Other
descriptions are identical to those for Figure 4.

(a)  (b)

Figure 20: RV orbits of HD 169889. Other descriptions are identical to those
for Figure 5.

(a)  (b)

(c)

Figure 21: Relative (top panel) and absolute (bottom panel) astrometry for the
HD 169889 system. Other descriptions are identical to those for Figure 6.

Parameter | Prior Distribution | Posteriors $\pm 1\sigma$
---|---|---
Fitted parameters
Primary mass ($M_{\odot}$) |  $0.980\pm 0.049$ | ${0.983}_{-0.049}^{+0.048}$
Companion mass ($M_{\odot}$) |  $1/M$ (log-flat) | ${0.526}_{-0.037}^{+0.039}$
Parallax (mas) |  $28.2794\pm 0.0259$ | $28.2795\pm 0.0079$
Semimajor axis $a$ (AU) |  $1/a$ (log-flat) | ${41}_{-10}^{+15}$
Inclination $i$ ($\degree$) |  $\sin{i}$ | ${101.7}_{-8.8}^{+12}$
$\sqrt{e}\sin{\omega}$ | Uniform | ${0.901}_{-0.050}^{+0.035}$
$\sqrt{e}\cos{\omega}$ | Uniform | ${-0.21}_{-0.18}^{+0.25}$
Mean longitude at $t_{\rm ref}=2455197.50$ JD ($\degree$) | Uniform | ${52}_{-23}^{+19}$
PA of the ascending node $\Omega$ ($\degree$) | Uniform | ${149}_{-20}^{+10}$
RV jitter $\sigma$ (m s-1) | Log-flat over [0,1000 m s-1] | ${4.3}_{-1.5}^{+2.6}$
Derived parameters
Orbital period (yr) | … | ${212}_{-75}^{+126}$
Semimajor axis (mas) | … | ${1154}_{-294}^{+419}$
Eccentricity $e$ | … | ${0.896}_{-0.088}^{+0.064}$
Argument of periastron $\omega$ $(\degree)$ | … | ${103}_{-16}^{+11}$
Time of periastron $T_{0}$ (JD) | … | ${2466913}_{-819}^{+1550}$
Mass ratio | … | ${0.536}_{-0.047}^{+0.048}$
Table 10: Posteriors of the HD 169889 system.