# The White Dwarf Binary Pathways Survey - IX. Three long period white dwarf
plus subgiant binaries

S. G. Parsons1, M. S. Hernandez2,3, O. Toloza2,3, M. Zorotovic4, M. R.
Schreiber2,3, B. T. Gänsicke5, F. Lagos5, R. Raddi6, A. Rebassa-Mansergas6,7,
J. J. Ren8 and D. Koester9
1 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK
2 Departamento de Física, Universidad Técnica Santa María, Avenida España
1680, Valparaíso, Chile
3 Millennium Nucleus for Planet Formation, NPF, Valparaıso, Av. España 1680,
Chile
4 Instituto de Física y Astronomía de la Universidad de Valparaíso, Av. Gran
Bretaña 1111, Valparaíso, Chile
5 Department of Physics, University of Warwick, Coventry CV4 7AL, UK
6 Departament de Física, Universitat Politècnica de Catalunya, c/Esteve
Terrades 5, E-08860 Castelldefels, Spain
7 Institute for Space Studies of Catalonia, c/Gran Capità 2-4, Edif. Nexus
201, E-08034 Barcelona, Spain
8 Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, P. R. China
9 Institut für Theoretische Physik und Astrophysik, University of Kiel, 24098
Kiel, Germany<EMAIL_ADDRESS>
(Accepted 2022 November 15. Received 2022 November 10; in original form 2022
July 15)

###### Abstract

Virtually all binaries consisting of a white dwarf with a non-degenerate
companion can be classified as either close post-interaction systems (with
orbital periods of a few days or less), or wide systems (with periods longer
than decades), in which both components have effectively evolved as single
stars. Binaries with periods between these two extremes can help constrain
common envelope efficiency, or highlight alternative pathways towards the
creation of compact binaries. To date such binaries have remained mostly
elusive. Here we present three white dwarfs in binaries with evolved subgiant
stars with orbital periods of 41, 52 and 461 d. Using Hubble Space Telescope
spectroscopy we find that all three systems contain low mass white dwarfs
($\leq$0.4 $\mathrm{M_{\odot}}$). One system, TYC 8394$-$1331$-$1, is the
inner binary of a hierarchical triple, where the white dwarf plus subgiant
binary is orbited by a more distant companion star. These binaries were likely
formed from a phase of stable but non-conservative mass transfer, as opposed
to common envelope evolution. All three systems will undergo a common envelope
phase in the future, but the two shorter period systems are expected to merge
during this event, while the longest period system is likely to survive and
create a close binary with two low mass white dwarfs.

###### keywords:

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

††pubyear: 2022††pagerange: The White Dwarf Binary Pathways Survey - IX. Three
long period white dwarf plus subgiant binaries–8

##  1 Introduction

Around one quarter of all solar-type stars are found in close binaries
(P${}_{\mathrm{orb}}<10^{4}$ days, Moe et al. 2019), many of which are
expected to interact with each other when the more massive member of the
binary evolves off the main-sequence, often leading to a common envelope phase
and a shrinking of the binary separation (Paczynski, 1976; Willems & Kolb,
2004). This evolutionary pathway is thought to lead to the creation of compact
binaries such as cataclysmic variables, double degenerate binaries and
thermonuclear supernovae (Webbink, 1984).

The common envelope phase itself, during which the core of the evolved star
and its binary companion orbit within a shared envelope of material, is a
brief but extremely complex process, which hydrodynamical models cannot yet
fully recreate (Ivanova et al., 2013). Instead, a simple energy equation with
an efficiency, $\alpha_{\mathrm{CE}}$, is typically used in binary population
models. High values of $\alpha_{\mathrm{CE}}$ imply efficient removal of the
envelope and hence a small loss of orbital energy from the binary, resulting
in relatively wide orbits after the common envelope. Conversely, low values of
$\alpha_{\mathrm{CE}}$ imply inefficient envelope removal, causing a
significant reduction in the binary separation and creating very close post-
common envelope binaries.

Studies of close white dwarf plus M dwarf or brown dwarf binaries have shown
that $\alpha_{\mathrm{CE}}$ is likely quite low for this class of systems,
with values of $\alpha_{\mathrm{CE}}\simeq$ 0.2-0.3 typically quoted
(Zorotovic et al., 2010; Zorotovic et al., 2014; Toonen & Nelemans, 2013;
Camacho et al., 2014; Zorotovic & Schreiber, 2022). However, while such a low
value of $\alpha_{\mathrm{CE}}$ is able to recreate this population of
binaries, it is unclear how universal this is, particularly at higher stellar
masses. Indeed, the common envelope phase in high-mass stars is often modelled
as highly efficient, $\alpha_{\mathrm{CE}}\simeq 1$ (e.g. Belczynski et al.,
2002). Moreover, low efficiencies are unable to reproduce populations of
double white dwarf binaries, since these systems need to remain at relatively
wide separations after the first mass transfer phase in order to survive the
second phase (Nelemans et al., 2000).

The challenge of reconstructing the evolution of double white dwarf binaries
with the classical energy balance approach led Nelemans et al. (2000) to
suggest the so-called $\gamma$-formalism, which considers angular momentum
balance (with some efficiency $\gamma$), as an alternative. This has the
advantage that in some cases the separation of the two stars does not change
much or can even increase as a result of mass transfer. However, this approach
is not without its problems, as it was specifically designed to address the
evolution of double white dwarf binaries it is unclear how applicable it is to
other types of binaries undergoing common envelope evolution. Moreover, the
$\gamma$-formalism appears to predict some systems that may violate energy
conservation during their formation if only orbital and thermal energies are
available (Ivanova et al., 2013). As an alternative to this approach Webbink
(2008) suggested that instead the first mass transfer event could be
dynamically stable but non-conservative (as opposed to common envelope
evolution, which is dynamically unstable and non-conservative), which can
occur if the original mass ratio of the binary is close to one (Ge et al.,
2020) and can lead to orbital expansion resulting in wider binaries (Kawahara
et al., 2018; Lagos et al., 2022). Subsequent modelling has shown that the
population of observed double white dwarf binaries can be recreated with a
phase of stable and non-conservative mass transfer (Woods et al., 2012;
Schreiber et al., 2022) and also appears to be important for the creation of
several other types of binary systems (e.g. Chen et al., 2017; Vos et al.,
2019). However, due to their large mass ratios, the progenitor systems of
close white dwarf plus M dwarf binaries are not expected to experience stable
and non-conservative mass transfer. White dwarfs with high mass companions are
needed in order to investigate this process.

While there is much observational data for close white dwarf plus M dwarf
binaries (e.g. Rebassa-Mansergas et al., 2010; Rebassa-Mansergas et al.,
2016), the situation is poorer for white dwarfs in close binaries with more
massive A, F, G and K type stars (hereafter WD+AFGK binaries). In these
binaries the white dwarfs are outshone by their companions at optical
wavelengths by factors of hundreds or even thousands, only revealing
themselves in the ultraviolet (Parsons et al., 2016). WD+AFGK binaries are
particularly important to study, since they represent the last common ancestor
for a wide variety of compact binaries. Depending upon the post-mass transfer
separation and stellar masses they may go on to form supersoft X-ray source
systems and cataclysmic variables if they emerge at short periods (Parsons et
al., 2015; Hernandez et al., 2021, 2022b), or symbiotic binaries and double
white dwarf systems if they emerge at longer periods (Zorotovic et al., 2014).
Moreover, WD+AFGK binaries allow us to probe the common envelope efficiency at
intermediate masses and investigate whether phases of stable but non-
conservative mass transfer may have occurred in some of these systems and why
this may be the case for some systems but not others.

The majority of WD+AFGK binaries studied to date have very short orbital
periods, consistent with the same low value of $\alpha_{\mathrm{CE}}$ seen in
white dwarf plus M dwarf binaries (O’Brien et al., 2001; Parsons et al., 2015;
Krushinsky et al., 2020; Hernandez et al., 2021, 2022b). However, a handful of
longer period systems have also been identified, most of which were found as
self-lensing binaries by the Kepler space mission (Kruse & Agol, 2014;
Kawahara et al., 2018; Masuda et al., 2019). The evolution of these longer
period systems cannot be reconstructed using the same low common envelope
efficiency (Zorotovic et al., 2014). While it is possible to include other
energy sources to help remove the envelope, such as recombination energy, it
is not clear how substantial this would be and even this additional energy
cannot help explain some of the widest systems.

In this paper we present 2MASS J18361702$-$5110583 (hereafter 2MASS
J1836$-$5110), TYC 6992$-$827$-$1 and TYC 8394$-$1331$-$1, three new white
dwarf binaries with orbital periods substantially longer than typical WD+AFGK
binaries (P${}_{\mathrm{orb}}>40$ days) and evolved subgiant star companions
(all three companions can be classified as K1$-$G9 IV stars). Their long
periods make them ideal laboratories to see if it is possible to create these
systems via standard, low efficiency common envelope evolution or whether a
stable but non-conservative mass transfer phase is required instead. With this
in mind, we measure the stellar and binary parameters and reconstruct the past
and future evolution of these systems.

##  2 Observations and their reduction

The three systems presented in this paper were first identified as candidate
WD+AFGK binaries by Parsons et al. (2016) who detected excess flux at
ultraviolet wavelengths based on a combination of optical Radial Velocity
Experiment (RAVE) data (Kordopatis et al., 2013) and ultraviolet Galaxy
Evolution Explorer (GALEX) data (Bianchi, 2014). Follow up ground-based
spectroscopy revealed radial velocity variations in these systems, indicating
binarity. Further ground-based spectroscopic observations were obtained in
order to determine the orbital periods, while space-based ultraviolet data
were taken in order to determine the properties of the white dwarfs. In this
section we outline these ground- and space-based observations.

###  2.1 Optical Spectroscopy

Ground-based data were collected with a range of telescopes and instruments
over multiple years and comprise data taken in both visitor and service mode.
Here we summarise the data collection and reduction procedure for each
instrument.

####  2.1.1 Du Pont echelle

We used the high resolution echelle spectrograph (1 arcsec slit, R$\simeq$40
000, covering the wavelength range 3700Å to 7000Å) on the 2.5-m Du Pont
telescope located at Las Campanas Observatory, Chile to obtain two spectra of
2MASS J1836$-$5110 (on 2014 June 1 and 2) and four spectra of TYC
6992$-$827$-$1 (on 2014 June 2, 2015 January 3, 4 and 5). Each science
observation was bracketed by ThAr spectra to correct for instrumental drift.
Standard image reductions were performed and the spectra optimally extracted
and wavelength calibrated using the Collection of Elemental Routines for
Echelle Spectra (CERES) package (Brahm et al., 2017).

####  2.1.2 FEROS

High resolution spectra for all three systems were obtained with the FEROS
echelle spectrograph (R$\simeq$48 000) on the 2.2-m Telescope at La Silla,
Chile (Kaufer & Pasquini, 1998). FEROS covers the wavelength range from
$\simeq$3500Å to $\simeq$9200Å. Observations were performed in Object-
Calibration mode where one fibre is placed on the target while the other feeds
light from a ThAr+Ne calibration lamp permitting velocity measurements to
extremely high precision ($\simeq$10$\mathrm{m\,s^{-1}}$) and allowed us to
correct for instrumental drift throughout the night. FEROS data were obtained
over multiple nights covering several years and all were reduced using the
CERES package.

####  2.1.3 CHIRON

Both TYC 6992$-$827$-$1 and TYC 8394$-$1331$-$1 were observed with the high
resolution echelle spectrometer CHIRON (Tokovinin et al., 2013) on the 1.5-m
SMARTS telescope at Cerro Tololo, Chile. We used $3\times 1$ binning resulting
in R$\simeq$40 000 covering a wavelength range of 4150Å to 8800Å. CHIRON
observations were performed in service mode over many nights in both 2014 and
2015. These data are automatically reduced by the CHIRON team using standard
reduction methods.

####  2.1.4 UVES

All three targets were observed with UVES (Dekker et al., 2000), a high
resolution echelle spectrograph mounted on the 8.2-m European Southern
Observatory Very Large Telescope at Cerro Paranal, Chile, as part of a poor
weather program. We used the dichroic 1 setup (390+564) with a 0.7 arcsec
slit, resulting in R$\simeq$50 000 and covering the wavelength range 3260Å to
6680Å with a small gap between 4540Å and 4580Å. Typical exposure times of 300
s were used and the data were reduced using the UVES data reduction pipeline
(version 5.8.2) using standard reduction methods within esoreflex.

####  2.1.5 X-Shooter

A single observation of TYC 8394$-$1331$-$1 was obtained with the medium
resolution spectrograph X-Shooter (Vernet et al., 2011), mounted on the 8.2-m
European Southern Observatory Very Large Telescope on the night of 2021 June
6. X-Shooter covers both the optical and near-infrared wavelength ranges using
three arms, the UVB (3000$-$5600 Å), VIS (5600$-$10,000 Å) and NIR (10
000$-$24 000 Å). Separate slit widths can be set for each arm and our
observations were performed with slit widths of 1.0, 0.9 and 0.9 arcsec in the
UVB, VIS and NIR arms respectively, giving R$\simeq$5 000. Observations were
performed in STARE mode, with exposure times of 120 s in the UVB and VIS arms
and 60 s in the NIR arm. The data were reduced using the X-shooter reduction
pipeline (version 3.5.0) within esoreflex.

###  2.2 Ultraviolet Spectroscopy

We obtained far-ultraviolet (FUV) spectroscopy of all three systems in order
to confirm the presence of a white dwarf companion and measure the stellar
parameters. We used the Space Telescope Imaging Spectrograph (STIS, Kimble et
al. 1998) on-board the Hubble Space Telescope (HST) as part of program GO
16224. TYC 8394$-$1331$-$1 was observed on 2021 May 27 over one spacecraft
orbit, with a total exposure time of 2325 s. TYC 6992$-$827$-$1 was observed
on 2022 January 10 for two orbits, for a total exposure time of 4980 s. 2MASS
J1836$-$5110 was observed on 2021 April 9 for two orbits, for a total exposure
time of 5244 s. Data were acquired with the MAMA detector and the G140L
grating. The spectra cover a wavelength range of 1150$-$1730Å with a resolving
power between 960$-$1440\. The spectra were reduced and wavelength calibrated
following the standard STIS pipeline (Sohn et al., 2019).

##  3 Fitting procedures

###  3.1 Radial velocity measurements

Radial velocities were computed from all our echelle spectra using cross-
correlation against a binary mask representative of a G2-type star, which is
the closest spectral type to our targets (K$0\pm 1$, i.e. K1$-$G9) available
(see Brahm et al. 2017 for more information). While this technique is
specifically designed to work with main-sequence stars, we have found that the
results are also reliable for subgiant stars, provided they are relatively
slow rotators (which is the case for all three systems presented in this
paper). In general this technique yields velocities to a precision of
$\sim$0.1 $\mathrm{km\,s^{-1}}$ or better. However, due to the unstable nature
of the Du Pont and CHIRON spectrographs, as well as the uncertain systematic
errors introduced in our UVES and X-shooter reductions, we placed lower limits
of 0.5 $\mathrm{km\,s^{-1}}$ on the precision of our velocity measurements
from these instruments. This limit is based on previous experience with these
instruments. The full list of radial velocity measurements for each system are
given in Tables 5, 6 and 7 in the appendix.

###  3.2 Orbital period measurements

Figure 1: Radial velocity fit for 2MASS J1836$-$5110\. The top panel shows the full data set and Keplerian orbit fit (Du Pont echelle – red dots, FEROS – green stars, UVES – blue squares). The periodogram is shown in the bottom-left and the phase-folded data are shown in the bottom-right panel (phase zero corresponds to the time of inferior conjunction of the subgiant star). Figure 2: Same as Figure 1 but for TYC 6992$-$827$-$1 (Du Pont echelle – red dots, FEROS – green stars, UVES – blue squares, CHIRON – magenta triangles). Figure 3: Radial velocity fit for TYC 8394$-$1331$-$1 using a combination of two Keplerian orbits. The top panel shows the full data set and fit ((FEROS – green stars, UVES – blue squares, CHIRON – magenta triangles, X-shooter – cyan diamonds). The centre-left panel shows the periodogram of the full dataset. The centre-right panel shows the data folded on the shorter period signal, with the longer period signal removed. The lower-left panel shows the periodogram with the strongest signal from the original periodogram subtracted off. The bottom-right panel shows the data folded on the longer period signal, with the shorter period signal removed. Table 1: Best fit Keplerian orbit parameters and uncertainties for 2MASS J1836$-$5110 and TYC 6992$-$827$-$1\. Parameters directly constrained from the data are indicated as "fitted", while parameters derived from these are listed as "derived". Parameter | 2MASS J1836$-$5110 | TYC 6992$-$827$-$1
---|---|---
Fitted:
Porb [days] | $461.48\pm 0.04$ | $41.45\pm 0.01$
Tconj [BJD] | $2458189.9\pm 0.1$ | $2457073.50\pm 0.06$
$\sqrt{e}\cos{\omega}$ | $-0.075\pm 0.005$ | $-0.06\pm 0.06$
$\sqrt{e}\sin{\omega}$ | $-0.151\pm 0.005$ | $0.08\pm 0.06$
$\ln{K}$ [$\mathrm{km\,s^{-1}}$] | $1.700\pm 0.001$ | $1.683\pm 0.006$
Derived:
$\gamma$ [$\mathrm{km\,s^{-1}}$] | $25.717\pm 0.004$ | $-41.969\pm 0.005$
$e$ | $0.028\pm 0.001$ | $0.013\pm 0.06$
$K$ [$\mathrm{km\,s^{-1}}$] | $5.476\pm 0.005$ | $5.38\pm 0.03$
Table 2: Best fit parameters and uncertainties for the double Keplerian orbit of TYC 8394$-$1331$-$1\. The subscripts 1 and 2 refer to the inner and outer orbits respectively. Parameter | value
---|---
Fitted:
Porb,1 [days] | $51.851\pm 0.009$
Tconj,1 [BJD] | $2458337.8\pm 0.3$
$\sqrt{e_{1}}\cos{\omega_{1}}$ | $0.04\pm 0.10$
$\sqrt{e_{1}}\sin{\omega_{1}}$ | $0.07\pm 0.08$
$\ln{K_{1}}$ [$\mathrm{km\,s^{-1}}$] | $1.86\pm 0.02$
Porb,2 [days] | $863\pm 3$
Tconj,2 [BJD] | $2458549\pm 6$
$\sqrt{e_{2}}\cos{\omega_{2}}$ | $-0.03\pm 0.10$
$\sqrt{e_{2}}\sin{\omega_{2}}$ | $0.1\pm 0.1$
$\ln{K_{2}}$ [$\mathrm{km\,s^{-1}}$] | $1.80\pm 0.03$
Derived:
$\gamma$ [$\mathrm{km\,s^{-1}}$] | $35.4\pm 0.1$
$e_{1}$ | $0.02\pm 0.02$
$K_{1}$ [$\mathrm{km\,s^{-1}}$] | $6.4\pm 0.1$
$e_{2}$ | $0.03\pm 0.02$
$K_{2}$ [$\mathrm{km\,s^{-1}}$] | $6.1\pm 0.2$

Initial estimates of the orbital periods were made by fitting a constant plus
sine wave to the velocity measurements over a range of periods and computing
the $\chi^{2}$ of the resulting fit at each period (i.e. a periodogram). This
approach works well for orbits that are very close to circular. These initial
estimates were then used as starting points for fitting the Keplerian orbits
to the measured radial velocities using the Python package radvel (Fulton et
al., 2018).

We allowed the eccentricity to vary (where $\sqrt{e}\cos{\omega}$ and
$\sqrt{e}\sin{\omega}$ are the fitted parameters, $e$ is the eccentricity and
$\omega$ is the argument of periapsis) and fitted the radial velocity as
$\ln{K}$, where $K$ is the semi-amplitude (i.e. the velocity varies over time
$t$ as $K\sin{t}$), which is particularly useful when $K$ is large compared to
the uncertainty on each measurement. Along with these parameters we also
fitted Porb, the orbital period and Tconj, the time of inferior conjunction of
the subgiant star. The systemic velocity, $\gamma$, was solved analytically
during the fit and so was not a free
parameter111http://cadence.caltech.edu/~bfulton/share/Marginalizing_the_likelihood.pdf.

The distributions of our model parameters were found using the Markov chain
Monte Carlo (MCMC) method as implemented within radvel. We placed a uniform
prior on the eccentricity to ensure it stayed between zero and one and a
uniform prior on Tconj to prevent it from changing by more than one orbital
cycle.

For TYC 8394$-$1331$-$1 it was immediately clear that a single Keplerian orbit
was insufficient to model the radial velocity data. Therefore, this system was
modelled with a combination of two orbits. An initial estimate for the period
of the outer orbit was made by subtracting the best fit sinusoid from the
radial velocity data and fitting the residuals with an additional sine wave
over a range of periods. The best fit value was then used as the starting
period for the outer orbit for the radvel fit. For this system both orbits
were then fitted simultaneously using the same parameters for each orbit
detailed above.

There were no clear additional orbits in either 2MASS J1836$-$5110 or TYC
6992$-$827$-$1\. The best fit parameters for these two systems are listed in
Table 1, while the best fit orbital parameters for TYC 8394$-$1331$-$1 are
given in Table 2. The periodograms, best fits and residuals are shown in
Figures 1, 2 and 3 for 2MASS J1836$-$5110, TYC 6992$-$827$-$1 and TYC
8394$-$1331$-$1 respectively.

###  3.3 Subgiant parameters

Figure 4: Example spectra (black) and model atmosphere fits (red) for all
three systems. In each case a single UVES spectrum is shown. The fits cover
the entire spectral range, but we plot just the region around the Mg i triplet
for clarity.

All three objects presented in this paper were originally selected as
ultraviolet excess main-sequence stars ($\log{g}>3.5$) from RAVE DR4
(Kordopatis et al., 2013), before any Gaia data releases (Parsons et al.,
2016). However, the Gaia DR2 parallaxes revealed that these objects are
slightly evolved and subsequent RAVE data releases (which factor in Gaia
parallaxes) have revised down the surface gravities of all these objects
(Steinmetz et al., 2020). Our extensive follow-up spectroscopy and the release
of Gaia DR3 allowed us to place more precise constraints on these stars than
the most recent RAVE DR6 values and in this section we detail how this was
achieved.

We determined the stellar parameters of the subgiant stars in all three
binaries in two steps. Firstly, we fitted the high resolution optical spectra
of each object to measure the effective temperature (Teff,SG, where SG refers
to subgiant), surface gravity ($\log{g}_{\mathrm{SG}}$), metallicity ([M/H])
and rotational broadening ($v_{\mathrm{SG}}\sin{i}$) using the spectral
analysis software ispec (Blanco-Cuaresma et al., 2014b; Blanco-Cuaresma,
2019). We decided to fit all of the UVES spectra for each object, since these
data are generally the highest signal-to-noise ratio spectra and they cover a
wide wavelength range.

We used the moog222http://www.as.utexas.edu/~chris/moog.html radiative
transfer code and the MARCS GES/APOGEE model atmosphere grid (Gustafsson et
al., 2008). ispec uses a least-squares algorithm to minimize the difference
between the synthetic and observed spectra. In each iteration, the algorithm
varies one free parameter at a time in order to determine in which direction
it should move (Blanco-Cuaresma et al., 2014b). In order to speed up the
fitting, ispec interpolates between the pre-computed MARCS GES/APOGEE models
if the values lie inside the grid, otherwise a spectrum is synthesised using
moog. Micro- and macro-turbulences were not fitted but were determined using
empirical relations (Blanco-Cuaresma et al., 2014a). The linear limb darkening
coefficient was fixed at 0.6 and the resolution was also fixed at 50,000. All
spectra were first corrected for radial velocity shifts based on the measured
values and the velocity was then fixed at zero during the spectral fitting.
Initial values for Teff,SG, $\log{g}_{\mathrm{SG}}$, [M/H] were set using the
RAVE DR6 parameters (Steinmetz et al., 2020), while $v_{\mathrm{SG}}\sin{i}$
was initiated at 5$\mathrm{km\,s^{-1}}$. Additional fits with the initial
parameters altered by T${}_{\mathrm{eff,SG}}\pm 100$ K,
$\log{g}_{\mathrm{SG}}\pm 0.25$ dex, [M/H]$\pm 0.1$ and
$v_{\mathrm{SG}}\sin{i}\pm 5$ $\mathrm{km\,s^{-1}}$ were also performed, to
ensure that the fits always converged to the same best fit values. Examples of
the best fitting spectral models are shown in Figure 4, in a small range
around the Mg i triplet (although the entire UVES spectral range was included
in the fit, with the exception of any wavelength ranges affected by telluric
lines).

The best fit parameters for a specific star varied from spectrum to spectrum
by an amount comparable to the uncertainties on the individual fits. We
therefore combined the results from all UVES spectral fits and adopted the
inverse-variance weighted mean and variance as the final best fit parameters
and their uncertainties. These are listed in Table 3 for all three systems.

The second step in determining the subgiant parameters was to fit their
spectral energy distributions (SEDs) in order to measure their radii (RSG)
and, when combined with $\log{g}_{\mathrm{SG}}$, hence their masses (MSG). We
obtained Gaia DR3 $G_{\mathrm{BP}}$, $G$ and $G_{\mathrm{RP}}$ data (Gaia
Collaboration et al., 2021), 2MASS $J$, $H$ and $K_{\mathrm{S}}$ band data
(Skrutskie et al., 2006) and WISE $W1$ and $W2$ photometry (Cutri et al.,
2012) for all three systems. We fitted these photometric data with solar
metallicity BT-Settl models (Allard et al., 2011).

Table 3: Best fit stellar parameters for the subgiant stars in our binaries Parameter | 2MASS J1836$-$5110 | TYC 6992$-$827$-$1 | TYC 8394$-$1331$-$1
---|---|---|---
From spectral fit:
Teff,SG [K] | $5050\pm 50$ | $5250\pm 50$ | $5150\pm 20$
$\log{g}_{\mathrm{SG}}$ [dex] | $3.48\pm 0.05$ | $3.48\pm 0.04$ | $3.06\pm 0.02$
$[\mathrm{M/H}]$ [dex] | $-0.05\pm 0.07$ | $-0.10\pm 0.10$ | $0.03\pm 0.15$
$v_{\mathrm{SG}}\sin{i}$ [$\mathrm{km\,s^{-1}}$] | $4.7\pm 0.5$ | $2.6\pm 0.2$ | $6.2\pm 0.6$
From Gaia DR3:
$\varpi$ [mas] | $1.09\pm 0.02$ | $1.98\pm 0.06$ | $1.44\pm 0.07$
From STILISM or Schlafly & Finkbeiner (2011)∗:
$E(B-V)$ [mag] | $0.06\pm 0.02$ | $<0.02^{*}$ | $0.04\pm 0.02$
From SED fit:
RSG [$\mathrm{R_{\odot}}$] | $3.54\pm 0.07$ | $3.45\pm 0.12$ | $5.57\pm 0.24$
Derived:
MSG [$\mathrm{M_{\odot}}$] | $1.38\pm 0.16$ | $1.31\pm 0.14$ | $1.31\pm 0.12$

Models were generated for a given $\log{g}_{\mathrm{SG}}$ and Teff,SG, the
model fluxes (which are $4\pi\times F_{\mathrm{Eddington}}$) were then scaled
by a factor of (RSG/D)2, where D is the distance, and were reddened by a
factor $E(B-V)$. The model flux in each photometric band was then calculated
and compared to the observed values. The fit was performed using the MCMC
method as implemented in the emcee python package (Foreman-Mackey et al.,
2013). We placed Gaussian priors on $\log{g}_{\mathrm{SG}}$ and Teff,SG based
on the fit to the high resolution spectra. A prior was also placed on the
reddening, based on the value for each system from the STILISM reddening map
(Capitanio et al., 2017), the reddening was also forced to be a positive
value. For TYC 6992$-$827$-$1 the maps do not extend far enough to cover this
object, so we place an upper limit on the reddening of this object based on
the maximum reddening set by Schlafly & Finkbeiner (2011). Finally, we used
the Gaia DR3 parallaxes to calculate the distances (via parallax inversion,
note that all three systems have $\varpi/\sigma_{\varpi}>20$) and placed a
Gaussian prior on the parallax based on the Gaia DR3 value.

Figure 5: SED fits to the subgiant stars in the three systems presented in
this paper. The blue points show the GALEX NUV and FUV, Gaia
$G_{\mathrm{BP}}$, $G$ and $G_{\mathrm{RP}}$, 2MASS $J$, $H$ and
$K_{\mathrm{S}}$ and WISE $W1$ and $W2$ band measurements (error bars are
plotted but are too small to be visible). The best fit model spectra to the
subgiant stars are shown in light grey and the model fluxes in the observed
bands are shown as red points (the GALEX data were not included in the fit).
We also show the HST/STIS spectra of the white dwarfs in dark grey and the
best fit white dwarf spectrum in red. The lower panels show the residuals to
both the white dwarf and subgiant star fits.

The best fit SEDs are shown in Figure 5 and the best fit reddening, radii and
masses (from $\log{g}_{\mathrm{SG}}$) are listed in Table 3. The MCMC
parameter distributions are also shown in Figure 10 in the Appendix. Based on
the fitted temperatures, all three objects have the spectral classification of
K$0\pm 1$ IV. However, we add a note of caution here, since we will see in the
next section that the white dwarf cooling ages in 2MASS J1836$-$5110 and TYC
6992$-$827$-$1 are comparable to the thermal timescale of the subgiant stars.
Therefore, these stars may not be fully relaxed to their equilibrium radii.
Since the theoretical models are based on stars in equilibrium, it is possible
that additional systematic errors exist on the subgiant parameters in these
two systems. Unfortunately, since we do not know how far from equilibrium
these stars are and theoretical models do not exist for such objects, this is
the best we can do at present.

The GALEX FUV and NUV measurements are also indicated in Figure 5, although
these were not included in the SED fits (since the white dwarf contributes a
non-negligible amount of flux at these wavelengths). In all three cases the
NUV measurements show a clear excess, even after accounting for the white
dwarf contribution. A similar effect has been observed in other WD+AFGK
binaries (Hernandez et al., 2022b, a) and was attributed to chromospheric
emission from the rapidly rotating and highly active main-sequence star
components. While there is no evidence of rapid rotation in the subgiant stars
presented in this paper, there is clear evidence of chromospheric activity in
the form of Ca ii H and K emission from the subgiants in all three systems
(see Figure 6). Chromospheric emission is known to cause excess flux at NUV
wavelengths. For example NUV excesses have been detected in rapidly rotating
red giants, Dixon et al. 2020, and even the subgiant star in the wide white
dwarf K subgiant binary UCAC2 46706450 (Werner et al., 2020), and since
chromospheric emission is not included in the BT-Settl models used to model
the SEDs of our subgiant stars this is likely the cause of our under predicted
NUV fluxes.

Chromospheric emission may also be the cause of the slight overestimate of the
Gaia $G_{\mathrm{BP}}$ fluxes seen in Figure 5. However, excluding these
measurements from the SED fit does not significantly change the subgiant
parameters.

Figure 6: UVES spectra around the Ca ii H and K lines, showing clear emission
lines in all three systems (which track the motion of the subgiant stars),
likely due to chromospheric activity.

###  3.4 White dwarf parameters

Figure 7: Fits to the HST/STIS white dwarf spectra of the three systems
presented in this paper. Note that the Ly$\alpha$ geocoronal emission has not
been removed, but was masked during the fitting.

We fitted the HST/STIS spectra to determine the effective temperatures,
masses, radii and ages of the systems. To that purpose we computed a grid of
synthetic white dwarf models with pure hydrogen atmospheres using the code of
Koester (2010). The grid spans T${}_{\mathrm{eff}}=12\,000-30\,000$ K (in
steps of 200 K) and $\log{g}=6.0-9.0$ (in steps of 0.1 dex). The convective
mixing length parameter was set to 0.8. Thus, the free parameters were Teff,
$\log{g}$, parallax, and reddening. Flat priors based on the edges of the grid
were applied to Teff and $\log{g}$, and Gaussian priors were imposed on the
parallax and reddening using Gaia DR3 parallax and reddening maps,
respectively. Similar to the the subgiant fits, STILISM (Capitanio et al.,
2017) was used for 2MASS J1836$-$5110 and TYC 8394$-$1331$-$1, whereas a flat
prior from zero to a maximum reddening set by Schlafly & Finkbeiner (2011) was
considered for TYC 6992$-$827$-$1\. We also account for the effects of
interstellar neutral hydrogen using equation 1 from Jenkins (1971), however,
this has a very minor effect on the final white dwarf parameters (comparable
to the statistical errors) for all our targets.

The errors on the fluxes reported from STIS/MAMA observations are
significantly underestimated because important noise sources were neglected,
most importantly the read noise is set to zero and the gain to
one333https://hst-docs.stsci.edu/stisdhb/chapter-2-stis-data-
structure/2-5-error-and-data-quality-array. Therefore, we artificially
increased the errors on our spectra by including an additional ten per cent
uncertainty on all the fluxes added in quadrature.

The ensemble sampler emcee was used (Foreman-Mackey et al., 2013), where 100
walkers sample the parameter space during 5000 steps which ensures independent
samples. In general, the autocorrelation time for each parameter was less than
100. These best-fit parameters combined with interpolation of cooling models
of Althaus et al. (2013) provide the masses and ages for the white dwarfs.

2MASS J1836$-$5110 and TYC 6992$-$827$-$1 were both observed for two HST
orbits and we independently fitted the spectrum from each orbit. In the case
of 2MASS J1836$-$5110 both spectra give consistent results and we take the
average values of the two fits. However, the two spectra of TYC 6992$-$827$-$1
give significantly different results. The second spectrum is noisier and the
fit is poorer. We suspect that the second spectrum may not have been centred
correctly or is possibly suffering from high dark current. Therefore, for this
object we report the parameters from fitting the first spectrum only. The
final results for all three systems are listed in Table 4 and the HST spectra
along with the best fitting models are shown in Figure 7. The uncertainties
reported in Table 4 should be considered purely statistical since they do not
include any systematic uncertainties in the white dwarf models themselves.
These are typically of the order of 1.5 per cent in Teff and 0.04 dex in
$\log{g}$ (e.g. Barstow et al., 2003; Gianninas et al., 2011), comparable to
the quoted statistical uncertainties.

Both TYC 8394$-$1331$-$1 and 2MASS J1836$-$5110 show narrow absorption
features in their HST/STIS spectra. These are extremely weak and narrow for
2MASS J1836$-$5110 and are likely interstellar in origin. However, TYC
8394$-$1331$-$1 shows features of C ii at 1335 Å, C iii at 1175 Å, the silicon
doublet 1260, 1265 Å and Si i and Si iii around 1300 Å. Several of these lines
are contaminated by interstellar absorption lines, in addition the region
around 1300 Å can be contaminated by geo-coronal airglow emission of O i. With
this caveat, we fitted the spectrum of TYC 8394$-$1331$-$1 in order to
estimate the abundances of carbon, silicon and oxygen. We computed atmospheric
models which included the opacities of these elements in addition to hydrogen
and compared these to the best-fit pure hydrogen model with the same
temperature and surface gravity. We found a flux difference of the continuum
of $<$1.5 per cent, with the largest difference at wavelengths bluer than Ly
$\alpha$. Individual abundances were also poorly constrained due to the
weakness of the features. Therefore, given the small flux difference and the
large uncertainties of the abundances, we adopt the best-fit parameters from
the pure hydrogen model.

With the masses of both stellar components in the binary as well as the
orbital fits we can now determine the semi-major axis, $a$, using Kepler’s
third law:

$a^{3}=\frac{G(M_{\mathrm{WD}}+M_{\mathrm{SG}})P_{\mathrm{orb}}}{4\pi^{2}},$
(1)

where $G$ is the gravitational constant. We can also calculate the orbital
inclination, $i$, via the binary mass function:

$\frac{M_{\mathrm{WD}}^{3}\sin^{3}{i}}{(M_{\mathrm{WD}}+M_{\mathrm{SG}})^{2}}=\frac{P_{\mathrm{orb}}K}{2\pi
G}.$ (2)

Both the semi-major axis and orbital inclination are listed in Table 4.

Table 4: Best fit stellar parameters for the white dwarfs presented in this paper and binary parameters derived from the masses and orbital fits. Parameter | 2MASS J1836$-$5110 | TYC 6992$-$827$-$1 | TYC 8394$-$1331$-$1
---|---|---|---
From spectral fit:
Teff,WD [K] | $22250\pm 250$ | $15750\pm 50$ | $19400\pm 100$
$\log{g}_{\mathrm{WD}}$ [dex] | $7.49\pm 0.03$ | $7.14\pm 0.02$ | $6.53\pm 0.03$
Derived:
MWD [$\mathrm{M_{\odot}}$] | $0.40\pm 0.01$ | $0.28\pm 0.01$ | $0.24\pm 0.01$
$\tau_{\mathrm{cool,WD}}$ [Myr] | $7.2\pm 0.1$ | $6.9\pm 0.1$ | $175\pm 21$
$a$ [au] | $1.42\pm 0.04$ | $0.27\pm 0.01$ | $0.31\pm 0.01$
$i$ [deg] | $47\pm 4$ | $26\pm 2$ | $39\pm 2$

##  4 Notes on individual systems

In this section we describe each system in detail. The locations of all three
systems in the Gaia Hertzsprung-Russell diagram are shown in Figure 8, along
with an 1.3$\mathrm{M_{\odot}}$ evolutionary track for reference, taken from
the Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011)
Isochrones and Stellar Tracks (MIST) model grids (Dotter, 2016; Choi et al.,
2016).

###  4.1 2MASS J1836$-$5110

With an orbital period of 461 days, 2MASS J1836$-$5110 has the longest period
of the three systems presented in this paper and is comparable to the orbital
periods of the Kepler self-lensing systems. Despite the low eccentricity
($e=0.028\pm 0.001$), the probability of falsely rejecting a circular fit is
low according to the proposal of Lucy & Sweeney (1971), hence we consider the
measured eccentricity to be accurate and the binary is not quite circular.
While the white dwarf mass is the highest of our three systems ($0.40\pm 0.01$
$\mathrm{M_{\odot}}$), it is still much lower than typically seen in detached
white dwarf binaries (e.g. Zorotovic et al., 2011), including the self-lensing
systems (with the exception of the very low mass white dwarf in KIC 8145411,
Masuda et al. 2019).

Figure 8: Gaia DR3 Hertzsprung-Russell diagram with the location of the three
systems presented in this paper indicated in red. The blue line shows a
1.3$\mathrm{M_{\odot}}$ MIST evolutionary track. A random sample of objects
within 500 pc is shown in black.

We note that 2MASS J1836$-$5110 has a slightly high renormalised unit weight
error (RUWE) of 1.6 in Gaia DR3 (Gaia Collaboration et al., 2021), which is
often indicative of binarity. Given the long orbital period of this system it
is likely that the binary motion causes some astrometric variability, leading
to this high value (the semi-major axis for this system is $1.42\pm 0.04$ au).
However, this system does not feature in any of the Gaia DR3 non-single star
catalogues (Gaia Collaboration, 2022). This is likely due to its relatively
large distance of 900 pc and the low mass of the white dwarf, although the
orbital inclination of only $47\pm 4$ degrees is favourable for astrometric
detection. There are no clear signs that this binary is part of a higher order
system.

###  4.2 TYC 6992$-$827$-$1

The orbital period of TYC 6992$-$827$-$1 is 41 days, placing it between the
short period (post-common envelope) systems and longer period self-lensing
systems. To date, all of the ultraviolet excess objects from Parsons et al.
(2016) with periods in this range have turned out to be contaminants, from
either active stars in binaries or triple systems, where the ultraviolet
excess arises from a distant white dwarf companion to a main-sequence binary
(Lagos et al., 2020a, 2022). However, in this case we can rule out stellar
activity, since our HST data clearly reveal that the ultraviolet excess is
caused by a white dwarf. Moreover, the extremely low mass for the white dwarf
($0.28\pm 0.01$ $\mathrm{M_{\odot}}$) must be the result of binary
interaction, since a single star could not produce such a low mass white dwarf
within a Hubble time, making it unlikely that the white dwarf is a distant
companion to a binary. Finally, we note that the contaminating main-sequence
binaries identified in Lagos et al. (2020a, 2022) all have high eccentricities
($e>0.26$), which is not the case for TYC 6992$-$827$-$1\. While our fit to
the radial velocities does favour a very small eccentricity, we cannot reject
the possibility of a circular orbit in this binary (Lucy & Sweeney, 1971), in
stark contrast to the Lagos et al. (2020a, 2022) systems.

Similar to 2MASS J1836$-$5110, TYC 6992$-$827$-$1 also has a very high RUWE
value in Gaia DR3 of 4.2 (Gaia Collaboration et al., 2021). This system is
somewhat closer at 500 pc, but the much shorter orbital period and lower
stellar masses makes the astrometric signal from this binary weaker (the semi-
major axis for this system is $0.27\pm 0.01$ au), although the low inclination
of this system ($26\pm 2$ degrees) will somewhat boost the astrometric signal.
Therefore, while it is likely that the binary itself is responsible for the
high RUWE value, it is possible that TYC 6992$-$827$-$1 is part of a higher
order system, with a distant companion adding to the high astrometric scatter.
The system is listed as a single-lined spectroscopic binary in Gaia DR3 (Gaia
Collaboration, 2022), with orbital parameters consistent with our fit
(P${}_{\mathrm{orb}}=41.32\pm 0.06$ d, $e=0.09\pm 0.07$, $K=5.5\pm 0.3$
$\mathrm{km\,s^{-1}}$) , but is not classified as an astrometric binary, hence
we cannot confirm if it is a higher order system.

###  4.3 TYC 8394$-$1331$-$1

TYC 8394$-$1331$-$1 stands out from the other two objects because it is
clearly a triple system. However, based on the radial velocity data we can
confidently conclude that the white dwarf must be part of the inner 52 day
binary. This is because the mass function implies a minimum companion mass of
0.14 $\mathrm{M_{\odot}}$ for the inner orbit and 0.39 $\mathrm{M_{\odot}}$
for the outer orbit. We know that the white dwarf has a mass of $0.24\pm 0.01$
$\mathrm{M_{\odot}}$ from our HST spectrum, therefore, it cannot be the outer
companion, since this would require an non-physical inclination (i.e. the
measured white dwarf mass and outer orbit parameters give $\sin{i}>1$ in
Equation 2). Using the inner orbit parameters gives an inclination for the
inner binary of $39\pm 2$ degrees. If the outer orbit is co-planar with this
then the mass of the outer companion is $0.62\pm 0.04$ $\mathrm{M_{\odot}}$.
If this outer companion is a main-sequence star then it can easily be hidden
in the glare of the luminous subgiant star, which would be consistent with the
lack of any clear infrared excess in the SED.

The inner binary has a semi-major axis of $0.31\pm 0.01$ au, while the outer
orbit has a semi-major axis of 2.1-2.2 au, depending upon the mutual
inclination of the orbits. Despite being a compact triple
($a_{\mathrm{out}}/a_{\mathrm{in}}\simeq 7$) the system is secularly stable in
its current configuration, due mainly to the very circular orbits (Toonen et
al., 2020). Indeed, given the uncertainties on our measurements we cannot
exclude the possibility that both the inner and outer orbits are circular
(Lucy & Sweeney, 1971). Our measured parameters place the system outside of
eccentric Lidov-Kozai (LK) regime (in which extreme eccentricities close to
one can be achieved), but it will still experience lower order LK cycles (i.e.
quadrupole order Lidov, 1962; Kozai, 1962; Naoz, 2016). Depending upon the
mutual inclination, this could push the eccentricities as high as 0.77.

It is worth noting that mass loss from the central binary, due to the
evolution of the white dwarf progenitor, is likely to have increased the size
of the outer orbit in the past. The original system would therefore have been
even more compact than at present and hence the LK timescales would have been
much shorter in the past. It is possible that eccentricity oscillations could
have played a role in triggering the mass transfer phase in the inner binary
before the white dwarf progenitor had a chance to evolve, thus forming the
very low mass white dwarf we see today (e.g. Toonen et al., 2020).

TYC 8394$-$1331$-$1 was identified as a non-single source in Gaia DR3 with
clear astrometric acceleration, but no period was measured (Gaia
Collaboration, 2022). This is likely due to the compact triple nature of the
system, since such high order orbits were not modelled in Gaia DR3. The period
of the outer orbit is also at the upper end of what can be detected with the
baseline of Gaia DR3.

##  5 Discussion

Figure 9: Orbital period and mass distributions for post-mass transfer white
dwarf binaries. The grey-dashed line is the maximum orbital period for a post-
common envelope system with a 1$\mathrm{M_{\odot}}$ companion assuming high
efficiency (i.e. $\alpha_{\mathrm{CE}}=1$), but no additional energy sources
(Zorotovic et al., 2014). The black line and grey shaded region is the
theoretical $P_{\mathrm{orb}}-M_{\mathrm{WD}}$ relation for stable mass
transfer (Rappaport et al., 1995) while the black dotted line a modified
version of this for shorter period systems (Lin et al., 2011). Large circles
show post-common envelope binaries found in the Sloan Digital Sky Survey
(SDSS) from Rebassa-Mansergas et al. (2010); Rebassa-Mansergas et al. (2016).
Triangles show self-lensing systems (Kruse & Agol, 2014; Kawahara et al.,
2018; Masuda et al., 2019). Diamonds are EL CVn systems from Maxted et al.
(2014) and van Roestel et al. (2018). Small circles are the EL CVn progenitor
systems from El-Badry & Rix (2022). Squares are blue stragglers from Gosnell
et al. (2019). Star symbols show WD+AFGK systems (Landsman et al., 1993;
O’Brien et al., 2001; Parsons et al., 2015; Krushinsky et al., 2020; Hernandez
et al., 2021, 2022b), including the three presented in this paper, which are
highlighted in bold.

With precise stellar and binary parameters in hand for all three systems, in
this section we reconstruct their past evolution and predict their future
fates. For completeness, we include a list of the stellar and binary
parameters for all of the systems discovered in The White Dwarf Binary
Pathways Survey to date (include the three presented in this paper) in Table 8
in the Appendix.

###  5.1 Past evolution

In Figure 9 we show the orbital period as a function of white dwarf mass for
the observed sample of close white dwarf binaries with non-degenerate
companions. The mass of the companion is also shown colour coded. The dashed
line is the maximum orbital period for post-common envelope systems assuming a
1$\mathrm{M_{\odot}}$ companion and a highly efficient use of the orbital
energy during the common-envelope phase, i.e. $\alpha_{\mathrm{CE}}=1$, but
with no contribution of any other energy sources (Zorotovic et al., 2014). The
three systems studied in this paper lie well above this line, which gives us a
first indication that they are most likely not the result of a common envelope
phase. We note that there are two systems above this line, KOI 3278 and IK
Peg, for which common envelope evolution remains the most likely explanation
for their current orbital configuration (Zorotovic et al., 2014). However,
both contain more massive white dwarfs ($>0.5$ $\mathrm{M_{\odot}}$) that
descend from progenitors on the asymptotic giant branch, i.e. with extended
envelopes, and therefore the inclusion of a very small fraction of
recombination energy in the binding energy of the envelope is sufficient to
explain their current orbital periods. Given the low mass of the white dwarfs
in the three systems studied here, which imply their progenitors were not
highly evolved, it does not seem likely that the inclusion of recombination
energy in the calculation of the binding energy will help us to explain their
past evolution assuming a common envelope phase.

In order to definitively rule out that these are post-common envelope
binaries, we decided to reconstruct the common envelope phase using the same
algorithm described in detail in Hernandez et al. (2021), based on the binary
star evolution (BSE) code from Hurley et al. (2002). As we expected, no
possible progenitors were found for any of the three systems, even assuming
the most efficient use of orbital energy during the common-envelope phase
($\alpha_{\mathrm{CE}}=1$). We then also allowed our reconstruction algorithm
to include a variable fraction of the hydrogen recombination energy that
contributed to expel the envelope ($\alpha_{\mathrm{rec}}$ in Zorotovic et al.
2014). Even in the unrealistic case of including 100 per cent of this energy
in the calculations of the envelope’s binding energy, i.e.
$\alpha_{\mathrm{CE}}=1$ and $\alpha_{\mathrm{rec}}=1$, we were not able to
find any possible progenitors. We therefore conclude that the three systems
most likely experienced a phase of dynamically stable but non-conservative
mass transfer, which is consistent with their location in Figure 9 close to
the theoretical $P_{\mathrm{orb}}-M_{\mathrm{WD}}$ relation for stable mass
transfer from Rappaport et al. (1995).

We also include EL CVn binaries (close pre-white dwarfs with A/F main-sequence
star companions) in Figure 9, which all sit close to the theoretical
$P_{\mathrm{orb}}-M_{\mathrm{WD}}$ relation for stable mass transfer (Maxted
et al., 2014; van Roestel et al., 2018). The location of these systems is
unsurprising given the very low (pre-) white dwarf masses in these systems,
which imply that mass transfer occurred when the originally more massive star
was at the end of the main-sequence or shortly afterwards on the subgiant
branch (Chen et al., 2017). For EL CVn systems the mass transfer is stable and
a common envelope is avoided. It is thought that the mass transfer is at least
partially conservative for EL CVn systems, given the high masses of the
companion stars (Chen et al., 2017). Given the lower companion masses in our
new long period systems the mass transfer was likely far less conservative in
these cases (if at all), but the overall evolution may well be quite similar.
In particular, Lagos et al. (2020b) showed that EL CVn systems are generally
the inner binaries of hierarchical triples. This is because of the need for
the original main-sequence binary to be close in order to start mass transfer
at the end of the main-sequence and virtually all known close main-sequence
binaries ($P_{\mathrm{orb}}<3$ days) are known to be the inner binaries of
hierarchical triple systems (Tokovinin et al., 2006). We know that TYC
8394$-$1331$-$1 is also the inner binary of a hierarchical triple and it is
possible that TYC 6992$-$827$-$1 is as well, hence there are clear
similarities between our systems and EL CVn binaries.

We also include the recently discovered EL CVn progenitor systems identified
by El-Badry & Rix (2022). These single-lined binaries were originally though
to be high-mass function systems, but were shown instead to be binaries
containing highly stripped low mass giant donors, the majority of which are
recently detached systems (although at least one system appears to still be
mass transferring). In figure 9 these systems sit between the EL CVn binaries
and our newly discovered systems, reinforcing the link between these
populations.

We also point out the similarity of our systems to the long period hot
subdwarf-B (sdB) binaries presented in Vos et al. (2019), which are thought to
be the result of stable non-conservative mass transfer. Our systems appear to
represent the extreme lower white dwarf/sdB mass and orbital period limit of
this sample and a possible link between these systems and EL CVn binaries. Our
systems also closely resemble blue stragglers (e.g. Carney et al., 2001;
Gosnell et al., 2019; Leiner et al., 2019), which are also thought to descend
from stable mass transfer, although they often have moderately high
eccentricities, in contrast to our systems. Two blue straggler systems with
spectroscopically confirmed white dwarf companions are also shown in Figure 9.

Until now, all of the WD+AFGK systems characterised by our survey have stellar
and binary parameters consistent with being typical post-common envelope
binaries that do not require a highly efficient envelope removal (Parsons et
al., 2015; Hernandez et al., 2021, 2022b), which is also the case for all
white dwarf plus M dwarf or brown dwarf binaries (Zorotovic & Schreiber,
2022). All of the longer period systems have either been contaminants (Lagos
et al., 2020a, 2022) or the descendants of stable mass transfer. This implies
that $\alpha_{\mathrm{CE}}$ may always be small and independent of the
companion mass. Long period post-common envelope systems (e.g. IK Peg or KOI
3278) are likely to be rare.

It is clear that the WD+AFGK systems consistent with common envelope evolution
(i.e. those with short orbital periods and more typical white dwarf masses)
descend from binaries that were originally wider, allowing the white dwarf
progenitor to evolve before mass transfer began. Hence the original binary
separation plays a key role in determining the type of mass transfer that
occurs and the nature of the resulting system. The original mass ratio is also
important, since large mass ratios exclude stable mass transfer, hence the
reason why all the close white dwarf plus M dwarf binaries are clearly post-
common envelope systems (Ge et al., 2022). SLB1, 2 and 3 all likely had
original orbits similar to the post-common envelope systems, but mass ratios
close to one and hence evolved through stable mass transfer to their current
long orbits. Therefore, to form the systems presented in this paper requires
an initially short period and an original mass ratio not too different from
one.

###  5.2 Future evolution

We also use the BSE code in order to predict the future evolution of the three
systems. According to the current orbital period and mass ratio, the three
will enter a common envelope phase when the subgiant companion evolves to the
first giant branch. Using our standard small efficiency
($\alpha_{\mathrm{CE}}=0.2$-0.3) and without recombination energy TYC
6992$-$827$-$1 and TYC 8394$-$1331$-$1 are expected to experience a merger
during the common-envelope phase (note that we did not include any influence
from the outer companion when simulating the future of TYC 8394$-$1331$-$1).
What the outcome of these mergers will be is not clear, but these two systems
will most likely end up as single white dwarfs, somewhat more massive than the
low-mass white dwarfs we see today. This may be a possible formation channel
for creating single low mass white dwarfs (Nelemans & Tauris, 1998). On the
other hand, the longer orbital period derived for 2MASS J1836$-$5110 implies
it will most likely survive the common envelope phase and become a close
double helium-core white dwarf system. The later is expected to emerge from
the common envelope phase with white dwarf masses of
$\sim$0.40$\mathrm{M_{\odot}}$ (the white dwarf we observe today) and
$\sim$0.45$\mathrm{M_{\odot}}$ at an orbital period of 0.37-0.67 days. A
number of close double white dwarf systems containing two low mass white
dwarfs are already known (e.g. Bours et al. 2014; Parsons et al. 2020, see
Schreiber et al. 2022 for a full list) and these binaries may well have
descended from similar systems to those presented in this paper. Indeed
Nelemans et al. (2000) noted that binaries containing two low mass white
dwarfs are difficult to reproduce via two common envelope events.

All three systems might also appear as symbiotic systems for a short period
($\sim$100$-$200 Myr) before entering the common envelope phase, while the
giants are still under-filling their Roche Lobes and mass transfer comes from
a wind.

Finally, we reiterate the fact that the cooling ages of the white dwarfs in
both 2MASS J1836$-$5110 and TYC 6992$-$827$-$1 are comparable (or even shorter
than) the thermal timescale of the subgiant stars and hence the stellar
parameters for the subgiants in these two systems may not be completely
accurate. While this does not affect our conclusions of the past evolution of
these systems (where the orbital period and white dwarf mass are far more
important), it may alter the future evolution of these systems from our
predictions if the companion stars have lower masses than our estimates. In
particular, if we have strongly overestimated their masses then it is possible
that these systems will not evolve for longer than a Hubble time and hence
would not be representative of current day double white dwarf systems.
Measuring the orbits astrometrically would place strong additional constraints
on the masses and hence future Gaia data releases may well resolve this issue.

##  6 Conclusions

We have discovered three low mass white dwarfs in binaries with subgiant stars
with orbital periods substantially longer than typical post-common envelope
binaries. Using both ground- and space-based spectroscopy we placed precise
constraints on the stellar and binary parameters of all three systems,
revealing one to be the inner binary of a hierarchical triple. The combination
of long orbital periods and low white dwarf masses mean that we are unable to
reproduce these systems via common envelope evolution, even assuming highly
efficient envelope ejection and the addition of recombination energy. We
therefore conclude that these systems must be the result of stable but non-
conservative mass transfer, which demonstrates that white dwarfs in binaries
with intermediate mass A, F, G and early-K companions can follow radically
different evolutionary pathways depending upon the initial separation and mass
ratio of the binary.

## Acknowledgements

SGP acknowledges the support of a Science and Technology Facilities Council
(STFC) Ernest Rutherford Fellowship. MSH, OT, and MRS acknowledge support by
ANID – Millennium Science Initiative Program – NCN19_171. OT was also
supported by FONDECYT (grant 321038). MRS and MZ were also supported by
FONDECYT (grant 1221059). BTG was supported by grant ST/T000406/1 from the
STFC. 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). RR has received funding from the
postdoctoral fellowship programme Beatriu de Pinós, funded by the Secretary of
Universities and Research (Government of Catalonia) and by the Horizon 2020
programme of research and innovation of the European Union under the Maria
Skłodowska-Curie grant agreement No 801370. ARM acknowledges support from
Grant RYC$-$2016$-$20254 funded by MCIN/AEI/10.13039/501100011033 and by ESF
Investing in your future, and from MINECO under the PID2020$-$117252GB$-$I00
grant. The results presented in this paper are based on observations collected
at the European Southern Observatory under programme IDs 096.D-0217,
098.D-0023 and 108.D-0122. 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 Statement

Raw and reduced FEROS, UVES and X-shooter data are available through the ESO
archive at http://archive.eso.org/cms.html. CHIRON and Du Pont data will be
shared on reasonable request to the corresponding author.

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##  Appendix A Radial velocity measurements

Table 5: Velocity measurements for 2MASS J1836$-$5110 BJD (mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$) | Instrument
---|---|---|---
2456809.76248753 | 26.235 | 0.500 | Du Pont
2456810.75213144 | 26.323 | 0.500 | Du Pont
2456819.74917433 | 26.756 | 0.200 | FEROS
2456819.76019754 | 26.716 | 0.200 | FEROS
2456819.73815273 | 26.821 | 0.200 | FEROS
2456827.75170523 | 27.362 | 0.013 | FEROS
2456828.62383469 | 27.431 | 0.010 | FEROS
2456835.60862554 | 27.925 | 0.010 | FEROS
2457189.68636435 | 20.912 | 0.010 | FEROS
2457472.81628360 | 27.562 | 0.010 | FEROS
2457587.66341401 | 20.571 | 0.010 | FEROS
2458290.66927933 | 31.068 | 0.017 | FEROS
2458290.75708253 | 31.052 | 0.020 | FEROS
2458290.83118133 | 31.096 | 0.021 | FEROS
2458291.72839868 | 31.092 | 0.015 | FEROS
2458292.72061165 | 31.032 | 0.014 | FEROS
2458292.89637992 | 31.274 | 0.023 | FEROS
2458293.69924433 | 31.052 | 0.020 | FEROS
2458293.80774890 | 31.105 | 0.028 | FEROS
2458600.70033571 | 22.235 | 0.025 | FEROS
2458601.72229306 | 22.333 | 0.022 | FEROS
2458602.85311495 | 22.350 | 0.021 | FEROS
2458602.88968531 | 22.255 | 0.019 | FEROS
2458603.77265516 | 22.477 | 0.022 | FEROS
2458603.83193249 | 22.378 | 0.024 | FEROS
2458606.77059273 | 22.657 | 0.019 | FEROS
2458606.77465320 | 22.578 | 0.020 | FEROS
2458606.77871538 | 22.644 | 0.018 | FEROS
2458606.78279148 | 22.569 | 0.018 | FEROS
2458606.78686883 | 22.663 | 0.019 | FEROS
2459530.52684574 | 22.539 | 0.500 | UVES
2459508.59217903 | 21.553 | 0.500 | UVES
2459647.88459790 | 30.384 | 0.500 | UVES
2459653.77530590 | 30.416 | 0.500 | UVES
Table 6: Velocity measurements for TYC 6992$-$827$-$1 BJD (mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$) | Instrument
---|---|---|---
2456810.88987164 | -45.535 | 0.500 | Du Pont
2456828.77035696 | -38.964 | 0.010 | FEROS
2456829.81668064 | -38.271 | 0.012 | FEROS
2456829.91031794 | -38.255 | 0.010 | FEROS
2456830.81378745 | -37.772 | 0.010 | FEROS
2456831.91639108 | -37.303 | 0.010 | FEROS
2457002.63027686 | -36.798 | 0.010 | FEROS
2457003.63339269 | -37.072 | 0.010 | FEROS
2457004.54995757 | -37.425 | 0.010 | FEROS
2457025.59357472 | -46.098 | 0.500 | Du Pont
2457026.53553894 | -46.043 | 0.500 | Du Pont
2457027.55362046 | -45.898 | 0.500 | Du Pont
2457188.82273257 | -47.183 | 0.010 | FEROS
2457250.71244946 | -36.994 | 0.500 | CHIRON
2457251.83661388 | -37.380 | 0.500 | CHIRON
2457252.79154637 | -37.701 | 0.500 | CHIRON
2457261.66273899 | -43.801 | 0.500 | CHIRON
2457269.64503261 | -47.782 | 0.500 | CHIRON
2457283.71222538 | -39.627 | 0.500 | CHIRON
2457294.66224794 | -37.917 | 0.500 | CHIRON
2457297.69522221 | -39.590 | 0.500 | CHIRON
2457303.73600960 | -44.364 | 0.500 | CHIRON
2457311.68670972 | -47.468 | 0.500 | CHIRON
2457313.63664922 | -47.656 | 0.500 | CHIRON
2457319.61130834 | -43.733 | 0.500 | CHIRON
2457329.61830650 | -36.986 | 0.500 | UVES
2457332.60726505 | -36.971 | 0.500 | CHIRON
2457340.53546990 | -40.643 | 0.500 | UVES
2457364.59907431 | -41.053 | 0.500 | CHIRON
Table 7: Velocity measurements for TYC 8394$-$1331$-$1 BJD (mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$) | Instrument
---|---|---|---
2456873.776669648 | 39.523 | 0.500 | CHIRON
2456888.587295888 | 30.472 | 0.500 | CHIRON
2456906.592794124 | 27.957 | 0.500 | CHIRON
2456911.562011322 | 31.399 | 0.500 | CHIRON
2456920.612289260 | 36.885 | 0.500 | CHIRON
2456924.559734249 | 37.633 | 0.500 | CHIRON
2456926.577372765 | 37.477 | 0.500 | CHIRON
2456931.537606544 | 35.435 | 0.500 | CHIRON
2456937.555521341 | 31.019 | 0.500 | CHIRON
2456941.553596461 | 27.688 | 0.500 | CHIRON
2456944.514096235 | 25.927 | 0.500 | CHIRON
2457189.715487490 | 37.899 | 0.010 | FEROS
2457222.548450931 | 33.548 | 0.500 | CHIRON
2457223.670184840 | 34.324 | 0.500 | CHIRON
2457224.662427607 | 35.183 | 0.500 | CHIRON
2457249.605290546 | 35.341 | 0.500 | CHIRON
2457250.611555887 | 33.873 | 0.500 | CHIRON
2457251.743665238 | 33.247 | 0.500 | CHIRON
2457252.701218139 | 32.589 | 0.500 | CHIRON
2457257.570658996 | 30.039 | 0.500 | CHIRON
2457267.494933716 | 31.393 | 0.500 | CHIRON
2457273.705291479 | 35.463 | 0.500 | CHIRON
2457277.662845165 | 38.429 | 0.500 | CHIRON
2457283.616742769 | 42.241 | 0.500 | CHIRON
2457293.603606648 | 41.915 | 0.500 | CHIRON
2457319.564909399 | 33.068 | 0.500 | CHIRON
2457333.536274699 | 43.315 | 0.500 | CHIRON
2457454.882325782 | 42.476 | 0.500 | UVES
2457635.574287844 | 35.884 | 0.500 | UVES
2457640.602255318 | 39.406 | 0.500 | UVES
2457645.614058001 | 42.386 | 0.500 | UVES
2457665.541105256 | 34.940 | 0.500 | UVES
2457687.515129791 | 33.677 | 0.500 | UVES
2459371.918669399 | 40.037 | 0.550 | X-SHOOTER
2459508.581898705 | 34.272 | 0.500 | UVES
2459530.539642073 | 30.267 | 0.500 | UVES
2459538.528942762 | 24.979 | 0.500 | UVES
2459541.542958050 | 24.058 | 0.500 | UVES
2459649.875344526 | 23.428 | 0.500 | UVES
2459651.892380544 | 24.322 | 0.500 | UVES
2459668.886111152 | 35.160 | 0.500 | UVES
2459672.809120230 | 36.119 | 0.500 | UVES
2459673.828855139 | 36.316 | 0.500 | UVES

##  Appendix B SED fit corner plots

Figure 10: Posterior probability distributions for model parameters obtained
through fitting the SEDs of 2MASS J1836$-$5110 (left), TYC 6992$-$827$-$1
(centre) and TYC 8394$-$1331$-$1 (right).

##  Appendix C WD+AFGK binaries from The White Dwarf Binary Pathways Survey

Table 8: Published WD+AFGK binaries with measured orbits discovered in The White Dwarf Binary Pathways Survey. References: (1) Parsons et al. (2015), (2) Hernandez et al. (2022b), (3) Hernandez et al. (2021), (4) Hernandez et al. (2022a), (5) this paper.  Name | Porb [d] | Teff,WD [K] |  $\log{g}_{\mathrm{WD}}$ [dex] | MWD [$\mathrm{M_{\odot}}$] | Teff,2 [K] |  $\log{g}_{\mathrm{2}}$ [dex] | M2 [$\mathrm{M_{\odot}}$] | R2 [$\mathrm{R_{\odot}}$] |  $[\mathrm{M/H}]$ [dex] |  $a$ [$\mathrm{R_{\odot}}$] |  $i$ [deg] | Distance [pc] | Reference
---|---|---|---|---|---|---|---|---|---|---|---|---|---
TYC 6760$-$497$-$1 | 0.49869(3) | $20250\pm 750$ | $7.95\pm 0.15$ | $0.60\pm 0.08$ | $6400\pm 100$ | $4.40\pm 0.08$ | $1.24\pm 0.02$ | $1.29\pm 0.11$ | $-0.28\pm 0.03$ | $3.24\pm 0.04$ | $38\pm 5$ | $295.2\pm 2.6$ | (1)
TYC 110$-$755$-$1 | 0.85805(1) | $16850\pm 35$ | $8.39\pm 0.01$ | $0.78\pm 0.03$ | $5560\pm 13$ | $4.24\pm 0.05$ | $0.80\pm 0.09$ | $1.114\pm 0.006$ | $-0.14\pm 0.14$ | $4.43\pm 0.09$ | $20\pm 2$ | $134.2\pm 0.3$ | (2)
TYC 4962$-$1205$-$1 | 1.2798(26) | - | - | $0.59-0.77$ | $5380\pm 30$ | $4.13\pm 0.03$ | $0.97\pm 0.06$ | $1.404\pm 0.012$ | $-0.42\pm 0.07$ | $5.4-5.6$ | $37-47$ | $77.1\pm 0.3$ | (3)
CPD$-$65 264 | 1.3704(1) | $24600\pm 50$ | $8.38\pm 0.01$ | $0.87\pm 0.01$ | $5950\pm 30$ | $4.39\pm 0.02$ | $1.00\pm 0.05$ | $1.06\pm 0.01$ | $-0.14\pm 0.05$ | $6.39\pm 0.06$ | $64\pm 1$ | $206.0\pm 0.5$ | (4)
TYC 1380$-$957$-$1 | 1.613(11) | - | - | $0.64-0.85$ | $5815\pm 65$ | $4.41\pm 0.05$ | $1.18\pm 0.15$ | $1.122\pm 0.014$ | $-0.03\pm 0.07$ | $7.1-7.3$ | $49-71$ | $163.1\pm 1.3$ | (3)
TYC 3858$-$1215$-$1 | 1.6422(8) | - | - | $0.21-0.68$ | $4410\pm 10$ | $4.55\pm 0.04$ | $0.61\pm 0.07$ | $0.679\pm 0.004$ | $+0.15\pm 0.37$ | $5.5-6.4$ | $25-87$ | $68.4\pm 0.9$ | (2)
TYC 4700$-$815$-$1 | 2.4667(87) | - | - | $0.38-0.44$ | $6040\pm 31$ | $3.92\pm 0.05$ | $1.45\pm 0.17$ | $2.190\pm 0.023$ | $-0.04\pm 0.13$ | $9.4-9.5$ | $63-90$ | $165.8\pm 1.2$ | (3)
TYC 6992$-$827$-$1 | 41.45(1) | $15750\pm 50$ | $7.14\pm 0.02$ | $0.28\pm 0.01$ | $5250\pm 50$ | $3.48\pm 0.04$ | $1.31\pm 0.14$ | $3.45\pm 0.12$ | $-0.10\pm 0.10$ | $58.6\pm 1.8$ | $26\pm 2$ | $500\pm 20$ | (5)
TYC 8394$-$1331$-$1 | 51.851(9) | $19400\pm 100$ | $6.53\pm 0.03$ | $0.24\pm 0.01$ | $5150\pm 20$ | $3.06\pm 0.02$ | $1.31\pm 0.12$ | $5.57\pm 0.24$ | $+0.03\pm 0.15$ | $67.7\pm 0.3$ | $39\pm 2$ | $680\pm 30$ | (5)
2MASS J1836$-$5110 | 461.48(4) | $22250\pm 250$ | $7.49\pm 0.03$ | $0.40\pm 0.01$ | $5050\pm 50$ | $3.48\pm 0.05$ | $1.38\pm 0.16$ | $3.54\pm 0.07$ | $-0.05\pm 0.07$ | $304\pm 9$ | $47\pm 4$ | $900\pm 20$ | (5)