# The Interferometric Binary $\epsilon$ Cnc in Praesepe: Precise Masses and
Age

Leslie M. Morales San Diego State University, Department of Astronomy, San
Diego, CA, 92182 USA  Eric L. Sandquist San Diego State University, Department
of Astronomy, San Diego, CA, 92182 USA  Gail H. Schaefer The CHARA Array of
Georgia State University, Mount Wilson Observatory, Mount Wilson, CA 91023,
USA  Christopher D. Farrington The CHARA Array of Georgia State University,
Mount Wilson Observatory, Mount Wilson, CA 91023, USA  Robert Klement The
CHARA Array of Georgia State University, Mount Wilson Observatory, Mount
Wilson, CA 91023, USA  Luigi R. Bedin Istituto Nazionale Astrofisica di
Padova—Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122
Padova, Italy  Mattia Libralato AURA for the European Space Agency (ESA), ESA
Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218, USA  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 Istituto Nazionale Astrofisica di
Padova—Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122
Padova, Italy  Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France  Jerome
A. Orosz San Diego State University, Department of Astronomy, San Diego, CA,
92182 USA  John D. Monnier Astronomy Department, University of Michigan, Ann
Arbor, MI 48109, USA  Stefan Kraus Astrophysics Group, Department of Physics &
Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK  Jean-
Baptiste Le Bouquin Institut de Planetologie et d’Astrophysique de Grenoble,
Grenoble 38058, France  Astronomy Department, University of Michigan, Ann
Arbor, MI 48109, USA  Narsireddy Anugu The CHARA Array of Georgia State
University, Mount Wilson Observatory, Mount Wilson, CA 91023, USA  Theo ten
Brummelaar  The CHARA Array of Georgia State University, Mount Wilson
Observatory, Mount Wilson, CA 91023, USA  Claire L. Davies Astrophysics Group,
Department of Physics & Astronomy, University of Exeter, Stocker Road, Exeter,
EX4 4QL, UK  Jacob Ennis Astronomy Department, University of Michigan, Ann
Arbor, MI 48109, USA  Tyler Gardner Astronomy Department, University of
Michigan, Ann Arbor, MI 48109, USA  Cyprien Lanthermann The CHARA Array of
Georgia State University, Mount Wilson Observatory, Mount Wilson, CA 91023,
USA

###### Abstract

We observe the brightest member of the Praesepe cluster, $\epsilon$ Cnc, to
precisely measure the characteristics of the stars in this binary system, en
route to a new measurement of the cluster’s age. We present spectroscopic
radial velocity measurements and interferometric observations of the sky-
projected orbit to derive the masses, which we find to be
$M_{1}/M_{\odot}=2.420\pm 0.008$ and $M_{2}/M_{\odot}=2.226\pm 0.004$. We
place limits on the color-magnitude positions of the stars by using
spectroscopic and interferometric luminosity ratios while trying to reproduce
the spectral energy distribution of $\epsilon$ Cnc. We re-examine the cluster
membership of stars at the bright end of the color-magnitude diagram using
Gaia data and literature radial velocity information. The binary star data are
consistent with an age of $637\pm 19$ Myr, as determined from MIST model
isochrones. The masses and luminosities of the stars appear to select models
with the most commonly used amount of convective core overshooting.

††facilities: CHARA (CLIMB, MIRC-X), KPNO:CFT, NOT (FIES), OHP
(ELODIE)††software: BF-rvplotter (https://github.com/mrawls/BF-rvplotter),
FIESTool

##  1 Introduction

Absolute age measurements for stars are fraught with difficulties because
comparisons with theoretical models are required, and because our knowledge of
important factors like the initial chemical composition, or the stellar
masses, or even the behavior of certain basic physical processes is often
poor. Often, even when we are able to precisely measure characteristics of
stars that can speak to a star’s age, these gaps in our understanding result
in substantial systematic errors. Star clusters provide an environment where
difficulties like chemical composition uncertainties can be mitigated and a
uniform age is usually a good assumption. The study of binary systems within
such clusters can give the opportunity to evaluate stellar masses in a precise
way.

The goals of this paper are to precisely characterize stars (including masses)
at the turnoff of the open cluster Praesepe, and derive a precise age for the
cluster from these stars in order to set up Praesepe as an age-calibration
standard. There are a number of astrophysically-interesting features of the
cluster that recommend it for attention. The rotation of the main sequence
stars has been studied extensively, among the A-type stars (McGee et al.,
1967; Abt & Willmarth, 1999; Debernardi et al., 2000; Fossati et al., 2007,
2008; Cummings et al., 2017), solar-type stars (Delorme et al., 2011; Kovács
et al., 2014), and low-mass stars (Scholz & Eislöffel, 2007; Agüeros et al.,
2011; Scholz et al., 2011; Douglas et al., 2017). These measurements establish
Praesepe as a gyrochronology age standard, although a more precise absolute
age would improve its utility. Among the A stars, there are a number of
interesting phenomena like $\delta$ Scuti pulsation (e.g. Breger et al., 2012;
Frandsen et al., 2001; Hernandez et al., 1998; Belmonte et al., 1994) and
surface chemical peculiarities (Abt, 1986; Hui-Bon-Hoa & Alecian, 1998;
Burkhart & Coupry, 1998; Fossati et al., 2007, 2008). The cluster has a fairly
large population of white dwarfs, with 12 known (Salaris & Bedin, 2019).
Almost all of the white dwarfs have spectroscopically-measured masses, and
they cover a fairly large range from about 0.7 to $0.9M_{\odot}$ (Cummings et
al., 2018; Salaris & Bedin, 2019), in contrast to the field white dwarf
population. These white dwarfs probably originate from progenitor stars in a
fairly narrow mass range, where the stars ignite core helium burning in a non-
degenerate gas but do not undergo a second dredge-up convection event that
helps limit the size of the core.

To these ends, we have conducted a thorough analysis of the known double-lined
spectroscopic binary star system $\epsilon$ Cnc (also known as Meleph, HD
73731, 41 Cnc, BD +20 2171). With the combined light of its stars, $\epsilon$
Cnc is one of the brightest members of Praesepe. Plaskett et al. (1921) were
the first to measure the velocity of the second component of the binary, but
Abt & Willmarth (1999) were the first to fit the spectroscopic orbit of the
system, finding a period of 35.202 days. Both stars in the binary are rotating
relatively modestly for A-type stars, and both stars appear to show surface
chemical peculiarities (Burkhart & Coupry, 1998). But most importantly, the
binary is bright enough, close enough, and has a wide enough orbit that the
orbit can now be resolved with interferometric methods and a precise
measurement of the masses of the component stars is within reach. Further, the
system’s color implies that both stars are on or near the very brightest part
of the main sequence, with the brighter star likely to be near the end of its
core hydrogen burning life. Because the brighter star is starting to evolve
more rapidly than its companion, its characteristics are sensitive functions
of age.

In section 2, we discuss the spectroscopic, photometric, and interferometric
data used, as well as the measurement of the radial velocities of the stars.
In section 3, we describe the fitting method used to determine the
characteristics of the stars, like the masses. In section 4, we discuss the
age measurement for Praesepe and its applications to other areas of stellar
astrophysics, like the white dwarf initial-final mass relation.

###  1.1 Reddening, Distance, and Metallicity

Taylor (2006) critically examined measurements of Praesepe’s reddening, and
settled on $E(B-V)=0.027\pm 0.004$. This is consistent with the reddening
derived by Gaia Collaboration et al. (2018a) from isochrone fitting. Zero
reddening is sometimes used for the cluster (e.g. Cummings et al., 2017,
2018), but we believe this is ruled out by comparisons with the Hyades
cluster. The chemical compositions and ages of these two clusters are
extremely similar, but without a significant reddening for Praesepe, the main
sequences do not align, and there are troubling offsets between the white
dwarf initial-final mass sequences (Salaris & Bedin, 2019). Douglas et al.
(2019) used a spectroscopic examination of FGK stars in Praesepe and Hyades,
and found a reddening value $E(B-V)=0.011\pm 0.004$ between zero and the
Taylor value.

The weighted average parallax for Praesepe members in Gaia DR2 was determined
as $5.371\pm 0.003$ mas (Gaia Collaboration et al., 2018b), or $5.361\pm
0.005$ mas (Cantat-Gaudin et al., 2018). There are a number of studies
indicating that Gaia parallaxes are systematically offset to smaller values
(and implying that objects are systematically closer than Gaia indicates;
Lindegren et al. 2018; Zinn et al. 2019), however. In this paper, we employ
the Gaia Early Data Release 3 (EDR3), which may have a smaller parallax offset
of $-17\mu$as than DR2 (Lindegren et al., 2021). Based on a simple selection
of members within $5\arcdeg$ of the cluster center, 6 mas yr-1 of the
cluster’s proper motion vector, and parallax within 0.4 mas of an iteratively-
determined average, we find $\bar{\omega}=5.424$ mas. This implies a distance
modulus $(m-M)_{0}=6.33\pm 0.02$.

Recent spectroscopic examinations of Praesepe stars agree well that the
cluster is quite metal-rich, similar to the Hyades. Most of the narrow-lined
A-type stars in the cluster show chemical peculiarities (e.g. Burkhart &
Coupry 1998), and so are not appropriate for indications of the mean cluster
abundances. However, giants and solar-type stars should have deep enough
surface convection zones that chemical diffusion effects would be mostly
erased. Yang et al. (2015a) find [Fe/H]$=+0.16\pm 0.06$ from four giant stars,
while Carrera & Pancino (2011) found [Fe/H]$=+0.16\pm 0.05$ from three giants.
Boesgaard et al. (2013) find [Fe/H]$=+0.12\pm 0.04$ from 11 solar-type stars,
while D’Orazi et al. (2020) find [Fe/H]$=+0.21\pm 0.01$ from 10 solar-type
dwarfs. Finally, Cummings et al. (2017) find [Fe/H]$=+0.156\pm 0.066$ from 39
F, G, and K dwarfs, and Vejar et al. (2021) find [Fe/H]$=+0.21\pm 0.02$ from 6
stars with spectral types between G5 and F8.

###  1.2 Review of Literature Ages

In the most recent studies of Praesepe, the age of the cluster has been
calculated to range from 590 Myrs to 800 Myrs. This is due to debates over the
cluster’s metallicity and the effects of stellar rotation. Gáspár et al.
(2009) tested different metallicities in isochrone modeling using PADOVA
isochrones, and derived an age of $757\pm 36$ Myr using [Fe/H]$=+0.2$.

Praesepe and Hyades are often studied together because it is widely understood
that these are nearly coeval clusters and because of their similarity in
characteristics such as metallicity. Brandt & Huang (2015) asserted the
importance of understanding stellar rotation in discussing these clusters
because it can cause chemical mixing in their brightest main sequence stars.
They argue that if stellar rotation is neglected, it creates inconsistencies
in age in the Bayesian color-magnitude technique that was used to date the
clusters using PARSEC models. The age of the cluster becomes more consistent
in the models when rotation is taken into account. They derived an older age
of $800$ Myr using [Fe/H]$=+0.12$.

On the other hand, Gossage et al. (2018) used MESA models to examine rotating
and non-rotating color-magnitude diagrams of Praesepe. In contrast to Brandt &
Huang (2015), they conclude that the best-fit isochrone is not one with
rotation but rather a younger non-rotating one, estimating an age of $590$ Myr
using [Fe/H]$=+0.24$.

The relative age difference between Praesepe and the Hyades can also be
estimated using gyrochronology of solar-type stars. Hyades stars rotate slower
on average than Praesepe stars do, leading to a determination that Prasesepe
is about 57 Myr younger than the Hyades (Douglas et al., 2019).

##  2 Observations and Analysis

###  2.1 Spectroscopy

The majority of the radial velocity measurements in this study are based on a
reanalysis of spectra taken as part of a survey of bright Praesepe stars by
Abt & Willmarth (1999). The spectra were obtained on the Kitt Peak National
Observatory (KPNO) 0.9m coudé feed telescope and spectrograph between 1991 and
1996. The spectra generally covered the wavelength range $436-474.2$ nm, at a
resolution of about 0.022 nm. Abt & Willmarth obtained a double-lined
spectroscopic orbit of $\epsilon$ Cnc at the time, although they noted that
the rotationally-broadened lines of the two stars are always blended. To
improve the accuracy of the radial velocities, we performed a new analysis of
the spectra, as detailed in section 3.

In addition, we extracted two spectra from the archive of the ELODIE
spectrograph at the Observatoire de Haute-Provence (OHP). The spectra
($R=42000$) were taken in 2004 by observer R. Monier, covering the range
$385-681$ nm.

Finally, we obtained four new spectra in 2020 with the Fibre-fed Echelle
Spectrograph (FIES) on the 2.56m Nordic Optical Telescope (NOT). Two spectra
each were obtained near predicted radial velocity extremes in order to better
resolve the stars individually, and improve the measurement of the velocity
amplitude for later determination of the masses. We used the medium-resolution
($R=46000$) fiber in our observations, covering the spectral range $370-830$
nm (Telting et al., 2014), although we only used $440-660$ nm in our
measurements. We processed the images and extracted one-dimensional spectra
using FIESTool (version 1.5.1). The wavelength calibration was done using
exposures of the ThAr lamp taken immediately before the science exposures.
Individual spectral orders were continuum normalized before being merged.

####  2.1.1 Literature and Newly-Measured Radial Velocities

Literature radial velocities are available for the primary star in $\epsilon$
Cnc from several sources, but many of these were rejected due to likely
effects of blending of the lines of the two stars. Plaskett et al. (1921)
report measurements at four epochs in 1919 and 1920 from the 1.83m Plaskett
Telescope at Dominion Astrophysical Observatory (DAO), including one claiming
detection of both stars. The detection of double lines occurred approximately
at one velocity extremum ($\phi\approx 0.14$), and the velocity separation of
the measurements agrees well with our modern solution if a $-6.5$ km
s${}^{-1}\;$offset is used. We therefore regard the detection as legitimate,
and utilize the measurement with the offset applied. The remaining
measurements were discarded as affected by blending.

Abt (1970) reported eight measurements taken at Mt. Wilson Observatory in
1923, although we only used five of these. The remaining three measurements
with phases $0.1<\phi<0.3$ show clear signs of blending. Finally, there were
10 epochs observed by Abt & Willmarth (1999) for which the original spectra
have not been recovered. In those cases, the reported velocities were still
used in the fitting of a velocity solution. These literature values are
reported along with our measurements in Tables 7 and 8. Julian dates reported
by Plaskett et al. (1921) and UT dates reported by Abt (1970) were converted
to heliocentric Julian dates.

Radial velocity measurements were made using broadening functions (BFs;
Rucinski 1992) derived from the spectra. We used the program BF-rvplotter
(Rawls, 2016) to calculate the broadening functions for each spectrum after
modifying the code to fit rotational broadening functions (rather than
Gaussians) to the two star profiles. The BFs are calculated through a
comparison with a non-rotating synthetic spectrum of similar temperature and
surface gravity, which in this case came from Coelho et al. (2007). Because of
the particular combination of radial velocity separations and rotational
broadening for the two stars, we found that the broadening profiles of the
stars overlapped substantially and produced significant correlations between
the measured rotation and radial velocities. As a result, we found it
necessary to determine rotational velocities for the two stars from spectra
with the highest signal-to-noise ratios and velocity separations. We
subsequently fixed the values of the rotational velocities and refitted all
spectra to derive the radial velocities we present. Heliocentric velocity
corrections were applied to all calculated velocity values.

Figure 1: Broadening profiles for the spectra with the most distinguishable
contributions from the two stars, with orbital phase given in the upper right
corner. The lines shows the fits to the primary star profile (green), the
secondary star profile (red), the normalized fit (blue) and the summation
(orange). The first four panels are from NOT FIES spectra and the last seven
panels are from KPNO spectra.

####  2.1.2 Rotational Velocity and Luminosity Ratio

Figure 1 shows the broadening functions calculated from the spectra with the
cleanest separation of the two stars’ peaks. These eleven spectra were taken
close to velocity extremes, and gave very consistent measurements for the
widths of the rotational broadening function fits (Table 1). We derived
weighted averages of the primary and secondary star’s velocity widths from
these spectra: $v_{A}\sin i=48.0\pm 0.2$ km s${}^{-1}\;$and $v_{B}\sin
i=76.5\pm 0.2$ km s-1. The stars are probably rotating considerably slower
than single stars of similar brightness in the cluster (see Table 5). If the
stars have aligned rotation axes, then the system cannot be close to
synchronism because the primary has a lower measured rotational velocity.

Table 1: Rotational Velocity and Luminosity Ratio Measurements for $\epsilon$ Cnc mJDaamJD = HJD - 2400000 | Phase $\phi$ |  $v_{A}\sin i$ (km/s) |  $v_{B}\sin i$ (km/s) | $L_{B}/L_{A}$ | Source
---|---|---|---|---|---
48638.824 | 0.217 | $47.23\pm 0.08$ | $76.26\pm 0.22$ | 0.680 | KPNO
49021.874 | 0.117 | $48.83\pm 0.08$ | $75.31\pm 0.19$ | 0.642 | KPNO
49021.931 | 0.119 | $48.94\pm 0.08$ | $75.56\pm 0.22$ | 0.639 | KPNO
49021.980 | 0.120 | $48.43\pm 0.08$ | $75.27\pm 0.25$ | 0.628 | KPNO
49080.622 | 0.789 | $48.77\pm 0.07$ | $77.56\pm 0.22$ | 0.593 | KPNO
49081.706 | 0.820 | $48.86\pm 0.08$ | $76.29\pm 0.25$ | 0.595 | KPNO
49094.698 | 0.190 | $48.29\pm 0.08$ | $76.57\pm 0.23$ | 0.683 | KPNO
58960.375 | 0.931 | $47.05\pm 0.07$ | $77.41\pm 0.26$ | 0.604 | NOT FIES
58960.379 | 0.931 | $47.05\pm 0.07$ | $77.49\pm 0.27$ | 0.600 | NOT FIES
58968.404 | 0.160 | $47.70\pm 0.07$ | $77.02\pm 0.21$ | 0.590 | NOT FIES
58968.408 | 0.160 | $47.15\pm 0.07$ | $76.96\pm 0.21$ | 0.585 | NOT FIES

As we will find in section 2.3, the stars of $\epsilon$ Cnc have similar
surface temperatures, and in that situation, the broadening function areas for
the two stars give an indication of the relative luminosities. We therefore
used our fits of the spectra in Table 1 to estimate the luminosity ratio of
the two stars in the visible wavelength range, complementing measurements in
the infrared from interferometry below. We restricted the rotational velocity
fitting to a $\pm 1\sigma$ range around the weighted averages determined
above, and redid the fits for the BF heights in order to derive the areas. The
results are given in Table 1, where the average ratio (with the error in the
mean) is $0.622\pm 0.010$. This is in agreement with the infrared ratios (see
Table 3), supporting the idea of very similar surface temperatures.

###  2.2 Interferometry

Although lunar occultations of $\epsilon$ Cnc have been observed several times
(Peterson & White, 1984; Peterson et al., 1989), the system was not previously
resolved. As we will see from the modeling below, this is largely due to a
confluence of aggravating factors: a high inclination, an orientation of the
orbit that strongly foreshortens the major axis, and a mostly north-south
alignment of the minor axis (putting it fairly close to perpendicular to the
ecliptic). These issues have made interferometric observations necessary to
resolve the orbit.

Our interferometric observations were obtained from the Center for High
Angular Resolution Astronomy (CHARA) at Mt. Wilson Observatory using two
different beam combiners called CLIMB and MIRC-X (ten Brummelaar et al.,
2005). The CLIMB beam combiner is a broad-band, single spectral channel
instrument that optimizes sensitivity, and is a three-beam expansion of the
CLASSIC two-beam combiner (Ten Brummelaar et al., 2013). The MIRC-X instrument
is an image-plane combiner that simultaneously combines the light from up to
six telescopes, and works well on faint binary companions (Anugu et al.,
2020). In the low-resolution ($R=50$) configuration we used, MIRC-X observes
in eight different spectral channels (Kraus et al., 2018). The CLIMB data were
reduced using the pipeline developed by J.D. Monnier, with the general method
described in Monnier et al. (2011) and extended to three-beams (e.g., Kluska
et al. 2018). The MIRC-X data were reduced using the standard MIRC-X pipeline
version 1.3.5 111 https://gitlab.chara.gsu.edu/lebouquj/mircx_pipeline.

The log of the observations is given in Table 2, along with the number of
fringe visibility (two-telescope interference) and closure phase (derived from
the phase shifts in the fringe patterns for three-telescope groups)
measurements. Fringe visibilities as a function of projected baseline provide
information about the separation and orientation of the two stars in the
binary, as well as their relative brightnesses. Visibilities are calibrated
using observations on the same night of unresolved single stars. The
calibrators are listed in the table, and the observed visibilities were
compared to predicted visibilities (based on model angular diameters for a
uniformly illuminated disk; Bourges et al. 2017a) to derive the normalization
for $\epsilon$ Cnc observations. The assumed diameters for the listed
calibrators can be found in the JMMC Stellar Diameters Catalog V2 (Bourges et
al., 2017b). To ensure that the calibrators are unresolved and fringe
visibility is at its maximum for atmospheric conditions, those with diameters
less than 0.5 mas were chosen. In almost all cases, calibration observations
were taken before and after observations of $\epsilon$ Cnc. In the CLIMB
observations, visibilities were measured simultaneously in several combined
output beams for the same pairs of telescopes, and we treated the resulting
visibility measurements as independent. Closure phases are observables that
provide information about asymmetry in the light distribution, and can provide
more powerful measures of the luminosity ratio for the stars in a binary.

Our CLIMB observations were conducted in $K$ band, and our MIRC-X observations
in $H$ band (in 8 subchannels), so they give us relative luminosity
information for the component stars in the two wavelength regions. Most of our
CLIMB observations were affected by marginal weather that made position
measurements difficult, but the interferometry still provides observational
limits of value to the orbit determination.

Table 2: Log of Interferometric Observations of $\epsilon$ Cnc UT Date | mJD StartaamJD = HJD - 2400000 at start of observation sequence. | Combiner | $N_{tel}$ | $N_{vis}$ | $N_{clos}$ | Calibrators
---|---|---|---|---|---|---
2017 Jan 31 | 57784.77 | CLIMB | 3 | 9 | 3 | HD 73785, 73819
2018 Feb 05 | 58154.76 | CLIMB | 3 | 42 | 6 | HD 72779, 74379
2018 Feb 06 | 58155.86 | CLIMB | 3 | 41 | 6 | HD 72779, 74379
2018 Nov 19 | 58441.99 | CLIMB | 3 | 21 | 3 | HD 72779
2018 Nov 25 | 58448.02 | CLIMB | 3 | 14 | 2 | HD 72779
2018 Dec 13 | 58465.94 | CLIMB | 3 | 28 | 4 | HD 73344, 73533, 74379
2018 Dec 24 | 58476.85 | CLIMB | 3 | 26 | 4 | HD 73344
2019 Nov 08 | 58795.85 | MIRC-X | 5 | 512 | 448 | HD 72779, 73533, 74379
2019 Dec 13 | 58830.98 | MIRC-X | 5 | 272 | 184 | HD 72779
2021 Feb 24 | 59269.78 | MIRC-X | 5 | 320 | 320 | HD 72779, 73533
2021 Mar 25 | 59298.75 | MIRC-X | 5 | 320 | 320 | HD 72779
2021 Mar 29 | 59302.69 | MIRC-X | 6 | 480 | 640 | HD 72779, 74379, 82394
2021 Apr 03 | 59307.67 | MIRC-X | 6 | 480 | 640 | HD 72779, 82394, 74379
Total |  |  |  | 2496 | 2570 |

###  2.3 Spectral Energy Distributions

$\epsilon$ Cnc does not have eclipses, and so to get additional information on
the component stars, we need to utilize photometry. There is an abundance of
calibrated photometry for the binary that we use below to characterize the
spectral energy distribution (SED) of the stars.

Ultraviolet: The ultraviolet is particularly important for characterizing
A-type stars at the turnoff of the Praesepe cluster. We obtained photometry
from the Galaxy Evolution Explorer (GALEX; Martin et al. 2005) archive in the
FUV ($1344-1786$ Å) passbands. The cluster was imaged in the FUV for 3329 s as
part of a guest investigator program (GI1 proposal 48, P.I. M. Burleigh).
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 magnitudes ($m_{FUV}=18.82$) and reference
fluxes ($1.40\times 10^{-15}$ erg s-1 cm2 Å-1, respectively) to convert to
flux.

The cluster was observed by the Astronomical Netherlands Satellite (ANS) in
five ultraviolet bands centered at 1550, 1800, 2200, 2500, and 3300 Å.
Magnitudes and measurement errors were taken from Wesselius et al. (1982), and
are directly related to monochromatic fluxes. The absolute flux calibration is
described in de Boer & Wesselius (1980).

Optical, Near Ultraviolet, and Near Infrared: There are several sources for
narrow- and broad-band photometry in and near the optical range. We obtained
narrow-band Strömgren $uvby$ photometry for cluster stars from Paunzen (2015),
and we employed reference fluxes from Gray (1998) to convert the magnitudes to
fluxes. Kazlauskas et al. (2006) presented seven-filter Vilnius photometry of
cluster stars, and we converted the reported magnitudes to fluxes using the
zeropoints from Mann & von Braun (2015). Observations in the Geneva system
were taken from the Catalogue of Stars Measured in the Geneva Observatory
Photometric System (Rufener, 1988, 4th edition). Observations in the Johnson
13-color system can be found in Johnson & Mitchell (1975), and reference
fluxes for the conversion from magnitudes to fluxes were taken from the
Spanish Virtual Observatory (SVO) Filter Profile Service (Rodrigo et al.,
2012; Rodrigo & Solano, 2020). Additionally, there are spectrophotometic
observations of many bright Praesepe stars in 16 very narrow passbands in
Clampitt & Burstein (1997). These observations are tabulated as colors
relative to measurements in their 5556 Å filter.

Photoelectric photometry in Johnson $UBVRI$ filters were taken from Morel &
Magnenat (1978) and the APASS survey (Henden, 2019, Data Release 10), which
also included Sloan $g^{\prime}r^{\prime}i^{\prime}$ filters. The $UBVRI$
measurements were converted to fluxes using reference magnitudes from Table A2
of Bessell et al. (1998), accounting for the known reversal of the zero point
correction rows for $f_{\lambda}$ and $f_{\nu}$. Observations on the four-
color $WBVR$ system came from Kornilov et al. (1991), with flux calibration
again making use of Mann & von Braun (2015) references. Photometry in the
Tycho filters $B_{T}$ and $V_{T}$ was taken from the Tycho Reference Catalogue
(Hog et al., 1998), with absolute flux calibration information taken from Mann
& von Braun (2015).

Due to the brightness of the binary and other cluster members, there are only
$u$, $g$, and $r$ measurements in the Sloan Digital Sky Survey (SDSS; York et
al. 2000). We used PSF magnitudes from Data Release 14, 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
APASS $g^{\prime}r^{\prime}i^{\prime}$ filters are on an AB system similar to
the SDSS, and were calibrated the same way.

Finally, Gaia has already produced high-precision photometry extending far
down the main sequence of the cluster as part of Early Data Release 3\. We
obtained the fluxes in the $G$, $G_{\rm BP}$, and $G_{\rm RP}$ bands from the
Gaia Archive.

Infrared: We have 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). We also used photometry in all four bands ($W1$, $W2$, $W3$,
$W4$) from the Wide Field Infrared Explorer (WISE; Wright et al. 2010), which
were also converted to fluxes using tabulated reference fluxes at zero
magnitude. An additional measurement in the S9W filter at 8.22 $\mu$m was
taken from AKARI satellite IRC all-sky survey data (Ishihara et al., 2010).

Some clues to the CMD positions of the two stars in the binary come from
luminosity ratios derived from the broadening function fits in the visible
part of the spectrum (see section 2.1.2) and from the interferometry in the
infrared $H$ and $K$ bands. These ratios all return values near 0.6, implying
very similar temperatures and clearly restricting the stars to the upper part
of the main sequence. If the spectroscopically-measured luminosity ratio
applies to the Gaia $G$ magnitude, then the two stars of $\epsilon$ Cnc should
have $G=6.784\pm 0.007$ and $G=7.324\pm 0.011$. The primary should therefore
be among the stars at the bright end of the CMD gap ($6.8<G<7.4$), and the
secondary should be approximately 0.2 mag brighter than the faint end of the
gap.

To put further limits the components of the binary, we can check whether
existing cluster members can act as proxies, such that the binary’s SED is
reproduced when the SEDs of the two stars are added together. To find the best
proxy for $\epsilon$ Cnc A, we considered the five cluster objects that are
closest in magnitude to $\epsilon$ Cnc A (HD 73210, 73575, 73712, 73785,
73819; see Fig. 2). As discussed in Appendix A.1, we first looked for signs of
binarity, which would affect the CMD position. The two redder stars (HD 73575
and the possible triple HD 73712) are too red to reproduce the color of
$\epsilon$ Cnc in combination with a main sequence companion. One of the
remaining stars grouped near $(G_{BP}-G_{RP})=0.27$ (HD 73210) is a double-
lined spectroscopic binary with a fairly extreme mass ratio ($q\approx 0.35$;
Abt & Willmarth 1999). The final two stars (HD 73785 and 73819) are the best
candidates for a proxy star.

To find the best proxy for $\epsilon$ Cnc B, we examined stars at the bright
end of the main sequence. We first checked for binarity, and considered
whether they could be blue stragglers that are beyond the end of the main
sequence in this cluster. The three brightest main sequence stars (HD 72846,
72942, and 73711) are discussed in Appendix A.1. Two of the stars (HD 72942
and 73711) show chemical peculiarities that are often associated with
binarity, but only HD 72942 has a clearly detected companion.

We added the SEDs of different bright and faint proxy star candidates together
to see if they could reproduce the SED of the binary, and found that the
combination of HD 73819 and 73711 agreed best. Because the spectroscopic
luminosity ratio implies that the fainter component of $\epsilon$ Cnc B is
brighter than any of the cluster turnoff stars, we scaled the SED of HD 73711
upward by 10% to account for this. Figure 2 shows the best match. The lower
panel shows that we are able to reproduce the binary’s SED to within about 5%
in filter bands stretching from the ultraviolet far into the infrared. Redder
stars at the brightness level of $\epsilon$ Cnc A produce stronger mismatches
as a function of wavelength, and can be ruled out. We believe this provides
sufficient localization of the two stars to make some use of CMD positions as
part of our analysis of $\epsilon$ Cnc.

Figure 2: Top: SED for the binary $\epsilon$ Cnc from photometry (red points),
and for the best-fit main sequence stars (HD 73819 with blue, and HD 73711
with green points increased in flux by a factor of 1.1). Horizontal error bars
represent the effective width of the filter. Bottom: Fractional difference
between the binary star photometry fluxes and the best-fit sum of main
sequence stars.

As shown in Figure 3, the brighter of the proxy stars (EP Cnc or HD 73819)
falls among a few stars that appear to be single stars near core hydrogen
exhaustion, while the fainter proxy star (HD 73711222Coincidentally, HD 73711
is cited as a distant companion to $\epsilon$ Cnc in the Washington Double
Star Catalog (Mason et al., 2001). We cannot ascribe any significance to the
apparent similarity between HD 73711 and the fainter component of $\epsilon$
Cnc though.) appears to be close to the populated bright end of the main
sequence. Clustering of stars in the CMD may indicate an evolutionary
slowdown. Thus there is an elevated probability that the primary star of
$\epsilon$ Cnc would be in such a phase.

Figure 3: Gaia EDR3 color-magnitude diagram for bright Praesepe members.
$\epsilon$ Cnc is shown with a large red pentagon, the best-fit main sequence
stars (HD 73819 and HD 73711) are outlined in green, the blue straggler
candidate HD 73666 is a large blue point, likely single faint main sequence
stars are small blue points, faint binary candidates are small black points,
and stars without measured rotational velocities are large black points.
Otherwise the bright stars are color coded by rotational velocity. $\times$
markers indicate binary systems. The green horizontal lines are inferred
brightnesses for $\epsilon$ Cnc A and B.

To get an indication of the properties of $\epsilon$ Cnc A, we fit the
photometric SED of HD 73819 with ATLAS9 models (Castelli & Kurucz,
2003)333Models were calculated using the ATLAS9 fortran code that employed
updated 2015 linelists and used temperatures between the published
gridpoints., as shown in Fig. 4. We used models with fixed values
[Fe/H]$=+0.2$ and $\log g=3.6$, where the surface gravity was chosen from a
MIST evolutionary track because there is some sensitivity to gravity in the
ultraviolet shortward of the Balmer jump. The best fit temperature (8060 K)
was chosen based on the minimization of trends in the fractional residuals
(bottom panel) as a function of wavelength. The fit returns a bolometric flux
$F_{bol}=5.39\times 10^{-8}$ erg cm-2 s-1 when corrected for wavelength-
dependent reddening using the relations from Cardelli et al. (1989). At the
distance of Praesepe, this corresponds to a luminosity $L=59.8L_{\odot}$.
These values are likely to be fairly good representations of $\epsilon$ Cnc A
because HD 73819’s $G$ magnitude is a good match for the prediction from the
spectroscopic luminosity ratio and because the SED of $\epsilon$ Cnc is well
matched with HD 73819’s SED (and it is the biggest contributor to the SED).
There are likely to be systematic uncertainties involved, and for this reason,
we do not quote uncertainties on these values.

Figure 4: Top: SED for HD 73819 from photometry (red points), compared to an
ATLAS9 model with $T_{\rm eff}=8060$ K and $\log g=3.6$ and synthetic
photometry calculated from the model (green points). Horizontal error bars
represent the effective width of the filter. Bottom: Fractional difference
between the observed photometry fluxes and best-fit synthetic photometry.

For $\epsilon$ Cnc B, we fit the photometric SED of HD 73711 with ATLAS9
models fixing the values [Fe/H]$=+0.2$ and $\log g=4.0$, as shown in Fig. 5.
The star has less ultraviolet data, but the optical part of the spectrum is
still well-sampled. The best fit temperature was approximately 8330 K, with
bolometric flux $F_{bol}=2.73\times 10^{-8}$ erg cm-2 s-1. At Praesepe’s
distance, this corresponds to luminosity $L=30.3L_{\odot}$. In this case, the
luminosity is likely to be a significant underestimation of the luminosity of
$\epsilon$ Cnc B based on the spectroscopic luminosity ratio. Scaling the SED
of HD 73711 upward by 10% in all bands improves the matching of the $\epsilon$
Cnc SED when added with that of HD 73819.

Figure 5: Top: SED for HD 73711 from photometry (red points), compared to an
ATLAS9 model with $T_{\rm eff}=8330$ K and $\log g=4.0$ and synthetic
photometry calculated from the model (green points). Horizontal error bars
represent the effective width of the filter. Bottom: Fractional difference
between the observed photometry fluxes and best-fit synthetic photometry.

Using the bolometric fluxes and effective temperatures, we estimate the
angular diameters of the two stars using $\theta^{2}/4=F_{bol}/\sigma T_{\rm
eff}^{4}$. We find $\theta_{1}=0.194$ mas (from the data for HD 73819) and
$\theta_{2}=0.136$ mas (from the data for HD 73711, including the 10% flux
correction).

##  3 Binary Star Modeling

With the observational data in hand, the next task is to fit the orbit of
$\epsilon$ Cnc and determine its parameters. As a first step, we conducted
fits to the radial velocity data alone, partly to derive more realistic a
posteriori measurement uncertainties using scatter around the best fits models
and partly to look at velocity offsets between datasets. We used the Eclipsing
Light Curve code (ELC; Orosz & Hauschildt, 2000) to fit eleven parameters:
period $P$, reference time of a periastron passage $t_{P}$, orbit velocity
amplitude of the primary star $K_{1}$, mass ratio $q=M_{2}/M_{1}$,
eccentricity $e$, argument of periastron $\omega_{1}$, and system velocities
$\gamma_{i}$ for five different subsets of data (velocities from KPNO, OHP,
and NOT spectra; and literature velocities from Abt & Willmarth 1999 and Abt
1970). Based on the initial fit using a priori measurement errors, we scaled
the uncertainties on each velocity dataset to return a reduced $\chi^{2}$
value of approximately 1, and then we refitted. These results are reported in
Table 3. The system velocities were in reasonable agreement, but there
appeared to be significant offsets that we then corrected for before the joint
interferometry-spectroscopy fitting. (The NOT zeropoint was used as the
reference.) We find $\Delta\gamma(\mbox{KPNO}-\mbox{NOT})=-1.32\pm 0.08$ km
s-1, $\Delta\gamma(\mbox{OHP}-\mbox{NOT})=-0.48\pm 0.15$ km s-1,
$\Delta\gamma(\mbox{MWO}-\mbox{NOT})=-0.34\pm 1.09$ km s-1, and
$\Delta\gamma(\mbox{A\&W99}-\mbox{NOT})=-2.06\pm 0.15$ km s-1.

For the fits to the combined radial velocity and interferometry (visibility
amplitudes and closure phases) dataset, we developed a program (which we call
interfRVorbit). In this case, we use a set of thirteen parameters to model the
orbits: period $P$, reference time of a periastron passage $t_{P}$, orbit
velocity amplitudes of both stars $K_{1}$ and $K_{2}$, systemic velocities for
both stars $\gamma_{1}$ and $\gamma_{2}$, eccentricity $e$, argument of
periastron $\omega$, argument of the ascending node $\Omega$, inclination $i$,
angular size of the binary semi-major axis $a$, and brightness ratios in $H$
and $K_{s}$ bands. The interferometric observations do not have much leverage
on the unresolved angular diameters of the stars $\theta_{1}$ and
$\theta_{2}$, and so we fix these at the values calculated from the SEDs (in
the previous section). Observable quantities are then calculated from these
parameters through a forward model. In the case of the interferometry, the
complex visibility for a uniform disk of angular diameter $\theta$ is

$V=\frac{2J_{1}(\pi B\theta/\lambda)}{\pi B\theta/\lambda}$

where $J_{1}$ is the first-order Bessel function and $B$ is the projected
baseline of the interferometer at the star’s sky position. For a binary
system, the observable squared visibility is calculated from

$V^{2}=\frac{V_{1}^{2}+r^{2}V_{2}^{2}+2\left|V_{1}\right|\left|V_{2}\right|r\cos\left[2\pi\vec{B}\cdot\vec{s}/\lambda\right]}{(1+r)^{2}}$

where $V_{1}$ and $V_{2}$ are the complex visibilities of the individual
stars, $r$ is the luminosity ratio in the observed band, $\vec{s}$ is the
vector of the angular separation of the two stars on the sky, and $\vec{B}$ is
the baseline vector for the interferometer. For the fits, we have assumed
uniformly illuminated disks for both stars, and used a single luminosity ratio
for all observations within a bandpass (in our case, $H$ or $K_{s}$).
Bandpasses are broken into smaller parts
($n=\mbox{int}(200\Delta\lambda/\lambda)$, with $n\geq 2$) and the calculated
visibilities are averaged

$V_{band}^{2}=\frac{1}{n}\sum_{i=1}^{n}V_{i}^{2}$

The closure phase can be calculated as the product of the complex visibilities
$V$ measured by each pair of telescopes around a closed triangle of baselines.
All of the interferometric astrometry measurements are referenced to the
brighter star in the binary, so proper motion does not play any role in the
fits.

We search the parameter space using a genetic algorithm (Charbonneau, 1995).
The goodness of the fit is based on the total $\chi^{2}$ value from comparing
model predictions to observed data. After an initial optimization to find a
best fit using a priori measurement uncertainties, we scaled the uncertainties
for each observational dataset (interferometric visibilities, closure phases,
and radial velocities separated by measured star and spectroscopic source) to
return a reduced $\chi^{2}$ of approximately 1. In this way, we make the
datapoint uncertainties consistent with the observed scatter around the best-
fit model and attempt to produce appropriate weightings for the different data
types in subsequent fits.

We subsequently conducted a 20000 generation run (with approximately 100 model
members per generation) to seek the best-fit model and to derive the parameter
uncertainties from trial models near the minimum. The best-fit parameters are
given in Table 3, where the spectroscopic orbit from Abt & Willmarth (1999) is
also provided for comparison. Uncertainty estimates were generated from the
full parameter range covered by models with $\chi^{2}$ within 1 of the minimum
(Avni, 1976).

Table 3: Orbit Fitting Results for $\epsilon$ Cnc

| Interf. & RVs (Equal L ratios) | RVs Only | A&W99
---|---|---|---
Parameter |  | $\sigma$ |  | $\sigma$ |  | $\sigma$
$P$ (d) | 35.14101 | 0.00005 | 35.14113 | 0.00012 | 35.202 | 0.033
$t_{P}-~{}2400000$ | 48314.598 | 0.016 | 48314.56 | 0.04 | 48313.5 | 0.7
$\gamma_{1}$ (km s-1) | 34.71 | 0.03 | 34.78aamJD = HJD $-2400000$ | 0.07 | 29.9 | 1.1
$K_{1}$ (km s-1) | 56.60 | 0.03 | 56.53 | 0.08 | 53.0 | 1.9
$K_{2}$ (km s-1) | 61.55 | 0.10 | 61.50 | 0.08 | 67.8 | 3.9
$q=M_{2}/M_{1}$ | 0.9196 | 0.0015 | 0.9192 | 0.0005 | 0.78bbThe published value in Abt & Willmarth (1999) does not follow from their measured $K_{1}$ and $K_{2}$, and we recalculate it here. |
$e$ | 0.4195 | 0.0003 | 0.4165 | 0.0021 | 0.32 | 0.04
$\omega_{1}(^{\circ})$ | 258.38 | 0.02 | 258.0 | 0.2 | 265 | 5
$\gamma_{2}$ (km s-1) | 34.88 | 0.09 |  |  |  |
$\Omega(^{\circ})$ | 356.069 | 0.014 |  |  |  |
$i(^{\circ})$ | 81.454 | 0.010 |  |  |  |
$L_{2}/L_{1}(H)$ | 0.61200 | 0.0007 |  |  |  |
$a$ (mas) | 1.9127 | 0.0004 (0.0047ccSystematic error due to absolute wavelength calibration uncertainty for MIRC-X instrument (Monnier et al., 2012).) |  |  |  |
$M_{1}/M_{\odot}$ | 2.420 | 0.008 |  |  |  |
$M_{2}/M_{\odot}$ | 2.226 | 0.004 |  |  |  |
aamJD = HJD $-2400000$

Value derived from measured Abt & Willmarth (1999) spectra.

Figure 6 compares the radial velocity measurements with the best-fit model as
a function of orbital phase. The eccentricity of the orbits are clearly seen.
Although we allowed the systematic velocities for the two stars to be
different to account for differences in gravitational redshifts or convective
blueshifts between the two stars, the velocities disagree by only about 0.2 km
s${}^{-1}\;$.

The best fitting model of the binary system’s orbit on the sky is shown in
Figure 7. Although we specifically fit interferometric observables (fringe
visibilities and closure phases), in the figure we show relative position
measurements for the epochs when they could be derived. These position
measurements are given in Table 4 and were computed using a binary grid search
procedure444https://www.chara.gsu.edu/analysis-software/binary-grid-search;
Schaefer et al. 2016. The position uncertainties include 0.5% uncertainty in
the MIRC-X wavelength scale (Anugu et al., 2020). These position measurements
are given in Table 4. The comparisons between interferometric observables and
model predictions are shown in Figures 9 and 10. For perspective, the closure
phases will be near zero (or $180\arcdeg$) for point-symmetric sources, but
the maximum excursions away from zero are good indicators of the luminosity
ratio of the stars in the binary. (Larger deviations reflect luminosity ratios
closer to 1.)

Figure 6: Radial velocity versus orbital phase for the components of $\epsilon$ Cnc. Star symbols represent the primary star and circles represent the secondary star. Figure 7:  Sky orbit for $\epsilon$ Cnc B relative to $\epsilon$ Cnc A (marked with an $\times$). Black points mark the model orbital position during the measurements with the CLIMB beam combiner at CHARA. Blue points are position measurements from the MIRC-X beam combiner. Red points are position measurements from the CLIMB beam combiner (triangles are values closest to the predicted position, and not the global best binary fit). Red ellipses show $5\sigma$ uncertainties. Secondary star motion is counterclockwise in this diagram. The inset panels are zooms on observations taken near the extremes of the orbit. In the insets, blue dashed lines connect the measured position to the model’s predicted position.  Figure 8: Squared visibilities as a function of interferometric baseline for the CLIMB data, separated by date of observation. Black dots are the predicted values and the colored dots are the observed values. Different colors in the panels represent measurements from different telescope pairs. Figure 9: Squared visibilities as a function of interferometic baseline for the MIRC-X data, separated by date of observation. Black dots are the predicted values and the colored dots are the observed values. Different colors in the panels represent measurements from different telescope pairs. Figure 10: Closure phases ($\Phi$) as a function of the interferometric baseline for CLIMB and MIRC-X data, separated by date of observation. Here the baseline is the sum of the baseline lengths for the first two legs of the triangle of telescopes. Black dots are the predicted values, and colored dots are observed values on different dates (CLIMB) or from different telescope pairs (MIRC-X). Table 4: Interferometric Relative Position Measurements for $\epsilon$ Cnc mJDaamJD = HJD $-2400000$ | $\rho$ | $\theta$ |  $\sigma_{maj}$bbThe size of the error ellipse major and minor axes, and the orientation $\varphi$ of the major axis relative to north. |  $\sigma_{min}$bbThe size of the error ellipse major and minor axes, and the orientation $\varphi$ of the major axis relative to north. |  $\varphi$bbThe size of the error ellipse major and minor axes, and the orientation $\varphi$ of the major axis relative to north. | $f$ | $\sigma_{f}$
---|---|---|---|---|---|---|---
| (mas) | (deg) | (mas) | (mas) | (deg) | $L_{1}/(L_{1}+L_{2})$ |
CLIMB CombinerccCLIMB entries give the binary fit positions most consistent
with the orbit.
57784.826 | 0.7313 | 30.61 | 0.0138 | 0.0091 | 151.5 | 0.619 | 0.008
58154.878 | 0.2027 | 292.23 | 0.0364 | 0.0208 | 169.4 | 0.620ddLuminosity ratio was assumed on nights when the interferometry did not give a reliable independent measurement. |
58155.860 | 0.6974 | 333.17 | 0.1142 | 0.0372 | 158.4 | 0.619 | 0.122
58441.988 | 1.9026 | 358.87 | 0.0543 | 0.0159 | 160.9 | 0.620ddLuminosity ratio was assumed on nights when the interferometry did not give a reliable independent measurement. |
58448.033 | 1.4136 | 8.84 | 0.0283 | 0.0125 | 163.5 | 0.620ddLuminosity ratio was assumed on nights when the interferometry did not give a reliable independent measurement. |
58465.939 | 1.4492 | 176.73 | 0.0249 | 0.0183 | 179.3 | 0.620ddLuminosity ratio was assumed on nights when the interferometry did not give a reliable independent measurement. |
58476.882 | 2.0120 | 358.87 | 0.0422 | 0.0104 | 165.7 | 0.620ddLuminosity ratio was assumed on nights when the interferometry did not give a reliable independent measurement. |
MIRC-X Combiner
58795.986 | 1.8559 | 2.28 | 0.0093 | 0.0041 | 132.3 | 0.613 | 0.001
58796.057 | 1.7933 | 359.35 | 0.0414 | 0.0087 | 55.1 | 0.572 | 0.052
58830.984 | 1.8869 | 1.96 | 0.0101 | 0.0082 | 112.3 | 0.617 | 0.008
59269.783 | 1.2874 | 164.14 | 0.0129 | 0.0057 | 66.2 | 0.617 | 0.012
59298.759 | 0.3933 | 87.66 | 0.0129 | 0.0085 | 46.6 | 0.641 | 0.021
59302.695 | 0.9283 | 154.57 | 0.0048 | 0.0023 | 113.6 | 0.617 | 0.002
59307.677 | 1.5710 | 170.74 | 0.0022 | 0.0011 | 138.5 | 0.608 | 0.011

When we allow the $H$ and $K$ band luminosity ratios to be fit independently,
the ratios are in rough agreement, but differ significantly. Because both
filters sample the long-wavelength side of the SEDs of these hot stars, it is
to be expected that the ratios would be nearly identical. We therefore
conducted a second fitting run where we forced the $H$ and $K$ luminosity
ratios to be the same. Predictably, the final ratio came out very close to the
previous $H$ ratio due to the much greater number of interferometric data in
that band. It does also lead to a slight shift in the fitted orbit, however,
because the choice affects the observational limits from the CLIMB data.

The masses of the primary and secondary stars were computed from the orbital
parameters for each run, and we compute $1\sigma$ uncertainties on the masses
using the range of masses producing $\chi^{2}$ values within 1 of the minimum
value.

Using the period $P$, orbit velocity amplitudes $K_{1}$ and $K_{2}$, and
inclination $i$ from Table 3, we can calculate the true semi-major axis of the
orbit. Used along with the angular size, we get a distance of $183.1\pm 0.2\pm
0.5$ pc.

(The second quoted error is systematic, deriving from uncertainty in absolute
wavelength calibration of the MIRC-X instrument.) This is in good agreement
with the geometric distance calculated using the Gaia EDR3 parallax for
$\epsilon$ Cnc ($d=184.4^{+1.8}_{-1.4}$ pc; Bailer-Jones et al. 2021).

##  4 Discussion

We can now use the estimated photometric characteristics of the stars of
$\epsilon$ Cnc along with the interferometrically measured masses and radii to
compare with models. One goal of these comparisons is to derive an age as
accurate as possible for the cluster, but we must be wary of the possibility
of physics errors in the modeling codes that introduce systematic errors into
derived ages. As a result, it is important to examine the cluster stars more
broadly in order to validate the coded physics. For our purposes, the
treatment of convective cores is critical, as the extent of the core in a star
directly determines the amount of fuel available for the main sequence phase.
The masses for both stars in $\epsilon$ Cnc put them in the plateau of the
amount of convective core overshooting embedded into codes, where a fixed
overshooting distance (often in terms of pressure scale height $H_{P}$) is
implemented. At lower masses (usually less than $2M_{\odot}$), a ”ramp” is
enforced in which the amount of core overshooting is decreased linearly with
decreasing mass down to zero for stars that no longer have convective cores.
So in principle, the stars of $\epsilon$ Cnc avoid uncertainties related to
the implementation of the ramp, but they remain a tough test of convective
overshooting because they are stars with precise masses at or near the end of
their main sequence lives, where the convective core history produces its most
noticeable effects.

In Table 5, we summarize photometry, rotational velocity, and multiplicity
information for the brightest stars in Praesepe. We primarily used Gaia EDR3
information on parallaxes, positions, and proper motions to construct a likely
member list. We selected stars based on a position within $5\arcdeg$ of
$(\alpha,\delta)=(130.10,19.667)$, a proper motion vector within 6 mas yr-1 of
$(\mu_{\alpha},\mu_{\delta})=(-36.047,-12.917)$, a parallax within 0.4 mas of
5.371 mas (Gaia Collaboration et al., 2018a). Because there is evidence of
tidal tails stretching much farther from the cluster center, we examined the
membership list of Röser & Schilbach (2019) for additional bright members.
While Röser & Schilbach do identify some bright high-probability members in
the tidal tails, we find that nearly all of them have radial velocities
inconsistent with cluster membership (deviating from the cluster mean by 10 km
s${}^{-1}\;$or more). All of the stars in the table have systematic radial
velocities within 5 km s${}^{-1}\;$of the cluster mean (sources: Abt &
Willmarth 1999; Raboud & Mermilliod 1998; de Bruijne & Eilers 2012; Yang et
al. 2015b; Cummings et al. 2017).

Table 5: Rotational Velocities and Multiplicity of Bright Stars ($G<9.0$) in
Praesepe

KW | HD/BD | $G$ |  $v\sin i$ (km/s) | Ref. | Multiplicity | Ref. | Notes
---|---|---|---|---|---|---|---
253 | 73665 | 6.15 | $<10$ | 3 | No | 3,14 | RC
283 | 73710 | 6.18 | $<10$ | 3 | No | 3,14,15 | RC
212 | 73598 | 6.39 | $<10$ | 3 | speck,occ | 14,16 | RGB?
| 72779 | 6.42 | 99 | 3 | No | 3,14 | SGB
428 | 73974 | 6.66 | $<10$ | 3 | SB1O,occ | 3,15 | SGB
265 | 73666 | 6.58 | 9/10 | 3,5 | No/occ | 3/14 | BSS
204 | 73575 | 6.61 | 150/129/127/140 | 1,3,5,11 | No | 3,14,15,16 |  $\delta$ Sct
50 | 73210 | 6.70 | 80/90/90,80 | 1,2,3 | SB2O | 3 |
284 | 73712 | 6.72 | 56:,61: | 3 | SB2O | 3,14,16 |  $\delta$ Sct, triple?
348 | 73819 | 6.74 | 152/180/79 | 1,3,11 | No | 3,14,15,16 |  $\delta$ Sct
328 | 73785 | 6.80 | 85/91 | 1,3 | No/SB2? | 3,15,16/7 |
150 | 73449 | 7.38 | 235/229 | 1,3 | No/SB? | 3,14,15/7 |  $\delta$ Sct
534 | 72942 | 7.46 | 62/73 | 1,5 | SB1O | 3 | A/Am hybrid
| 72846 | 7.46 | $119\pm 6$ | 6 | No? | 14 |
229 | 73619 | 7.50 | 20,18/$11.2\pm 0.5$ | 3,4 | SB2O | 3,7 | Am
276 | 73711 | 7.52 | 60/66/$66\pm 11$/62 | 1,3,4,5 | No/SB2? | 3,14,15/7,9 | Am
207 | 73576 | 7.63 | 200/204/$208\pm 11$ | 1,3,10 | No/SB1? | 3,14/7 |  $\delta$ Sct
279 | 73709 | 7.67 | 20/$17.3\pm 0.3$/10 | 3,4,5 | SB1O | 3,7 | quadruple,Am
40 | 73174 | 7.74 | 11/$7.3\pm 1.1$/$<$5 | 3,4,5 | SB1O,SB1O | 3 | triple,Am
323 | 73763 | 7.77 | $200\pm 3$ | 10 | No? | 14 |  $\delta$ Sct
449 | 74050 | 7.86 | 150/$188\pm 8$/145 | 1,6,11 | No? | 14 |  $\delta$ Sct
445 | 74028 | 7.92 | 160/$150\pm 9$ | 1,6 | No? | 14,15,16 |  $\delta$ Sct
203 | 73574 | 7.92 | 108/108/$102\pm 4$ | 1,3,6 | No/bin | 3/14,15 |
286 | 73730 | 7.98 | 30/$34.4\pm 3.4$/29/35 | 1,4,5,8 | No? | 14,15 | Am
| 74656 | 7.99 | $25\pm 1$ | 6 | No? | 13 | Am
385 | 73890 | 8.00 | 165/141 | 1,11 | No? | 14 |  $\delta$ Sct
114 | 73345 | 8.11 | 98/85/$85\pm 4$/90 | 1,2,6,8 | No? | 14,15 |  $\delta$ Sct
| 74740 | 8.16 |  |  | No? | 12,13 |
45 | 73175 | 8.21 | 180/$163\pm 9$/175 | 1,6,8 | No? | 14 |  $\delta$ Sct
143 | 73430 | 8.28 | 82/73/80 | 1,5,8 | No?/occ | 14/15 |
375 | 73872 | 8.29 | 180/135 | 1,8 | No?/SB1? | 14,15/7 |  $\delta$ Sct
| 74718 | 8.33 | $155\pm 7$ | 6 | No? | 17 |
| 72757 | 8.37 | $179\pm 10$ | 6 |  |  |
| 74720 | 8.38 |  |  | speck | 18 | no rv measure?
| 74589 | 8.40 | $127\pm 5$ | 6 | No? | 13 | pulsator
340 | 73798 | 8.43 | 175/$200\pm 11$/200 | 1,6,8 | No? | 14 |  $\delta$ Sct
154 | 73450 | 8.45 | 138/140/$138\pm 7$/166 | 1,2,6,8 | No? | 14,15 |  $\delta$ Sct
| 74587 | 8.46 | $90\pm 4$ | 6 | No? | 13 |  $\delta$ Sct
429 | 73993 | 8.48 | 200/$240\pm 14$/210 | 1,6,8 | No? | 14 |
318 | 73746 | 8.60 | 110/87/$95\pm 4$/95 | 1,2,6,8 | No? | 14 |  $\delta$ Sct
38 | 73161 | 8.63 | 160/190 | 1,8 | No? | 14,15 |
350 | 73818 | 8.65 | 85/81/66/65 | 1,4,5,8 | SB1 | 5 | Am
| 70297 | 8.70 |  |  |  |  | no rv measure?
271 | +20 2161 | 8.76 | 85/69/78 | 1,2,8 | No? | 14 |
| 74135 | 8.77 | $100\pm 4$ | 6 |  |  | pulsator
226 | 73616 | 8.83 | 125/110/115 | 1,2,8 | No? | 14,15 | pulsator
124 | 73397 | 8.92 | 100/86/91 | 1,2,8 | SB1O | 7 | pulsator
370 | 73854 | 8.95 | 137/68/70 | 1,2,8 | No? | 14,15 | pulsator
| 73620 | 9.04 |  |  |  |  |

††footnotetext: NOTE: KW identifications are from Klein Wassink (1927).
References: 1: McGee et al. (1967). 2: Rachford (1998). 3: Abt & Willmarth
(1999). 4: Debernardi et al. (2000). 5: Fossati et al. (2007). 6: Fossati et
al. (2008). 7: Raboud & Mermilliod (1998). 8. Cummings et al. (2017). 9:
Burkhart & Coupry (1998). 10: Royer et al. (2002). 11: Bush & Hintz (2008).
12. Evans et al. (1985). 13. Grenier et al. (1999). 14. Mason et al. (1993a).
15. Peterson & White (1984). 16. Peterson et al. (1989). 17. Evans & Edwards
(1983). 18. Horch et al. (2017).

Figures 11 – 13 show comparisons between theoretical isochrones and Gaia
photometry for the cluster. In Figures 11 and 12, evolution tracks for the
measured masses of the $\epsilon$ Cnc stars are shown on top of the
isochrones. Our most important limits come from the luminosity ratios and the
total photometry for the binary. While metallicity affects the luminosity of
the stars in the last stages of core hydrogen burning (in the sense that more
metal poor models are more luminous), the amount of convective core
overshooting appears to be an important factor in the models shown here.
PARSEC models have the largest amount of core overshooting ($\sim 0.25H_{P}$
for $M>1.5M_{\odot}$) among the models examined here, and the predicted
luminosities of the $\epsilon$ Cnc components appear to be too high to
reproduce $\epsilon$ Cnc. (Alternately, if the luminosities are forced to
match the observations, the primary star is forced to be too red, and the
color of the binary can’t be matched.) The MIST and BaSTI-IAC evolution tracks
for the masses of the $\epsilon$ Cnc components pass through the observed
magnitude levels toward the end of core hydrogen burning, which occur near the
red kinks in each track. The redward end of the kink approximately matches the
CMD positions of the bright single main sequence stars in Praesepe as well. As
shown in Figure 14, MIST models match the inferred $G$ magnitudes of both
components to within the uncertainties in the range $632-641$ Myr if the
masses are as measured and the cluster metallicity roughly matches the value
from the literature ([Fe/H]$=+0.14$). When the model photometry values are
combined, the measured Gaia photometry for the system is also matched.

Uncertainties on the measured masses and cluster metallicity extend the range
of acceptable ages. Lower primary star masses and higher metallicity models
require older ages. The primary star mass uncertainty results in about 15 Myr
of age uncertainty. At the ends of the mass and metallicity uncertainty
ranges, color changes in the primary star lead to unsatisfactory matches to
the observed color of the binary. This is a stronger effect for the
metallicity, with [Fe/H] values more than about $\pm 0.03$ dex (approximately
$0.5\sigma$) away from the spectroscopic value producing stellar colors that
are too blue or red. A $0.5\sigma$ uncertainty in metallicity results in an
age uncertainty of about 10 Myr. Adding these uncertainties in quadrature, we
therefore quote an age uncertainty of 19 Myr. Perhaps more importantly than
the statistical uncertainty, we believe the observational limits reduce
systematic errors in the age by ruling out some isochrone sets (and the model
physics they utilize) as unable to reproduce the characteristics of $\epsilon$
Cnc. Smaller changes to the amount of convective core overshooting could still
produce model stars consistent with the characteristics of the components of
$\epsilon$ Cnc at different ages (with larger amounts of overshooting
resulting in younger ages). We also note that the initial He abundance of
these metal-rich Praesepe stars carries some uncertainties that affect the
detailed comparisons with models, including the color of the main sequence and
the main sequence lifetime of turnoff stars. (For a discussion of this issue
in the even more metal-rich cluster NGC 6791, see Brogaard et al. (2011,
2012).) Evaluation of these contributions to the age uncertainty is beyond the
scope of this study, however.

A MIST isochrone of approximately 635 Myr and the spectroscopically-measured
cluster metallicity can simultaneously match the blue edge of the distribution
of bright main sequence stars, the red extent of the red kink at the cluster
turnoff, and the magnitudes of the components of $\epsilon$ Cnc at their
measured masses.

Figure 11: MIST isochrones (Choi et al., 2016) for ages 600 (red), 650 (cyan),
700 (yellow), and 750 (orange) Myrs. MIST evolutionary tracks are also shown
(black) for masses of the primary and secondary star. ($M_{p}=2.420M_{\odot}$,
$M_{s}=2.226M_{\odot}$). $\times$ markers indicate binary systems. To graph
the evolution tracks and isochrones, a distance modulus $(m-M)_{0}=6.33$,
initial rotation $v/v_{crit}=0$, iron abundance [Fe/H]$=+0.16$, and reddening
$E(G_{BP}-G_{RP})=0.073$ were used.  Figure 12: BaSTI-IAC isochrones (Hidalgo
et al., 2018) for ages 600 (red), 650 (cyan), 700 (yellow), and 750 (orange)
Myrs. BaSTI-IAC evolutionary tracks are also shown (black) for masses of the
primary and secondary stars ($M_{p}=2.420M_{\odot}$, $M_{s}=2.226M_{\odot}$).
$\times$ markers indicate binary systems. In the graph, distance modulus
$(m-M)_{0}=6.33$, iron abundance [Fe/H]$=+0.16$, and reddening
$E(G_{BP}-G_{RP})=0.073$ were used. Overshooting and diffusion were also
allowed under available grids.  Figure 13: PARSEC isochrones (Bressan et al.,
2012) for ages 600 (red), 650 (cyan), 700 (yellow), and 750 (orange) Myr. In
the graph, distance modulus $(m-M)_{0}=6.33$, iron abundance [Fe/H]$=+0.16$,
and reddening $E(G_{BP}-G_{RP})=0.073$ were used. $\times$ markers indicate
binary systems.The colored $\times$ markers indicate where our calculated
masses in table 3 land on the isochrone. Figure 14: MIST isochrones (Choi et
al., 2016) for ages 630 (green), 635 (light green), 640 (yellow), and 645
(orange) Myr. Distance modulus $(m-M)_{0}=6.33$, iron abundance
[Fe/H]$=+0.14$, and reddening $E(G_{BP}-G_{RP})=0.073$ were used. Colored dots
show the model positions of stars with the measured masses of the components
of $\epsilon$ Cnc, and blue rectangles show the $G$ magnitude and uncertainty
of the components as inferred from spectroscopy.

While it is gratifying that models reproduce several important features of the
CMD, a careful examination raises some questions. The isochrone does not match
the photometry of the star HD 72779 in the Hertzsprung gap. (See Appendix A.1
for evidence that the star is single, and representative of where the subgiant
portion of the isochrones should be.) This may indicate some mismatch in the
hydrogen abundance profile near the core at hydrogen exhaustion. The
distribution of stars in the range $6.5<G<7.5$ is also difficult to interpret
with conventional stellar models (Fig. 11-13). The stars at $G\approx 6.7$ and
7.5 are consistent with a single age of around 635 Myr, with the first group
of stars at the turnoff and the second group near the reddest point in the
isochrone’s kink (Fig. 11). According to the evolutionary timescales, both
parts of the isochrone should be fairly well populated. However, the range
$6.8<G<7.4$ should also be occupied as it should still contain stars burning
core hydrogen, and we find no stars in this range. Additionally, most of the
bright main sequence stars in Praesepe ($7.5<G<10$; $1.2<M_{G}<3.7$) reside
brighter or redder than the model isochrones, although the amount of
disagreement varies between model sets and can be affected by chemical
composition (metallicity or He abundance). In addition, there appears to be a
jag in Praesepe main sequence at $G\approx 9.1$ ($M_{G}\approx 2.8$), and
slightly fainter stars seem to deviate noticeably from the models. An
examination of the Hyades (below) does not give us clear evidence for a
similar feature. The Praesepe feature appears to be about a magnitude too
bright to be related to the Kraft break (Kraft, 1967) in stellar rotation
rates.

The Hyades cluster provides a useful comparison, as it has nearly identical
age and chemical composition, and a similar richness of stars. Bright Hyades
members are plotted in Figure 15, with the plot ranges chosen to cover the
ranges in the Praesepe plots. There are no stars found at the luminosity level
of the brighter group ($G\approx 6.7$ in Praesepe, $M_{G}\approx 0.4$), and a
handful of stars are scattered above the bright end of the main sequence in
Praesepe ($G\approx 7.5$, $M_{G}\approx 1.2$). Most of these bright stars have
shown some evidence of binarity in the literature, but we discuss in Appendix
A.2 why we believe the photometry for most of the stars with $M_{G}<1.6$ and
$(G_{BP}-G_{RP})<0.3$ is probably representative of single stars. Similarly to
the situation for Praesepe, there appear to be almost no stars present in what
should be the last stages of core hydrogen burning, between the turnoff and
the red kink.

Figure 15: Hyades members in an color-absolute magnitude diagram using Gaia
EDR3 photometry and parallaxes. Color coding for individual stars is based on
measured rotational velocity $v_{rot}\sin i$, and $\times$ symbols indicate
binary systems. The blue point is the blue straggler $\delta^{3}$ Tau. MIST
isochrones (Choi et al., 2016) for [Fe/H]$=+0.14$ and zero reddening are
shown, with ages 635 (blue), 740 (green), and 850 (red) Myr shown.

In summary, the component stars of $\epsilon$ Cnc, and the two groups of late
main sequence stars push for an age around $637\pm 19$ Myr, but the single
star in the Hertzsprung gap and the gap in the star distribution may imply a
statistical fluke or possibly an error in the stellar model physics or
chemical composition. In addition, the PARSEC isochrones generally predict too
high a luminosity for a star of $\epsilon$ Cnc A’s mass in the cluster, while
the MIST and BaSTI-IAC isochrones reproduce the luminosity better. As a
result, we believe our analysis puts limits on the amount of convective core
overshooting used in the isochrones, and therefore reduces systematic
uncertainties in the cluster age. We encourage work to analyze additional
stars in Praesepe in order to identify the cause of these inconsistencies.

###  4.1 White Dwarf Progenitor Masses

A primary result of the measurements of $\epsilon$ Cnc is the precise
measurement of the mass of a star near the end of core hydrogen burning. This
significantly reduces the uncertainties in the predicted masses of more
evolved stars, and more cleanly separates those predicted masses from
uncertainties surrounding the cluster age. While there are still some
uncertainties surrounding subsequent evolutionary phases, this can improve the
determination of white dwarf progenitor masses in the cluster.

For a single-age cluster, Praesepe contains a sizable sample of white dwarfs
with measured masses covering a fairly large range from about 0.7 to
$0.9M_{\odot}$ (Cummings et al., 2018; Salaris & Bedin, 2019). These white
dwarfs are in a transitional range between the masses of common field white
dwarfs with CO compositions and massive white dwarfs above $1M_{\odot}$
presumed to have ONe compositions. The progenitor stars are believed to be
ones that ignite helium burning in a non-degenerate core, but do not
experience a second convective dredge-up in their asymptotic giant stage. The
second dredge-up limits the growth of the hydrogen-depleted cores in more
massive stars, and results in slower growth of white dwarf masses with
increasing initial star mass. More precise determinations of the initial
masses for Praesepe’s white dwarfs can produce a strong lower mass limit for
stars that undergo the second dredge up, as well as a stronger test of our
understanding of the initial-final mass relation (IFMR) in this transition
range.

To determine initial masses for these white dwarfs, we started with isochrones
for which a star of $\epsilon$ Cnc A’s mass passed through the range of $G$
magnitudes we limited the star to earlier in this paper. We then subtracted
cooling times for the Praesepe WDs from Cummings et al. (2018) from the
isochrone ages, and calculated isochrones for those younger ages. The
progenitor masses for the white dwarfs are then the initial masses of stars
ending the asymptotic giant branch at that earlier time. Although the initial
masses derived in this way are still somewhat subject to uncertainties in
evolutionary phases after the main sequence (most notably during the
relatively long core helium burning phase), the progenitor masses are more
explicitly tied to masses measured for present-day stars. We based our
calculations on PARSEC models exclusively because of the difficulties other
isochrone sets had in reproducing the CMD position of $\epsilon$ Cnc A, and
these difficulties probably result from differences in convective core
overshooting algorithms. Based on the new age estimation of $637\pm 19$ Myr,
the new initial masses are given in Table 6.

Table 6: Initial Mass Determinations for Praesepe WDs ID |  $M_{WD}/M_{\odot}$aaCummings et al. (2018). |  $t_{cool}$ (Myr)aaCummings et al. (2018). | $M_{i}/M_{\odot}$
---|---|---|---
WD0833+194 | $0.813\pm 0.027$ | $364^{+33}_{-30}$ | $3.71^{+0.19}_{-0.14}$
WD0836+199 | $0.830\pm 0.028$ | $352^{+34}_{-31}$ | $3.66^{+0.19}_{-0.11}$
WD0837+189 | $0.870\pm 0.029$ | $402^{+42}_{-39}$ | $3.92^{+0.29}_{-0.23}$
WD0837+199 | $0.757\pm 0.025$ | $190^{+16}_{-15}$ | $3.16^{+0.06}_{-0.06}$
WD0840+190 | $0.895\pm 0.030$ | $425^{+48}_{-44}$ | $4.07^{+0.43}_{-0.30}$
WD0840+200 | $0.752\pm 0.027$ | $233^{+19}_{-18}$ | $3.27^{+0.07}_{-0.07}$
WD0843+184 | $0.898\pm 0.028$ | $423^{+38}_{-41}$ | $4.05^{+0.32}_{-0.28}$

In their examination of the IFMR, Cummings et al. (2018) derived ages for
Praesepe of $705\pm 25$ Myr and $685\pm 25$ Myr from non-rotating PARSEC and
MIST isochrones, respectively for [Fe/H]$=+0.15$ but zero reddening. Salaris &
Bedin (2019) used the age derived by Gaia Collaboration et al. (2018a):
$708^{+140}_{-90}$ Myr for [Fe/H]$=+0.16$ and reddening $E(B-V)=0.027$. Even
with high-quality Gaia photometry, there is definite ambiguity in fitting the
CMD with isochrones due to the small number of bright stars and potentially
confounding factors like binarity and high rotation rates. Our lower age
results in higher initial masses for the cluster white dwarfs by
$0.14-0.31M_{\odot}$ relative to the MIST results

from Cummings et al. For all but the youngest white dwarfs, the cooling time
uncertainty is the largest contributor to the uncertainty in the initial
masses. Salaris & Bedin (2019) derived cooling times from fitting tracks to
Gaia DR2 photometry of the white dwarfs, and found values that were in most
cases smaller (by $\sim 80-160$ Myr) than those of Cummings et al. (2018).
Cummings et al. derived cooling times from spectroscopic measurements of
$T_{\rm eff}$ and $\log g$. As a result, it should be remembered that
systematic uncertainties in the cooling times could have a larger effect on
the initial masses than our revision of the cluster age.

We believe that our determinations represent an improvement in how realistic
the statistical uncertainties are (because Cummings et al. appear to have only
included cooling time uncertainties in their quotations of initial mass
uncertainties) and in reducing systematic uncertainties (because the measured
star masses allow us to identify isochrones that match masses as well as CMD
properties).

##  5 Conclusion

Our overall goal was to determine the age of the open star cluster Praesepe
using the binary star system $\epsilon$ Cnc. Using radial velocities and
interferometric measurements, we have been able to calculate the primary
star’s mass as $2.420\pm 0.008M_{\odot}$ and the secondary star’s mass as
$2.226\pm 0.004M_{\odot}$, as well as put observational limits on the
brightness of the two stars. With this information, we find that MIST models
with the spectroscopic cluster metallicity are able to successfully reproduce
the inferred CMD positions of the stars of $\epsilon$ Cnc at their masses and
the position of the pre-core hydrogen exhaustion kink. While this gives
evidence in favor of the commonly used amount of convective core overshooting
in the MIST models, there are other features (the luminosity of a cluster
subgiant, and an unusual distribution of stars brighter than the cluster
turnoff) that may be indicating issues with model physics that should be
investigated further. The MIST models indicate an age of $637\pm 19$ Myr for
the cluster.

The direct measurement of masses for stars nearing core hydrogen exhaustion
also allows us to re-examine the likely progenitor masses for the cluster
white dwarfs. Our measurements imply an upward shift to the initial masses for
white dwarfs in the $0.7-0.9M_{\odot}$ range, although systematic
uncertainties in cooling ages need to be addressed and could have a
substantial effect.

We gratefully acknowledge the assistance of Daryl W. Willmarth in locating and
providing Kitt Peak spectra of $\epsilon$ Cnc from more than 25 years ago, and
support from the National Science Foundation under grant AAG 1817217 to E.L.S.
This research made use of observations from the SIMBAD database, operated at
CDS, Strasbourg, France; the WEBDA database, operated at the Institute for
Astronomy of the University of Vienna; data products from the AAVSO
Photometric All Sky Survey (APASS), funded by the Robert Martin Ayers Sciences
Fund and the National Science Foundation (grant AST-1412587). We use
observations made with the Nordic Optical Telescope, owned in collaboration by
the University of Turku and Aarhus University, and operated jointly by Aarhus
University, the University of Turku and the University of Oslo, representing
Denmark, Finland and Norway, the University of Iceland and Stockholm
University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of
the Instituto de Astrofisica de Canarias. This work is based upon observations
obtained with the Georgia State University Center for High Angular Resolution
Astronomy Array at Mount Wilson Observatory. The CHARA Array is supported by
the National Science Foundation under Grant No. AST-1636624 and AST-2034336.
Institutional support has been provided from the GSU College of Arts and
Sciences and the GSU Office of the Vice President for Research and Economic
Development. Time at the CHARA Array was granted through the NOIRLab community
access program (NOAO PropIDs: 2017A-0273, 2018A-0286, 2018B-0218, 2020A-0379;
NOIRLab PropID: 2021A-0276; PI: E. Sandquist). MIRC-X received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant No. 639889). J.D.M. acknowledges
funding for the development of MIRC-X (NASA-XRP NNX16AD43G, NSF-ATI 1506540,
NSF-AST 1909165). S.K. acknowledges support from an ERC Consolidator Grant
(Grant Agreement No. 101003096). A.L. received support from a Science
Technology and Facilities Council (STFC) studentship (No. 630008203). LRB
acknowledge support by MIUR under PRIN program $\\#$2017Z2HSMF.

Table 7: Literature Radial Velocity Measurements for $\epsilon$ Cnc A  mJDaamJD = HJD $-2400000$ | $v_{r,A}$ | $\sigma_{v}$ | Phase $\phi$ | Source
---|---|---|---|---
| km s${}^{-1}\;$ |  |
19849.3bbEstimated, as time of observation is not available. | 17 | 5 | 0.971 | Potsdam
19852.3bbEstimated, as time of observation is not available. | 73 | 5 | 0.057 | Potsdam
23451.941 | 32.6 | 9.6 | 0.489 | MWO
23534.651 | $-21.3$ | 2.5 | 0.843 | MWO
23539.721 | 19.3 | 16.9 | 0.987 | MWO
23563.646 | 14.4 | 3.5 | 0.668 | MWO
23573.664 | $-16.4$ | 12.4 | 0.953 | MWO
48309.755 | $-28.5$ | 3.5 | 0.853 | KPNO
48312.724 | $-17.9$ | 3.5 | 0.938 | KPNO
48330.863 | 44.2 | 3.5 | 0.454 | KPNO
48580.917 | 27.5 | 3.5 | 0.569 | KPNO
48673.845 | 78.3 | 3.5 | 0.214 | KPNO
48676.736 | 65.3 | 3.5 | 0.296 | KPNO
48694.761 | $-17.6$ | 3.5 | 0.809 | KPNO
48695.800 | $-22.4$ | 3.5 | 0.839 | KPNO
48969.909 | 16.9 | 3.5 | 0.639 | KPNO
49107.740 | 32.4 | 7 | 0.561 | KPNO
Table 8: Radial Velocity Measurements for $\epsilon$ Cnc A and B

mJDaamJD = HJD $-2400000$ | $v_{r,A}$ | $\sigma_{v}$ | $v_{r,B}$ | $\sigma_{v}$ | Phase $\phi$ | Source
---|---|---|---|---|---|---
| km s${}^{-1}\;$ | km s${}^{-1}\;$ |  |
22384.753 | 85.4 | 3 | $-20.1$ | 3 | 0.121 | DAO
48613.92562 | 36.5 | 1.5 | 24.7 | 2.8 | 0.509 | KPNO
48638.82381 | 79.9 | 1.1 | $-17.4$ | 2.7 | 0.217 | KPNO
48639.93300 | 76.3 | 1.1 | $-12.5$ | 2.7 | 0.249 | KPNO
48674.83253 | 73.4 | 1.0 | $-18.4$ | 2.9 | 0.242 | KPNO
48696.76600 | $-27.9$ | 0.6 | 95.6 | 2.2 | 0.866 | KPNO
48967.99971 | 25.5 | 1.3 | 36.4 | 3.0 | 0.584 | KPNO
49019.89793 | 70.1 | 1.1 | $-9.1$ | 2.9 | 0.061 | KPNO
49021.87432 | 84.6 | 0.9 | $-19.9$ | 2.5 | 0.117 | KPNO
49021.93111 | 85.3 | 0.9 | $-19.7$ | 2.4 | 0.119 | KPNO
49021.98021 | 85.6 | 1.0 | $-19.5$ | 2.3 | 0.120 | KPNO
49030.84515 | 58.7 | 1.3 | 6.3 | 3.7 | 0.373 | KPNO
49080.62158 | $-13.8$ | 1.1 | 83.6 | 2.8 | 0.789 | KPNO
49081.70621 | $-19.2$ | 1.0 | 90.5 | 2.6 | 0.820 | KPNO
49082.71360 | $-23.8$ | 1.0 | 94.9 | 2.8 | 0.849 | KPNO
49094.69748 | 83.5 | 1.0 | $-17.8$ | 2.3 | 0.190 | KPNO
49095.69961 | 79.4 | 1.0 | $-17.1$ | 2.7 | 0.218 | KPNO
49096.69919 | 76.3 | 1.1 | $-12.3$ | 3.0 | 0.247 | KPNO
49097.71513 | 72.9 | 1.0 | $-4.7$ | 3.0 | 0.276 | KPNO
49106.71287 | 33.8 | 1.4 | 35.3 | 2.7 | 0.532 | KPNO
49344.96554 | 67.7 | 1.2 | $-3.0$ | 3.1 | 0.311 | KPNO
49345.98097 | 64.2 | 1.3 | 2.8 | 3.2 | 0.340 | KPNO
49381.90408 | 60.6 | 0.9 | 9.0 | 3.6 | 0.363 | KPNO
49382.80479 | 56.7 | 1.4 | 10.4 | 3.5 | 0.388 | KPNO
49383.77115 | 53.2 | 1.5 | 14.6 | 4.1 | 0.416 | KPNO
49384.87950 | 46.3 | 1.4 | 25.2 | 3.1 | 0.447 | KPNO
50800.02879 | 1.1 | 1.3 | 62.4 | 3.7 | 0.717 | KPNO
53009.47942 | 23.9 | 0.1 | 45.6 | 2.2 | 0.590 | OHP
53014.62148 | $-1.8$ | 0.1 | 75.0 | 1.9 | 0.736 | OHP
58960.37513 | $-19.8$ | 0.1 | 94.3 | 1.3 | 0.931 | NOT
58960.37905 | $-19.8$ | 0.1 | 94.1 | 1.3 | 0.931 | NOT
58968.40424 | 86.1 | 0.1 | $-20.4$ | 1.5 | 0.160 | NOT
58968.40826 | 86.0 | 0.1 | $-20.5$ | 1.5 | 0.160 | NOT

bbfootnotetext: The normalized digitized spectra for KPNO and NOT (30 spectra)
are available as the Data behind the Figure.

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##  Appendix A Bright Main Sequence and Subgiant Stars in Praesepe and the
Hyades

In order to make the cleanest CMD comparison between the two clusters, we need
to understand whether the photometry of the most age-sensitve stars is
affected by binarity. Most of the bright stars have have been claimed to be
close binaries, but in many cases the evidence is rather marginal. We discuss
the current evidence below, proceeding from most to least luminous stars.

###  A.1 Praesepe Members

35 Cnc / HD 72779 / VL 133: This star is a fairly rapidly rotating subgiant
star in the red half of the Hertzsprung gap. Radial velocity monitoring by Abt
& Willmarth (1999) indicate a constant radial velocity, while lunar
occultation observations (Blow et al., 1982; Eitter & Beavers, 1974) and
speckle interferometry (Mason et al., 1993a) have not shown evidence of a
companion.

38 Cnc / BT Cnc / HD 73575 / KW 204: Abt & Willmarth (1999) report that this
star has no velocity variability, but does show line profile variations. This
may relate to its $\delta$ Sct variability. Breger et al. (2012) finds light
variations with full amplitude of about 0.06 mag. No companion was detected
during lunar occultations (Peterson & White, 1984; Peterson et al., 1989) or
speckle interferometry (Mason et al., 1993a).

HD 73210 / KW 50: Abt & Willmarth (1999) derive a double-line spectroscopic
orbit with a 12 day period for this star. The detection of both stars implies
substantial light blending, so that the CMD position will not be
representative of single stars. However, the binary has a fairly extreme mass
ratio ($q\approx 0.35$) that means that the CMD position of the brighter star
does not reside too far from that of the binary. Lunar occultation (Blow et
al., 1982) and speckle interferometry (Mason et al., 1993a) did not reveal a
companion.

HD 73712 / KW 284: Abt & Willmarth (1999) observe a sharp-lined pair of stars
in spectra, superimposed on a third component with broad lines. They derive a
spectroscopic orbit with a 48 day period for the sharp-lined pair with a mass
ratio fairly close to one. Needless to say, the CMD position does not
adequately represent any of the component stars, and all are likely to be near
the main sequence. Speckle interferometry (Mason et al., 1993a; McAlister et
al., 1989; Hartkopf et al., 1994, 1997, 2000; Tokovinin, 2017) and lunar
occultation (Peterson et al., 1989) probably show orbital motion of the third
star, approximately 1.6 mag fainter with about $0\farcs 1$ angular separation.

EP Cnc / HD 73819 / KW 348: Similar to 35 Cnc, Abt & Willmarth (1999) report
that this star has no velocity variability but rapid line profile variations.
This may also relate to its $\delta$ Sct variability. The amplitude of the
stars’s photometric variations is relatively small ($\sim 0.01$ mag; Breger et
al. 2012), however. No companion was detected during lunar occultations
(Peterson & White, 1984; Peterson et al., 1989).

42 Cnc / HD 73785 / KW 328: Abt & Willmarth (1999) report no velocity
variation in their extensive observations, in contradiction to an earlier
report by Raboud & Mermilliod (1998) that it was an SB2 based on older
observations. No companion was detected during lunar occultations (Peterson &
White, 1984; Peterson et al., 1989).

HD 73449 / KW 150: This star has long been known to rotate at greater than 200
km s${}^{-1}\;$, and it is considerably redder than other cluster stars at the
same brightness. A lunar occultation (Peterson & White, 1984) and speckle
interferometry (Mason et al., 1993a) did not reveal evidence of a companion
star. Raboud & Mermilliod (1998) identified it as an SB1 on the basis of old
radial velocity measurements, but Abt & Willmarth (1999) found it to have
constant radial velocity. The star has been identified as photometrically
variable in the older literature, but recent studies have used it as a
constant photometric comparison star for $\delta$ Sct studies (Hernandez et
al., 1998; Pena et al., 1998).

HD 72942 / KW 534: HD 72942 is a known single-lined spectroscopic binary with
an orbital period of 131.96 d (Abt & Willmarth, 1999), indicating a lack of
interaction between the stars. The fact that the binary’s color is bluer than
any main sequence star also implies that the primary star must be one of the
bluest stars in the cluster. The star has an interesting hybrid surface
composition between normal A stars and Am stars (Fossati et al., 2007). In
addition, the primary star is rotating at a rate that puts it toward the high
end of the range for the cluster’s Am stars (Fossati et al., 2008). These
items leave a somewhat ambiguous picture of the star’s nature, but a blue
straggler history cannot be ruled out. A lunar occultation (Peterson & White,
1984) and speckle interferometry observations (Mason et al., 1993a) were
negative for a companion. We conclude that this star probably should not be
considered a representative of single cluster stars.

HD 72846:  This star had a composition study by Fossati et al. (2008), who
found it to be consistent with normal main sequence abundances in the cluster.
No evidence of binarity for this star has yet been found in speckle
interferometry observations (Mason et al., 1993a), and a small number of
radial velocity measurements are consistent with a constant value (Abt, 1970).

HD 73711 / KW 276: This star has abundances consistent with chemically-
peculiar Am stars (Fossati et al., 2007; Burkhart & Coupry, 1998), and as with
many of those, there are tentative signs of binarity in spectra. Burkhart &
Coupry (1998) identified the star as SB2 based on line profiles, while Raboud
& Mermilliod (1998) identified it as SB1 based on variation in older radial
velocities. No velocity variations were found in the observations by Abt &
Willmarth (1999), although they saw evidence of asymmetries in the line
profiles. Lunar occultation (Peterson & White, 1984) and speckle
interferometry (Mason et al., 1993a) did not reveal a companion.

###  A.2 Hyades Members

$\delta$ Cas / 37 Cas / HD 8538: This star is apart from the main body of the
cluster, and was recently identified as a member of the preceding tidal tail
of the cluster (Röser et al., 2019). Its position in the Hertzsprung gap makes
it interesting for isochrone fitting. The star does not appear to be velocity
variable (Abt, 1965). Although it is identified as an eclipsing variable in
SIMBAD, there is no strong evidence that it is photometrically variable
(Samus’ et al., 2017). To date, the evidence is that it is a single star,
although there is not high-resolution imaging or interferometry in the
literature.

$\theta^{2}$ Tau / 78 Tau / HD 28319 / vB 72: This is a very well-known
spectroscopic and interferometrically-resolved binary. The most recent
analyses of the binary are in Torres et al. (2011) and Armstrong et al.
(2006). Although the photometry of the two components can potentially be
disentangled, this is beyond the scope of this work.

$\kappa^{1}$ Tau / 65 Tau / HD 27934 / vB 54: This star is identified as
probably velocity constant in Abt (1965). The only reference to binarity is a
report from a visual observation of a lunar occultation (Dunham, 1974) that
reported the minimum resolvable separation ($0\farcs 1$) and equal brightness
components. Attempts at speckle interferometry detections have been negative
(Patience et al., 1998; Mason et al., 1993b; Hartkopf & McAlister, 1984). We
conclude that there is not significant evidence of binarity for this star.

$\delta^{3}$ Tau A / 68 Tau / HD 27962 / vB 56: This star is identified as a
cluster blue straggler based on its very blue color, as well as an Am star,
and so we do not consider it in discussing the cluster age. It is a velocity
variable (Borgniet et al., 2019), but there have only been negative detections
of a close companion in lunar occultations (Trunkovsky, 2013) and speckle
interferometry (Patience et al., 1998; Mason et al., 1993b; Hartkopf &
McAlister, 1984).

$\iota$ Tau / 102 Tau / HD 32301 / vB 129: 102 Tau was shown to have constant
radial velocity in Abt (1965). Speckle interferometry observations have not
revealed a close companion (Hartkopf & McAlister, 1984; Mason et al., 1993b;
Patience et al., 1998). There is a report of a lunar occultation in 1956 that
indicated two equal-brightness stars at $0\farcs 4$ separation (Hoffleit &
Jaschek, 1991), although this seems to contradict the negative speckle
results. At present, the evidence for a companion star with significant
brightness is minimal.

c Tau / 90 Tau / HD 29388 / vB 104: We ignore the wide common proper motion
components in this discussion. A faint companion was detected
interferometrically by Marion et al. (2014) at a separation of 11 mas,
contributing only about 3% in $H$ band. 90 Tau was reported early on as a
spectroscopic binary (Barrett, 1910), but is labeled as a ”constant:” velocity
star in Abt (1965). Based on the evidence, we expect the interferometric
companion to have little effect on the system photometry in the optical, but
it is unclear whether there is a closer companion affecting the velocities.

$\sigma^{2}$ Tau / 92 Tau / HD 29488 / vB 108: 92 Tau was identified by Lee
(1909) as a double-lined spectroscopic binary, but later shown by Abt (1965)
to have constant velocity. Borgniet et al. (2019) identify low-level velocity
variations ($\sim 1$ km s-1) from more recent data covering 1370 days, and
Lagrange et al. (2009) identify it as velocity variable but not as a binary
candidate. Speckle interferometric observations have not detected a companion
(Patience et al., 1998; Mason et al., 1993b). We conclude that there may be a
companion to the star, but that it is likely to be faint based on the lack of
spectroscopic detections.

$\lambda$ Gem / 54 Gem / HD 56537: This is a star identified in the trailing
tail of the Hyades (Röser et al., 2019). It was identified early as a
spectroscopic binary with a period of $4-5$ days by Hnatek (1913), and again
by Frost (1924), with other indications of velocity variations in Abt (1970).
Lagrange et al. (2009) did not identify it as a binary candidate from more
recent radial velocity measurements, nor did Becker et al. (2015). While there
is a visual companion at $10\arcsec$ separation, a closer companion has been
resolved by lunar occultation: initially by Dunham (1977) and then by Richichi
et al. (1999) following several negative results. Dunham initially identified
the closer companion as visually fainter by 1 mag, but Richichi et al. found
it fainter in $K$ by about 3.4 mag. Both studies found separations of tens of
milli-arcseconds. The companion was not detected in other ground-based
interferometer measurements (Marion et al., 2014). Based on the evidence, we
expect the companion to have a small effect on the system photometry.

$\theta$ Cas / 33 Cas / HD 6961: This star is in the preceding tidal tail of
the cluster (Röser et al., 2019), and was identified as a long-period single-
lined spectroscopic binary by Abt (1965) and Abt & Levy (1974). More recently,
Borgniet et al. (2019) found no significant radial velocity variation over a
1417 d baseline. Speckle interferometry studies have not revealed a close
companion (Hartkopf & McAlister, 1984). The star is used as a calibrator for
interferometric studies, and there have been no reports of binarity. We
conclude that there may be low-level contamination of the light of the primary
star.

HD 30210 / vB 112: We will ignore two wide commmon proper motion companions in
this discussion. This star was identified in the trailing tail of the cluster
(Röser et al., 2019). It is also a known Am star, which generally have higher
probabilities of binarity than single stars. Abt & Levy (1974) identify it as
a single-lined spectroscopic binary with uncertain orbital elements. Speckle
observations (Mason et al., 1993b) gave negative results for a companion.
Based on the potential color effects of being an Am star and the chance that
the spectroscopic companion might affect the photometry, we leave it out of
consideration for age purposes.

$\delta^{2}$ Tau / 64 Tau / HD 27819 / vB 47: This star was identified as a
likely spectroscopic binary by Lee (1909), but was identified as velocity
constant in Abt (1965) with a chance of slow variation. Borgniet et al. (2019)
find evidence of low amplitude velocity variations over a 1399 day span, but
inconsistent with the earlier Lee measurements. A lunar occultation
observation (Africano et al., 1978) and speckle interferometry (Hartkopf &
McAlister, 1984; Mason et al., 1993b) were negative for a companion. We
conclude that there may be low-level effects on the photometry of the primary
star.

HD 28527 / vB 82: This star is labelled as a $\delta$ Sct variable in SIMBAD,
but it appears to be a constant star (Kovacs & Paparo, 1989). It was
identified as a possible binary by Barrett (1910), but later labeled as
velocity constant in Abt (1965). Multiple lunar occultation observations
(White, 1979; Radick & Lien, 1980; Fekel et al., 1980; Evans & Edwards, 1981;
Radick et al., 1982; Richichi et al., 1999) have been negative for a
companion, while Peterson et al. (1981) provide ”strong, but not definitive”
evidence of a companion with fairly large magnitude difference ($\sim 3$ mag)
at very small separation. We conclude that the weight of the evidence is
against a bright, close companion.