# Fundamental effective temperature measurements for eclipsing binary stars –
III. SPIRou near-infrared spectroscopy and CHEOPS photometry of the benchmark
G0V star EBLM J0113+31

P. F. L. Maxted,1 N. J. Miller,1 S. Hoyer,2 V. Adibekyan,3 S. G. Sousa,3 N.
Billot,4 A. Fortier,5,6 A. E. Simon,5 A. Collier Cameron,7 M. I. Swayne,1 P.
Gutermann,2,8 A. H. M. J. Triaud,9 J. Southworth,1 Y. Alibert,5 R.
Alonso,10,11 G. Anglada,12,13 T. Bárczy,14 D. Barrado y Navascues,15 S. C. C.
Barros,3,16 W. Baumjohann,17 M. Beck,4 T. Beck,5 W. Benz,5,6 X. Bonfils,18 A.
Brandeker,19 C. Broeg,5,6 M. Buder,20 J. Cabrera,20 S. Charnoz,21 C. Corral
van Damme,22 Sz. Csizmadia,20 M. B. Davies,23 M. Deleuil,2 L. Delrez,24,25 O.
Demangeon,3,16 B.-O. Demory,6 D. Ehrenreich,4 A. Erikson,20 L. Fossati,17 M.
Fridlund,26,27 D. Gandolfi,28 M. Gillon,24 M. Güdel,29 K. Heng,6,30 J. E.
Hernández Leon,31 K. G. Isaak,32 L. L. Kiss,33,34 J. Laskar,35 A. Lecavelier
des Etangs,36 M. Lendl,4 C. Lovis,4 D. Magrin,37 M. Munari,38 V. Nascimbeni,37
G. Olofsson,19 R. Ottensamer,39 I. Pagano,38 E. Pallé,10 G. Peter,40 G.
Piotto,37,41 D. Pollacco,30 D. Queloz,42,43 R. Ragazzoni,37,41 N. Rando,22 H.
Rauer,20,44,45 I. Ribas,12,13 N. C. Santos,3,16 G. Scandariato,38 D.
Ségransan,4 A. M. S. Smith,20 M. Steinberger,17 M. Steller,17 Gy. M.
Szabó,34,46 N. Thomas,5 S. Udry,4 V. Van Grootel,25 N. Walton47
(Affiliations are listed at the end of the paper) E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

EBLM J0113+31 is moderately bright (V=10.1), metal-poor ([Fe/H] $\approx-0.3$)
G0V star with a much fainter M dwarf companion on a wide, eccentric orbit
(=14.3 d). We have used near-infrared spectroscopy obtained with the SPIRou
spectrograph to measure the semi-amplitude of the M dwarf’s spectroscopic
orbit, and high-precision photometry of the eclipse and transit from the
CHEOPS and TESS space missions to measure the geometry of this binary system.
From the combined analysis of these data together with previously published
observations we obtain the following model-independent masses and radii:
$M_{1}=1.029\pm 0.025M_{\odot}$, $M_{2}=0.197\pm 0.003M_{\odot}$,
$R_{1}=1.417\pm 0.014R_{\odot}$, $R_{2}=0.215\pm 0.002R_{\odot}$. Using
$R_{1}$ and the parallax from Gaia EDR3 we find that this star’s angular
diameter is $\theta=0.0745\pm 0.0007$ mas. The apparent bolometric flux of the
G0V star corrected for both extinction and the contribution from the M dwarf
($<0.2$ per cent) is ${\mathcal{F}}_{\oplus,0}=(2.62\pm 0.05)\times 10^{-9}$
erg cm-2 s-1. Hence, this G0V star has an effective temperature $T_{\rm
eff,1}=6124{\rm\,K}\pm 40{\rm\,K\,(rnd.)}\pm 10{\rm\,K\,(sys.)}$. EBLM
J0113+31 is an ideal benchmark star that can be used for “end-to-end” tests of
the stellar parameters measured by large-scale spectroscopic surveys, or
stellar parameters derived from asteroseismology with PLATO. The techniques
developed here can be applied to many other eclipsing binaries in order to
create a network of such benchmark stars.

###### keywords:

techniques: spectroscopic, binaries: eclipsing, stars: fundamental parameters,
stars: solar-type

††pubyear: 2021††pagerange: Fundamental effective temperature measurements for
eclipsing binary stars – III. SPIRou near-infrared spectroscopy and CHEOPS
photometry of the benchmark G0V star EBLM J0113+31–Fundamental effective
temperature measurements for eclipsing binary stars – III. SPIRou near-
infrared spectroscopy and CHEOPS photometry of the benchmark G0V star EBLM
J0113+31

##  1 Introduction

Benchmark stars have properties that have been directly and accurately
measured to good precision. They play a fundamental role in stellar
astrophysics because we can only ascertain the accuracy and reliability of
stellar models by comparing their predictions to the observed properties of
real stars. Benchmark stars can also be used to establish empirical relations
between stellar properties, e.g. colour – effective temperature ($T_{\rm
eff}$) relations (Boyajian et al., 2013; Van Belle et al., 2021; Huang et al.,
2015), or equations to estimate the mass or radius of a main-sequence star
from $T_{\rm eff}$, $\log g$ and [Fe/H] (Torres et al., 2010). Empirical
relations are particularly useful in cases where stellar structure models are
known to be unreliable, e.g. for low-mass stars, where stellar models tend to
under-predict the radius and over-predict $T_{\rm eff}$ (Spada et al., 2013;
Cassisi & Salaris, 2019; Zhou et al., 2014; Berger et al., 2006).

Considerable effort has put into calibrating the $T_{\rm eff}$ scale for FGK-
type dwarf stars. In recent years, this effort has been partly driven by the
need for accurate $T_{\rm eff}$ estimates for planet host stars in order to
estimate their masses and radii using stellar models (Boyajian et al., 2015;
Baines et al., 2009). Much of the progress in characterising exoplanets in
recent years has been due to the improved precision in measuring stellar
masses and radii (Jontof-Hutter, 2019).

Benchmark FGK-type stars are also essential to calibrate the level of
systematic and random uncertainties in massive spectroscopic surveys such as
RAVE (Steinmetz et al., 2020), the Gaia-ESO survey (Gilmore et al., 2012),
LAMOST (Deng et al., 2012), GALAH (Buder et al., 2018), etc. (Blanco-Cuaresma
et al., 2014a; Heiter et al., 2015; Jofré et al., 2018). These surveys aim to
reconstruct the formation history of the Galaxy by studying the pattern of
elemental abundances in stars as a function of their mass, age and kinematics.
Jofré et al. (2019) in their recent comprehensive review of “industrial scale”
stellar abundance measurements suggest that it is today not possible to know
the temperature of a star better than an accuracy of 50 K. This uncertainty
has a direct impact on reliability of trends observed in stellar abundance
patterns between different stellar populations. Errors in $T_{\rm eff}$ are
the dominant source of uncertainty when estimating the mass, radius,
composition and age of a star (Valle et al., 2018; Jofré et al., 2019).

Validation and calibration of $T_{\rm eff}$ estimates for FGK-type dwarf stars
currently rely on angular diameter measurements for a small sample of very
bright stars such as Procyon, $\tau$ Cet, 18 Sco, $\alpha$ Cen A, etc.
(Karovicova et al., 2022; Heiter et al., 2015; Boyajian et al., 2013).
Repeated measurements of the angular diameter for the same star often show
differences much larger than the quoted uncertainties, with systematic errors
of 5 per cent or more being quite common. For example, the 15 repeated
measurements provided in Table 9 of Karovicova et al. (2022) for 7 G-type
dwarf stars require an additional “external error” of about 0.04 mas to be
added to the quoted uncertainties to achieve $\chi^{2}_{r}=1$ for a fit of a
constant offset to these difference. Tayar et al. (2022) show that current
uncertainties on measured interferometric angular diameters and bolometric
fluxes set a systematic uncertainty floor of $\approx 2$ per cent in $T_{\rm
eff}$ for solar-type exoplanet host stars, i.e. $\pm 120$ K at $T_{\rm
eff}=6000$ K.

Very low-mass stars (VLMSs, $\loa 0.2M_{\odot}$) are attractive targets for
exoplanet studies because of the possibility to detect and characterise the
atmospheres of terrestrial planets in the habitable zones of these stars
(Sebastian et al., 2021). There are very few well-characterised VLMSs because
they are intrinsically very faint and small. For example, the recent empirical
colour – $T_{\rm eff}$, colour – luminosity and colour – radius relations
published by Boyajian et al. (2012) are based on a sample that contains only
one star with a spectral type later than M4V ($M\approx 0.2M_{\odot}$, Mann et
al., 2019).

The EBLM project (Triaud et al., 2013) aims to improve our understanding of
VLMSs by studying eclipsing binaries with low-mass companions that have been
found by the WASP survey (Pollacco et al., 2006). These eclipsing binaries
typically have a late-F- to mid-G-type primary star with an M-dwarf that
contributes $\ll 1$ per cent of the flux at optical wavelengths. The light
curves of these EBLM systems look very similar to those of transiting hot
Jupiters, which are the main targets for the WASP survey. As a result, dozens
of these EBLM systems have been identified in the WASP survey. The analysis of
the light curve combined with a spectroscopic orbit for the primary star and
an estimate for its mass provides a direct estimate for the mass and radius of
the M-dwarf companion (Von Boetticher et al., 2019; Gill et al., 2019). With
very high quality photometry it is possible to detect the eclipse of the
M-dwarf and, hence, its surface brightness relative to the primary star. This
surface brightness ratio can then be used to infer $T_{\rm eff}$ for the
M-dwarf given an estimate of $T_{\rm eff}$ for the primary star and a surface
brightness – $T_{\rm eff}$ relation for the stars, either empirical (Graczyk
et al., 2021) or based on stellar model atmospheres. The first results from an
on-going programme to measure mass, radius and $T_{\rm eff}$ for the M-dwarf
in a sample of EBLM systems using ultrahigh-precision photometry obtained as
part of the guaranteed time observations (GTO) with the CHEOPS mission (Benz
et al., 2021) have been published by Swayne et al. (2021). Most of the targets
for this programme were selected from a sample of over 100 EBLM systems with
spectroscopic orbits published by Triaud et al. (2017).

The first study of EBLM J0113+31, the target for this study, was published by
Gómez Maqueo Chew et al. (2014, GMC+2014 hereafter). They used ground-based
photometry of the eclipse in the J-band to infer $T_{\rm eff}\approx 3900$ K
for the very low-mass companion, much higher than expected given their
estimate for its mass ($M_{2}=0.186\pm 0.010M_{\odot}$). Subsequent analysis
of the TESS light curve for this binary system by Swayne et al. (2020) found
no evidence for a very hot companion. Their value of $T_{\rm eff,2}=3208\pm
43$ K is similar to that for other VLMSs. They conclude that the anomalous
result from GMC+2014 was due to systematic errors in the J-band photometry,
illustrating the need for very high-quality space-based photometry to make
reliable measurements of $T_{\rm eff,2}$ in EBLM systems.

Here we present new photometry of the transit and eclipse in EBLM J0113+31
obtained with CHEOPS, and high-resolution, phase-resolved spectroscopy
obtained with the near-infrared échelle spectrograph SPIRou on the Canada-
France-Hawaii telescope. We have used the SPIRou spectroscopy to directly
measure the semi-amplitude of the M-dwarf’s spectroscopic orbit. We have used
this measurement combined with the analysis of the new light curves and other
published data to directly and accurately measure the mass, radius and $T_{\rm
eff}$ of both stars in this binary system. We discuss the use of the
techniques developed here to determine fundamental stellar properties for
stars in EBLM systems, and conclude that it is now feasible to establish a
network of well-studied EBLM systems that will be an ideal set of benchmark
stars for future surveys.

##  2 Observations

###  2.1 CHEOPS photometry

CHEOPS is a telescope with an effective aperture of 30 cm in low Earth orbit
that is designed to obtain ultrahigh precision broadband photometry of bright
stars (Benz et al., 2021). To achieve this, the instrument has been purposely
defocused to produce a point-spread function (PSF) with a diameter of
approximately 32″. We observed two transits and one eclipse of EBLM J0113+31
with CHEOPS (Table 1). There are gaps in the observations due to occultation
of the target by the Earth and passages of the satellite though the South
Atlantic Anomaly.

The raw data were processed using version 13.1.0 of the CHEOPS data reduction
pipeline (DRP, Hoyer et al., 2020). The DRP corrects the images for
environmental and instrumental effects before performing aperture photometry
of the target. The contamination of the photometric aperture during the
exposure by nearby stars is estimated using simulations of the field of view
based on the Gaia DR2 catalogue (Gaia Collaboration et al., 2018). The
instrument response function for CHEOPS is very similar to the Gaia G band.
The detector used on the CHEOPS instrument is a frame-transfer charge-coupled
device (CCD), so the DRP must also account for the “smear” trails from bright
stars produced during the frame transfer. Both of these effects (contamination
and smear) vary from image to image because the satellite rotates continuously
during its 99-minute orbit.

Aperture photometry is extracted by the DRP using three different fixed
aperture sizes labelled “RINF”, “DEFAULT” and “RSUP” (at radii of 22.5, 25.0
and 30.0 pixels, respectively) and a further “OPTIMAL” aperture whose size is
determined for each visit dependent upon the amount of contamination. The
observed and processed data are made available on the Data Analysis Center for
Exoplanets (DACE) web platform111The DACE platform is available at
http://dace.unige.ch. We downloaded our data from DACE using
pycheops222https://pypi.org/project/pycheops/, a python module developed for
the analysis of data from the CHEOPS mission (Maxted et al., 2021).

There are three stars that are 5–6 magnitudes fainter than EBLM J0113+31
within 1′ of the target (Fig. 1). As a result, the OPTIMAL aperture is set to
its maximum allowed value by the DRP (40 pixels = 40″). Although this
maximises the contamination of the aperture by these nearby stars, it
minimises the noise due to the variations in this contamination due to changes
in the fraction of the stars’ PSFs inside the photometric aperture as the
field of view rotates.

Figure 1: Simulated images of the CHEOPS field of view. Upper panel: all the stars in the field of view including the target. Lower panel: The target has been removed to show only the background stars in the field of view. Black circles show the DEFAULT (25 pixel) and OPTIMAL (40 pixel) apertures and the red cross shows the location of the target star. Table 1: Log of CHEOPS observations. Effic. is the fraction of the observing interval covered by valid observations of the target. The variables in final column are spacecraft roll angle, $\phi$, and aperture contamination, contam.  File key | Event | Start date [UTC] | Duration [s] | $N_{\rm obs}$ | Effic. [%] | Decorrelation parameters
---|---|---|---|---|---|---
CH_PR100037_TG011601_V0200 | Transit | 2020-11-24T15:41:07 | 48 682 | 429 | 52.8 |  contam, $\sin\phi$, $\cos 3\phi$
CH_PR100037_TG017101_V0200 | Transit | 2021-10-19T00:20:09 | 48 983 | 519 | 63.5 |  contam, $\sin\phi$, $\cos\phi$,$\sin 2\phi$
CH_PR100037_TG017201_V0200 | Eclipse | 2021-09-28T03:09:09 | 34 936 | 338 | 57.9 |  contam, $\sin\phi$, $\cos 2\phi$,$\sin 3\phi$

###  2.2 CFHT SPIRou spectroscopy

SPIRou is a fibre-fed, cross-dispersed échelle spectrograph mounted on the
Canada-France-Hawaii telescope (CFHT) on Maunakea, Hawaii. The spectrograph
provides spectra covering the entire wavelength range from 0.95 to 2.35
microns at a spectral resolving power $R\approx 75,000$ (Donati et al., 2020).
22 spectra of EBLM J0113+31 with a signal-to-noise per pixel between 77 and
103 near 1 micron were obtained on separate nights between 2020-02-05 and
2020-08-01.

We used spectra extracted from the raw data provided by the observatory using
data reduction system (DRS) version 0.6.131. We dealt with the data order-by-
order, selecting only those orders with little contamination due to telluric
features. The selected wavelength regions are listed in Table 2. The
corrections for the échelle blaze function and telluric absorption provided by
the DRS were applied.

Table 2: SPIRou échelle orders used in this analysis. Features typically visible in the spectra of M dwarfs from Jones et al. (1994) are listed with wavelengths in nm in the final column. Order |  $\lambda_{\rm min}$ [nm] |  $\lambda_{\rm max}$ [nm] | Notes
---|---|---|---
32 | 2363 | 2437 |
33 | 2291 | 2363 |
34 | 2224 | 2294 |
35 | 2160 | 2228 | Na I 2206, 2209
36 | 2100 | 2166 |
37 | 2043 | 2108 |
44 | 1718 | 1772 |
45 | 1680 | 1733 |
46 | 1643 | 1695 | Al 1676, 1677
47 | 1608 | 1659 |
48 | 1575 | 1624 |
58 | 1303 | 1344 |
59 | 1281 | 1321 | Ca I 1313
62 | 1219 | 1257 | K I 1243, 1252
63 | 1199 | 1237 |
64 | 1181 | 1218 | VO 1200
65 | 1162 | 1199 | K I 1169, 1177, 1178
66 | 1145 | 1181 |
72 | 1049 | 1082 |
73 | 1035 | 1067 |
74 | 1021 | 1053 |
75 | 1007 | 1039 |
78 | 968 | 999 | FeH 990
79 | 956 | 986 |

###  2.3 TESS photometry

Figure 2: A typical image of EBLM J0113+31 from the TESS target pixel file
showing the pixels used to extract the light curve (red hatching).

One transit and two secondary eclipses of EBLM J0113+31 were observed at 120 s
cadence by the Transiting Exoplanet Survey Satellite (TESS, Ricker et al.,
2015) in Sector 17 of the primary mission. The TESS target pixel files were
downloaded from the Mikulski Archive for Space
Telescopes333https://archive.stsci.edu/ (MAST) and processed to produce a
light curve using the package Lightkurve 2.0 (Lightkurve Collaboration et al.,
2018).444https://docs.lightkurve.org/ The pixels used to extract the
photometry from the target pixel file are shown in Fig. 2. Instrumental noise
was removed using the cotrending basis vectors (CBVs) provided by the TESS
Science Processing Operations Center (SPOC) (Jenkins et al., 2016). We used 16
“Single-Scale” and 7 “Spike” CBVs to model trends present in all targets on
the same CCD as EBLM J0113+31. The amplitude of each CBV was determined using
only data outside the eclipses and transit. We set the L2-norm penalty to
$\alpha=0.1$ to achieve a balance between over-fitting the data and
effectively removing instrumental trends.

##  3 Analysis

###  3.1 Radial velocity measurements from the SPIRou data

We use synthetic spectra taken from Husser et al.
(2013)555http://phoenix.astro.physik.uni-goettingen.de/ to produce a template
for the spectrum of the G0V primary star, using linear interpolation to create
a spectrum appropriate for $T_{\rm eff}=6150$ K, ${\rm[Fe/H]}=-0.4$, $\log
g=4.15$ and $[\alpha/{\rm Fe}]=0.0$. We then measure the position of the peak
in the cross-correlation against this template for the observed spectra order-
by-order. Low-frequency noise in the data for each order was removed prior to
cross-correlation using a 5-th order high-pass Butterworth filter with a
critical frequency of 16/4096 pixels-1. We then reject measurements more than
5 km s-1 from the median before calculating the mean and standard error in the
mean given in Table 3.

Table 3: Radial velocity measurements for EBLM J0113+31 A measured from the SPIRou spectra of EBLM J0113+31. The number of orders used to calculate the mean and standard error on the mean is given in the final column. Exposure number | BJDTDB |  $V_{r}$ [km/s] | $N$
---|---|---|---
2503696 | 2459063.1144 | $16.72\pm 0.14$ | 24
2502923 | 2459059.1322 | $27.01\pm 0.14$ | 24
2469680 | 2458896.7054 | $-1.79\pm 0.18$ | 24
2499300 | 2459038.1105 | $3.12\pm 0.16$ | 24
2498079 | 2459033.1244 | $22.56\pm 0.11$ | 24
2493617 | 2459011.1184 | $-1.84\pm 0.13$ | 24
2498553 | 2459035.1050 | $14.54\pm 0.13$ | 23
2499489 | 2459039.0948 | $-0.06\pm 0.08$ | 24
2468747 | 2458885.7312 | $5.73\pm 0.15$ | 22
2502578 | 2459056.1340 | $-1.91\pm 0.13$ | 24
2469510 | 2458895.7054 | $2.27\pm 0.16$ | 23
2497677 | 2459031.1037 | $27.73\pm 0.12$ | 24
2498357 | 2459034.1213 | $18.36\pm 0.16$ | 23
2499115 | 2459037.1309 | $6.50\pm 0.17$ | 24
2499760 | 2459040.1279 | $-2.89\pm 0.16$ | 23
2498783 | 2459036.1298 | $10.66\pm 0.15$ | 23
2468572 | 2458884.7115 | $-2.25\pm 0.18$ | 24
2497879 | 2459032.1204 | $25.75\pm 0.14$ | 24
2499953 | 2459041.1116 | $-3.52\pm 0.13$ | 23
2503110 | 2459060.0927 | $27.40\pm 0.15$ | 23
2469883 | 2458897.7056 | $-3.43\pm 0.14$ | 23
2470074 | 2458898.7057 | $-3.21\pm 0.12$ | 24

###  3.2 Pre-processing of the SPIRou data

The M dwarf contributes less than 2 per cent of the flux at 1 $\mu$m so we
removed the spectral features in the SPIRou data due to the G0V primary star
prior to our attempt to detect the faint companion in these spectra. We use
the spectroscopic orbit from GMC+2014 to shift the template spectrum for the
primary star to the radial velocity corresponding to the time of mid-exposure
for each SPIRou spectrum and then divide the observed spectrum by the shifted
model spectrum.

The correction for telluric absorption in the observed spectra will be
imperfect so we mask pixels where the telluric absorption is greater than 50
per cent. We also mask all pixels in order 47 at wavelengths $>1616$ nm
because there is a strong telluric absorption band at these wavelengths. The
removal of spectral features from the primary star will also be imperfect so
we mask pixels where absorption lines in the template spectrum are deeper than
50 per cent. We then flatten the spectrum by dividing the data by a 16th-order
polynomial fit by least-squares to the unmasked data in each order.

Outliers due to cosmic ray hits on the detector and other image anomalies were
then identified and removed by flagging pixels more than 4 times the inter-
quartile range from the mean in 10 blocks of data per order.

The signal-to-noise is similar for each spectrum but varies quite strongly
with wavelength so we use 1.25$\times$ the mean absolute deviation of the data
across the observed spectra to assign a standard error to the pixels at each
wavelength.

###  3.3 Detection of the M dwarf in the SPIRou spectra

The signal from the M dwarf is too weak to be detected in the individual
SPIRou spectra, but it is possible to measure the semi-amplitude of M dwarf’s
spectroscopic orbit, $K_{2}$, by calculating the average cross-correlation
function against a suitable template spectrum after shifting these CCFs to the
rest frame of the M dwarf assuming a range of $K_{2}$ values. The barycentric
radial velocity of the M dwarf at the time of mid-exposure for each spectrum
is

$V_{r,2}=K_{2}\,[\cos(\nu+\omega_{2})+e\cos(\omega_{2})]$ (1)

The value of the eccentricity $e$ and the longitude of periastron
$\omega_{2}=\omega_{1}+\pi$ are known accurately from the spectroscopic orbit
of the primary star with longitude of periastron $\omega_{1}$, taken from
GMC+2014. Similarly, the true anomaly at the time of mid-exposure, $\nu$, can
be accurately predicted from the values of $T_{0}$ (time of mid-transit), $P$
(orbital period), $e$ and $\omega_{1}$, also taken from GMC+2014.

We use synthetic spectra taken from Husser et al. (2013) as a template for the
spectra of the M dwarf, using linear interpolation to create a spectrum
appropriate for $T_{\rm eff}=3300$ K, ${\rm[Fe/H]}=-0.4$, $\log g=5.0$ and
$[\alpha/{\rm Fe}]=0.0$. The cross-correlation function is calculated order-
by-order. Low-frequency noise in the data for each order was removed prior to
cross-correlation using a 5-th order high-pass Butterworth filter with a
critical frequency of 32/4096 pixels-1. The data are apodized using a Gaussian
filter with a standard deviation of 64 pixels applied to the data at each end
of the order. The correlation coefficient for each order is calculated after
shifting the template according to radial velocity computed with equation (1)
includes the weights calculated from the estimated standard errors on each
pixel. This is repeated for a uniform grid of $K_{2}$ values. The average CCF
over all orders and all exposures as a function of $K_{2}$ (“stacked CCF”) is
shown in Fig. 3. Gómez Maqueo Chew et al. (2014) estimate that $K_{2}=80.3\pm
1.5$ km s-1. There is indeed a peak in the stacked CCF near this value of
$K_{2}$. To measure the position of this peak we model the stacked CCF as a
Gaussian process (GP) plus a Gaussian profile. We use the celerite package
(Foreman-Mackey et al., 2017) to compute the likelihood for a GP with a kernel
of the form $k(\tau)=a_{j}\,e^{-c_{j}\,\tau}$ and the affine-invariant Markov
chain Monte Carlo sampler emcee (Foreman-Mackey et al., 2013) to sample the
posterior probability distribution for the model parameters. Based on this
analysis, the peak in the stacked CCF occurs at $K_{2}=82.9\pm 0.7$ km s-1 and
has a width of $5.7\pm 0.6$ km s-1.

The broad peak in the stacked CCF around $K_{2}=0$ is due to imperfect removal
of telluric features and spectral features from the G0V primary star. We
compared the stacked CCF to the average CCF computed with negative values of
$K_{2}$ plotted against $|K_{2}|$, i.e. the mirror image of the stacked CCF.
As can be seen in Fig. 3, there is no corresponding peak at $K_{2}\approx-83$
km s-1. This reassures us that the peak at $K_{2}\approx+83$ km s-1 is
unlikely to be due to imperfect removal of telluric features or spectral
features from the G0V primary star. We used a fit to the stacked CCF done in
the same way as above but excluding data around the peak at $K_{2}=83$ km s-1
to estimate the statistical significance of this feature. Based on the GP
prediction of the correlated noise in this region shown in Fig. 3, we estimate
that the peak height corresponds to a detection with a significance $\approx
4$-$\sigma$. We also verified that the height of the peak in the stacked CCF
is very close to the height expected for an M-dwarf companion given the flux
ratio $\ell_{T}\approx 0.00155$ inferred from the depth of the secondary
eclipse in the TESS light curve. We subtracted a scaled version of the
template M-dwarf spectrum from the spectra used to compute the stacked CCF
based on this flux ratio in the TESS band and re-computed the stacked CCF. As
can be seen in Fig. 3, the resulting stacked CCF has no peak near $K_{2}=83$
km s-1. Based on these three tests, we are confident that our detection of the
M-dwarf is robust and that the measurement of $K_{2}$ is reliable.

Figure 3: Mean cross-correlation function of EBLM J0113+31 after shifting to
the rest frame of EBLM J0113+31 B assuming a range of $K_{2}$ values. Upper
panel: Gaussian process fit (orange) of a Gaussian profile to the peak near
$K_{2}=83$ km/s in the stacked CCF (black points). The maximum-likelihood
Gaussian profile is plotted in dark blue and 50 samples from the posterior
probability distribution are plotted in light grey. Upper middle panel: the
stacked CCF (solid line) and its reflection about $K_{2}=0$ (dashed line). The
estimated value of $K_{2}=80.3$ km/s from GMC+2014 is indicated by a vertical
dotted line. Lower middle panel: Gaussian process fit to the stacked CCF
excluding the peak near $K_{2}=83$ km/s. The orange shaded region shows
$2\times$ the standard error range on the predicted values of the Gaussian
process. Lower panel: the stacked CCF (solid line) and its reflection about
$K_{2}=0$ (dashed line) computed for spectra with the signature of the M dwarf
removed using a model spectrum with $T_{\rm eff}=3300$ K.

###  3.4 Initial assessment of the CHEOPS data

We used the software package pycheops (Maxted et al., 2021) to make an initial
assessment of the light curve data from each of the three CHEOPS visits to the
target listed in Table 1. We excluded data from the analysis where the
background level in the images due to scattered light is more than 20 per cent
larger than the median value during the visit. The data file provided by the
data reduction pipeline (DRP, Hoyer et al., 2020) includes a quantity
LC_CONTAM that is an estimate of the contamination of the photometric aperture
by nearby stars. This quantity varies during the orbit because of the rotation
of the spacecraft and the strongly asymmetric point spread function of the
instrument. This calculation of LC_CONTAM is based on a simulation of the
field of view using the mean G-band magnitudes of the target and nearby stars
from Gaia DR2. Fig. 1 shows the results of this simulation for one image. This
contamination estimate does not account for variability of target itself, so
we added a new function to pycheops version 1.0.2 that corrects the measured
flux (FLUX) by subtracting the value ${\tt LC\\_CONTAM}\times
10^{-0.4(G-G_{0})}$ from FLUX. The zero-point $G_{0}$ is calculated from the
average value of

$-2.5\log[({\tt LC\\_CONTAM}+1)\times 10^{-0.4G}\times f_{\rm frac}/{\tt
FLUX}],$

where $f_{\rm frac}$ is the fraction of the total flux from the target in the
photometric aperture, $G$ is the mean G-band magnitude of the target, and the
average is taken over data points outside of transit and eclipse.

Based on the simulated image of the field of view shown in Fig. 1, we decided
to use the OPTIMAL aperture with a radius of 40 pixels for our analysis. This
maximises the contamination of the photometric aperture but minimises the
uncertainty in this quantity due to errors in measuring the positions of the
stars in the image and, hence, the fraction of the flux from each star that is
contained in the aperture. We repeated our analysis using the DEFAULT aperture
with a radius of 25 pixels and found that the results are entirely consistent
with those presented here. For each visit we calculate a best-fit for a
transit or eclipse model to the light curve including linear decorrelation
against LC_CONTAM to account for small errors in estimating the amplitude of
the variations in this quantity. We then calculate the best-fit light curves
including each of the other available decorrelation parameters and add them
one-by-one if the Bayes factor for the parameter exceeds 1. The decorrelation
parameters selected by this method are listed in Table 1. The fit to the data
from a typical visit including these detrending parameters in shown in Fig. 4.

Figure 4: CHEOPS light curve from one visit to observe the transit of EBLM
J0113+31. Upper panel: The observed light curve is displayed in cyan. The dark
blue points are the data points binned over 0.01 phase units. The full model
including instrumental trends is shown in brown and the transit model without
trends is shown in green. Lower panel: Residuals obtained after subtraction of
the best-fit model.

###  3.5 Updated transit time ephemeris

The two times of mid-transit measured from the CHEOPS data during the initial
assessment of the data described above are listed in Table 4 together with the
time of mid-transit from GMC+2014 and one new time of mid-transit from a
least-squares fit to the TESS light curve using the transit model from
pycheops. From a least-squares fit to these data we obtain the following
updated ephemeris for the times of mid-transit in EBLM J0113+31:

${\rm BJD}~{}T_{\rm mid}=2459107.068051(45)+14.27684012(73)\,{\rm E}$ (2)

There is no evidence for any change in orbital period greater than
$|\dot{P}/P|\approx 1\times 10^{-5}$ from these times of mid-transit.

Table 4: Times of mid-transit for EBLM J0113+31. Residuals from the linear ephemeris given in section3.5 are given in the second column. BJD$-2450000$ |  $({\rm O}-{\rm C})$ [s] | Source
---|---|---
$6023.27063\pm 0.00036$ | $3.9$ | GMC+2014
$8778.70047\pm 0.00042$ | $-22.3$ | TESS
$9178.45224\pm 0.00017$ | $-1.0$ | CHEOPS
$9506.81960\pm 0.00013$ | $2.2$ | CHEOPS

###  3.6 Combined analysis of light curve and radial velocity data

We used the light curve model ellc (Maxted, 2016) to calculate synthetic light
curves in the TESS and CHEOPS bands, and the spectroscopic orbit of the
primary star. This model gives us more flexibility in choosing the level of
numerical noise in these synthetic light curves than is possible with the
qpower2 algorithm used in pycheops (Maxted & Gill, 2019). For the analysis
presented here we used the “default” grid size for the primary star and the
“very_sparse” grid size for the companion, which gives numerical noise of only
a few ppm at most orbital phases and everywhere less than 10 ppm. We also
tested for the impact of the gravitational distortion of the stars by their
mutual gravity on the light curve. This is less than 1.5 ppm through the
transit so we assumed spherical stars for our analysis in order to speed-up
the calculation.

The parameters of the binary star model are: the radii of the stars in units
of the semi-major axis (fractional radii), $r_{1}=R_{1}/a$ and
$r_{2}=R_{2}/a$; the surface brightness ratios in the TESS and CHEOPS bands,
$S_{\rm T}$ and $S_{\rm C}$, respectively; the orbital inclination, $i$; the
time of mid-transit, $T_{0}$; the orbital period, $P$;
$f_{s}=\sqrt{e}\sin(\omega)$ and $f_{c}=\sqrt{e}\cos(\omega)$, where $e$ is
the orbital eccentricity and $\omega$ is the longitude of periastron for the
primary star; the semi-amplitude of the primary star’s spectroscopic orbit,
$K_{1}$; the limb-darkening parameters assuming a power-2 limb-darkening law,
$h_{\rm 1,T}$ and $h_{\rm 2,T}$ in the TESS band, and $h_{\rm 1,C}$ and
$h_{\rm 2,C}$ in the CHEOPS band. The ephemeris for the time of mid-transit
derived in Section 3.5 is very accurate so we fix $T_{0}$ and $P$ at these
values in our analysis. The curvature of the light curve between the second
and third contact points is very clearly seen in the CHEOPS and TESS light
curves, and is almost directly related to the parameters $h_{\rm 1,C}$ and
$h_{\rm 1,T}$, respectively, so we leave this as a free parameter in the
analysis. The parameters $h_{\rm 2,C}$ and $h_{\rm 2,T}$ will have a much more
subtle influence on the light curve that is almost entirely confined to the
ingress and egress phases so we impose priors on these parameters based on the
tabulated values of $h_{2}$ in the TESS and CHEOPS bands from Maxted (2018).
The width of the priors is based on the comparison of these tabulated values
to the observed values of this parameter from an analysis of the Kepler light
curves of transiting exoplanet systems in the same study.

Prior to the analysis of the CHEOPS data combined with the other data sets, we
applied a correction for hot pixels in the photometric aperture.
Quantitatively, we define hot pixels as pixels with dark current above 3 e-
s-1. Since the beginning of the mission, hot pixels have appeared regularly in
the CHEOPS CCD at a rate of $\sim$100 new hot pixels per day. The CHEOPS
Instrument Team monitors closely the number and location of hot pixels.
Approximately once per week, “dark images” are acquired for that purpose (10
full frame images obtained observing a region of the sky void of stars). These
images are used to produce the reference files that track the location and
dark current of hot pixels. These reference file are available from the CHEOPS
data archive.666https://cheops-archive.astro.unige.ch/archive_browser/ We used
hot pixel maps generated about 2 days after each visit to EBLM J0113+31 to
calculate the contribution of these hot pixels to the count rate in the
photometric aperture. The hot-pixel contamination is $\approx 0.6$ per cent in
the OPTIMAL aperture for the visit in 2020 and $\approx 1.2$ per cent for the
visits in 2021. The hot-pixel contamination in the DEFAULT aperture is
$\approx 0.3$ per cent for all visits. The hot pixel contamination is
calculated for every image but the variation in this quantity is small ($\loa
10$ per cent of its value) so we apply the correction by subtracting the mean
value of the contamination during the visit from the count rate.

Our model includes the parameter $\ell_{\rm 3,C}$ that is a constant added to
the synthetic CHEOPS light curve to account for contamination of the
photometric aperture. We applied a correction to the light curves for
contamination prior to the combined analysis so, to account for uncertainties
in these corrections, we assign a Gaussian prior to $\ell_{\rm 3,C}$ with a
mean value of 0 and a standard error equal to 50 per cent of the total
contamination estimate. Similarly, the parameter $\ell_{\rm 3,T}$ accounts for
the contamination of the photometric aperture shown in Fig. 2 used to extract
the TESS light curve. We noticed that the entry TIC 400048098 in the TESS
input catalogue (TIC, Stassun et al., 2019) has no counterpart in GAIA EDR3
(Gaia Collaboration et al., 2016, 2021) so we assume that this is a spurious
entry and do not include it in our calculation of the contamination. The star
TIC 400048094 appears near the edge of the default photometric aperture
provided with the target pixel file. We added one pixel to this aperture so
that there is no ambiguity over whether this star should be included in the
calculation of the contamination or not. From the $T$ magnitudes listed the
TIC we estimate $\ell_{\rm 3,T}=0.0030$. We allow this parameter to vary in
the fit but assign a Gaussian prior to it, assuming an arbitrary uncertainty
of 50 per cent.

The light curves produced by CHEOPS are known to have very low levels of
instrumental noise after decorrelation. Similarly, the TESS light curve
following correction for instrumental trends that we calculated with
lightkurve shows little sign of residual instrumental noise or stellar
variability. We therefore adopt a white noise model for our analysis and
assume that the standard deviation per point in the TESS and CHEOPS light
curves – $\sigma_{\rm T}$ and $\sigma_{\rm C}$, respectively – are the same
for all data points from the same instrument. The logarithm of these standard
errors are included as a hyperparameters in our analysis by correctly
normalising the calculation of the posterior probability distribution. We only
include data from the TESS light curves at orbital phases near the transit and
eclipses in this analysis. For both the CHEOPS and TESS data, each section of
data around a transit or eclipse is divided by a straight line fit to the data
either side of the transit or eclipse prior to analysis.

We use all the radial velocities published by GMC+2014 plus the new radial
velocities from Table 3 in our analysis. We see no evidence for excess noise
in the radial velocities so we use their standard errors as quoted for the
calculation of the posterior probability distribution.

In total, we are using nine sets of data, each of which has an uncertain zero-
point that should be included in the analysis. Additionally, there are eleven
basis functions that are used for the removal of instrumental noise from the
CHEOPS data, each with its own coefficient that should be varied independently
during the fit to the data. To avoid explicitly calculating these nuisance
parameters we use the procedure described by Luger et al. (2017), in which the
likelihood for any proposed set of model parameters marginalised over a set of
nuisance parameters for a linear model can be calculated by modifying the
covariance matrix.

We used emcee (Foreman-Mackey et al., 2013), a python implementation of an
affine invariant Markov chain Monte Carlo (MCMC) ensemble sampler, to
calculate the posterior probability distribution of the model parameters. The
maximum-likelihood model fit to the data is shown in Fig. 5. The mean and
standard error of the posterior probability distributions for each of the
model parameters and various derived parameters are given in Table 5.

The parameters in Table 5 can be combined with our measured value of $K_{2}$
from the analysis of the SPIRou spectra to determine the masses and radii of
both stars with no additional model input. To account for the correlations
between parameters, we do this using the sampled posterior probability
distribution for the relevant parameters generated by emcee together with a
sample of $K_{2}$ values assuming a Gaussian distribution for this parameter.
The masses and radii derived are given in Table 6.

Figure 5: Top panel: radial velocity measurements for EBLM J0113+31 (points)
and our maximum-likelihood model (line) based on a fit to the combined radial-
velocity and light-curve data. Middle panel: TESS (red points) and CHEOPS
(blue points) photometry of the transit and eclipse in EBLM J0113+31. The
maximum-likelihood models based on a fit to the combined radial-velocity and
light-curve data is also shown (lines). Data obtained during around the
eclipses are plotted as function of orbital phase $-0.5323$. Data and models
have been offset vertically for clarity. The CHEOPS data have been corrected
for instrumental noise calculated as part of the analysis. Lower panel:
Residuals from the maximum likelihood models plotted in the middle panel.
Table 5: Fit to RV and LC data. ${\cal N}(\mu,\sigma)$ denotes a Gaussian
prior applied to a parameter with mean $\mu$ and standard deviation $\sigma$.

Parameter | Value | Notes
---|---|---
$R_{1}/a$ | $0.05348\pm 0.00031$ |
$R_{2}/a$ | $0.008111\pm 0.000063$ |
$i\>[^{\circ}]$ | $89.110\pm 0.041$ |
$f_{s}$ | $-0.54885\pm 0.00043$ |
$f_{c}$ | $0.08693\pm 0.00026$ |
$S_{\rm T}$ | $0.0675\pm 0.0033$ |
$S_{\rm C}$ | $0.0384\pm 0.0023$ |
$K_{1}$ [km/s] | $15.861\pm 0.010$ |
$h_{\rm 1,C}$ | $0.7683\pm 0.0038$ |
$h_{\rm 2,C}$ | $0.720\pm 0.036$ | ${\cal N}(0.409,0.045)$
$h_{\rm 1,T}$ | $0.8008\pm 0.0074$ |
$h_{\rm 2,T}$ | $0.779\pm 0.022$ | ${\cal N}(0.379,0.045)$
$\ell_{\rm 3,C}$ | $0.007\pm 0.009$ |  ${\cal N}(0.000,0.012)$, 1
$\ell_{\rm 3,T}$ | $0.019\pm 0.010$ |  ${\cal N}(0.030,0.015)$, 2
$\ln\sigma_{\rm C}$ | $-7.80\pm 0.02$ |
$\ln\sigma_{\rm T}$ | $-7.04\pm 0.02$ |
Derived parameters
$e$ | $0.30879\pm 0.00045$ |
$\omega\>[^{\circ}]$ | $279.000\pm 0.031$ |
$\sin i$ | $0.99988\pm 0.00001$ |
$R_{2}/R_{1}$ | $0.15164\pm 0.00073$ |
$\ell_{\rm T}$ | $0.00155\pm 0.00008$ | Flux ratio, TESS
$\ell_{\rm C}$ | $0.00088\pm 0.00005$ | Flux ratio, CHEOPS
$\sigma_{\rm T}$ [ppm] | $874\pm 18$ |
$\sigma_{\rm C}$ [ppm] | $410\pm 8$ |

1: After correction for contamination of the photometric aperture by nearby
stars and hot pixels. 2: Including flux from other stars in the photometric
aperture.

Table 6: Mass, radius, effective temperature and derived parameters for the stars in EBLM J0113+31. The metallicity [M/H] is estimated from our analysis of the spectrum of EBLM J0113+31 A. N.B. $T_{\rm eff,1}$ and $T_{\rm eff,2}$ are subject to additional systematic uncertainty of 10 K and 7 K, respectively. Parameter | Value | Error |
---|---|---|---
$M_{1}/{\mathcal{M}^{\rm N}_{\odot}}$ | 1.029 |  $\pm$ 0.025 | [2.4%]
$M_{2}/{\mathcal{M}^{\rm N}_{\odot}}$ | 0.197 |  $\pm$ 0.003 | [1.5%]
$R_{1}/{\mathcal{R}^{\rm N}_{\odot}}$ | 1.417 |  $\pm$ 0.014 | [1.0%]
$R_{2}/{\mathcal{R}^{\rm N}_{\odot}}$ | 0.215 |  $\pm$ 0.002 | [1.1%]
$T_{\rm eff,1}$ [K] | $6124$ |  $\pm$ 40 | [0.6%]
$T_{\rm eff,2}$ [K] | $3375$ |  $\pm$ 40 | [1.3%]
$\rho_{1}/{\rho^{\rm N}_{\odot}}$ | 0.362 |  $\pm$ 0.006 | [1.7%]
$\rho_{2}/{\rho^{\rm N}_{\odot}}$ | 19.9 |  $\pm$ 0.5 | [2.4%]
$\log g_{1}$ [cgs] | 4.148 |  $\pm$ 0.006 | [1.5%]
$\log g_{2}$ [cgs] | 5.068 |  $\pm$ 0.006 | [1.5%]
$\log L_{1}/{\mathcal{L}^{\rm N}_{\odot}}$ | 0.406 |  $\pm$ 0.014 | [3.2%]
$\log L_{2}/{\mathcal{L}^{\rm N}_{\odot}}$ | $-2.267$ |  $\pm$ 0.024 | [5.5%]
[M/H] | $-0.3$ |  $\pm$ 0.1 |

###  3.7 Direct measurement of the stellar effective temperature

The effective temperature for a star with Rosseland radius $R$ and total
luminosity $L$ is defined by the equation

$L=4\pi R^{2}\sigma_{\rm SB}{\rm T}_{\rm eff}^{4},$

where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant. For a binary star at
distance $d$, i.e. with parallax $\varpi=1/d$, the flux corrected for
extinction observed at the top of Earth’s atmosphere is

$f_{0,b}=f_{0,1}+f_{0,2}=\frac{\sigma_{\rm SB}}{4}\left[\theta_{1}^{2}{\rm
T}_{\rm eff,1}^{4}+\theta_{2}^{2}{\rm T}_{\rm eff,2}^{4}\right],$

where $\theta_{1}=2R_{1}\varpi$ is the angular diameter of star 1, and
similarly for star 2. All the quantities are known or can be measured for EBLM
J0113+31 provided we can accurately integrate the observed flux distributions
for the two stars independently. This is possible because photometry of the
combined flux from both stars is available from ultraviolet to mid-infrared
wavelengths, and the flux ratio at wavelengths where the majority of the flux
is emitted by the primary star has been measured from the TESS and CHEOPS
light curves. Although we have no direct measurement of the flux ratio at
infrared wavelengths, we can make a reasonable estimate for the small
contribution of the M-dwarf to the measured total infrared flux using
empirical colour – $T_{\rm eff}$ relations. The M-dwarf contributes less than
0.2 per cent to the total flux so it is not necessary to make a very accurate
estimate of the M-dwarf flux distribution in order to derive an accurate value
of $T_{\rm eff}$ for the G0-type primary star.

The photometry used in this analysis is given in Table 7. The NUV and FUV
magnitudes are taken from GALEX data release GR7 (Bianchi et al., 2014) with a
correction to the IUE flux scale based on the results from Camarota & Holberg
(2014). We assume that the flux from the M-dwarf at ultraviolet wavelengths is
negligible. The Gaia photometry is from Gaia data release EDR3. J, H and Ks
magnitudes are from the 2MASS survey (Skrutskie et al., 2006). WISE magnitudes
are from the All-Sky Release Catalog (Cutri & et al., 2012) with corrections
to Vega magnitudes made as recommended by Jarrett et al. (2011). Photometry in
the PanSTARRS-1 photometry system is taken from Tonry et al. (2018). Details
of the zero-points and response functions used to calculate synthetic
photometry from an assumed spectral energy distribution are given in Miller et
al. (2020).

To estimate the reddening towards EBLM J0113+31 we use the calibration of
E(B$-$V) versus the equivalent width of the interstellar Na I D1 line by
Munari & Zwitter (1997). To measure EW(Na I D1) we used 11 spectra obtain with
the FIES spectrograph on the Nordic Optical Telescope used in medium
resolution mode ($R=46,000$). We first shifted these spectra into the rest
frame of the primary star and then took the median value at each wavelength to
obtain a high signal-to-noise spectrum of the G0V primary star. We then
divided each observed spectrum by this spectrum of the G0V primary star after
shifting it back to the barycentric rest frame. We then took the median of
these residual spectra to obtain a high signal-to-noise spectrum of the
interstellar features. The equivalent width of the Na I D1 line measured by
numerical integration is EW(Na I D${}_{1})=77.1\pm 6.0$ mÅ. This value is less
than the values of EW(Na I D1) for all the stars in the calibration sample of
Munari & Zwitter (1997). To estimate the uncertainty on the value of E(B$-$V)
for EBLM J0113+31 we take the sample standard deviation for the 5 stars in the
calibration sample with the lowest values of EW(Na I D${}_{1})\approx 250$ mÅ.
Based on this analysis we obtain the estimate E(B$-$V$)=0.002\pm 0.012$. We
use this as a Gaussian prior in our analysis but exclude negative values of
E(B$-$V).

To establish colour – $T_{\rm eff}$ relations suitable for dwarf stars with
$3100\,{\rm K}<{\rm T}_{\rm eff}<3500\,{\rm K}$ we use a robust linear fit to
the stars listed in Table 6 of Fouqué et al. (2018) within this $T_{\rm eff}$
range. Photometry for these stars is taken from the TESS input catalogue. To
estimate a suitable standard error for a Gaussian prior based on this fit we
use 1.25$\times$ the mean absolute deviation of the residuals from the fit.
Colour – $T_{\rm eff}$ relations suitable for the primary G0V star were
calculated in similar way based on stars selected from the Geneva-Copenhagen
survey (Holmberg et al., 2009; Casagrande et al., 2011) with $5950\,{\rm
K}<{\rm T}_{\rm eff}<6250\,{\rm K}$, $E({\rm B}-{\rm V})<0.01$ and $3.5<\log
g<4.5$. The results are given in Table 8.

Table 7: Observed apparent magnitudes for EBLM J0113+31 and predicted values based on our synthetic photometry. The predicted magnitudes are shown with error estimates from the uncertainty on the zero-points for each photometric system. The pivot wavelength for each band pass is shown in the column headed $\lambda_{\rm pivot}$. The magnitudes of the primary G0V star alone corrected for the contribution to the total flux from the M-dwarf are shown in the column headed $m_{1}$. The flux ratio in each band is shown in the fina column. Band |  $\lambda_{\rm pivot}$ [nm] | Observed | Computed | $\rm O-\rm C$ | $m_{1}$ |  $\ell$ [%]
---|---|---|---|---|---|---
FUV | 154 | $20.01\pm 0.54$ | $20.74\pm 0.13$ | $-0.73\pm 0.55$ | $20.01\pm 0.54$ | 0.00
NUV | 230 | $14.28\pm 0.71$ | $14.41\pm 0.15$ | $-0.13\pm 0.73$ | $14.28\pm 0.71$ | 0.00
G | 622 | $9.920\pm 0.003$ | $9.919\pm 0.003$ | $+0.002\pm 0.004$ | $9.922\pm 0.003$ | 0.09
BP | 511 | $10.197\pm 0.003$ | $10.202\pm 0.003$ | $-0.005\pm 0.004$ | $10.197\pm 0.003$ | 0.04
RP | 777 | $9.477\pm 0.004$ | $9.475\pm 0.004$ | $+0.002\pm 0.005$ | $9.479\pm 0.004$ | 0.17
gP1 | 485 | $10.249\pm 0.020$ | $10.234\pm 0.005$ | $+0.015\pm 0.021$ | $10.249\pm 0.020$ | 0.03
rP1 | 620 | $9.961\pm 0.024$ | $9.911\pm 0.005$ | $+0.050\pm 0.025$ | $9.962\pm 0.024$ | 0.07
iP1 | 754 | $9.868\pm 0.021$ | $9.820\pm 0.005$ | $+0.048\pm 0.022$ | $9.870\pm 0.021$ | 0.16
J | 1241 | $8.982\pm 0.024$ | $8.973\pm 0.005$ | $+0.009\pm 0.025$ | $8.987\pm 0.024$ | 0.45
H | 1650 | $8.692\pm 0.029$ | $8.713\pm 0.005$ | $-0.021\pm 0.029$ | $8.699\pm 0.029$ | 0.60
Ks | 2164 | $8.620\pm 0.024$ | $8.652\pm 0.005$ | $-0.032\pm 0.025$ | $8.628\pm 0.024$ | 0.73
W1 | 3368 | $8.590\pm 0.023$ | $8.613\pm 0.002$ | $-0.023\pm 0.023$ | $8.600\pm 0.023$ | 0.91
W2 | 4618 | $8.629\pm 0.020$ | $8.619\pm 0.002$ | $+0.010\pm 0.020$ | $8.642\pm 0.020$ | 1.18
W3 | 12073 | $8.633\pm 0.021$ | $8.617\pm 0.002$ | $+0.016\pm 0.021$ | $8.651\pm 0.021$ | 1.64
W4 | 22194 | $8.38\pm 0.22$ | $8.674\pm 0.002$ | $-0.29\pm 0.22$ | $8.40\pm 0.22$ | 1.64
Table 8: Colour-$T_{\rm eff}$ relations used to establish Gaussian priors on the flux ratio at infrared wavelengths for EBLM J0113+31. The dependent variables are $X_{1}=T_{\rm eff,1}-6.1\,{\rm kK}$ and $X_{2}=T_{\rm eff,2}-3.3\,{\rm kK}$.  Colour | Primary | Secondary
---|---|---
V$-$J | $1.048-0.4257\,X_{1}\pm 0.015$ | $4.187-2.762\,X_{2}\pm 0.11$
V$-$H | $1.288-0.5568\,X_{1}\pm 0.019$ | $4.776-2.552\,X_{2}\pm 0.15$
V$-$Ks | $1.357-0.5926\,X_{1}\pm 0.016$ | $5.049-2.776\,X_{2}\pm 0.12$
V$-$W1 | $1.405-0.5829\,X_{1}\pm 0.027$ | $5.207-2.720\,X_{2}\pm 0.12$
V$-$W2 | $1.411-0.5753\,X_{1}\pm 0.045$ | $5.365-2.957\,X_{2}\pm 0.11$
V$-$W3 | $1.355-0.5919\,X_{1}\pm 0.022$ | $5.477-3.091\,X_{2}\pm 0.13$
V$-$W4 | $1.397-0.5812\,X_{1}\pm 0.045$ | $5.620-3.248\,X_{2}\pm 0.23$

The method we have developed to measure $T_{\rm eff}$ for eclipsing binary
stars is described fully in Miller et al. (2020). Briefly, we use emcee
(Foreman-Mackey et al., 2013) to sample the posterior probability distribution
$P(\Theta|D)\propto P(D|\Theta)P(\Theta)$ for the model parameters $\Theta$
with prior $P(\Theta)$ given the data, $D$ (observed apparent magnitudes and
flux ratios). The model parameters are

$\Theta=\left({\rm T}_{\rm eff,1},{\rm T}_{\rm
eff,2},\theta_{1},\theta_{2},{\rm E}({\rm B}-{\rm V}),\sigma_{\rm
ext},\sigma_{\ell},c_{1,1},\dots,c_{2,1},\dots\right).$

The prior $P(\Theta)$ is calculated using the angular diameters $\theta_{1}$
and $\theta_{2}$ derived from the radii $R_{1}$ and $R_{2}$ and the parallax
$\varpi$, the priors on the flux ratio at infrared wavelengths based on the
colour – T${\rm eff}$ relations in Table 8, and the Gaussian prior on the
reddening described above. The hyperparameters $\sigma_{\rm ext}$ and
$\sigma_{\ell}$ account for additional uncertainties in the synthetic
magnitudes and flux ratio, respectively, due to errors in zero-points,
inaccurate response functions, stellar variability, etc. The parallax is taken
from Gaia EDR3 with corrections to the zero-point from Flynn et al. (2022).

To calculate the synthetic photometry for a given value of $T_{\rm eff}$ we
used a model spectral energy distribution (SED) multiplied by a distortion
function, $\Delta(\lambda)$. The distortion function is a linear superposition
of Legendre polynomials in log wavelength. The coefficients of the distortion
function for star 1 are $c_{1,1},c_{1,2},\dots$, and similarly for star 2. The
distorted SED for each star is normalized so that the total apparent flux
prior to applying reddening is $\sigma_{\rm SB}\theta^{2}{\rm T}_{\rm
eff}^{4}/4$. These distorted SEDs provide a convenient function that we can
integrate to calculate synthetic photometry that has realistic stellar
absorption features, and where the overall shape can be adjusted to match the
observed magnitudes from ultraviolet to infrared wavelengths, i.e. the
effective temperatures we derive are based on the integrated stellar flux and
the star’s angular diameter, not SED fitting.

For this analysis we use model SEDs computed from BT-Settl model atmospheres
(Allard et al., 2013) obtained from the Spanish Virtual
Observatory.777http://svo2.cab.inta-csic.es/theory/newov2/index.php?models=bt-
settl We use linear interpolation to obtain a reference SED for the G0V star
appropriate for $T_{\rm eff,1}=6130$ K, $\log g_{1}=4.15$, $[{\rm Fe/H}]=-0.3$
and $[{\rm\alpha/Fe}]=0.0$. For the reference SED for the M dwarf companion we
assume $T_{\rm eff,1}=3380$ K, $\log g_{1}=5.0$, and the same composition. We
experimented with distortion functions with 1, 2, 3, 4 coefficients per star
and found the results to be very similar in all cases. The results presented
here use two distortion coefficient per star because there is no improvement
in the quality if the fit if we use a larger number of coefficients. The
predicted apparent magnitudes including their uncertainties from errors in the
zero-points for each photometric system are compared to the observed apparent
magnitudes in Table 7. The posterior probability distribution for the model
parameters is summarised in Table 9 and the spectral energy distribution is
plotted in Fig. 6.

The random errors quoted in Table 9 do not allow for the systematic error due
to the uncertainty in the absolute calibration of the CALSPEC flux scale
(Bohlin et al., 2014). This additional systematic error is 10 K for the G0V
primary star and 7 K for the M-dwarf companion.

Table 9: Results from our analysis to obtain the effective temperatures for both stars in EBLM J0113+31. N.B. $T_{\rm eff,1}$ and $T_{\rm eff,2}$ are subject to additional systematic uncertainty of 10 K and 7 K, respectively. Parameter | Value | Error | Units
---|---|---|---
$T_{\rm eff,1}$ | $6124$ | $\pm 40$ | K
$T_{\rm eff,2}$ | $3375$ | $\pm 40$ | K
$\theta_{1}$ | $0.0745$ | $\pm 0.0007$ | mas
$\theta_{2}$ | $0.0113$ | $\pm 0.0001$ | mas
E(B$-$V) | $0.010$ | $\pm 0.007$ |
$\sigma_{\rm ext}$ | $0.014$ | $\pm 0.011$ |
$\sigma_{\ell}$ | $0.0002$ | $\pm 0.0001$ |
$c_{1,1}$ | $0.06$ | $\pm 0.03$ |
$c_{1,2}$ | $-0.08$ | $\pm 0.05$ |
$c_{2,1}$ | $0.3$ | $\pm 0.2$ |
$c_{2,2}$ | $-0.3$ | $\pm 0.2$ |
Figure 6: The spectral energy distribution (SED) of EBLM J0113+31. The
observed fluxes are plotted with open circles and the predicted fluxes for the
mean of the posterior probability distribution (PPD) integrated over the
response functions shown in grey are plotted with filled symbols. The SED
predicted by the mean of the PPD is plotted in dark blue and light blue shows
the SEDs produced from 100 random samples from the PPD. The contribution to
the total SED from the M dwarf (barely visible) is shown in orange. The W3 and
W4 mid-infrared bands also used in the analysis are not shown here.

###  3.8 Abundance analysis

We have used the H-band spectrum of EBLM J0113+31 A to estimate this star’s
metallicity. For this abundance analysis we used the observed SPIRou spectra
merged into 1-dimensional spectra provided by the observatory. We first
subtracted the model spectrum for the M-dwarf companion described in Section
3.3 from each of these 1-dimensional spectra, scaled such that the flux ratio
in the TESS band matched the value measured from the depth of the secondary
eclipse in the light curve. We then co-added the spectra in the rest frame of
the primary star to produce a high signal-to-noise spectrum of the G0V star
with negligible contamination from the M-dwarf.

For the analysis of this spectrum we used iSpec (Blanco-Cuaresma et al.,
2014b; Blanco-Cuaresma, 2019) with the APOGEE line list for atomic and
molecular data in the wavelength range 1500 – 1700 nm (Shetrone et al., 2015).
We followed Sarmento et al. (2020) in selecting Turbospectrum (Alvarez & Plez,
1998; Plez, 2012) for the spectrum synthesis assuming a micro-turbulent
velocity $v_{\rm mic}=1.06$ km s-1 with model atmospheres from the MARCS grid
(Gustafsson et al., 2008) and solar abundances from Grevesse et al. (2007). We
excluded from the fit $\pm 4$ nm around the two Brackett series lines at
1681.11 nm and 1641.17 nm, and also some instrumental features that occur near
the ends of the échelle orders at 1657 – 1659 nm and 1622- 1624 nm. We fixed
the value of $T_{\rm eff}=6124$ K and $\log g=4.15$. For the macro-turbulent
velocity we used the calibration by Valenti & Fischer (2005) to estimate
$v_{\rm mac}=4.67$ km s-1. We included the rotational broadening parameter
$v\sin i$ as a free parameter in the least-squares fit with a linear limb-
darkening coefficient of 0.5 in the H-band based on the results from Claret
(2018). We attempted a least-squares fit including the $\alpha$-element
abundance as a free parameter but found that the value obtain is not accurate
enough to be useful so we fixed $[\alpha/{\rm Fe}]=0$ in the least-squares
fit. From this least-squares fit we obtained $[{\rm M/H}]=-0.33\pm 0.01$ and
$v\sin i=6.6\pm 0.3$ km s-1. There are several additional sources of
uncertainty in this analysis, e.g. inaccurate normalisation, errors in atomic
data, approximations in the stellar atmosphere models, etc., so the accuracy
of our metallicity estimate will be much worse than the precision estimated
from the least-squares fitting algorithm (Blanco-Cuaresma, 2019; Jofré et al.,
2019). Based on the results from independent analyses of APOGEE spectra by
Jönsson et al. (2018), we assume an accuracy of 0.15 dex, i.e. $[{\rm
M/H}]=-0.33\pm 0.15$. The fit to the spectrum is shown in Fig. 7.

We used the co-added FIES spectra of the star to determine the stellar
atmospheric parameters ($T_{\mathrm{eff}}$, $\log{g}$, micro-turbulence, and
[Fe/H]) and chemical abundances following the methodology described in our
previous works (Sousa, 2014; Santos et al., 2013; Adibekyan et al., 2012;
Adibekyan et al., 2015). In brief, we make use of the equivalent widths (EW)
of spectral lines, as measured using the ARES v2 code888The last version of
ARES code (ARES v2) can be downloaded at http://www.astro.up.pt/~sousasag/ares
(Sousa et al., 2015), and we assume ionization and excitation equilibrium. The
process makes use of a grid of Kurucz model atmospheres (Kurucz, 1993) and the
radiative transfer code MOOG (Sneden, 1973).

For the stellar spectroscopic parameters we obtained $T_{\rm eff}=6025\pm 50$
K, $\log g=4.10\pm 0.05$, $V_{\mathrm{tur}}=1.07\pm 0.06$ km s-1 and $[{\rm
Fe/H}]=-0.31\pm 0.04$. Within the uncertainties, these values are in agreement
with those presented in Table|9. In order to be consistent, and because of
higher accuracy, we fixed the values of effective temperature and surface
gravity to $T_{\mathrm{eff}}=6124\pm 40$ K and $\log g=4.148\pm 0.006$ when
determining the abundances of individual elements. Our derivation of three
$\alpha$-elements ($[{\rm Mg/H}]=-0.18\pm 0.09$, $[{\rm Si/H}]=-0.26\pm 0.04$,
$[{\rm Ti/H}]=-0.22\pm 0.07$) indicates that EBLM J0113+31 is not an
$\alpha$-enhanced star ($[\alpha/{\rm Fe}]=0.09\pm 0.08$) which is typical for
stars in the Galactic thin-disk population (Adibekyan et al., 2011).

Figure 7: The H-band spectrum of EBLM J0113+31 A (blue) and a synthetic
spectrum fit by least-squares using iSpec (red). Residuals from the synthetic
spectrum fit are shown in green offset vertically by 1.05 units.

Using the astrometric data from Gaia EDR3 and the radial velocity of the
system ($11.179\pm 0.004$ km s-1, GMC+2014) we calculated the Galactic space
velocity components $(U,V,W)=(-17,16,21)$ km s-1 with respect to the local
standard of rest (Schönrich et al., 2010). Based on these velocities, adopting
the characteristics parameters of Galactic stellar populations of Reddy et al.
(2006), and following Adibekyan et al. (2012) we estimated a probability of
99% that the star belongs to the Galactic thin disk, which is in agreement
with our conclusion based on the composition of the star.

Based on the results from the analysis of the SPIRou and FIES spectra we adopt
the value $[{\rm M/H}]=-0.3\pm 0.1$ for the metallicity of EBLM J0113+31. The
co-added SPIRou spectra corrected for the contribution from the M-dwarf and
the co-added FIES spectrum are available from the supplementary online
information that accompanies this article.

##  4 Discussion

###  4.1 Astrometric noise due to binary orbital motion

The projected semi-major axis of the G0V star’s orbit is $\alpha_{1}=a_{1}/d$
= 0.11 mas, so we expect excess noise in the Gaia astrometry $\approx 0.1$ mas
due to the orbital motion of the primary star. Indeed, the astrometric excess
noise in the Gaia EDR3 catalogue for EBLM J0113+31 is 0.163 mas. This is
higher than expected for a good fit to the data for a single star with ${\rm
G}\approx 10$, and consistent with the noise expected from the orbital motion
of the G0V star. This will only lead to a systematic error in the parallax if
the position angle of the binary at the times of observation are not randomly
distributed around the binary star orbit. This can be checked using the
parameter ipd_gof_harmonic_amplitude provided in the EDR3 catalogue (Lindegren
et al., 2021). For EBLM J0113+31, this parameter takes the value 0.014, which
is less than the median value of this statistic for stars with 6-parameter
solutions in the magnitude range G=9-12 (0.020). Although the detection of the
astrometric noise is statistically significant, it is a small contribution to
the uncertainties on the parallax. The renormalised unit weight error for EBLM
J0113+31 is RUWE=1.154, which is only slightly higher than the median value
for stars with 6-parameter solutions in the magnitude range G=9-12
(RUWE=1.127), and is close to the expected value $\approx 1$ for “for well
behaved sources”.

We can therefore be confident that the orbital motion of the G0V star does not
produce a systematic error in the measured Gaia parallax.

###  4.2 Comparison to stellar evolution models

The mass, radius and effective temperature for both stars in EBLM J0113+31 are
given in Table 6, together with the derived surface gravity, mean stellar
density and luminosity for both stars.

We used the software package bagemass (Maxted et al., 2015) to compare the
parameters of the primary star, EBLM J0113+31 A, to a grid of stellar models
computed with the garstec stellar evolution code (Weiss & Schlattl, 2008). The
methods used to calculate the stellar model grid are described in Serenelli et
al. (2013). bagemass uses a Markov-chain Monte-Carlo method to explore the
posterior probability distribution (PPD) for the mass and age of a star based
on its observed $T_{\rm eff}$, luminosity, mean stellar density and surface
metal abundance [Fe/H]. We find a very good fit to the observed parameters of
EBLM J0113+31 A for an age of $6.7\pm 0.5$ Gyr, as can be seen in Fig. 8. More
the 99 per cent of samples from the PPD correspond to models where EBLM
J0113+311 A is a post main-sequence star that has exhausted all the hydrogen
in its core. The garstec model grid accounts for diffusion so the initial
metal abundance for this star is inferred to be ${\rm[Fe/H]}=-0.2\pm 0.1$.
Isochrones for the same age and initial metal abundance from the Dartmouth
stellar evolution database (Dotter et al., 2008) and the MESA Isochrones &
Stellar Tracks (MIST, Choi et al., 2016) are also shown in Fig. 8. There is
very good agreement between these different stellar evolution codes, as might
be expected given that the properties of EBLM J0113+31 A are similar to the
Sun and all three grids of stellar models are calibrated to match the observed
properties of the Sun.

Figure 8: EBLM J0113+31 A in the mass-radius and Hertzsprung-Russell diagrams
compared to isochrones for an age of $6.7\pm 0.7$ Gyr assuming an initial
metal abundance $[{\rm Fe/H}]=-0.2$ interpolated from a grid of garstec
stellar models. The ellipses show 1-$\sigma$ and 2-$\sigma$ confidence regions
on the parameters of EBLM J0113+31 A. Also shown are isochrones for the same
age and initial metal abundance from the Dartmouth stellar evolution database
(cyan dotted line) and MIST (green dashed line).

The same isochrones from the Dartmouth and MIST stellar model grids are
compared to the properties of EBLM J0113+31 B in Fig. 9. Our grid of garstec
models does not extend to these very low masses. The agreement between the
models and observations is reasonably good, which is somewhat surprising given
the long-standing observation that stellar models tend to under-predict the
radius and over-predict $T_{\rm eff}$ for low-mass stars (Spada et al., 2013;
Cassisi & Salaris, 2019; Zhou et al., 2014; Berger et al., 2006; Hoxie, 1973;
Lacy, 1977). This can be seen from the mass, radius and $T_{\rm eff}$
measurements for six other very low-mass stars in the same figure. These six
stars are members of three eclipsing binaries with orbital periods less 2
days. This complicates the interpretation of their properties in the light of
the so-called “radius inflation” problem because these stars will be forced to
rotate much faster than most single M-dwarf stars by tidal forces in these
short-period binaries. EBLM J0113+31 B is a valuable addition to the small
sample of well-characterised VLMSs because we have an independent estimate of
its age and initial metal abundance based on observations of the G0V primary
star to add to the accurate mass, radius and $T_{\rm eff}$ measurements.

Figure 9: EBLM J0113+31 B in the mass-radius and Hertzsprung-Russell diagrams
compared to isochrones for ages of 6.8 Gyr assuming $[{\rm Fe/H}]=-0.2$ from
the Dartmouth stellar evolution database (cyan dotted line) and MIST (green
dashed line). The ellipses show 1-$\sigma$ and 2-$\sigma$ confidence regions
on the parameters of EBLM J0113+31 B. Parameters for very low mass stars shown
as error bars in blue are taken from DEBCat (Southworth, 2015).

###  4.3 EBLM systems as benchmark stars

Benchmark FGK dwarf stars with direct $T_{\rm eff}$ measurements based on
angular diameters measured by interferometry typically have apparent
magnitudes ${\rm V}=1$ – 6 (Jofré et al., 2014). This is 5 – 10 magnitudes
brighter than the magnitude limits for large-scale spectroscopic surveys, so
special observing modes must be employed to obtain spectra of these benchmark
stars. These bright benchmark stars also tend to be single stars, so there are
often no direct measurements of their mass or surface gravity. It is difficult
to extend this sample because new candidates for benchmark stars will
necessarily be more distant than the existing sample, i.e. they will have
smaller angular diameters than the existing benchmark stars. For example, a
nominal Sun-like star at distance of 10 pc will have an angular diameter
$\theta=0.465$ mas, so a systematic error of only 0.04 mas, which is typical
for existing measurements (Karovicova et al., 2022), implies a systematic
error of 250 K in the measured value of $T_{\rm eff}$ for such a star.

In contrast, EBLM J0113+31 is within the magnitude range of recent large-scale
spectroscopic surveys, e.g. the TESS-HERMES survey ($10<{\rm V}<13.1$, Sharma
et al., 2018), LAMOST “VB mode” observations ($9.0\leq J\leq 12.5$, Luo et
al., 2015), and stars in open clusters observed as part of the Gaia-ESO survey
($9<{\rm V}<16.5$, Bragaglia et al., 2021). This makes it feasible to observe
EBLM J0113+31 and other EBLM binaries in exactly the same way as other stars
observed by these survey instruments as part of their routine operations. The
contribution of the M-dwarf to the total flux at optical wavelengths is $\loa
0.2$ per cent for EBLM binaries, so the M-dwarf will have a completely
negligible effect on the atmospheric parameters derived from the analysis of
the optical spectrum. This makes it possible to make an “end-to-end” test of
the accuracy of parameters derived by the combination of these survey
instruments plus their data processing and analysis pipelines. Even at near-
infrared wavelengths used by surveys such as APOGEE (Jönsson et al., 2018) the
contribution from the M-dwarf is $\loa 1$ per cent, so the results of any
analysis that includes a correction for this small contribution to the total
flux will be insensitive to the details of how this correction is done.

Many EBLM binaries in the magnitude range $10\loa{\rm V}\loa 12$ have been
identified and have well-determined spectroscopic orbits that have been
published (Triaud et al., 2017) or that are in preparation thanks to the EBLM
project and BEBOP survey (Standing et al., 2022). High-quality space-based
photometry is already available for many of these stars from the TESS survey
and/or from our on-going CHEOPS GTO programme. Several échelle spectrographs
that can provide high-resolution spectroscopy at near-infrared wavelengths are
currently operational on 4 – 10 m telescopes, e.g. CARMENES on the Calar Alto
Observatory 3.5-m telescope (Quirrenbach et al., 2016), NIRPS on the ESO 3.6-m
telescope (Grieves et al., 2021), CRIRES+ on the ESO 8.2-m VLT (Kaeufl et al.,
2004), and IRD on the 8.2-m Subaru telescope (Kotani et al., 2014). We can
also look forward to high-quality spectrophotometry and improved parallax
measurements for these EBLM systems in future data releases from the Gaia
mission.999https://www.cosmos.esa.int/web/gaia/release In summary, the
instrumentation, data and targets needed to create a network of moderately-
bright FGK dwarf stars covering both hemispheres that are ideal benchmark
stars for on-going large-scale spectroscopic surveys are all now available.

Apart from their utility as benchmarks stars for large-scale spectroscopic
surveys, follow-up observations of additional EBLM systems will also provide
valuable data on the properties of very low-mass stars. With observations
similar to those presented here we can create a sample VLMSs with precise and
accurate $T_{\rm eff}$, mass and radius measurements. These EBLM binaries will
have independent estimates for their age and initial metallicity based on the
observed properties of the primary stars in these systems. It is not feasible
to obtain a direct spectrum for these very faint companion stars, but it
should be possible given sufficiently high-quality data to estimate the
projected rotational velocity of the star from the width of the peak in the
stacked-CCF. Data of this quality will be very useful for testing and
calibrating models of very low mass stars that include additional physics to
account for the radius inflation problem (Mullan et al., 2018; Feiden &
Chaboyer, 2014).

Many of these EBLM binary systems will also be ideal benchmark stars for the
upcoming PLATO mission (Rauer et al., 2014) if we can measure model-
independent masses for the primary star using the techniques presented in this
study. The PLATO mission will focus on bright stars (4 – 11 mag) with the aim
to detect and characterize planets down to Earth-size by photometric transits.
Asteroseismology will be performed for these bright stars to obtain stellar
parameters, including masses and ages. The PLATO Definition Study
Report101010https://sci.esa.int/science-e/www/object/doc.cfm?fobjectid=59251
(“red book”) specifies that PLATO must be capable of delivering accurate
stellar ages with a precision of 10 per cent. Some corrections for systematic
errors in the current generation of stellar models will be needed to reach
this accuracy in stellar ages (Goupil, 2017). The planned observing strategy
includes a step-and-stare phase that will cover about 50 per-cent of the sky.
EBLM binaries can be used to perform “end-to-end” tests of the PLATO data
analysis to ensure that the mass estimates delivered for these stars are
accurate, and to calibrate the next generation of stellar models using direct
mass, radius, and $T_{\rm eff}$ measurements combined with asteroseismology.

##  5 Conclusions

We have derived precise and accurate masses, radii and effective temperatures
for both stars in the eclipsing binary system EBLM J0113+31. These data can be
used to validate and calibrate stellar models, empirical relations for stellar
properties, and to test data analysis techniques. With the techniques
established here, it is feasible to create a network of moderately-bright FGK
dwarf stars covering both hemispheres that are ideal benchmarks for on-going
large-scale spectroscopic surveys and for the upcoming PLATO mission.

## Acknowledgements

CHEOPS is an ESA mission in partnership with Switzerland with important
contributions to the payload and the ground segment from Austria, Belgium,
France, Germany, Hungary, Italy, Portugal, Spain, Sweden, and the United
Kingdom. The CHEOPS Consortium would like to gratefully acknowledge the
support received by all the agencies, offices, universities, and industries
involved. Their flexibility and willingness to explore new approaches were
essential to the success of this mission.

Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT)
which is operated from the summit of Maunakea by the National Research Council
of Canada, the Institut National des Sciences de l’Univers of the Centre
National de la Recherche Scientifique of France, and the University of Hawaii.
The observations at the Canada-France-Hawaii Telescope were performed with
care and respect from the summit of Maunakea which is a significant cultural
and historic site. Based on observations obtained with SPIRou, an
international project led by Institut de Recherche en Astrophysique et
Planétologie, Toulouse, France. PM is grateful to the observatory staff at the
CHFT for their help with the planning, execution and data reduction of the
SPIRou data used for this study.

Based on 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 paper includes data collected by the TESS mission, which is publicly
available from the Mikulski Archive for Space Telescopes (MAST) at the Space
Telescope Science Institure (STScI). Funding for the TESS mission is provided
by the NASA Explorer Program directorate. STScI is operated by the Association
of Universities for Research in Astronomy, Inc., under NASA contract NAS
5–26555. We acknowledge the use of public TESS Alert data from pipelines at
the TESS Science Office and at the TESS Science Processing Operations Center.

The authors would like to thank the anonymous referee for their useful
comments that have helped to improve this manuscript.

This research made use of Lightkurve, a Python package for Kepler and TESS
data analysis (Lightkurve Collaboration et al., 2018).

PM acknowledges support from UK Science and Technology Facilities Council
(STFC) research grant number ST/M001040/1.

NM is supported by STFC grant number ST/S505444/1.

SH gratefully acknowledges CNES funding through the grant 837319.

S.G.S. acknowledge support from FCT through FCT contract nr.
CEECIND/00826/2018 and POPH/FSE (EC).

ACC acknowledges support from STFC consolidated grant numbers ST/R000824/1 and
ST/V000861/1, and UKSA grant number ST/R003203/1.

YA and MJH acknowledge the support of the Swiss National Fund under grant
200020_172746.

MIS acknowledges support from STFC grant number ST/T506175/1.

We acknowledge support from the Spanish Ministry of Science and Innovation and
the European Regional Development Fund through grants ESP2016-80435-C2-1-R,
ESP2016-80435-C2-2-R, PGC2018-098153-B-C33, PGC2018-098153-B-C31,
ESP2017-87676-C5-1-R, MDM-2017-0737 Unidad de Excelencia Maria de Maeztu-
Centro de Astrobiología (INTA-CSIC), as well as the support of the Generalitat
de Catalunya/CERCA programme. The MOC activities have been supported by the
ESA contract No. 4000124370.

S.C.C.B. acknowledges support from FCT through FCT contracts
nr.IF/01312/2014/CP1215/CT0004.

XB, SC, DG, MF and JL acknowledge their role as ESA-appointed CHEOPS science
team members.

ABr was supported by the SNSA.

This project was supported by the CNES.

The Belgian participation to CHEOPS has been supported by the Belgian Federal
Science Policy Office (BELSPO) in the framework of the PRODEX Program, and by
the University of Liège through an ARC grant for Concerted Research Actions
financed by the Wallonia-Brussels Federation.

L.D. is an F.R.S.-FNRS Postdoctoral Researcher.

This work was supported by FCT - Fundação para a Ciência e a Tecnologia
through national funds and by FEDER through COMPETE2020 - Programa Operacional
Competitividade e Internacionalizacão by these grants: UID/FIS/04434/2019,
UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-AST/32113/2017 &
POCI-01-0145-FEDER- 032113, PTDC/FIS-AST/28953/2017 &
POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT.

B.-O.D. acknowledges support from the Swiss National Science Foundation
(PP00P2-190080). This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (project Four Aces. grant agreement No 724427). It has also been
carried out in the frame of the National Centre for Competence in Research
PlanetS supported by the Swiss National Science Foundation (SNSF). DE
acknowledges financial support from the Swiss National Science Foundation for
project 200021_200726.

MF and CMP gratefully acknowledge the support of the Swedish National Space
Agency (DNR 65/19, 174/18).

DG gratefully acknowledges financial support from the CRT foundation under
Grant No. 2018.2323 “Gaseous or rocky? Unveiling the nature of small worlds”.

M.G. is an F.R.S.-FNRS Senior Research Associate.

KGI is the ESA CHEOPS Project Scientist and is responsible for the ESA CHEOPS
Guest Observers Programme. She does not participate in, or contribute to, the
definition of the Guaranteed Time Programme of the CHEOPS mission through
which observations described in this paper have been taken, nor to any aspect
of target selection for the programme.

This work was granted access to the HPC resources of MesoPSL financed by the
Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01)
of the programme Investissements d’Avenir supervised by the Agence Nationale
pour la Recherche.

ML acknowledges support of the Swiss National Science Foundation under grant
number PCEFP2_194576.

GSc, GPi, IPa, LBo, VNa and RRa acknowledge the funding support from Italian
Space Agency (ASI) regulated by “Accordo ASI-INAF n. 2013-016-R.0 del 9 luglio
2013 e integrazione del 9 luglio 2015 CHEOPS Fasi A/B/C”.

This work was also partially supported by a grant from the Simons Foundation
(PI Queloz, grant number 327127).

IR acknowledges support from the Spanish Ministry of Science and Innovation
and the European Regional Development Fund through grant PGC2018-098153-B-
C33, as well as the support of the Generalitat de Catalunya/CERCA programme.

GyMSz acknowledges the support of the Hungarian National Research, Development
and Innovation Office (NKFIH) grant K-125015, a PRODEX Institute Agreement
between the ELTE Eötvös Loránd University and the European Space Agency
(ESA-D/SCI-LE-2021-0025), the Lendület LP2018-7/2021 grant of the Hungarian
Academy of Science and the support of the city of Szombathely.

V.V.G. is an F.R.S-FNRS Research Associate.

## Data Availability

The data underlying this article are available in the following repositories:
The Data & Analysis Center for Exoplanets (DACE) – https://dace.unige.ch/
(CHEOPS); Canadian Astronomy Data Centre – https://www.cadc-ccda.hia-iha.nrc-
cnrc.gc.ca/ (CFHT/SPIRou); Mikulski Archive for Space Telescopes –
https://archive.stsci.edu/ (TESS).

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1 Astrophysics Group, Keele University, Keele, Staffordshire ST5 5BG, UK
2 Aix Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388
Marseille, France
3 Instituto de Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP,
Rua das Estrelas, 4150-762 Porto, Portugal
4 Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51,
Versoix, Switzerland
5 Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012
Bern, Switzerland
6 Center for Space and Habitability, Gesellsschaftstrasse 6, 3012 Bern,
Switzerland
7 Centre for Exoplanet Science, SUPA School of Physics and Astronomy,
University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
8 Division Technique INSU, CS20330, 83507 La Seyne sur Mer cedex, France
9 School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham B15 2TT, UK
10 Instituto de Astrofisica de Canarias, 38200 La Laguna, Tenerife, Spain
11 Departamento de Astrofisica, Universidad de La Laguna, 38206 La Laguna,
Tenerife, Spain
12 Institut de Ciencies de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n,
08193 Bellaterra, Spain
13 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
14 Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary
15 Depto. de Astrofisica, Centro de Astrobiologia (CSIC-INTA), ESAC campus,
28692 Villanueva de la Cañada (Madrid), Spain
16 Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do
Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal
17 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,
A-8042 Graz, Austria
18 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
19 Department of Astronomy, Stockholm University, AlbaNova University Center,
10691 Stockholm, Sweden
20 Institute of Planetary Research, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany
21 Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005
Paris, France
22 ESTEC, European Space Agency, 2201AZ, Noordwijk, NL
23 Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund,
Sweden
24 Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C,
B-4000 Liège, Belgium
25 Space sciences, Technologies and Astrophysics Research (STAR) Institute,
Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium
26 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The
Netherlands
27 Department of Space, Earth and Environment, Chalmers University of
Technology, Onsala Space Observatory, 43992 Onsala, Sweden
28 Dipartimento di Fisica, Universita degli Studi di Torino, via Pietro Giuria
1, I-10125, Torino, Italy
29 University of Vienna, Department of Astrophysics, Türkenschanzstrasse 17,
1180 Vienna, Austria
30 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry
CV4 7AL, United Kingdom
31 Airbus defence and Space, Avenida de Aragon 404, 28022 Madrid, Spain
32 Science and Operations Department - Science Division (SCI-SC), Directorate
of Science, European Space Agency (ESA), European Space Research and
Technology Centre (ESTEC), Keplerlaan 1, 2201-AZ Noordwijk, The Netherlands
33 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121
Budapest, Konkoly Thege Miklós út 15-17, Hungary
34 ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány
1/A, 1117 Budapest, Hungary
35 IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77
av. Denfert-Rochereau, 75014 Paris, France
36 Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie
Curie, 98bis blvd. Arago, 75014 Paris, France
37 INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122
Padova, Italy
38 INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania,
Italy
39 Department of Astrophysics, University of Vienna, Tuerkenschanzstrasse 17,
1180 Vienna, Austria
40 Institute of Optical Sensor Systems, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany
41 Dipartimento di Fisica e Astronomia "Galileo Galilei", Universita degli
Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy
42 ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2, CH-8093
Zurich, Switzerland
43 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK
44 Center for Astronomy and Astrophysics, Technical University Berlin,
Hardenberstrasse 36, 10623 Berlin, Germany
45 Institut für Geologische Wissenschaften, Freie Universität Berlin, 12249
Berlin, Germany
46 MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent Imre h. u. 112,
Hungary
47 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, United Kingdom