11institutetext: Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, c/
Can Magrans s/n, E-08193 Bellaterra, Barcelona, Spain
11email<EMAIL_ADDRESS>22institutetext: Institut d’Estudis Espacials de
Catalunya (IEEC), c/ Gran Capità 2-4, E-08034 Barcelona, Spain
33institutetext: Centro de Astrobiología (CSIC-INTA), Ctra. Ajalvir, km 4.
E-28850 Torrejón de Ardoz, Madrid, Spain 44institutetext: Department of
Physics, Ariel University, 40700 Ariel, Israel 55institutetext: Hamburger
Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany
66institutetext: Dept. Física Quàntica i Astrofísica, Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1,
E-08028 Barcelona, Spain 77institutetext: Leibniz-Institut für Astrophysik
Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany 88institutetext:
Instituto de Astrofísica de Canarias, Vía Láctea S/N, E-38205 La Laguna,
Tenerife, Spain 99institutetext: Universidad de La Laguna, Departamento de
Astrofísica, E-38206 La Laguna, Tenerife, Spain

# Absolute dimensions and apsidal motion of the eclipsing binaries V889 Aql
and V402 Lac

D. Baroch 1122 A. Giménez 33 J. C. Morales 1122 I. Ribas 1122 E. Herrero 1122
V. Perdelwitz 4455 C. Jordi 6622 T. Granzer 77 C. Allende Prieto 8899

(Received 16 June 2022 / Accepted 24 June 2022)

###### Abstract

Context. Double-lined eclipsing binaries allow the direct determination of
masses and radii, which are key to test stellar models. With the launch of the
TESS mission, many well-known eclipsing binaries have been observed at higher
photometric precision, permitting the improvement of the absolute dimensions
determinations.

Aims. Using TESS data and newly-obtained spectroscopic observations, we aim at
determining the masses and radii of the eccentric eclipsing binary systems
V889 Aql and V402 Lac, together with their apsidal motion parameters.

Methods. We modelled simultaneously radial velocity curves and times of
eclipse for each target to precisely determine the orbital parameters of the
systems, which we used to analyse the light curves and then obtain their
absolute dimensions. We compared the obtained values with those predicted by
theoretical models.

Results. We determined masses and radii of the components of both systems with
relative uncertainties lower than 2 %. V889 Aql is composed of two stars with
masses $2.17\pm 0.02$ M⊙ and $2.13\pm 0.01$ M⊙ and radii $1.87\pm 0.04$ R⊙ and
$1.85\pm 0.04$ R⊙. We found conclusive evidence of the presence of a third
body orbiting V889 Aql with a period of 67 years. Based on the detected third
light and the absence of signal in the spectra, we suggest that this third
body could in turn be a binary composed by two $\sim$1.4 M⊙ stars. V402 Lac is
composed by two stars with masses $2.80\pm 0.05$ M⊙ and $2.78\pm 0.05$ M⊙ and
radii $2.38\pm 0.03$ R⊙ and $2.36\pm 0.03$ R⊙. The times of minimum light are
compatible with the presence of a third body for this system too, although its
period is not yet fully sampled. In both cases we have found a good agreement
between the observed apsidal motion rates and the model predictions.

###### Key Words.:

binaries: eclipsing – binaries: spectroscopic – techniques: radial velocities
– techniques: photometric – techniques: spectroscopic – eclipses

##  1 Introduction

Well-detached double-lined eclipsing binaries are the basic source of
information for stellar dimensions, such as masses and radii, to high
precision (Andersen 1991; Torres et al. 2010). Moreover, eccentric systems
showing apsidal motion, i.e., the precession of the line of apsides, yields
additional unique information about the stellar internal density distribution
through the $\log k_{2}$ parameter. This can be only accurately determined for
this type of eclipsing binaries, where the masses and radii of their
components can be measured to a precision better than $\sim 2$% (Claret et al.
2021).

Observed apsidal motion rates have two main components: a secular variation
due to the non-spherical shape of the stars, called the quadrupole or
classical term, and another contribution due to the effect in the two-body
equations of motion given by General Relativity, the relativistic term. Both
components have recently been studied in Baroch et al. (2021) and Claret et
al. (2021) for those systems with precise times of minimum light
determinations based on photometric measurements obtained with the TESS space
mission (Ricker et al. 2015), which we also use in this paper. An additional
element of the apsidal motion rate can be caused by the presence of a
perturbing third body Borkovits et al. (2007, 2019). Although this term is
generally negligible when compared to the other terms, it may become relevant
in precise determinations of the apsidal motion rate or when the classical
term is relatively small.

Two well-known cases of eclipsing binaries showing apsidal motion and with
TESS data were not included in Baroch et al. (2021) and Claret et al. (2021)
because of the lack of sufficiently precise absolute dimensions. These are
V889 Aql and V402 Lac. Both systems show quite eccentric orbits, with $e\sim
0.4$, but different orbital periods. This leads to different dominant terms of
the apsidal motion: relativistic for V889 Aql and classical for V402 Lac.

The system V889 Aql was identified 4 decades ago as the first case of
relativistic apsidal motion (Giménez & Scaltriti 1982), based on the
comparison of the displacement of the secondary eclipse compared with a
previous light curve by Semeniuk (1967). A value of $\dot{\omega}$ =
0.00046$\pm$0.00015 deg cycle-1 was obtained adopting an orbital eccentricity
of 0.37 deduced from the light curve analysis. Khaliullin & Khaliullina (1989)
obtained a new light curve and confirmed the apsidal motion rate given by
Giménez & Scaltriti (1982). Nevertheless, a deeper analysis of the light curve
yielded a refined determination of the orbital eccentricity, the presence of
significant third light ($L_{3}=$18.5% of the total light) as well as
indications of the presence of a third body. This was later confirmed, with
more precise times of minimum light, by Wolf et al. (2005) establishing a
period for the third body orbit of 52 years. Nevertheless, the estimated mass
of the third body, assuming a co-planar orbit, could not explain the observed
$L_{3}$ value. In any case, the comparison of the observed apsidal motion rate
with theoretical predictions requires a good radial velocity curve, allowing a
precise determination of the stellar masses and eccentricity. A more recent
study by Kıran et al. (2019) included spectroscopic measurements but could not
achieve the required precision. Moreover, the authors used an incomplete light
curve, with no consideration of $L_{3}$. A new, more precise light curve, with
no phase gaps, as well as a better radial velocity curve, are clearly needed.

The system V402 Lac was studied using Hipparcos photometry by Bulut & Demircan
(2008), who obtained initial results with the EBOP light curve analysis tool.
Herrero (2010) presented new photometry and analysed the available radial
velocity curves but the degeneracy between eccentricity, flux ratio and ratio
of radii could not be solved and the presence of third light was not
considered. New photometry by Hoyman et al. (2018) produced an additional
light curve, and analysed the times of eclipse available, leading to an
apsidal motion period of 213 years and the detection of a third body, with an
orbital period of 20.5 years around the eclipsing binary. Unfortunately a
constraint of $L_{2}/L_{1}$ from spectroscopy could not be established and the
presence of third light in the crowded field of view was not analysed. To
address these issues, TESS measurements provide additional times of minima and
a precise light curve while new spectroscopic observations allowed
constraining the possible solutions.

In this work, new spectroscopic measurements have been obtained for both stars
and their TESS light curves, as well as new eclipse timings, have been
analysed. As a result, we present new absolute dimensions and orbital
parameters. We show the general properties and sidereal orbital periods
($P_{\rm s}$) of the two objects in Table 1.

Table 1: General properties of V889 Aql and V402 Lac.

Parameter | Star | Ref.
---|---|---
V889 Aql | V402 Lac
$\alpha$ (J2016) | 19:18:49.84 | 22:09:15.20 | (1)
$\delta$ (J2016) | +16:15:00.4 | +44:50:47.3 | (1)
$\varpi$ [mas] | 2.96$\pm$0.02 | 3.54$\pm$0.09 | (1)
$d$ [pc] | 338$\pm$2 | 283$\pm$8 | (1)
$G$ [mag] | 8.532$\pm$0.003 | 6.723$\pm$0.003 | (1)
Sp. T | B9.5 V | B8 V | (2, 3)
$P_{\rm s}$ [d] | 11.120757$\pm$0.000004 | 3.782043$\pm$0.000005 | (4)

(1): Gaia Collaboration (2020); (2): Abt (2004); (3): Wolf et al. (2010); (4):
This work.

Section 2 presents the observational data, both from spectroscopy and
photometry. Section 3 describes the analysis of the radial velocity curves and
the times of eclipse, leading to the orbital parameters. Section 4 presents
the modelling of the TESS light curves, which yield the corresponding absolute
dimensions of the system. Finally, Section 5 compares the results with
theoretical predictions and summarises the conclusions.

##  2 Observations

###  2.1 Spectroscopy

We obtained new spectroscopic observations of V899 Aql and V402 Lac with four
different spectrographs. Observations of the two targets between September and
October 2020 were acquired with CAFE (Aceituno et al. 2013), a high resolution
échelle spectrograph mounted on the 2.2 m telescope at Calar Alto observatory,
Spain, which covers a wavelength range between 365 nm and 980 nm with a
spectral resolution of 70 000. V889 Aql was also observed between March and
April 2021 with the échelle spectrograph mounted on the 1.2 m telescope TIGRE
(Schmitt et al. 2014) at la Luz observatory, Mexico. The spectrograph at TIGRE
has a spectral resolution of 20 000 and covers a wavelength range between 350
nm and 880 nm in two channels, the blue channel between 350 nm and 560 nm, and
the red channel between 580 nm and 880 nm. Here we only use data from the blue
channel of TIGRE, since it is closer to the emission peak of V889 Aql.

Additionally, we also observed V402 Lac between October 2001 and August 2006,
and between September and November 2009 with the Robert G. Tull Coudé
Spectrograph (Tull et al. 1995) and with the STELLA échelle spectrograph
(Strassmeier et al. 2001), respectively. The Tull spectrograph is mounted on
the 2.7 m telescope at the McDonald Observatory, USA, and covers a wavelength
range between 375 nm and 1020 nm at a resolving power of about 60 000. The
STELLA échelle spectrograph is mounted on the two 1.2 m telescopes at Izana
Observatory, Spain, and covers a wavelength range between 390 nm and 870 nm
with a spectral resolution of about 55 000. Data from these two instruments
were already used in the analysis of Herrero (2010). We summarise the number
of available spectra from each instrument in Table 2, together with the time
span of each dataset.

Table 2: Number and time span of the available spectra for each target coming from different instruments, with the number of RVs used in the analysis in parenthesis. | V889 Aql | V402 Lac
---|---|---
| N(Nused) |  $\Delta t$ [d] | N(Nused) |  $\Delta t$ [d]
CAFE | 27(19) | 46 | 39(27) | 47
TIGRE | 15(15) | 29 | $\cdots$ | $\cdots$
STELLA | $\cdots$ | $\cdots$ | 48(38) | 79
Tull | $\cdots$ | $\cdots$ | 33(30) | 1771
Total | 42(34) | 227 | 120(95) | 6944

The radial velocities (hereafter RVs) of the components of each system were
computed by searching for their signature in the spectra using todmor (Zucker
et al. 2003), an implementation for multi-order spectra of the two-dimensional
cross-correlation method todcor (Zucker & Mazeh 1994). In this technique, two
different template spectra are used for each component, scaled according to
their flux ratio in the wavelength range of the observations. The two spectra
are simultaneously Doppler-shifted, constructing a two-dimensional cross-
correlation function (CCF) map from which the RVs of the two components are
computed. As templates for the calculation of the CCFs, we employed synthetic
Phoenix stellar models (Husser et al. 2013) for the observations of V889 Aql.
However, since the expected temperature of V402 Lac is close to the limit of
the Phoenix models, 12 000 K (Herrero 2010; Hoyman et al. 2018), we instead
decided to use models from the Coelho synthetic stellar library (Coelho 2014),
which covers temperatures up to 25 000 K, and allowed us to explore a wider
grid of template temperatures. We adopted solar metallicities for all the
spectra. For each target and instrument, we checked all the échelle orders of
the spectra with the highest signal-to-noise ratio (SNR) with todmor, and
discarded those with no signals in their CCFs due to telluric contamination or
lack of spectral lines.

The parameters of the best templates were found by exploring a grid of values
for effective temperatures ($T_{\rm eff}$), luminosity ratios ($L_{2}/L_{1}$),
and spectral line broadening ($v_{\rm b}$) due to rotation, spectral
resolution, and other effects such as macroturbulence, seeking the combination
that maximised the CCF peak of each spectrum. In the case of V889 Aql, a
luminosity ratio of $L_{2}/L_{1}=0.98\pm 0.06$ was derived, together with the
$T_{\rm eff}$ and $v_{\rm b}$ values for each instrument given in Table 3, and
with the weighted mean values listed in the last row. In the case of V402 Lac,
however, we found strong degeneracies between $L_{2}/L_{1}$ and $v_{\rm b}$,
probably due to the lack of clear and deep lines in the spectra caused by the
large rotational velocities of the components. Therefore, we decided to fix
the luminosity ratio to 1 based on the results found in Sects. 3.2 and 4.2,
which point towards two equal-mass stars of similar luminosities. $T_{\rm
eff}$ and $v_{\rm b}$ values derived for V402 Lac, with the adopted
$L_{2}/L_{1}=1$, are listed in Table 4.

Finally, RV curves were computed consistently with todmor using as templates
for each instrument the ones with their parameters closer to those listed in
Tables 3 and 4. The RVs of spectra taken during any of the eclipses or at
orbital phases close to conjunction, where the spectral lines of the
components could not be separated due to rotational broadening, were not
further considered. The final number of RVs used in this analysis is shown in
parenthesis in Table 2. We list the derived RVs of each object in Tables 12
and 13.

Concerning the possibility to check rotational velocities using the Rossiter-
McLaughlin effect (Rossiter 1924; McLaughlin 1924), we could obtain eight CAFE
spectra within the primary eclipse of V889 Aql but not for the shorter period
case of V402 Lac. Observations showed that the rotational effect on the radial
velocities during eclipse could not provide any indication of misalignment of
the rotational axes of V889 Aql, although more observations during both
eclipses are needed to confirm the actual alignment of all components.

Table 3: Template parameters of V889 Aql found from the maximisation of the CCF peak of all available spectra with todmor and their weighted means with uncertainties. Instrument | $T_{\rm eff,1}$ | $T_{\rm eff,2}$ | $v_{\rm b,1}$ | $v_{\rm b,2}$
---|---|---|---|---
[K] | [K] | [km s-1] | [km s-1]
CAFE | $9240\pm 100$ | $9200\pm 90$ | $15.8\pm 0.5$ | $20.1\pm 0.8$
TIGRE | $9040\pm 100$ | $8920\pm 150$ | $19.9\pm 0.4$ | $23.4\pm 1.0$
Mean | $9140\pm 100$ | $9130\pm 110$ | $18.1\pm 0.5$ | $21.4\pm 0.9$
Table 4: Same as Table 3 for V402 Lac. Instrument | $T_{\rm eff,1}$ | $T_{\rm eff,2}$ | $v_{\rm b,1}$ | $v_{\rm b,2}$
---|---|---|---|---
[K] | [K] | [km s-1] | [km s-1]
CAFE | $11400\pm 200$ | $11200\pm 200$ | $108\pm 6$ | $111\pm 8$
Tull | $11200\pm 300$ | $11100\pm 200$ | $109\pm 5$ | $108\pm 5$
STELLA | $11600\pm 400$ | $11200\pm 200$ | $112\pm 12$ | $111\pm 12$
Mean | $11400\pm 300$ | $11200\pm 200$ | $109\pm 7$ | $109\pm 7$

###  2.2 Photometry

The two systems studied in this work have been observed by the Transiting
Exoplanet Survey Satellite (TESS) mission (Ricker et al. 2015), which is an
all-sky photometric survey obtaining densely-covered light curves at very high
precision of the nearest and brightest stars, aimed at detecting exoplanets.
The excellent quality of TESS data make them ideal for the study of stellar
eclipse events (Baroch et al. 2021; Claret et al. 2021; Southworth 2021b),
which are much more prominent than exoplanet transits.

V402 Lac was one of the TESS targets included in Sector 16, which was observed
during September 2019. We collected the 2-minute cadence simple aperture
photometry (SAP) produced by the Science Process Operation Centre (SPOC,
Jenkins et al. 2016), available at the Mikulski Archive for Space
Telescopes111https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html
(MAST). We show in Fig. 1 the mask used for the computation of the SAP and the
brightest neighbours to V402 Lac. As it can be seen in the Figure, the mask
contains a star (BD+44 4060, label #2) with Gaia magnitude $G=9.87$, which we
estimate to contribute 5.2% of the total flux in the Gaia band. In addition,
the bright visual companion to V402 Lac, HD 210387 (label #3, $G=6.72$), has a
PSF extending several pixels inside the mask, which would also be
significantly contributing to the total flux through an undetermined amount.

Figure 1: TESS target pixel file of V402 Lac in Sector 16 (created with
tpfplotter, Aller et al. 2020), with the colour code indicating the flux
measured by each pixel. We show the pixels included in the computation of the
SAP as orange bordered pixels. The size of the red circles indicates the TESS
magnitudes of the three brightest stars contained in the field, with V402 Lac
marked with a white cross.

On the other hand, V889 Aql was not included as a TESS target in any of the
sectors, though it fell within the TESS field of view during Sectors 14 and
40, which were observed between July and August 2019 and between June and July
2021, with a cadence of 30 and 10 minutes, respectively. Therefore, we
downloaded the TESS full frame images (FFIs) from MAST, and used the public
TESS aperture photometry tool Eleanor222https://adina.feinste.in/eleanor/
(Feinstein et al. 2019) to extract the SAP of V889 Aql with a mask defined by
the pixels shown in Fig. 2. The mask was modified to avoid the addition of a
visual companion (label #2, $G=11.21$), which was included in the default mask
computed by Eleanor.

Figure 2: Portion of the TESS FFI containing V889 Aql in Sector 40 (created
with tpfplotter, Aller et al. 2020), with the colour code indicating the flux
measured by each pixel. We show the pixels used to extract the photometry as
orange bordered pixels. The size of the red circles indicates the TESS
magnitudes of the two brightest stars contained in the field, with V889 Aql
marked with a white cross.

The out-of-eclipse TESS photometry of both stars, shown in Figs. 3 and 4,
present long term trends produced by effects such as, e.g., pointing jitter
and scattered light from the Earth and Moon (Hattori et al. 2021), which can
bias the resulting orbital parameters and measured times of minimum light. To
mitigate the effects of these trends, we employed the Python package george
(Ambikasaran et al. 2015) to model the out-of-eclipse photometry using a
Gaussian process correlated-noise model with a squared-exponential covariance
function (see e.g. Gibson et al. 2012; Aigrain et al. 2016, for more details).
The length-scale hyperparameter was set to a value of 2 days for V889 Aql,
while for V402 Lac was set to a value of 20 days to avoid fitting the
modulations produced by geometric effects, observed with a periodicity of 3.78
d, the orbital period of the system. The resulting model, shown as red lines
in the upper panels in Figs. 3 and 4, were used to normalise the entire light
curve, including the eclipses.

In the photometric analysis presented in Sect. 4, in order to speed up the
optimisation of the orbital parameters, we reduced the number of photometric
points by only using the last part (i.e., after intermediate data downlink) of
the TESS lightcurve in Sector 16 for V889 Aql and Sector 40 in V402 Lac, which
in both cases seem to be the portions less affected by TESS systematics. The
data used in the analysis represent $\sim$37% of the available data for V889
Aql, containing one pair of primary and secondary eclipses, and a $\sim$50 %
of the available data for V402 Lac, containing 3 pairs of primary and
secondary eclipses.

Figure 3: Model (red) used to detrend the out-of-eclipse TESS photometry
(black dots) of V402 Lac in the second part of Sector 16. Figure 4: Model
(red) used to detrend the out-of-eclipse TESS photometry (black dots) of V889
Aql in the second part of Sector 40.

###  2.3 Times of minimum light

Following the same approach as in Baroch et al. (2021), we determined the time
of minimum light of the TESS photometry using the Kwee & van Woerden (1956)
method (KvW), considering only eclipses where both the ingress and egress are
well sampled, and using the same orbital phase interval for all primary and
secondary eclipses for consistency. We discarded eclipses with strong
asymmetries in their bisectors (see Baroch et al. 2021, for more details),
which could be indicative of biased timing determinations due to, e.g.,
systematic trends remaining in the data after our modelling. As a result of
this procedure, we obtained 7 useful eclipse timings of V889 Aql (4 primary
and 3 secondary) and 12 of V402 Lac (6 primary and 6 secondary).

In addition, we collected numerous photoelectric times of eclipse of V889 Aql
and V402 Lac that have been measured over the last decades, and listed them in
Tables 5 and 6. In total, 40 eclipse timings (20 primary and 20 secondary) are
available for V889 Aql and 50 (31 primary and 19 secondary) for V402 Lac.

Preliminary attempts to fit the ephemeris of the eclipse timings of V402 Lac
yielded residuals with a median deviation ten times larger than the published
uncertainties, with some timings deviating up to 100 times the stated error.
Although several works have shown that in the presence of systematic
deviations from an ideal eclipse the KvW method may produce exceedingly
optimistic errors (Breinhorst et al. 1973; Mikulášek et al. 2014; Torres et
al. 2017), the deviations are much larger that those usually observed (cf. the
timings of V889 Aql have a median deviation $\sim$1.5 times larger than their
errors). For this reason, we decided to use errors of 0.005 and 0.010 days to
primary and secondary timings, respectively, and errors of 0.0001 and 0.0004
days to primary and secondary TESS timings, respectively. These values are set
to approximately match the dispersion shown by the residuals of the fit
described in Sect. 3.2.

Table 5: Times of eclipse for V889 Aql.

BJD | Error | Type | Reference
---|---|---|---
$(+2400000)$ | [d]
38164.4901 | 0.0005 | 1 | Semeniuk (1967)
38242.3352 | 0.0006 | 1 | Semeniuk (1967)
38257.4204 | 0.0010 | 2 | Semeniuk (1967)
40088.396 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | de Bernardi & Scaltriti (1978)
41927.306 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | de Bernardi & Scaltriti (1978)
42568.359 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | de Bernardi & Scaltriti (1978)
42583.429 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | de Bernardi & Scaltriti (1978)
42594.550 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | de Bernardi & Scaltriti (1978)
42961.549 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | de Bernardi & Scaltriti (1978)
43246.731 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | de Bernardi & Scaltriti (1978)
43250.694 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | de Bernardi & Scaltriti (1978)
44047.433 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | Giménez & Scaltriti (1982)
44062.507 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Giménez & Scaltriti (1982)
44340.535 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Giménez & Scaltriti (1982)
44492.270 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | Giménez & Scaltriti (1982)
44785.364 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Khaliullin & Khaliullina (1989)
44803.656 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | Skillman (1982)
45515.391 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | Khaliullin & Khaliullina (1989)
46264.442 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Diethelm (1985)
48106.537 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 1 | Diethelm (1990)
48755.483 | 0.002 | 2 | Hegedus et al. (1996)
50312.388 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Diethelm (1996)
53255.3928 | 0.0010 | 1 | Biro et al. (2007)
53993.2958 | 0.0005 | 2 | Bozkurt (2011)
54645.4737 | 0.0006 | 1 | Brát et al. (2008)
54938.5513 | 0.0010 | 2 | Brát et al. (2009)
55012.4554 | 0.0006 | 1 | Brát et al. (2011)
55835.3860 | 0.0004 | 1 | Hoňková et al. (2013)
56072.8595 | 0.0008 | 2 | Diethelm (2012)
56491.5066 | 0.0003 | 1 | Diethelm (2014)
56495.442 | 0.005aa$a$Timing with no published error, adopted to be 0.005 days. | 2 | Braune (2014)
57240.536 | 0.006 | 2 | Hubscher (2017)
57592.456 | 0.004 | 1 | Hubscher (2017)
58686.22397 | 0.00016 | 2 | This work
58693.41033 | 0.00007 | 1 | This work
58704.5310 | 0.0002 | 1 | This work
58708.46522 | 0.00011 | 2 | This work
59394.01831 | 0.00005 | 1 | This work
59409.07196 | 0.00005 | 2 | This work
59416.25984 | 0.00002 | 1 | This work
333Primary and secondary eclipses are labelled as 1 and 2 in the Type column,
respectively.

Table 6: Times of eclipse for V402 Lac.

BJD | Error | Type | Reference
---|---|---|---
$(+2400000)$ | [d]
50955.501 | 0.005 | 1 | Herrero (2010)
51366.561 | 0.010 | 2 | Herrero (2010)
51371.520 | 0.005 | 1 | Herrero (2010)
51839.304 | 0.010 | 2 | Hegedus et al. (2003)
52115.405 | 0.010 | 2 | Bulut & Demircan (2003)
52116.573 | 0.005 | 1 | Bulut & Demircan (2003)
53900.505 | 0.010 | 2 | Herrero (2010)
53905.533 | 0.005 | 1 | Herrero (2010)
55051.503 | 0.005 | 1 | Hubscher et al. (2010)
55081.760 | 0.005 | 1 | Herrero (2010)
55085.544 | 0.005 | 1 | Herrero (2010)
55118.310 | 0.010 | 2 | Herrero (2010)
55123.363 | 0.005 | 1 | Herrero (2010)
55380.544 | 0.005 | 1 | Badenas-Agustí (2012)
55433.494 | 0.005 | 1 | Brát et al. (2011)
55443.571 | 0.010 | 2 | Badenas-Agustí (2012)
55443.562 | 0.010 | 2 | Brát et al. (2011)
55448.622 | 0.005 | 1 | Badenas-Agustí (2012)
55496.503 | 0.010 | 2 | Badenas-Agustí (2012)
55500.289 | 0.010 | 2 | Zasche et al. (2011)
55500.295 | 0.010 | 2 | Zasche et al. (2011)
55505.354 | 0.010 | 1 | Badenas-Agustí (2012)
55806.612 | 0.010 | 2 | Badenas-Agustí (2012)
55815.482 | 0.005 | 1 | Zasche et al. (2011)
55825.534 | 0.010 | 2 | Badenas-Agustí (2012)
56163.434 | 0.005 | 1 | Zasche et al. (2014)
56511.383 | 0.005 | 1 | Zasche et al. (2014)
56564.335 | 0.005 | 1 | Hoyman et al. (2018)
56821.516 | 0.005 | 1 | Zasche et al. (2014)
56874.462 | 0.005 | 1 | Hoyman et al. (2018)
56893.374 | 0.005 | 1 | Hoyman et al. (2018)
57179.482 | 0.010 | 2 | Zasche et al. (2017)
57203.506 | 0.005 | 1 | Zasche et al. (2017)
57237.544 | 0.005 | 1 | Hubscher (2017)
58330.562 | 0.005 | 1 | Pagel (2020)
58402.422 | 0.005 | 1 | Paschke (2019)
58731.463 | 0.005 | 1 | Pagel (2020)
58739.02667 | 0.00010 | 1 | This work
58741.4330 | 0.0004 | 2 | This work
58742.80886 | 0.00010 | 1 | This work
58745.2140 | 0.0004 | 2 | This work
58746.59120 | 0.00010 | 1 | This work
58748.9973 | 0.0004 | 2 | This work
58752.7785 | 0.0004 | 2 | This work
58754.15512 | 0.00010 | 1 | This work
58756.5605 | 0.0004 | 2 | This work
58757.93735 | 0.00010 | 1 | This work
58760.3428 | 0.0004 | 2 | This work
58761.71929 | 0.00010 | 1 | This work
444Primary and secondary eclipses are labelled as 1 and 2 in the Type column,
respectively. We assigned uncertainties of 0.005 and 0.010 days to primary and
secondary literature timings, and 0.0001 and 0.0004 days to primary and
secondary TESS timings, see text.

##  3 Study of the orbital parameters

For the determination of the orbital parameters from RVs we usually need good
knowledge of the orbital period, which is generally constrained to high
precision from the times of minimum light. At the same time, the analysis of
the eclipse timings in eccentric orbits require good determination of the
orbital eccentricity, which can be highly correlated with the apsidal motion
(Torres et al. 2015). Therefore, a combined solution of the radial velocities
and the times of minimum was obtained, allowing a precise simultaneous
determination of the anomalistic orbital period and the eccentricity. This
procedure has been already successfully followed in the somewhat similar
binary systems V501 Mon (Torres et al. 2015) and V541 Cyg (Torres et al.
2017).

For the RV model we assumed a Keplerian orbital motion, and solved for the
sidereal period ($P_{\rm s}$), the semi-amplitudes of the components ($K_{\rm
1}$ and $K_{\rm 2}$), the eccentricity ($e$), the apsidal motion
($\dot{\omega}$), the argument of periastron ($\omega_{0}$) and time of
periastron passage ($T_{\rm peri}$) defined at a reference time of the primary
eclipse $T_{\rm I}$, and the systemic velocity of each instrument
($\gamma_{\rm Inst.}$). Additionally, we also allowed an adjustable RV jitter
term for each instrument (JitInst.), which was added in quadrature to the
errors as described in Baluev (2009). Simultaneously, the times of eclipse
were modelled using Eq. 11 in Giménez & Bastero (1995), solving for $P_{\rm
a}$, $T_{\rm I}$, $e$, $\omega_{0}$ and $\dot{\omega}$. We also modelled the
light travel time effect over the times of eclipse due to a potential third
body with Eq. 2 of Irwin (1959). This additional effect yields the orbital
parameters of the third body, as well as the time semi-amplitude ($\tau$),
from which we can estimate the minimum mass.

The optimisation of all orbital parameters and computation of uncertainties
was done by sampling the posterior probability distribution with the emcee
sampler (Foreman-Mackey et al. 2013), an implementation of the affine-
invariant ensemble sampler Markov chain Monte Carlo (Goodman & Weare 2010).

Table 7: Parameters of the best joint fit to the radial velocities and eclipse timings of V889 Aql. Parameter | Value
---|---
Inner binary
$P_{\rm a}$ [d] | $11.120771\pm 0.000004$
$P_{\rm s}$ [d] | $11.120757\pm 0.000004$
$T_{\rm I}$ | $2459416.2598\pm 0.00002$
$T_{\rm II}$ | $2459420.19276\pm 0.00006$
$T_{\rm peri}$ | $2459416.752\pm 0.006$
$e$ | $0.3750\pm 0.0013$
$\omega_{0}$ [deg] | $126.1\pm 0.2$
$e\sin\omega_{0}$ | $0.303\pm 0.002$
$e\cos\omega_{0}$ | $-0.22107\pm 0.00013$
$\dot{\omega}$ [deg cycle-1] | $0.00046\pm 0.00002$
$K_{\rm A}$ [km s-1] | $82.7\pm 0.3$
$K_{\rm B}$ [km s-1] | $84.6\pm 0.3$
$M_{\rm A}\sin^{3}i$ [M⊙] | $2.17\pm 0.02$
$M_{\rm B}\sin^{3}i$ [M⊙] | $2.125\pm 0.014$
$q$ | $0.978\pm 0.005$
$a\sin i$ [au] | $0.1585\pm 0.0003$
$\gamma_{\rm CAFE}$ [km s-1] | $-20.9\pm 0.2$
$\gamma_{\rm TIGRE}$ [km s-1] | $-19.1\pm 0.2$
JitCAFE [km s-1] | $0.2\pm 0.2$
JitTIGRE [km s-1] | $0.3\pm 0.3$
Third body
$P_{\rm 3}$ [a] | $67\pm 4$
$T_{\rm peri,3}$ | $2447700\pm 1200$
$\tau_{\rm 3}$ [d] | $0.047\pm 0.002$
$e_{\rm 3}$ | $0.24\pm 0.04$
$\omega_{\rm 3}$ [deg] | $100\pm 20$
$f(m_{\rm 3})$ [M⊙] | $0.122\pm 0.012$
$a_{\rm 3}\sin i$ [au] | $8.2\pm 0.4$

###  3.1 V889 Aql

We used the model described above to simultaneously fit the RVs in Table 12
and eclipse timings in Table 5. We show the best fits in Figs. 5 and 6. The
parameters of the best joint fit are given in Table 7. The value of the
orbital inclination, needed to fit the eclipse timings with eq. 11 in Giménez
& Bastero (1995), was adopted from the preliminary solution of the light curve
to be 89 deg (see section 4.1), and the fitted parameters were found to be
very robust to variations of the adopted value. As expected, the simultaneous
fit of both eclipse timings and RVs allowed us to obtain tight constrains on
the anomalistic orbital period and the eccentricity, with $P_{\rm
a}=11.120771\pm 0.000004$ d and $e=0.3750\pm 0.0013$, which otherwise would be
impossible to determine without fixing one of the values. The velocity semi-
amplitude values obtained in the best fit indicate that the two stars have
very similar masses, but not identical, with a mass ratio $q=0.978\pm 0.005$.
In addition, we also obtained a precise determination of the apsidal motion,
of $\dot{\omega}=0.00046\pm 0.00002$ deg cycle-1, in agreement with previous
determinations (Giménez & Scaltriti 1982; Khaliullin & Khaliullina 1989; Wolf
et al. 2005) and consistent within mutual uncertainties with the value of
$0.000487\pm 0.000013$ deg cycle-1 obtained by (Kıran et al. 2019), who
modelled the eclipse timings with a free value for the eccentricity. The
effect of the apsidal motion on the eclipse timings is clearly seen in Fig. 6.
In the upper-left panel, we show the eclipse timings once the linear ephemeris
(i.e. the first three terms in eq. 11 in Giménez & Bastero 1995) are
subtracted, leaving only the effects of the apsidal motion and the third body,
while the lower-left panel show the effect of the apsidal motion alone (i.e.
after the linear ephemeris and the third body effect are subtracted).

Similarly, in the upper-right panel in Fig. 6 we show the eclipse timings
after linear ephemeris and apsidal motion contribution are subtracted, leaving
a clear indication of the presence of a third body around V889 Aql, as already
found by Wolf et al. (2005) and Kıran et al. (2019). The best-fit parameters
indicate a third body orbital period of $67\pm 4$ a and an eccentricity of
$0.24\pm 0.04$. The third body induces a semi-amplitude over the eclipse
timing residuals from a linear ephemeris of $0.047\pm 0.002$ d, from which we
derive a mass function of $0.122\pm 0.012$ M⊙, or equivalently, a minimum mass
of $\sim 1.6$ M⊙.

As a sanity check to confirm that the presence of a third body is not
affecting the measured value of the apsidal motion term, we used eq. 7 in
Baroch et al. (2021) to compute the apsidal motion from a linear fit to the
difference between secondary and primary eclipse timings ($T_{2}-T_{1}$), from
which we derive an apsidal motion of $\dot{\omega}=0.00047\pm 0.00002$ deg
cycle-1, confirming the value found previously.

Figure 5: Best fit to the CAFE (circles) and TIGRE (triangles) RVs of V889
Aql, phase-folded to the orbital period. Blue and red colours correspond to
the primary and secondary components, respectively. The bottom panels show the
residuals of the best fit. Figure 6: Best fit to the primary (blue) and
secondary (red) times of minimum light of V889 Aql, after subtracting linear
ephemeris (top-left), after subtracting linear ephemeris and the effect of the
third body (bottom-left), and after subtracting linear ephemeris and the
apsidal motion (top-right). The panel at the bottom-right shows the residuals
from all the fits.

###  3.2 V402 Lac

Following the same approach as for V889 Aql, we simultaneously modelled the
RVs and eclipse timings of V402 Lac, listed in Tables 13 and 6, respectively.
In this case, however, the RV observations span almost 20 years, covering a
non-negligible portion of the apsidal motion period, which we expect to cause
a significant deviation of the RVs from a Keplerian orbit due to periastron
precession. For this reason, we modelled the RVs with a Keplerian orbit, but
allowing the argument of periastron, $\omega$, to vary according to the
apsidal motion rate $\dot{\omega}$ as $\omega=\omega_{0}+\dot{\omega}\cdot N$,
with $N$ representing the number of cycles since $T_{\rm I}$ and with
$\omega_{0}$ defined at the same reference time of primary eclipse $T_{\rm I}$
used in the fit to the eclipse timings. We list the obtained best fitting
parameters of the joint model in Table 8. We obtained an anomalistic orbital
period $P_{\rm a}=3.782138\pm 0.000006$ d with an eccentricity of $e=0.376\pm
0.003$. The obtained values of the semi-amplitudes are compatible within the
uncertainties, indicating that the two component stars are nearly
indistinguishable, with a mass ratio of $q=0.995\pm 0.008$. The improvement in
the precision of the eccentricity value with respect to other works (Bulut
2013; Hoyman et al. 2018) allows us to precisely determine the apsidal motion
rate of the system, yielding a value of $\dot{\omega}=0.0090\pm 0.0003$ deg
cycle-1.

Table 8: Parameters of the best joint fit to the radial velocities and eclipse timings of V402 Lac. Parameter | Value
---|---
$P_{\rm a}$ [d] | $3.782138\pm 0.000006$
$P_{\rm s}$ [d] | $3.782043\pm 0.000005$
$T_{\rm I}$ | $2458761.71931\pm 0.00006$
$T_{\rm II}$ | $2458764.1245\pm 0.0002$
$T_{\rm peri}$ | $2458761.579\pm 0.008$
$e$ | $0.376\pm 0.003$
$\omega$ [deg] | $57.9\pm 0.4$
$e\sin\omega$ | $0.319\pm 0.004$
$e\cos\omega$ | $0.1996\pm 0.0003$
$\dot{\omega}$ [deg cycle-1] | $0.0090\pm 0.0003$
$K_{\rm A}$ [km s-1] | $128.5\pm 0.8$
$K_{\rm B}$ [km s-1] | $129.2\pm 0.8$
$M_{\rm A}\sin^{3}i$ [M⊙] | $2.67\pm 0.04$
$M_{\rm B}\sin^{3}i$ [M⊙] | $2.66\pm 0.04$
$q$ | $0.995\pm 0.008$
$a\sin i$ [au] | $0.0830\pm 0.0003$
$\gamma_{\rm CAFE}$ [km s-1] | $-4.8\pm 1.0$
$\gamma_{\rm Tull}$ [km s-1] | $1.9\pm 0.7$
$\gamma_{\rm STELLA}$ [km s-1] | $-18.3\pm 0.7$
JitCAFE [km s-1] | $4.0\pm 1.0$
JitTull [km s-1] | $3.1\pm 0.7$
JitSTELLA [km s-1] | $3.2\pm 0.7$
Third body
$P_{\rm 3}$ [a] | $>20$
$\tau_{\rm 3}$ [d] | $>0.004$

As can be seen in Fig. 7, which shows the best fit to the RVs phase-folded to
the orbital period, the relatively fast apsidal motion rate has noticeable
effects over the available RVs, arising from the change of the argument of
periastron with time. The upper-left panel in Fig. 8 shows the best fit to the
times of eclipse of V402 Lac, with the bottom-left and upper-right panels
showing the apsidal motion and third body effects separately, as in Fig. 6.
While the apsidal motion effect produces a clear deviation of the eclipse
timings from a linear ephemeris, the effect of a hypothetical third body is
more dubious. The relatively low precision of the secondary minimum timings,
the low amplitude of the modulation produced by the third body, and a period
at least as long as the baseline of the observations, makes it impossible to
derive precise and non-degenerate orbital parameters. For this reason, we
decided to fix the eccentricity to zero, and give only a lower limit to the
orbital period of the third body. Although the best fit yields an orbital
period of 46 years and an amplitude of 0.015 d, which is shown in the upper-
right panel in Fig. 8, we obtain comparable fits for periods and amplitudes as
low as 20 years and 0.004 d, respectively. This lower limit to the orbital
period of the third body is comparable to the period determined by Hoyman et
al. (2018), of 20.54 years, although they obtained an amplitude five times
larger than the one we obtain fixing the same period. Finally, to check the
significance of the third body signal and the effect that it may have over the
determined orbital parameters, we repeated the analysis of the eclipse timings
without including the light travel time effect in the model. We obtained a
best-fit with a log-likelihood 20 units lower than the obtained with the full
model, giving significant evidence in favour of the presence of a third body.
Furthermore, the orbital parameters derived with the model without the effect
of a third body do not differ more than one sigma from those of the full
model.

Figure 7: Best fit to the CAFE (circles), STELLA (triangles) and Tull
(squares) RVs of V402 Lac, phase folded to the anomalistic period. The
argument of periastron precessing according to $\dot{\omega}$. Blue hues and
red hues correspond to the primary and secondary components, respectively,
while the colour code indicates the date of the measurements and fit. The
bottom panels show the residuals from the best fit. Figure 8: Same as Fig. 6,
but for V402 Lac.

##  4 Light curve analysis and absolute dimensions

As mentioned before, in this type of eclipsing binaries with similar
components there is an intrinsic degeneracy between the orbital parameters and
the ratio of radii. Therefore, in the following analysis of the light curves,
we adopt the orbital eccentricity and the sidereal period derived in Sect. 3
from the study of the radial velocity curves and the eclipse timings.

###  4.1 V889 Aql

In order to obtain reliable photometric parameters of V889 Aql, we have
analysed the TESS photometry with the
jktebop555http://www.astro.keele.ac.uk/jkt/codes/jktebop.html code, a modern
implementation written by J. Southworth of the Nelson–Davis–Etzel (NDE)
eclipsing binary model used in the code ebop (Nelson & Davis 1972; Etzel 1981;
Popper & Etzel 1981). In our analysis, we adopted the sidereal period and
orbital eccentricity given in Table 7, and solved for the reference time of
the primary eclipse ($T_{\rm I}$), the sum of the fractional radii normalised
to the semi-major axis ($r_{\rm A}+r_{\rm B}$) and their ratio ($k\equiv
r_{\rm B}/r_{\rm A}$), the orbital inclination ($i$), argument of periastron
($\omega$), the central surface brightness ratio of the two stars ($J$), the
fraction of light from a third body divided by the total light ($L_{3}/L$),
and a magnitude zero-point ($m_{0}$).

As shown by other authors, a well-known and common problem in the analysis of
partial eclipses with similar components is that the solutions for the ratio
of radii, $k$, and the surface flux ratio, $J$, are highly degenerated, and
yielding solutions above and below $k=1$ Torres et al. (2015); Southworth
(2021a). This situation is more acute in the case of a significant amount of
third light. To avoid such correlations, given that the spectroscopic analysis
provided us with information about the luminosity ratio between the
components, we used the resulting value, $0.98\pm 0.06$, as a prior in the
light curve analysis with jktebop. The gravity darkening and quadratic limb
darkening coefficients were adopted from Claret (2017), specifically computed
for the TESS wavelength range for identical components with an effective
temperature of 9200 K and $\log g=4.20$. To compute the errors, we generated
10000 synthetic light curves with jktebop by adding random noise according to
the observational uncertainties. We then computed the best fit and determined
the standard deviation of the parameters. In addition, to determine more
realistic uncertainties, we repeated the analysis fixing $e$ at the upper and
lower value of the error bar in Table 7, computed the dispersion of the
results, and added them quadratically to the uncertainties determined with
jktebop. This process had the highest impact over the uncertainties of $k$,
$i$, and $\omega$, which were increased by a factor of $\sim 10$. We show the
best fit parameters and uncertainties in Table 9, and show the corresponding
best fit in Fig. 9 together with the original TESS measurements. We computed
the individual fractional radii from the values of $k$ and $r_{\rm A}+r_{\rm
B}$, obtaining $r_{\rm A}=0.0549\pm 0.0012$ and $r_{\rm A}=0.0544\pm 0.0012$.
The value of the argument of periastron we find, of $126.1\pm 0.2$ deg, is in
perfect agreement with the 126.14 deg value obtained in Sect. 3.1.

It should be noted that the $L_{3}$ value of 19.1% confirms that found in the
light curve analysis of Khaliullin & Khaliullina (1989), who also obtained an
eccentricity in perfect agreement with our value in Table 7. However, our
value may be overestimated due to the large pixel size of TESS, which may be
including light from other close sources. Assuming a mass-luminosity relation
of $L\propto M^{3.5}$, the measured third light would correspond to a third
body with a mass of $\sim 1.75$ M⊙, which is slightly higher than the minimum
mass found from the light travel time effect analysis. This could suggest that
the inclination of the third body orbit is close to that of the eclipsing
system. However, with a third light of 19.1% of the total light in the TESS
band, we expect a third light of around 14% in the band at which we obtained
the spectra with CAFE. With this contribution to the total light, the spectral
lines of the third body should be detected by analysing the spectra with
todmor. However, we could not see any additional signal that could be
attributed to the third body. This could suggest that the light from the third
body is actually coming from two stars in a binary system. In this case, the
outer binary system should be composed of two stars with masses $\sim$1.4 M⊙,
each one contributing about 9.5% to the total light in the TESS band, or $\sim
5$% in the CAFE band, which should be challenging to detect with todmor. In
this case, the mutual inclination between the inner and the outer binary would
be close to 50 deg, thus increasing the effect that the outer system may have
over the orbital parameters and apsidal motion of the inner system Borkovits
et al. (2019) through the Lidov-Kozai mechanism (Lidov 1962; Kozai 1962).
Nevertheless the time scale of the Lidov-Kozai induced oscillations, around
50000 years, would not have a significant effect in our measurement of the
orbital eccentricity.

Figure 9: Top: TESS photometry of V889 Aql from the second orbit of Sector 40 (black dots), phase-folded to the sidereal period. The best fit computed with jktebop is shown as a red line. Bottom: Details of the primary (left) and secondary (right) eclipses. The residuals of the fits are shown in the bottom part of all panels. Table 9: Parameters of the fit to the TESS light curve of V889 Aql. Parameter | Value
---|---
$P_{\rm s}$ [d] | 11.120757 (fixed)
$T_{\rm I}$ [BJD] | $2459416.25979\pm 0.00004$
$e$ | 0.3750 (fixed)
$\omega$ [deg] | $126.1\pm 0.2$
$r_{\rm A}+r_{\rm B}$ | $0.10932\pm 0.00005$
$k$ | $0.99\pm 0.04$
$i$ [deg] | $89.06\pm 0.02$
$J$ | $0.971\pm 0.002$
$L_{3}/L$ | $0.191\pm 0.002$
$m_{0}$ [mag] | $0.00062\pm 0.00002$
Derived parameters
$r_{\rm A}$ | $0.0549\pm 0.0012$
$r_{\rm B}$ | $0.0544\pm 0.0012$

Using the results presented in Tables 7 and 9, the absolute dimensions of the
eclipsing components of V889 Aql are given in Table 11.

###  4.2 V402 Lac

As in the case of V889 Aql, we have modelled the TESS light curves of V402 Lac
adopting the sidereal period and the orbital eccentricity from the study in
Section 3.2. Given the larger stellar deformation of the components of V402
Lac, apparent in the out-of-eclipse photometry, and the relatively long
duration of the eclipses, we used the same jktebop code only for preliminary
solutions, while the final parameters were computed using the Wilson-Devinney
(hereafter WD, Wilson & Devinney 1971) code instead. This code represents the
stars using Roche geometry and takes into account reflection and proximity
effects accordingly. We fixed the effective temperature to the values
indicated in Table 4, and adopted the quadratic limb-darkening coefficients by
Claret (2017) corresponding to these temperatures. We also adopted the mass
ratio from the spectroscopic solution, and fixed the bolometric albedo and the
gravity darkening exponent to 1, as usual for radiative envelopes. The
rotation of the components was assumed to be pseudo-synchronised at periastron
(Hut 1981), although we also checked that the results obtained assuming the
broadening values listed in Table 4 did not change significantly. We solved
for the time of primary eclipse, the argument of periastron, the orbital
inclination, the gravitational potentials of each component ($\Omega_{\rm A}$
and $\Omega_{\rm B}$), the luminosity of the component A (as a parameter
depending on the light curve normalisation), and the third light contribution.
Unlike for V889 Aql, we could not fix the value of $L_{\rm B}/L_{\rm A}$ from
the spectroscopic analysis, since we could not find a precise value different
from 1. Therefore, we opted for constraining its value by fixing the effective
temperature of star B and assuming radiative coupling between the components
for the $I$-band in the WD code. $L_{\rm B}/L_{\rm A}$ is finally reported in
the table, and justifies the equal-luminosity assumption we made in the
spectroscopic analysis.

We evaluated the uncertainties following the same process done for V889 Aql.
We reanalysed the light curve with the eccentricity fixed at the value found
in the spectroscopic solution and at the upper and lower values of the error
bar in Table 7, computed the dispersion of the solutions, and added them
quadratically to the formal uncertainties determined with the WD code. We
obtained errors much larger than the formal uncertainties computed by the WD
code, as expected. We list in Table 10 the parameters of the best solution and
the corresponding uncertainties, and shown in Fig. 10 the best fit to the
data.

As can be seen in the residuals of the best fit, there is a systematic
deviation of the model at the beginning of the primary eclipse. This
systematic effect is repeated when using the two parts of the TESS light curve
individually, which rules out that it is produced by trends in the data
processing. We attribute this systematic deviation to a lack of accuracy in
the WD code in modeling the stellar deformations producing the reflection
effect. It should be mentioned that the phase of periastron coincides with the
beginning of the primary eclipse in the light curve of V402 Lac. Therefore, to
prevent the underestimation of the errors due to any bias that may arise from
this systematic deviation, we inflated the errors by a factor of two. In any
case, all solutions agree with a system of almost identical components.

We obtained an argument of periastron of $58.1\pm 1.4$ d, fully compatible
with the value corresponding to the same reference time derived by the joint
fit to RVs and times of minima. From the values of the gravitational potential
we derive individual fractional radii of $r_{\rm A}=0.1315\pm 0.0018$ and
$r_{\rm B}=0.1304\pm 0.0014$. The luminosity ratio as derived with the WD code
is $0.99\pm 0.02$, which is compatible with the value of 1 that we fixed in
the spectroscopic analysis with todmor. In addition, we detect a significant
third light contribution of $15\pm 4$%. Although some of this light may be
coming from a third companion, as seen in by the mask used by TESS shown in
Fig. 1, a significant part of it might have its origin in the star (label #2)
inside the TESS mask and in the bright star (label #3) close to V402 Lac.

Figure 10: Top: TESS photometry of V402 Lac from the second orbit of Sector 16 (black dots), phase-folded to the sidereal period. The best fit computed with the WD code is shown as a red line. Bottom: Details of the primary (left) and secondary (eclipses). The residuals of the fits are shown in the bottom part of all panels. Table 10: Parameters of the fit to the TESS light curve of V402 Lac. Parameter | Value
---|---
$P_{\rm s}$ [d] | 3.782043 (fixed)
$T_{\rm I}$ [BJD] | 2458761.7191$\pm$0.0014
$e$ | 0.376 (fixed)
$\omega$ [deg] | $58.1\pm 1.4$
$i$ [deg] | $79.99\pm 0.12$
$L_{\rm B}/L_{\rm A}$ | $0.99\pm 0.02$
$L_{3}/L$ | $0.15\pm 0.04$
$\Omega_{\rm A}$ | $9.28\pm 0.10$
$\Omega_{\rm B}$ | $9.29\pm 0.04$
Derived parameters
$r_{\rm A}$ | $0.1315\pm 0.0018$
$r_{\rm B}$ | $0.1304\pm 0.0014$
Table 11: Absolute dimensions of V889 Aql and V402 Lac. | V889 Aql | V402 Lac
---|---|---
$M_{\rm A}$ [M⊙] | $2.17\pm 0.02$ | $2.80\pm 0.05$
$M_{\rm B}$ [M⊙] | $2.13\pm 0.02$ | $2.78\pm 0.05$
$R_{\rm A}$ [R⊙] | $1.87\pm 0.04$ | $2.38\pm 0.03$
$R_{\rm B}$ [R⊙] | $1.85\pm 0.04$ | $2.36\pm 0.03$
$\log g_{\rm A}$ | $4.23\pm 0.02$ | $4.132\pm 0.013$
$\log g_{\rm B}$ | $4.23\pm 0.02$ | $4.136\pm 0.014$
Derived from models
$\log k_{\rm 2,A}$ | $-2.40\pm 0.02$ | $-2.40\pm 0.02$
$\log k_{\rm 2,B}$ | $-2.41\pm 0.02$ | $-2.40\pm 0.02$
$T_{\rm eff,A}$ [K] | $9210\pm 110$ | $11330\pm 250$
$T_{\rm eff,B}$ [K] | $9075\pm 110$ | $11280\pm 240$
$t_{\rm A}$ [Ma] | $200\pm 40$ | $230\pm 20$
$t_{\rm B}$ [Ma] | $210\pm 40$ | $230\pm 20$
$L_{\rm bol,A}$ [L⊙] | $22.6\pm 1.4$ | $85\pm 8$
$L_{\rm bol,B}$ [L⊙] | $20.9\pm 1.4$ | $82\pm 7$

##  5 Discussion and conclusions

Absolute dimensions of the components of V889 Aql and V402 Lac, as derived
from the above orbital elements and light curve solutions, are given in Table
11, with relative errors between 0.8 and 2.1 %. These absolute dimensions are
sufficiently precise for a meaningful comparison with theoretical models and
the analysis of the observed apsidal motion rates. Nevertheless, a reliable
comparison with the distance given by Gaia, as shown in Table 1, was not
possible due to the uncertainties in the contribution of the third light
components measured with TESS in the $V$-band photometry and the computation
of the extinction coefficients, expected to be relatively large. We could only
verify that a reasonable combination of adopted third light and extinction
could reproduce the Gaia measurements.

We computed the theoretical values of the $k_{2}$ using the theoretical models
published by Claret (2019), which are based on the Modules for Experiments in
Stellar Astrophysics package (MESA; Paxton et al. 2011, 2013, 2015). These
models assume convective core overshooting following the semi-empirical
dependence on mass given by Claret & Torres (2018). We homogeneously derived
$k_{2}$ values and their errors from interpolation in the models, using $M$
and $\log g$ as input parameters. However, since there are no previous
determinations of the metallicity of any of the two systems, we tuned them
until the derived mean effective temperature derived was compatible with the
mean effective temperatures listed in Tables 3 and 4. This was achieved by
assuming a metallicity of [Fe/H]$=+0.10\pm 0.02$ for V889 Aql and
[Fe/H]$=0.00\pm 0.05$ for V402 Lac, as can be seen in Figs. 11 and 12.
Additional cross-checks were carried out by comparing the derived ages of the
components two components, obtaining compatible values in both systems, thus
providing additional reassurance. The uncertainties of the theoretical values
were computed from the standard deviations of the results obtained from
$10^{5}$ random resamplings of the input parameters, $M$, $\log g$, and
$[Fe/H]$, employing their quoted uncertainties.

Figure 11: $T_{\rm eff}$-$\log g$ relationship for the primary (blue) and
secondary (red) components of V889 Aql, together with the theoretical models
from Claret (2019) corresponding to the mass of each component and
metallicities of 0.0 (dashed lines) and 0.1 (dotted lines). Figure 12: $T_{\rm
eff}$-$\log g$ relationship for the primary (blue) and secondary (red)
components of V402 Lac, together with the theoretical models from Claret
(2019) corresponding to the mass of each component and metallicities of 0.0
(dashed lines) and 0.1 (dotted lines).

Using the absolute dimensions and orbital parameters of V889 Aql, we obtain a
theoretical apsidal motion rate of $\dot{\omega}=0.000426\pm 0.000007$ deg
cycle-1, with a classical term of $\dot{\omega}=0.000075\pm 0.000007$ deg
cycle-1 and a relativistic term of $\dot{\omega}=0.000336\pm 0.000002$ deg
cycle-1. Additionally, the third body companion, hypothetically composed of
two stars of 1.4 M⊙ and with a mutual inclination of 50 deg, contributes with
an estimated value of $\dot{\omega}=0.000030$ deg cycle-1 (Borkovits et al.
2019), which increases the theoretical apsidal motion value to
$\dot{\omega}=0.00046$ deg cycle-1. This value agrees well with the observed
value of $\dot{\omega}=0.00046\pm 0.00002$ deg cycle-1, although the
assumption on the third body mass and inclination remains a source of
significant uncertainty.

Using the absolute dimensions and orbital parameters of V402 Lac, we obtain a
theoretical apsidal motion of $\dot{\omega}=0.0088\pm 0.0005$ deg cycle-1,
with a classical term of $\dot{\omega}=0.0080\pm 0.0005$ deg cycle-1 and a
relativistic term of $\dot{\omega}=0.000822\pm 0.000007$ deg cycle-1. Despite
not considering the effect of the potential third body, which is nevertheless
not expected to be significant given the large classical term, the predicted
value is compatible with the observed $\dot{\omega}=0.0090\pm 0.0003$ deg
cycle-1 within errors.

In conclusion, both systems are now part of the select group of eclipsing
binaries with precise absolute dimensions, necessary for the improvement of
empirical calibrations (Torres et al. 2010) or the test of apsidal rates
(Baroch et al. 2021). The case of V889 Aql is a clear example of apsidal
motion dominated by the relativistic term, 75% of the total, with a
significant third body contribution. On the other hand, V402 Lac shows a
rather fast apsidal motion dominated by the classical term, with negligible
relativistic and third body contributions. In the light curves, significant
third light contributions were identified as well as the presence of a third
body through the variations of the eclipse timings, definitely in the case of
V889 Aql.

###### Acknowledgements.

This article has made use of observations collected at the Centro Astronómico
Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Junta de
Andalucía and the Instituto de Astrofísica de Andalucía (CSIC). Data were
partly obtained with the TIGRE telescope, located at La Luz observatory,
México. TIGRE is a collaboration of the Hamburger Sternwarte, the Universities
of Hamburg, Guanajuato and Liège. This paper includes data collected by the
TESS mission. Funding for the TESS mission is provided by the NASA’s Science
Mission Directorate. STELLA was made possible by funding through the State of
Brandenburg (MWFK) and the German Federal Ministry of Education and Research
(BMBF). The facility is a collaboration of the AIP in Brandenburg with the IAC
in Tenerife. The construction of the 2.7 m telescope and Tull spectrograph was
supported by contract NASr-242 with the National Aeronautics and Space
Administration. This work has made use of data from the European Space Agency
(ESA) mission Gaia. Gaia data are being processed by the Gaia Data Processing
and Analysis Consortium (DPAC). Funding for the DPAC is provided by national
institutions, in particular the institutions participating in the Gaia
MultiLateral Agreement (MLA). The Gaia mission website is
https://www.cosmos.esa.int/gaia. The Gaia archive website is
https://archives.esac.esa.int/gaia. This publication has been made possible by
grant PGC2018-098153-B-C33 funded by MCIN/AEI/10.13039/501100011033 and by
“ERDF A way of making Europe”, by the program Unidad de Excelencia María de
Maeztu CEX2020-001058-M, and by the Generalitat de Catalunya/CERCA programme.

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##  Appendix A RV data

Tables 12 and 13 list the RVs used in this work, as computed with todmor. The
RVs from each instrument are labelled as C (CAFE), T (TIGRE), Tu (Tull), and S
(STELLA).

Table 12: Radial velocities of V889 Aql. BJD | RVA | RVB | Ins.
---|---|---|---
| [km s-1] | [km s-1] |
2459102.3675 | $42.469\pm 0.325$ | $-85.62\pm 0.42$ | C
2459103.3310 | $40.454\pm 0.301$ | $-83.30\pm 0.39$ | C
2459103.3826 | $39.666\pm 0.278$ | $-82.56\pm 0.37$ | C
2459114.3370 | $41.624\pm 0.319$ | $-84.50\pm 0.43$ | C
2459114.3744 | $41.201\pm 0.445$ | $-84.13\pm 0.56$ | C
2459115.3202 | $14.186\pm 0.394$ | $-55.87\pm 0.53$ | C
2459116.3237 | $-71.968\pm 0.366$ | $30.90\pm 0.50$ | C
2459116.3702 | $-76.775\pm 0.903$ | $36.40\pm 1.33$ | C
2459121.3077 | $-5.061\pm 0.298$ | $-37.28\pm 0.38$ | C
2459121.3253 | $-4.744\pm 0.293$ | $-37.76\pm 0.38$ | C
2459122.2978 | $14.273\pm 0.299$ | $-56.48\pm 0.40$ | C
2459122.3335 | $15.184\pm 0.351$ | $-56.34\pm 0.47$ | C
2459132.3424 | $-6.924\pm 0.354$ | $-36.18\pm 0.47$ | C
2459134.3353 | $28.252\pm 0.365$ | $-71.86\pm 0.50$ | C
2459134.3504 | $28.721\pm 0.408$ | $-72.33\pm 0.54$ | C
2459135.3316 | $39.934\pm 0.317$ | $-83.20\pm 0.42$ | C
2459135.3644 | $39.968\pm 0.312$ | $-83.53\pm 0.41$ | C
2459146.2947 | $38.837\pm 0.886$ | $-81.91\pm 1.16$ | C
2459148.2882 | $31.171\pm 0.615$ | $-72.87\pm 0.80$ | C
2459300.9837 | $29.097\pm 0.819$ | $-65.96\pm 1.03$ | T
2459303.9885 | $32.721\pm 0.756$ | $-69.75\pm 1.02$ | T
2459307.9894 | $-66.936\pm 0.583$ | $33.51\pm 0.80$ | T
2459309.9585 | $-11.892\pm 0.891$ | $-25.16\pm 1.24$ | T
2459311.9485 | $25.763\pm 0.606$ | $-64.17\pm 0.77$ | T
2459313.9349 | $44.741\pm 0.615$ | $-84.86\pm 0.87$ | T
2459315.9387 | $-13.346\pm 0.820$ | $-23.41\pm 1.14$ | T
2459317.9266 | $-110.733\pm 0.685$ | $76.74\pm 0.90$ | T
2459319.9227 | $-42.241\pm 0.638$ | $8.53\pm 0.90$ | T
2459321.9157 | $7.355\pm 0.707$ | $-44.38\pm 0.85$ | T
2459322.9755 | $25.088\pm 0.678$ | $-62.02\pm 0.93$ | T
2459323.9769 | $39.146\pm 0.575$ | $-76.52\pm 0.84$ | T
2459325.9019 | $40.406\pm 0.617$ | $-77.65\pm 0.89$ | T
2459327.9296 | $-97.977\pm 0.828$ | $62.12\pm 1.06$ | T
2459329.9394 | $-78.204\pm 0.856$ | $40.71\pm 1.20$ | T
Table 13: Radial velocities of V402 Lac. BJD | RVA | RVB | Ins.
---|---|---|---
| [km s-1] | [km s-1] |
2452205.6795 | $-23.2\pm 1.4$ | $23.6\pm 1.4$ | Tu
2452206.6948 | $134.9\pm 1.7$ | $-129.4\pm 1.7$ | Tu
2452206.7861 | $151.5\pm 1.8$ | $-149.3\pm 1.9$ | Tu
2452207.5727 | $-34.4\pm 1.3$ | $38.8\pm 1.3$ | Tu
2452207.5842 | $-36.1\pm 1.3$ | $40.0\pm 1.3$ | Tu
2452208.8099 | $-69.8\pm 1.3$ | $72.8\pm 1.4$ | Tu
2452209.5928 | $-8.8\pm 2.3$ | $12.5\pm 2.3$ | Tu
2452254.5742 | $-45.1\pm 1.1$ | $43.5\pm 1.1$ | Tu
2452254.5857 | $-43.4\pm 1.0$ | $42.7\pm 1.0$ | Tu
2452255.5780 | $81.5\pm 1.2$ | $-70.9\pm 1.1$ | Tu
2452256.7221 | $-30.8\pm 1.2$ | $31.9\pm 1.2$ | Tu
2452256.7350 | $-31.4\pm 1.1$ | $30.1\pm 1.2$ | Tu
2452257.5633 | $-84.5\pm 1.1$ | $91.7\pm 1.1$ | Tu
2452257.6743 | $-81.6\pm 1.2$ | $88.8\pm 1.3$ | Tu
2453218.7628 | $-63.9\pm 1.2$ | $62.2\pm 1.2$ | Tu
2453307.6607 | $151.7\pm 1.3$ | $-153.0\pm 1.4$ | Tu
2453307.7799 | $113.0\pm 1.1$ | $-107.6\pm 1.2$ | Tu
2453308.7670 | $-91.9\pm 1.0$ | $97.0\pm 1.1$ | Tu
2453310.7328 | $74.3\pm 1.3$ | $-66.9\pm 1.3$ | Tu
2453889.9096 | $166.6\pm 3.9$ | $-165.4\pm 4.3$ | Tu
2453889.9579 | $168.6\pm 4.0$ | $-166.6\pm 4.5$ | Tu
2453890.9115 | $-73.8\pm 3.8$ | $91.1\pm 4.5$ | Tu
2453890.9613 | $-76.4\pm 3.8$ | $95.5\pm 4.4$ | Tu
2453891.8249 | $-63.0\pm 7.4$ | $66.5\pm 9.0$ | Tu
2453891.9420 | $-56.4\pm 5.1$ | $56.6\pm 5.2$ | Tu
2453972.9724 | $146.9\pm 3.4$ | $-139.7\pm 3.7$ | Tu
2453973.9370 | $-60.3\pm 3.0$ | $64.1\pm 3.5$ | Tu
2453976.8437 | $158.6\pm 3.5$ | $-160.0\pm 3.5$ | Tu
2453976.9196 | $165.7\pm 3.4$ | $-165.2\pm 3.6$ | Tu
2453976.9762 | $167.6\pm 3.9$ | $-165.0\pm 5.0$ | Tu
2455077.4986 | $144.8\pm 1.7$ | $-177.2\pm 1.8$ | S
2455077.5114 | $139.8\pm 1.9$ | $-180.3\pm 1.7$ | S
2455077.5241 | $143.4\pm 1.7$ | $-177.6\pm 1.7$ | S
2455078.7390 | $-117.9\pm 2.1$ | $78.7\pm 2.1$ | S
2455079.3836 | $-92.8\pm 3.0$ | $49.9\pm 2.8$ | S
2455079.6449 | $-69.1\pm 3.3$ | $26.5\pm 3.2$ | S
2455079.6576 | $-68.5\pm 2.9$ | $29.6\pm 2.7$ | S
2455081.4869 | $123.0\pm 1.6$ | $-166.0\pm 1.5$ | S
2455081.4997 | $122.5\pm 1.6$ | $-154.1\pm 1.9$ | S
2455082.5519 | $-110.1\pm 2.2$ | $80.3\pm 2.6$ | S
2455139.5204 | $-109.9\pm 1.6$ | $72.5\pm 1.5$ | S
2455139.5514 | $-104.5\pm 1.8$ | $73.9\pm 1.7$ | S
2455143.3322 | $-106.4\pm 1.4$ | $74.8\pm 1.5$ | S
2455143.3449 | $-108.5\pm 1.5$ | $70.7\pm 1.7$ | S
2455143.3576 | $-105.6\pm 1.7$ | $74.1\pm 2.0$ | S
2455143.3712 | $-105.4\pm 1.6$ | $75.8\pm 1.7$ | S
2455147.3229 | $-102.3\pm 1.5$ | $65.2\pm 1.5$ | S
2455147.3357 | $-101.2\pm 1.4$ | $62.8\pm 1.3$ | S
2455147.3484 | $-95.8\pm 1.6$ | $65.6\pm 1.5$ | S
2455147.4530 | $-87.1\pm 1.6$ | $61.9\pm 1.4$ | S
2455147.4657 | $-90.4\pm 1.6$ | $53.4\pm 1.5$ | S
2455147.4787 | $-90.2\pm 1.5$ | $55.3\pm 1.4$ | S
2455149.3138 | $138.5\pm 1.6$ | $-175.0\pm 1.5$ | S
2455149.3266 | $143.9\pm 2.0$ | $-175.8\pm 1.8$ | S
2455149.3393 | $139.6\pm 1.7$ | $-176.3\pm 1.6$ | S
2455149.4421 | $138.1\pm 2.7$ | $-183.7\pm 2.5$ | S
2455149.4549 | $133.3\pm 3.8$ | $-176.6\pm 5.3$ | S
2455149.4677 | $133.8\pm 2.2$ | $-175.5\pm 2.3$ | S
2455149.4805 | $138.7\pm 2.2$ | $-176.3\pm 2.4$ | S
2455149.4933 | $138.2\pm 2.1$ | $-177.5\pm 2.0$ | S
2455149.5060 | $134.9\pm 2.2$ | $-177.1\pm 2.4$ | S
2455149.5190 | $131.4\pm 1.9$ | $-176.0\pm 1.9$ | S
2455150.3125 | $-101.5\pm 2.3$ | $55.6\pm 2.4$ | S
2455150.3252 | $-101.9\pm 3.9$ | $67.6\pm 3.8$ | S
Table 13: Continued. BJD | RVA | RVB | Ins.
---|---|---|---
| [km s-1] | [km s-1] |
2455150.3379 | $-99.6\pm 2.0$ | $64.6\pm 2.1$ | S
2455150.4817 | $-106.6\pm 4.3$ | $64.2\pm 6.1$ | S
2455156.4852 | $70.0\pm 3.5$ | $-101.2\pm 4.3$ | S
2455156.5117 | $73.4\pm 3.8$ | $-119.5\pm 4.0$ | S
2459102.3310 | $-54.3\pm 2.1$ | $43.0\pm 2.2$ | C
2459102.6343 | $-104.4\pm 2.6$ | $98.2\pm 3.2$ | C
2459102.6482 | $-107.1\pm 2.2$ | $98.8\pm 2.8$ | C
2459103.3466 | $-92.5\pm 1.7$ | $72.0\pm 2.0$ | C
2459103.3712 | $-89.6\pm 1.9$ | $69.4\pm 2.3$ | C
2459103.6092 | $-68.4\pm 2.2$ | $51.7\pm 2.2$ | C
2459114.3461 | $-107.7\pm 1.3$ | $89.9\pm 1.7$ | C
2459114.5400 | $-83.6\pm 3.2$ | $92.0\pm 3.9$ | C
2459114.5535 | $-92.0\pm 4.8$ | $78.1\pm 5.8$ | C
2459116.3337 | $90.3\pm 2.1$ | $-103.4\pm 2.5$ | C
2459116.3609 | $100.1\pm 4.1$ | $-110.6\pm 3.8$ | C
2459121.3176 | $-76.2\pm 2.5$ | $66.3\pm 2.4$ | C
2459121.4795 | $-101.3\pm 1.7$ | $86.5\pm 1.9$ | C
2459122.3248 | $-86.7\pm 2.7$ | $71.1\pm 2.5$ | C
2459122.5446 | $-63.4\pm 2.8$ | $61.5\pm 2.7$ | C
2459122.5625 | $-65.3\pm 3.0$ | $58.9\pm 2.7$ | C
2459128.2973 | $138.4\pm 2.3$ | $-144.6\pm 2.6$ | C
2459133.3468 | $-100.4\pm 2.2$ | $89.7\pm 2.4$ | C
2459133.5187 | $-92.3\pm 2.6$ | $82.7\pm 3.0$ | C
2459135.3393 | $107.8\pm 1.8$ | $-111.0\pm 1.7$ | C
2459135.3726 | $106.5\pm 1.6$ | $-115.5\pm 1.4$ | C
2459135.5200 | $132.5\pm 1.8$ | $-135.6\pm 2.1$ | C
2459146.3043 | $49.2\pm 7.3$ | $-48.8\pm 7.0$ | C
2459146.4732 | $85.6\pm 4.9$ | $-82.9\pm 4.9$ | C
2459147.4269 | $61.7\pm 3.5$ | $-67.6\pm 4.3$ | C
2459148.3252 | $-96.7\pm 5.8$ | $96.5\pm 5.5$ | C
2459148.4267 | $-106.5\pm 2.9$ | $98.5\pm 3.1$ | C