# Testing White Dwarf Age Estimates using Wide Double White Dwarf Binaries
from Gaia EDR3

Tyler M. Heintz Department of Astronomy & Institute for Astrophyiscal
Research, Boston University, 725 Commonwealth Ave, Boston, MA, 02215, USA  J.
J. Hermes Department of Astronomy & Institute for Astrophyiscal Research,
Boston University, 725 Commonwealth Ave, Boston, MA, 02215, USA  Kareem El-
Badry Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Harvard Society of Fellows, 78 Mount Auburn Street,
Cambridge, MA 02138  Charlie Walsh  Department of Astronomy & Institute for
Astrophyiscal Research, Boston University, 725 Commonwealth Ave, Boston, MA,
02215, USA  Jennifer L. van Saders Institute for Astronomy, University of
Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  C. E. Fields Feynman
Fellow  Center for Theoretical Astrophysics, Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  Computer, Computational, and Statistical Sciences
Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA  X
Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM
87545, USA  Detlev Koester Institut für Theoretische Physik und Astrophysik,
University of Kiel, 24098 Kiel, Germany

###### Abstract

White dwarf (WD) stars evolve simply and predictably, making them reliable age
indicators. However, self-consistent validation of the methods for determining
WD total ages has yet to be widely performed. This work uses 1565 wide ($>100$
au) WD+WD binaries and 24 new triples containing at least two WDs to test the
accuracy and validity of WD total age determinations. For these 1589 wide
double-WD binaries and triples, we derive total ages of each WD using
photometric data from all-sky surveys, in conjunction with Gaia parallaxes and
current hydrogen-atmosphere WD models. Ignoring initial-to-final-mass
relations and considering only WD cooling ages, we find that roughly $21-36\%$
of the more massive WDs in a system have a shorter cooling age. Since more
massive WDs should be born as more massive main-sequence stars, we attribute
this unphysical disagreement as evidence of prior mergers or the presence of
an unresolved companion, suggesting that roughly $21-36\%$ of wide WD+WD
binaries were once triples. Among the 423 wide WD+WD pairs that pass high-
fidelity cuts, we find that 25% total age uncertainties are generally
appropriate for WDs with masses $>0.63$ $M_{\sun}$ and temperatures
$<12{,}000\,K$, and provide suggested inflation factors for age uncertainties
for higher-mass WDs. Overall, WDs return reliable stellar ages, but we detail
cases where total ages are least reliable, especially for WDs $<0.63$
$M_{\sun}$.

white dwarfs, binary stars, ages

##  1 Introduction

White dwarf (WD) stars are the end stages of all stars with initial masses
less than roughly $8-10$ $M_{\sun}$ (Doherty et al., 2015) and have been used
for decades as stellar chronometers (see review by Fontaine et al. 2001).
Since WDs no longer undergo core fusion, their evolution is generally a
cooling problem. The development of robust cooling models (e.g., Bergeron et
al. 1995) enables WDs to act as precise and accurate age indicators. However,
to determine the total age of a WD, the time from the zero-age main sequence
(ZAMS) to the current state of the WD is required. The progenitor lifetimes
from ZAMS to the WD phase are determined through the use of a semi-empirical
initial-final mass relation (IFMR, Catalán et al. 2008, Kalirai et al. 2008,
Salaris et al. 2009, Cummings et al. 2018) in conjunction with stellar
evolution model grids, such as MESA (Dotter, 2016) and PARSEC (Bressan et al.,
2012).

The total ages of WDs can be used to date a variety of astronomical objects,
including wide binary companions, such as M dwarfs (e.g., Fouesneau et al.
2019; Qiu et al. 2021, Kiman et al. 2021), brown dwarfs (Zhang et al., 2020),
and a growing number of planet-host stars (Martin et al., 2021). WDs can also
age-date larger populations, such as stars in the disk (e.g., Winget et al.
1987), or even the halo of our Galaxy (Kilic et al., 2019; Torres et al.,
2021). WD ages can also be used to study the star formation history of the
Milky Way (e.g., Tremblay et al. 2014). An understanding of the limitations
and reliability of WD total ages is crucial to the inferences derived from
these studies.

The total age derived for a WD is strongly dependent on the IFMR used,
especially for the lowest-mass WDs ($M<0.7$ $M_{\sun}$). However, the low-mass
region of the IFMR is poorly sampled because nearby clusters containing at
least one WD are not old enough. Attempts have been made to alleviate this
problem by using wide binary companions to WDs (Zhao et al., 2012; Barrientos
& Chanamé, 2021) and the distribution of WDs in the color-magnitude diagram
(El-Badry et al., 2018), but this still results in a large range of predicted
progenitor masses across different IFMRs. Further, the IFMR may show evidence
of a kink where the mass-loss relationship is no longer monotonic (Marigo et
al., 2020). The IFMR is likely also sensitive to the metallicity and initial
rotation of the progenitor star (Cummings et al., 2019). These parameters are
not easy to determine, and thus assumptions for their values must be made.

Widely separated double WD binaries (WD+WD binaries) are ideal systems for
checking the viability of our existing methods for determining WD total ages.
It is safe to assume the two WDs bound to each binary system formed from the
same collapsing molecular cloud at nearly the same time with the same chemical
composition (White & Ghez, 2001; Kraus & Hillenbrand, 2009). Wide WD+WD
binaries that are sufficiently separated ($>100$ au) have not experienced any
significant mass transfer events or other interactions between the two stars
during their lifetimes. At intermediate separations (1-100 au) where tidal
forces, mass transfer, and accretion are not a significant source of
interaction, it has been shown that the interaction between a companion and
the circumstellar disk can affect the angular momentum evolution of the star
(Morgan et al., 2012). In the absence of any interactions between the two
stars, any methods for determining the total age of a WD should independently
return the same total age for both components of the binary.

Since the launch of the Gaia spacecraft (Gaia Collaboration et al. 2016, Gaia
Collaboration et al. 2018a, Gaia Collaboration et al. 2021), the sample of
wide WD+WD binary systems has increased by an order of magnitude (Andrews et
al. 2015, El-Badry & Rix 2018, Tian et al. 2020, El-Badry et al. 2021). This
increase in systems presents the opportunity for a large statistical study
into the accuracy of the models and methods used to determine the total ages
of WDs, which is the focus of this work. In Section 2, we present the sample
selection of resolved WD+WD binary systems, including 23 new WD+WD+MS systems
and one bound WD+WD+WD system. Section 3 discusses the techniques used to
determine their ages. In Section 4, we discuss comparisons of the total ages
in each binary. We end with a summary of the findings and results in Section
5.

Figure 1: Basic properties of the full sample of 3179 WDs in wide WD+WD
binaries from El-Badry et al. (2021) used in this work, including 23 of our
newly discovered wide WD+WD+MS triples and one WD+WD+WD triple: distribution
in galactic coordinates, color-magnitude diagrams for the primaries and
secondaries, apparent magnitude distributions, magnitude difference,
distances, and projected angular and physical separations. The primaries are
defined to be the more massive WD component. A large range of the WD cooling
tracks is covered by the sample, which allows us to test WD total ages across
a representative sample of WD masses and effective temperatures. Cooling
tracks for a 0.6 $M_{\sun}$ and 0.9 $M_{\sun}$ WD model from Bédard et al.
(2020) are shown for reference on the color-magnitude diagrams, as well as
dots representing WD cooling ages of 0.5, 1, 2, 4, and 6 Gyr for each. The
separation distribution of pairs with $\texttt{R\\_chance\\_align}>0.1$ is
shown in cyan for reference in the bottom right panels. The upturn in the
distribution at high separations is caused by these possible chance
alignments.

##  2 Sample Selection

Obtaining a large sample of wide double WD pairs is crucial for this work. The
improved parallaxes and proper motions for more than 1 billion sources all-sky
with the release of Gaia EDR3 (Gaia Collaboration et al. 2021) significantly
increased the potential yield of searches for wide common-proper-motion pairs.
With this larger sample we can delve into the systematic errors present in WD
total age determinations.

###  2.1 Binary Sample

For this work, the base of our sample is composed of the 1565 wide WD+WD
binaries from El-Badry et al. (2021). Candidate pairs in this work were
selected to have projected separations less than a parsec, parallaxes that are
consistent within $3\sigma$, and proper motions consistent within $2\sigma$ of
a bound orbit. This sample is restricted to objects with fractional parallax
uncertainties less than 20% for both components, with most systems within 300
pc. The sample covers a wide range of magnitudes, spanning $13<G<21$ mag, with
roughly 68% of the 3130 individual WDs in the sample fainter than $G>19$ mag.

El-Badry et al. (2021) calculate a chance alignment factor (R_chance_align)
for each candidate binary pair through the use of a shifted Gaia catalog.
Conducting a search for binaries in the shifted catalog only produces chance
alignments, and thus can be used as a way to determine the likelihood that a
binary pair is a chance alignment (see their Sec 3). This parameter can be
used to filter out low-fidelity wide binaries.

###  2.2 Triple Sample

To further increase the sample of wide double WDs, we conduct a search for
wide, resolved triple star systems containing at least two WDs. In El-Badry et
al. (2021), all triple systems are intentionally removed from the sample by
the neighboring pairs cut (see their Sec 2.1). In this work, we take the
sample of binary pairs from El-Badry et al. (2021) before the cleaning of the
sample and remove the neighboring pairs cut, but retain the cut on neighboring
sources of $N<30$, where a neighboring source is defined, following El-Badry
et al. (2021), as a source with (i) projected on-sky separation less than 5
parsecs; (ii) proper motion difference less than 5 $\mathrm{km\ s^{-1}}$ with
a $2\sigma$ tolerance; (iii) parallaxes consistent within $2\sigma$.

Next, we make a list of systems of various sizes by combining all binaries
with at least one common source between them. We iterate this process until
each system is independent of each other.

To clean this new sample of possible wide systems, we use the classifier for
spurious astrometric solutions provided in Rybizki et al. (2021). We require
each component of the triple to have the fidelity_v1 parameter greater than
0.5. This assures that each source has a reliable astrometric solution. We are
only concerned with finding systems with three components in this extended
list and throw out any systems with a size not equal to three. (Repeating this
exercise for $N=4,5,\ldots$, we do not find any systems with a size greater
than $3$ in this extended list of wide systems with at least two WDs.)

With these cuts, we identify 23 new WD+WD+MS triples and one WD+WD+WD triple.
This list includes the first resolved triple WD system found by Perpinyà-
Vallès et al. (2019). All WD+WD+MS and WD+WD+WD triple systems are detailed in
Table 3. All but three of the triples in our sample are hierarchical, where we
have defined a hierarchical triple as one in which the largest separation is
greater than five times the smallest separation. An excess of hierarchical
triples is expected since the hierarchy improves dynamical stability in the
system (Hut & Bahcall, 1983).

The total ages on the non-hierarchical systems (WDWDMS EDR3
3681220906103896192, WDWDMS EDR3 6756233901661753472, WDWDMS EDR3
929001808377561216) can provide interesting constraints on the timescales of
dissipation of such systems. In both WDWDMS EDR3 3681220906103896192 and
WDWDMS EDR3 6756233901661753472, one of the WDs does not have a total age
estimate since it has a mass of $0.2$$M_{\odot}$and $0.35$$M_{\odot}$,
respectively, but the total ages of the other WDs in these triples are
$92^{+3}_{-5}$ Myr and $2.9^{+0.4}_{-0.2}$ Gyr, respectively. The weighted
mean total age of the WDs in WDWDMS EDR3 929001808377561216 is
$3.3^{+3.1}_{-1.3}$ Gyr. Further information is found in Table 3.

Although two of these non-heirarchical wide triples seem too old ($>2.5$ Gyr)
to still be bound, it has recently been shown that the unstable phase of these
triples on average lasts $10^{3}-10^{4}$ crossing times, and can even last as
long as $10^{6}-10^{7}$ (Toonen et al., 2021). Assuming the crossing time is
of similar magnitude to the outermost orbital period, the two older systems
would have a crossing time of $\sim{10^{4}}$ years, indicating the older
systems have experienced $\sim{10^{5}}$ crossing times. Of the 21 systems that
are hierarchical, 19 systems have an inner pair of two WDs, although this is
likely an observational bias caused by the high luminosity ratio of a MS star
to a WD.

Our search is sensitive to wide triple WD systems, but we do not find any new
wide WD+WD+WD systems. We do find a possible candidate for a new WD+WD+WD
system (Gaia EDR3 source IDs: 261249666477351552, 261249670772776832,
261249636413169920) but one component has a fidelity_v1 parameter from Rybizki
et al. (2021) of 0.01. Still, all three WDs have significant parallaxes and
proper motions that are in better than $2\sigma$ agreement. With all
components $G>20$ mag, this potential new WD+WD+WD is significantly fainter
than the first published triple WD system. This limits the parallaxes of each
WD to 10%-15% uncertainties. The system lies somewhere between $100-130$ pc,
with the weighted mean distance being $118\pm 8$ pc. This possible WD triple
has a similar hierarchy as the first such identified system in Perpinyà-Vallès
et al. (2019); the system has an inner pair separated by $\approx$$170$ au and
an outer component separated by $\approx$$6500$ au. (The first triple WD
system has an inner separation of $\approx$$300$ au and outer separation of
$\approx$$6400$ au.) Follow-up analysis is needed to confirm the true nature
of this possible WD+WD+WD system.

###  2.3 Full Sample

Various parameter distributions of the full sample, including a color-
magnitude diagram (CMD) of all wide double WDs, is shown in Figure 1. Our full
sample of wide WD+WD pairs spans a large range of the WD cooling sequence,
covering many masses and effective temperatures. The sample also spans many
different evolutionary stages. In fact, we expect roughly 430 WDs (13%) to be
mostly crystallized, based on the 80% crystallization boundary from Tremblay
et al. (2019). However, many of our targets are quite faint; the majority of
sources are fainter than $G>19.5$ mag (see Figure 1). The separation
distribution shows an excess of high separation pairs. These pairs are likely
chance alignments and can be removed with a cut on R_chance_align from El-
Badry et al. (2021), as discussed in Section 4.

In preparation for the eventual total age determination, we gather additional
photometric data on each WD in order to sample a larger range of wavelengths
than just the three broad photometric bands in Gaia EDR3. We use the
crossmatches provided by the Gaia team for EDR3 (Marrese et al. 2021) to get
photometric data from the Sloan Digital Sky Survey (SDSS), the Panoramic
Survey Telescope and Rapid Response System (PanSTARRS), the SkyMapper Southern
Survey, and the Two Micron All Sky Survey (2MASS). We find that 32%, 70%, 23%,
and 4% of the WDs in our sample have photometry in SDSS, PanSTARRS, Skymapper,
and 2MASS, respectively.

##  3 Total Age Determination

WD total ages have two necessary components that need to be determined: the
time the star has spent as a WD (its cooling age) and its progenitor main-
sequence and giant branch lifetimes. To calculate the cooling age of a WD,
atmospheric parameters of the WD must be determined, which can be derived
using either spectroscopic or photometric data in conjunction with model
atmospheres. Fundamentally, the WD surface gravity and temperature must be
determined.

Spectroscopic observations are available in the literature for roughly 300 of
the WDs in our sample, so we lack spectroscopy for the vast majority of our
3179 objects. Therefore, we resort to conducting spectral energy distribution
(SED) fitting of the average fluxes in various photometric bands from all sky
surveys. In practice, SED fitting is sensitive to the radius and temperature
of a WD. This radius can be converted into a surface gravity through the use
of the mass-radius relationship for WDs (Bédard et al., 2020).

Once a surface gravity and temperature are determined, an estimate of the mass
of the WD can be calculated using the same mass-radius relationship. This
estimate of the mass, with the use of an IFMR and stellar evolutionary grids,
can be used to generate an estimate of the lifetime of the ZAMS progenitor. We
discuss this process in more detail below.

###  3.1 WD Atmospheric Parameters

To determine masses and effective temperatures of the WDs in our sample, we
use available photometry from Gaia EDR3 and the all-sky surveys discussed in
Section 2.3, in conjunction with a fitting technique similar to the
photometric technique described in Bergeron et al. (2019).

Reddening effects from interstellar gas and dust can be important for the WDs
in our sample (the majority are at a distance beyond 150 pc). Thus, the
observed magnitudes in each band are dereddened following the process outlined
in Gentile Fusillo et al. 2019, which we summarize here. We use reddening maps
from Schlafly & Finkbeiner (2011) to determine the extinction coefficient in
the Johnson $V$ band, $A_{{\rm V}}$. Then, the $A_{{\rm V}}$ values are
converted to extinction coefficients in the other bands using the appropriate
$R$ values. We assume the material is concentrated along the galactic plane
and has a scale height of 200 pc. When parameterised this way, the dereddened
magnitude in a given band $x$ is determined by,

$M_{{\rm x}}=M_{{\rm x,obs}}-A_{{\rm
x}}\left(1-\exp\left(-\frac{\mathrm{sin\left(|b|\right)}}{200\omega}\right)\right)$
(1)

where $M_{{\rm x,obs}}$ is the observed magnitude, $A_{{\rm x}}$ is the
extinction coefficient, $b$ is the galactic latitude, and $\omega$ is the
parallax in arcseconds. A more sophisticated reddening determination can be
used similar to what is done in Gentile Fusillo et al. (2021), but in
comparisons with atmospheric parameters derived in their work, we find no
systematic shifts and the parameters are in good agreement (as demonstrated
below).

The observed, dereddened magnitudes are converted to average fluxes using the
appropriate zero points for each bandpass. We apply the $G$-band magnitude
correction for all sources with a six-parameter astrometric solution provided
in Gaia Collaboration et al. (2021). To account for the offset of SDSS
magnitudes from the AB magnitude standard, we apply an offset of $-$0.04 and
0.02 mags in the $u$ band and $z$ band, respectively (Eisenstein et al.,
2006). For each survey, we remove photometry that has been flagged for various
issues. For SDSS, we remove photometry that has been flagged with EDGE,
PEAKCENTER, SATUR, and NOTCHECKED. We remove photometry from PanSTARRS that
uses rank detections less than one. For SkyMapper, we only use photometry that
has no raised flags. We also enforce a minimum uncertainty in each bandpass of
0.03 mag to account for systematic errors in the conversion of magnitudes to
average fluxes. This also gives equal weight to each bandpass and one point
with a small error will not dominate the behavior of the fits.

We generate synthetic averages fluxes in a given bandpass using synthetic
hydrogen atmosphere WD (spectral type DA) spectra from Koester (2010),
convolved with the appropriate filter transmission profiles. The observed and
synthetic fluxes are adjusted to a distance of 10 pc using the weighted mean
of the parallaxes of the two components of the binary, or the weighted mean of
all three components for our triples. To adjust the synthetic fluxes to 10 pc,
the radius of the WD must be calculated from a mass-radius relation. We use
the WD evolutionary models from Bédard et al. (2020) to convert the
temperature and surface gravity of a given model to a radius. The photometric
uncertainties are then added in quadrature with the uncertainty on this
weighted mean parallax. The use of the weighted parallax in these systems
allows for more precise determinations of the atmospheric parameters due to
its higher precision. For the sample of triples, the precision of the parallax
is six times better on average, because the MS companions are much brighter.
This leads to the WDs in these triples having smaller mass uncertainties than
what is expected given their brightness (an average improvement of 40% for the
brighter WD component and 15% for the fainter WD component).

Due to the lack of spectral information for the majority of our objects, we
assume all of our WDs are DAs. Although fitting a non-DA WD with a DA model
can introduce systematic mass errors on the order of $10-15$% (Giammichele et
al., 2012), we are forced to choose a model. A DA model is the simplest
assumption, considering $>$65% of WDs in magnitude-limited samples within Gaia
are DAs (Kleinman et al., 2013).

We use the python package emcee (Foreman-Mackey et al., 2013), which uses an
affine invariant Markov Chain Monte Carlo (Goodman & Weare, 2010), to get the
posterior range on the temperature and surface gravity for each WD in the
sample. We use the $50^{\mathrm{th}}$ percentile of these posteriors as an
estimate of the best fit. We use flat priors on both effective temperature and
surface gravities with bounds defined from the limits of the models
($3000-40{,}000$ K and $\log{g}$ of $6.0-9.5$ dex) and assume Gaussian errors.
Using the WD evolutionary models from Bédard et al. (2020), which assume thick
H layers, we convert the surface gravities and temperatures to masses and
cooling ages. The lower and upper errors on these parameters are taken as the
16th and 84th percentiles, respectively. To account for any unknown
systematics, we set a lower limit on the uncertainty of the atmospheric
parameters derived from the fits of 0.038 dex for surface gravity and 1.2% in
temperature (Liebert et al., 2005). An example of a fit can be seen in Figure
2.

Figure 2: An example fit to get atmospheric parameters of our WDs. The top two
panels shows the SED, including all observed photometry and a representative
atmospheric model, as well as the residual from the fit. The posterior
distributions from the MCMC fitting code used in this work are shown in the
bottom part of the figure.

As a check on our methods, we compare the atmospheric parameters we derive to
the parameters from the 100-pc WD sample of Kilic et al. (2020), and to the
full Gaia EDR3 WD catalog from Gentile Fusillo et al. (2021), which were also
determined from the photometric technique we employ. The comparison to the
100-pc WD sample from Kilic et al. (2020) reveals no significant systematic
differences between the two works. The atmospheric parameters on average agree
within $0.9\sigma$ (100 K) and $0.6\sigma$ (0.03 dex) for effective
temperature and surface gravity, respectively. We observe some small
dispersion at higher temperatures, which we expect since our work incorporates
interstellar reddening whereas Kilic et al. (2020) does not. Comparing to the
Gaia EDR3 WD catalog, we find that the atmospheric parameters on average agree
within $0.4\sigma$ (300 K) and $0.35\sigma$ (0.1 dex) for effective
temperature and surface gravity, respectively. The quoted uncertainties we
find are significantly smaller than the ones in Gentile Fusillo et al. (2021)
due to the use of more photometric points in the SED and the weighted
parallax. Although we have smaller uncertainties and use different methods to
determine the amount of interstellar reddening, we do not find any systematic
shifts in derived WD atmospheric parameters.

We also search for spectral types of our objects that have been published in
the literature, finding 305 objects with previously reported spectra. The
sources for these spectral types can be found in the catalog of WD pairs
described in Table 2.

We supplement these previously published spectral types with 57 spectra taken
on the DeVeny Spectrograph mounted on the 4.3-m Lowell Discovery Telescope
(LDT, Bida et al. 2014) in Happy Jack, Arizona. Using a 300 line
$\mathrm{mm}^{-1}$ grating we obtain a roughly 4.5 Å resolution, and rotate
the position angle of the slit to capture both WDs simultaneously in our
observations. We extracted the spectra using a quick-look pipeline written
using the ccdproc and specutils packages from the AstroPy package (Astropy
Collaboration et al., 2018), and wavelength calibrated the spectra with a
HgArNe lamp. We do not attempt to fit these spectra and only conduct a visual
inspection for the presence of hydrogen absorption features, and include their
spectral type in the sp_type column described in Table 2. Future work will
further analyze atmospheric parameters derived from this spectroscopy.

Comparing the derived atmospheric parameters of the previously identified DAs
with the parameters derived in this work, we find that the effective
temperatures agree well but that the determination of the surface gravities
from spectroscopic observations vary from our own values derived from
photometric information. We find that the atmospheric parameters of our work
and the spectroscopic parameters on average agree within $0.9\sigma$ (300 K)
and $1.3\sigma$ (0.1 dex) for effective temperature and surface gravity,
respectively. We also find that the spectroscopic fits find higher surface
gravities for the WDs on average and thus hotter temperatures as well.
Although the absolute differences are similar to the other comparisons, the
quoted uncertainties from the spectroscopic parameters are smaller and result
in a stronger disagreement in surface gravity ($1.3\sigma$ compared to
$0.9\sigma$ and $0.35\sigma$).

###  3.2 Initial Masses and Progenitor Lifetimes

To get the total ages of the WDs in our sample, we need an estimate of the
lifetime of the ZAMS progenitor. The first step towards this is to determine
an initial main-sequence mass for each of the WDs in our sample. To get these
masses, an initial-final mass relation is needed. In this work, we use the
theoretical IFMR from Fields et al. (2016), which was generated by running a
suite of solar metallicity stellar evolution models from the Modules for
Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011, Paxton et al.
2013, Paxton et al. 2015) starting at the pre-MS phase through the thermally
pulsing asymptotic giant branch to the final WD phase. This allows us to
remain self-consistent within both our IFMR and our determination of ZAMS
progenitor lifetimes using the same open-source stellar evolution code and
allows us to use a more theoretically motivated shape for the IFMR.

The theoretical IFMR (shown as the dashed line in Figure 3) lies below what is
observed for WDs in near solar metallicity clusters, an effect that is still
not fully understood, though might involve uncertainties in the amount of
convective overshoot that is used in the models (Fields et al., 2016) or
uncertainties in nuclear reaction rates that can affect the core mass (e.g.,
deBoer et al. 2017). To account for this systematic effect, we fit a positive
offset to the IFMR using the same WDs in solar metallicity clusters. We find
the best-fitting offset to apply to the final WD mass is $+0.0636$ $M_{\sun}$.
The points of this shifted IFMR can be found in Table 4.

Using this modified IFMR (the solid line in Figure 3), we generate progenitor
MS masses for all the WDs in our sample. We convert these MS masses into
progenitor lifetimes by interpolating the solar metallicity, zero rotation
MESA evolutionary tracks of Dotter (2016) and Choi et al. (2016) and summing
the main-sequence and giant branch lifetimes.

The uncertainties on the progenitor main-sequence masses and progenitor
lifetimes are determined by taking the $1\sigma$ uncertainties on the WD mass,
determined from the model-atmosphere fitting described in Section 3.1, which
determine an upper- and lower-limit on the MS mass. Then the uncertainty on
the progenitor lifetime is quoted as the difference between these $1\sigma$
bounds and the determined value. Total ages are then determined through the
sum of the cooling age and the progenitor lifetime, and the uncertainties on
total age are calculated by adding the uncertainties on cooling age and
progenitor lifetime in quadrature. A table of all double WD pairs with
atmospheric parameters and total ages can be accessed at
https://sites.bu.edu/buwd/files/2022/05/Table_A1.csv; a description of the
table columns and data can be seen in Table 2.

As a check, we compare the progenitor lifetimes from our MESA IFMR to ones
derived from the three-piece, cluster-calibrated IFMR from Cummings et al.
(2018). We find that the progenitor lifetimes from both IFMRs are within 25%
of each other on average. The difference between the total ages is on average
10% when comparing the two IFMRs. Overall, we find that the use of the IFMR
from Cummings et al. (2018) does not significantly change the conclusions
discussed in subsequent sections.

Figure 3: The initial-to-final mass relation (IFMR) used in this work to
connect a WD mass to a single ZAMS progenitor mass. The dashed line shows the
theoretical IFMR from MESA calculated by Fields et al. (2016). The
observational data points are WDs in solar metallicity clusters along with
Sirius B from Catalán et al. (2008), Cummings et al. (2015) and Cummings et
al. (2016). The solid line shows our adopted IFMR, which include a constant
y-axis offset of $+0.0636$ $M_{\sun}$ found as a best-fit to the cluster
observations (the values for which are in Table 4).

###  3.3 Precision of Total Ages

The WD mass is the dominant factor for determining the total age of the WDs in
our sample, and subsequently, the uncertainties in the WD mass strongly
influence the degree of precision on the total ages.

Figure 4: Visualization of how the precision on the ZAMS progenitor lifetime
changes as a function of the WD mass uncertainty for a $1.0$ $M_{\sun}$ WD
(left) and $0.6$ $M_{\sun}$ WD (right). The grey shaded regions represent the
1$\sigma$ uncertainties on the progenitor lifetimes. The axes are scaled to
show the relative size of the uncertainties. The ZAMS progenitor lifetimes are
given in the top left of each plot. The dashed vertical lines show the average
relative uncertainty on the WD mass at a given $G$-band magnitude. The
uncertainty on the WD mass is also affected by the precision of the parallax,
and thus, for a given $G$-band magnitude there is some dispersion where more
precise parallaxes give better mass constraints.

This is especially true for the lower-mass WDs. For a typical $0.6$ $M_{\sun}$
WD (Kepler et al., 2007; Tremblay et al., 2016), the initial progenitor MS
mass is around $1.2$ $M_{\sun}$ which corresponds to a progenitor lifetime of
$\approx 6$ Gyr. A $5\%$ decrease in the WD mass results in a ZAMS progenitor
mass of $0.88$ $M_{\sun}$ and progenitor lifetime around 17 Gyr. Thus, for WDs
near this mass range, it is extremely difficult to constrain the total age
without very precise WD mass estimates. This problem is compounded by the fact
that the lower-mass region of the IFMR is the least constrained. Depending on
the IFMR used, initial mass values for a 0.6 $M_{\odot}$ WD can range between
$1.2-2$ $M_{\odot}$ which corresponds to progenitor lifetimes between $1.4-6$
Gyr. This problem is also exaggerated when considering that for these low-mass
WDs, the progenitor lifetimes are a major fraction of the total age.

We stress that low-mass WDs are still useful in providing a lower limit on
total ages. A $0.6$ $M_{\sun}$ WD with a 5% error on its mass can provide a
$1\sigma$ lower limit of 1.4 Gyr on the progenitor lifetime, modulo
uncertainties on the IFMR used. This, in conjunction with a precise cooling
age, can still provide useful information on these WDs and their companions.

Fortunately, the opposite is true for the higher-mass WDs. Higher-mass WDs
($>0.8$ $M_{\sun}$) come from massive progenitors, which have short ZAMS
progenitor lifetimes. Thus, even larger mass uncertainties for high-mass WDs
do not result in large errors on the progenitor lifetime (see Figure 4). Put
another way, high-mass WDs can have large relative uncertainties in their
progenitor lifetimes but still have relatively precise total ages.
Additionally, the IFMR of these high-mass WDs are better empirically
constrained from cluster WDs.

##  4 Coeval Analysis

Before engaging in any analysis on the total ages, we remove systems where we
know our methods described in the previous section return unreliable total
ages. We apply cuts on the sample to remove all systems with on-sky separation
$<$2″, as well as systems with R_chance_align $<$ 0.1. We also remove systems
with at least one component with $\mathrm{T_{eff}}<3200$ K, as well as those
with an inferred total age $>14$ Gyr.

The cut to on-sky separation removes systems with significant blending of the
photometry, which confuses the SED fits (154 of the WD pairs have on-sky
separations $<2$″). We adopt the cut on the R_chance_align parameter provided
in El-Badry et al. (2021); cutting at 0.1 ensures the remaining systems have a
high probability of being bound ($>90$%). Although we cut on a $<10$% chance
alignment probability, we emphasize that the expected contamination fraction
from chance alignments is much less than 10%, because most of the pairs in the
sample have chance alignment probabilities that are much lower than 10%. From
the actual chance alignment probabilities of the pairs in our sample, we
estimate that there are $4\pm 2$ chance alignments in the sample,
corresponding to a contamination fraction of about 0.3%. The wide WD+WD sample
contains 1390 high fidelity pairs with R_chance_align $<0.1$ (see Figure 1).

The cut on effective temperature is employed because our model grid stops at
3000 K. Since our fitting routine does not attempt to extrapolate, objects
with reported temperatures near the cutoff will have underestimated
uncertainties and their temperatures will be artificially hotter than their
true temperatures. We also ignore all systems with a total age greater than
that of the Universe. Many of these systems come from overluminous sources in
the Gaia CMD (see Figure 1). These sources will have low WD masses that, if
they are actually so low mass, could not have formed from single-star
evolution. At lower temperatures, the observed WD sample does not follow the
cooling model trends which may result in these lower unreliable masses (see
Figure 1). There are a total of 670 WDs with inferred total ages $>14$ Gyr,
which results in 548 pairs being removed.

After these cuts, the high-fidelity sample used for analysis contains 1272 WDs
in 636 wide double WD pairs. The mean magnitude of this surviving sample is
$G=19.3$ mag.

###  4.1 Age Comparison

The first test of the total ages made is to check the absolute age difference
in each pair. Since we assume the stars in each wide pair in our sample are
coeval, we expect them to return identical total ages. This absolute age
difference is plotted in Figure 5. The total age difference between each
component is typically close to zero until one of the components masses is
below a value of $\sim 0.63M_{\odot}$. These systems do not provide very
precise ages, and it is evident that the systematic uncertainty in these
systems is too large to provide statistically significant comparisons.

This leads us to remove any systems that have a WD with a mass less than
$<0.63\ M_{\odot}$ from any further analysis. These systems can still be
useful to provide lower limits on the total age, as discussed in Section 3.3,
but we are not able to assess their reliability due to the large systematics
in conjunction with their large uncertainties. After removing these systems,
the number of wide WD+WD pairs is reduced to 423, with a total of 846 WDs.

Figure 5: The total age difference in each binary as a function of the mass of
the secondary in each binary (the least massive component) for our 1272
systems surviving the cuts outlined in Section 4. The red line shows the cut
used to remove the poorly behaved low mass systems, whose behavior change
drastically at $M<0.63M_{\odot}$. Since the secondary is always the least
massive component, the cut shown by the red line removes all systems with WDs
$<0.63M_{\odot}$.

The second test of the accuracy of the total ages is to compare the ages of
the WDs in each binary in relation to the uncertainties on those total ages.
To do this, we explore distributions of age agreement, which we define as

$\mathbf{\sigma_{1-2}=}\frac{\Delta\tau}{\sigma_{\Delta\tau}}=\frac{\tau_{1}-\tau_{2}}{\sqrt{\sigma_{\tau_{1}}^{2}+\sigma_{\tau_{2}}^{2}}}$
(2)

where $\tau_{1}$ and $\sigma_{\tau_{1}}$ are the total age and uncertainty on
the total age of the primary (the most massive component of each binary) and
$\tau_{2}$ and $\sigma_{\tau_{2}}$ are the total age and uncertainty on the
total age of the less massive component. Systems with an age agreement
($\mathbf{\sigma_{1-2}}$) equal to 5 have individual component ages that
disagree at a $5\sigma$ level.

We show in Figure 6 the distribution of the $\sigma_{1-2}$ disagreement for
the 423 high-fidelity pairs where both WDs have $M>0.63M_{\odot}$. The
distribution exhibits a long tail out to large absolute values and has a
notable bias towards negative values. Fewer than half (47%) of the systems
have total age agreement within $1\sigma$. The slight tendency towards
negative $\sigma_{1-2}$ disagreement values shows that the more massive
component of each pair has a younger total age on average compared to its
companion. Possible causes of this asymmetry are further explored in Section
4.2. The long tail of the distribution reveals that there is a population of
at least 60 systems (14%) with values of the $\sigma_{1-2}$ disagreement
greater than five where the systematic errors ($\Delta\tau$) are non-
negligible compared to the random errors reported ($\sigma_{\tau}$).

Figure 6: The distribution of $\sigma_{1-2}$ disagreement for the 423 wide
WD+WD pairs in our sample that pass the cuts discussed in Section 4. The red
dashed line shows the expected distribution for a Gaussian with $\sigma=1$ for
reference. The distribution is skewed towards negative values, where the
primary is the more massive of the two components. The black dashed line
represents the $50^{th}$ percentile and the lower and upper errors come from
the $16^{th}$ and $84^{th}$ percentiles respectively.

###  4.2 Ill-Behaved Systems

Substantial uncertainty in the total ages in these WDs comes from the process
of converting the WD mass to a ZAMS progenitor lifetime. We remove the most
model-dependent aspect in the determination of the WD’s progenitor lifetime by
simply considering that the IFMR is monotonic and increasing, such that a
higher-mass WD comes from a higher-mass progenitor. This assumption may not
hold for all WD masses, as some evidence of a kink is seen in the IFMR for a
range of WD masses around $0.65$ $M_{\odot}$ (Marigo et al. 2020).

Still, for the vast majority of cases, the more massive WD in the binary comes
from a more massive main-sequence star, and it’s progenitor lifetime should be
smaller than the progenitor lifetime of its companion. For a coeval binary
pair, the cooling age of the more massive WD must be longer than the cooling
age of its companion. Any systems where the more massive WD has a shorter
cooling age than its lower mass companion are discrepant before invoking any
theoretical IFMR or other process for determining a progenitor lifetime.

Our sample contains many systems where this is the case, as shown by Figure 7.
The figure includes systems that pass the temperature, separation, and chance
alignment cuts discussed in Section 4, but we do not apply the total age and
mass cut described in that section for this comparison. Roughly 43% lie in the
negative region of the plot, where the more massive WD has a shorter cooling
age. This statistic requires we properly define the more massive WD (see
representative uncertainty in Figure 7); 21% of our systems are more than
1$\sigma$ away from having the more massive component with a shorter cooling
age. This is a conservative estimate for the number of these ill-behaved
systems because the masses of the two WDs in each pair have a positive
correlation. This is due to the fact that the two WDs are effectively at the
same distance, and the uncertainty on the distance affects the determined
uncertainties on the masses. If in fact the system is closer and therefore
brighter, then both WDs would be brighter and the derived masses would both be
lower, leaving the determination of the more-massive primary unchanged.

Figure 7: A comparison of the cooling age difference versus mass difference
for the full wide binary sample (1259 grey points) and a subset of pairs where
both WDs are known to be DA (79 red diamonds). A representative error bar is
provided in the bottom right of the figure. A significant fraction of objects
(43% for the full sample, 35% for the DA+DA systems) sit in the negative
region, which implies the more massive component has a shorter cooling age,
which is unphysical and likely implies a large fraction of wide WD+WD binaries
were once triple systems.

Thus, somewhere between 21-43% of the wide WD+WD systems have the more massive
WD with a shorter cooling age. Given a monotonic and increasing IFMR, these
systems will have large $\sigma_{1-2}$ disagreement values. The systems that
have negative cooling age differences shown in Figure 7 show no bias towards
any mass or temperature range. Removal of the 187 ill-behaved systems that
pass the cuts described in Section 4 results in a substantial fraction of
systems in the long negative tail of the $\sigma_{1-2}$ disagreement
distribution being removed (see Figure 8).

Figure 8: The distribution of $\sigma_{1-2}$ disagreement for the 423 wide
binaries that pass the cuts outlined in Section 4 (black) compared to the
distribution of $\sigma_{1-2}$ disagreement for the sample of systems where
the more massive WD has a shorter cooling age (red). These systems can account
for the bias towards negative values including the most discrepant systems.

One possible explanation for this discrepancy is that we are assuming all of
our WDs have a hydrogen dominated atmosphere. Applying a DA (hydrogen-
dominated atmosphere) model to a non-DA WD can cause systematic errors on the
order of 10-15% on the WD mass (Giammichele et al. 2012) which can lead to the
wrong determination of the primary. The objects in the sample that are known
to have two pure DA WDs still show objects in the negative region where the
higher mass WD has a shorter cooling age (see Figure 7). The amount of wide
DA+DAs that are ill-behaved is around 36% of 74 systems (with 21% being more
than 1$\sigma$ discrepant). Systems where one of the WDs is known from
spectroscopy to be a non-DA have 63% out of 73 systems in the negative region
(with 52% being more than 1$\sigma$ discrepant).

Our spectroscopic follow-up using the LDT was designed to specifically target
systems in the negative region of Figure 7 and thus biases our results to this
region. An alternative way to determine if there is an excess of non-DA
systems in the ill-behaved region of Figure 7 without having to worry about
the bias introduced by our follow-up observations is to compare relative rates
of spectral types within the sample of the ill-behaved objects. There are 101
systems in the negative region and 156 of the 202 WDs have a spectral type
determined. 66% of these WDs are DAs. In magnitude limited samples, we expect
$\approx 65$% of the WDs to be DAs (Kleinman et al., 2013). For the systems
that are more than $1\sigma$ discrepant, 59% of the WDs with known spectral
types are DAs.

Although there is some evidence for an overabundance of non-DAs in the
negative region of Figure 7, we conclude it is unlikely that a large
population of non-DAs is the full explanation for the number of these ill-
behaved systems. To avoid any contamination from non-DAs, we will refer to the
values for the DA+DA subsample (21-36%) when discussing the percentage of ill-
behaved systems.

Instead, we propose that most of the ill-behaved systems are the descendants
of triples that have an inner pair that either experienced a merger that
effectively reset the age of the more massive WD or is currently unresolved.
Through binary population modelling, Temmink et al. (2020) find that assuming
single star evolution for a merger remnant can result in underestimating the
age by $3-5$ times on average, which could manifest as an anomalous cooling
age like those in Figure 7. They also find $10-30$% of isolated WDs have
experienced a merger in their past, usually before the WD phase is reached,
which is similar to the percentage of ill-behaved systems in our sample
($21-36$%). While that value is for single WDs once in binaries, it is a
valuable number to anchor expectations.

If all the systems in the negative region of Figure 7 are or were previously
triples, this implies that roughly 21-36% of the wide WD+WD pairs in our
sample started as a triple system and have either undergone a merger or have a
close inner pair that is unresolved. This range is consistent with the
expected binary and triple fraction ($30-35$% and $10-20$%, respectively) for
WD progenitor stars that are $2-4$ $M_{\sun}$ on the ZAMS (Moe & Di Stefano,
2017). We also find that 50% of the double WD pairs in the triple systems are
in the negative region of Figure 7. This would imply that these systems are
descended from quadruple systems.

At least one well-studied merger remnant is included in our sample: the
massive ($>1.1$ $M_{\sun}$, Külebi et al. 2010), strongly magnetic (up to 800
MG, Burleigh et al. 1999), rapidly rotating (725 s, Vennes et al. 2003) object
REJ0317-853 (Ferrario et al., 1997). This WD is in a system with a
$\sigma_{1-2}$ disagreement value of $-17$ and its effective temperature is at
least twice that of its lower mass companion. We conclude that the majority of
these ill-behaved systems are likely merger remnants or unresolved triples.
Spectroscopic follow-up of these systems can help to disentangle the subset of
these objects that are the result of a merger from the unresolved triples
which can place empirical constraints on the triple fraction of WD
progenitors.

###  4.3 Dependence on Mass and Temperature

The dependence of the age disagreement on the WD mass and temperature can
provide insights into where our models are insufficient. A treatment and
correction for these dependencies is crucial if WD ages are to be used on
large scales to age various populations.

We explore the dependence of the age disagreement on the WD mass and
temperature by splitting the systems into individual WDs, and inspecting the
median of the absolute value of the age disagreement of each system in various
bins of effective temperature and mass. In this analysis, we do not remove any
of the ill-behaved systems discussed in Section 4.2, since it is not usually
feasible to separate these possible merger remnants when looking at an
isolated WD or WD+MS system in the field.

Figure 9: The absolute value of the total age disagreement and the resulting
total age error inflation factors as a function of WD mass. Each point
represents the median of the absolute value of the total age disagreement of a
bin centered at the point. The total age error inflation factors are
calculated by dividing this median by the expectation for a folded Gaussian
distribution. The black points represent the full sample of 423 systems that
survive the cuts described in Section 4. The grey background points represent
the median age disagreement for a sub-sample of systems where both WDs have
masses contained in each bin. Each bin contains 20% of the sample, which is
roughly 170 individual WDs per bin for the black points (the grey points are
around $3-5$ times fewer). The red line is a fit to the black points that is
described in Section 4.3 to define an inflation factor; a coarse mapping for
this fit can be found in Table 1.

We find that the age disagreement is effectively constant as a function of
temperature, but shows a strong trend as a function of mass. This can be seen
in Figure 9. The highest masses ($>0.8$$M_{\odot}$) have a median
$|\sigma_{1-2}|$ disagreement value of $\approx{1.5}$ while the lowest mass
bin ($<0.67$$M_{\odot}$) has a median value of $\approx{0.6}$. The
uncertainties in the medians shown are found through a bootstrapping method
with replacement. To ensure that this trend is the true underlying trend, we
calculate the median age disagreement for a sub-sample of systems where the
mass of both WDs in the system are in the same bin. This removes cross
contamination from systems that have WDs with very different masses. The grey
points in Figure 9 show the trend for this sub-sample, and are consistent with
the points for the whole sample.

This trend in mass can be caused by larger systematic errors in the total ages
of massive WDs causing larger relative age differences in binaries containing
them, or by an underestimation of the total age uncertainties of high mass
WDs. We find that the relative age difference is essentially constant
($\approx{25\%}$) as a function of WD mass. This leads us to conclude that the
uncertainties on the total ages of the high mass WDs in the sample are
underestimated.

To determine the amount of inflation of the total age errors needed, we take
the median age disagreement and divide by the expected median value of a
folded Gaussian distribution with a standard deviation of one. These values
can be seen as a second y-axis in Figure 9. Next, we fit these points with a
sigmoid function, defined as,

$f_{\mathrm{inflate}}\left(M_{\odot}\right)=\frac{a}{1+\mathrm{e}^{-c(M-b)}}$
(3)

where $M$ is the mass of the WD and $a$, $b$, and $c$ are all constants. We
fit this function to the inflation factor points shown in Figure 9. The best
fit parameters are $a=2.2$, $b=0.68$, and $c=22$. The resultant fit can be
seen in Figure 9 as the red dashed line, and select points are given in Table
1. The total age error inflation factor for the lowest mass bin is 0.85, but
we do not recommend deflating any total age errors. The uncertainty makes the
point consistent with a total age error inflation factor of 1.0. Thus, we
recommend using the fit function only for WDs with masses greater than
$0.67M_{\odot}$.

Table 1: Total Age Error Inflation Factors WD Mass $\left[M_{\odot}\right]$ | $f_{\mathrm{inflate}}$ | WD Mass $\left[M_{\odot}\right]$ | $f_{\mathrm{inflate}}$
---|---|---|---
0.67 | 1.00 | 0.74 | 1.74
0.68 | 1.10 | 0.75 | 1.81
0.69 | 1.20 | 0.76 | 1.88
0.70 | 1.34 | 0.77 | 1.90
0.71 | 1.45 | 0.78 | 2.00
0.72 | 1.55 | 0.80 | 2.05
0.73 | 1.65 |  $>$0.90 | 2.20

To further understand where WD ages are the most reliable, we split the
parameter space into 25 bins in temperature and mass. For each bin, we check
the median total age, absolute age difference, and total age error inflation
factor needed. The results of this are shown in Figure 10. The most
problematic areas of parameter space are the higher masses, which is what is
seen in Figure 9, and especially at the hotter temperatures where the cooling
age contribution to the total age is minimized. These hot, high-mass regions
of parameter space are where we would expect the merger remnants to be at
their highest contamination rate. This is also where the Q-branch is located,
which was first seen in the HR diagram in _Gaia_ DR2 (Gaia Collaboration et
al., 2018b; Cheng et al., 2019), as well as the onset of crystallization
(Tremblay et al., 2019). The cooling models we use do include crystallization,
but not other exotic cooling effects, such as sedimentation of neutron-rich
ions (e.g., Bildsten & Hall 2001; Bauer et al. 2020; Blouin et al. 2021).

Figure 10: The inflation factor and median total age and age difference for 25
bins in mass and temperature of roughly similar size. Each box is colored by
the inflation factor and the white box shows the median total age (Gyr, upper
left), median total age difference (Gyr, upper right), the total age error
inflation factor (lower left), and the number of WDs in the bin (lower right).
The upper right bin also includes the parameters represented in the white
boxes. For example, the 50 WDs in the bottom left bin ($4000-6350$ K,
$0.63-0.67$ $M_{\odot}$) have a median total age of 4.6 Gyr, their ages differ
from their companions by 1.4 Gyr on average, and their total age uncertainties
need to be inflated by a factor of 1.2 for their ages to come into 1$\sigma$
agreement.

The typical percent age difference across all of the bins is around $25\%$,
ranging from 60% in the worst case ($T_{\mathrm{eff}}>12000$ K, $0.8$
$M_{\odot}$$<M<0.9$ $M_{\odot}$) and 15% in the best case ($7550$ K
$<T_{\mathrm{eff}}<9100$ K, $0.73$ $M_{\odot}$$<M<0.8$ $M_{\odot}$). The WDs
with the hottest temperatures show larger percent age differences on the order
of 35%-60% (0.2-0.5 Gyr) compared to the rest of parameter space. Thus, 25%
total age uncertainties are a representative approximation for most WD age
estimates on an individual WD for WDs $>0.63$ $M_{\odot}$ and temperatures
cooler than 12,000 K. This median WD age precision compares well to similar
results from an independent analysis of many of the same objects (Qiu et al.,
2021). It should be noted that with additional information through
spectroscopy we can achieve better age precision (Moss et al., 2022).

##  5 Conclusions

In this work, we constructed a sample of 1565 wide WD+WD binaries as well as
24 mostly new wide triple WD+WD+X systems to test the accuracy and precision
of WD total age determinations when no spectral type information is known. We
measured effective temperatures, surface gravities, and masses for all WDs in
our sample by using hydrogen-dominated atmosphere models and fitting
photometry from a variety of all-sky surveys. Using these atmospheric
parameters, we determined the total ages of each WD through the use of WD
cooling models, a theoretically motivated and observationally calibrated
initial-to-final-mass relation, and stellar evolution model grids.

Using a high-fidelity sample of wide WD+WD pairs with uncontaminated
photometry described in Section 4, the main conclusions of this work can be
summarized as follows:

  * $\cdot$

We find 24 mostly new widely separated triples with at least two WDs, and
possibly the second ever resolved triple-WD system. We find 21 of the 24
triples have a hierarchical structure. The three triples not in a hierarchical
structure will likely dissipate due to dynamical instability, but the ages of
the WDs in these systems can provide useful lower limits on the dissipation
timescale. We find that two of the non-heirarchical triple systems are
$\approx{3}$ Gyr old, while the other is quite young at $\approx{90}$ Myr old.

  * $\cdot$

We find that the total ages of the lowest mass WDs in the sample
($<0.63M_{\odot}$) are too uncertain to be used in the age comparisons
conducted in this work. The absolute age difference of systems containing at
least one lower-mass WD quickly diverges towards high values of a few Gyr. A
lower limit on the total age can still be determined for these WDs, but we do
not attempt to test the absolute reliability of their total ages. This is
unfortunate, since the mean mass of field WDs is roughly 0.63 $M_{\odot}$
(Tremblay et al., 2016; Kepler et al., 2016, 2019).

  * $\cdot$

When comparing the total ages of each component of the WD pairs, we find a
significant fraction of systems ($21-43$%) where the more massive WD has a
shorter cooling age. We find that this trend holds even for a subsample of
DA+DA systems, with 21-36% having the more massive WD with a shorter cooling
age. This is the opposite of what is expected for a monotonic and increasing
initial-to-final-mass relation. We attribute this discrepancy to the presence
of many merger remnants or unresolved triples in the sample, such that roughly
$21-36$% of our sample of currently wide WD+WD binaries started as bound
triple systems. These WDs have large age disagreement values and can cause
problems when determining ages for field WDs, but unresolved companions or a
merger history is a problem for many techniques for determining stellar ages.

  * $\cdot$

We find that the level of age disagreement strongly depends on the mass of the
WD. Wide binaries containing at least one higher-mass WD ($>$0.8 $M_{\odot}$)
generally have a lower level of agreement compared to those with lower-mass
WDs, reflecting the fact that the formal age uncertainties are often
underestimated for higher-mass WDs. We fit this dependence to determine the
amount of error inflation on the total ages needed to improve the agreement.
We find that errors on the total ages of the highest-mass WDs ($>$0.9
$M_{\odot}$) need to be inflated, up to a factor of 2.2, when assuming that
the WD is a DA. Fortunately, higher-mass WDs also have more precisely
determined ages (median total age uncertainties of $\approx{0.2}$ Gyr), so
they remain very useful chronometers.

  * $\cdot$

Looking at 25 bins in mass and effective temperature space in Figure 9, we
find that total ages are roughly accurate at the 25% level for WDs with masses
$>$0.63 $M_{\odot}$ using only photometry. The WDs with the hottest
temperatures have the largest inaccuracy, with uncertainties of 35%-60%, but
these are relatively small absolute errors on the order of 0.3 Gyr.

Due to the large sample size, we assumed that all the WDs in our sample are
DA. When a spectral type is known, a more accurate total age can be determined
and the recommendations discussed above are not immediately applicable. A
large sample of wide double WDs with known spectral types needs to be
investigated to determine the accuracy of WD total ages when the spectral type
is known.

Future work is planned to quantify how atmospheric parameters from
spectroscopy can alleviate the systematic errors mentioned above.
Spectroscopic observations do not suffer from the same problems as the
photometric methods employed in this work, and should be more sensitive to the
WD mass. A spectrum can also provide important clues as to whether a WD has a
prior merger history or is part of an unresolved binary (for example, if it is
strongly magnetic, or if it is radial-velocity variable). With our follow-up
efforts using the LDT, we continue to increase the sample of wide WD+WD pairs
with available spectra in hopes of gathering a large sample of wide double WDs
with spectroscopically determined atmospheric parameters.

In general, we find that WD total ages agree very well in wide double WDs, but
the uncertainties occasionally need to be inflated to compensate for
systematics on the total ages, the presence of an unresolved companion, or
merger history. Given the massive increase in the number of systems with a
wide WD companion thanks to Gaia, we hope this result provides a first road
map to providing more reliable stellar ages using these wide binaries.

##  6 Acknowledgements

We would like to acknowledge Jeff Andrews for helpful discussions. T.M.H,
J.J.H., and C.W. acknowledge support from the National Science Foundation
under Grant No. AST-1908119. J.v.S. acknowledges support from the National
Science Foundation under Grant No. AST-1908723. We thank Phil Muirhead and
Adam Samuels for support with the quick-look spectroscopic tool.

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. These results made use of
the Lowell Discovery Telescope (LDT) at Lowell Observatory. Lowell is a
private, non-profit institution dedicated to astrophysical research and public
appreciation of astronomy and operates the LDT in partnership with Boston
University, the University of Maryland, the University of Toledo, Northern
Arizona University and Yale University. The upgrade of the DeVeny optical
spectrograph has been funded by a generous grant from John and Ginger Giovale
and by a grant from the Mt. Cuba Astronomical Foundation.

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##  Appendix A Double WD Pair Catalog and Triples Catalog

The full catalog of wide double WD pairs with total ages and atmospheric
parameters can be accessed at
https://sites.bu.edu/buwd/files/2022/05/Table_A1.csv. Table 2 gives a
description of the columns of the catalog. Table 3 shows the 23 WD+WD+MS and
WD+WD+WD triple found in this work.

Table 2:  Double WD Pair Catalog Description

Column | Units | Description
---|---|---
System_Name |  | name of the system; determined by the source_id of the brightest component
source_id |  |  _Gaia_ EDR3 source identifier
ra | deg | right ascension (J2016)
dec | deg | declination (J2016)
pmra | mas $yr^{-1}$ | proper motion in the direction of right ascension
pmra_error | mas $yr^{-1}$ | uncertainty on the proper motion in the direction of right ascension
pmdec | mas $yr^{-1}$ | proper motion in the direction of declination
pmdec_error | mas $yr^{-1}$ | uncertainty on the proper motion in the direction of declination
parallax | mas | parallax
parallax_error | mas | uncertainty on the parallax
phot_g_mean_mag | mag |  _Gaia_ EDR3 G-band magnitude
phot_g_mean_flux | e- s-1 |  _Gaia_ EDR3 G-band flux
phot_g_mean_flux_error | e- s-1 | uncertainty on the _Gaia_ EDR3 G-band flux
phot_bp_mean_mag | mag |  _Gaia_ EDR3 GBP-band magnitude
phot_bp_mean_flux | e- s-1 |  _Gaia_ EDR3 GBP-band flux
phot_bp_mean_flux_error | e- s-1 | uncertainty on the _Gaia_ EDR3 GBP-band flux
phot_rp_mean_mag | mag |  _Gaia_ EDR3 GRP-band magnitude
phot_rp_mean_flux | e- s-1 |  _Gaia_ EDR3 G-band flux
phot_rp_mean_flux_error | e- s-1 | uncertainty on the _Gaia_ EDR3 GRP-band flux
u_sdss | mag | SDSS u-band magnitude
g_sdss | mag | SDSS g-band magnitude
r_sdss | mag | SDSS r-band magnitude
i_sdss | mag | SDSS i-band magnitude
z_sdss | mag | SDSS z-band magnitude
e_u_sdss | mag | uncertainty on the SDSS u-band magnitude
e_g_sdss | mag | uncertainty on the SDSS g-band magnitude
e_r_sdss | mag | uncertainty on the SDSS r-band magnitude
e_i_sdss | mag | uncertainty on the SDSS i-band magnitude
e_z_sdss | mag | uncertainty on the SDSS z-band magnitude
u_smap | mag | SkyMapper u-band magnitude
v_smap | mag | SkyMapper v-band magnitude
g_smap | mag | SkyMapper g-band magnitude
r_smap | mag | SkyMapper r-band magnitude
i_smap | mag | SkyMapper i-band magnitude
z_smap | mag | SkyMapper z-band magnitude
e_u_smap | mag | uncertainty on the SkyMapper u-band magnitude
e_v_smap | mag | uncertainty on the SkyMapper v-band magnitude
e_g_smap | mag | uncertainty on the SkyMapper g-band magnitude
e_r_smap | mag | uncertainty on the SkyMapper r-band magnitude
e_i_smap | mag | uncertainty on the SkyMapper i-band magnitude
e_z_smap | mag | uncertainty on the SkyMapper z-band magnitude
g_pstarr | mag | PanSTARRS g-band magnitude
r_pstarr | mag | PanSTARRS r-band magnitude
i_pstarr | mag | PanSTARRS i-band magnitude
z_pstarr | mag | PanSTARRS z-band magnitude
y_pstarr | mag | PanSTARRS y-band magnitude
e_g_pstarr | mag | uncertainty on the PanSTARRS g-band magnitude
e_r_pstarr | mag | uncertainty on the PanSTARRS r-band magnitude
e_i_pstarr | mag | uncertainty on the PanSTARRS i-band magnitude
e_z_pstarr | mag | uncertainty on the PanSTARRS z-band magnitude
e_y_pstarr | mag | uncertainty on the PanSTARRS y-band magnitude
j_tmass | mag | 2MASS J-band magnitude
h_tmass | mag | 2MASS H-band magnitude
ks_tmass | mag | 2MASS Ks-band magnitude
e_j_tmass | mag | uncertainty on the 2MASS J-band magnitude
e_h_tmass | mag | uncertainty on the 2MASS H-band magnitude
e_ks_tmass | mag | uncertainty on the 2MASS Ks-band magnitude
LoggH | [cm s-2] | log of the surface gravity from a DA model with a thick H layer
e_LoggH_upper | [cm s-2] | upper uncertainty on the log of the surface gravity from a DA model
e_LoggH_lower | [cm s-2] | lower uncertainty on the log of the surface gravity from a DA model
TeffH | K | effective temperature from a DA model
e_TeffH_upper | K | upper uncertainty on the effective temperature from a DA model
e_TeffH_lower | K | lower uncertainty on the effective temperature from a DA model
MassH | $M_{\odot}$ | mass from a DA model
e_MassH_upper | $M_{\odot}$ | upper uncertainty on the mass from a DA model
e_MassH_lower | $M_{\odot}$ | lower uncertainty on the mass from a DA model
cool_ageH | Gyr | cooling age for a DA model
e_cool_ageH_upper | Gyr | upper uncertainty on the cooling age for a DA model
e_cool_ageH_lower | Gyr | lower uncertainty on the cooling age for a DA model
init_mass | $M_{\odot}$ | initial zero-age main-sequence mass
e_init_mass_upper | $M_{\odot}$ | upper uncertainty on the initial mass
e_init_mass_lower | $M_{\odot}$ | lower uncertainty on the initial mass
tot_age | Gyr | total age of the WD
e_tot_age_upper | Gyr | upper uncertainty on the total age
e_tot_age_lower | Gyr | lower uncertainty on the total age
sp_type |  | spectral type of the WD, if available
sp_type_source |  | source of spectral type**Spectral types in this work are collected from Sion et al. (1991); Finley & Koester (1997); McCook & Sion (1999); Croom et al. (2004); Koester et al. (2009); Külebi et al. (2010); Tremblay et al. (2011); Bergeron et al. (2011); Gianninas et al. (2011); Dobbie et al. (2012); Kleinman et al. (2013); Limoges et al. (2013); Baxter et al. (2014); Andrews et al. (2015); Kepler et al. (2015); Ferrario et al. (2015); Guo et al. (2015); Kepler et al. (2016); Gentile Fusillo et al. (2019); Kilic et al. (2020); Tremblay et al. (2020)
wtd_par | mas | weighted mean of the parallax of the system
e_wtd_par | mas | uncertainty on the weighted mean of the parallax of the system
sep_AU | AU | projected on-sky separation for the pair
R_chance_align |  | chance alignment variable from El-Badry et al. (2021)

Note. — Columns dealing with properties of the individual WDs will have “_1”
or “_2” at the end to denote the primary and secondary, respectively, where
the primary has been defined as the more massive WD. The final 26 rows contain
the wide double WD pairs from the triples found in this paper.

Table 3: WD+WD+X Triples System Name | source_id | RA | Dec | $\mu_{RA}$ | $\mu_{Dec}$ | $\omega$ | G | Stellar | $\theta_{12}$ | $\theta_{13}$ | $\theta_{23}$
---|---|---|---|---|---|---|---|---|---|---|---
| (EDR3) | (J2016 ∘) | (J2016 ∘) | (mas yr-1) | (mas yr-1) | (mas) | (mag) | Type | (AU) | (AU) | (AU)
WDWDMS | 105617338913173888 | 30.6841382 | 26.0846896 | $-21.52\pm 0.08$ | $9.97\pm 0.08$ | $3.75\pm 0.08$ | 15.4 | MS |  |  |
EDR3 | 105617304552984576 | 30.6768988 | 26.0867710 | $-21.55\pm 0.39$ | $9.83\pm 0.38$ | $3.93\pm 0.34$ | 19.4 | WD | $6550$ | $6330$ | $386$
105617338913173888 | 105617338913173760 | 30.6772592 | 26.0870098 | $-21.64\pm 0.48$ | $9.80\pm 0.44$ | $3.56\pm 0.38$ | 19.5 | WD |  |  |
WDWDMS | 1463032750563461248 | 198.5937866 | 30.8450953 | $-47.76\pm 0.01$ | $26.49\pm 0.01$ | $7.17\pm 0.02$ | 11.9 | MS |  |  |
EDR3 | 1463032819282938112 | 198.5894261 | 30.8474817 | $-48.30\pm 0.09$ | $27.43\pm 0.08$ | $7.16\pm 0.11$ | 17.8 | WD | $2230$ | $2010$ | $385$
1463032750563461248 | 1463032819282938240 | 198.5902718 | 30.8477281 | $-48.02\pm 0.11$ | $25.96\pm 0.10$ | $7.26\pm 0.14$ | 18.1 | WD |  |  |
WDWDMS | 1932475596596345088 | 340.4531174 | 40.4721954 | $47.54\pm 0.01$ | $140.41\pm 0.01$ | $12.50\pm 0.01$ | 12.4 | MS |  |  |
EDR3 | 1932475596593097344 | 340.4527195 | 40.4734028 | $43.98\pm 0.08$ | $141.16\pm 0.09$ | $12.41\pm 0.10$ | 17.6 | WD | $359$ | $5380$ | $5340$
1932475596596345088 | 1932475562233361536 | 340.4758520 | 40.4792377 | $45.46\pm 0.15$ | $140.12\pm 0.19$ | $12.55\pm 0.21$ | 18.9 | WD |  |  |
WDWDMS | 2063683029262340608 | 311.2472539 | 39.4521779 | $-18.82\pm 0.17$ | $-34.76\pm 0.20$ | $5.74\pm 0.17$ | 18.9 | MS |  |  |
EDR3 | 2063684201794288512 | 311.1961772 | 39.5036734 | $-18.42\pm 0.21$ | $-34.48\pm 0.23$ | $5.25\pm 0.20$ | 19.2 | WD | 43200 | 43400 | 400
2063683029262340608 | 2063684201788243712 | 311.1954031 | 39.5037395 | $-18.65\pm 0.23$ | $-35.06\pm 0.26$ | $5.06\pm 0.22$ | 19.4 | WD |  |  |
WDWDMS | 2477316902742801152 | 22.1010462 | -8.3944721 | $-60.79\pm 0.02$ | $-170.47\pm 0.01$ | $9.41\pm 0.02$ | 9.2 | MS |  |  |
EDR3 | 2477317211979980544 | 22.1036359 | -8.3822745 | $-62.280.31$ | $-171.74\pm 0.16$ | $10.75\pm 0.28$ | 18.6 | WD | 4765 | 4879 | 128
2477316902742801152 | 2477317211980446208 | 22.1038481 | -8.3820147 | $-61.01\pm 0.86$ | $-170.16\pm 0.17$ | $9.74\pm 0.39$ | 18.8 | WD |  |  |
WDWDMS | 3197507047683918976 | 61.4674880 | -5.3325341 | $15.68\pm 0.54$ | $-21.82\pm 0.57$ | $3.95\pm 0.59$ | 20.1 | WD |  |  |
EDR3 | 3197506983261219328 | 61.4666405 | -5.3345423 | $17.00\pm 0.74$ | $-22.07\pm 0.65$ | $5.79\pm 0.77$ | 20.3 | WD | 1700 | 18000 | 17000
3197507047683918976 | 3197506944605295488 | 61.4588283 | -5.3547354 | $15.25\pm 0.78$ | $-22.32\pm 0.73$ | $4.70\pm 0.85$ | 20.5 | MS |  |  |
WDWDMS | 3218914298562400640 | 88.5970332 | 0.6795522 | $-3.25\pm 0.22$ | $-12.71\pm 0.19$ | $3.64\pm 0.22$ | 18.8 | WD |  |  |
EDR3 | 3218914298560464768 | 88.5960159 | 0.6820507 | $-3.39\pm 0.45$ | $-12.21\pm 0.39$ | $4.50\pm 0.45$ | 19.7 | WD | 2600 | 36000 | 38000
3218914298562400640 | 3218912889811507072 | 88.6192839 | 0.6487874 | $-2.52\pm 0.48$ | $-12.21\pm 0.41$ | $3.78\pm 0.49$ | 19.7 | MS |  |  |
WDWDMS | 3389527411923077376 | 82.8666421 | 13.4102532 | $56.74\pm 0.02$ | $-19.96\pm 0.01$ | $6.29\pm 0.02$ | 9.3 | MS |  |  |
EDR3 | 3389527338906751744 | 82.8642176 | 13.3994875 | $56.72\pm 0.52$ | $-19.40\pm 0.33$ | $6.70\pm 0.49$ | 19.9 | WD | 6300 | 6690 | 390
3389527411923077376 | 3389527343202017280 | 82.8640568 | 13.3988239 | $55.58\pm 0.65$ | $-19.04\pm 0.40$ | $6.33\pm 0.59$ | 20.1 | WD |  |  |
WDWDMS | 338979267341825408 | 35.0462274 | 41.0692488 | $16.16\pm 0.02$ | $-43.19\pm 0.02$ | $6.11\pm 0.02$ | 13.1 | MS |  |  |
EDR3 | 338979301700437120 | 35.0674487 | 41.0816494 | $16.58\pm 0.71$ | $-44.35\pm 0.68$ | $6.91\pm 0.57$ | 20.1 | WD | 11920 | 12370 | 460
338979267341825408 | 338979297405358848 | 35.0681580 | 41.0822196 | $15.83\pm 0.94$ | $-44.05\pm 0.88$ | $5.67\pm 0.72$ | 20.4 | WD |  |  |
WDWDMS | 3681220906103896192 | 188.0529329 | -4.1563234 | $-150.63\pm 0.02$ | $0.21\pm 0.02$ | $11.30\pm 0.02$ | 8.8 | MS |  |  |
EDR3 | 3681220906102862720 | 188.0547401 | -4.1571420 | $-153.55\pm 0.07$ | $1.45\pm 0.06$ | $11.35\pm 0.07$ | 15.7 | WD | 630 | 750 | 208
3681220906103896192 | 3681220906103896064 | 188.0547785 | -4.1577956 | $-151.04\pm 0.06$ | $3.74\pm 0.05$ | $11.33\pm 0.06$ | 15.7 | WD |  |  |
WDWDMS | 4438147044497125248 | 243.7861861 | 6.3673258 | $-55.58\pm 0.05$ | $21.32\pm 0.04$ | $3.82\pm 0.06$ | 16.2 | MS |  |  |
EDR3 | 4438147418158270848 | 243.7956106 | 6.3658639 | $-55.44\pm 0.65$ | $20.13\pm 0.52$ | $3.41\pm 0.61$ | 20.2 | WD | 8900 | 9500 | 554
4438147044497125248 | 4438147422454430464 | 243.7961101 | 6.3655519 | $-54.09\pm 0.81$ | $20.39\pm 0.60$ | $3.74\pm 0.66$ | 20.4 | WD |  |  |
WDWDMS | 4987518647789031040 | 13.6994498 | -40.0797643 | $10.22\pm 0.01$ | $32.19\pm 0.01$ | $5.27\pm 0.01$ | 12.8 | MS |  |  |
EDR3 | 4987518682147617024 | 13.6869636 | -40.0802902 | $10.16\pm 0.42$ | $31.98\pm 0.49$ | $5.93\pm 0.51$ | 20.2 | WD | 6540 | 6390 | 285
4987518647789031040 | 4987518682148769280 | 13.6872776 | -40.0806311 | $10.64\pm 0.53$ | $32.40\pm 0.68$ | $5.08\pm 0.69$ | 20.5 | WD |  |  |
WDWDMS | 5041268170632888704 | 20.2704431 | -23.8818893 | $81.43\pm 0.02$ | $-84.14\pm 0.01$ | $8.48\pm 0.02$ | 14.1 | MS |  |  |
EDR3 | 5041268200697001600 | 20.2798805 | -23.8794056 | $82.22\pm 0.45$ | $-83.36\pm 0.33$ | $9.04\pm 0.41$ | 19.8 | WD | 3810 | 4080 | 278
5041268170632888704 | 5041268204992049664 | 20.2805981 | -23.8793989 | $82.86\pm 0.64$ | $-85.21\pm 0.46$ | $8.88\pm 0.62$ | 20.3 | WD |  |  |
WDWDMS | 5250998123862710272 | 143.2102271 | -61.9153951 | $91.80\pm 0.07$ | $-118.74\pm 0.05$ | $8.43\pm 0.04$ | 16.4 | MS |  |  |
EDR3 | 5250998119556563200 | 143.2106552 | -61.9129685 | $92.89\pm 0.23$ | $-117.90\pm 0.15$ | $8.54\pm 0.14$ | 18.9 | WD | 1038 | 18650 | 17670
5250998123862710272 | 5251022416193428096 | 143.1869479 | -61.8730341 | $92.32\pm 0.69$ | $-117.36\pm 0.47$ | $8.89\pm 0.41$ | 20.2 | WD |  |  |
WDWDMS | 5455352324194328192 | 158.0412243 | -29.9751592 | $72.86\pm 0.01$ | $-5.94\pm 0.02$ | $7.89\pm 0.02$ | 10.5 | MS |  |  |
EDR3 | 5455352702151342592 | 158.0283938 | -29.9535991 | $72.95\pm 0.06$ | $-6.34\pm 0.10$ | $7.78\pm 0.10$ | 17.4 | WD | 11080 | 10020 | 1077
5455352324194328192 | 5455352697853121280 | 158.0291298 | -29.9558703 | $73.17\pm 0.23$ | $-5.65\pm 0.41$ | $7.42\pm 0.41$ | 19.7 | WD |  |  |
WDWDMS | 5560517931332969600 | 111.8889790 | -41.3676372 | $-154.05\pm 0.01$ | $-4.86\pm 0.01$ | $9.65\pm 0.01$ | 11.5 | MS |  |  |
EDR3 | 5560518000052459520 | 111.9127938 | -41.3679211 | $-153.72\pm 0.06$ | $-5.43\pm 0.06$ | $9.61\pm 0.06$ | 17.1 | WD | 6671 | 6676 | 559
5560517931332969600 | 5560518000048274304 | 111.9127590 | -41.3664248 | $-153.02\pm 0.79$ | $-6.17\pm 0.91$ | $9.91\pm 0.76$ | 20.5 | WD |  |  |
WDWDMS | 5874024705460836992 | 224.4952433 | -63.2877651 | $-33.67\pm 0.02$ | $-54.39\pm 0.02$ | $8.60\pm 0.02$ | 8.1 | MS |  |  |
EDR3 | 5874024769842933760 | 224.5481622 | -63.2928712 | $-34.04\pm 0.06$ | $-54.37\pm 0.06$ | $8.49\pm 0.06$ | 16.8 | WD | 10190 | 10440 | 299
5874024705460836992 | 5874024774146311808 | 224.5496572 | -63.2926298 | $-33.33\pm 0.09$ | $-53.16\pm 0.09$ | $8.75\pm 0.09$ | 17.5 | WD |  |  |
WDWDMS | 6087696442178470400 | 202.4274557 | -44.0483557 | $-14.23\pm 0.08$ | $-5.96\pm 0.06$ | $3.89\pm 0.07$ | 16.8 | MS |  |  |
EDR3 | 6087695613244135936 | 202.4327137 | -44.0750059 | $-14.20\pm 0.20$ | $-6.22\pm 0.17$ | $3.48\pm 0.21$ | 18.9 | WD | 25200 | 25800 | 840
6087696442178470400 | 6087695617537911040 | 202.4318915 | -44.0756854 | $-15.18\pm 0.33$ | $-5.69\pm 0.26$ | $3.53\pm 0.31$ | 19.6 | WD |  |  |
WDWDMS | 6145125411962111104 | 184.8137056 | -42.3393993 | $-25.88\pm 0.02$ | $9.91\pm 0.02$ | $2.88\pm 0.03$ | 15.4 | MS |  |  |
EDR3 | 6145125411959049856 | 184.8084702 | -42.3439378 | $-25.86\pm 0.26$ | $10.15\pm 0.20$ | $3.35\pm 0.32$ | 19.5 | WD | 7450 | 7930 | 482
6145125411962111104 | 6145125411962108416 | 184.8080984 | -42.3442087 | $-25.93\pm 0.34$ | $9.64\pm 0.31$ | $3.03\pm 0.48$ | 19.7 | WD |  |  |
WDWDMS | 6756233901661753472 | 288.8382875 | -32.3347021 | $15.70\pm 0.02$ | $4.10\pm 0.01$ | $6.05\pm 0.02$ | 11.8 | MS |  |  |
EDR3 | 6756233905958975360 | 288.8369795 | -32.3330247 | $15.28\pm 0.36$ | $4.63\pm 0.29$ | $6.67\pm 0.34$ | 19.4 | WD | 1195 | 2191 | 1757
6756233901661753472 | 6756233905958976512 | 288.8339392 | -32.3344810 | $16.28\pm 0.52$ | $2.76\pm 0.44$ | $5.71\pm 0.47$ | 19.9 | WD |  |  |
WDWDMS | 763543514763336832 | 169.8371661 | 37.0721947 | $90.75\pm 0.01$ | $-53.81\pm 0.02$ | $8.82\pm 0.02$ | 12.0 | MS |  |  |
EDR3 | 763543312899793664 | 169.8497518 | 37.0715253 | $90.80\pm 0.48$ | $-54.38\pm 0.58$ | $9.27\pm 0.51$ | 19.8 | WD | 4108 | 3937 | 423
763543514763336832 | 763543308609191936 | 169.8490934 | 37.0706329 | $89.68\pm 0.48$ | $-54.14\pm 0.59$ | $8.93\pm 0.50$ | 19.9 | WD |  |  |
WDWDMS | 779305808516516864 | 161.4150757 | 40.0952785 | $-72.91\pm 0.03$ | $21.17\pm 0.03$ | $6.99\pm 0.03$ | 14.9 | MS |  |  |
EDR3 | 779305808516517376 | 161.4055376 | 40.0933527 | $-74.03\pm 0.22$ | $21.61\pm 0.25$ | $7.23\pm 0.27$ | 18.8 | WD | 3880 | 4050 | 183
779305808516516864 | 779305808516231552 | 161.4050805 | 40.0934117 | $-71.23\pm 0.25$ | $21.35\pm 0.24$ | $7.36\pm 0.29$ | 18.9 | WD |  |  |
WDWDMS | 929001808377561216 | 121.6861312 | 44.7481737 | $3.01\pm 0.03$ | $-3.41\pm 0.03$ | $4.99\pm 0.03$ | 15.5 | MS |  |  |
EDR3 | 929001804082135168 | 121.6837451 | 44.7508697 | $3.08\pm 0.15$ | $-3.38\pm 0.11$ | $4.93\pm 0.14$ | 18.2 | WD | 2230 | 2710 | 1023
929001808377561216 | 929001804082135296 | 121.6818686 | 44.7503870 | $3.46\pm 0.21$ | $-2.99\pm 0.16$ | $5.00\pm 0.20$ | 18.8 | WD |  |  |
WDWDWD | 4190499986125543168 | 298.3999651 | -10.3254921 | $-11.48\pm 0.07$ | $-16.52\pm 0.04$ | $7.75\pm 0.06$ | 16.3 | WD |  |  |
EDR3 | 4190499986125543296 | 298.4001112 | -10.3248602 | $-10.92\pm 0.07$ | $-15.78\pm 0.04$ | $7.66\pm 0.06$ | 16.4 | WD | 302 | 6280 | 6490
4190499986125543168 | 4190500054845023488 | 298.3879711 | -10.3319738 | $-10.96\pm 0.11$ | $-16.14\pm 0.06$ | $7.76\pm 0.10$ | 17.3 | WD |  |  |

##  Appendix B Shifted IFMR

The table used for the initial-to-final-mass relation in Figure 3 is detailed
here in Table 4, adapted from Fields et al. (2016). The WD mass is
$M\textsubscript{f}$ and the ZAMS mass is $M\textsubscript{i}$.

Table 4: Shifted MESA IFMR $M\textsubscript{i}$ $\left[M_{\odot}\right]$ |  $M\textsubscript{f}$ $\left[M_{\odot}\right]$ |  $M\textsubscript{i}$ $\left[M_{\odot}\right]$ |  $M\textsubscript{f}$ $\left[M_{\odot}\right]$
---|---|---|---
1.0 | 0.583 | 3.5 | 0.802
1.1 | 0.594 | 3.6 | 0.828
1.2 | 0.601 | 3.7 | 0.848
1.3 | 0.608 | 3.8 | 0.868
1.4 | 0.613 | 3.9 | 0.880
1.5 | 0.615 | 4.0 | 0.888
1.6 | 0.619 | 4.1 | 0.895
1.7 | 0.621 | 4.2 | 0.901
1.8 | 0.624 | 4.3 | 0.906
1.9 | 0.625 | 4.4 | 0.913
2.0 | 0.629 | 4.5 | 0.919
2.1 | 0.634 | 4.6 | 0.927
2.2 | 0.640 | 4.7 | 0.930
2.3 | 0.645 | 4.8 | 0.939
2.4 | 0.651 | 4.9 | 0.955
2.5 | 0.657 | 5.0 | 0.960
2.6 | 0.664 | 5.1 | 0.957
2.7 | 0.672 | 5.2 | 0.965
2.8 | 0.682 | 5.3 | 0.974
2.9 | 0.697 | 5.4 | 0.983
3.0 | 0.709 | 5.5 | 0.994
3.1 | 0.724 | 5.6 | 1.006
3.2 | 0.743 | 5.7 | 1.020
3.3 | 0.763 | 5.9 | 1.023
3.4 | 0.780 | 6.0 | 1.033