# The white dwarf binary pathways survey - X. Gaia orbits for known UV excess
binaries.

J. A. Garbutt,1 S. G. Parsons,1 O. Toloza,2,3 B. T. Gänsicke,4 M. S.
Hernandez,2,3 D. Koester,5 F. Lagos,4 R. Raddi,6 A. Rebassa-Mansergas,6,7 J.
J. Ren,8 M. R. Schreiber,2,3 M. Zorotovic9
1Department of Physics & Astronomy, University of Sheffield, Sheffield S3 7RH,
UK
2Departamento de Física, Universidad Técnica Federico Santa María, Avenida
España 1680, Valparaíso, Chile
3Millennium Nucleus for Planet Formation, NPF, Valparaıso, Av. España 1680,
Chile
4Department of Physics, University of Warwick, Coventry CV4 7AL, UK
5Institut für Theoretische Physik und Astrophysik, University of Kiel, 24098
Kiel, Germany
6Departament de Física, Universitat Politècnica de Catalunya, c/Esteve
Terrades 5, E-08860 Castelldefels, Spain
7Institute for Space Studies of Catalonia, c/Gran Capità 2-4, Edif. Nexus 201,
E-08034 Barcelona, Spain
8Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beĳing 100101, P. R. China
9Instituto de Física y Astronomía, Universidad de Valparaíso, Av. Gran Bretaña
1111, Valparaíso, Chile E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

White dwarfs with a F, G or K type companion represent the last common
ancestor for a plethora of exotic systems throughout the galaxy, though to
this point very few of them have been fully characterised in terms of orbital
period and component masses, despite the fact several thousand have been
identified. Gaia data release 3 has examined many hundreds of thousands of
systems, and as such we can use this, in conjunction with our previous UV
excess catalogues, to perform spectral energy distribution fitting in order to
obtain a sample of 206 binaries likely to contain a white dwarf, complete with
orbital periods, and either a direct measurement of the component masses for
astrometric systems, or a lower limit on the component masses for
spectroscopic systems. Of this sample of 206, four have previously been
observed with Hubble Space Telescope spectroscopically in the ultraviolet,
which has confirmed the presence of a white dwarf, and we find excellent
agreement between the dynamical and spectroscopic masses of the white dwarfs
in these systems. We find that white dwarf plus F, G or K binaries can have a
wide range of orbital periods, from less than a day to many hundreds of days.
A large number of our systems are likely post-stable mass transfer systems
based on their mass/period relationships, while others are difficult to
explain either via stable mass transfer or standard common envelope evolution.

###### keywords:

binaries: close – stars: evolution – stars: solar-type – stars: white dwarfs

††pubyear: 2023††pagerange: The white dwarf binary pathways survey - X. Gaia
orbits for known UV excess binaries.–A

##  1 Introduction

White Dwarfs (hereafter, WDs) are the last stage of the evolution of low to
intermediate mass stars, of initial mass around $\sim 0.85-9$ M$\odot$ (Kepler
et al., 2007; Cummings et al., 2018). It has been found through observations
(e.g. Holberg 2009; Toonen et al. 2017), that around 18-26 per cent of WDs are
in a binary system, meaning that the WD evolved in proximity to a companion.

If the initial binary system had an orbital period, $P$, of less than roughly
$10^{4}$ days, then it is likely that the two stars interacted as the WD
progenitor evolved off the main-sequence (Willems, B. & Kolb, U., 2004). A
possible interaction would be a common envelope phase (Paczynski, 1976). Here,
as the more massive star’s envelope expands and overflows its Roche lobe, it
begins unstably shedding matter. In this process, the less massive star also
overfills its Roche lobe, resulting in the formation of a common envelope
around the core of the donor and the less massive companion. This causes them
to spiral inward. If enough orbital energy is transferred to the envelope to
unbind and eject it, a post-common envelope system with a shorter period is
formed (Rebassa-Mansergas et al., 2008; Nebot Gómez-Morán et al., 2011).

The specific dynamics of the common envelope phase are difficult to reproduce
with hydrodynamical modelling (Passy et al., 2012; Ohlmann et al., 2016;
Ondratschek, Patrick A. et al., 2022; Moreno et al., 2022). As such, typically
a simplified equation involving the common envelope efficiency,
$\alpha_{\mathrm{CE}}$, is used instead - where $\alpha_{\mathrm{CE}}$ is the
fraction of the change in orbital energy used to unbind the envelope.
Therefore, a lower efficiency implies a greater reduction in the orbital
period. Zorotovic et al. (2010) has found a value of
$\alpha_{\mathrm{CE}}\approx 1/3$ through observations of WD + dM binaries,
with similarly low efficiencies being found for WD + BD binaries (Zorotovic &
Schreiber, 2022) and close low mass WD + WD binaries (Scherbak & Fuller, 2023)
\- though it is uncertain if such a value is universal. If this is the case,
then most WD + FGK systems would be expected to emerge from the common
envelope phase with periods too short for them to survive a second common
envelope phase, meaning that forming double-degenerate systems via this
pathway would be extremely challenging. Indeed, it was this issue that lead to
the creation of the so-called $\gamma$ formalism (Nelemans et al., 2000).
Gamma formalism, also called “common envelope without spiral in", is stable
but highly non-conservative mass transfer. However, there are alternative
channels to the creation of longer period WD + FGK binaries that do not
involve a common envelope phase, such as the stable mass transfer channel
(Webbink, 2008).

Stable mass transfer, as proposed by Webbink (2008), is thought to occur when
the masses of the two stars are very similar, or if the system comes into
contact when the donor star is on, or has just left, the main sequence. In
this scenario, the less massive star is likely to successfully accrete the
mass overflowing from the more massive companion at a steady rate, which can
lead to a widening of the binary if the transfer is non-conservative
(Podsiadlowski, 2014). This would leave the system with a wide enough period
to survive a common envelope phase without merging when the lower mass star
evolves, which can lead to double-degenerate systems, a progenitor for
thermonuclear supernovae. It is not yet understood how conservative this
mechanism is in practice, with Kawahara et al. (2018) suggesting the mass
transfer is usually non-conservative, whilst Podsiadlowski (2014) suggests a
near fully conservative transfer.

Until recently, the majority of WD + FGK binaries with known orbital periods
(e.g. Parsons et al. 2015; Hernandez et al. 2020, 2022a; Lagos et al. 2022;
Hernandez et al. 2022b) were short period systems of around $\sim\,0.5-2.5$
days, which can be reproduced with the same common envelope efficiency as used
for WD + dM binaries ($\alpha_{\mathrm{CE}}\approx 1/3$). These short period
systems can go on to become cataclysmic variables or supersoft X-ray source
systems. Longer period systems had proven more elusive, though there were a
small number published (Kruse & Agol, 2014a; Kawahara et al., 2018; Masuda et
al., 2019; Parsons et al., 2023; Yamaguchi et al., 2023), with Shahaf et al.
(2023) identifying a few thousand WD + FGK candidates in Gaia data release 3,
with periods around $\approx 100-1000$ days - though this sample has yet to be
explored in great detail. These longer period systems are likely the
progenitors of double degenerate systems and symbiotic binaries (Zorotovic et
al., 2014). Thus far, these systems have not been able to be reconstructed
with the common envelope efficiency of $1/3$ as found by Zorotovic et al.
(2010), or even with $\alpha_{\mathrm{CE}}=1$ although see Belloni et al.
(2024) for a potential solution to this. It is possible for some these to be
post-stable mass transfer systems (Parsons et al., 2023), but many of these
longer period systems are a challenge to produce even by this channel. Given
that WD + FGK binaries are the last common ancestor to a number of exotic
phenomena, such as the before-described cataclysmic variable and double
degenerate systems - they are thus important to study and understand. It is
possible to find WD + FGK binaries by looking for sources that appear to be F,
G or K type stars in the optical, but have a flux excess in the UV, as a WD
would be fully obscured by a F, G or K binary companion in optical
wavelengths, but in turn outshine them in the UV owing to their high residual
temperatures. Given the importance of such objects, The White Dwarf Binary
Pathways Survey has set about trying to catalogue the titular systems and
determine their formation channels.

Gaia data release 3 (hereafter Gaia) can be used as an important tool for
getting a large number of binary parameters, such as the orbital period, with
little effort, as it possesses accurate orbital period measurements for a
large number of systems, bypassing the need for extensive follow-up
observations to identify and characterise WD + FGK binaries. In this paper we
probe Gaia for previously established UV excess binaries (Parsons et al. 2016;
Rebassa-Mansergas et al. 2017; Ren et al. 2020) to determine their binary and
stellar parameters, most crucially their orbital periods and component masses,
so that we can investigate their past and future evolution.

##  2 Target Selection

In order to ensure a relatively clean sample of WD + FGK binaries, we make
cuts to remove as many sources of contamination as possible so that we are
only working with systems where a WD + FGK binary is likely, as our method of
identifying these systems by their UV excess is only valid for these systems,
where the luminous companion completely dominates in optical wavelengths but
where the WD causes a notable excess in the UV.

Taking the RAVE sample of 430 candidates from Parsons et al. (2016), 1549
candidates from the LAMOST sample of Rebassa-Mansergas et al. (2017) and the
sample of 775 candidates from the TGAS sample of Ren et al. (2020), we matched
these 2754 candidates to the Gaia source catalogue using positional
crossmatching. These samples were constructed using RAVE data release 4,
LAMOST data release 4 and TGAS; crossmatching them with GALEX UV data in order
to identify candidate UV excess sources.

To this point, these samples may contain poor Gaia matches and stars that are
not of the F, G or K spectral classes. To resolve this, we implemented a cut
to our selection criteria - removing systems that were a magnitude bluer than
the main sequence track as defined by the
MIST111https://waps.cfa.harvard.edu/MIST/ isochrone (Dotter, 2016; Choi et
al., 2016; Paxton et al., 2011; Paxton et al., 2013, 2015) for a solar
metalicity star using Gaia G and GBP \- GRP magnitudes, which removed 6
systems from the RAVE sample of Parsons et al. (2016), 204 from the LAMOST
sample of Rebassa-Mansergas et al. (2017) and 1 systems from the TGAS sample
of Ren et al. (2020). For the astrometric binaries, we used the parallax
values from the gaiadr3.nss_two_body_orbit catalogue throughout this paper.
This removed sources where the compact object may contribute to the optical
flux, such as WD + dM or hot subdwarf + FGK binaries. Whilst measures were
taken by Parsons et al. (2016), Rebassa-Mansergas et al. (2017) and Ren et al.
(2020) to remove M stars, some have since been flagged by Gaia, while no
efforts were taken to remove hot subdwarf systems in the original studies.

Following the removal of these contaminants, along with the removal of
duplicates within each survey, we crossmatched the surveys with the
gaiadr3.nss_two_body_orbit catalogue, which provided us with orbital solutions
for systems identified as astrometric, spectroscopic or eclipsing binaries.
The optical colour-magnitude diagrams of each of the surveys, colour-coded by
their orbital period, can be observed in Figure 1. It is worth noting that the
presence of giant stars (the outcrop of points on the luminous, redder end of
the main sequence track) in the RAVE and LAMOST samples was somewhat
unexpected, as each had taken measures to avoid giant stars, with the other
points sitting between $\sim\,0.3$ and $\sim\,1.8$ in the
$G_{\mathrm{BP}}\,-\,G_{\mathrm{RP}}$ domain, roughly where we would expect
for F, G and K stars along the main sequence track. We can see that, in
contrast to previous results, there are a large number of long-period
systems222We refer to systems with orbital periods in the order of $\sim$
100-1000 days as long-period binaries, as opposed to wide binaries which have
periods of many years and will not have had any binary interactions. found
across all three samples, with TGAS especially being dominated by periods in
excess of 100 days. It can also be observed that the TGAS sample does not
stretch as far down the main sequence track as much as either the RAVE or
LAMOST samples, which could be down to selection bias, as Ren et al. (2020)
was conducting a sample of WD + AFGK binaries opposed to just WD + FGK
binaries, which may have skewed their selection away from the fringe cases of
late-K stars, or owing to the initial selection that used DR1 as the original
sample.

Figure 1: Optical colour-magnitude diagrams using Gaia $G$, $BP$ and $RP$
magnitudes, split by the three surveys of Parsons et al. (2016), Rebassa-
Mansergas et al. (2017) and Ren et al. (2020). The points are colour-coded by
the log10 of their period. The background points are a randomly selected
sample of stars from Gaia within 250 pc and with a parallax-over-error greater
than 20. The larger, circular points represent candidate hot subdwarf binary
systems (see Section 5.5).

The solutions found within the gaiadr3.nss_two_body_orbit catalogue are:
‘Orbital’, ‘OrbitalAlternative’, ‘OrbitalTargetedSearch’,
‘OrbitalTargetedSearchValidated’ (all four of which are astrometric fits, and
will hereafter be referred to as astrometric systems), ‘EclipsingBinary’,
‘EclipsingSpectro’ (which are both visually eclipsing systems), ‘SB1’, ‘SB2’,
‘SB1C’, ‘SB2C’ ( which are all spectroscopic systems - the number referring to
how many resolved lines are present, and the C indicating a circular orbit),
and ‘AstroSpectroSB1’ (which is a combined astrometric and single-line
spectroscopic model, but for our purposes can be counted as solely
astrometric). The breakdown of how many systems belonged to each solution type
is detailed in Table 1, along with the number of eclipsing binaries found by
cross-matching with the gaiadr3.vari_classifier_result catalogue.

Table 1: A table breaking down the different solution types found across Parsons et al. (2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020) (given the short-hand form of of RAVE, LAMOST and TGAS respectively). The numbers given outside the brackets are the numbers in our final sample of WD + FGK candidates, whilst those given in the brackets represent the number found within the gaiadr3.nss_two_body_orbit catalogue (or flagged as ECL in the gaiadr3.vari_classifier_result catalogue. The total number of candidates here listed is 211, with the discrepancy between this number and our quoted number of 206 candidates arising from duplicates between the samples. Additionally, the total number of objects stated here from the gaiadr3.nss_two_body_orbit catalogue occur before our listed quality control cuts. Solution / Systems | RAVE | LAMOST | TGAS
---|---|---|---
Orbital | 4 (7) | 24 (30) | 18 (26)
OrbitalTargetedSearch | 0 | 0 | 1 (2)
OrbitalTargetedSearchValidated | 0 (1) | 0 | 0
SB1 | 25 (37) | 25 (33) | 56 (73)
SB2 | 0 | 0 | 0 (6)
SB2C | 0 | 0 | 0 (4)
AstroSpectroSB1 | 12 (14) | 8 (9) | 38 (42)
Eclipsing Binaries | 0 (0) | 0 (1) | 0 (4)
Total | 41 (59) | 57 (72) | 113 (153)

With the solution types known, we can make further cuts in accordance with
Gaia Collaboration et al. (2022). Which is to say ‘Orbital’ systems were
excluded if;

  * •

phot_g_mag $>$ 19,

  * •

ipd_frac_multi_peak $\geq$ 2,

  * •

ipd_gof_harmonic_amplitude $>$ 0.1,

  * •

visibility_periods_used $<$ 11,

  * •

$C^{*}$ $<$ 1.645$\sigma_{\mathrm{C^{*}}}$333Here, $C^{*}$ is the corrected
$BP$ and $RP$ flux excess and $\sigma_{\mathrm{C^{*}}}$ is its associated
uncertainty, as defined by Riello et al. (2021).,

where ipd_frac_multi_peak is the percent of successful Image Parameter
Determination (IPD) with more than one peak, ipd_gof_harmonic_amplitude is the
amplitude of the the IPD goodness of fit vs the postition angle of the
associated scan, and visibility_periods_used is the number of visibility
periods used in the astrometric solution. For ‘SB1’ and ‘SB1C’, the criteria
for exclusion were;

  * •

rv_renormalised_gof < 4,

  * •

rv_nb_transits < 11,

  * •

3875 > rv_template_teff > 8125,

where rv_renormalised_gof is the renormalised goodness of fit of the radial
velocity measurements, rv_nb_transits is the number of transits used in the
calculation of the radial velocity, and rv_template_teff is the effective
temperature ($T_{\mathrm{eff}}$ of the template used in the radial velocity
calculations.

Since we are interested in systems where the optical flux comes exclusively
from one star, systems flagged as ‘EclipsingBinary’, ‘EclipsingSpectro’, ‘SB2’
or ‘SB2C’ were dropped from the samples, as the Eclipsing systems by their
nature contain stars bright enough to show an eclipse (which would not be
possible except for very hot WDs with very late-K stars), and the double-lines
spectroscopic by their nature have two optically luminous components, which
would not be the case for a WD + FGK binary. Additionally, systems which were
flagged as ‘ECL’ (shorthand for eclipsing) in the
gaiadr3.vari_classifier_result were likely dropped, for the same reason as
stated for the eclipsing systems within gaiadr3.nss_two_body_orbit. After
these systems are removed and the above astrometric and spectroscopic cuts are
applied, we are left with 55 systems from the RAVE sample of Parsons et al.
(2016), 71 from the LAMOST sample of Rebassa-Mansergas et al. (2017) and 128
from the TGAS sample of Ren et al. (2020). It should be noted that there are
systems that lie across multiple of these surveys, with there being a total of
246 systems after this criterion is taken into account.

For the astrometric binaries, we used the python nss
tools444https://gitlab.obspm.fr/gaia/nsstools (Halbwachs et al., 2022) in
order to transform the given orbital Thiele-Innes constants, denoted as $A$,
$B$, $F$ and $G$, and defined by;

$A=a_{0}\left[\cos(\omega)\cos(\Omega)-\sin(\omega)\sin(\Omega)\cos(i)\right],$
(1)
$B=a_{0}\left[\cos(\omega)\sin(\Omega)+\sin(\omega)\cos(\Omega)\cos(i)\right],$
(2)
$F=-a_{0}\left[\sin(\omega)\cos(\Omega)+\cos(\omega)\sin(\Omega)\cos(i)\right],$
(3)
$G=-a_{0}\left[\sin(\omega)\sin(\Omega)-\cos(\omega)\sin(\Omega)\cos(i)\right],$
(4)

where $a_{0}$ is the angular semi-major axis of the system, $\omega$ is the
argument of periapsis, $\Omega$ is the longitude of the ascending node and $i$
is the inclination; into the Campbell parameters ($a_{0}$, $\omega$, $\Omega$
and $i$), the most pertinent of which being the angular semi-major axis,
$a_{0}$ of the system.

##  3 Stellar Parameters

With our samples cleaned and orbital parameters acquired, we can set about
obtaining the parameters of the luminous star555Luminous star here referring
to the more optically luminous companion, which can be either a main sequence
or an evolved star. and then the WD. This will allow us to probe the evolution
of these systems, along with flagging additional contaminants.

###  3.1 Luminous star parameters

Whilst Gaia does give estimates for stellar parameters in their
gaiadr3.astrophsical_parameters, gaiadr3.astrophsical_parameters_supp and
gaiadr3.binary_masses catalogues, these are not necessarily accurate and are
not available for all systems. As such, we obtain properties for the luminous
component of the binary system using spectral energy distribution (SED)
fitting with the python Isochrones
package666https://github.com/timothydmorton/isochrones (Morton, 2015). We
perform fitting using synthetic $u$, $g$, $r$, $i$, $z$ magnitudes from
gaiadr3.synthetic_photometry_gspc where available, $g$, $r$, $i$, $z$ if there
was no $u$ measurement, or $G$, $BP$ and $RP$ as a last resort if there was no
synthetic photometry available - alongside $J$, $H$, $Ks$, $W1$ and $W2$
magnitude measurements taken from 2MASS (Skrutskie et al., 2006) and AllWISE
(Cutri et al., 2021) Gaia match catalogues (gaiadr1.tmass_original_valid with
gaiadr3.tmass_psc_xsc_best_neighbour and gaiadr1.allwise_original_valid with
gaiadr3.allwise_best_neighbour respectively). SED fitting is not without its
drawbacks - it relies on assumptions about its inputs and our results without
$u$ band synthetic photometry in particularly may not be able to constrain the
effective temperature, $T_{\mathrm{eff,LS}}$ very well. Ultimately,
spectroscopic data is needed to fix these parameters properly.

We place priors on our fit, these include a Gausian prior on parallax based on
Gaia measurements, a flat prior on extinction, based on the values and
uncertainties from the 3d dustmaps of stilism Capitanio et al. (2017), with a
Jeffreys prior, as defined by

$P(E(B-V)\propto E(B-V)^{1/2}.$ (5)

where we place a lower bound of $10^{-4}$ and an upper bound at the maximum
value found in the system’s direction within stilism. We also used the
Casagrande et al. (2011) metalicity prior. We then fitted our data to mist
isochrones (Dotter, 2016; Choi et al., 2016; Paxton et al., 2011; Paxton et
al., 2013, 2015) using a Markov Chain Monte Carlo (MCMC) method with 3000 live
points per fit. For our initial fits we assumed that all the flux comes from
the optically luminous star (hence why we did not include UV measurements in
our fit), however there may be cases where the WD component could contribute
to the optical flux, particularly at the shortest wavelengths. Using the WD
parameters derived in Section 3.2, we can estimate what the WD contribution is
likely to be at optical wavelengths and found that this reaches around 5 per
cent at most. We ran additional fits to re-derive the luminous star parameters
accounting for the WD contribution and found that even in the worst cases the
resulting difference in the parameters was far smaller than the uncertainties
on the original measurements.

Some non-WD companions (usually low mass active stars) can contaminate UV
excess samples such as the ones we are dealing with. These contaminants can be
found by looking for infrared excesses, as a low-mass star will contribute a
non-negligible amount of flux at longer wavelengths. To identify these kinds
of contaminants, we fitted an SED with a two component fit using the
Isochrones package. We then performed a log-likelihood ratio test on every
system comparing their single and double star model fits, flagging systems
where the double fit was more likely (comprising 23 systems) as likely
contaminants. We also note that it is possible to hide relatively high mass
main-sequence companions (e.g. A-type stars) next to giants, which may result
in the detection of a UV excess, while having little effect on the optical
colours of the system. However, these stars have masses typically much higher
than the Chandrasekhar mass and so can be removed with a simple upper limit on
the companion mass. Ultimately, of the combined sample of 246 systems, we have
43 contaminants, giving us a contamination rate of $\sim$ 17 per cent, which
is fairly consistent with the previous works in this series.

This gave us values describing the luminous companion - its effective
temperature, the log of its surface gravity, log($g$)LS, its radius,
$R_{\mathrm{LS}}$, and most importantly, its mass, $M_{\mathrm{LS}}$, all of
which can be seen in appendix A. It also provided values for metallicity,
log($Z$)LS, though the SED is insensitive to this parameter and is in many
cases unconstrained, so we do not include it here.

###  3.2 White dwarf parameters

By using the parameters of the luminous companion determined by SED fitting
with the Gaia orbital parameters, we can determine the properties of the
hidden companion, assumed to be a WD. This is done using a different approach
depending on wether the system is an astrometric binary or a spectroscopic
binary, so they will be discussed separately.

####  3.2.1 Astrometric binaries

We can determine the mass of the unseen companion within an astrometric binary
using the astrometric mass function, $\mathcal{A}$, described by Shahaf et al.
(2019),

$\mathcal{A}\equiv\frac{q}{(1+q)^{\frac{2}{3}}}\left(1-\frac{\mathcal{S}(1+q)}{q(1+\mathcal{S})}\right)=\frac{a_{0}}{\varpi}{M_{\mathrm{LS}}^{-\frac{1}{3}}P^{-\frac{2}{3}}},$
(6)

in which $a_{0}$ is the angular semi-major axis, $\varpi$ is the parallax, $P$
is the period, $q$ is the mass ratio of the system, such that

$q=\frac{M_{\mathrm{WD}}}{M_{\mathrm{LS}}},$ (7)

and $\mathcal{S}$ is the ratio of the intensities of the components, which we
take as zero as our underlying assumption is that 100 per cent of the optical
flux is contributed by the luminous companion.

It is thus straightforward to extract the mass of the WD by solving the
astrometric mass function. We also propagated all uncertainties through this
function to estimate errors on the WD masses. Any systems where the unseen
companion has a mass clearly above the Chandrasekhar mass limit were flagged
as contaminants.

####  3.2.2 Spectroscopic binaries

Spectroscopic binary systems are instead determined by the binary mass
function determined from Kepler’s laws,

$\frac{{M_{\mathrm{WD}}^{3}\sin^{3}(i)}}{(M_{\mathrm{WD}}+M_{\mathrm{LS}})^{2}}=\frac{K^{3}P}{2\pi
G}(1-e^{2})^{\frac{3}{2}},$ (8)

where $K$ is the radial velocity semi-amplitude and $e$ is the eccentricity.
It should be noted that in general the inclination, $i$, is unknown for
spectroscopic systems. This limits us to only getting a minimum WD mass
estimate for these systems by assuming $\sin{i}=1$, which will limit the
conclusions that can be drawn from these systems.

####  3.2.3 White dwarf temperature estimates

With the masses known (or lower limits in the case of the spectroscopic
systems), and $FUV$ magnitudes measured by GALEX (Martin et al., 2005), we can
set about obtaining estimates of the effective temperature of the WD
$T_{\mathrm{eff,WD}}$ (and consequently the log of its surface gravity,
log($g$)WD) and its cooling age, $\tau_{\mathrm{cool}}$. This allows us to
estimate how long it has been since the last mass transfer phase, which also
tells us if the luminous star is likely to be out of thermal equilibrium (i.e.
if its thermal timescale is longer than the WD cooling age). The WD cooling
age is particularly important for properly understanding the evolution of the
shortest period systems, where additional angular momentum loss may have
significantly altered their orbital period from what it was after the last
mass transfer phase. It is important to correct for this effect when
reconstructing their evolution (e.g. Zorotovic et al., 2010).

In order to obtain these values, we matched our de-reddened $FUV$ values
(obtained by taking the $FUV$ magnitude from GALEX and de-reddening by our
calculated reddening values) and our WD mass values against interpolated
values from models. For WDs with a mass measured below 0.5 M$\odot$ we used
the helium core models of Althaus et al.
(2013)777http://evolgroup.fcaglp.unlp.edu.ar/TRACKS/newtables.html, whilst for
more massive WDs (above 0.5 M$\odot$) we used the carbon/oxygen core models of
Bédard et al.
(2020)888https://www.astro.umontreal.ca/~bergeron/CoolingModels/, assuming a
standard mass-radius relationship. Note that, as our spectroscopic masses are
only lower limit estimates, this likewise puts a lower limit on our
$T_{\mathrm{eff,WD}}$ values (since if the WD has a higher mass, hence a
smaller radius, its temperature must increase to match the UV flux). Since
some of our spectroscopic systems have minimum WD masses below 0.2 M$\odot$ we
are not able to get temperature estimates for all our systems.

With our stellar and binary parameters established (which can be seen in
appendix A), we can compare our values to those found by other means to check
the integrity of our method, and investigate the implications of what our
parameters tell us about the evolution of these systems.

Based on our $T_{\mathrm{eff,WD}}$ values, we can estimate the WD flux
contributions in various passbands. We find that in the $u$ band for example
the vast majority of WDs contribute less than 5 per cent of the observed flux,
and in most cases less than 1 per cent. This is true in the $G$ band, which
Gaia uses for astrometric measurements, and is therefore relevant to equation
6, in particular $\mathcal{S}$, the ratio of intensities. However, even
accounting for a worst case scenario of a 5 per cent contribution in the $G$
band ($\mathcal{S}=0.05$), the resulting increase in $M_{\mathrm{WD}}$ is
smaller than the uncertainties from our method.

##  4 Hubble Space Telescope Observations

Eight of our systems have been observed spectroscopically at UV wavelengths
with the Hubble Space Telescope (HST). These data allow us to confirm the
origin of the UV excess in these objects and in the cases where this is due to
a WD, we can determine their physical parameters by fitting the spectrum. Here
we present HST data for four new objects comprising three WD + FGK binaries
and an active star (TYC 6086-1317-1, TYC 9151-303-1, TYC 6434-457-1 and CPD-82
849), and refer the reader to previous publications for the other four objects
(see section 5.1), though a breakdown of all eight systems is given briefly in
Table 2.

Table 2: The eight systems within our sample which have HST UV spectra, with the Cosmic Origins Spectrograph (COS) using the G130M grating centred on 1291 Å and the Space Telescope Imaging Spectrograph (STIS) using the MAMA detector and the G140L grating centred on 1425 Å. Listed are the proposed source of their UV excess, the compiled exposure time and the source of publication. Name | UV Source | Observation Date | Exposure time [s] | Source
---|---|---|---|---
TYC 9151-303-1 | WD in binary | 2015 / 07 / 14 | 2441 | This paper
CPD-82 849 | Active star | 2021 / 04 / 05 | 8718 | This paper
TYC 6086-1317-1 | WD in binary | 2021 / 03 / 01 | 2196 | This paper
TYC 6434-457-1 | WD in binary | 2020 / 12 / 31 | 4934 | This paper
2MASS J06281844-7621467 | Active star | 2021 / 04 / 21 | 11731 | Lagos et al. (2022)
TYC 7218-934-1 | WD companion to MS + MS binary | 2015 / 04 / 29 | 4384 | Lagos et al. (2020a)
TYC 6996-449-1 | WD companion to MS + MS binary | 2014 / 12 / 03 | 2418 | Lagos et al. (2022)
TYC 6992-827-1 | WD in binary | 2022 / 01 / 10 | 4980 | Parsons et al. (2023)

For the four HST spectra clearly containing a WD we fitted synthetic DA WD
models in order to determine the stellar parameters. We followed the procedure
outlined in Parsons et al. (2023), whereby we used a pre-generated model grid
created using the code of Koester (2010), spanning a range of effective
temperatures of 12000–30000 K in steps of 200 K and surface gravities of
6.0-9.0 in steps of 0.1 dex, and interpolate between these grid points.
Additionally, we included the effects of reddening and interstellar neutral
hydrogen (although the latter has a very minor effect) and scale the model
flux based on parallax. Thus the free parameters were effective temperature,
surface gravity, parallax and reddening. WD masses, radii and cooling ages are
then determined via a theoretical mass-radius relationship (Bédard et al. 2020
for masses above 0.5 M⊙ or Althaus et al. 2013 for masses below 0.5 M⊙).

We used the Markov Chain Monte Carlo (MCMC) method to fit the spectra,
implemented using the ensemble sampler emcee (Foreman-Mackey et al., 2013),
with 4000 steps and 100 walkers, where the autocorrelation time for each
parameter was generally found to be less than 100 steps, and a burn-in period
of 5 times that of the autocorrelation time was used. We applied Gaussian
priors to the parallax and reddening based on measurements from Gaia and the
3D reddening map of stilism (Capitanio et al., 2017), respectively. We also
add in quadrature systematic errors of 1.5 per cent in $T_{\mathrm{eff}}$ and
0.04 dex in log($g$) (Parsons et al., 2023). For the astrometric binary
systems we also generated model spectra based on the dynamically determined WD
parameters.

##  5 Discussion

###  5.1 Comparison of dynamical white dwarf masses to spectroscopic masses
from HST UV spectroscopy

Of the eight systems previously mentioned to have been observed with HST
spectroscopy, four (2MASS J06281844-7621467, TYC 7218-934-1, TYC 6996-449-1
and CPD-82 849) were identified as contaminants, with CPD-82 849 having not
yet been published. Its UV spectrum (as seen in Figure 2) clearly identifies
it as an active star.

Figure 2: The HST/STIS spectrum of CPD-82 849. The spectrum lacks any clear indication of a WD, but does show strong emission lines typically seen in active stars, which are likely the origin of the UV excess in this system. Table 3: A comparison of the WD parameters found through orbital analysis and SED fitting of Gaia data ("dynamical") against those from fitting HST UV spectra ("spectroscopic"). The analysis of the UV spectrum of TYC 6992-827-1 was presented in Parsons et al. (2023) and we list their values here. Both TYC 6992-827-1 and TYC 6434-457-1 are spectroscopic binaries, hence the dynamical constraints represent lower limits (i.e. assuming $\sin(i)=1$), which for TYC 6992-827-1 gives a WD mass so low that we are unable to estimate its temperature or surface gravity. The two methods give consistent results to within 2-3 $\sigma$ in all cases. Name | Porb [d] | Method |  $T_{\mathrm{eff,WD}}$ [K] | log($g$)WD [dex] |  $\tau_{\mathrm{cool}}$ [Myr] |  $M_{\mathrm{WD}}$ [M$\odot$]
---|---|---|---|---|---|---
TYC 6992-827-1 | 41.45 $\pm$ 0.01 | Dynamical | - | - | - |  $>$0.13 $\pm$ 0.01
|  | Parsons et al. (2023) | $15750\pm 50$ | 7.14 $\pm$ 0.02 | 6.9 $\pm$ 0.1 | 0.28 $\pm$ 0.01
TYC 6086-1317-1 | 524.2 $\pm$ 3.3 | Dynamical | 20100 $\pm$ 400 | 7.29 $\pm$ 0.04 | 40 $\pm$ 6 | 0.38 $\pm$ 0.01
|  | Spectroscopic | 19850 $\pm$ 350 | 7.12 $\pm$ 0.05 | 20 $\pm$ 12 | 0.34 $\pm$ 0.02
TYC 9151-303-1 | 797.8 $\pm$ 9.6 | Dynamical | 15700 $\pm$ 300 | 7.51 $\pm$ 0.03 | 190 $\pm$ 13 | 0.42 $\pm$ 0.01
|  | Spectroscopic | 14900 $\pm$ 250 | 7.35 $\pm$ 0.06 | 184 $\pm$ 26 | 0.36 $\pm$ 0.03
TYC 6434-457-1 | 930.2 $\pm$ 9.8 | Dynamical |  $>$14800 $\pm$ 1500 |  $>$7.78 $\pm$ 0.12 |  $>$234 $\pm$ 26 |  $>$0.51 $\pm$ 0.06
|  | Spectroscopic | 13600 $\pm$ 300 | 7.93 $\pm$ 0.12 | 252 $\pm$ 23 | 0.58 $\pm$ 0.03

The other four systems with HST spectra - TYC 6992-827-1, TYC 6086-1317-1, TYC
9151-303-1 and TYC 6434-457-1 - clearly contain a WD, with the best-fit
parameters shown in Table 3, and are each discussed in more detail below. For
the following discussion we refer to the parameters measured via the Gaia
orbital solution plus GALEX/FUV fit as "dynamical" measurements, while those
measured via fitting the HST UV spectrum are referred to as "spectroscopic"
measurements. It is also worth noting that TYC 9151-303-1 was flagged by our
SED fitting method as a contaminant, with the two-star model SED fitting the
system better - which could indicate either that our SED contamination
detection method may not always be reliable or that the SED is genuinely
contaminated, either by a background source or a distant companion to the
binary.

####  5.1.1 TYC 6992-827-1

Despite the large difference in mass between our dynamical mass and that of
Parsons et al. (2023) shown in Table 3, it should be noted that this system is
a spectroscopic binary, and thus this is only a minimum mass estimate. If we
are to apply the inclination of $26\pm 2$ °from Parsons et al. (2023), we
instead get a mass of $0.30\pm 0.02$ M⊙, which is consistent with their value,
and is illustrated in Figure 3. This is unsurprising given that the Gaia
orbital fit is consistent with the ground-based one presented in Parsons et
al. (2023).

Figure 3: A comparison of the dynamical white dwarf mass from Gaia orbits
(x-axis) vs those found through HST spectral fitting (y-axis).

####  5.1.2 TYC 6086-1317-1

TYC 6086-1317-1 was flagged as an astrometric binary system by Gaia, and so
its mass estimates are more accurate, since the inclination is known. The
dynamical and spectroscopic WD parameters are consistent to within 2-3
$\sigma$ demonstrating that the dynamical method can give reliable results. To
further highlight this, The HST spectrum of this system is shown in the top
panel of Figure 4 with both the best fit model spectrum and a model spectrum
computed at the dynamically determined WD parameters. Both methods indicate
the WD has a low mass and very short cooling age (see Table 3.

Figure 4: UV spectra of the astrometric binary systems TYC 6086-1317-1 (top,
STIS) and TYC 9151-303-1 (bottom, COS). Overplotted are the best fit models in
red, while models with the parameters fixed at the "dynamical" values are
shown as green dashed lines.

####  5.1.3 TYC 9151-303-1

TYC 9151-303-1 was flagged as an astrometric binary system by Gaia. Just as
with TYC 6086-1317-1 the dynamical and spectroscopic parameters are consistent
to within 2-3 $\sigma$. This system also contains a low mass WD, albeit with a
slightly longer cooling age than TYC 6086-1317-1. The UV spectrum of this
source is shown in the bottom panel of Figure 4.

####  5.1.4 TYC 6434-457-1

TYC 6434-457-1 is a single-lined spectroscopic binary in Gaia, which has an
orbital period comparable to many of the astrometric binary systems. Assuming
an edge-on orbit gives a minimum mass for the WD of $0.50\pm 0.06$ M$\odot$,
which is somewhat lower than the spectroscopically determined mass of $0.58\pm
0.03$M$\odot$, implying that the system has an inclination of around 60
degrees. In contrast to the two previous objects, TYC 6434-457-1 appears to
host a fairly typical mass WD. The HST spectrum of this object is shown in
Figure 5.

Figure 5: The HST/STIS spectrum of the spectroscopic binary system TYC
6434-457-1. The best fit model spectrum in shown in red, we do not plot a
model based on the "dynamical" measurements, since this assumes an edge-on
orbit and hence under predicts the WD mass.

###  5.2 Comparing to previous works

Each of the three surveys of Parsons et al. (2016), Rebassa-Mansergas et al.
(2017) and Ren et al. (2020) calculated specific stellar parameters for the
luminous companion, with which we can compare our values against. Given that
the dynamical WD masses are entirely dependent on the masses of the luminous
stars, it is vital that these values are as reliable as possible. Parsons et
al. (2016) and Rebassa-Mansergas et al. (2017) each determined values of
$T_{\mathrm{eff,LS}}$ and log($g$)LS, whilst Ren et al. (2020) measured values
for $T_{\mathrm{eff,LS}}$ and $R_{\mathrm{LS}}$, the comparison of these
values to those found by our dynamical modelling can be observed in Figure 6.

Figure 6: A comparison of our parameters against those of Parsons et al.
(2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020). Note that for
Parsons et al. (2016) and Rebassa-Mansergas et al. (2017) these were made
before any Gaia data release, hence the miss-identification of many evolved
stars in those studies.

####  5.2.1 RAVE and LAMOST

We find that, compared to the values of Parsons et al. (2016) and Rebassa-
Mansergas et al. (2017), our temperatures were mostly consistent.

The comparison of our values of log($g$)LS is quite telling. There are many
systems across both surveys (particularly those of RAVE) in which our values
were considerably lower than those found by either Parsons et al. (2016) or
Rebassa-Mansergas et al. (2017), which is consistent with our earlier
discovery of giant stars in their samples which they had identified as main
sequence stars.

####  5.2.2 TGAS

Both the temperatures and stellar radii from the survey of Ren et al. (2020)
match very closely with the values we found. This is unsurprising, as Ren et
al. (2020) likewise obtained their values through SED fitting, though they
used parallaxes from Gaia data release 2, whilst we used the updated versions
from Gaia data release 3.

###  5.3 Comparing luminous star masses with Gaia

Another useful source to compare against is the gaiadr3.binary_masses
catalogue, which contains mass estimates for many systems which Gaia has
flagged as being binaries. We should note that, of our sample of 246 systems,
only 124 had a mass estimates in the Gaia binary masses table, as Gaia
seemingly does not generate parameters for such systems where the optically
luminous companion has evolved off the main sequence. Given that this is a
crucial parameter to determine the WD mass, there is a clear need to derive
these ourselves or WD masses would not be constrained for a significant
fraction of our sample. In general, our masses show good agreement with the
Gaia values, as can be observed in Figure 7, with the more extreme outliers
being systems which we have flagged as contaminants.

Figure 7: A comparison between our dynamical masses of the more luminous
companion against their respective masses found in the gaiadr3.binary_masses
catalogue. Only around 50 per cent of our sample had masses in the Gaia
catalogue, with evolved stars missing from Gaia.

It should be noted that, of the systems absent from the gaiadr3.binary_masses
catalogue, all except 2 can be found as more evolved stars along the giant
branch. When using the Gaia binary masses table one should be aware of this
bias.

###  5.4 Evolutionary states

In Figure 8, we show the orbital period vs the white dwarf mass distribution,
with two panels - one each for the more robust astrometric data, and for the
lower mass estimates from the spectroscopic data. As can be observed, we have
a wide range of orbital periods from our candidate systems. We can see that
for our astrometric binaries (top panel) our systems lie near the region for
post-stable mass transfer systems as proposed by Rappaport et al. (1995) (the
grey shaded region), with few outliers. A similar result was recently
presented by Shahaf et al. (2023) using a purely astrometrically-selected
sample of systems. We note that this relation is based on donor stars on the
Red Giant Branch (RGB), which appears to be the case for many of our systems
where the WD mass is $<0.5$ M⊙, but this evolutionary channel may not be able
to explain the origin of systems with higher mass WDs that likely reached the
Asymptotic Giant Branch (AGB). A small but significant number of our systems
sit just below the stable mass transfer region, with similar parameters to the
self-lensing systems discovered by the Kepler mission (Kruse & Agol, 2014b;
Kawahara et al., 2018). Given their long periods it is still challenging to
explain them as a result of common envelope evolution, even assuming very
efficient envelope removal. Interestingly, the luminous star in many of these
systems is somewhat evolved itself, generally on the subgiant branch. This is
unsurprising if these systems are the result of stable mass transfer, since
this requires an initial mass ratio close to unity and therefore similar main-
sequence lifetimes for the two components. Given their current stellar and
binary parameters these systems are likely to undergo a common envelope phase
at some point in the future. Should both components survive this additional
mass transfer phase, this would lead to the creation of a double WD binary,
and could represent a significant formation channel for such systems.

Figure 8: $M_{\mathrm{WD}}$/$P$ relationships for our candidate WD binary
systems, colour-coded by the mass of the luminous companion, and split by
their solution type (top panel: astrometrically detected systems, bottom
panel: spectroscopically detected systems). Of note, the WD mass is only a
minimum mass estimate for the spectroscopic systems. The solid black line and
shaded grey region represent the theoretical region for conservative post-
stable mass transfer binaries from Rappaport et al. (1995), with the dotted
line representing an update on this from Lin et al. (2011). Systems above this
region may be explained by non-conservative stable mass transfer.

The spectroscopically identified systems (bottom panel of Figure 8) show much
wider distributions for both period and WD mass. Given that we only have lower
limits on the WD masses the picture is less clear for these systems. There are
some clear post-common envelope systems with periods less than a couple of
weeks and minimum WD masses around 0.5 M$\odot$, similar to other systems
found in the white dwarf binary pathways survey. However, there is a
significant number of systems with very low minimum WD masses, covering a wide
range of periods. While these could all be low inclination post-common
envelope systems, there is also a possibility that there is a population of
post-stable mass transfer systems ranging from periods of a day up to 1000
days. Indeed the shortest periods systems, EL CVn binaries, are already known
and intermediate period post-stable mass transfer binaries have now also been
found (Parsons et al., 2023). Given the low WD masses, these may well be the
progenitors of double WD systems containing at least one extremely low mass
WD.

We also note that there are a number of systems in both the astrometric and
spectroscopic samples with clearly high mass WDs and orbital periods longer
than a few weeks. The high WD masses make these systems challenging to explain
as the result of stable mass transfer, but equally their long orbital periods
are challenging to explain via standard common envelope evolution. Recently
Yamaguchi et al. (2023) discovered 5 such systems occupying this parameter
space (albeit with slightly shorter orbital periods than many of our systems)
which, combined with the roughly 25 such systems we find, clearly indicates
that there is a population of WD binaries in this region of parameter space.
Yamaguchi et al. (2023) proposed that these systems could be the result of
common envelope evolution if internal (recombination) energy also aids the
expulsion of the envelope. Whether this can also explain the systems we find
with orbital periods of 100s of days is unclear.

One clear result appears to be that a far wider range of periods is viable for
WD + FGK systems compared to WD + dM binaries, which generally have periods
averaging around 10.3 hours (Nebot Gómez-Morán et al., 2011), with a very
small sample of WD + dM binaries from Shahaf et al. (2023) having relatively
long periods being the current major exception, whilst WD + FGK systems can
span periods from around half a day to almost 1000 days. This is likely a
result of the multiple different formation pathways possible for WD + FGK
systems, whereas the extreme initial mass ratios of WD + dM systems mean that
they are all likely the result of standard common envelope evolution.

###  5.5 Systems with candidate hot subdwarf stars

We made a somewhat empirical cut to determine if the system was a hot subdwarf
candidate, by flagging systems with an absolute $FUV$ magnitude brighter than
5.2 mag as being hot subdwarf binary candidate systems, opposed to WD binary
systems. Doing so, we have found three systems in our sample that are
candidate hot subdwarf binaries, as can be observed in Figure 1, given their
extreme UV luminosities. Hot subdwarf stars are core helium burning stars that
have been stripped of their envelope (Heber, 2009), with masses in the domain
of $\sim$ 0.4-0.5 M$\odot$ (Vos et al., 2018), and are bright in the
ultraviolet, similar to WDs - though they are brighter at optically blue
wavelengths in comparison. This means that the cut to remove systems a
magnitude bluer than the main sequence track made in Parsons et al. (2016),
Rebassa-Mansergas et al. (2017), Ren et al. (2020) and our present study
should remove such systems in a binary with a dwarf late F, G or K companion.
However, such a cut would not remove hot subdwarfs with a more evolved, hence
luminous companion, as a similar cut was not made along the giant branch. Hot
subdwarf binaries typically come in two groups; those with low mass companions
in short period systems (such as those discussed in Kupfer et al. 2015), and
those with higher mass companions and a longer orbital period (as discussed in
Vos et al. 2018).

Our three hot subdwarf candidates are in binaries with evolved companions (see
Figure 1), and are found spectroscopically. Two of our three systems, UCAC2
15655859 and TYC 4889-1238-1, possess periods that are consistent with the
established period domain of hot subdwarfs with FGK stars from Vos et al.
(2018) (having periods of $\sim$ 996 and 1079 days respectively, and minimum
masses of $\sim$ 0.19 M$\odot$). The third candidate, TYC 6472-1267-1, has a
period of around 56 days, which is unusual, as there is a dearth of hot
subdwarf binary systems with such periods (ignoring the systems with WD
companions, which have passed through two mass transfer phases and so are not
directly comparable). It is worth noting that this system is a candidate
triple system based on the findings of Tian et al. (2020), who flagged the
system as potentially having a common proper motion companion. If true (which
is somewhat dubious given the separation of 55302 au found by Tian et al.
2020), then this would be consistent with the findings of Lagos et al. (2020b)
and Parsons et al. (2023), that these intermediate period post-stable mass
transfer systems are generally found in triples, since they need to have very
short initial separations. We do note however, that extremely low mass WDs can
also have similar UV luminosities to hot subdwarf stars, so, while these are
clearly not standard mass WDs, they may not be hot subdwarf stars either and
follow up data are required to properly classify these objects.

###  5.6 UV Parameter Space

Given that both the RAVE and LAMOST samples were created before any Gaia data
releases, it is useful to see where these systems sit in a UV colour-magnitude
diagram, both the genuine WD + FGK systems as well as the contaminants. This
may aid future selections of these kinds of objects. Figure 9 shows such a UV
colour-magnitude diagram, separated by the systems found by Parsons et al.
(2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020).

Figure 9: UV colour-magnitude diagrams using GALEX $FUV$ and $NUV$ magnitudes,
split by the three surveys of Parsons et al. (2016), Rebassa-Mansergas et al.
(2017) and Ren et al. (2020). The points are colour-coded by the log10 of
their period. A random sample of stars within 250 pc are shown in the
background.

Most systems sit between the WD and main-sequence track as expected for these
kinds of objects. In general, there is more contamination towards fainter
objects, with almost all sources fainter than $M_{FUV}\sim 15$ mag seen to be
contaminants, likely active stars. However, there are also brighter
contaminants that sit right in the main population of WD + FGK systems,
including some that we have observed with HST, confirming that they are indeed
contaminants. In general, these tend to be triple systems where the WD is a
wide companion to a main-sequence binary. We also note that there appears to
be some genuine WD + FGK systems that show very minor UV excesses (those that
sit close to the main-sequence), which are likely systems containing cool
and/or high mass WDs. However, there is also significant contamination close
to the main-sequence, so caution is needed when searching for WD binaries in
this region.

We note that the largest source of contamination is generally due to active
stars, i.e. binaries consisting of an FGK star plus a lower mass companion,
where either one or both components have strong chromospheric emission in the
UV. Despite this, we advise against removing active stars from catalogues
before searching for UV excess systems, because many genuine WD + FGK systems
contain an active FGK star, since they are often spun up as a result of mass
transfer and/or tidal locking.

###  5.7 Astrometric Parameter Space

It is now possible to identify WD + FGK systems directly as astrometric
binaries without the need for any UV data, by means of their astrometric mass
function, $\mathcal{A}$, as described in Shahaf et al. (2019), and removing
systems likely to be either a MS + MS or a hierarchical triple system. Shahaf
et al. (2023) recently used this technique to identify several thousand
candidate WD + FGK systems in Gaia. Given that our selection criteria are
significantly different to theirs it is worth comparing the two methods. In
particular, in Figure 10 we plot the astrometric mass function for our sample
and indicate the typical region that a WD + FGK system is likely to reside.
The Shahaf et al. (2023) method relies on ruling out the possibility of a
binary being composed of either two luminous stars, or the companion being a
binary itself (i.e. a triple system), which generally requires placing a
fairly tight limit of $\mathcal{A}$ greater than around 0.4 (see Shahaf et al.
2023 for the exact cut). Applying the same cut to our samples would exclude a
significant fraction of all our systems. This is due to a combination of many
of our WD masses being significantly lower than the canonical WD mass
(possibly as a result of stable mass transfer) and our sample containing a
large number of more massive luminous stars, since the Shahaf et al. (2023)
sample cuts off at around 1.3 M$\odot$, where contamination from normal
binaries overwhelms the WD systems. Many of our luminous stars are also
giants, which were not included in the Shahaf et al. (2023) sample. Therefore,
the UV-excess method appears to be the only way to identify systems with very
low mass WDs and/or higher mass luminous stars at present. We do note that
some of our systems sit quite outside the region expected for WD companions
(i.e. the purple region in Figure 10) and while some of these objects have
been identified as contaminants, it is possible that our candidates
significantly above and below the WD companion region may also no be genuine
WD + FGK systems.

Figure 10: The relationship between the astrometric mass function of Shahaf et
al. (2019), $\mathcal{A}$, and the mass of the luminous companion. The purple
highlighted region represents the region where the hidden companions mass is
between 0.4-0.75 M$\odot$, a typical mass range for WDs in these systems. The
full sample of astrometric binaries in Gaia is shown in the background.

As one could imagine, our sample is bias towards systems with a large UV
excess - namely hotter, lower mass WDs, whilst our cut removing systems bluer
than the main sequence as discussed in Section 2 is bias against lower mass
luminous companions where the WD may contribute a significant percentage of
the optical flux. There are further more complex selection effects in each of
the parent samples contributing to our sample, see Parsons et al. (2016),
Rebassa-Mansergas et al. (2017) and Ren et al. (2020) for more information on
each. However, our bias towards hot, low mass WDs is complimentary to the
selection effects of the purely astrometric method of Shahaf et al. (2023),
which is somewhat bias towards higher mass WDs, so a combination of both
methods could lead to a more complete sample.

##  6 Conclusions

We have confirmed the binary nature of 246 candidate WD + FGK binaries from
Parsons et al. (2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020)
using data from Gaia and combined this with modelling the SED of the luminous
star in order to constrain the mass of the unseen WD. Once systems which have
been flagged as likely contaminants are set aside, we have a sample of 206 WD
+ FGK binary candidates, four of which have been confirmed through HST
spectroscopic observations in the ultraviolet and one of which has been
published before (Parsons et al., 2023).

We find that our astrometric systems occupy a region of parameter space that
is incompatible with a post-CE origin, given their low WD masses, but they do
lie near the relationship proposed by Rappaport et al. (1995) for post-stable
mass transfer binaries, indicating these systems likely underwent a period of
stable, non-conservative mass transfer earlier in their evolution. Many of
these systems are likely to be the progenitors of double WD binaries.

The analysis of the spectroscopically identified systems is complicated by the
fact that we can only set a lower limit on the WD parameters. Nevertheless,
they confirm the existence of WD+FGK binaries at a wide range of orbital
periods, from less than a day, to more than 1000 days, in contrast to the
population of WD + dM binaries. We also identify a group of systems with
orbital periods of more than a few weeks containing high mass WDs, which are
difficult to reproduce either via stable mass transfer or standard
$\alpha_{\mathrm{CE}}\sim 0.3$ common envelope evolution.

## Acknowledgements

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. SGP acknowledges the support
of a Science and Technology Facilities Council (STFC) Ernest Rutherford
fellowship. This work has been partially supported by the Spanish MINECO grant
PID2020-117252GB-I00 and by the AGAUR/Generalitat de Catalunya grant
SGR-386/2021. RR acknowledges support from Grant RYC2021-030837-I funded by
MCIN/AEI/ 10.13039/501100011033 and by “European Union NextGeneration
EU/PRTR”. MZ and MRS acknowledge support from Fondecyt (grant 1221059). This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 101020057). For the purpose of open access, the author has
applied a creative commons attribution (CC BY) licence to any author accepted
manuscript version arising.

## Data Availability

The previous data samples which this paper built on are upon are available
publicly online. Likewise Gaia data release 3 is available publicly online at
https://www.cosmos.esa.int/web/gaia/dr3. Raw and reduced Hubble data are
available through MAST at
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html

## References

  * Althaus et al. (2013) Althaus L. G., Miller Bertolami M. M., Córsico A. H., 2013, A&A, 557, A19
  * Bédard et al. (2020) Bédard A., Bergeron P., Brassard P., Fontaine G., 2020, ApJ, 901, 93
  * Belloni et al. (2024) Belloni D., Zorotovic M., Schreiber M. R., Parsons S. G., Garbutt J. A., 2024, arXiv e-prints, p. arXiv:2401.07692
  * Bédard et al. (2020) Bédard A., Bergeron P., Brassard P., Fontaine G., 2020, The Astrophysical Journal, 901, 93
  * Capitanio et al. (2017) Capitanio L., Lallement R., Vergely J. L., Elyajouri M., Monreal-Ibero A., 2017, A&A, 606, A65
  * Casagrande et al. (2011) Casagrande L., Schönrich R., Asplund M., Cassisi S., Ramírez I., Meléndez J., Bensby T., Feltzing S., 2011, A&A, 530, A138
  * Choi et al. (2016) Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ, 823, 102
  * Cummings et al. (2018) Cummings J. D., Kalirai J. S., Tremblay P.-E., Ramirez-Ruiz E., Choi J., 2018, The Astrophysical Journal, 866, 21
  * Cutri et al. (2021) Cutri R. M., et al., 2021, VizieR Online Data Catalog, p. II/328
  * Dotter (2016) Dotter A., 2016, ApJS, 222, 8
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Gaia Collaboration et al. (2022) Gaia Collaboration et al., 2022, arXiv e-prints, p. arXiv:2206.05595
  * Halbwachs et al. (2022) Halbwachs J.-L., et al., 2022, arXiv e-prints, p. arXiv:2206.05726
  * Heber (2009) Heber U., 2009, ARA&A, 47, 211
  * Hernandez et al. (2020) Hernandez M. S., et al., 2020, Monthly Notices of the Royal Astronomical Society, 501, 1677
  * Hernandez et al. (2022a) Hernandez M. S., et al., 2022a, Monthly Notices of the Royal Astronomical Society, 512, 1843
  * Hernandez et al. (2022b) Hernandez M. S., et al., 2022b, Monthly Notices of the Royal Astronomical Society, 517, 2867
  * Holberg (2009) Holberg J. B., 2009, Journal of Physics: Conference Series, 172, 012022
  * Kawahara et al. (2018) Kawahara H., Masuda K., MacLeod M., Latham D. W., Bieryla A., Benomar O., 2018, The Astronomical Journal, 155, 144
  * Kepler et al. (2007) Kepler S. O., Kleinman S. J., Nitta A., Koester D., Castanheira B. G., Giovannini O., Costa A. F. M., Althaus L., 2007, Monthly Notices of the Royal Astronomical Society, 375, 1315
  * Koester (2010) Koester D., 2010, Mem. Soc. Astron. Italiana, 81, 921
  * Kruse & Agol (2014a) Kruse E., Agol E., 2014a, Science, 344, 275
  * Kruse & Agol (2014b) Kruse E., Agol E., 2014b, Science, 344, 275
  * Kupfer et al. (2015) Kupfer T., et al., 2015, A&A, 576, A44
  * Lagos et al. (2020a) Lagos F., et al., 2020a, MNRAS, 494, 915
  * Lagos et al. (2020b) Lagos F., Schreiber M. R., Parsons S. G., Gänsicke B. T., Godoy N., 2020b, Monthly Notices of the Royal Astronomical Society: Letters, 499, L121
  * Lagos et al. (2022) Lagos F., Schreiber M. R., Parsons S. G., Toloza O., Gänsicke B. T., Hernandez M. S., Schmidtobreick L., Belloni D., 2022, MNRAS, 512, 2625
  * Lin et al. (2011) Lin J., Rappaport S., Podsiadlowski P., Nelson L., Paxton B., Todorov P., 2011, The Astrophysical Journal, 732, 70
  * Martin et al. (2005) Martin D. C., et al., 2005, ApJ, 619, L1
  * Masuda et al. (2019) Masuda K., Kawahara H., Latham D. W., Bieryla A., Kunitomo M., MacLeod M., Aoki W., 2019, The Astrophysical Journal Letters, 881, L3
  * Moreno et al. (2022) Moreno M. M., Schneider F. R. N., Röpke F. K., Ohlmann S. T., Pakmor R., Podsiadlowski P., Sand C., 2022, A&A, 667, A72
  * Morton (2015) Morton T. D., 2015, isochrones: Stellar model grid package, Astrophysics Source Code Library, record ascl:1503.010 (ascl:1503.010)
  * Nebot Gómez-Morán et al. (2011) Nebot Gómez-Morán A., et al., 2011, A&A, 536, A43
  * Nelemans et al. (2000) Nelemans G., Verbunt F., Yungelson L. R., Portegies Zwart S. F., 2000, A&A, 360, 1011
  * Ohlmann et al. (2016) Ohlmann S. T., Röpke F. K., Pakmor R., Springel V., 2016, ApJ, 816, L9
  * Ondratschek, Patrick A. et al. (2022) Ondratschek, Patrick A. Röpke, Friedrich K. Schneider, Fabian R. N. Fendt, Christian Sand, Christian Ohlmann, Sebastian T. Pakmor, Rüdiger Springel, Volker 2022, A&A, 660, L8
  * Paczynski (1976) Paczynski B., 1976, in Eggleton P., Mitton S., Whelan J., eds, Proceedings of the Symposium Vol. 73, Structure and Evolution of Close Binary Systems. p. 75
  * Parsons et al. (2015) Parsons S. G., et al., 2015, Monthly Notices of the Royal Astronomical Society, 452, 1754
  * Parsons et al. (2016) Parsons S. G., Rebassa-Mansergas A., Schreiber M. R., Gänsicke B. T., Zorotovic M., Ren J. J., 2016, MNRAS, 463, 2125
  * Parsons et al. (2023) Parsons S. G., et al., 2023, MNRAS, 518, 4579
  * Passy et al. (2012) Passy J.-C., et al., 2012, ApJ, 744, 52
  * Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton B., et al., 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
  * Podsiadlowski (2014) Podsiadlowski P., 2014, Accretion Processes in Astrophysics, p. 45
  * Rappaport et al. (1995) Rappaport S., Podsiadlowski P., Joss P. C., Di Stefano R., Han Z., 1995, Monthly Notices of the Royal Astronomical Society, 273, 731
  * Rebassa-Mansergas et al. (2008) Rebassa-Mansergas A., et al., 2008, Monthly Notices of the Royal Astronomical Society, 390, 1635
  * Rebassa-Mansergas et al. (2017) Rebassa-Mansergas A., et al., 2017, MNRAS, 472, 4193
  * Ren et al. (2020) Ren J. J., et al., 2020, ApJ, 905, 38
  * Riello et al. (2021) Riello M., et al., 2021, A&A, 649, A3
  * Scherbak & Fuller (2023) Scherbak P., Fuller J., 2023, MNRAS, 518, 3966
  * Shahaf et al. (2019) Shahaf S., Mazeh T., Faigler S., Holl B., 2019, MNRAS, 487, 5610
  * Shahaf et al. (2023) Shahaf S., Hallakoun N., Mazeh T., Ben-Ami S., Rekhi P., El-Badry K., Toonen S., 2023, arXiv e-prints, p. arXiv:2309.15143
  * Skrutskie et al. (2006) Skrutskie M. F., et al., 2006, AJ, 131, 1163
  * Tian et al. (2020) Tian H.-J., El-Badry K., Rix H.-W., Gould A., 2020, ApJS, 246, 4
  * Toonen et al. (2017) Toonen S., Hollands M., Gänsicke B. T., Boekholt T., 2017, A&A, 602, A16
  * Vos et al. (2018) Vos J., Vučković M., Chen X., Han Z., Boudreaux T., Barlow B. N., Østensen R., Németh P., 2018, Monthly Notices of the Royal Astronomical Society, 482, 4592
  * Webbink (2008) Webbink R. F., 2008, in Milone E. F., Leahy D. A., Hobill D. W., eds, Astrophysics and Space Science Library Vol. 352, Astrophysics and Space Science Library. p. 233 (arXiv:0704.0280), doi:10.1007/978-1-4020-6544-6_13
  * Willems, B. & Kolb, U. (2004) Willems, B. Kolb, U. 2004, A&A, 419, 1057
  * Yamaguchi et al. (2023) Yamaguchi N., et al., 2023, arXiv e-prints, p. arXiv:2309.15905
  * Zorotovic & Schreiber (2022) Zorotovic M., Schreiber M., 2022, MNRAS, 513, 3587
  * Zorotovic et al. (2010) Zorotovic M., Schreiber, M. R. Gänsicke, B. T. Nebot Gómez-Morán, A. 2010, A&A, 520, A86
  * Zorotovic et al. (2014) Zorotovic M., Schreiber M. R., García-Berro E., Camacho J., Torres S., Rebassa-Mansergas A., Gänsicke B. T., 2014, A&A, 568, A68

##  Appendix A Tables of Parameters

Table 4: Table of astrometric binary systems, separated by their original survey. Here, RAVE, LAMOST and TGAS are used as shorthand for Parsons et al. (2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020) respectively. The full table consisting of all 20 RAVE astrometric binaries, 39 LAMOST astrometric binaries and 64 TGAS astrometric binaries, with a more complete list of properties, will be made available online upon publication. Name |  $P_{\mathrm{orb}}$ [d] |  $T_{\mathrm{eff,LS}}$ [K] | log($g$)LS [dex] |  $R_{\mathrm{LS}}$ [R$\odot$] |  $M_{\mathrm{LS}}$ [M$\odot$] |  $T_{\mathrm{eff,WD}}$ [K] |  $M_{\mathrm{WD}}$ [M$\odot$] | Contaminant | Original Survey
---|---|---|---|---|---|---|---|---|---
BD-13 6521 | 459.7 $\pm$ 2.0 | 6248.8 $\pm$ 37.8 | 4.424 $\pm$ 0.022 | 1.037 $\pm$ 0.016 | 1.039 $\pm$ 0.036 | 25525.6 $\pm$ 895.8 | 0.479 $\pm$ 0.024 | True | RAVE
TYC 4670-766-1 | 547.7 $\pm$ 0.598 | 5372.5 $\pm$ 29.8 | 4.409 $\pm$ 0.007 | 0.974 $\pm$ 0.011 | 0.888 $\pm$ 0.021 | 11018.0 $\pm$ 94.8 | 0.456 $\pm$ 0.009 | False | RAVE
TYC 5202-162-1 | 826.5 $\pm$ 61.9 | 5888.8 $\pm$ 43.5 | 3.498 $\pm$ 0.021 | 3.821 $\pm$ 0.112 | 1.674 $\pm$ 0.033 | 31222.2 $\pm$ 5208.8 | 0.421 $\pm$ 0.072 | False | RAVE
UCAC2 28312072 | 442.9 $\pm$ 0.5 | 5154.5 $\pm$ 14.0 | 4.593 $\pm$ 0.007 | 0.777 $\pm$ 0.004 | 0.865 $\pm$ 0.010 | 12014.4 $\pm$ 227.5 | 0.578 $\pm$ 0.005 | True | RAVE
TYC 6086-1317-1 | 524.2 $\pm$ 3.3 | 6763.8 $\pm$ 44.5 | 4.106 $\pm$ 0.017 | 1.679 $\pm$ 0.018 | 1.310 $\pm$ 0.035 | 20844.7 $\pm$ 378.6 | 0.377 $\pm$ 0.012 | False | RAVE
… | … | … | … | … | … | … | … | … | …
Lamost J111853.41-084457.3 | 718.0 $\pm$ 4.4 | 6070.9 $\pm$ 40.1 | 4.361 $\pm$ 0.015 | 1.234 $\pm$ 0.019 | 1.271 $\pm$ 0.023 | 12940.0 $\pm$ 231.2 | 0.492 $\pm$ 0.018 | True | LAMOST
Lamost J035311.09+290033.4 | 1484.3 $\pm$ 134.1 | 5172.5 $\pm$ 48.6 | 2.874 $\pm$ 0.041 | 8.467 $\pm$ 0.144 | 1.958 $\pm$ 0.141 | 52999.1 $\pm$ 10745.2 | 1.109 $\pm$ 0.162 | False | LAMOST
TYC 1749-1463-1 | 718.3 $\pm$ 3.2 | 6098.2 $\pm$ 50.2 | 4.139 $\pm$ 0.021 | 1.433 $\pm$ 0.017 | 1.030 $\pm$ 0.038 | 28958.9 $\pm$ 1506.8 | 0.561 $\pm$ 0.017 | False | LAMOST
Lamost J084109.22+254236.1 | 464.1 $\pm$ 1.7 | 5738.6 $\pm$ 27.3 | 4.494 $\pm$ 0.012 | 0.951 $\pm$ 0.008 | 1.028 $\pm$ 0.020 | 14748.5 $\pm$ 333.1 | 0.515 $\pm$ 0.010 | False | LAMOST
Lamost J083358.62+333406.8 | 530.6 $\pm$ 5.7 | 6780.1 $\pm$ 69.6 | 4.248 $\pm$ 0.020 | 1.400 $\pm$ 0.024 | 1.261 $\pm$ 0.030 | 26649.8 $\pm$ 1476.5 | 0.438 $\pm$ 0.033948 | False | LAMOST
… | … | … | … | … | … | … | … | … | …
TYC 6465-1734-1 | 1285.0 $\pm$ 45.1 | 6196.1 $\pm$ 29.1 | 4.419 $\pm$ 0.016 | 1.063 $\pm$ 0.007 | 1.081 $\pm$ 0.030 | 19320.6 $\pm$ 1955.3 | 0.547 $\pm$ 0.041 | False | TGAS
TYC 9148-665-1 | 1319.1 $\pm$ 56.4 | 5547.2 $\pm$ 23.5 | 4.568 $\pm$ 0.010 | 0.815 $\pm$ 0.004 | 0.894 $\pm$ 0.016 | 15323.5 $\pm$ 2813.9 | 0.596 $\pm$ 0.039 | False | TGAS
TYC 9344-137-1 | 164.2 $\pm$ 1.1 | 4751.0 $\pm$ 30.9 | 3.119 $\pm$ 0.038 | 4.626 $\pm$ 0.052 | 1.028 $\pm$ 0.085 | 5081.9 $\pm$ 526.8 | 0.184 $\pm$ 0.023 | True | TGAS
TYC 8823-1109-1 | 820.0 $\pm$ 8.1 | 6455.1 $\pm$ 41.0 | 4.253 $\pm$ 0.016 | 1.394 $\pm$ 0.013 | 1.265 $\pm$ 0.029 | 29190.5 $\pm$ 1464.8 | 0.528 $\pm$ 0.010 | False | TGAS
TYC 8384-1121-1 | 793.4 $\pm$ 12.0 | 6680.4 $\pm$ 75.3 | 4.200 $\pm$ 0.019 | 1.506 $\pm$ 0.025 | 1.308 $\pm$ 0.030 | 24888.5 $\pm$ 901.7 | 0.496 $\pm$ 0.019 | False | TGAS
… | … | … | … | … | … | … | … | … | …
Table 5: Table of spectroscopic binary systems, separated by their original survey. Here, RAVE, LAMOST and TGAS are used as shorthand for Parsons et al. (2016), Rebassa-Mansergas et al. (2017) and Ren et al. (2020) respectively. The full table consisting of all 35 RAVE spectroscopic binaries, 32 LAMOST spectroscopic binaries and 64 TGAS spectroscopic binaries, with a more complete list of properties, will be made available online upon publication. Name |  $P_{\mathrm{orb}}$ [d] |  $T_{\mathrm{eff,LS}}$ [K] | log($g$)LS [dex] |  $R_{\mathrm{LS}}$ [R$\odot$] |  $M_{\mathrm{LS}}$ [M$\odot$] |  $T_{\mathrm{eff,WD}}$ [K] |  $M_{\mathrm{WD}}$ [M$\odot$] | Contaminant | Original Survey
---|---|---|---|---|---|---|---|---|---
TYC 6992-827-1 | 41.3 $\pm$ 0.1 | 5249.1 $\pm$ 28.1 | 3.520 $\pm$ 0.027 | 3.615 $\pm$ 0.113 | 1.578 $\pm$ 0.086 | - | 0.128 $\pm$ 0.010 | False | RAVE
TYC 6419-603-1 | 37.5 $\pm$ 0.0 | 4409.2 $\pm$ 13.9 | 2.647 $\pm$ 0.016 | 8.014 $\pm$ 0.124 | 1.039 $\pm$ 0.032 | 5800.6 $\pm$ 780.1 | 0.228 $\pm$ 0.020 | False | RAVE
TYC 5263-340-1 | 5.7 $\pm$ 0.0 | 4813.7 $\pm$ 14.0 | 4.527 $\pm$ 0.010 | 0.823 $\pm$ 0.004 | 0.833 $\pm$ 0.015 | 5526.2 $\pm$ 170.4 | 0.227 $\pm$ 0.004 | False | RAVE
BD-16 210 | 19.5 $\pm$ 0.0 | 4541.0 $\pm$ 9.9 | 2.669 $\pm$ 0.017 | 7.557 $\pm$ 0.108 | 0.972 $\pm$ 0.024 | - | 0.132 $\pm$ 0.004 | False | RAVE
UCAC3 161-284029 | 2.1 $\pm$ 0.0 | 4059.4 $\pm$ 16.6 | 3.812 $\pm$ 0.063 | 1.088 $\pm$ 0.008 | 0.283 $\pm$ 0.036 | - | 0.054 $\pm$ 0.005 | True | RAVE
… | … | … | … | … | … | … | … | … | …
Lamost J043826.34+394739.7 | 5.1 $\pm$ 0.0 | 5457.9 $\pm$ 74.9 | 2.138 $\pm$ 0.053 | 25.040 $\pm$ 1.935 | 3.112 $\pm$ 0.112 | 5908.7 $\pm$ 2260.8 | 0.150 $\pm$ 0.031 | False | LAMOST
Lamost J053056.93+460345.7 | 60.1 $\pm$ 0.1 | 5441.3 $\pm$ 214.3 | 2.665 $\pm$ 0.154 | 10.600 $\pm$ 0.345 | 1.983 $\pm$ 0.556 | - | 1.818 $\pm$ 0.281 | True | LAMOST
Lamost J054307.98+502435.6 | 45.2 $\pm$ 0.2 | 6019.7 $\pm$ 48.0 | 2.293 $\pm$ 0.087 | 11.482 $\pm$ 0.395 | 0.884 $\pm$ 0.205 | 29063.6 $\pm$ 12979.4 | 0.318 $\pm$ 0.056 | False | LAMOST
Lamost J051005.99+492321.9 | 146.3 $\pm$ 0.5 | 5436.0 $\pm$ 76.9 | 2.046 $\pm$ 0.055 | 25.608 $\pm$ 1.801 | 2.651 $\pm$ 0.184 | - | 0.954 $\pm$ 0.050 | False | LAMOST
Lamost J050413.97+530343.8 | 3.2 $\pm$ 0.0 | 5215.0 $\pm$ 137.4 | 2.495 $\pm$ 0.108 | 11.344 $\pm$ 0.460 | 1.515 $\pm$ 0.403 | 5903.1 $\pm$ 2285.2 | 0.164 $\pm$ 0.036 | False | LAMOST
… | … | … | … | … | … | … | … | … | …
TYC 2562-1312-1 | 16.3 $\pm$ 0.0 | 6084.4 $\pm$ 40.7 | 4.418 $\pm$ 0.021 | 1.030 $\pm$ 0.009 | 1.012 $\pm$ 0.036 | - | 0.172 $\pm$ 0.008 | True | TGAS
TYC 1241-181-1 | 1.7 $\pm$ 0.0 | 6397.8 $\pm$ 79.3 | 3.634 $\pm$ 0.028 | 2.314 $\pm$ 0.026 | 0.867 $\pm$ 0.035 | 19321.1 $\pm$ 810.8 | 0.456 $\pm$ 0.020 | False | TGAS
TYC 1218-1456-1 | 1.3 $\pm$ 0.0 | 7729.5 $\pm$ 56.0 | 3.995 $\pm$ 0.029 | 2.133 $\pm$ 0.028 | 1.639 $\pm$ 0.092 | - | 0.167 $\pm$ 0.009 | False | TGAS
TYC 1783-665-1 | 18.0 $\pm$ 0.0 | 5708.7 $\pm$ 14.2 | 4.538 $\pm$ 0.010 | 0.818 $\pm$ 0.012 | 0.845 $\pm$ 0.008 | - | 0.130 $\pm$ 0.011 | False | TGAS
TYC 2803-978-1 | 1035.9 $\pm$ 218.5 | 6210.1 $\pm$ 40.1 | 4.118 $\pm$ 0.038 | 1.589 $\pm$ 0.078 | 1.203 $\pm$ 0.049 | 10159.4 $\pm$ 3025.5 | 0.291 $\pm$ 0.049 | True | TGAS
… | … | … | … | … | … | … | … | … | …