11institutetext: Department of Physics, Faculty of Science, University of
Zagreb, 10 000 Zagreb, Croatia
11email<EMAIL_ADDRESS>22institutetext: European Southern Observatory,
Karl-Schwarzschild-Str. 2, 85748 Garching, Germany 33institutetext: Institute
of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
44institutetext: Department of Astronomy Space Sciences, Science Faculty,
Erciyes University, 38030 Melikgazi, Kayseri, Turkey 55institutetext:
Astronomy and Space Sciences Observatory and Research Center, Erciyes
University, 38281 Talas, Kayseri, Turkey 66institutetext: U.S. Naval
Observatory, Flagstaff Station, 10391 W. Naval Obs. Rd., Flagstaff, AZ 86001,
USA 77institutetext: Central Michigan University, Department of Physics, Mt.
Pleasant, MI 48859, USA 88institutetext: Akdeniz University, Faculty of
Science, Department of Space Sciences and Technologies, 07058, Antalya, Turkey

# Dynamical parallax, physical parameters and evolutionary status of the
components of the bright eclipsing binary $\alpha$ Draconis ††thanks: Based on
observations made with the Mercator Telescope, operated on the island of La
Palma by the Flemish Community, at the Spanish Observatorio del Roque de los
Muchachos of the Instituto de Astrofísica de Canarias.

K. Pavlovski 11 C. A. Hummel 22 A. Tkachenko 33 A. Dervişoğlu 4455 C. Kayhan
44 R. T. Zavala 66 D. J. Hutter 77 C. Tycner 77 T. Şahin 88 J. Audenaert 33 R.
Baeyens 33 J. Bodensteiner 3322 D. M. Bowman 33 S. Gebruers 33 N. E. Jannsen
33 J. S. G. Mombarg 33

###### Abstract

Aims.  Altough both components of the bright eclipsing binary $\alpha$ Dra
having been resolved using long baseline interferometry and the secondary
component shown to contribute some 15% of the total flux, a spectroscopic
detection of the companion star was so far unsuccessful. We aim for a firm
spectroscopic detection of the secondary component of $\alpha$ Dra using
state-of-the-art spectroscopic analysis methods for very-high quality
spectroscopic observations. This will allow the determination of fundamental
and atmospheric properties of the components in the system with high precision
and accuracy.

Methods.  To achieve our goals, we use a combined data set from interferometry
with the Navy Precision Optical Interferometer (NPOI), photometry with the
TESS space observatory, and high-resolution spectroscopy with the HERMES
fibre-fed spectrograph at the La Palma observatory. We use the method of
spectral disentangling to search for the contribution of a companion star in
the observed composite HERMES spectra, to separate the spectral contributions
of both components, and to determine orbital elements of the $\alpha$ Dra
system. TESS light curves are analysed in an iterative fashion with
spectroscopic inference of stellar atmospheric parameters to determine
fundamental stellar properties and their uncertainties. Finally, NPOI
interferometric measurements are used for determination of the orbital
parameters of the system and angular diameters of both binary components.

Results.  We report the first firm spectroscopic detection of the secondary
component in $\alpha$ Dra and deliver disentangled spectra of both binary
components. The components’ masses and radii are inferred with high precision
and accuracy, and are $M_{\rm A}=3.186\pm 0.044$ M⊙, $R_{\rm A}=4.932\pm
0.036$ R⊙, and $M_{\rm B}=2.431\pm 0.019$ M⊙, $R_{\rm B}=2.326\pm 0.052$ R⊙
for the primary and secondary components, respectively. Combined astrometric
and spectroscopic analysis yields the semi-major axis of the system, which is
ultimately used to derive the dynamical parallax of $\pi=11.48\pm 0.13$ mas,
and the distance $d=87.07\pm 1.03$ pc to the $\alpha$ Dra system. Evolutionary
analysis of both binary components with mesa stellar structure and evolution
models suggests the primary is an evolved post-TAMS A-type star, while the
companion is a main-sequence A-type star with a convective core mass of
$M_{cc}=0.337\pm 0.011$ M⊙. Positions of both binary components in the Kiel-
and HR-diagrams suggest a value of the convective core overshooting parameter
$f_{\rm ov}$ well below 0.010 $H_{\rm p}$, and we infer the age of the system
to be 310$\pm$25 Myr.

Conclusions.  The inferred near-core mixing properties of both components do
not support a dependence of the convective core overshooting on the stellar
mass. At the same time, the $\alpha$ Dra system provides extra support to
hypothesise that the mass discrepancy in eclipsing spectroscopic double-lined
binaries is associated with inferior atmospheric modelling of intermediate-
and high-mass stars, and less so with the predictive power of stellar
structure and evolution models as to the amount of near-core mixing and mass
of the convective core.

###### Key Words.:

stars: fundamental parameters – stars: evolution – binaries: eclipsing –
binaries: spectroscopic – techniques: interferometric – stars: individual:
$\alpha$ Dra

##  1 Introduction

It is a well-known fact that a considerable fraction of stars are born in
binary or higher-order multiple systems. In the case of hot massive OB-stars,
binarity or higher multiplicity is found to have a strong effect on their
evolution as over 50% of those stars are found in binaries or higher-order
multiple systems whose components will interact one or multiple times in the
course of their lifetime (e.g. Sana et al. 2012, 2013). Moreover,
observational evidence was recently presented that the binary fraction among
massive stars is independent of their environment. As such, Almeida et al.
(2017) demonstrate that O-type stars in the 30 Dor region of the Large
Magellanic Cloud (LMC) show the observed binary fraction of some 55-60% that
is similar to the existing Galactic samples of massive stars. Furthermore,
Bodensteiner et al. (2021) and Banyard et al. (2021) report similar binary
fraction of 50-55% among SMC and Galactic B-type stars, respectively. At the
same time, Luo et al. (2021) report a somewhat lower binary fraction of some
40% among Galactic OB-stars from LAMOST low-resolution multi-epoch
spectroscopy. Finally, Cohen et al. (2020) report the binary fraction among
solar-type stars to be about 42% suggesting that binarity is a key player in
the evolution and fate of stars, independently of their initial mass.

Given the large binary fraction among solar-type (Duchêne & Kraus 2013), and
more massive stars (Sana et al. 2014), binarity can become a serious obstacle
for interpretation of astrophysical results when it is not properly accounted
for. For example, detailed chemical composition analysis of stars may reveal
complex and spurious abundance patterns when a faint and spectroscopically
undetected companion is not accounted for in the spectroscopic analysis
(Schöller et al. 2010). Furthermore, observationally undetected stellar pairs
can lead to the appearance of multiple parallel main sequences in the
Hertzsprung-Russell (HR) diagrams from photometric surveys (Kroupa et al.
1993; Elson et al. 1998). Moreover, ignorance of the orbital motion of the two
stars in a binary system also leads to a large bias in the astrometric
measurement of the stellar parallax (Makarov & Kaplan 2005; Frankowski et al.
2007; Kervella et al. 2019). Finally, classification of a star as an X-ray
source or a flaring object can be erroneous when a possible companion star
remains undetected (Schröder & Schmitt 2007; Stelzer et al. 2011).

On the other hand, when the subject of targeted research, spectroscopic
double-lined or eclipsing binaries, if spatially resolved, become a principal
source of the fundamental stellar quantities such as masses and radii of
stars. With the advent of observational instruments and techniques, such as
high-precision and high-resolution spectroscopy, high-angular resolution
interferometry, and micro-magnitude precision and high duty cycle space-based
photometry, stellar masses and radii are more routinely measured than ever
before, with accuracy better than 3% (Torres et al. 2010; Serenelli et al.
2021). High-precision and accuracy measurements of fundamental and atmospheric
properties of stars, on the one hand, put theory of stellar structure and
evolution (SSE) at a challenge, and offer a powerful tool for the improvement
of the SSE theory, on the other hand. Over the last two decades, spectroscopic
double-lined eclipsing binaries have been extensively exploited to study the
effects of the stabilization of thermal convection by magnetic fields and of
atomic diffusion in low-mass stars (e.g., Feiden & Chaboyer 2012, 2013, 2014;
Torres et al. 2014; Higl & Weiss 2017), probe the levels of near-core mixing
(typically in the form of convective core overshooting) in intermediate- and
high-mass stars (e.g., Pols et al. 1997; Guinan et al. 2000; Claret & Torres
2016, 2017, 2018, 2019; Martinet et al. 2021), investigate chemical element
transport from accurate measurements of surface abundances in massive stars
(Pavlovski et al. 2018), and measure convective core masses in intermediate
and high-mass BA-type stars (e.g., Johnston et al. 2019a; Tkachenko et al.
2020).

Nowadays, binary stars are often found to be synergistic with intrinsically
variable stars, where the variability is typically caused by stellar
pulsations, rotational modulation of surface inhomogeneities, and activity of
cool companion stars. Hence, combining complementary research approaches
(e.g., asteroseismology and binarity) and/or observational techniques (e.g.,
spectroscopy, astrometry, and photometry) allows us to attack the problem of
uncertainties in SSE models from various angles, where SB2 EBs play a crucial
role.

In this work, we study $\alpha$ Dra (also known as Thuban), a bright ($V$ =
3.7 mag) binary system that was first reported as such by Harper (1907).
Following the recent detection of eclipses in the photometric data recorded
with the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2014,
2015) by Bedding et al. (2019) and the possible detection of a faint companion
star in high-resolution spectra by Hey et al. (2021, in review), we acquired a
new set of high-resolution optical spectra, complementing already existing
high-angular resolution interferometric data to enable a high-precision study
of the system. In Sect. 2, we provide a short overview of the previous studies
of the $\alpha$ Dra system. Results of the analysis of Mark III and NPOI
interferometric, HERMES high-resolution optical spectroscopic, and TESS space-
based photometric data are presented in Sects. 3, 4, and 5, respectively. We
assess the evolutionary status of both binary components in Sect. 6, while the
dynamical parallax measurement is provided in Sect. 7 along with discussion on
all the so far available distance estimates to the system. We present a
general discussion of the obtained results in Sect. 8 and close the paper with
conclusions and an outline of future work provided in Sect. 9.

##  2 An overview of $\alpha$ Dra

Since the first determination of the orbital parameters by Harper (1907),
$\alpha$ Dra (11 Dra, HD 123299) was frequently observed spectroscopically as
a SB1 system. A high degree of consistency is observed for orbital solutions
reported by different research groups, yet some of the reported differences
are statistically significant as they are larger than the quoted
uncertainties. For example, Elst & Nelles (1983) report the RV semi-amplitude
of $K=49.8\pm 0.3$ km s-1 for the visible component, while more recent studies
by Kallinger et al. (2004) and Bischoff et al. (2017) find $K=48.488\pm 0.080$
km s-1 and $K=47.48\pm 0.21$ km s-1, respectively. Since $\alpha$ Dra was not
recognised as a SB2 system so far, all previous inferences of RV semi-
amplitude are likely to be affected by the broad spectral features of an (at
that time) undetected companion star. No appreciable changes in the longitude
of periastron $\omega$ indicative of an apsidal motion are reported in the
literature; the system is found to reside in a highly eccentric orbit,
$e=0.426\pm 0.004$ (Bischoff et al. 2017). The star was also first resolved
with the Navy Precision Optical Interferometer (NPOI), and an estimate of the
brightness difference of $\Delta m=1.83\pm 0.07$ mag at $\lambda$ 7000 Å was
derived by Hutter et al. (2016).

Adelman et al. (2001) performed a detailed abundance analysis of $\alpha$ Dra
based on DAO spectrograms and classified the system as a $\lambda$ Bootis star
owing to the detected metal deficiencies. However, more recently, $\alpha$ Dra
has been withdrawn from a list of $\lambda$ Boo candidate stars by Murphy et
al. (2015). A historical disagreement as to the true value of the projected
rotational velocity of the star was finally resolved thanks to the advent of
electronic detectors and high spectral resolution of the instruments. In doing
so, Gray (2014) measured $v\sin i$ of 26.2 km s-1 from a high-resolution and
high signal-to-noise spectrum using the Fourier transform technique. The
measurement is in a good agreement with the findings by Royer et al. (2002)
and Shorlin et al. (2002) who report $v\sin i$ of $25\pm 2$ km s-1 and $28\pm
2$ km s-1, respectively.

A dedicated spectroscopic observational campaign was organised by Kallinger et
al. (2004) to search for possible intrinsic variability of the star through a
study of line profile variations. Their data set consists of 140 echelle
spectra secured during 10 nights in January 2003 at the Observatoire de Haute-
Provence (OHP) and 1150 time resolved high-resolution spectra acquired during
45 nights in 2003-2004 at the Türinger Landessternwarte Tautenburg. Kallinger
et al. (2004) report the detection of periodic RV variations at frequency of
26.5 d-1 and with variable, orbital phase-dependent amplitude. The authors
conclude that the observed periodic RV variability is caused by the
interaction between tides in the system and a pulsation driving mechanism
intrinsic to the star.

In the HR diagram $\alpha$ Dra is positioned in between the instability strips
of $\delta$ Sct and Slowly Pulsating B-type (SPB) stars. This fact triggered a
speculation that $\alpha$ Dra might belong to a loosely defined class of “Maia
variable stars”, named after the B8-type prototype Maia star in the Pleiades
(Struve 1955). Although the variability of Maia itself was disproved by Struve
et al. (1957), the speculation about a new class of intrinsically variable
stars was reintroduced by Mowlavi et al. (2013) who reported the detection of
36 stars in NGC 3766 that were found to reside in between the $\delta$ Sct and
SPB instability strips and whose variability was attributed to stellar
pulsations. Moreover, a recent study of the seven brightest B-type stars in
the Pleiades by White et al. (2017) concludes that six of those objects are
SPB stars while Maia itself exhibits photometric variability with a period of
about 10 days that is caused by rotational modulation of a large chemical
spot. Absence of rapid line profile variations is also reported by Monier
(2021) based on the analysis of FEROS and NARVAL high-resolution spectra.
Finally, Balona et al. (2015) presented a study of a large sample of B-type
stars in the $Kepler$ field and classified all stars with high-frequency
variability that are cooler than 20 000 K as “Maia variables”. It is worth
noting that the study by Balona et al. (2015) does not provide a solid proof
of the existence of “Maia variables”, instead it once again reintroduced the
speculation about an unknown class of variables among B-type stars.

Bedding et al. (2019) investigated two sectors of TESS space-based photometric
data of $\alpha$ Dra and report the detection of primary and secondary
eclipses separated by some 38.5 d. The authors also inspect the TESS light
curve for signatures of intrinsic variability and find no evidence for it at
the level of 10 parts per million. This result is in contradiction with the
findings by Kallinger et al. (2004) who reported spectroscopic variability on
the timescales of 1 hr. More recently, Hey et al. (2021, in review) performed
a study of $\alpha$ Dra based on the combined TESS photometry and newly
obtained high-resolution SONG spectroscopy. The authors find a hint for a
faint secondary component in the strongest absorption lines in the archival
SOPHIE spectra of the system, yet spectral disentangling proved difficult
owing to the low flux contribution from and a rapid rotation ($v\sin i\sim
200$ km s-1) of a companion star. Based on the results of a combined light-
and RV-curve analysis, and using MIST isochrones, Hey et al. (2021, in review)
constrain $T_{\rm eff}$ and the mass of the primary to be 9975$\pm$125 K and
3.7$\pm$0.1 M⊙, respectively. From the measured inclination and mass function
of the system, the authors report a minimum mass of the secondary component to
be 2.5$\pm$0.1 M⊙. In addition, their chemical composition analysis of the
high signal-to-noise ratio co-added SOPHIE spectrum reveals a complex
abundance pattern for the primary component, and the authors confirm the claim
of Bedding et al. (2019) on the absence of intrinsic variability in the TESS
photometric data.

##  3 High-angular resolution interferometry

###  3.1 Observations, data reduction, and calibration

$\alpha$ Dra was observed both with the Mark III
interferometer111Decommissioned in 1992. (Shao et al. 1988) and the NPOI
(Armstrong et al. 1998). The observation log is given in Table 7 with
information about the baseline lengths and calibrators used. The Mark III
data, recorded in narrow-band channels centered at 500 nm, 550 nm, and 800 nm,
were reduced and the visibilities calibrated as described by Mozurkewich et
al. (2003). Even though baselines of different lengths could be configured,
the Mark III operated only a single baseline during a night, so that no
closure phases were obtained.

The reduction of early NPOI data using a 3-way combiner was described by
Hummel et al. (1998), and for the 6-way combiner (data from 2002) by Hummel et
al. (2003). The data consist of visibilities and closure phases.

The calibrators were taken from a list of single stars maintained at NPOI with
diameters estimated from $V$ and $(V-K)$ using the surface brightness relation
published by Mozurkewich et al. (2003) and van Belle et al. (2009). Values for
$E(B-V)$ were derived from comparing the observed colours to theoretical
colors as a function of spectral type by Schmidt-Kaler in Aller et al. (1982).
Values for the extinction derived from $E(B-V)$ were compared to estimates
based on maps by Drimmel et al. (2003), and used to correct $V$ if they agreed
within 0.5 magnitudes. Even though the surface brightness relationship based
on $(V-K)$ colours is to first order independent of reddening, we included
this small correction. The minimum (squared) visibility amplitudes
corresponding to the diameter estimates are given in Table 1 for any NPOI
observation performed and show that the calibrator stars are mostly unresolved
or only weakly resolved.

Table 1: List of NPOI calibrators. $V^{2}_{\rm min}$ is minimum (squared) visibility on the longest baselines used. HD | Sp. | $V$ | $V-K$ | $E(B-V)$ | $\theta_{V-K}$ | $V^{2}_{\rm min}$
---|---|---|---|---|---|---
| type | [mag] | [mag] | [mag] | [mas] |
FK5 423 | A2V | 3.34 | 0.26 | -0.06 | 0.86 | 0.88
FK5 447 | A0V | 2.44 | 0.01 | 0.02 | 1.11 | 0.81
FK5 456 | A3Vv | 3.31 | 0.21 | 0.00 | 0.84 | 0.90
FK5 472 | B6IIIp | 3.87 | 0.05 | 0.02 | 0.59 | 0.89
HR 5062 | A5V | 4.01 | 0.87 | 0.01 | 0.67 | 0.95
FK5 509 | B3V | 1.86 | -0.41 | 0.01 | 1.11 | 0.81
HR 5329 | A8IV | 4.54 | 0.44 | -0.05 | 0.56 | 0.79
FK5 622 | O9.5V | 2.60 | -0.08 | 0.29 | 0.00 | 1.00
FK5 668 | A0V | 3.75 | 0.13 | 0.06 | 0.65 | 0.93
FK5 677 | B5Ib | 3.97 | -0.03 | 0.12 | 0.50 | 0.97
FK5 913 | A1Vn | 4.36 | 0.10 | 0.01 | 0.48 | 0.99

Figure 1:  The measured (squared) visibility amplitudes plotted against $uv$
radius. Only data with errors less than 4% are shown for clarity. The
decreasing upper envelope of the visibility amplitudes indicates that the
primary component is well resolved, while the weaker trend of the lower
envelope indicates a smaller diameter for the secondary. The strong variation
of the amplitude in between is due to the interference pattern caused by the
binarity of the source structure.

###  3.2 Orbital analysis

Modelling of the (squared) visibility amplitudes (Fig. 1) and closure phases
was performed with OYSTER222http://www.eso.org/~chummel/oyster/oyster.html.
Each night’s data were fit using a binary model parameterized with the angular
diameters of the two stars, their magnitude difference, and their relative
position. The stellar parameters were not free to vary in this step, but were
constrained with the orbital elements replacing the relative positions in a
fit directly to all visibility data. To better constrain the orbital period,
we added radial velocity measurements from historical sources (Pearce 1957;
Elst & Nelles 1983; Adelman et al. 2001; Bischoff et al. 2017). The smaller
diameter of the secondary, which is not well constrained in this fit, was
estimated in a second fit by adopting the effective temperatures and surface
gravities given in Table 6 and the diameter of the primary. The astrometric
results are summarized in Table 8, and the apparent orbit of resolved binary
system of $\alpha$ Dra is shown in Fig. 2. The astrometric error ellipses were
derived from fitting the $4\sigma$ contour of the minimum of the
$\chi^{2}$-surface around the secondary’s position, normalized to unity.

Table 2: Orbital elements and component parameters for $\alpha$ Dra. Ascending node is for epoch J2000.0, and the periastron angle is that of the primary.  Parameter | Unit | Value $\pm$ | Error
---|---|---|---
Period $P$ | [d] |  $51.417350$ $\pm$ | $0.00034$
Periastron epoch $T_{\rm pp}$ | [JD] |  $2453498.8$ $\pm$ | $0.1$
Eccentricity $e$ |  |  $0.43$ $\pm$ | $0.01$
Periastron angle $\omega$ | [deg] |  $22.4$ $\pm$ | $0.5$
Ascending node $\Omega$ | [deg] |  $252.6$ $\pm$ | $0.4$
Inclination $i$ | [deg] |  $85.4$ $\pm$ | $0.5$
Semi-major axis $a^{\prime\prime}$ | [mas] |  $5.52$ $\pm$ | $0.06$
Magnitude difference $\Delta R$ | [mag] |  $1.79$ $\pm$ | $0.02$
Primary’s diameter $D_{\mathrm{A}}$ | [mas] |  $0.62$ $\pm$ | $0.05$
Secondary’s diameter $D_{\mathrm{B}}$ | [mas] |  $0.28$ $\pm$ | $0.05$
Figure 2:  The apparent orbit of $\alpha$ Draconis. The solid straight line
indicates the periastron, and the dashed line the ascending node. The two
small circles at/near the origin indicate the interferometric sizes of the
primary and secondary components, respectively, at inferior conjunction.

##  4 High-resolution spectroscopy

###  4.1 New HERMES high-resolution spectroscopy

New high-resolution echelle spectra were obtained from 10 February to 1 April
2021 at the 1.2-m Mercator telescope at the Observatorio del Roque de los
Muchachos, La Palma, Canary Islands, Spain. The spectra were secured with the
HERMES fibre-fed high-resolution ($R=85\,000$) spectrograph (Raskin et al.
2011). The HERMES spectra cover the entire optical spectral range from 3770 to
9000 Å in 55 spectral orders. Depending on zenith distance, exposure times
ranged between 40 sec and 120 sec for a typical signal-to-noise (S/N) ratio of
over 200. In total 50 spectra were collected, of which 4 were discarded for an
inferior S/N.

The reduction of the raw observations was performed with a dedicated HERMES
data reduction pipeline. The individual reduction steps include background and
bias subtraction, flat fielding, wavelength calibration using a ThArNe lamp,
and echelle order merging. Reduced spectra are resampled in constant velocity-
bins preserving the size of the detector pixels. Continuum normalisation is
done manually by fitting a spline function to a carefully selected set of
pseudo-continuum points.

Figure 3: Two observed composite HERMES spectra taken at quadrature in the
region of the Mg ii 4481 Å spectral line. A broad and shallow spectral feature
of the secondary component is clearly seen at about 1% level of the continuum.
A fiducial pseudo-continuum is indicated with the dashed line for reference.
The spectra were obtained on BJD 2 459 286.64516 (red line), and BJD 2 459
305.62250 (blue line), and correspond to the phases of the orbital cycle 0.486
and 0.856, respectively.

###  4.2 Spectroscopic detection of the secondary component

Figure 4:  Small portions of the disentangled spectra of the primary (blue
line) and secondary (black line) components of the $\alpha$ Dra system. From
left to right: Mg ii 4481 Å doublet, the Fe ii 4549.5/Ti ii 4549.7 Å spectral
blend, and the Fe ii 5316.6/Fe ii 5316.8 Å spectral blend. The secondary’s
disentangled spectrum is arbitrary shifted for clarity.

As discussed in detail in Sect. 2, $\alpha$ Dra has so far been classified as
a SB1 system, and only recently Hey et al. (2021, in review) found a hint for
a companion star in a high S/N archival SOPHIE spectrum. According to the
interferometric measurements, the magnitude difference between the two stars
is $\Delta m=1.81\pm 0.02$ mag, such that a companion star contributes some
15% to the total light. Therefore, we suspect that the lack of detection of
signatures of the secondary component in the observed composite spectra of the
system so far is probably associated with its rapid rotation, just as it is
also hypothesised by Hey et al. (2021, in review). Therefore, our HERMES
spectroscopic campaign was specifically tuned to a detection of broad and
shallow lines of a companion star, hence the requirement for high S/N and full
orbital phase coverage. A visual inspection of the normalised HERMES spectra
confirms the suspicion: a broad and shallow “depression” in the continuum is
clearly visible around prominent metal lines in the composite spectra of the
system. This is illustrated in Fig. 3 where two spectra taken at quadrature
are shown in the region of Mg ii 4481 Å spectral line.

Spectral disentangling is an efficient method for extracting the individual
components’ spectra while simultaneously optimising the orbital elements of a
binary (or higher-order multiple) system (Simon & Sturm 1994). A time-series
of the observed spectra, preferably of high-resolution and uniformly
distributed with the orbital phase is required for the task. The method has
proven to be very efficient in revealing even very faint companions,
contributing as little as 1-2 percent to the total light of the system (e.g.,
Mayer et al. 2013; Torres et al. 2014; Kolbas et al. 2015; Themeßl et al.
2018).

We employ spectral disentangling in the Fourier space as proposed by Hadrava
(1995) and implemented in the FDBinary software package (Ilijic et al. 2004).
A complete procedure was worked-out in Hensberge et al. (2000), and elaborated
and refined in detail in Pavlovski et al. (2018). We apply the method to our
dataset of HERMES high-resolution spectra, starting with prominent metal lines
such as those of the Mg ii 4481 Å doublet. This procedure leads us to
unambiguously detect the spectral contribution of the secondary component;
Fig. 4 shows a portion of the disentangled spectra of both binary components
in the regions of the Mg ii 4481 Å doublet (left panel), the spectral line
blends of Fe ii 4549.5/Ti ii 4549.7 Å (middle panel) and Fe ii 5316.6/Fe ii
5316.8 Å (right panel). Spectral lines of the secondary component are shallow,
with depths at the level of 1% of the continuum, owing to its high rotational
velocity. We note that the detection of such a faint contribution is only
possible when several conditions are simultaneously met: (i) a high resolving
power of the instrument, (ii) a high S/N of the observed composite spectra,
and (iii) optimally extracted and normalised spectra. The latter aspect is
particularly important for the successful reconstruction of the individual
spectra of (faint) binary components. Furthermore, a proper selection of
spectral segments for Fourier-based disentangling is essential, where
particular attention should be paid to placing the edges of selected spectral
intervals in the (pseudo-)continuum regions that are free of spectral lines.
Violation of the latter condition leads to large-amplitude continuum
undulations in the disentangled spectra, which affects the line depths.

Interferometric and photometric analyses of the $\alpha$ Dra system presented
in Sects. 3 and 5, respectively, unambiguously identify the star as a detached
binary system. Both components are found to be well-bounded within their Roche
lobes, and there is barely any light variation detected outside the partial
eclipses in the TESS light curve. In the cases when changes in the light ratio
between the two components are absent and/or eclipse spectra are missing, an
ambiguity occurs in the spectral disentangling: Since the zeroth-order term in
the Fourier expansion is indeterminable, the line depths in the disentangled
spectra are not uniquely determined (Pavlovski & Hensberge 2005). In that
case, the best approach is to perform spectral disentangling in the so-called
“pure separation” mode (Pavlovski & Hensberge 2010), of which the net result
is the individual components’ separated spectra that are diluted according to
the light ratio of the two stars. Two options are available for further
analysis of the separated spectra: (i) they are re-normalised to their
individual continua using the light ratio inferred from the light curve
analysis; or (ii) the light ratio can be used as a free parameter in
spectroscopic analysis along with atmospheric parameters of the star.

An important aspect of spectral disentangling is a S/N gain in the resulting
disentangled spectra. Spectral disentangling is acting as co-addition of the
observed spectra, so the expected gain is proportional to $\sqrt{N}$, where
$N$ is the number of observed composite spectra. The gain in S/N is
distributed to both disentangled spectra proportional to the fractional light
contribution of a component to the total light of the system. For $\alpha$
Dra, we obtain S/N of $\sim$1470 and $\sim$260 for the primary and secondary
component, respectively.

Final separation of the components’ spectra of $\alpha$ Dra from our newly
obtained set of HERMES spectra was performed with the orbital parameters
determined in Sect. 4.3. The disentangled spectra cover a wavelength range
from 4000 Å to 5650 Å, including the prominent H$\beta$, H$\gamma$, and
H$\delta$ Balmer lines. These spectra are used for our detailed spectrum
analysis in Sect. 4.4.

Table 3: Orbital parameters of the $\alpha$ Dra system obtained with the method of spectral disentangling from 46 high-resolution, high S/N HERMES spectra. The orbital period was fixed to the value determined from the observed primary minima (cf. Sect. 5). The uncertainties in the orbital parameters were calculated with the bootstrapping method from 10 000 samples. Parameter | Unit | Value
---|---|---
$P$ | [d] | 51.41891 (fix)
$T_{\rm pp}$ | [d] | 2 451 441.804$\pm$0.014
$e$ |  | 0.4229$\pm$0.0012
$\omega$ | [deg] | 21.28$\pm 0.13$
$K_{\rm A}$ | [km s-1] | 48.512$\pm$0.054
$K_{\rm B}$ | [km s-1] | 63.58$\pm$0.41
$q$ |  | 0.7642$\pm$0.0064
$M_{\rm A}\,\sin^{3}i$ | $[M_{\odot}]$ | $3.167\pm 0.044$
$M_{\rm B}\,\sin^{3}i$ | $[M_{\odot}]$ | $2.416\pm 0.019$
$A\sin i$ | [R⊙] | $103.19\pm 0.38$
Figure 5: RVs computed from the final orbital solution; blue and red dots
refer to the primary and secondary component, respectively. The obtained
spectroscopic orbital solution is indicated with the black solid line.

###  4.3 Spectroscopic orbit from spectral disentangling

In the spectral disentangling method, orbital elements are optimised based on
a time-series of the observed spectra in a self-consistent manner, along with
the reconstruction of the individual components’ spectra (Simon & Sturm 1994;
Hadrava 1995). In this way, the determination of RVs from the individual
exposures is not needed. Spectra of both components of the $\alpha$ Dra system
are dominated by strong and broad hydrogen lines. Those are not the best
diagnostic lines for determination of orbital parameters of the system,
irrespective of whether the parameters are inferred from one of the RV
determination methods or from the method of spectral disentangling. The main
reasons why Balmer lines in A-type stars are not suitable for the task are:
(i) owing to their strengths and widths, hydrogen lines are typically covered
by two echelle orders which make them subject to systematic uncertainties in
the order merging and normalisation to the local continuum; and (ii) hydrogen
lines in the individual components’ spectra are rarely sufficiently well
separated in velocity space leading to a more uncertain orbital solution. In
the case of the $\alpha$ Dra system, both of the aforementioned obstacles are
present, hence we do not use the Balmer lines in the determination of the
orbital parameters of the system.

To ensure that a sufficient number of metal lines are considered for the
determination of orbital parameters of the system, and to make the best use of
the high quality of the extracted and normalised HERMES spectra, we opted for
the spectral disentangling in large spectral segments of about 350 Å, from
4400 Å to 4750 Å, and about 500 Å, from 5000 Å to 5500 Å. Our final solution
is given in Table 3; RVs computed from the finally adopted spectroscopic
orbital parameters are shown in Fig. 5 and are representative of the obtained
orbital phase coverage.

Figure 6: Comparison between the best fit synthetic (blue lines) and the
observed disentangled (red lines) spectra of the primary component in the
regions of H$\gamma$ (upper profile) and H$\beta$ (lower profile) Balmer
lines. Both profiles are shown relative to their central wavelengths; the
corresponding best fit atmospheric parameters are listed in Table 4 (Run 2b).
The spectra centered on H$\gamma$ are arbitrary shifted for better visibility.

###  4.4 Atmospheric parameters

Table 4: Atmospheric parameters of both components of the $\alpha$ Dra system as inferred from their disentangled spectra with the GSSP_single module. The asterisk symbol (∗) refers to the parameters whose values are fixed to those inferred from the TESS light curve. The finally accepted set of the parameters is highlighted in boldface. Parameter | Value$\pm$Error
---|---
Run 1 | Run 2a | Run 2b
| Primary
$T_{\rm eff}$ (K) | 10190 $\pm$ | 125 | 10205 $\pm$ | 100 |  10225 $\pm$ | 100
$\log\,g$ (dex) | 3.52 $\pm$ | 0.09 | 3.5511∗ $\pm$ | 0.0081 |  3.5511∗ $\pm$ | 0.0081
${\rm[M/H]}$ (dex) | -0.03 $\pm$ | 0.05 | 0.01 $\pm$ | 0.05 |  0.01 $\pm$ | 0.05
$\xi$ (km s-1) | 1.06 $\pm$ | 0.21 | 1.26 $\pm$ | 0.30 |  1.34 $\pm$ | 0.30
$v\,\sin\,i$ (km s-1) | 25.4 $\pm$ | 0.7 | 25.4 $\pm$ | 0.9 |  25.4 $\pm$ | 0.9
light factor | 0.848 $\pm$ | 0.012 | 0.83 $\pm$ | 0.01 |  0.825∗ $\pm$ | 0.010
| Secondary
$T_{\rm eff}$ (K) | —– | 10150 $\pm$ | 300 |  10165 $\pm$ | 250
$\log\,g$ (dex) | —– | 4.095∗ $\pm$ | 0.023 |  4.095∗ $\pm$ | 0.023
${\rm[M/H]}$ (dex) | —– | -0.12 $\pm$ | 0.15 |  -0.01 $\pm$ | 0.09
$\xi$ (km s-1) | —– | 0.1 $\pm$ | 0.9 |  0.75 $\pm$ | 0.75
$v\,\sin\,i$ (km s-1) | —– | 168 $\pm$ | 10 |  168 $\pm$ | 11
light factor | —– | 0.20 $\pm$ | 0.02 |  0.175∗ $\pm$ | 0.011

Figure 7: Quality of the fit (blue) to the disentangled spectra (red) of the
primary (top panels) and secondary (bottom panels) in two metal line regions:
$\lambda\lambda$ 4495 to 4570 Å (left column) and $\lambda\lambda$ 5010 to
5065 Å (right column). We note the difference in the Y-axis between the top
and bottom panels owing to the small relative light contribution and rapid
rotation of the secondary component.

For the analysis of the disentangled spectra, we employ the Grid Search in
Stellar Parameters (GSSP; Tkachenko 2015) software package. The method is
based on the comparison of a grid of synthetic spectra to the observed
spectrum of the star and the quality of the fit is judged on the $\chi^{2}$
merit function. GSSP allows for simultaneous optimisation of five atmospheric
and line-broadening parameters, namely effective temperature $T_{\rm eff}$,
surface gravity $\log\,g$, metallicity [M/H], microturbulent velocity $\xi$,
and projected rotational velocity $v\,\sin\,i$ of the star. Optionally, the
effect of light dilution on the disentangled spectrum can also be taken into
account, by either assuming that the light dilution factor is wavelength
independent (as implemented in the GSSP_single module; Tkachenko 2015, their
Eq. (1)), or by replacing it with the ratio of the radii of the two stars
where the wavelength dependence of the light dilution effect is introduced via
the continuum ratio of the binary components (as implemented in the
GSSP_binary module, Tkachenko 2015, their Eq. (4)). The latter approach is
justified in the cases where binary components have significantly different
effective temperatures, otherwise the assumption of the wavelength-independent
light dilution factor is robust. Given that our spectroscopic and photometric
analyses of the $\alpha$ Dra system both show that the binary components have
nearly identical effective temperatures, we limit ourselves to the analysis
with the GSSP_single module where the light dilution factor is assumed to be
independent of wavelength.

The GSSP package employs a grid of atmosphere models pre-computed with the
LLmodels program of Shulyak et al. (2004) (for the summary of the grid
properties, see the Table 1 of Tkachenko 2015), while theoretical spectra are
computed “on-the-fly” with the SynthV (Tsymbal 1996) spectrum synthesis code.
The $\chi^{2}$ merit function is computed for each pair of the observed-
theoretical spectra in the grid, and 1$\sigma$ uncertainties are computed from
$\chi^{2}$ statistics by projecting all $\chi^{2}$ values on the parameter in
question. That way, we take into account possible correlations between the
free parameters in our model.

The spectroscopic analysis is performed iteratively with the analysis of the
TESS light curve (see Sect. 5). We start with the analysis of the disentangled
spectrum of the primary component only, and optimise the following six
parameters: $T_{\rm eff}$, $\log\,g$, [M/H], $\xi$, $v\,\sin\,i$, and the
light dilution factor. The results of this analysis are summarised in Table 4
(column designated as “Run 1”) and suggest the primary component is a somewhat
evolved A-type star with a close-to-solar metallicity. A light curve solution
can then be obtained with the $T_{\rm eff}$ value of the primary component
fixed to 10 190 K, in turn delivering photometric values of the surface
gravity and light dilution factor for both binary components.

In the next step of the spectroscopic analysis, we analyse the disentangled
spectra of both binary components by fixing the $\log\,g$ values for both
stars to their photometric values while keeping all other parameters free. It
is important to note that we exclude broad Balmer lines from the analysis of
the disentangled spectrum of the faint secondary component as those are found
to be non-negligibly distorted in the disentangling process. This issue is not
present in the disentangled spectrum of the primary component thanks to its
dominance in the observed composite spectra of the $\alpha$ Dra system. The
results of the aforementioned spectroscopic analysis are reported in Table 4
(column designated as “Run 2a”). One can see that the parameters of the
primary component barely change compared to the previous iteration (columns
“Run 1” vs. “Run 2a”). Furthermore, we find a good agreement between the
spectroscopic and photometric $T_{\rm eff}$ values for the secondary
component.

On the other hand, we obtain a slightly larger value for the light dilution
factor of the secondary as compared to its photometric value, and the
secondary appears to be slightly metal deficient compared to the primary
component. Because we fit a region of the secondary’s spectrum free of
hydrogen lines, there is a significant degeneracy between the light dilution
factor and the metallicity parameter of the star. Therefore, in the final
iteration of the spectroscopic analysis, we decided to fix the light factors
of both binary components to the values inferred from the light curve
solution, in addition to the previously fixed values of $\log\,g$. Our finally
accepted spectroscopic parameters of both binary components are listed in
Table 4 (column designated as “Run 2b”), and suggest that, within the quoted
1$\sigma$ uncertainties, both stars have equal effective temperatures and
metallicities. On the other hand, the primary component is found to be more
evolved and to have substantially lower projected rotational velocity than its
companion star. The quality of the fit of the disentangled spectra
corresponding to the finally accepted atmospheric parameters is illustrated in
Figs 6 and 7.

Adopting the light ratio determined in the TESS light curve analysis
($\ell_{\rm B}/\ell_{\rm A}=0.212\pm 0.014$) in the final iteration of the
optimal fit of disentangled spectra needs justification since the TESS
passband covers the red, and NIR part of the spectrum, from about
$\lambda\lambda$ 6000 - 10 000 Å, with the effective wavelength about
$\lambda=7865$ Å (the TESS passband is centered at the standard Cousinns I
passband as given in Ricker et al. 2014). The optimal fitting of disentangled
spectra are performed on 500 Å long spectral segment centered at about 5250 Å,
and 350 Å long spectral segment centered at about 4570 Å. We did calculation
of the light ratio as a function of the wavelength using the spectral energy
distribution for both stars. It was found that changes from spectral region
centered at the 5200 Å, to the spectral range covered by the TESS passband is
within assigned uncertainties, and barely about 1%. Therefore, we conclude
that our assumption on the ’constant’ light ratio for the entire spectral
range analysed is justified.

Figure 8:  The Target Pixel File (TPF) of $\alpha$ Dra in Sector 15. Our
custom mask (red) and bright pixels of the target (yellow) are easily seen.
The aperture to obtain the light curve that is represented with our custom
mask (red) covers all the bright pixels of the target. To obtain the light
curve, all pixels outside the mask that are background and nearby star light
contamination are subtracted.

Figure 9: TESS photometric measurements of $\alpha$ Dra: normalised flux as a
function of time, where $t=0$ is for BDJ 2 458 680.0 d. The light curve
analysis presented in Sect. 5.2 was performed separately for the observations
in Sectors 14, 15, and 16 (part 1, in blue), and Sectors 21, and 22 (part 2,
in red).

##  5 TESS space photometry

###  5.1 Observations

$\alpha$ Dra (TIC 165991532) was observed by the Transiting Exoplanet Survey
Satellite (TESS, Ricker et al. 2014, 2015) at five different Sectors with a
2-min cadence. At the time of this study, Sectors 14, 15, 16, 21 and 22 of the
TESS data were available for $\alpha$ Dra. To extract the light curves, we
used the Target Pixel Files (TPFs) from the Mikulski Archive for Space
Telescopes (MAST) database. Because of the brightness of $\alpha$ Dra
($V=3.68$ mag), the reduction methods utilised to construct the Simple
Aperture Photometry (SAP) or Pre-Search Data Conditioning Simple Aperture
Photometry light curves from MAST (PDCSAP, Smith et al. 2012) proved to be not
ideal and led to anomalies in the extracted light curves. Instead, we employ
the lightkurve software package (Lightkurve Collaboration et al. 2018) to
produce a target custom mask, and to estimate and subtract the background in
each TESS sector. For that purpose, we utilise the median flux value computed
across the TPF and the standard deviation estimated from the Median Absolute
Deviation (MAD) of the data set. The MAD and the standard deviation are
related through a scale factor $k$ that is taken to be 1.4826, assuming
normally distributed data. All pixels whose flux values are found to exceed a
certain threshold (computed as a number of standard deviations from the median
flux value) are considered to belong to the target custom aperture while
pixels with the flux values below the threshold are considered as background.
Here, we find through a set of experiments the threshold value of nearly 30
percent of the maximum flux amplitude pixel to be the optimal one and consider
all pixels with fluxes below (above) as background (target custom aperture).
The background flux is estimated and subtracted per pixel over time which
allows us to minimize the noise contribution and to remove light contamination
from the nearby (background) stars. Figure 8 shows an example of the TESS TPF
and our custom aperture selection (red), used to obtain a light curve of
$\alpha$ Dra from Sector 15 of the TESS data. The same method is used to
extract light curves from other sectors of the available TESS data.

Altogether, from the aforementioned five TESS Sectors, we obtained 91173
photometric measurements that cover three (two) primary (secondary) minima and
out-of-eclipse phases of the $\alpha$ Dra system.

Figure 10: Fits to the TESS light curve of $\alpha$ Dra for the primary (left)
and secondary (right) minima. The normalised TESS measurements (upper panels)
and O-C residuals (lower panels) are shown with red symbols while the best fit
obtained with the wd code is illustrated with blue solid line. See text for
definition of the “part 1” and “part 2” light curves. The phases are
calculated from the ephemeris determined from the observed primary minima,
$T_{\rm pr.\leavevmode\nobreak\ min}=(2458696.01948\pm 0.00003)+(51.41891\pm
0.00011)\,E$. The light curves and best fit, and the residuals, for the Part 2
are arbitrary shifted upward for better visibility. Figure 11: Same as Fig.
10, but in the portion of the light curve preceding the primary minimum.
Brightening due to the proximity effect (reflection) is well below 1 millimag
level, and is detected only thanks to the superb high-precision space
photometry acquired by the TESS mission. The light curves and best fit for the
Part 2 are arbitrary shifted upwards for better visibility.

###  5.2 The light curve analysis

We performed the light curve analysis using the Wilson-Devinney (WD) code,
first formulated by Wilson & Devinney (1971) and subsequently generalised to
eccentric binaries, also implementing complementary RVs analysis (Wilson
1979). The WD code employs Roche equipotential surfaces to describe the shapes
of stars forming a binary system and is commonly used for the analysis of
various configurations of binary systems. In our analysis, we utilised the
latest public version of the WD code with designation WD2015 (Wilson & Van
Hamme 2014), through a user-friendly designed gui as well as a Python backend
wrapper developed by Güzel & Özdarcan (2020).

We started to set up our initial binary model by taking into account the
orbital and atmospheric parameters derived in previous sections and reported
in Tables 3 and 4. According to the iterative process described in Sec. 4.4,
we fixed the precisely determined temperature of the primary star ($T_{\rm
eff,A}$), the mass ratio ($q$) and the projected separation of the orbit
($A\sin i$). Without prior knowledge of these parameters, it is impossible to
break the degeneracy of the binary solutions which are only sensitive to the
ratio of the effective temperatures and radii as well as the relative
separation. An initial set of orbital parameters to be adjusted was prepared
from the astrometric and spectroscopic orbital solutions (the period $P$, time
of periastron passage $T_{\rm pp}$, eccentricity $e$, and an angle of
periastron $\omega$), and the effective temperature of the secondary component
$T_{\rm eff,B}$. The surface gravities of both components $\log g_{\rm A,B}$
are also used for initial guesses of the surface potentials of the components,
$\Omega_{\rm A,B}$. Since in light curve modelling only the ratio of the
effective temperatures, and not temperatures themselves, can be determined,
the effective temperature of the primary component $T_{\rm eff,A}$ was fixed.
This component is dominant in the spectra of $\alpha$ Dra, contributing the
majority of the total flux of the binary system, and consequently its
effective temperature is determined with high precision from the atmospheric
analysis. In our initial runs, we optimised a total of nine free parameters:
$P$, $T_{\rm pp}$, $e$, $\omega$, $T_{\rm eff,B}$, $\Omega_{\rm A}$,
$\Omega_{\rm B}$, fractional light contribution of primary star $L_{\rm A}$,
and inclination angle of the binary orbit $i$. Because WD2015 does not contain
the TESS photometric bandpass, the Cousin Ic bandpass was used instead since
its transmission is very close to that of TESS. The linear limb-darkening law
was used with the coefficients interpolated from the tables of van Hamme
(1993) with a WD2015 built-in subroutine. The gravity-brightening coefficient
0.25 for radiative atmospheres was applied after seminal derivation by von
Zeipel (1924).

Table 5: The WD2015 best fit solutions for the TESS light curves of the $\alpha$ Dra system. The finally adopted solution is reported in the last column and represents the mean of the two individual solutions. See text for the definition of the “Part 1” and “Part 2” data sets.  Parameter | Unit | Part 1 | Part 2 | Adopted
---|---|---|---|---
|  | Sectors 14,15,16 | Sectors 21,22 | solution
Orbital elements |  |  |  |
Orbital period $P$ | [d] | $51.4198\pm 0.0031$ | $51.4200\pm 0.0032$ | $51.4199\pm 0.0031$
Time of periastron $T_{\rm pp}$ | [d] | $2458753.107\pm 0.019$ | $2458753.118\pm 0.026$ | $2458753.112\pm 0.023$
Eccentricity $e$ |  | $0.4230\pm 0.0013$ | $0.4217\pm 0.0026$ | $0.4223\pm 0.0021$
Argument of periastron $\omega$ | [deg] | $20.74\pm 0.53$ | $20.27\pm 0.78$ | $20.51\pm 0.67$
Orbital inclination $i$ | [deg] | $86.345\pm 0.024$ | $86.358\pm 0.026$ | $86.352\pm 0.025$
Stars |  |  |  |
Surface potential of star A $\Omega_{\rm A}$ |  | $22.22\pm 0.14$ | $22.32\pm 0.16$ | $22.27\pm 0.15$
Surface potential of star B $\Omega_{\rm B}$ |  | $36.21\pm 0.80$ | $35.90\pm 0.95$ | $36.06\pm 0.88$
Fractional radius of star A $r_{\rm A}$ |  | $0.0479\pm 0.0003$ | $0.0476\pm 0.0004$ | $0.0477\pm 0.0003$
Fractional radius of star B $r_{\rm B}$ |  | $0.0224\pm 0.0005$ | $0.0226\pm 0.0006$ | $0.0225\pm 0.0005$
Eff. temperature of star A $T_{\rm eff,A}$ | [K] |  $10\,225$ (fix) |  $10\,225$ (fix) | 10 225 (fix)
Eff. temperature of star B $T_{\rm eff,B}$ | [K] | $10\,031\pm 248$ | $9\,837\pm 375$ | $9930\pm 328$
Light of star A $L_{\rm A}$ |  | $101.61\pm 1.44$ | $100.61\pm 1.78$ | $101.11\pm 1.62$
Light of star B $L_{\rm B}$ |  | $21.57\pm 1.24$ | $21.27\pm 1.49$ | $21.42\pm 1.37$
Fractional light of star A $\ell_{\rm A}$ |  | $0.8249\pm 0.0172$ | $0.8255\pm 0.0214$ | $0.8252\pm 0.0194$
Fractional light of star B $\ell_{\rm B}$ |  | $0.1751\pm 0.0104$ | $0.1745\pm 0.0126$ | $0.1748\pm 0.0115$
Light ratio $\ell_{\rm B}/\ell_{\rm A}$ |  | $0.2123\pm 0.0125$ | $0.2114\pm 0.0152$ | $0.2118\pm 0.0139$

$\alpha$ Dra is a highly eccentric system ($e=0.423$) with a long orbital
period resulting in very short eclipses which cover only about 2.5 % of the
orbital cycle. In such cases, outside-of-eclipse measurements would have too
large a weight in the light curve modelling. To suppress this unwanted effect,
the TESS photometric observations were resampled; only each 10th measurement
in the out-of-eclipses portions of the light curves is stored whilst all the
observations in the eclipses are used. This reduces significantly the number
of observations that are employed in the light curve modelling. In total, for
all five sectors of the TESS observations of $\alpha$ Dra (Fig. 9), 15187
measurements survived our selection which constitute some 16.5% of the
initially extracted photometric measurements. We also increased the surface
resolution to 60$\times$60 (i.e. N1 and N2 parameters of WD2015) to make sure
the required level of precision is achieved for modelling small-amplitude
variations outside the eclipses (see below).

In the initial analysis, we detect sub-millimag level discrepancies in the
descending branches of the secondary eclipses. Since the ascending and
descending branches of the primary eclipses are in perfect agreement except
for the depth in the last observed (third) primary eclipse, which is about 1
millimag deeper than in the previous two, we concluded that the orbital period
is determined correctly. At present, it is not clear if the effect seen in the
secondary eclipses is real or rather associated with some instrumental and/or
data processing effects. To avoid any mix-up, we decided to perform the light
curve analysis based on two “observing parts” separately, so that the results
can ultimately be examined for consistency. This way, the nearly continuous
observations in Sectors 14, 15, and 16 comprise the “Part 1” data set (9042
measurements) while the observations in Sectors 21 and 22 comprise the “Part
2” data set (6145 measurements). This division is also indicated in Fig. 9
which shows raw TESS observations for all 5 sectors separately.

$\alpha$ Dra is a partially eclipsing system as is obvious from the
characteristic V-shape of its light curve minima. Therefore, degeneracies and
correlations of the parameters in a multidimensional parameter space are
expected (Conroy et al. 2020). To explore the parameter space, as well as to
obtain error estimates, many researchers utilize Markov-Chain Monte-Carlo
(MCMC; Goodman & Weare 2010) based optimisers (e.g. Lehmann et al. 2020,
Dervişoǧlu et al. 2018). For the purpose of optimising the aforementioned nine
free parameters, we developed a Python framework (LCMCMC) which uses PyWD2015
(Güzel & Özdarcan 2020) as a WD backend and the widely used emcee package
(Foreman-Mackey et al. 2013) as an ensemble sampler.

To obtain a good fit, we sampled parameter priors with a normal distribution
around 10 % sigma. To guarantee the convergence, we used the integrated auto-
correlation time module of emcee to assure that enough number iteration have
been achieved. Our final MCMC solution yielded approximately 200 000
iterations per each part of the photometric data through 64 walkers.

In Table 5, we report the results of the posterior probability distributions
of the optimised parameters. We calculated the relative radii $r_{\rm A,B}$,
surface gravities $\log g_{\rm A,B}$, and the light ratio $\ell_{\rm
B}/\ell_{\rm A}$, accordingly. The reported uncertainties and mean values were
calculated from the 16th, 50th, and 84th percentiles of each distribution. For
the finally adopted values, we assumed symmetric errors which is also the core
assumption of the normal distributions (the last column in Table 5).

The best fit models are shown for the primary and secondary minima in both
“Part 1” and “Part 2” data sets in Fig. 10. We also illustrate the accuracy of
our models which successfully fitted the 1 millimag level of the out-of-
eclipse brightening as shown in Fig. 11. The corner plots of the samples in
the marginalised posterior distributions of each part of the photometric data
are also shown in Appendix (Figs. 14 and 15). The obvious correlation between
the optimised parameters demonstrates the importance of adopting stochastic
methods such as Bayesian inference for the multi-parameter model estimations.

It is evident from Figs. 10 and 11 that the decision to analyse the TESS light
curves from the Sectors 14, 15, and 16 (“Part 1”), and Sectors 21 and 22
(“Part 2”) separately is fully justified. There are no systematics in the
residuals between the best fitting calculated light curves and observations in
the secondary minima whilst the residuals are in sub-millimag level (right
panel in Fig. 10). The same is true for the proximity (reflection) light
excess at the orbital phases from 0.80 to 0.99 (Fig. 11). The residuals from
fitting 2 (1) primary minima in “Part 1” (“Part 2”) are somewhat larger, yet
the amplitude does not exceed 1 millimag level (see left panel in Fig. 10).
Comparison between the “Part 1” and “Part 2” light curve solutions reveals a
high degree of consistency within the quoted 1$\sigma$ uncertainties for all
inferred parameters (see Table 5).

Orbital parameters inferred from interferometric, spectroscopic, and
photometric data are listed in Tables 2, 3, and 5, respectively. Four
parameters are in common between these determinations: the orbital period $P$,
time of periastron passage $T_{\rm pp}$, orbital eccentricity $e$, and
argument of periastron $\omega$. The former three parameters are found to be
in good agreement within the quoted 1$\sigma$ uncertainties, while $\omega$
inferred from the interferometric data is 2$\sigma$ away from the respective
values deduced from the spectroscopic and photometric data. Furthermore, the
orbital inclination obtained in the astrometric analysis is also 2$\sigma$
away from the value inferred from photometry. However, we note that the
astrometric value has an order of magnitude larger uncertainty. For that
reason, we use the orbital inclination value of $i=86^{\circ}.352\pm
0^{\circ}.025$ as inferred from the light curve solution for the final
determination of physical properties of the stars as well as for the dynamical
parallax of the system.

Figure 12: Top: positions of the primary (circle) and secondary (squares)
components of $\alpha$ Dra in the Kiel (left column) and HRD (right column)
diagrams. The black/blue square indicates the position of the secondary
component corresponding to its photometric/spectroscopic value of $T_{\rm
eff}$. MESA evolutionary tracks are shown with lines, the exact values of the
stellar mass (in M⊙ units) are indicated in the plot. Black solid lines
correspond to the dynamical masses of the stars and are computed for the
overshooting parameter $f_{\rm ov}$=0.005 $H_{\rm p}$. The blue track is the
best fit to the spectroscopic $T_{\rm eff}$ of the secondary; the red line
indicates a MESA evolutionary track corresponding to the dynamical mass of the
primary component and computed with $f_{\rm ov}$=0.010 $H_{\rm p}$. Bottom:
close ups for the primary (left) and secondary (right) components of the
binary systems.

##  6 Evolutionary models

The combined spectroscopic, photometric, and astrometric analysis described in
the previous sections allows us to put both components of the $\alpha$ Dra
system in the $\log\,T_{\rm eff}$-$\log\,g$ Kiel diagram and assess its
evolutionary status. We closely followed the procedure outlined in Tkachenko
et al. (2020) that makes use of a grid of mesa stellar structure and evolution
models (Paxton et al. 2011, 2013, 2015, 2018, 2019) as presented in Johnston
et al. (2019b). The grid employs the Ledoux criterion for convection and
assumes a chemical mixture after Przybilla et al. (2008) and Nieva & Przybilla
(2012) (also known as the cosmic abundance standard). We set the initial
helium and hydrogen fractions to Y = 0.276 and X = 0.71, respectively, and the
mixing-length parameter $\alpha_{\rm mlt}$ to the solar calibrated value of
1.8. In addition, the grid employs a mixing profile in the radiative envelope
as determined by Rogers & McElwaine (2017) from 2D hydrodynamical simulations
of internal gravity waves and implemented in mesa by Pedersen et al. (2018).
Overall, the grid of Johnston et al. (2019b) is optimised for intermediate-
mass stars born with a convective core which makes it an excellent tool for
the evolutionary analysis of the $\alpha$ Dra system.

Table 6: Stellar properties of the components in the binary system $\alpha$
Dra determined from the combined spectroscopic and light curve analysis, and
evolutionary calculations.

Parameter | Unit | Primary | Secondary
---|---|---|---
$M$ | [M⊙] | $3.186\pm 0.044$ | $2.431\pm 0.019$
$R$ | [R⊙] | $4.932\pm 0.036$ | $2.326\pm 0.052$
$\log g$ | [dex] | $3.555\pm 0.006$ | $4.090\pm 0.019$
$T_{\rm eff}$ | [K] | $10\,225\pm 100$ | $9\,930\pm 330$
$\log L$ | [L⊙] | $2.380\pm 0.018$ | $1.677\pm 0.059$
$M_{\rm bol}$ | [mag] | $-1.200\pm 0.045$ | $0.557\pm 0.147$
$v\,\sin\,i$ | [km s-1] | $25.4\pm 0.9$ | $168\pm 11$
Age | [Myr] | 280$\pm$10 | 345$\pm$ 25
$M_{cc}$ | [M⊙] | 0.01$\pm$0.04 | 0.337$\pm$0.011
Xc |  | 0.00$\pm$0.005 | 0.41$\pm$0.03
$f_{\rm ov}$ |  | 0.007$\pm$0.002 | 0.005∗
333∗ secondary component is not sufficiently evolved to deduce its
overshooting parameter from the Kiel or HRD diagram, hence $f_{\rm ov}$ is
fixed to 0.005 $H_{\rm p}$.

Figure 12 shows the results of fitting mesa evolutionary tracks to the
positions of both binary components in the Kiel (left column) and HR (right
column) diagrams. The positions of both components (black circle and square
for the primary and secondary, respectively) are well reproduced within
1$\sigma$ uncertainties of the measured effective temperatures, surface
gravities, and luminosities of the stars (see Table 6 for numerical values)
and assuming their dynamical masses (solid black tracks in Fig. 12). For the
secondary component, we also considered a second scenario corresponding to its
spectroscopic value of $T_{\rm eff}$ (cf. Table 4, column “2b” for numerical
value and see blue square in Fig. 12). The best fit solution (solid blue track
in Fig. 12) is obtained for the stellar mass that is some 3% larger than the
inferred dynamical mass of the star. Yet, owing to the large spectroscopic
$\Delta T_{\rm eff}$ of 250 K, the corresponding position of the secondary in
the Kiel/HR diagram is also consistent within 1$\sigma$ uncertainty of the
dynamical mass track of the star. For the primary component, we find that a
small amount of convective core overshooting cannot be excluded (solid red
track in Fig. 12) although the $f_{\rm ov}$=0.005 track (corresponding to the
lower boundary in the employed grid of mesa models) is also consistent with
the stellar position in the Kiel and HR diagrams.

In Table 6, we list the final accepted parameters (age, mass of the convective
core, and overshooting parameter) inferred from our evolutionary analysis. We
note that the parameters are based on the effective temperature and luminosity
of the secondary as inferred from the TESS light curve. As discussed in detail
in Sect. 4.4, owing to a small contribution of the secondary component to the
composite spectrum of the binary system, its disentangled spectrum is
significantly more uncertain than that of the primary component. In
particular, we found that the Balmer lines were considerably affected in the
disentangling process and hence could not be used in the spectroscopic
analysis. That being said, the quality of the spectroscopic information that
we have at our disposal for the secondary component is inferior to the quality
of the TESS light curve of the system, hence we put more trust in the
photometrically inferred $\log T_{\rm eff}$ / $\log\ L$ pair for the secondary
component. According to our findings, the primary component is an evolved
post-TAMS444Terminal Age Main Sequence A-type star that has a tiny convective
core ($M_{cc}=0.01\pm 0.04$). On the other hand, the secondary component is a
relatively unevolved star whose convective core mass constitutes some 14% of
the stellar mass. The age of the system is estimated by us to be about 300
Myr, with the individual ages being 280$\pm$10 and 345$\pm$25 Myr for the
primary and secondary component, respectively. We note that the individual
ages are in a good agreement with each other within 2$\sigma$ uncertainties,
and the difference between them reduces to within 1$\sigma$ when the
spectroscopic $T_{\rm eff}$ value of 10165$\pm$250 K is assumed for the
secondary (age and the convective core mass of the star are respectively
315$\pm$22 Myr and $M_{cc}=0.354\pm 0.010$ M⊙ in that case).

Another test could be made to verify consistency between the light ratio
determined from the observables ($\ell_{\rm B}/\ell_{\rm A}=0.212\pm 0.014$)
in the TESS passband, and the light ratio inferred from the evolutionary
tracks for determined dynamical masses of the components. The mist interactive
tool (Dotter 2016; Choi et al. 2016; Paxton et al. 2011, 2013, 2015, 2018,
2019) provided the synthetic photometry for the mesa models for the various
photometric passbands. First we compared the evolutionary tracks for the
components from our grid, and the mist calculations, and found an ecellent
matches. Then for the TESS passband the calculations is giving the light ratio
$({\ell_{\rm B}/\ell_{\rm A}})_{\rm MESA}=0.221\pm 0.015$ which is within
1-$\sigma$ uncertainty to the light ratio determined in the light curve
analysis.

In this context, we should mention that the calculation of the light ratio in
B, and V, passbands (the spectral range in which our spectroscopic analysis
was performed) corroborate our previous assumption that changes with the
wavelength is small, and within the uncertainties of our analysis: $(\ell_{\rm
B}/\ell_{\rm V})_{\rm B}=0.213\pm 0.018$, and $(\ell_{\rm B}/\ell_{\rm
V})_{\rm V}=0.221\pm 0.022$.

##  7 Distance to $\alpha$ Dra

Prior to the launch of astrometric space missions, the trigonometric parallax
for $\alpha$ Dra was very uncertain. The entry for its parallax in the “The
General Catalogue of trigonometric [stellar] parallaxes” (van Altena et al.
1995) reads $\pi=14.8\pm 7.5$ mas. On the other hand, the Hipparcos parallax
from the first 1997 reduction (Perryman et al. 1997) is $\pi_{\rm
HIP}=10.56\pm 0.52$ mas and provides a considerable improvement over the
previously published measurement. Moreover, the improved Hipparcos reduction
from 2007 results in a slight increase of the absolute value of the parallax
for $\alpha$ Dra and in a significant increase in the precision of the
measurement: $\pi_{\rm HIP}=10.76\pm 0.17$ mas (van Leeuwen 2007). Finally,
the Gaia DR2 (Gaia Collaboration et al. 2018) reported trigonometric parallax
of $\pi_{Gaia}=12.18\pm 0.32$ mas is somewhat larger than the 2007 Hipparcos
value putting $\alpha$ Dra at a shorter distance ($d_{Gaia}=82.1\pm 2.1$ pc).

The Gaia EDR3 parallax for $\alpha$ Dra is $\pi_{\rm EDR3}=12.516\pm 0.203$
mas (Gaia Collaboration et al. 2021), slightly larger than quoted in the Gaia
DR2. The parallax uncertainty may be underestimated by $\approx 30\%$ as
described in §6.3.1 of Gaia Collaboration et al. (2021). The distance derived
from the EDR3 parallax and published uncertainty is $d_{\rm EDR3}=79.9\pm 1.3$
pc, and considering the increased uncertainty it is
$d_{GAIA}=79.9^{+4.1}_{-3.7}$ pc. As seen in Fig. 13 this distance places
$\alpha$ Dra closer than other distance determinations. The solution in EDR3
is a 5-parameter solution, and the Gaia does not yet provide solutions for
binary stars. There are hints within the statistics that $\alpha$ Dra might be
resolved. The renormalized unit weight error ruwe is 2.4 and this is greater
than the 1.4 level that may indicate a resolved double. However, the related
statistic ipd_gof_harmonic_amplitude is 0.056 which is less than the 0.1 that
is associated with resolved doubles (Gaia Collaboration et al. 2021). If
$\alpha$ Dra were resolved by Gaia the PSF should be elongated along the
position angle of the binary at epoch 2016.0, the epoch of EDR3 (Gaia
Collaboration et al. 2021; Fabricius et al. 2021). Statistic
ipd_gof_harmonic_phase is 108∘.091. For the orbital elements in Table 2 the
predicted ephemeris gives a position angle of 252∘.76 and a separation of 7.4
mas. As the phase of the PSF elongation is modulo 180∘ the rotated ephemeris
position angle to consider is 72∘.76. This is offset from the
ipd_gof_harmonic_amplitude by 31∘. Until a binary solution from Gaia is
provided it is speculative to associate this 31∘ offset for a 7 mas binary
with the flux ratio determined in this work. In summary, there are two
statistics suggestive of a resolved binary: ruwe and ipd_gof_harmonic_phase.
Although, it is not clear that the orientation of the phase supports agreement
with the position angle of the binary at this time. The
ipd_gof_harmonic_amplitude, which should be considered with the ruwe, does not
suggest a resolved binary. We eagerly await future releases from Gaia that
include results for resolved doubles.

Our complementary observations of the $\alpha$ Dra system enable direct
determination of its distance through the measurements of the angular semi-
major axis of the orbit $a^{\prime\prime}$ (see Table 2) and the semi-major
axis of the spectroscopic orbit (see Table 3). With $A\,\sin i=103.37\pm 0.49$
R⊙, $a^{\prime\prime}=5.52\pm 0.06$ mas, and $i=86.35\pm 0.03$ deg inferred
from interferometry, spectroscopy, and combined spectroscopic and photometric
analysis, respectively, we obtain a distance to the $\alpha$ Dra binary system
of $d=87.07\pm 1.03$ pc, or expressed as the dynamical parallax $\pi_{\rm
dyn}=11.48\pm 0.13$ mas. Figure 13 provides a graphical representation of the
comparison of the distance derived in this work with the distances inferred
from the Hipparcos and Gaia parallaxes.

The distance to a binary system can also be derived from stellar luminosities
and the measured apparent brightness, provided the fractional light
contribution of each component to the total light of the binary system is
known. Since the individual radii, effective temperatures and fractional light
contributions of the components are determined in the present analysis, we can
also provide alternative distance estimates to the system. Oja (1993) measured
the $BV$ brightness of $\alpha$ Dra to be $B=3.640\pm 0.007$ mag and
$V=3.680\pm 0.009$ mag. Absolute bolometric magnitudes, $M_{\rm bol}$, also
given in Table 6, are transformed to absolute visual magnitudes, $M_{\rm V}$,
hence the bolometric correction BCV are applied. Here, we use the bolometric
corrections from four different sources: empirical tabulation from Code et al.
(1976), and Flower (1996)555See corrections given in Torres (2010), and
theoretical estimates from model atmospheres for given photometric bandpasses
by Bessell et al. (1998) and Girardi et al. (2002). We use an estimate of the
colour excess for $\alpha$ Dra $E(B-V)=0.025$ from Zorec et al. (2009) who
compute it as an average from several independent determinations with
different methods. In Kervella et al. (2019), a somewhat lower colour excess
of $E(B-V)=0.007$ is reported using an interactive code stilsm which
implements 3D maps of the local ISM (Lallement et al. 2014, 2018). Our
calculations are performed with the code jktabsdim (Southworth et al. 2005)
where in addition to the parameters listed above we also assume interstellar
absorption. Taking interstellar attenuation of light into account resulted in
shorter distances compared to the case when the effect is ignored.

Figure 13: Dynamical parallax of the binary system $\alpha$ Dra determined in
this work compared to the trigonometric parallaxes measured by dedicated space
astrometric missions, Hipparcos, and Gaia, respectively. Distance
determinations based on the geometric and radiative properties of the stars in
$\alpha$ Dra using different bolometric corrections are shown, too. The latest
Gaia distance from the Gaia EDR3 is shown in coloured strip for easy
comparison.

Calculations of the distance to $\alpha$ Dra using the stellar properties of
its components determined in the present work eventually give two groups of
the results, clustered at a “short” distance of $d\sim 90$ pc, and at a “long”
distance of $d\sim 100$ pc. None of our estimates are in agreement with the
gaia parallax-based distance measurement, while some overlap within 1-$\sigma$
uncertainties is obtained with the Hipparcos parallax-based distance
determinations (see Fig. 13). We note that among the different calibrations of
the bolometric corrections, the empirical tabulation given by Flower (1996)
gives the distance $d_{\rm FLO,V}=87.8\pm 2.5$ pc which is in a good agreement
within the quoted 1-$\sigma$ uncertainty with the distance measured from the
dynamical parallax in this work. We also note that the Gaia parallax
measurement did not take into account the fact that $\alpha$ Dra is a binary
system, hence the corresponding distance estimate is expected to be uncertain.

A group of the “long” distance determinations comprises distances determined
from the theoretical bolometric corrections: those of Girardi et al. (2002)
being close to distance of 100 pc, $d_{\rm GIR,B}=95.8\pm 2.6$ pc, and $d_{\rm
GIR,V}=98.7\pm 2.7$ pc, while application of the theoretical bolometric
corrections from Bessell et al. (1998) gives the largest distance of all,
$d_{\rm BES,V}=99.9\pm 2.7$ pc. The largest distance came for the empirical
calibration given by Code et al. (1976), $d_{\rm COD,V}=99.8\pm 3.7$ pc, also
with the largest uncertainty. Distances quoted are for the calculations of the
binary system as a whole. With a prevailing flux coming from the primary
component, less precisely determined physical properties of the secondary
component have no decisive role in the error budget in distance
determinations.

The mist interactive tool using the mesa models we use in this work also
provide synthetic photometry with built-in bolometric corrections. These
enables calculations of the absolute magnitudes in a given photometric
passband. The absolute magnitudes in B, and V, passbands calculated for the
MESA models with the physical quantities given in Table 6 are $M_{\rm
B}=-1.195$ mag, and $M_{\rm V}=-1.134$ mag. Using the photometry for $\alpha$
Dra from Oja (1993), and $E(B-V)=0.075$ from Zorec et al. (2009), the inferred
distance to $\alpha$ Dra is $d_{\rm MESA,B}=88.4\pm 2.3$ pc, and $d_{\rm
MESA,V}=88.6\pm 2.4$ pc. This is rather close to the distance we determined
from dynamical parallax, as well to the distance calculated with the
bolometric correction from Flower (1996), as represented in Fig. 13.

##  8 Summary and Discussion

The detailed analysis of the newly obtained high-resolution, high S/N HERMES
optical spectra unambiguously reveals the spectral contribution of an A-type
secondary component in the observed composite spectra of the $\alpha$ Dra
system. The companion star is found to contribute some 17.5% of the total flux
of the system, however, its rapid rotation implied by $v\sin i$ of $168\pm 11$
km s-1 results in the apparent spectral contribution of only 1% of the
continuum level in the regions of metal lines. Therefore, it does not come as
a surprise that $\alpha$ Dra has for long been classified in the literature as
a SB1 system. Our newly obtained data set also allows us to achieve the
precision of $\sim$0.1% in the measurement of the primary’s RV semi-amplitude.
A similar precision has been achieved by Kallinger et al. (2004) and Hey et
al. (2021, in review) while other measurements available in the literature are
significantly inferior (e.g. Elst & Nelles 1983; Bischoff et al. 2017).

The combined analysis of the HERMES high-resolution spectroscopic and the
high-quality TESS space-based photometric data enabled us to determine the
fundamental and atmospheric properties of both components of the $\alpha$ Dra
system with high confidence. The precision achieved for the mass, radius, and
effective temperature of the primary (secondary) component are some 1.4%
(0.8%), 1.1% (3.1%), and 1.0% (3.0%). We find the mass and radius of the
primary component to be respectively some 14% and 15% lower than the values of
3.7$\pm$0.1 M⊙ and 5.8$\pm$0.1 R⊙ reported by Hey et al. (2021, in review). We
note that the $\alpha$ Dra system will be re-observed by the TESS mission
during Sectors 48 and 49666$\alpha$ Dra was also observed by the TESS mission
during Sector 41, however, only out-of-eclipse phases were covered during the
respective 27.4 days of observations. of the cycle 4 observations, offering a
prospect for further improvement of the aforementioned precision, and in
particular for the radius of the secondary component.

The primary component of the $\alpha$ Dra system is an evolved A-type star,
which is an extremely rare case when it comes to eclipsing double-lined
binaries. Querying DEBCat777https://www.astro.keele.ac.uk/jkt/debcat/, an on-
line catalogue of detached eclipsing binaries with the masses and radii
determined with a precision better than 2% (maintained and regularly updated
by Southworth 2015), results in only one binary system with an evolved A-type
star, namely $\psi$ Cen. This binary system comprises a pair of A0 IV and A1 V
stars that reside in a 38.8 d orbit. Fundamental and atmospheric properties of
both components of the $\psi$ Cen system have been determined with high
precision, thanks to the available WIRE space-based photometric (Bruntt et al.
2006) and ground-based high-resolution spectroscopic data (Mantegazza et al.
2010; Gallenne et al. 2019). For example, Gallenne et al. (2019) report the
mass and radius of the primary component of $\psi$ Cen to be $M=3.187\pm
0.031$ M⊙ and $R=3.814\pm 0.007$ R⊙. Therefore, the primary components of the
$\psi$ Cen and $\alpha$ Dra are strikingly similar in terms of their masses,
while the difference of about 1 R⊙ in their radii (see Table 6 for the
parameters of $\alpha$ Dra) indicates the primary component of $\alpha$ Dra is
evolutionary more advanced. This in turn suggests a slight difference in ages
of the two systems, and indeed, we find $\alpha$ Dra to be $310\pm 25$ Myr
old, which is to be compared to the age determinations of $\psi$ Cen of
$\tau\,({\rm\psi\,Cen\,A})=280\pm 10$ Myr (Gallenne et al. 2019) and
$\tau\,({\rm\psi\,Cen})\sim 290$ Myr (Bruntt et al. 2006). Looking for “a
replica” of the secondary component of $\alpha$ Dra in terms of mass and age,
the closest matches are the TZ Men A (Andersen et al. 1987) and V541 Cyg A
(Torres et al. 2017) systems. Their masses and surface gravities (used as a
proxy for age) are matching the respective quantities of $\alpha$ Dra B within
0.1 M⊙ and 0.1 dex

Royer et al. (2007) report a bi-modal distribution of the projected rotational
velocity $v\,\sin i$ in early A-type stars, with “slowly” and “rapidly”
rotating stars having distributions centered at 45 km s-1 and 200 km s-1,
respectively. It is thus not unexpected to find a binary system composed of
two early A-type stars with significantly different projected rotational
velocities, especially when one of the components is at an advanced
evolutionary stage. Our measurement of $v\,\sin i$ for the primary component
is in excellent agreement with the value of $v\,\sin i=25\pm 2$ km s-1
inferred by Royer et al. (2002) with the Fourier method. The projected
rotational velocity of the secondary component of 168$\pm$11 km s-1 determined
by us corroborates well the speculation by Hey et al. (2021, in review) that a
companion star must be a rapidly rotating A-type star with $v\,\sin i\sim 200$
km s-1. A-type stars with discordant projected rotational velocities measured
from different spectral lines were identified in observational studies by
Zverko et al. (2011, 2018) and Vaňko et al. (2020), and references therein.
Systematic spectroscopic observations revealed that this disparity in
$v\,\sin\,i$ is a consequence of binarity itself, and the $\alpha$ Dra system
adds extra value to that argument.

The angular diameters of both components of the $\alpha$ Dra binary system
were resolved in our high-angular resolution interferometric observations with
the NPOI. The measured angular diameters for the primary and secondary
components are $\theta_{\rm A}=0.62\pm 0.05$ mas and $\theta_{\rm B}=0.28\pm
0.05$ mas, respectively. Combining angular diameters with the dynamical
parallax of $\pi=11.48\pm 0.13$ mas ($d=87.07\pm 1.03$ pc; see Sect. 7), we
obtain linear radii of $R_{\rm lin,A}=5.80\pm 0.47$ R⊙ and $R_{\rm
lin,B}=2.62\pm 0.47$ R⊙ for the primary and secondary component, respectively.
We note that uncertainties in the interferometric measurements are the main
contributor to the total error budget of the linear radii. The aforementioned
values are within 2-$\sigma$ in agreement with the radius for the primary, and
1-$\sigma$ for the radius of the secondary component inferred from the
combined spectroscopic and light curve analysis.

The high precision achieved in the measurement of the dynamical parallax of
the system in turn leads to the 1.1% uncertainty on its distance. The distance
from Gaia EDR3 trigonometric parallax is about 7 pc shorter than the value
derived from our dynamical parallax. Due to the brightness of $\alpha$ Dra,
the uncertainties of Gaia parallax when corrected are large and within
2$\sigma$ in agreement with our dynamical parallax. At the same time,
inference of the distance to $\alpha$ Dra from physical properties of the
components derived in this work (radii, effective temperatures, and accounting
for the light ratio between the two components) and bolometric corrections
gave a broad range of results. The closest distance to our measurement of the
dynamical parallax is obtained from the empirical bolometric correction by
Flower (1996), $d_{\rm FLO}=87.9\pm 2.5$ pc. On the other hand, using
theoretically calculated bolometric corrections from Bessell et al. (1998),
and Girardi et al. (2002) both yield a distance that is about 10 pc further
away than the distance obtained in this work (see Fig. 13). We attribute the
difference between the Gaia and dynamical parallaxes to the (current) neglect
of binarity in the Gaia solution, where the orbital movement produces changes
in the photocenter of the star thus affects the determination of the
trigonometric parallax, as well produces anomalous proper motion.

Inspired by the results of Hey et al. (2021, in review) who reported a complex
abundance pattern for the primary component of $\alpha$ Dra, we inferred
abundances of helium, silicon, and scandium to check whether they are
consistent with the solar metallicity of the star inferred from its
disentangled spectrum. We focus here on He, Si, and Sc as these three elements
are reported by Hey et al. (2021, in review) to show some of the largest
under-abundances, while a detailed chemical composition analysis of both
components of $\alpha$ Dra is the subject of a forthcoming paper. We confirm a
slight under-abundance of He (though consistent with the solar composition
within the quoted errors) at the level of 0.1$\pm$0.1 dex, whereas abundances
of Si and Sc are found to be in excellent agreement with the deduced solar
metallicity of the star. Hence, we speculate that the complex abundance
pattern reported by Hey et al. (2021, in review) for the primary component
from the analysis of a co-added SOPHIE spectrum is a consequence of
unaccounted spectral contribution of the secondary component to the observed
composite spectrum of the system. Analysis of the spectrum of $\alpha$ Dra as
if it was a single star is also likely the reason why Andersen et al. (1987)
and Adelman et al. (2001) classified the system as a $\lambda$ Boo chemically
peculiar star. The latter are known for their metal deficiency, an effect that
is easily mimicked by light dilution from an unseen companion in spectroscopic
data.

Finally, from an evolutionary analysis of the system, we find that the primary
component of $\alpha$ Dra is an evolved post-TAMS A-type star. The secondary
component is shown to be relatively unevolved, whose convective core mass
constitutes some 14% of the apparent mass of the star. Fundamental and
atmospheric properties of both binary components are consistent with low
values of the convective core overshooting, with $f_{\rm ov}$ just below 0.010
$H_{\rm p}$.

##  9 Conclusion and future work

The analysis of $\alpha$ Dra presented in this work builds upon two
fundamental findings in the literature: (i) spatially resolving the binary
components based on the high-resolution NPOI interferometric data (Hutter et
al. 2016), and (ii) detection of eclipses in TESS space-based photometric data
(Bedding et al. 2019). Both these data sets suggest that $\alpha$ Dra is
unlikely to be a SB1 system as previously reported in the literature, where
Hutter et al. (2016) give an estimate of the light ratio of the two stars of
about 0.19. Intrigued by the fact that such a rare system with an evolved
A-type primary component has been spectroscopically misclassified as a SB1 so
far, due to a low apparent contribution of the secondary, we collected a new
data set of high-resolution, high S/N optical spectroscopic data to attempt
for spectroscopic detection of a companion star, and for detailed analysis of
the system based on combined spectroscopic, interferometric, and photometric
data sets. We report the first unambiguous detection of the secondary
component in the $\alpha$ Dra system which makes it an eclipsing spectroscopic
double-lined (SB2) binary.

The main results and conclusions of our work are:

  * •

the companion star is found to contribute some 17% of the total light of the
system, with the apparent spectroscopic line depths of about 1% in metal line
regions, owing to the high rotational velocity of the star;

  * •

dynamical masses and radii of the primary and secondary components are found
to be $M_{\rm A}=3.186\pm 0.044$ M⊙ and $R_{\rm A}=4.932\pm 0.036$ R⊙, and
$M_{\rm B}=2.431\pm 0.019$ M⊙ and $R_{\rm B}=2.326\pm 0.052$ R⊙, respectively.
High precision of the above-mentioned parameters, coupled with precise
measurements of the effective temperature for both stars, allows us to
constrain the evolutionary status of both components and the age of the system
as a whole;

  * •

we confirm that the primary component is an evolved post-TAMS early A-type
star. The companion is, on the other hand, a fairly unevolved main-sequence
star whose convective core mass constitutes some 14% of its apparent total
mass. The age of the system is estimated to be 310$\pm$25 Myr;

  * •

the high quality of our interferometric observations with an optimal
distribution over the orbital cycle, allow us to infer the angular semi-major
axis with a precision of about 1%. This high precision in turn allows for the
inference of a precise dynamical parallax of $\pi=11.48\pm 0.13$ mas,
ultimately leading to the distance estimate of $d=87.07\pm 1.03$ pc.

Both components of the $\alpha$ Dra system are interesting targets to consider
in the context of the mass discrepancy and near-core mixing levels in detached
eclipsing double-lined binaries. The term mass discrepancy refers to the
difference between the measured dynamical mass of the star and its mass that
is inferred from fitting evolutionary models to the position of the star in
the Kiel- or HR-diagram. Until recently, it was common to associate the mass
discrepancy with large amounts of the near-core mixing in the form of
overshooting (e.g. Guinan et al. 2000; Pavlovski et al. 2009), hence the two
effects are not totally disconnected. Claret & Torres (2019) report a sharp
increase in the $f_{\rm ov}$ overshooting parameter up to a mass of some 2 M⊙
and a subsequent levelling off at the value of some 0.017 $H_{\rm p}$ beyond 2
M⊙ and at least up to 4.4 M⊙. On the other hand, Tkachenko et al. (2020) find
that (i) the mass discrepancy is a strong function of the surface gravity of
the star while it does not show any dependence on the stellar mass itself;
(ii) the mass discrepancy can be only partially explained by insufficient
near-core mixing and convective core mass predicted by evolutionary models;
(iii) there is evidence that the mass discrepancy is strongly connected with
incorrect spectroscopic inference of the effective temperature of the star due
to ignoring the contribution of the turbulent and radiative pressure in the
stellar atmosphere. We note, however, that the study by Tkachenko et al.
(2020) concerned generally higher mass stars, in the mass range between some
4.5 M⊙ and 17 M⊙. Our evolutionary analysis of the $\alpha$ Dra system does
not support the conclusion of Claret & Torres (2019) as to the dependence of
the convective core overshooting parameter on stellar mass. We find low
overshooting values well below $f_{\rm ov}=0.010\,H_{\rm p}$ for both binary
components, while according to the dynamical masses and empirical findings by
Claret & Torres (2016) the stars should show significantly higher levels of
core overshooting. Perfect agreement between the dynamical and evolutionary
mass for the evolved primary component of $\alpha$ Dra is also seemingly in
contradiction with the findings of Tkachenko et al. (2020) that evolved stars
tend to show large values of the mass discrepancy. However, we stress that
both components of $\alpha$ Dra have substantially lower masses than the bulk
sample studied by Tkachenko et al. (2020), hence we do not expect their
$T_{\rm eff}$ inference be significantly influenced by the turbulent and/or
radiative pressure in the stellar atmosphere. Therefore, $\alpha$ Dra provides
an extra argument for the mass discrepancy in eclipsing binaries to be
strongly connected with inferior modelling of stellar atmospheres in certain
mass regimes, and probably less so with the predictive power of evolutionary
models as to the convective core mass and amount of near-core mixing.

Finally, by looking at abundances of He, Si, and Sc in the atmosphere of the
primary component of $\alpha$ Dra, we find the former to be slightly under-
abundant, while the other two elements show abundances consistent with the
solar metallicity of the star. This result suggests the primary component
might be a chemically ordinary A-type star, however a more in-depth chemical
composition analysis is required before any firm conclusions can be drawn.
Detailed chemical composition analysis of both components of $\alpha$ Dra and
subsequent comparison of the inferred abundance patterns with predictions from
stellar structure and evolution models are a subject of a forthcoming paper
(Sahin et al., in prep.).

###### Acknowledgements.

We would like to thank the referee for a thorough review of the paper and for
very useful comments that helped us to improve the manuscript. We acknowledge
Daniel Hey for kindly communicating his manuscript before publication. Based
on observations obtained with the HERMES spectrograph, which is supported by
the Research Foundation - Flanders (FWO), Belgium, the Research Council of KU
Leuven, Belgium, the Fonds National de la Recherche Scientifique
(F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de
Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.
The Navy Precision Optical Interferometer is a joint project of the Naval
Research Laboratory and the US Naval Observatory, in cooperation with Lowell
Observatory and is funded by the Office of Naval Research and the
Oceanographer of the Navy. The authors thank Dr. J.A. Benson and the NPOI
observational support staff whose efforts made this project possible. This
article made use of the Smithsonian/NASA Astrophysics Data System (ADS), of
the Centre de Données astronomiques de Strasbourg (CDS) and of the Jean-Marie
Mariotti Center (JMMC). The research leading to these results has (partially)
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement
N∘670519: MAMSIE), from the KU Leuven Research Council (grant C16/18/005:
PARADISE), from the Research Foundation Flanders (FWO) under grant agreement
G0H5416N (ERC Runner Up Project), and a PhD Fellowship to R.B., and S.G. under
contract No. 11E5620N, as well as from the BELgian federal Science Policy
Office (BELSPO) through PRODEX grant PLATO. DMB gratefully acknowledges a
senior postdoctoral fellowship from the Research Foundation Flanders (FWO)
with grant agreement no. 1286521N. J.B. acknowledges support from the FWO
Odysseus program under project G0F8H6N. A.D. and C.K. acknowledge support by
Erciyes University Scientific Research Projects Coordination Unit under grant
number MAP-2020-9749. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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##  Appendix A Interferometric observations of $\alpha$ Dra

Table 7: Mark III (1989 - 1992) and NPOI (1997 - 2015) observations UT Date | Julian Year | Triangles and baselines | $B_{\rm min}[m]$ | $B_{\rm max}[m]$ | Calibrators
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
1989 Apr 18 | 1989.2945 | NF-SC | 16 | 17 | FK5 509
1991 Jun 19 | 1991.4627 | NF-SC | 14 | 17 | FK5 913
1992 Feb 28 | 1992.1584 | NF-SC | 13 | 17 | FK5 447 FK5 456
1992 Apr 28 | 1992.3225 | NF-SD | 19 | 20 | HR 5062 FK5 447 FK5 509
1992 May 02 | 1992.3337 | NC-SC | 8 | 10 | HR 5062 HR 5329 FK5 456
1992 Jun 08 | 1992.4349 | NF-SC | 12 | 17 | HR 5062 FK5 456
1992 Jul 06 | 1992.5113 | NA-SA | 9 | 10 | HR 5062
1997 Mar 19 | 1997.2125 | AC-AE-AW | 18 | 37 | FK5 472
1997 Mar 26 | 1997.2318 | AC-AE-AW | 16 | 37 | FK5 447 FK5 472
1997 Apr 15 | 1997.2864 | AC-AE-AW | 17 | 37 | FK5 423 FK5 472
1997 Apr 18 | 1997.2946 | AC-AE-AW | 17 | 37 | FK5 472
1997 May 08 | 1997.3493 | AC-AE-AW | 17 | 37 | FK5 423 FK5 472 FK5 668
1997 May 29 | 1997.4067 | AC-AE-AW | 16 | 37 | FK5 423 FK5 668
1997 Jun 19 | 1997.4640 | AC-AE-AW | 16 | 37 | FK5 447 FK5 456 FK5 472 FK5 509 FK5 668
1997 Jun 21 | 1997.4695 | AC-AE-AW | 17 | 37 | FK5 447 FK5 472 FK5 509
1997 Jun 22 | 1997.4723 | AC-AE-AW | 14 | 37 | FK5 447 FK5 472 FK5 509 FK5 668
1997 Jun 26 | 1997.4834 | AC-AE-AW | 15 | 36 | FK5 472 FK5 509 FK5 668
1997 Jun 27 | 1997.4859 | AC-AE-AW | 17 | 37 | FK5 472 FK5 509
1997 Jul 04 | 1997.5051 | AC-AE-AW | 14 | 37 | FK5 447 FK5 509
1997 Jul 08 | 1997.5159 | AC-AE-AW | 17 | 37 | FK5 509 FK5 668
1997 Jul 09 | 1997.5188 | AC-AE-AW | 15 | 36 | FK5 509
1997 Jul 13 | 1997.5298 | AC-AE-AW | 16 | 37 | FK5 509
1997 Jul 14 | 1997.5325 | AC-AE-AW | 15 | 37 | FK5 509
1997 Jul 18 | 1997.5435 | AC-AE-AW | 14 | 35 | FK5 509
2013 Mar 05 | 2013.1743 | AW-AC | 19 | 53 | FK5 0472
2013 Mar 06 | 2013.1769 | AW-AC | 19 | 53 | FK5 0472
2013 Mar 07 | 2013.1797 | AW-AC | 19 | 53 | FK5 472
2013 May 24 | 2013.3931 | AW-AC | 15 | 50 | FK5 472
2013 May 25 | 2013.3956 | AC-AW-E6 | 16 | 53 | FK5 0472
2013 May 26 | 2013.3985 | AW-AC | 16 | 50 | FK5 472
2013 May 27 | 2013.4012 | AC-AW-E6 | 16 | 53 | FK5 472
2013 May 30 | 2013.4094 | AC-AW-E6 | 15 | 53 | FK5 472
2013 May 31 | 2013.4121 | AC-AW-E6 | 16 | 53 | FK5 472
2013 Jun 01 | 2013.4148 | AC-AW-E6 | 16 | 53 | FK5 472
2013 Jun 03 | 2013.4203 | AC-AW-E6 | 16 | 53 | FK5 472
2013 Jun 04 | 2013.4230 | AC-AW-E6 | 16 | 53 | FK5 472
2013 Jun 05 | 2013.4258 | AC-AW-E6 | 15 | 53 | FK5 472
2013 Jun 07 | 2013.4315 | AW-AC | 15 | 50 | FK5 472
2013 Jun 08 | 2013.4341 | AW-AC | 16 | 50 | FK5 423 FK5 472
2013 Jun 10 | 2013.4395 | AW-AC | 15 | 50 | FK5 423 FK5 472
2013 Jun 11 | 2013.4423 | AW-AC | 15 | 50 | FK5 423 FK5 472
2013 Jun 13 | 2013.4477 | AW-AC | 17 | 51 | FK5 423 FK5 472
2013 Jun 15 | 2013.4532 | AW-AC | 16 | 51 | FK5 423 FK5 472
2013 Jun 16 | 2013.4559 | AW-AC | 16 | 51 | FK5 423 FK5 472
2013 Jun 17 | 2013.4586 | AW-AC | 16 | 50 | FK5 423 FK5 472
2014 Jun 24 | 2014.4772 | AE-AW-E6 AW-E6-W7 | 15 | 79 | HR 5329
2014 Jun 25 | 2014.4799 | AE-AW-E6 AW-E6-W7 | 15 | 79 | HR 5329
2014 Jun 26 | 2014.4826 | AE-AW-E6 AW-E6-W7 | 15 | 79 | HR 5329
2015 Mar 28 | 2015.2360 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 622
2015 Mar 29 | 2015.2387 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 622
2015 Apr 21 | 2015.3015 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 622
2015 Apr 28 | 2015.3206 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 622
2015 Apr 30 | 2015.3261 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 622
2015 May 20 | 2015.3808 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 677
2015 May 28 | 2015.4026 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 677
2015 Jun 01 | 2015.4136 | AC-AW-E03 | 10 | 32 | HR 5329 FK5 423 FK5 677
Table 8: Astrometric results UT Date | Julian year | Number of vis. | $\rho$ | $\theta$ | $\sigma_{\rm maj}$ | $\sigma_{\rm min}$ | $\phi$ | $O-C_{\rho}$ | $O-C_{\theta}$
---|---|---|---|---|---|---|---|---|---
|  |  | mas | deg. | mas | mas | deg. | mas | deg.
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
Apr 18 | 1989.2946 | 44 | 3.47 | 70.28 | 0.371 | 0.079 | 102.4 | 0.26 | -0.3
Jun 19 | 1991.4629 | 59 | 7.22 | 251.89 | 0.208 | 0.098 | 65.3 | -0.19 | -0.6
Feb 28 | 1992.1583 | 39 | 6.98 | 253.18 | 0.198 | 0.075 | 112.3 | 0.23 | 1.8
Apr 28 | 1992.3226 | 75 | 8.27 | 254.54 | 0.287 | 0.119 | 84.1 | 0.74 | 0.4
May 02 | 1992.3336 | 61 | 7.47 | 256.13 | 0.327 | 0.166 | 83.5 | 0.60 | 0.7
Jun 08 | 1992.4349 | 87 | 6.39 | 249.32 | 0.222 | 0.062 | 62.6 | 0.26 | -1.3
Mar 19 | 1997.2123 | 549 | 4.62 | 251.34 | 0.264 | 0.192 | 10.1 | 0.02 | 2.6
Mar 26 | 1997.2315 | 549 | 7.25 | 252.08 | 0.226 | 0.131 | 135.0 | 0.04 | -0.0
May 08 | 1997.3492 | 337 | 3.73 | 247.60 | 0.409 | 0.156 | 121.7 | -0.00 | 0.3
Jun 19 | 1997.4642 | 234 | 3.12 | 77.49 | 0.345 | 0.136 | 135.0 | -0.01 | 4.0
Jun 21 | 1997.4697 | 316 | 2.08 | 77.18 | 0.414 | 0.183 | 154.8 | -0.10 | -1.1
Jun 22 | 1997.4724 | 450 | 1.47 | 76.79 | 0.345 | 0.136 | 135.0 | 0.02 | -6.9
Jun 26 | 1997.4833 | 237 | 2.27 | 250.11 | 0.478 | 0.181 | 116.5 | 0.28 | 8.4
Jun 27 | 1997.4861 | 328 | 2.82 | 236.87 | 0.264 | 0.192 | 10.1 | 0.07 | -8.0
Jul 04 | 1997.5052 | 656 | 6.26 | 248.55 | 0.226 | 0.131 | 135.0 | -0.21 | -2.5
Jul 08 | 1997.5162 | 246 | 7.42 | 254.26 | 0.453 | 0.117 | 135.0 | 0.01 | 1.8
Jul 09 | 1997.5189 | 246 | 7.39 | 252.07 | 0.740 | 0.186 | 122.1 | -0.14 | -0.7
Jul 13 | 1997.5299 | 137 | 7.85 | 257.55 | 0.789 | 0.268 | 135.0 | 0.28 | 3.6
Jul 14 | 1997.5326 | 410 | 7.32 | 252.20 | 0.414 | 0.183 | 115.2 | -0.15 | -2.1
Mar 05 | 2013.1740 | 420 | 6.25 | 256.07 | 0.226 | 0.131 | 135.0 | -0.01 | -0.3
Mar 06 | 2013.1767 | 360 | 5.62 | 256.83 | 0.292 | 0.143 | 157.7 | -0.31 | 0.0
Mar 07 | 2013.1794 | 330 | 5.39 | 256.09 | 0.582 | 0.226 | 22.4 | -0.18 | -1.2
May 24 | 2013.3930 | 445 | 2.62 | 242.84 | 0.226 | 0.131 | 135.0 | -0.01 | -1.7
May 25 | 2013.3957 | 1208 | 3.43 | 244.31 | 0.264 | 0.192 | 10.1 | 0.09 | -2.2
May 26 | 2013.3985 | 480 | 4.11 | 245.67 | 0.374 | 0.221 | 121.5 | 0.11 | -2.1
May 27 | 2013.4012 | 1530 | 4.53 | 250.81 | 0.292 | 0.143 | 157.7 | -0.07 | 2.1
May 30 | 2013.4094 | 1649 | 6.04 | 251.26 | 0.226 | 0.131 | 135.0 | -0.01 | 0.7
May 31 | 2013.4122 | 1829 | 6.30 | 253.41 | 0.264 | 0.192 | 10.1 | -0.12 | 2.4
Jun 01 | 2013.4149 | 1620 | 6.66 | 253.59 | 0.226 | 0.131 | 135.0 | -0.07 | 2.2
Jun 03 | 2013.4204 | 1304 | 7.27 | 252.46 | 0.292 | 0.143 | 157.7 | 0.05 | 0.4
Jun 04 | 2013.4231 | 1920 | 7.40 | 253.14 | 0.264 | 0.192 | 10.1 | 0.01 | 0.7
Jun 05 | 2013.4259 | 1608 | 7.60 | 253.22 | 0.226 | 0.131 | 135.0 | 0.09 | 0.5
Jun 07 | 2013.4313 | 60 | 7.56 | 255.34 | 1.058 | 0.490 | 115.7 | -0.07 | 2.0
Jun 08 | 2013.4341 | 339 | 7.63 | 257.33 | 0.576 | 0.317 | 143.8 | 0.00 | 3.7
Jun 10 | 2013.4395 | 450 | 7.63 | 255.59 | 0.264 | 0.192 | 10.1 | 0.14 | 1.4
Jun 11 | 2013.4423 | 457 | 7.72 | 256.67 | 0.414 | 0.183 | 154.8 | 0.35 | 2.1
Jun 13 | 2013.4478 | 171 | 7.15 | 258.39 | 0.385 | 0.243 | 162.3 | 0.13 | 3.2
Jun 15 | 2013.4532 | 300 | 6.77 | 258.07 | 0.688 | 0.171 | 161.5 | 0.26 | 2.1
Jun 16 | 2013.4560 | 300 | 6.38 | 256.03 | 0.345 | 0.136 | 135.0 | 0.17 | -0.4
Jun 17 | 2013.4587 | 359 | 5.95 | 256.65 | 0.483 | 0.197 | 136.6 | 0.08 | -0.2
Jun 24 | 2014.4772 | 4248 | 0.80 | 12.46 | 0.226 | 0.131 | 135.0 | 0.14 | -15.8
Jun 25 | 2014.4799 | 4186 | 1.23 | 41.58 | 0.226 | 0.131 | 135.0 | 0.07 | -9.7
Jun 26 | 2014.4827 | 4334 | 1.81 | 57.29 | 0.226 | 0.131 | 135.0 | 0.09 | -2.6
Mar 28 | 2015.2356 | 875 | 5.39 | 247.62 | 0.226 | 0.131 | 135.0 | -0.04 | -2.2
Mar 29 | 2015.2383 | 620 | 5.84 | 246.62 | 0.483 | 0.197 | 133.4 | -0.04 | -3.7
Apr 21 | 2015.3013 | 895 | 4.11 | 256.71 | 0.344 | 0.126 | 135.0 | 0.18 | -3.5
May 20 | 2015.3807 | 476 | 5.93 | 249.06 | 0.514 | 0.261 | 177.8 | -0.18 | -1.6
May 28 | 2015.4026 | 1065 | 7.46 | 249.77 | 0.345 | 0.136 | 135.0 | -0.17 | -3.6
Jun 01 | 2015.4135 | 630 | 7.23 | 251.26 | 0.483 | 0.197 | 136.6 | -0.12 | -3.3

##  Appendix B EMCEE calculations of the uncertainties in the light curve
analysis

Figure 14:  The MCMC determination of the light curve parameters and
accompanying uncertainties for TESS observations of $\alpha$ Dra in Part 1
(Sectors 14, 15, and 16). A strong correlation between parameters is present,
as expected for partial (almost grazing) eclipses. The posterior distribution
densities are plotted (solid contours). The histogram distributions (solid
lines) are plotted across the associated quantity with 16th, 50th, and 84th
percentiles levels (dashed lines).  Figure 15:  Same as Fig. 14 but for TESS
observations of $\alpha$ Dra in Part 2 (Sectors 21, and 22).