# The K2 M67 Study: Precise Mass for a Turnoff Star in the Old Open Cluster
M67

Eric L. Sandquist San Diego State University, Department of Astronomy, San
Diego, CA, 92182 USA  David W. Latham Center for Astrophysics — Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Robert D. Mathieu
University of Wisconsin-Madison, Department of Astronomy, Madison, WI 53706,
USA  Emily Leiner Center for Interdisciplinary Exploration and Research in
Astrophysics, Northwestern University, Evanston, IL 60208, USA  University of
Wisconsin-Madison, Department of Astronomy, Madison, WI 53706, USA  NSF
Astronomy and Astrophysics Postdoctoral Fellow  Andrew Vanderburg Harvard-
Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA  Dennis Stello
School of Physics, The University of New South Wales, Syndey NSW 2052,
Australia  Sydney Institute for Astronomy (SIfA), School of Physics,
University of Sydney, NSW, 2006, Australia  Stellar Astrophysics Centre,
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120,
DK-8000 Aarhus C, Denmark  Jerome A. Orosz  San Diego State University,
Department of Astronomy, San Diego, CA, 92182 USA  Luigi R. Bedin  Istituto
Nazionale Astrofisica di Padova - Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, I-35122 Padova, Italy  Mattia Libralato Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Istituto
Nazionale Astrofisica di Padova - Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, I-35122 Padova, Italy  Dipartimento di Fisica e
Astronomia ‘Galileo Galilei’, Università di Padova, Vicolo dell’Osservatorio
3, Padova I-35122, Italy  Luca Malavolta  Istituto Nazionale Astrofisica di
Padova - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5,
I-35122 Padova, Italy  Dipartimento di Fisica e Astronomia ‘Galileo Galilei’,
Università di Padova, Vicolo dell’Osservatorio 3, Padova I-35122, Italy
Domenico Nardiello  Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
Istituto Nazionale Astrofisica di Padova - Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, I-35122 Padova, Italy Eric L. Sandquist
<EMAIL_ADDRESS>
###### Abstract

We present a study of the bright detached eclipsing main sequence binary WOCS
11028 (Sanders 617) in the open cluster M67. Although the binary has only one
eclipse per orbital cycle, we show that the masses of the stars can be derived
very precisely thanks to a strong constraint on the orbital inclination:
$M_{A}=1.222\pm 0.006M_{\odot}$ and $M_{B}=0.909\pm 0.004M_{\odot}$. We use a
spectral energy distribution fitting method to derive characteristics of the
component stars in lieu of the precise radii that would normally be derived
from a doubly-eclipsing binary. The deconvolution of the SEDs reveals that the
brighter component of the binary is at the faint turnoff point for the cluster
— a distinct evolutionary point that occurs after the convective core has been
established and while the star is in the middle of its movement toward lower
surface temperature, before the so-called hook at the end of main sequence.
The measurements are in distinct disagreement with evolution models at solar
metallicity: higher metal abundances are needed to reproduce the
characteristics of WOCS 11028 A. We discuss the changes to model physics that
are likely to be needed to address the discrepancies. The clearest conclusions
are that diffusion is probably necessary to reconcile spectroscopic abundances
of M67 stars with the need for higher metallicity models, and that reduced
strength convective overshooting is occuring for stars at the turnoff. At
super-solar bulk metallicity, various indicators agree on a cluster age
between about 3.5 and 4.0 Gyr.

††facilities: MMT, FLWO: 1.5m, TNG (HARPS-N), Sloan (APOGEE)††software:  IRAF
(Tody, 1986, 1993), HARPS Data Reduction Software (v3.7), ELC (Orosz &
Hauschildt, 2000)

##  1 Introduction

As part of the K2 M67 Study (Mathieu et al., 2016), we are analyzing eclipsing
binary stars to provide a precise mass scale for the cluster stars as an aid
to comparisons with theoretical models. The K2 mission has uncovered eclipsing
binaries that would have been difficult to identify from the ground, and has
made it possible to precisely study even those systems with shallow eclipses.
Our goal here is to measure masses and radii to precisions of better than 1%
in order to be comparable to results from the best-measured binaries in the
field (Andersen, 1991; Torres et al., 2010; Southworth, 2015). We have
previously analyzed the binary WOCS 12009 (Sandquist et al., 2018), although
we found that at least the primary star showed signs of having been part of a
stellar merger. The brighter binary HV Cnc has recently been reanalyzed by
Gökay et al. (2020), but a proper analysis is complicated by an extremely
faint secondary star and a possibly associated third star.

WOCS 11028 (also known as Sanders 617, EPIC 211411112;
$\alpha_{2000}=08^{\mbox{h}}50^{\mbox{m}}26\fs 99$, $\delta_{2000}=+11\arcdeg
48\arcmin 31\farcs 3$) in M67 was first reported as a double-lined
spectroscopic binary star by Geller et al. (2015), although the system had
been monitored previously for about 7 years by D. Latham and collaborators,
including determination of orbital parameters. The K2 M67 Study collaboration
detected an eclipse in Kepler K2 observations for campaign 5. The
spectroscopic parameters ($P=62.593$ d, $e=0.625$, $\omega=236\arcdeg$) showed
that this meant that the system has only one eclipse per orbit, which is not
uncommon for eccentric binaries. However, because the orbital separation is
relatively large and the eccentricity is not too extreme, the orbital
inclination can still be fairly tightly constrained between two limits:
inclinations that give two eclipses per orbit (one a grazing eclipse), and
ones that give no eclipses. For this binary, the singly-eclipsing range only
covers inclinations from $86\fdg 2$ to $88\fdg 8$, and light-curve modeling
makes it possible to constrain this range even further. We will show below
that we can therefore derive precise masses for the two stars in this binary.
In the cluster’s color-magnitude diagram (CMD), the binary sits near the
cluster turnoff, which is an indication that the stars are relatively far
along in their core hydrogen burning. Thus, there is hope that these stars can
significantly constrain the age of the cluster they reside in.

In section 2, we describe the photometric and spectroscopic data that we
collected and analyzed for the binary. In section 3, we describe the modeling
of the binary system. In section 4, we discuss the results and the
interpretation of the system.

##  2 Observations and Data Reduction

###  2.1 K2 Photometry

M67 was observed during Campaigns 5, 16, and 18 of the K2 mission. WOCS 11028
was observed during all of the campaigns in a custom aperture with long
cadence (30 min) exposures. Single eclipses were observed during campaigns 5
and 16, having about a 2.7% decrease in flux. (No eclipse was observed in
campaign 18 data, and the ephemeris indicates that one was not expected.) The
eclipse times were measured to be at BJD $2457178.9531\pm 0.00088$, and
$2458117.8789\pm 0.00075$ using the bisector method of Kwee & van Woerden
(1956). The radial velocity curve shows that these are secondary eclipses (of
the cooler, less massive star). In the K2 light curves, there is no sign of
primary eclipses at the four epochs predicted by the orbital ephemeris (see
Figure 1).

Figure 1: K2 photometry from the EVEREST pipeline for the three campaigns that
observed WOCS 11028. The two observed eclipses are the largest flux drops in
C5 and C16, and the predicted times of the non-eclipsing conjunction are shown
with arrows.

Because the K2 camera has relatively large pixels ($3\farcs 98$ on a side), we
briefly discuss the possibility of contaminating light from unrelated stars.
Although WOCS 11028 is a member of M67, it is in the outskirts of the cluster.
As shown in Figure 2, the nearest bright star is WOCS 6030 / Sanders 619 at a
distance of $54\arcsec$, and about 0.3 mag fainter in Gaia $G$ band. There is
a Gaia source (indicated in Figure 2) $8\arcsec$ away, but nearly 8 $G$
magnitudes fainter. There are three other Gaia sources between 27 and
$34\arcsec$ away, but all are at least 5.3 mag fainter in $G$. In the light
curves described below, the photometric apertures did not extend more than 5
pixels away from the photocenter, and so avoided the brighter stars. As a
result, we assume contamination of the K2 apertures used for WOCS 11028 is
negligible.

Figure 2: Pan-STARRS1 $y$ image centered on WOCS 11028. The image is
$90\arcsec$ tall, and the sky orientation is shown.

Because of the incomplete gyroscopic stabilization during the K2 mission,
systematic effects on the light curve are known to be substantial. We used two
different light curves for WOCS 11028 as part of our investigation of
systematic error sources. The first light curve we experimented with utilized
the K2SFF pipeline (Vanderburg & Johnson, 2014; Vanderburg et al., 2016) that
involved stationary aperture photometry along with correction for correlations
between the telescope pointing and the measured flux. The light curve that
resulted still retained small trends over the long term, so these were removed
by fitting the out-of-eclipse points with a low-order polynomial and dividing
the fit. The second light curve came from version 2.0 of the EVEREST pipeline
(Luger et al., 2018), which uses pixel-level decorrelation to remove
instrumental effects. We used the detrended and co-trending basis vector-
corrected light curve. However, the eclipse depth is about 4% deeper in the
EVEREST light curve than in the K2SFF version, so we need to gauge the effects
on measurements of the binary characteristics. Because the K2 light curves
have high signal-to-noise, they have a substantial influence on best-fit
binary star parameter values (see section 3.4). The scatter among the best-fit
parameter values when comparing runs with different K2 light curve reductions
is sometimes larger than the statistical uncertainties. Because there is no
clearly superior method for processing the K2 light curves, we will take the
scatter in the binary model parameter values as a partial indicator of
systematic error resulting from the processing.

###  2.2 The Spectral Energy Distribution

Because the eclipses of the WOCS 11028 binary provide minimal information on
the sizes of the individual stars, we need other well-measured characteristics
of the binary’s stars in order to extract age information for the cluster.
Fortunately, M67 has been heavily observed over a wide wavelength range,
making it possible to construct well-sampled spectral energy distributions for
the binary and similarly bright stars in the cluster. In this section, we
describe the spectroscopy and photometry we have assembled, and the efforts to
put the observations on a consistent flux scale.

Ultraviolet: We obtained photometry and a spectrum from the Galaxy Evolution
Explorer (GALEX; Martin et al. 2005) archive for the NUV passband ($1771-2831$
Å). WOCS 11028 was imaged twice, for 1691.05 s (GI1 proposal 94, P.I. W.
Landsman) and 5555.2 s (GI1 proposal 55. P.I. K. Honeycutt). As the archived
magnitudes are based on count rates with minimal background contributions, we
computed a final magnitude and flux based on the average count rate for both
observations. Morrissey et al. (2007) describes the characteristics of the
GALEX photometry and its calibration to flux. GALEX magnitudes are on an AB
system (Oke & Gunn, 1983), and we used the zero point magnitude
($m_{NUV}=20.08$) and reference flux ($2.06\times 10^{-16}$ erg s-1 cm2 Å-1)
to convert to flux.

A NUV grism spectrum was taken as part of GI1 proposal 94 (PI: W. Landsman) on
2005 Feb. 18 with a total exposure time of 27653 s. The NUV spectrum covers a
wider wavelength range than the NUV photometry filter, but there was
effectively no signal detected for WOCS 11028 at wavelengths less than about
2000 Å. The spectrum was obtained from MAST, and the flux calibration from the
pipeline reduction was used.

Observations were taken of a smaller portion of M67 (including WOCS 11028)
using the UVOT telescope on the Swift satellite. We collected UV fluxes in the
$uvw1$, $uvm2$, and $uvw2$ bands from the Swift UVOT Serendipitous Source
Catalogue (version 1.1; Page et al. 2014). The three bands mostly cover the
same wavelength range as the GALEX NUV filter, but with somewhat finer
resolution. Flux correction factors from Poole et al. (2008) were applied to
go from the gamma-ray burst spectrum calibration given in the archive to one
utilizing Pickles (1998) library stars.

Near-ultraviolet, Optical and Near-infrared: M67 has been frequently observed
from the ground for the purposes of photometric calibration. For the purposes
of an SED, narrow-band filters are particularly useful, and we discuss these
first. Balaguer-Núñez et al. (2007) presented Strömgren $uvby$ photometry for
the cluster. Because M67 stars are commonly used as standards in Strömgren
photometry (Nissen et al., 1987), we can be fairly assured that the
observations are tied to the standard system. We employed reference fluxes
from Gray (1998) to convert the magnitudes to fluxes.

Fan et al. (1996) conducted wide-field observations of M67 in a series of
narrow-band filters (“BATC”) covering from 3890 to 9745 Å using bandpasses
avoiding most important sky lines. The wavelength coverage of the filters is
better in the near-infrared, making them a good complement to Strömgren
photometry. The BATC survey goal was spectrophotometry at the 1% level, and
the study largely achieved that, judging from the low scatter in their color-
magnitude diagrams. Their reported magnitudes are on the Oke & Gunn AB system.
We found that fluxes from these magnitudes were systematically higher than
similar observations in other systems. We collected magnitudes in a larger set
of filters from BATC Data Release 1111http://vizier.u-strasbg.fr/viz-
bin/VizieR-3?-source=II/262/batc, and found that these were more consistent,
probably due to improved calibration (Zhou et al., 2003).

Narrow-band photometry is also available in the Vilnius filter system for many
stars in M67 (although not WOCS 11028) in the study of Laugalys et al. (2004).
We converted the reported magnitudes to fluxes using the zeropoints from Mann
& von Braun (2015). The filters in the Vilnius system are somewhat denser in
the blue portions of the optical, and these again complement the
spectrophotometry in the BATC system.

We have Johnson-Cousins photometry in $BVI_{C}$ from Sandquist (2004) and
Yadav et al. (2008), in $UBVR_{C}I_{C}$ from Montgomery et al. (1993), in
$BVRI$ from Nardiello et al. (2016b), and in $BV$ from Data Release 9 of the
AAVSO Photometric All-Sky Survey (APASS; Henden et al. 2015). These
measurements are on a Vega magnitude system, and so have been converted to
fluxes using reference magnitudes from Table A2 of Bessell et al. (1998), and
accounting for the known reversal of the zero point correction rows for
$f_{\lambda}$ and $f_{\nu}$. The absolute calibrations of each of these
studies are unavoidably different for the same filter bands, and this will
contribute to the noise in the SEDs. However, this does not affect their use
in determining the relative contributions of the two stars in the WOCS 11028
binary (see §2.2.1). (The $R$ and $I$ filter observations given in Nardiello
et al. (2016b) were actually taken in SDSS $r$ and $i$ filters, and have been
recalibrated to the Sloan DR12 system (D. Nardiello, private communication).
These data were not used in fits to SEDs.)

There are several additional surveys that provide calibrated broad-band
photometric observations. We used PSF magnitudes from Data Release 14 of the
Sloan Digital Sky Survey (SDSS; York et al. 2000), calibrated according to
Finkbeiner et al. (2016). The SDSS is nearly on the AB system, with small
offsets in $u_{SDSS}$ and $z_{SDSS}$ that we have also corrected for here. The
Pan-STARRS1 survey (Kaiser et al., 2010) contains photometry in 5 filters, and
we use their mean PSF magnitudes here. Zero points for its AB magnitude system
are given in Schlafly et al. (2012). The APASS survey also observed the
cluster in Sloan $g^{\prime}r^{\prime}i^{\prime}$ filters, and their
photometry was flux calibrated from the AB system.

Finally, Gaia has already produced high-precision photometry extending far
down the main sequence of M67 as part of Data Release 2\. We obtained the
fluxes in the $G$, $G_{BP}$, and $G_{RP}$ bands from the Gaia Archive.

Infrared: We obtained Two-Micron All-Sky Survey (2MASS; Skrutskie et al. 2006)
photometry in $JHK_{s}$ from the All-Sky Point Source Catalog, and have
converted these to fluxes using reference fluxes for zero magnitude from Cohen
et al. (2003). Many stars also have photometry in three bands from the Wide
Field Infrared Explorer (WISE; Wright et al. 2010), which were also converted
to fluxes using tabulated reference fluxes at zero magnitude.

####  2.2.1 Photometric Deconvolution

A benefit of the binary’s membership in the M67 cluster is that it should be
possible to describe the binary’s light as the sum of the light of two single
cluster stars. To that end, we compiled a database of photometric measurements
from likely single main sequence stars in M67, and sought a combination of
stars whose summed fluxes most closely match the fluxes of WOCS 11028. For our
sample of probable single stars, we selected likely members based on Gaia
proper motions, parallaxes, and photometry, as well as radial velocity
membership and binarity information from Geller et al. (2015). Likely binaries
were rejected by restricting the sample to those classified as radial velocity
SM (single members) with Gaia photometry placing them within about 0.03 mag of
the blue edge of the main sequence band in the $G_{BP}-G_{RP}$ color. Selected
stars are shown in Figure 3.

Figure 3: Color-magnitude diagrams of likely single stars near the turnoff of
M67 in photometric datasets with the highest signal-to-noise. ($BV$ and
$VI_{c}$: Sandquist 2004; $vy$: Balaguer-Núñez et al. 2007; $PV$: Laugalys et
al. 2004; $m_{3890}m_{5795}$: Fan et al. 1996). The red points are combined
photometry of the WOCS 11028 binary, green points are the proxy photometry
values for the components of the binary, yellow points are solar twins, and
blue points are stars identified as being at critical points in the
evolutionary sequence.

The benefit of this procedure for constraining the SEDs of the binary’s stars
is that it is a relative comparison using other cluster stars with the same
distance, age, and chemical composition. As such, it is independent of
distance and reddening (as long as these are the same for the binary and
comparison stars), the details of the filter transmission curves (as long as
the same filter is used for observations of the different M67 stars), and flux
calibration of any of the filters (as long as the calibration is applied
consistently). We can also avoid systematic errors associated with theoretical
models or with the consistency of the different parts of empirical SEDs
compiled from spectra.

We tested two ways of doing the decomposition: using actual M67 stars as
proxies and checking all combinations of likely main sequence stars; and
fitting all main sequence stars with photometry in a given filter as a
function of Gaia $G$ magnitude in order to derive SEDs that could be combined.
When using actual M67 stars, we are somewhat at the mercy of the photometry
that is available for each star (and the binary) and of the stellar sampling
of the main sequence. The use of fits allows for finer examination of the main
sequence, although there is some risk of diverging from the photometry of real
stars. Even for a relatively rich cluster like M67, parts of the main sequence
are not well populated, and we believe that the main-sequence fitting method
gives better results in that case. We present the results of both analyses
below, however.

To judge the agreement between the summed fluxes of a pair of stars and the
binary photometry, we looked for a minimum of a $\chi^{2}$-like parameter
involving fractional flux differences in the different filter bands:

$\sum_{i}\left(\frac{F_{i,bin}-(F_{i,1}+F_{i,2})}{\sigma_{i,bin}\cdot
F_{i,bin}}\right)^{2}$

where $\sigma_{i,bin}=10^{-(\sigma_{i,m}+0.01)/2.5}-1$ and $\sigma_{i,m}$ is
the magnitude uncertainty in the $i$th filter band for the binary. The
addition of 0.01 mag somewhat deweights photometry with very low uncertainties
(Gaia and GALEX NUV) that results partly from their very wide filter
bandpasses.

When using M67 stars as proxies, the results can be affected by the selection
of filters that could be used. We examined solutions excluding the Swift/UVOT
and/or Strömgren photometry because they covered the smallest portion of the
cluster field, and excluding them allowed us to use larger sets of stars. The
sample sizes were 52 stars having photometry in all of the filter bands, 109
with UVOT excluded, and 125 with UVOT and Strömgren photometry excluded. In
all cases, the best fits involved WOCS 6018 (Sanders 763) as the brighter
star, while the preferred fainter star was either WOCS 10027/S1597 (with UVOT
photometry excluded) or WOCS 16013/S795 (for all photometry, or with UVOT and
Strömgren excluded). Solutions involving WOCS 6018 with different faint stars
(WOCS 17021/S820, WOCS 21046/S1731, WOCS 16018/S814) were the next-best fits,
and all of these stars reside in similar positions on the main sequence.
Figure 4 shows a comparison of the SED of WOCS 16013 with WOCS 6018, giving an
indication of the significant optical and infrared excess for the binary. The
bottom panel shows the residuals between the binary flux and the summed fluxes
of WOCS 6018 and WOCS 10027\. A potential limiting factor is the stellar
sampling available near the brighter star, but we have stars within 0.007 $G$
mag on the bright side and within 0.016 mag on the faint side.

Figure 4: Top: SED for the binary WOCS 11028 (photometry: red points, GALEX
spectrum: cyan lines) and for the main sequence stars WOCS 6018 (blue points)
and WOCS 16013 (green points). Horizontal error bars represent the effective
width of the filter. Bottom: Fractional difference between the binary star
photometry fluxes and the result of the main sequence fitting procedure.

The main-sequence fitting procedure can be employed in any filter with a
sufficient sample of stars covering the range of brightnesses for the binary’s
stars. In our case, this eliminated the $R_{C}$, Swift $uvm2$, and WISE W3 and
W4 filters from consideration. Our fit statistic had a minimum value of 50.0
for the selection of 46 filters. We estimated the $2\sigma$ uncertainty in the
fit based on where the goodness-of-fit statistic reached a value of 4 above
the minimum value. For example, this returns $2\sigma(G_{A})=0.035$ and
$2\sigma(G_{B})=0.15$. As expected, there is an anti-correlation between
values for the primary and secondary stars because of the need to match the
binary fluxes (see Fig. 5).

Figure 5: Goodness-of-fit contours in goodness-of-fit statistic $\chi^{2}$
versus Gaia $G$ magnitudes for the two components of WOCS 11028 out to
approximately $2\sigma$ away from the best fit.

For an additional check on the brighter star, we compared the GALEX
ultraviolet spectrum of WOCS 11028 with that of WOCS 6018, as shown in Fig. 6.
Although our minimization procedure encourages agreement in near-UV filters,
it doesn’t require agreement of the spectra. In spite of this, the overall
shape and flux level of the two stars agree well, with the binary becoming
consistently higher on the long wavelength end. The slope of the GALEX
spectrum is one of the more sensitive indicators of temperature for stars on
the upper main sequence of M67, and this combination of stars does a better
job of reproducing that than two equal-mass stars, for example. The
enhancement in flux at the long-wavelength end of the spectrum can be
attributed to a small contribution from the faint star in the binary. The four
photometric observations in the near ultraviolet (from GALEX and Swift/UVOT)
indicate that the binary is slightly brighter than WOCS 6018 in all of the
filter bands.

Figure 6: Top: Comparisons of GALEX NUV spectra for the binary WOCS 11028
(red) and likely main sequence star WOCS 6018 (cyan). Photometry from GALEX
(NUV filter) and Swift/UVOT is shown as points, with the approximate widths of
the filters shown as horizontal bars. Bottom: Fractional difference between
GALEX spectra of WOCS 11028 and WOCS 6018.

The above fits to the binary’s photometry provide luminosity ratios in
different bands independent of models, and we use some of these as constraints
in modeling the radial velocity and eclipse light curve data in §3.4. These
ratios are provided in Table The K2 M67 Study: Precise Mass for a Turnoff Star
in the Old Open Cluster M67.

####  2.2.2 Effective Temperatures and Bolometric Fluxes

With SEDs in hand for stars that we believe are good proxies for the binary’s
components, we can derive additional properties via fits with theoretical
models. The models can introduce systematic errors in the quantities we try to
measure, although we will try to mitigate them. We tested models from ATLAS9
(Castelli & Kurucz, 2004)222The models were calculated using the ATLAS9
fortran code that employed updated 2015 linelists, and at temperatures between
the published gridpoints. and Coelho (2014), but the results were very
similar, and we primarily discuss ATLAS9 model fits below. Models were
adjusted to account for the interstellar reddening of M67 [$E(B-V)=0.041\pm
0.004$; Taylor 2007] using the Cardelli et al. (1989) extinction curve. Model
photometry was calculated using the IRAF333IRAF is distributed by the National
Optical Astronomy Observatory, which is operated by the Association of
Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation. routine sbands. Bolometric flux was
derived from the median multiplicative factor needed to bring the best-fit
model photometry into agreement with each observation, and we quote an
uncertainty on the median that is based on the range covered by observations
that are within $\sqrt{N}/2$ entries of the middle in the ordered list. We
find that the SEDs of WOCS 6018 and the bright component from our main
sequence fit method are in good agreement with models of 6200 K (see Fig. 7),
while all of the candidate fainter components indicate a temperature of about
5500 K (see Fig. 8).

Figure 7: Top: SED for the proxy for the bright star in the WOCS 11028 binary
(red points for photometry) and a fitted ATLAS9 model for $T_{eff}=6185$ K and
$\log g=4.25$ (solid line, and green points for integrations over filter
response curves). Horizontal error bars represent the effective width of the
filter. A GALEX spectrum of WOCS 6018 is shown in cyan. Bottom: Fractional
difference between the stellar fluxes and the best-fit model fluxes for ATLAS9
(green)  Figure 8: Top: SED for the proxy for the faint star in the WOCS 11028
binary (red points) and a fitted ATLAS9 model for $T_{eff}=5500$ K (solid
line, and green points for integrations over filter response curves).
Horizontal error bars represent the effective width of the filter. Bottom:
Fractional difference between the stellar fluxes and the best-fit model fluxes
for ATLAS9 (green) models.

To try to achieve greater precision, we also calculated temperatures using the
infrared flux method (Blackwell & Shallis, 1977, IRFM;). Briefly, this method
exploits the difference in temperature sensitivity between the bolometric flux
and monochromatic fluxes in the infrared on the Rayleigh-Jeans portion of the
spectrum. With the available photometry databases, we have measurements of
fluxes covering the majority of the stellar energy emission. The ratio of the
bolometric and infrared fluxes (we use fluxes in 2MASS bands) can be compared
to theoretical values (where we again use ATLAS9 models):

$\frac{\mathcal{F}_{\rm
bol}(\mbox{Earth})}{\mathcal{F}_{\lambda_{IR}}(\mbox{Earth})}=\frac{\sigma
T_{\rm eff}^{4}}{\mathcal{F}_{\lambda_{IR}}(\mbox{model})}$

We used the 2MASS flux calibration of Casagrande et al. (2010) in this case,
in part because it produced greater consistency between the temperatures
derived in the three bands — full ranges between 15 and 40 K. Starting from a
solar-metallicity ATLAS9 model that produced a good fit by eye, we adjusted
the temperature of the synthetic spectrum until it matched the average IRFM
temperature from the three 2MASS bands. We find temperatures of 6185 K and
5500 K for the two stars. The model surface gravity was chosen from the
eclipsing binary results or from MESA models, although changes had little
effect. Uncertainty in the reddening (which modifies the shape of the
theoretical model we fit to the data) and metal content of the models affects
the measured temperature at about the 15 K level. Overall, we estimate that
there is an uncertainty of about 50 K and 75 K in the temperatures.

For comparison, there are spectroscopic temperatures available for stars near
the position of the brighter star in the CMD from recent surveys looking at
abundance differences as a function of evolutionary phase. Liu et al. (2019)
found temperatures by forcing excitation and ionization balance in their
modeling of Fe I and II lines, and for three stars slightly brighter in $G$,
they derived temperatures between about 6100 and 6150 K. Souto et al. (2019)
found temperatures from ASPCAP fits to infrared APOGEE spectra and the three
closest stars in $G$ covered a range from 6050 to 6110 K (in their
“calibrated” ASPCAP values). Gao et al. (2018) derived temperatures from fits
to first and second ionization states of Sc, Ti, and Fe lines, and two H
Balmer lines, and for the stars nearest in $G$ magnitude, they found a range
from about 6090 to 6140 K. (Our best fit M67 star, WOCS 6018, had a
temperature of 6127 K.) Bertelli Motta et al. (2018) used temperatures from
the Gaia-ESO Survey, and observed four stars slightly fainter than WOCS 6018
that returned a range between 6000 and 6060 K. Önehag et al. (2014) derived
temperatures from photometry, but also examined ionization and excitation
temperatures. For five stars slightly fainter than our fit, they found
temperatures between about 6130 and 6200 K, with the closest matches in
brightness being at the high end of that range. The agreement of our IRFM
temperature with these spectroscopic measurements is quite good, and if
anything, our temperature is higher than spectroscopic values by $\sim 50-100$
K.

We can also derive bolometric fluxes at Earth from the model fits to the SEDs.
The SEDs for both stars are very well sampled with photometry, and a GALEX
spectrum for the best bright star proxy can be employed as well. The results
are summarized in Table The K2 M67 Study: Precise Mass for a Turnoff Star in
the Old Open Cluster M67. The SED model fits can be combined to provide a
constraint on the radius ratio for the two stars that is independent of the
distance and reddening of the cluster via

$\frac{R_{2}}{R_{1}}=\sqrt{\frac{F_{bol,2}}{F_{bol,1}}\cdot\left(\frac{T_{1}}{T_{2}}\right)^{4}}$

Because the single grazing eclipse per orbit gives us a constraint on the sum
of the stellar radii, this radius ratio will allow us to compute the
individual stellar radii.

###  2.3 Spectroscopy

We have measured radial velocities for the WOCS 11028 binary using three
spectroscopic datasets. The first and largest set we employ here comes from
the CfA Digital Speedometers (Latham, 1985, 1992) on the 1.5 m Tilinghast
reflector at Fred Lawrence Whipple Observatory and the MMT. These observations
were recordings of a single echelle order covering 516.7 to 521 nm around the
Mg I b triplet using an intensified photon-counting Reticon detector. The
spectra were taken as part of a larger monitoring campaign of M67 stars.

We made two observations of the binary using the HARPS-N spectrograph
(Cosentino et al., 2012) on the 3.6 m Telescopio Nazionale Galileo (TNG).
HARPS-N is a fiber-fed echelle that has a spectral resolving power $R=115000$,
covering wavelengths from 383 to 693 nm. The spectra were processed,
extracted, and calibrated using the Data Reduction Software (version 3.7)
provided with the instrument.

We also obtained five archival spectra from the APOGEE database (Holtzman et
al., 2015) that were taken between January and April 2014. These are $H$-band
infrared ($1.51-1.70\mu$m) spectra with a spectral resolution $R\sim 22500$
taken on the 2.5 m Sloan Foundation Telescope. We used APOGEE flags to mask
out portions of the spectrum that were strongly affected by sky features, and
continuum normalized the spectra using a median filter.

The radial velocities were measured using the spectral separation algorithm
described in González & Levato (2006). In the first iteration step, a master
spectrum for each component is isolated by aligning the observed spectra using
trial radial velocities for that component and then averaging. This
immediately de-emphasizes the lines of the non-aligned stellar component, and
after the first determination of the average spectrum for each star, the
contribution of the non-aligned star is subtracted before the averaging in
order to better clean each spectrum. The radial velocities can also be
remeasured from spectra with one component subtracted. This procedure is
repeated for both components and continued until a convergence criterion is
met. We measure the broadening functions (Rucinski, 2002) to determine the
radial velocities using narrow-lined synthetic spectra as templates. Radial
velocity measurements from broadening functions improve accuracy in cases when
the lines from the two stars are moderately blended (Rucinski, 2002).

The CfA spectra covered a wide range of orbital phases that made it possible
to use a large number of spectra in calculating average spectra for the two
stars. In most cases, spectra were left out due to low signal-to-noise ratio.
We used 17 out of 28 spectra in determining the average spectrum for the
primary, but only used the 13 spectra with the most clearly detected secondary
component to determine its average spectrum. With good average spectra, it was
possible to subtract each component out of observed spectra even when they
were taken close to crossing points (phases $\phi\approx 0.05$ and 0.6). After
some experimentation, we found that synthetic templates (from the grid of
Coelho et al. (2005)) of 5250 K optimized the detection of broadening function
peaks for the secondary star. A template with 6250 K was used for the primary
star. For the CfA velocities only, we used run-by-run corrections derived from
observations of velocity standards. Run-by-run offsets measured for the CfA
spectra ranged from $-1.52$ to $+2.21$ km s-1. We initially assigned velocity
uncertainties derived from the spectral separation analysis, but scaled these
to ensure that the scatter around a best-fit model for this dataset was
consistent with the measurement errors. (In other words, we forced the reduced
$\chi^{2}$ value to 1.) The average uncertainty for the primary star
velocities was 1.25 km s${}^{-1}\;$, and for the secondary star it was 4.41 km
s-1.

The APOGEE spectra were analyzed separately, and detection of the secondary
star features were considerably more secure. The synthetic spectral templates
were taken from ATLAS9/ASS$\epsilon$T models (Zamora et al., 2015) calculated
for the APOGEE project. The uncertainty estimate for each APOGEE velocity was
generated from the rms of the velocities derived from the three wavelength
bands in the APOGEE spectra. This was typically around 0.15 km s${}^{-1}\;$for
the primary star and 0.65 km s${}^{-1}\;$for the secondary star. The velocity
was corrected to the barycentric system using values calculated by the APOGEE
pipeline. For all five spectra, the broadening function peaks for the two
stars were resolvable thanks to the high resolution of the spectra and
fortuitous phases of observation.

Because we only had two observations using the HARPS-N spectrograph and one of
them was taken at a phase very near a crossing point, we determined velocities
using broadening functions alone. To get an internal measure of the velocity
uncertainties, we measured the velocities in four 30 nm subsections of the
spectra and computed the error in the mean.

Because the synthetic spectra used in our broadening function measurements do
not account for gravitational redshifts of the stars, our velocities should
have an offset relative to velocities derived using an observed solar
template. For the CfA spectra, the run-by-run corrections were computed using
dawn and dusk sky exposures, and this defines the native CfA velocity system
used in Geller et al. (2015), incorporating the gravitational redshift of the
Sun. For consistency, we subtracted the combined gravitational redshift of the
Sun and gravitational blueshift due to the Earth (0.62 km s-1) from the APOGEE
and HARPS-N velocities we tabulate. Based on the derived masses and radii for
the stars in the binary, they should have slightly larger gravitational
redshifts than the Sun (0.66 and 0.69 km s-1, versus 0.64 km s${}^{-1}\;$for
the Sun) but we did not correct for these smaller differences. In our later
model fits, we did, however, allow the system velocities for the two stars to
differ to account for effects like this and convective blueshifting that could
produce small systematic shifts. These could affect the measured radial
velocity amplitudes $K_{1}$ and $K_{2}$ (and the stellar masses) if not
modeled.

The measured radial velocities are presented in Table 1, and the phased radial
velocity measurements are plotted in Fig. 9. The model fits are described in
Section 3.4, but the parameters that were used to fit the velocity data were
the velocity semi-amplitude of the primary star $K_{1}$, mass ratio
$q=M_{2}/M_{1}=K_{1}/K_{2}$, eccentricity ($e$), argument of periastron
($\omega$), and systematic radial velocities $\gamma_{1}$ and $\gamma_{2}$. In
the end, we do not see noticeable differences between the systematic
velocities.

Table 1: Radial Velocity Measurements mJDaamJD = BJD - 2400000. | $v_{A}$ | $\sigma_{A}$ | $v_{B}$ | $\sigma_{B}$ | mJDaamJD = BJD - 2400000. | $v_{A}$ | $\sigma_{A}$ | $v_{B}$ | $\sigma_{B}$
---|---|---|---|---|---|---|---|---|---
| (km s-1) | (km s-1) |  | (km s-1) | (km s-1)
CfA Observations | 49678.02714 | 54.71 | 0.59 | 6.53 | 3.23
48347.73537 | $-13.31$ | 3.84 | 98.95 | 6.47 | 49700.00054 | 38.07 | 0.89 | 29.41 | 2.87
48618.00088 | 53.87 | 0.59 | 0.83 | 3.95 | 49705.98524 | 31.68 | 1.77 |  |
48635.87968 | 38.55 | 1.48 | 24.19 | 3.95 | 50535.69747 | $-1.66$ | 1.18 | 76.62 | 1.44
48676.79118 | 56.54 | 0.59 | $-4.62$ | 5.03 | 50536.76047 | $-3.37$ | 1.77 | 84.15 | 2.87
48701.71198 | 32.68 | 0.89 | 36.16 | 4.67 | 50885.78158 | 41.13 | 0.89 | 27.03 | 4.67
48704.66488 | 32.50 | 0.59 | 34.70 | 1.80 | 50913.75637 | $-10.46$ | 0.89 | 88.80 | 3.59
48726.74348 | $-13.03$ | 3.25 | 84.47 | 7.90 | 50916.72187 | $-18.69$ | 0.89 | 97.47 | 2.16
48754.67588 | 44.44 | 1.77 | 14.28 | 11.86 | 50918.72237 | 4.69 | 1.48 | 68.75 | 3.59
48942.01949 | 44.13 | 0.30 | 25.07 | 5.39 | 50920.71047 | 37.55 | 0.59 | 25.23 | 6.11
48966.93949 | 14.32 | 1.48 | 58.96 | 9.70 | APOGEE Observations
48988.95240 | 57.20 | 0.89 | $-0.02$ | 1.08 | 56672.86368 | $-11.57$ | 0.06 | 91.22 | 0.09
49019.89560 | 29.34 | 1.18 | 37.60 | 3.95 | 56677.88083 | 13.45 | 0.15 | 57.87 | 0.64
49049.81820 | 56.23 | 1.48 | 3.07 | 5.03 | 56700.76762 | 46.59 | 0.15 | 12.91 | 0.17
49057.71970 | 52.50 | 0.89 | 6.60 | 2.16 | 56734.66627 | $-7.61$ | 0.13 | 85.78 | 1.07
49077.68900 | 34.51 | 0.59 | 29.15 | 1.44 | 56762.63514 | 46.73 | 0.35 | 13.13 | 0.64
49111.72870 | 59.12 | 2.36 | $-2.48$ | 4.67 | HARPS-N Observations
49328.01792 | 34.75 | 0.30 | 22.47 | 3.23 | 57752.76085 | 55.83 | 0.08 | 1.04 | 0.30
49347.86222 | $-5.88$ | 0.89 | 88.10 | 8.62 | 57780.74729 | 31.82 | 0.03 |  |

It is always prudent to check for systematic differences in velocity when
utilizing datasets from more than one observational set-up. Among the measured
primary star velocities, mean residuals (observed minus computed) were $-0.14$
km s${}^{-1}\;$for APOGEE, $+0.02$ km s${}^{-1}\;$for CfA, and $0.02$ km
s${}^{-1}\;$for HARPS-N. For the secondary star, we found mean residuals of
$+0.78$ km s${}^{-1}\;$for CfA, $+0.10$ km s${}^{-1}\;$for APOGEE, and $+0.09$
km s${}^{-1}\;$for HARPS-N (one measurement). Of these, only the average
residual for the CfA secondary velocities was more than one standard devation
(0.20 km s-1) away from zero. Because the APOGEE and HARPS-N spectra have
higher signal-to-noise than the CfA spectra, they are the main contributors to
measurement of the radial velocity amplitude of the secondary star, which
critically affects the calculated mass of the primary star.

Figure 9: Phased radial velocities (minus system velocity) for WOCS 11028,
along with the best-fit model. Measurements for the primary and secondary
stars are shown with stars and circles, respectively. The lower panels show
the observed minus computed values with error bars scaled to give a reduced
$\chi^{2}=1$ (see §3.4).

##  3 Analysis

###  3.1 Cluster Membership

WOCS 11028 is a fairly large distance (about $13\farcm 7$) from the center of
M67, but Carrera et al. (2019) find that half of the cluster turnoff stars
fall within $12\farcm 5$ of the cluster center. As a result, its position is
not strong evidence of being a field star.

All ground-based proper motion studies (Sanders, 1977; Girard et al., 1989;
Zhao et al., 1993; Yadav et al., 2008; Nardiello et al., 2016a) indicate a
high probability ($\geq 90$%) of cluster membership for WOCS 11028, with the
exception of Krone-Martins et al. (2010), who give 10%). Gaia observations
from Data Release 2 (Gaia Collaboration et al., 2018a) have produced much more
precise proper motions recently, and the binary still has a proper motion
vector ($\mu_{\alpha}=-11.08\pm 0.04$ mas yr-1, $\mu_{\delta}=-3.14\pm 0.04$
mas yr-1) safely residing among other likely members (centered at
$\mu_{\alpha}=-10.97$ mas yr-1, $\mu_{\delta}=-2.94$ mas yr-1; Gaia
Collaboration et al. 2018b). Similarly, the Gaia parallax ($\omega=1.111\pm
0.024$ mas) is within $1\sigma$ of the cluster mean ($\bar{\omega}=1.132$
mas).

Geller et al. (2015) published membership probabilities based on their radial
velocity survey by comparing the velocity distributions of cluster members and
field stars. They classified WOCS 11028 as a binary member (98% probability)
based on their system velocity ($32.91\pm 0.17$ km s-1) measured from the CfA
observations and carefully placed on the zeropoint of other stars in the
field. The best-fit system velocity from our measurements ($\gamma=32.37$ km
s-1) is slightly off from the mean cluster velocity of Geller et al. (33.64 km
s${}^{-1}\;$with a radial velocity dispersion of $0.59^{+0.07}_{-0.06}$ km
s-1). However, even with our lower $\gamma$ velocity, the radial velocity
membership probability is still 95%.

Based on the three-dimensional kinematic evidence, the binary is a high
probability cluster member.

###  3.2 Distance Modulus

With the release of Gaia DR2 parallaxes, the M67 distance modulus should be
revisited, as it will play a large role in comparisons with models later in
the article. The mean parallax derived for cluster members from Gaia DR2
measurements ($\bar{\omega}=1.1325\pm 0.0011$ mas; Gaia Collaboration et al.
2018b) is very precise, but possibly affected by a zeropoint uncertainty.
Lindegren et al. (2018) finds an offset of about $-30\>\mu$as (with tabulated
parallaxes being smaller than reality), while systematic variations with
position, magnitude, and color were below $10\>\mu$as. At the other extreme,
Stassun & Torres (2018) found an offset of $-82\pm 33\>\mu$as using a sample
of eclipsing binaries. Zinn et al. (2019) find offsets of $-52.8$ and
$-50.2\>\mu$as using comparisons with asteroseismic giants and red clump stars
in the Kepler field. Schönrich et al. (2019) derived an offset of $-54\pm
6\>\mu$as from an analysis of all stars (more than 7 million) in the radial
velocity sample of Gaia DR2.

These indications are consistent with the situation for M67. Without
correction, the Gaia cluster mean distance modulus would be
$(m-M)_{0}=9.730\pm 0.002$, but this is comparable to what has been derived
previously for the distance modulus with extinction. For example, Pasquini et
al. (2008) derive $(m-M)_{V}=9.76\pm 0.06\pm 0.05$ (statistical and systematic
uncertainties) from ten solar analogs, Sandquist (2004) derived $9.72\pm 0.05$
using metal-rich field dwarfs with Hipparcos parallaxes, and Stello et al.
(2016) found $9.70\pm 0.04$ from K2 asteroseismology of more than 30 cluster
red giants. With a $V$-band extinction $A_{V}\approx 0.12$ expected, this is a
significant discrepancy. For subsequent calculations in this article, we will
assume a $-54\>\mu$as offset in the Gaia parallaxes and include a systematic
uncertainty of $30\>\mu$as to conservatively account for disagreement in the
parallax offsets in the literature. The resulting distance modulus is
$(m-M)_{0}=9.63\pm 0.06$.

###  3.3 Li Abundance and Stellar Rotation

Even though the WOCS 11028 binary has a relatively large orbital separation,
our analysis of another wide binary in M67 (WOCS 12009; Sandquist et al.,
2018) indicated that the brighter star was likely the product of an earlier
merger. A key piece of evidence for that binary was the lack of detectable Li.
Single main sequence stars of similar brightness in the cluster inhabit the Li
plateau, a broad maximum in the Li abundance. The lack of detectable Li was an
indication of nuclear processing, consistent with surface material having
originally been in a lower-mass main sequence star with a deeper surface
convection zone.

To have greater assurance that the stars of WOCS 11028 have evolved as
isolated single stars and can be used to constrain the cluster age, we can
also use the Li abundance here. If the brighter star in WOCS 11028 was
unmodified since the cluster’s formation, its mass ($\sim 1.2M_{\odot}$) would
place it toward the bright end of the cluster’s Li plateau with abundances
$A(\mbox{Li})\approx 2.5$ (Pace et al., 2012). On the other hand, the
secondary star (mass near $0.9M_{\odot}$) should not have detectable Li. Stars
in the plateau have outer convective zones that do not transport Li nuclei
down to temperatures where nuclear reactions can burn them, and additional
mixing processes must have little, if any effect. Even with the diluting
effects of the secondary star’s light on the Li lines, Li should be detectable
if it is present with the plateau abundance.

We are able to detect Li in our HARPS-N spectra as shown in Fig. 10. If the
star had a history of interaction (mass transfer, or a merger involving lower
mass stars), it is unlikely that the Li would be detectable. If the brighter
star formed in a merger, the more massive merging star could not have been
more than about $0.2M_{\odot}$ less massive than the current star or else a
surface convection zone would have had a chance to consume its Li during its
main sequence evolution. In addition, the less massive merging star would take
up residence in the core of the merger remnant because of its lower entropy,
rebooting its nuclear evolution with gas having higher hydrogen abundance.
After some adjustment on a thermal timescale, it would have been bluer than
other cluster stars of the same mass. A precise determination of the Li
abundance is complicated by the secondary star’s light, and is beyond the
scope of this work. However, the strong Li detection coupled with strong
indications that WOCS 11028 A has a color consistent with its fellow cluster
members effectively rules out anything except a very early merger and/or a
merger with a very low mass object. Effectively, this would make today’s star
the same as an unmodified star of the same mass.

Figure 10: A HARPS-N spectrum for WOCS 11028 in the vicinity of the Li I
resonance doublet, shifted to account for the primary star velocity.

We do not see evidence of rotational spot modulation in the K2 light curves
due to spots, but our HARPS-N spectra allow us to constrain the rotational
speeds of the stars in the binary via the widths of the broadening function
peaks. We are unable to detect rotation with an upper limit $v_{\rm rot}\sin
i<5$ km s-1. If the primary was pseudo-synchronized, its rotation velocity
would be around this limit (rotation period near 14 d). However, the timescale
calculated for this (Hut, 1981) is far longer than the cluster age, thanks to
the substantial separation even at periastron. Considered as a single star,
the primary has probably spun down somewhat, although not to the same degree
as lower mass stars with deeper surface convection zones. Angus et al. (2015)
calibrated their gyrochronological models against the rotation of
asteroseismic targets in the Kepler field, and their sample probably gives the
best indication of typical rotation rates for old stars more massive than the
Sun. Their isochrones predict a rotation period near 16 day, and this would
imply a rotation velocity below our ability to detect it spectroscopically. As
an old cluster, M67 is expected to have slowly rotating stars, but this does
put a limit on how long ago mass transfer or a merger could have happened if
either star was somehow modified. A rapidly rotating star with the mass of our
primary star would require approximately 3 Gyr to reach our detection limit
according to the rotational isochrones of Angus et al. (2015), and this is the
majority of the cluster’s age.

###  3.4 Binary Star Modeling

We used the ELC code (Orosz & Hauschildt, 2000) to simultaneously model the
ground-based radial velocities and Kepler K2 photometry for the WOCS 11028
binary. The code is able to use a variety of optimizers to search the complex
multi-dimensional space of parameters used in the binary star model. In our
particular case, we used a set of 15 parameters. Two of the parameters were
the orbital period $P$, and the reference time of periastron $t_{P}$. Six
additional parameters mostly characterize the orbit: the velocity semi-
amplitude of the primary star $K_{1}$, mass ratio $q=M_{2}/M_{1}=K_{1}/K_{2}$,
systematic radial velocities444We allow for the possibility of differences for
the two stars that could result from differences in convective blueshifts or
gravitational redshifts. $\gamma_{1}$ and $\gamma_{2}$, eccentricity ($e$),
and argument of periastron ($\omega$). For a typical double-lined
spectroscopic and eclipsing binary, both the radial velocities and eclipse
light curves constrain $e$ and $\omega$, with the phase spacing of the eclipse
playing a large role. But because this binary only has one eclipse per cycle,
these parameters are constrained by the radial velocities almost exclusively.

The light curve has a large role in constraining the inclination parameter
$i$, but there are potentially correlations between inclination and choices
for radius and temperature parameters. For radii, we used the sum of the
stellar radii $R_{1}+R_{2}$ and their ratio $R_{1}/R_{2}$ as parameters. For
this binary, the sum of radii is constrained by the measurements of the single
grazing eclipse in each cycle along with the well-determined spectroscopic
parameters of the orbit. The radius ratio can be constrained by luminosity
ratios we have derived from our SED analysis (§2.2) in concert with
temperature information. For a typical eclipsing binary, the temperature ratio
would be constrained by the relative depths of the eclipses, but that is not
measurable here. We have separate constraints on the temperatures of the stars
from SED fits to the proxy stars, and so we use these as fit parameters:
temperatures of the primary $T_{1}$ and secondary $T_{2}$.

Finally, there is some dependence of the fits on the limb darkening, although
relatively little because only the near-limb of the secondary star is probed
by the grazing eclipses we see. So we fit for two quadratic limb darkening law
coefficients ($q_{1B},q_{2B}$) for the secondary star (the only star that is
eclipsed) in the Kepler bandpass. Our runs of the ELC code employed the
Kipping (2013) algorithm, which involves a search over a triangular area of
parameter space of physically realistic values. This forces the model star to
a) darken toward the limb, and b) have a concave-down darkening curve. In the
final analysis, we find that the limb darkening coefficients are virtually
unconstrained, but by allowing the code to systematically explore potential
values, the uncertainties will be incorporated in our uncertainties of the
other stellar parameters. Owing to the substantial orbital separation, we
assumed that the stars are spherical. Given this, we used the algorithm given
in Short et al. (2018) as implemented in ELC to rapidly compute the eclipses.
Finally, because there is little or no out-of-eclipse light modulation, we did
not see a need to model spots.

For the K2 light curve modeling, we only included observations that were
within about $\pm 0.01$ in phase of the eclipse, and a similar section around
where the other eclipse would have been if the binary was closer to edge-on.
In this way, our models “see” that there is only one eclipse per orbital
cycle. In doing this, we leave out the majority of observations, which may be
affected by systematics due to spacecraft pointing corrections and other
instrumental signals. Because the K2 light curves use long cadence data with
30 minute exposures, they also effectively measure the average flux during
that time. To account for this, the ELC code integrates the computed light
curves in each observed exposure window.

The quality of the model fit was quantified by an overall $\chi^{2}$ derived
from comparing the radial velocity and light curve data to the models, as well
as from how well a priori constraints were matched. For this binary, we used
stellar temperatures derived from SED fits and luminosity ratios in
$BVI_{C}JHK_{s}$ derived from the modeling of the binary SED with single
cluster stars. To illustrate the effect of these constraints, we conducted a
run without the luminosity ratios, as shown in Table 2.

To try to ensure that different datasets were given appropriate weights in the
models, we empirically scaled uncertainties on different datasets to produce a
reduced $\chi^{2}$ of 1 relative to a best-fit model for that particular
dataset. For spectroscopic velocity measurements, this meant scaling
differently according to the source spectrograph and the star (primary and
secondary). The K2 photometric light curve uncertainties were scaled
independently.

We used a differential evolution Markov Chain optimizer (Ter Braak, 2006) for
seeking the overall best-fit model and exploring parameter space around that
model to generate a posterior probability sampling. Approximate $1\sigma$
parameter uncertainties were derived from the parts of the posterior
distributions containing 68.2% of the remaining models from the Markov chains.
Gaussians were good approximations to the posterior distributions for all
parameters except limb darkening coefficients in the run where luminosity
constraints were applied. The results of the parameter fits are provided in
Table 2, and the posterior distributions are shown in Figure 11. A comparison
of the K2 light curve with the best-fit model is shown in Figure 12, and a
comparison of the radial velocity measurements with models are shown in Figure
9.

Table 2: Best-Fit Model Parameters for WOCS 11028 Parameter | No SED Constraint | SED Luminosity Constraints
---|---|---
Pipeline | K2SFF | K2SFF | EVEREST
Constraints: |  |  |
$T_{1}$ (K) | $6200\pm 100$ | $6200\pm 100$ | $6200\pm 100$
$T_{2}$ (K) | $5450\pm 100$ | $5450\pm 100$ | $5450\pm 100$
$L_{2}/L_{1}(B)$ |  | $0.186\pm 0.030$ | $0.186\pm 0.030$
$L_{2}/L_{1}(V)$ |  | $0.227\pm 0.030$ | $0.227\pm 0.030$
$L_{2}/L_{1}(I_{C})$ |  | $0.267\pm 0.030$ | $0.267\pm 0.030$
$L_{2}/L_{1}(J)$ |  | $0.306\pm 0.030$ | $0.306\pm 0.030$
$L_{2}/L_{1}(H)$ |  | $0.343\pm 0.030$ | $0.343\pm 0.030$
$L_{2}/L_{1}(K)$ |  | $0.349\pm 0.030$ | $0.349\pm 0.030$
$\gamma$ (km s-1) | $32.38\pm 0.02$ | $32.37\pm 0.02$ | $32.37\pm 0.02$
$P$ (d) | 62.59484 | 62.59484 | 62.59482
$\sigma_{P}$ (d) | 0.00003 | 0.00003 | 0.00002
$t_{P}$ | 49416.060 | 49416.058 | 49416.061
$\sigma_{t_{P}}$ | 0.013 | 0.012 | 0.012
$i$ ($\arcdeg$) | $86.70\pm 0.15$ | $86.84\pm 0.07$ | $86.86\pm 0.07$
$q$ | $0.743\pm 0.003$ | $0.743\pm 0.002$ | $0.743\pm 0.003$
$e$ | $0.6227\pm 0.0019$ | $0.6225\pm 0.0017$ | $0.6222\pm 0.0017$
$\omega$ ($\arcdeg$) | $235.82\pm 0.18$ | $235.70\pm 0.18$ | $235.78\pm 0.18$
$K_{1}$ (km s-1) | $37.54\pm 0.14$ | $37.54\pm 0.13$ | $37.55\pm 0.13$
$K_{2}$ (km s-1) | $50.49\pm 0.18$ | $50.51\pm 0.18$ | $50.48\pm 0.17$
$R_{1}/R_{2}$ |  | $1.58\pm 0.05$ | $1.57\pm 0.05$
$(R_{1}+R_{2})/R_{\odot}$ | $2.37\pm 0.07$ | $2.34\pm 0.03$ | $2.34\pm 0.03$
$M_{1}/M_{\odot}$ | $1.225\pm 0.006$ | $1.222\pm 0.006$ | $1.222\pm 0.006$
$M_{2}/M_{\odot}$ | $0.911\pm 0.004$ | $0.909\pm 0.004$ | $0.909\pm 0.004$
$R_{1}/R_{\odot}$ | $1.44^{+0.08}_{-0.10}$ | $1.43\pm 0.03$ | $1.43\pm 0.03$
$R_{2}/R_{\odot}$ | $0.86^{+0.15}_{-0.11}$ | $0.904\pm 0.015$ | $0.908\pm 0.015$
$\log g_{1}$ (cgs) | $4.21^{+0.07}_{-0.05}$ | $4.209\pm 0.019$ | $4.217\pm 0.019$
$\log g_{2}$ (cgs) | $4.51^{+0.10}_{-0.12}$ | $4.485\pm 0.014$ | $4.481\pm 0.014$
Figure 11: Posterior distributions (normalized to the peak bin) for the fitted
and calculated parameters for the EVEREST K2 data with SED luminosity
constraints.  Figure 12: K2 eclipse photometry (K2SFF pipeline in magenta, and
EVEREST pipeline in blue) compared with the best-fit binary models (for the
EVEREST pipeline). Orbital phases near the non-eclipsing conjunction are shown
in the right panel.

As can be expected, when we do not enforce luminosity ratio constraints from
SED fitting, there is no information on the ratio of radii in the data, and as
a result, the radii of the stars are very poorly constrained. The mass
determinations are essentially unaffected, however, because that information
is contained in the radial velocity curves and the orbital inclination
(derived from the light curve).

##  4 Discussion

M67 has long been studied because of similarities in chemical composition and
age with the Sun, but stars at the cluster turnoff are approximately 20-30%
more massive than the Sun, with substantial differences in internal physics.
Stars with masses of $1.1-1.3M_{\odot}$ may generate convective cores late in
their main sequence evolution, and this plays a critical role in their
evolution.

In the discussion below, we will leverage new information from two sources —
precise measurements of the stars in the WOCS 11028 binary, and additional
characterization of stars around the turnoff of the cluster — to identify
conflicts with models, and to try to understand the physics that might be
causing those conflicts.

For clarity in the discussion, we define the upper turnoff to be at the global
minimum in color at the blue end of the subgiant branch ($G\approx 12.74$);
the turnoff gap slightly fainter than this ($12.75\lesssim G\lesssim 12.95$);
and the lower turnoff to be at the fainter local color minimum ($G\approx
13.45$). Because WOCS 11028 A is nearly exactly at the lower turnoff based on
our SED decomposition (see Figure 13), it provides an excellent mass
calibration point for the cluster.

###  4.1 The Cluster Composition

As is usually the case, the chemical composition is an important component of
interpreting the measurements and understanding the implications for the
cluster age and for the stellar physics in the models we use. We discussed
high-precision cluster [Fe/H] measurements relative to the Sun in Sandquist et
al. (2018) in order to minimize systematic errors due to composition. Since
that paper though, it has become clearer that diffusion is playing a role in
the measured surface compositions of M67 stars. Significant trends in
abundance are seen as a function of the evolutionary state of the stars, with
subgiants having higher measured abundances than turnoff stars (Önehag et al.,
2014; Bertelli Motta et al., 2018; Liu et al., 2019; Souto et al., 2019).
Because subgiants should have deep enough convection zones to homogenize
surface layers that were affected by heavy element settling, the subgiant
abundances should be more representative of the initial bulk heavy element
composition of the stars. The spectroscopic studies generally come to the
conclusion that initial composition of the cluster members was [Fe/H]$=+0.05$
to $+0.10$. In the model comparisons below, we will consider initial metal
compositions between solar and [Fe/H]$=+0.10$.

Figure 13: Gaia color-magnitude diagram for the turnoff and main sequence of
M67. Small points are probable single-star members and yellow circles are
solar analogs identified by Pasquini et al. (2008). The system photometry for
WOCS 11028 is shown with a red circle, and deconvolved photometry of the
component stars are green circles. Theoretical isochrones have been shifted
according to a corrected Gaia parallax [$(m-M)_{0}=9.63$], and reddening
$E(B-V)=0.041$ (Taylor, 2007). PARSEC and MIST isochrones have ages 1.0
(dashed line), 3.2, 3.6, and 4.0 Gyr (solid blue, green, and red lines).
BaSTI-IAC plots show an isochrone with no overshooting or diffusion (black
dashed line), and with overshooting (solid red line) for 4.2 Gyr (top, right)
and 4.0 Gyr age (bottom, right). The theoretical predictions for stars with
masses equal to the primary star of WOCS 11028, the Sun, and the secondary
star of WOCS 11028 are shown with horizontal lines.

###  4.2 The Characteristics and Evolution Status of WOCS 11028 A and B

The verticality of the cluster CMD at the position of WOCS 11028 A constrains
the present direction of its evolution track, and we illustrate the discussion
with MESA model tracks (version r11701; Paxton et al. 2011, 2013, 2015, 2018,
2019) in the largest panel of Figure 14. The models predict that for the first
2 Gyr of the star’s life, it was evolving mostly upward in luminosity, with
slowly increasing temperature before hitting a maximum and starting to
decrease. The star must have started evolving more rapidly toward cooler
temperature in order to change the local slope of the main sequence isochrone
from increasing temperature to constant temperature with increasing mass.
Comparing the markers showing common age for three stars of slightly different
mass, the triplet becomes vertical near the midpoints of the tracks during
this coolward movement. For greater age, the triplets have decreasing
temperature with higher mass, and in M67 this corresponds to stars between the
lower turnoff and the faint end of the gap.

Figure 14: Evolutionary tracks for WOCS 11028 A in the HR diagram using the
MESA code. The red point shows the properties of WOCS 11028 A. In most plots,
the dotted line track is the baseline model shown in the upper left panel
($M=1.222M_{\odot}$, $Z=0.0162$, Asplund et al. 2009 composition mix), and the
colored tracks illustrate changes to the model physics. Blue and red tracks in
the upper right panel illustrate $1\sigma$ differences in mass from the best
fit value. Black points show ages from 2.5 to 4 Gyr in intervals of 0.25 Gyr.

The precise point where the vertical slope is produced will depend on details
of the fuel consumption in this phase, but we can be assured that the star
must still be evolving faster toward lower temperature than slightly lower
mass stars (in other words, accelerating) or else it would not be able to
match the temperature of those stars that should be evolving at a very similar
speed. This in turn assures us that convection must be present in the core of
the star — low-mass stars that do not establish convective cores on the main
sequence only accelerate toward lower temperature when they are past central
hydrogen exhaustion while on the subgant branch. So the mass measurement for
WOCS 11028 A gives us an observational lower limit for stars that establish
convective cores on the main sequence.

In the evolution tracks, the turn toward lower temperature occurs when the
extent of the convective core increases in the latter half of core hydrogen
burning. The CNO cycle overtakes the $pp$ chain as the dominant energy
generating reaction network while central temperature increases and hydrogen
abundance declines. So not only do we have good measurements of the WOCS 11028
A star (described below), but we have an indication of physical conditions
occuring within the star. A complete model of the star would match all of the
measurements and evolutionary constraints with a unique set of parameters,
while mismatches would indicate systematic errors.

####  4.2.1 Mass and Radius

Although the radii we derive from our analysis of the binary are not as
precise as they could have been if the system had been doubly eclipsing, they
can still provide some indication of the age independent of other stars in the
cluster if the stars have not undergone mass transfer or mergers in their
history555The more massive star in the binary WOCS 12009, analyzed in
Sandquist et al. (2018) and mentioned in section 3.3, had a much smaller
radius than expected for reasonable cluster ages, providing support for the
idea of a stellar merger in that case. The radius of the less massive star may
not have been involved in the merger, but its radius is also unlikely to be
significantly different even if it was.. In addition, these comparisons are
independent of uncertainties in quantities like distance and reddening.

The comparison with several sets of solar-metallicity isochrone models is
shown in Figure 15. Taken at face value, WOCS 11028 A is smaller than
predicted for typically quoted M67 ages ($3.5-4$ Gyr) by $2\sigma$ or more,
and WOCS 11028 B is larger than reasonable models by a similar amount. Note
that if we are systematically underestimating the luminosity ratio used in our
binary star modeling (as one might wonder from the discussion of the SED fits
in §2.2), this would exacerbate the conflict with models by decreasing the
primary star radius and increasing the secondary star radius.

Figure 15: Mass-radius plot for measured members of M67 (WOCS 11028 A and B,
and WOCS 12009 B; red), with $2\sigma$ uncertainties indicated by the error
ellipses. The probable straggler WOCS 12009 A is marked with a red dashed
ellipse, and measured eclipsing stars in the younger cluster NGC 6819 are blue
ellipses. Models use $Z=0.0152$, 0.0188, 0.0142, and 0.0162, respectively for
PARSEC (Bressan et al., 2012), Victoria-Regina (VandenBerg et al., 2006), MIST
(Dotter, 2016; Choi et al., 2016), and YaPSI (Spada et al., 2017) isochrones.
The solid black lines connect the lower turnoff points on the different
isochrones.

When compared with stars from well-measured binaries (Southworth 2015; see
Figure 16), the secondary stars of the two measured M67 binaries (WOCS 11028 B
and WOCS 12009 B) do not look out of the ordinary. PARSEC models generally
underpredict radii for stars with $0.6\leq M/M_{\odot}\leq 0.95$ like WOCS
11028 B, and this is not sensitive to the assumed age or metal content $Z$.
The disagreement with models leads to some justifiable concerns whether a
model age can be trusted if the characteristics of the relatively unevolved
stars cannot be predicted accurately.

Among the stars in the Southworth (2015) sample are several from the younger
but similarly metal-rich cluster NGC 6819 ($\sim 2.2$ Gyr; Brewer et al.,
2016). One of the stars is WOCS 40007 A, which has a mass ($1.218\pm
0.008M_{\odot}$) that is nearly identical to that of WOCS 11028 A. As another
indicator of M67’s relative age, WOCS 11028 A is at M67’s turnoff, while WOCS
40007 A is approximately 0.85 mag fainter than NGC 6819’s turnoff in $V$. WOCS
40007 A has a radius ($1.367\pm 0.003R_{\odot}$) that is smaller than WOCS
11028 A, although WOCS 11028 A’s radius is measured with about 10 times lower
precision. So while WOCS 11028 A appears to be somewhat small relative to the
model expectations for M67’s age, it does appear to be larger and older (by
about 0.4 Gyr) than the corresponding stars of NGC 6819.

Figure 16: Mass-radius plot for measured members of M67 (WOCS 11028, red; WOCS
12009 black) with $2\sigma$ uncertainties indicated by the error ellipses.
Precisely measured eclipsing binary stars (DEBCat, retrieved 2019; Southworth,
2015) are shown with $2\sigma$ error bars (primary stars blue, secondary stars
red). PARSEC (Bressan et al., 2012) isochrones are shown for $Z=0.0152$.

An absolute determination of the age requires comparisons to models though,
and the characteristics of WOCS 11028 A return a value of $3.0\pm 0.3$ Gyr for
isochrones having $Z\approx 0.015$. Radius is fairly insensitive to one of the
more important model uncertainties — bulk metal content $Z$ — and so should be
given extra attention in considering the age. From experiments with MESA
models, we find that $R\propto Z^{-1/4}$ for stars near WOCS 11028 A’s mass,
whereas $L\propto Z^{-1}$. By the same token, helium abundance $Y$ generally
affects radius in the opposite direction, and if one assumes, as most
isochrone sets do, that galactic chemical evolution increases $Y$ and $Z$
together (in other words, $\Delta Y/\Delta Z$ is a constant), radius appears
even less sensitive to composition changes. We will return to this point in
Section 4.2.3.

We should also consider the evolutionary status of the brighter star WOCS
11028 A though. If models agree with our mass-radius measurements at a
particular age but have the star at the wrong evolutionary state, it would be
an indication of a systematic error in model physics. In Figure 15, we have
connected isochrone points corresponding to the lower turnoff at each age. We
find that, for all of the model sets we could check, the mass-radius
combination for WOCS 11028 A is consistent with model predictions for that
evolutionary stage.

####  4.2.2 The Color-Magnitude Diagram (CMD)

To date, the primary method for measuring the age of M67 has been isochrone
fits to the CMD. Examples of results include 3.5 Gyr to 4.0 Gyr (Sarajedini et
al., 2009) and 3.6 to 4.6 Gyr (VandenBerg & Stetson, 2004). A large portion of
these age ranges result from model physics and chemical composition
uncertainties that impact this cluster, and new information may help reduce
the age uncertainty by reducing these modeling uncertainties. For example,
Stello et al. (2016) used an asteroseismic analysis of the masses of pulsating
giants in M67 to derive an age of about 3.5 Gyr, although that age still has
uncertainties due to model physics.

The modeling of the eclipsing binary star gives us very precise masses for the
component stars, and our SED analysis places restrictions on the CMD positions
of the stars. These are the most precise data that can be extracted from the
binary at present. When utilized with precise photometry for other cluster
members, isochrone fitting can provide us with an estimate of the age. It is
important to remember that masses are generally unavailable for traditional
CMD isochrone fitting, and by employing them here, we can start to address
systematic uncertainties on the derived age.

The Gaia photometric dataset is one of the most precise available, so in
Figure 13 we compare the M67 photometry with theoretical isochrones. The
isochrones were shifted based on the mean Gaia parallax for the cluster
corrected for the Schönrich et al. (2019) zeropoint offset [for a distance
modulus $(m-M)_{0}=9.63$), along with the Taylor (2007) reddening for the
cluster [giving extinction $A_{G}=0.111$ and $E(G_{BP}-G_{RP})=0.060$]. The
comparison indicates that the position of the faint component of WOCS 11028 is
consistent with the theoretical predictions for its mass, especially
considering an $1\sigma$ uncertainty in $G_{B}$ of about $\pm 0.075$ mag (see
section 2.2). There is significant disagreement between the position of WOCS
11028 A and the model predictions for its mass, however. The theoretical
predictions are at least about 0.17 mag too bright for the youngest models
under consideration, and closer to the brightness level of the binary. It
should be remembered as a practical matter that the magnitude of the brighter
star is much more certain than the fainter star because of the need to match
the brightness constraint of the binary, assuming that both stars are
radiating like normal single star cluster members. Our earlier analysis puts
the $1\sigma$ fit uncertainty at around $\pm 0.018$ mag in $G$, so that an
explanation beyond statistical uncertainty seems necessary.

The fact that model predictions fall near or brighter than the combined
brightness of the binary reduces the range of possible explanations: either
our assumption that the binary’s individual stars are emitting like unmodified
single main sequence stars is wrong (and the binary’s light should not be
modeled using stars on the main sequence), or there is a significant
systematic error with model predictions. We have not seen any evidence to
support the idea that the primary star is different from similar main sequence
stars: there is detectable Li and the rotational velocity is low and
consistent with a normally-evolving single star (section 3.3), and the
ultraviolet emission from the binary can be reproduced very well by the
emission of other cluster main sequence stars (section 2.2.1 and Figure 6). If
the primary star had been modified earlier in its history, it should appear
fainter and bluer than single main sequence stars that evolved uneventfully
with the cluster.

We can also compare with stars from field eclipsing binaries having well-
measured luminosity ratios in common filter bands. For this purpose, we
examined stars from the DEBCat database (Southworth, 2015) with stellar masses
within about $0.05M_{\odot}$ of that of WOCS 11028 A, and kept those stars
with precise Gaia parallaxes. The stars in our comparison are given in Table
5, and a comparison of the absolute magnitudes is shown in Figure 17. WOCS
11028 A is generally at the bright end of the range covered by stars with the
most similar masses, which is to be expected given the age of the cluster it
resides in. Even so, its magnitudes are quite comparable to several other
stars: UX Men A and WZ Oph A and B agree extremely well with WOCS 11028 A in
Strömgren filters, and WOCS 40007 A from the cluster NGC 6819 agrees quite
well in $BVI_{C}$ magnitudes if we make use of a distance modulus derived from
the eclipsing binaries in the cluster (as it is too distant for a good Gaia
parallax). We conclude that WOCS 11028 A is not unusually faint for its mass,
and this cannot explain the disagreement with model isochrones.

Figure 17: Absolute magnitudes for main sequence stars with masses near that
of WOCS 11028 A (green points). Blue points are primary stars of detached
eclipsing binaries, and red points are secondary stars.

####  4.2.3 Temperature and Luminosity

With the variety of photometric measurements made for M67 stars, the SEDs
provide us with excellent leverage on the temperatures of the binary star
components as well as other cluster stars for comparisons. We find that both
components of the binary agree well with stars from the DEBCat database, and
the temperature of the primary star matches well with spectroscopic
temperatures (see §2.2.1). While WOCS 11028 B matches the model temperature
for its mass (see Figure 18), WOCS 11028 A is generally around 100 K cooler
than the model that has its lower turnoff at the appropriate mass.

Figure 18: Mass-temperature plot for measured members of M67 (red), with
$2\sigma$ uncertainties indicated by the error ellipses. Models use
$Z=0.0152$, 0.0188, 0.0142, and 0.0162, respectively for PARSEC (Bressan et
al., 2012), Victoria-Regina (VandenBerg et al., 2006), MIST (Dotter, 2016;
Choi et al., 2016), and YaPSI (Spada et al., 2017) isochrones. The solid black
lines connect the lower turnoff points on the different isochrones.

We can compute the luminosities for the stars in two different ways. The first
uses the bolometric flux derived from SED fitting along with the Gaia-based
distance and the inverse square law. The bolometric flux is computed assuming
that any flux that is not covered by photometric measurements can be accounted
for using an appropriate synthetic spectrum. The luminosities are found to be
$2.43\pm 0.08$ and $0.61\pm 0.02L_{\odot}$ for the proxies of the two stars in
the binary. The quoted uncertainties include contributions from the parallax
and the bolometric flux fit. Alternately, we can compute luminosities from the
radii derived in the binary star analysis and the temperatures derived from
the SED fits. Here the results are $2.72\pm 0.22$ and $0.68\pm
0.056L_{\odot}$, respectively. These estimates are marginally consistent with
the values computed from the bolometric fluxes, but are less precise.

As seen in Figure 19, WOCS 11028 B is in general agreement with model
luminosity predictions at solar metallicity, but WOCS 11028 A is about 25%
lower than model values. This is a stronger statement than it might seem at
first because we have the additional constraint on the evolutionary state of
WOCS 11028 A. While it is possible to find a match with the models in
luminosity and mass, this occurs at young ages when a star of WOCS 11028 A’s
mass is not precisely at the lower temperature maximum at the cluster turnoff
in the cluster CMD (see §4.2). The discrepancy is potentially a serious issue
for age determination because it implies a significant systematic error. Below
we discuss factors that impact the luminosity disagreement, and the difficulty
of finding a clear explanation.

For comparison, we show exploratory MESA evolution models in Figure 14. For
these experiments, the baseline inputs were a solar-calibrated Asplund et al.
2009 composition ($Y=0.2703$, $Z=0.0162$) and mixing length parameter
($\alpha=1.80$), nuclear reaction rates from the JINA compilation, diffusion,
and disabled core convective overshooting. Before proceeding, we note that
convection or diffusion parameters produce minimal effects on the luminosity
of the star as it moves away from the main sequence toward its temporary
temperature minimum shortly before core hydrogen exhaustion. These factors
play more of a role when discussing the rest of the turnoff in §4.3.

  * •

Measured mass: A stellar mass approximately $0.05M_{\odot}$ lower than
measured would allow the redward evolution of the model star to pass through
the middle of the HR diagram error bars. This would be a $10\sigma$ error
according to our calculations, however, and seems unlikely. We note that a
star of $1.17M_{\odot}$ still produces a significant convective core at the
end of its main sequence evolution in the MESA models.

  * •

Measured luminosity: If there is a systematic error in our decomposition of
the binary’s light or in the distance of the system, it could account for some
of the luminosity discrepancy. If the distance is underestimated (due to an
overestimate in the parallax), the luminosity would be larger. The Gaia
parallax correction would be a suspect because it is acting in this direction.
The corrections being discussed (§3.2) would produce a maximum effect of 0.04
dex in $\log_{10}(L/L_{\odot})$, which is not enough to explain the full
discrepancy, and even so, smaller distances become more inconsistent with CMD-
based distance measures. As for the bolometric flux, we note that the star-
based decomposition of the SED (instead of the main sequence fit) leads to a
lower $F_{\rm bol}$, and so would exacerbate the discrepancy if true.

  * •

Nuclear reaction rates: CNO cycle rates in a stellar evolution code affect the
strength of the energy generation. Larger rates lead to an earlier initiation
of the convective core and lower luminosity for the star during its redward
evolution. The CNO cycle bottleneck reaction 14N(p,$\gamma$)15O is the most
important to consider. The reaction rate was revised downward by nearly 50% at
stellar energies as a consequence of a reduced contribution of capture to the
ground state of 15O, making capture to the 6.792 MeV excited state the
dominant channel. Summaries of relevant experimental measurements can be found
in Adelberger et al. (2011) and Wagner et al. (2018). The NACRE reaction rate
tabulation (Angulo et al., 1999) uses the older, higher astrophysical $S$
factor, while recent tabulations (Xu et al., 2013; Cyburt et al., 2010) use
values consistent with the recent experimental results (Formicola et al.,
2004; Imbriani et al., 2005). While there appears to be convergence on
experimental details of the 14N(p,$\gamma$)15O reaction, the extrapolation of
the $S$ factor to stellar energies now contributes the most to the uncertainty
at the level of about 8% (Wagner et al., 2018). Although significant for other
purposes (such as using solar neutrinos to address solar composition
uncertainties), this level of uncertainty is too small to account for the
luminosity discrepancy unless there is an unknown and more substantial
systematic error in the model reaction rate.

The bottleneck reaction in CNO cycle II is 17O$(p,\alpha)^{14}$N, and this
rate has recently been revised upward by a factor of 2 at stellar energies by
Bruno et al. (2016). This affects the flow of 16O into the main CNO cycle, and
so has a small effect on the luminosity. But again, within the likely
uncertainties, reaction rates also fall far short of explaining the size of
the luminosity discrepancy.

  * •

CNO and heavy element abundances: CNO element abundances affect the CNO cycle
energy generation through their role as catalysts, and larger abundances allow
the convective core to be established earlier in the main sequence evolution
and at lower luminosity. As an illustration in the middle right panel of
Figure 14, the use of a GS98 solar abundance mix moves the redward evolution
to lower luminosity because that mix has a larger proportion of CNO elements
and because solar calibration with this composition leads to a larger
metallicity. However, even a nearly 25% change in the CNO elements is not
enough to explain the luminosity discrepancy for WOCS 11028 A. As seen in the
upper right panel of Figure 14, increased metal content also results in a
reduced luminosity during the redward evolution. As discussed in section 4.1,
evidence from recent spectroscopic studies suggests that subgiants (which
should have deep enough outer convective zones to erase diffusion effects that
occurred on the main sequence) have higher surface metal abundances than
turnoff stars.

Reasonable changes to the bulk heavy element abundance can bring the models
into agreement with the measured luminosity and temperature of WOCS 11028 A at
its measured mass. However, if we assume that composition changes are
responsible, then this affects the agreement of other well-measured stars with
the models. In particular, the models become too low in luminosity to match
WOCS 11028 B.

Our conclusion is that a larger bulk heavy element abundance for M67 stars
(previously masked by diffusion effects) is the most significant effect on the
agreement between models and the characteristics of WOCS 11028 A that we can
identify. Figure 19 compares the characteristics of WOCS 11028 A with MESA
models for $Z=0.0152$ and $0.0195$. In the $T_{\rm eff}$ plot, the faint
cluster turnoff occurs at the age where the three tracks cross, indicating a
constant temperature in the mass range shown. With the larger bulk metal
content, the radius, effective temperature, and luminosity of the star are
approximately in agreement with models for ages roughly between 3.5 and 4.0
Gyr. Further adjustment of $Z$ would throw either the model $T_{\rm eff}$ or
$L$ out of agreement with the observations, although the radius is less
sensitive to metal content.

Figure 19: Stellar characteristics versus age for MESA models. Gray boxes show
$1\sigma$ uncertainties on the measured radius, temperature, and luminosity of
WOCS 11028 A (left) and B (right). MESA evolution tracks for $Z=0.0195$ (solid
lines) and $Z=0.0152$ (dashed lines) are shown, with masses $M=1.222M_{\odot}$
(black) in the left column and $0.910M_{\odot}$ in the right column. Tracks
for masses $1\sigma$ higher (blue) and lower (red) are also shown. Black
points in the left middle panel show lower turnoff points (where stars of
similar mass hit the same temperature).

If the same large metal abundance is applied to models of WOCS 11028 B, they
are pushed farther from agreement with the observations, however. The radius
in particular should not be sensitive to the metal content, but the models are
significantly lower than the observations for reasonable ages. As seen in
Figure 16, the measurements of WOCS 11028 B agree with those for other well-
measured eclipsing binary star components. As a result, we think that there
are still unidentified systematic errors that are playing a role.

###  4.3 The Turnoff of M67 and Convective Cores

With the new data on the massive star in the WOCS 11028 binary, there is an
opportunity to re-examine the cluster turnoff and the constraints it places on
the age and stellar physics. We can make a more precise estimate of star
masses at the turnoff than has been possible before by employing models that
are capable of reproducing the characteristics of WOCS 11028 A, reducing the
importance of systematic errors by making a relative comparison. The observed
properties of the CMD gap and the upper turnoff are strongly influenced by the
extent of the convective core, which in turn affects the amount of accessible
fuel and the main sequence lifetimes.

A major uncertainty about convective core extent is overshooting of the
convective boundary. The stars at M67’s turnoff are uniquely interesting
because they are close to the minimum mass for having convective cores, and
the amount of overshooting could play an outsized role in stars with small
cores. Some studies of eclipsing binaries in the $1.2-2.0M_{\odot}$ mass range
(Claret & Torres, 2017, 2018) indicate that convective overshooting is
consistent with zero at the mass of WOCS 11028 A, but rapidly increasing to a
plateau for masses greater than about $2M_{\odot}$. On the other hand, the
models of Higl & Weiss (2017) also show little or no need for overshooting in
binaries with main sequence stars having masses near $1.2M_{\odot}$, but some
weak evidence for overshooting in BG Ind, having a somewhat more massive
($1.42M_{\odot}$) and more evolved primary star. Constantino & Baraffe (2018)
found little evidence for a mass-dependent overshooting for masses between 1.2
and $2.0M_{\odot}$. A major problem with the binary star samples in most of
these studies is that overshooting reveals itself most clearly very close to
core hydrogen exhaustion, and unless stars are in a very specific evolutionary
phase, their conventional stellar properties (like $R$, $L$, and $T_{\rm
eff}$) will not be sensitive to the overshooting. This disagreement in the
literature reinforces the value of using the information contained in M67,
with its larger sample of stars and mass inferences from the WOCS 11028
binary.

More recently, a number of asteroseismic studies have also attempted to
constrain convective core overshooting with low-degree p modes. Studies like
Viani & Basu (2020) find positive trends in overshooting amount with mass,
while others like Angelou et al. (2020) find overshooting amounts that are
consistent with what is found from binary and cluster calibrations but without
evidence of a clear mass dependence. These issues highlight that there are
systematics and degeneracies in the fitting of parameters in asteroseismic
studies, and that they have not fully clarified the overshoot issue.
Asteroseismic masses have significantly lower precisions than binary system
masses as well. All of these factors call out for additional precise
constraints on the overshooting in this mass range.

Overshooting did not play a significant role in our interpretation of the WOCS
11028 binary because the primary star is well before core hydrogen exhaustion,
but it does more strongly affect the CMD positions of stars around the turnoff
gap and beginning of the subgiant branch. Models indicate that a star like
WOCS 11028 A reaches a maximum in surface temperature when the convective core
starts expanding as the core temperature increases and the CNO cycle becomes
more dominant. After the star passes its surface temperature maximum, models
show that the convective core is maintained and the star evolves mostly in
$T_{\rm eff}$ until shortly before core hydrogen exhaustion. At that time the
convective core rapidly shrinks and disappears, and the star executes a quick
movement to higher temperatures and luminosities in the HR Diagram. There is
still a residue of hydrogen in the core and a steep gradient in hydrogen
abundance outside that, so the evolution of the star slows down again in the
early subgiant branch. Overall, this evolution track is a hybrid, with the
early core hydrogen burning evolution similar to low-mass stars, and the late
evolution similar to more massive stars.

The extent and lifetime of the convective core in stars slightly more massive
than WOCS 11028 A is imprinted on the shape of the turnoff in the CMD and in
the number of stars present. The noticeable gap seen at $12.7\lesssim
G\lesssim 13$ in M67 is a demonstration of the rapid evolution that results
from the disappearance of the convective core shortly before central hydrogen
exhaustion. Similarly, the fairly large density of stars present just brighter
than the gap shows that gas with large hydrogen abundance is residing just
outside the exhausted center of the star, and the burning of this hydrogen
allows the evolution to slow down again.

To define the shape of the turnoff, we derived bolometric fluxes and surface
temperatures from SED fits for probable single stars at critical points in the
CMD. We identified WOCS 7006 / Sanders 1003 as a likely single cluster member
at the bright end of the gap based on CMD position and lack of radial velocity
variation (Geller et al., 2015), and WOCS 8011 / Sanders 1219 as a likely
single member at the faint end. We also examined WOCS 3003 / Sanders 1034 as a
representative of the end of the most heavily populated part of the early
subgiant branch. The stars are shown in CMDs in Figure 3, and their derived
properties are given in Table 3.

Table 3: SED Fit Results for Turnoff Stars | WOCS 3003 | WOCS 7006 | WOCS 8011 | WOCS 11028A | WOCS 11028B
---|---|---|---|---|---
$F_{bol}$ ($10^{-11}$ erg cm-2 s-1) | $25.65^{+0.04}_{-0.23}$ | $21.51\pm 0.07$ | $16.23\pm 0.08$ | $10.89\pm 0.03$ | $2.73\pm 0.02$
$L$ ($L_{\odot}$) | $5.70\pm 0.20$ | $4.78\pm 0.16$ | $3.62\pm 0.12$ | $2.42\pm 0.08$ | $0.68\pm 0.06$
$V$ | $12.656\pm 0.002$ | $12.824\pm 0.002$ | $13.129\pm 0.003$ | 13.568 | 15.167
$b-y$ | $0.381\pm 0.001$ | $0.359\pm 0.001$ | $0.376\pm 0.001$ | 0.372 | 0.486
$c_{1}$ | $0.408\pm 0.003$ | $0.440\pm 0.004$ | $0.341\pm 0.004$ | 0.394 | 0.321
H$\beta$ | $2.599\pm 0.009$ | $2.611\pm 0.008$ | $2.637\pm 0.016$ |  |
$T_{\rm eff}(b-y)$ (K) |  | 6225 | 6150 | 6225 |
$T_{\rm eff}$(IRFM) (K) | 6010 | 6170 | 6080 | 6185 | 5500

As was shown in Figure 14, several model physics parameters play a role in
setting the shape of the turnoff and subgiant branch, and most affect core
convection. As a result, there is some degeneracy in the parameters that can
be used to model the turnoff. For our purposes here, we will focus on
convective core overshooting. Overshooting has the most substantial effects on
evolution tracks shortly before the time of core hydrogen exhaustion. Core
hydrogen exhaustion is delayed by overshooting, which allows the redward
evolution to extend to lower temperature, giving isochrones a deeper redward
kink (see bottom middle panel of Figure 14). In addition, the morphology of
the upper turnoff changes in response to overshooting, with increasing amounts
resulting in a more horizontal early subgiant branch. (See Figure 13, and
compare the BaSTI-IAC isochrones with and without overshooting, or younger
PARSEC isochrones to older ones that have less overshoot.) Both the morphology
of the upper turnoff and the luminosity and temperature positions of the stars
measured near the gap in M67 are indicative of weaker convective overshooting
than is used in the MIST isochrones. This is a means of calibrating overshoot
for stars with masses around the gap (approximately $1.32M_{\odot}$, based on
models that match WOCS 11028 A). The exact amount depends somewhat on other
physics, like diffusion rates, CNO cycle reaction rates, and CNO element
abundances.

It has long been known that the turnoff of M67 is not well matched by
isochrones (VandenBerg et al., 2007; Magic et al., 2010). Currently published
theoretical isochrones should not be used to calibrate the overshooting or
read an age because they do not reproduce the characteristics of WOCS 11028 A
(see Section 4.2.3), and would therefore have clear systematic errors. There
are unfortunately several composition and physics factors that can affect the
size (and even the presence) of a convective core in models, and relatively
small changes in the extent of this core can strongly affect the amount of
hydrogen fuel the core gets, and therefore, the length of the star’s life. But
by using models that are at least able to match WOCS 11028 A, we should be
able to reduce the size of the errors.

Figure 20 shows isochrones from MESA models having $Z=0.0195$ and $Y=0.2703$,
with convective core overshooting set to half the default value, or $f=0.008$.
Larger amounts of overshooting move the gap to lower temperatures, and appear
to be ruled out. With this set of input parameters, we are able to
approximately reproduce the HR diagram positions of the three stars (described
above) that reside at identifiable points in the cluster CMD using an age
around 3.75-4 Gyr. In addition, Stello et al. (2016) found an average giant
star mass of $1.36\pm 0.01M_{\odot}$ in their asteroseismic analysis of M67
giant stars using K2 data. This measurement is also consistent with an age
near 4 Gyr for the MESA models used in Figure 20. The agreement with all of
these observations implies that the models are approximately reproducing the
hydrogen abundance profile in the stars at core exhaustion.

Figure 20: Isochrones for ages between 2.5 and 4.0 Gyr (0.25 Gyr spacing) from
MESA models for $Z=0.0195$ and half strength convective core overshooting. The
dashed green line connects models for the measured mass of WOCS 11028 A, with
$\pm 1\sigma$ uncertainties. Measured characteristics of WOCS 11028 A (red)
and stars at color-magnitude diagram turning points (blue) are shown with
points.

##  5 Conclusions

We present high-precision measurements of the masses, radii, and photometry
for stars in the bright detached eclipsing binary system WOCS 11028 in the
cluster M67. This cluster is commonly used as a testbed for stellar evolution
models of solar metallicity, and the addition of new observables for stars in
the cluster will further test the fidelity of the physics we use to model
them.

Despite orbits that only produce one eclipse per cycle, we have shown that
high-precision masses for the stars in WOCS 11028 are possible. We have used a
decomposition of the SED of the binary to constrain the temperatures, radii,
and luminosities of the stars as well. The brighter star in the binary sits
precisely at the fainter turnoff point for the cluster, where stars with
convective cores are accelerating toward the red in the later stages of core
hydrogen burning. The known mass of a star at an identifiable evolutionary
point constrains the age of the cluster, although uncertainties in model
physics reduce the precision that is attainable.

Systematic errors in the model physics remain a difficult problem for stars at
the turnoff of M67. Recent spectroscopic studies seem to make a strong case
for the inclusion of diffusion in models, and a larger bulk metallicity for
cluster stars can explain the characteristics of WOCS 11028 A. However, the
cost of this is a disgreement between fainter stars (solar analogs and less
massive stars) and models. We are forced to the conclusion that there appears
to be an unidentified systematic error that affects the luminosities of stars
between (and possibly including) 1 and $1.2M_{\odot}$. Our examination of
physics uncertainties like nuclear reaction rates, convection, and chemical
composition has not revealed an obvious culprit. With the assumption of a
larger bulk metallicity, age indicators are somewhat consistent in returning
ages between about 3.5 and 4 Gyr. Future adjustments to model physics will
hopefully provide greater agreement between the measures.

With a star of known mass at the lower cluster turnoff, we can more reliably
calibrate models of stars at core hydrogen exhaustion, which are sensitive to
the size of the convective core. The morphology of M67’s turnoff implies that
convective overshooting in turnoff mass stars is present, but to an extent
that is smaller than used in standard isochrones. After identifying
composition and physics parameters that allow us to match the characteristics
of WOCS 11028 A, we can estimate the mass of stars at the turnoff of M67. We
find that stars near the gap and upper turnoff should have masses of $1.32\pm
0.02M_{\odot}$. With these same model parameters, we can approximately match
the average asteroseismic giant mass of $1.36\pm 0.01M_{\odot}$ measured by
Stello et al. (2016) at an age near 4 Gyr. More precise comparisons between
observed masses and theory will require a better understanding of the core
convection physics because this critically affects the timing of a star’s
divergence from the main sequence. Precise and direct measurements of masses
in the eclipsing system HV Cnc could improve the comparison as well because
the brightest star resides at the upper turnoff. The difficulties in
completing such a study are the faintness of the companion and the
contaminating effects of an unresolved third star (Sandquist & Shetrone,
2003). However, this is one of the few ways of further identifying and
eliminating the systematic errors in our modeling of these stars and our
determination of their ages.

E.L.S. gratefully acknowledges support from the National Science Foundation
under grant AAG 1817217, and thanks K. Brogaard for providing the original
version of the spectral separation code used in this work. R.D.M. acknowledges
funding support from NSF AST 1714506 and the Wisconsin Alumni Research
Foundation. D.S. is the recipient of an Australian Research Council Future
Fellowship (project number FT1400147). This paper includes data collected by
the K2 mission, and we gratefully acknowledge support from NASA under grant
NNX15AW69G to R.D.M. Funding for the K2 mission is provided by the NASA
Science Mission Directorate. This research made use of the SIMBAD database,
operated at CDS, Strasbourg, France; the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI: 10.26093/cds/vizier; Ochsenbein et al. 2000) data
provided by the High Energy Astrophysics Science Archive Research Center
(HEASARC), which is a service of the Astrophysics Science Division at
NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian
Astrophysical Observatory; the WEBDA database, operated at the Institute for
Astronomy of the University of Vienna; and the Mikulski Archive for Space
Telescopes (MAST). STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST
was provided by the NASA Office of Space Science via grant NNX09AF08G and by
other grants and contracts. Funding for SDSS-III has been provided by the
Alfred P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, and the U.S. Department of Energy Office of Science. The
SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the
Astrophysical Research Consortium for the Participating Institutions of the
SDSS-III Collaboration including the University of Arizona, the Brazilian
Participation Group, Brookhaven National Laboratory, Carnegie Mellon
University, University of Florida, the French Participation Group, the German
Participation Group, Harvard University, the Instituto de Astrofisica de
Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns
Hopkins University, Lawrence Berkeley National Laboratory, Max Planck
Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics,
New Mexico State University, New York University, Ohio State University,
Pennsylvania State University, University of Portsmouth, Princeton University,
the Spanish Participation Group, University of Tokyo, University of Utah,
Vanderbilt University, University of Virginia, University of Washington, and
Yale University.

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  * Zinn et al. (2019) Zinn, J. C., Pinsonneault, M. H., Huber, D., et al. 2019, ApJ, 878, 136

Table 4: Photometry of the Binary WOCS 11028 and Proxy Stars

|  |  |  | Bright Component A | Faint Component B | Ratios |
---|---|---|---|---|---|---|---
|  | WOCS 11028 | WOCS 6018 | MS Fit | WOCS 10027 | MS Fit | Stars | MS Fit |
Filter |  $\lambda_{eff}$ (Å) | $m_{\lambda}$ | $\sigma_{m}$ | $m_{\lambda}$ | $\sigma_{m}$ | $m_{\lambda}$ | $m_{\lambda}$ | $\sigma_{m}$ | $m_{\lambda}$ | $F_{B}/F_{A}$ | $F_{B}/F_{A}$ | Ref.
$uvw2$ | 2030 | 16.715 | 0.058 | 16.898 | 0.040 | 16.886 |  |  | 19.389 |  | 0.100 |
$uvm2$ | 2231 | 16.655 | 0.072 | 16.757 | 0.050 |  |  |  |  |  |  |
NUV | 2315.7 | 18.164 | 0.006 | 18.230 | 0.008 | 18.221 | 21.047 | 0.049 | 21.277 | 0.075 | 0.060 |
$uvw1$ | 2634 | 15.228 | 0.030 | 15.390 | 0.021 | 15.382 |  |  | 17.753 |  | 0.113 |
$U_{V}$ | 3450 |  |  | 15.953 | 0.018 |  |  |  | 18.030 |  | 0.145 | 1
$u$ | 3520 | 15.283 | 0.004 | 15.476 | 0.004 | 15.386 | 17.183 | 0.016 | 17.470 | 0.208 | 0.147 | 2
$u_{SDSS}$ | 3551 | 14.851 | 0.009 | 15.025 | 0.014 | 14.993 | 16.825 | 0.029 | 17.096 | 0.191 | 0.144 |
$U$ | 3663 |  |  | 14.144 |  | 14.202 | 15.962 |  | 16.308 | 0.187 | 0.144 | 3
$P_{V}$ | 3740 |  |  | 15.475 | 0.013 | 15.449 |  |  | 17.590 |  | 0.139 | 1
BATC2 ($b$) | 3890 | 14.39 | 0.01 | 14.59 | 0.02 | 14.539 | 16.45 | 0.05 | 16.695 | 0.180 | 0.137 | 4
$X_{V}$ | 4054 |  |  | 14.882 | 0.011 | 14.854 |  |  | 16.825 |  | 0.163 | 1
$v$ | 4100 | 14.308 | 0.003 | 14.522 | 0.002 | 14.466 | 16.155 | 0.014 | 16.401 | 0.222 | 0.168 | 2
$B$ | 4361 |  |  |  |  | 14.125 | 15.739 | 0.006 | 15.930 |  | 0.190 | 5
$B$ | 4361 |  |  | 14.149 |  | 14.149 | 15.758 | 0.015 | 15.960 | 0.227 | 0.189 | 3
$B$ | 4361 | 13.971 | 0.068 | 14.164 | 0.031 | 14.118 | 15.786 | 0.096 | 15.955 | 0.224 | 0.184 | 6
$B$ | 4361 | 13.952 |  | 14.183 |  | 14.151 | 15.627 |  | 15.960 | 0.264 | 0.189 | 7
$B$ | 4361 | 13.881 | 0.013 |  |  | 14.123 | 15.675 | 0.004 | 15.912 |  | 0.193 | 8
BATC4 ($d$) | 4532 | 13.67 | 0.01 | 13.92 | 0.01 | 13.879 | 15.41 | 0.01 | 15.620 | 0.254 | 0.201 | 4
$Y_{V}$ | 4665 |  |  | 14.148 | 0.008 | 14.117 |  |  | 15.827 |  | 0.207 | 1
$g^{\prime}$ | 4640 | 13.759 | 0.386 | 13.979 | 0.307 | 13.807 | 15.360 | 0.176 | 15.481 | 0.280 | 0.214 | 6
$g_{SDSS}$ | 4686 | 13.592 | 0.001 | 13.825 | 0.008 | 13.816 | 15.292 | 0.010 | 15.491 | 0.259 | 0.214 |
$b$ | 4688 | 13.720 | 0.002 | 13.979 | 0.002 | 13.940 | 15.454 | 0.011 | 15.653 | 0.257 | 0.207 | 2
$g_{P1}$ | 4810 | 13.560 | 0.001 | 13.784 | 0.001 | 13.743 | 15.224 | 0.003 | 15.425 | 0.266 | 0.212 |
BATC5 ($e$) | 4916 | 13.53 | 0.01 | 13.78 | 0.01 | 13.710 | 15.17 | 0.01 | 15.375 | 0.278 | 0.216 | 4
$G_{BP}$ | 5051.5 | 13.5113 | 0.0013 | 13.7714 | 0.0011 | 13.7289 | 15.1898 | 0.0016 | 15.3801 | 0.271 | 0.219 |
$Z_{V}$ | 5162 |  |  | 13.815 | 0.008 | 13.777 |  |  | 15.451 |  | 0.214 | 1
BATC6 ($f$) | 5258 | 13.38 | 0.01 | 13.65 | 0.01 | 13.585 | 15.03 | 0.02 | 15.225 | 0.281 | 0.221 | 4
$V_{V}$ | 5442 |  |  | 13.622 | 0.006 | 13.582 |  |  | 15.182 |  | 0.229 | 1
$V$ | 5448 | 13.337 | 0.001 | 13.597 | 0.001 | 13.564 | 14.953 | 0.002 | 15.152 | 0.287 | 0.232 | 5
$V$ | 5448 |  |  | 13.598 |  | 13.597 | 15.010 | 0.010 | 15.183 | 0.275 | 0.227 | 3
$V$ | 5448 | 13.366 | 0.042 | 13.619 | 0.011 | 13.566 | 14.977 | 0.047 | 15.158 | 0.286 | 0.231 | 6
$V$ | 5448 | 13.391 |  | 13.675 |  | 13.597 |  |  | 15.174 |  | 0.234 | 7
$V$ | 5448 | 13.246 | 0.004 |  |  | 13.539 | 14.924 | 0.007 | 15.123 |  | 0.232 | 8
$y$ | 5480 | 13.349 | 0.003 | 13.616 | 0.003 | 13.568 | 14.974 | 0.008 | 15.167 | 0.286 | 0.229 | 2
BATC7 ($g$) | 5785 | 13.24 | 0.02 | 13.55 | 0.02 | 13.473 | 14.84 | 0.03 | 15.013 | 0.305 | 0.242 | 4
BATC8 ($h$) | 6069 | 13.16 | 0.02 | 13.44 | 0.02 | 13.380 | 14.70 | 0.03 | 14.901 | 0.313 | 0.246 | 4
$r^{\prime}$ | 6122 | 13.194 | 0.019 | 13.491 | 0.019 | 13.440 | 14.737 | 0.041 | 14.952 | 0.317 | 0.248 | 6
$r_{SDSS}$ | 6166 | 13.166 | 0.001 | 13.487 | 0.017 | 13.410 | 14.769 | 0.010 | 14.911 | 0.307 | 0.251 |
$r_{P1}$ | 6170 |  |  | 13.482 | 0.013 | 13.426 | 14.756 | 0.001 | 14.926 | 0.309 | 0.251 |
$G$ | 6230.6 | 13.1974 | 0.0002 | 13.4816 | 0.0002 | 13.4363 | 14.8044 | 0.0003 | 14.9717 | 0.296 | 0.243 |
$S_{V}$ | 6534 |  |  | 13.112 | 0.008 | 13.062 |  |  | 14.546 |  | 0.255 | 1
BATC9 ($i$) | 6646 | 13.13 | 0.02 | 13.44 | 0.01 | 13.394 | 14.66 | 0.03 | 14.859 | 0.325 | 0.259 | 4
BATC10a ($j$) | 7055 | 13.08 | 0.01 | 13.39 | 0.01 | 13.327 | 14.66 | 0.03 | 14.786 | 0.334 | 0.261 | 4
$i^{\prime}$ | 7440 | 13.264 | 0.369 | 13.528 | 0.305 | 13.374 | 14.545 | 0.072 | 14.790 | 0.392 | 0.271 | 6
$i_{SDSS}$ | 7480 |  |  | 13.357 | 0.001 | 13.308 | 14.602 | 0.021 | 14.742 | 0.318 | 0.267 |
$i_{P1}$ | 7520 | 13.225 |  | 13.465 |  | 13.337 | 14.618 | 0.003 | 14.748 | 0.346 | 0.273 |
BATC10b ($k$) | 7545 | 13.06 | 0.01 | 13.37 | 0.01 | 13.308 | 14.53 | 0.04 | 14.740 | 0.344 | 0.267 | 4
$G_{RP}$ | 7726.2 | 12.7237 | 0.0007 | 13.0354 | 0.0006 | 12.9865 | 14.2558 | 0.0014 | 14.4074 | 0.325 | 0.270 |
$I_{C}$ | 7980 | 12.655 | 0.001 | 12.957 | 0.002 | 12.919 | 14.132 | 0.002 | 14.325 | 0.339 | 0.274 | 5
$I_{C}$ | 7980 |  |  | 12.920 |  | 12.888 | 14.176 |  | 14.325 | 0.314 | 0.266 | 3
$I_{C}$ | 7900 |  |  | 12.981 | 0.008 | 12.934 |  |  | 14.355 |  | 0.270 | 1
$I_{C}$ | 7900 | 12.584 | 0.005 |  |  | 12.874 | 14.107 | 0.002 | 14.311 |  | 0.266 | 8
BATC11 ($m$) | 8020 | 13.04 | 0.01 | 13.35 | 0.01 | 13.283 | 14.48 | 0.02 | 14.697 | 0.353 | 0.272 | 4
BATC12 ($n$) | 8483 | 13.01 | 0.01 | 13.34 | 0.01 | 13.289 | 14.49 | 0.01 | 14.679 | 0.347 | 0.278 | 4
$z_{P1}$ | 8660 | 13.058 | 0.004 | 13.387 | 0.003 | 13.339 | 14.579 | 0.002 | 14.715 | 0.334 | 0.282 |
$z_{SDSS}$ | 8932 | 13.010 | 0.016 | 13.350 | 0.011 | 13.291 | 14.602 | 0.021 | 14.661 | 0.344 | 0.283 |
BATC13 ($o$) | 9180 | 12.97 | 0.01 | 13.30 | 0.01 | 13.246 | 14.41 | 0.01 | 14.610 | 0.360 | 0.285 |
$y_{P1}$ | 9620 | 13.037 | 0.005 | 13.366 | 0.007 | 13.318 | 14.527 | 0.010 | 14.667 | 0.343 | 0.289 |
BATC14 ($p$) | 9736 | 12.99 | 0.01 | 13.31 | 0.01 | 13.281 | 14.43 | 0.01 | 14.630 | 0.360 | 0.289 |
$J$ | 12350 | 12.177 | 0.018 | 12.546 | 0.022 | 12.492 | 13.609 | 0.036 | 13.767 | 0.376 | 0.309 |
$H$ | 16620 | 11.903 | 0.020 | 12.260 | 0.020 | 12.237 | 13.276 | 0.042 | 13.391 | 0.392 | 0.346 |
$K_{s}$ | 21590 | 11.829 | 0.020 | 12.239 | 0.023 | 12.177 | 13.205 | 0.024 | 13.311 | 0.411 | 0.352 |
$W1$ | 33526 | 11.799 | 0.023 | 12.211 | 0.024 | 12.131 | 13.015 | 0.030 | 13.264 | 0.477 | 0.352 |
$W2$ | 46028 | 11.839 | 0.021 | 12.250 | 0.415 | 12.165 | 13.095 |  | 13.311 | 0.459 | 0.348 |
$T$ (K) |  |  |  | 6200 | 100 | 6200 | 5600 | 100 | 5500 | 0.88 | 0.89 |
$F_{bol}$ ($10^{-11}$ erg / cm2 s) |  |  |  | 10.47 |  | 10.89 | 3.21 |  | 2.73 | 0.310 | 0.248 |
$R$ ($R_{\odot}$) |  |  |  |  |  |  |  |  |  | 0.719 | 0.643 |

References. — 1: Laugalys et al. (2004). 2: Balaguer-Núñez et al. (2007). 3:
Montgomery et al. (1993). 4: Zhou et al. (2003). 5: Sandquist (2004). 6:
Henden et al. (2015). 7: Nardiello et al. (2016b). 8: Yadav et al. (2008).

Table 5: Detached Eclipsing Binary Stars with Masses Similar to WOCS 11028 A Name | $M/M_{\odot}$ | $E(B-V)$ |  $\omega$ (mas) | Filter | Magnitude | Refs.aaReferences: 1: Clausen et al. (2008). 2: Clausen et al. (2010). 3: Lacy et al. (2014). 4: Lacy & Fekel (2014). 5: Vos et al. (2012). 6: Hełminiak et al. (2009); Andersen et al. (1989). 7: Hełminiak et al. (2019); Popper et al. (1986). 8: Graczyk et al. (2016). 9: Brewer et al. (2016).
---|---|---|---|---|---|---
AD Boo B | $1.209\pm 0.006$ | 0.034 | $5.118\pm 0.073$ | $y$ | $10.721\pm 0.013$ | 1
|  |  |  | $b$ | $11.092\pm 0.007$ |
|  |  |  | $v$ | $11.661\pm 0.011$ |
|  |  |  | $u$ | $12.570\pm 0.016$ |
BK Peg B | $1.257\pm 0.005$ | 0.061 | $3.139\pm 0.053$ | $y$ | $11.080\pm 0.014$ | 2
|  |  |  | $b$ | $11.440\pm 0.008$ |
|  |  |  | $v$ | $11.579\pm 0.016$ |
|  |  |  | $u$ | $12.514\pm 0.017$ |
AP And A | $1.277\pm 0.004$ | 0.058 | $2.866\pm 0.041$ | $V$ | 11.845 | 3
AP And B | $1.251\pm 0.004$ | 0.058 | $2.866\pm 0.041$ | $V$ | 11.963 | 3
VZ Hya A | $1.271\pm 0.009$ | 0.028 | $6.765\pm 0.049$ | $y$ | $9.442\pm 0.008$ | 1
|  |  |  | $b$ | $9.723\pm 0.005$ |
|  |  |  | $v$ | $10.169\pm 0.008$ |
|  |  |  | $u$ | $11.035\pm 0.009$ |
V501 Her A | $1.2690\pm 0.0035$ | 0.048 | $2.309\pm 0.022$ | $V$ | $11.623\pm 0.048$ | 4
V501 Her B | $1.2113\pm 0.0032$ | 0.048 | $2.309\pm 0.022$ | $V$ | $12.188\pm 0.048$ | 4
EF Aqr A | $1.244\pm 0.008$ | 0. | $5.765\pm 0.060$ | $y$ | $10.015\pm 0.022$ | 5
|  |  |  | $b$ | $10.353\pm 0.010$ |
|  |  |  | $v$ | $10.874\pm 0.010$ |
|  |  |  | $u$ | $11.778\pm 0.015$ |
WZ Oph A | $1.227\pm 0.007$ | 0.045 | $6.372\pm 0.033$ | $y$ | $9.856\pm 0.017$ | 1
|  |  |  | $b$ | $10.218\pm 0.005$ |
|  |  |  | $v$ | $10.729\pm 0.008$ |
|  |  |  | $u$ | $11.609\pm 0.011$ |
WZ Oph B | $1.220\pm 0.006$ | 0.045 | $6.372\pm 0.033$ | $y$ | $9.841\pm 0.017$ | 1
|  |  |  | $b$ | $10.211\pm 0.005$ |
|  |  |  | $v$ | $10.722\pm 0.008$ |
|  |  |  | $u$ | $11.615\pm 0.012$ |
UX Men A | $1.2229\pm 0.0015$ | 0.028 | $9.644\pm 0.025$ | $y$ | $8.89\pm 0.01$ | 6
|  |  |  | $b$ | $9.249\pm 0.005$ |
|  |  |  | $v$ | $9.769\pm 0.005$ |
|  |  |  | $u$ | $10.660\pm 0.005$ |
UX Men B | $1.1878\pm 0.0015$ | 0.028 | $9.644\pm 0.025$ | $y$ | $9.07\pm 0.01$ | 6
|  |  |  | $b$ | $9.438\pm 0.005$ |
|  |  |  | $v$ | $9.990\pm 0.005$ |
|  |  |  | $u$ | $10.899\pm 0.005$ |
FL Lyr A | $1.2102\pm 0.0076$ | 0.007 | $7.406\pm 0.025$ | $V$ | $9.58$ | 7
LL Aqr A | $1.1959\pm 0.0007$ | 0.018 | $7.269\pm 0.051$ | $V$ | $9.71\pm 0.012$ | 8
|  |  |  | $B$ | $10.224\pm 0.032$ |
|  |  |  | $U$ | $10.160\pm 0.062$ |
WOCS 40007 A | $1.218\pm 0.008$ | 0.16 | 0.420bbDistance modulus from Brewer et al. (2016) translated back into an equivalent parallax. | $V$ | $16.053\pm 0.02$ | 9
|  |  |  | $B$ | $16.669\pm 0.006$ |
|  |  |  | $I_{C}$ | $15.307\pm 0.025$ |