11institutetext: Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004,
18080 Granada, Spain 22institutetext: Dept. Física Teórica y del Cosmos,
Universidad de Granada, Campus de Fuentenueva s/n, 10871, Granada, Spain
33institutetext: Centro de Astrobiologia (CSIC-INTA), E-28850 Torrejón de
Ardoz, Madrid, Spain 44institutetext: International Space Science Institute
(ISSI), Hallerstrasse 6, CH-3012 Bern, Switzerland 55institutetext: Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, c/ Can Magrans s/n, E-08193
Bellaterra, Barcelona, Spain 66institutetext: Institut d’Estudis Espacials de
Catalunya (IEEC), c/ Gran Capità 2-4, E-08034 Barcelona, Spain

# Analysis of apsidal motion in eclipsing binaries using TESS data

II. A test of internal stellar structure

A. Claret 1122 A. Giménez 3344 D. Baroch 5566 I. Ribas 5566 J. C. Morales 5566
G. Anglada-Escudé 5566

(Received; accepted; )

###### Abstract

Context. The measurement of apsidal motion rates in eccentric eclipsing
binaries is a unique way to gain insight into the internal structure of stars
through the internal density concentration parameter, $k_{2}$. High-quality
physical parameters of the stellar components, together with precise
measurements of the advance of the periastron, are needed for the comparison
with values derived from models.

Aims. As a product of the Transiting Exoplanet Survey Satellite (TESS)
mission, high-precision light curves of a large number of eclipsing binaries
are now available. Using a selection of well-studied, double-lined eccentric
eclipsing binary systems, we aim to determine their apsidal motion rates and
place constraints on the internal density concentration and compare with the
predictions from state-of-the-art theoretical models.

Methods. We computed times of minimum light using the TESS light curves of 34
eclipsing binaries with precise absolute parameters. We used the changing
difference over time between primary and secondary eclipse timings to
determine the apsidal motion rate. To extend the time baseline, we combined
the high-precision TESS timings with reliable archival data. On the other
hand, for each component of our sample of double-lined eclipsing binaries, we
computed grids of evolutionary stellar models for the observed stellar mass
exploring ranges of values of the overshooting parameter $f_{\rm ov}$, the
mixing-length parameter, and the metallicity. To find the best solution for
the two components we adopted a $\chi^{2}$ statistic to infer the optimal
values of the overshooting parameter and the mixing-length parameter. The
theoretical internal structure constants to be compared with the observed
values were calculated by integrating the differential equations of Radau for
each stellar model.

Results. We have determined the apsidal motion rate of 27 double-lined
eclipsing binaries with precise physical parameters. The obtained values,
corrected for their relativistic contribution, yield precise empirical
parameters of the internal stellar density concentration. The comparison of
these results with the predictions based on new theoretical models shows very
good agreement. Small deviations are identified but remain within the
observational uncertainties and the path for a refined comparison is
indicated.

###### Key Words.:

stars: binaries: close; stars:evolution; stars:interiors; stars: fundamental
parameters; stars:rotation;

††offprints: A. Claret<EMAIL_ADDRESS>
##  1 Introduction

Double-lined eclipsing binaries (DLEBs) have demonstrated to be the basic
source of information about fundamental stellar properties, such as masses and
radii (Andersen, 1991). The comparison of observed physical properties with
theoretical models has been used to perform critical tests of stellar
structure and evolution models (Ribas et al., 2000b; Torres et al., 2010;
Claret & Torres, 2019). Precise masses and radii are needed for a solid and
reliable comparison between observations and theory, essentially better than
3%, and this generally calls for the use of DLEBs. Furthermore, the binary
systems have to be well detached, for instance, with the radii of both
components being much smaller than their Roche limits, to ensure that the
components represent the behavior of single stars with the same physical
properties.

Eccentric eclipsing binaries offer further opportunities to characterize the
internal structure of stars through the measurement of the precession rate of
the line of the apses of the orbit, for instance, the apsidal motion rate.
Such secular motion can be understood as the sum of two terms, one classical
and one relativistic. The classical, or Newtonian, effect is related to the
quadrupole interactions that depend on the internal mass distribution of the
stellar components (basically the degree of mass concentration toward the
center). The second term is a contribution arising from general relativity
(GR), the best known example of which is the advance of the perihelion of
Mercury. Of course, if there is a third body gravitationally bound to the
close binary, an additional term in the apsidal motion has to be considered.

Precise determinations of apsidal motion require long-term monitoring of the
times of eclipse, generally spanning several decades, with high-quality
measurements, although apsidal motion rates can also be derived from long time
series of radial velocity data (Schmitt et al., 2016). The Transiting
Exoplanet Survey Satellite (TESS) mission to study exoplanets through
photometric transits (Ricker et al., 2015), with its nearly full sky coverage,
provides, as a bonus, precise photometry of a large sample of eclipsing binary
systems with a time baseline of at least 27 days and up to two years in some
cases. Precise monitoring of binary light curves is thus possible from space,
without the disturbing day/night effect, and accurate eclipse timings can be
derived thanks to the uniform sampling. Equipped with this new tool, we have
established a program to make precise apsidal motion determinations in
eccentric eclipsing binaries with accurate absolute dimensions, in some cases
for the first time, and compared them with theoretical models.

The first results of our program were presented in Baroch et al. (2021),
hereafter referred to as paper I. This paper addressed eclipsing binary
systems with accurate dimensions and with apsidal motion rates dominated by
the relativistic contribution, with a limit set to be at least 60% of the
total apsidal motion rate. This allowed us to perform a stringent test of the
predictions of GR, which revealed excellent agreement between observations and
theory. In the present paper, we focus on the systems where the classical term
is dominant (for instance, GR contribution being less than 60% of the total).
For such cases, we calculate the GR apsidal motion rate analytically (which
paper I shows to be accurate) and we subsequently subtract it from the
observed rate to determine the observational classical term. We can then
compare with stellar model predictions and provide constraints on interior
structure (for instance, Claret & Giménez, 1993, 2010, and references
therein).

This paper is structured as follows. Section 2 is dedicated to describe the
observational sample and the measurements of eclipse timings. Section 3
describes the apsidal motion determinations, the methodology and the results,
with further details for each individual system in an appendix. Section 4 is
dedicated to describe the stellar evolutionary models, the differential
equations used to obtain theoretical values of the apsidal motion constants,
and the methodology employed to compare with the physical dimensions of the
component stars. Section 5 is devoted to the comparison between observed and
theoretical values of log k2 and, finally, in Section 6 we present our
conclusions.

##  2 The observational sample

For a useful interpretation of the apsidal motion rate observed in eccentric
eclipsing binaries, it is essential to have a precise knowledge of the
physical properties of the component stars, essentially masses and radii. Some
of the equations, for example, have a strong dependence on the relative radii
as they contribute to the fifth power. For this reason, we have limited our
dynamical study using eclipse timings to those cases where the masses and
radii of the components are known with an accuracy better than 3%. A list of
well-detached eccentric eclipsing binaries with good absolute dimensions was
published by Torres et al. (2010), and we have further added a number of
systems from the DEBCAT catalog of Southworth (2015), which is updated
permanently. We have only considered systems with TESS measurements of both
primary and secondary eclipses, thus permitting the determination of the
timing difference. Due to the expected amplitude of the apsidal motion
variations, we also set a lower limit to the orbital eccentricity at 0.01.

Our analysis is restricted to systems with an expected relativistic
contribution below 60% of the total apsidal motion. Those with a larger
relativistic contribution were discussed in paper I and are less useful for
the study of internal structure due to the larger relative uncertainty of the
observed classical term. The systems analyzed in the present paper are listed
in Table LABEL:tab:sample, together with their main physical parameters and
the corresponding references, sorted by decreasing mass of the primary
component. In addition to orbital period, masses and radii of the component
stars, we also provide the effective temperatures and the projected rotational
velocities, necessary for the computation of the apsidal motion rates, as
described in section 4.

The systems V380 Cyg (B1.1 III and B2.5/3 V), V636 Cen (G0 V and G0 V) and CM
Dra (M4.5 V and M4.5 V) are not considered due to the difficulties found with
standard theoretical stellar models in reproducing their observed physical
parameters, namely their masses, radii and effective temperatures,
independently of the apsidal motion results. This is a requisite of our
methodology to obtain theoretical apsidal motion parameters, as described in
Section 4. Each of these systems has some characteristics that push it beyond
the boundaries of our studied parameter space: the evolved stage and the
proximity to the Roche limit of the primary component of V380 Cyg, the strong
chromospheric activity of V636 Cen, or the very low masses and magnetic
activity of CM Dra. Investigating these systems will require detailed
individual studies of the observational data and model input physics, which is
left for subsequent publications. On the other hand, we have considered three
systems that should have been included in paper I due to an expected
relativistic contribution above 60%, but did not have either sufficient TESS
data at the time of publication or a reliable apsidal motion determination.
These systems are V1022 Cas, EW Ori, and BF Dra, and they are included in
Table LABEL:tab:sample and discussed in Section 3.

For all systems in Table LABEL:tab:sample we have analyzed the available TESS
photometric information retrieved from Sectors 1 to 34. To compute the time of
minimum light of the eclipses, we first normalized the TESS light curves using
the out-of-eclipse phases. We selected well-sampled individual eclipses using
the same orbital phase interval for all primary and secondary eclipses, and
computed their time of minimum light employing the widely-used Kwee & van
Woerden (1956) method. We then computed the corresponding difference between
primary and secondary eclipse timings, $T_{2}-T_{1}$, expressed in days, and
listed the resulting values in Table 1, which is available electronically. The
Table gives the values of $T_{2}-T_{1}$ determined from the individual
timings, together with their separation in orbital cycles (dN), that we have
limited to $\pm$1.

##  3 Determination of the apsidal motion rates

In Paper I we performed the determination of the apsidal motion rate from the
analysis of the time-variation of the difference between primary and secondary
eclipses, $T_{2}-T_{1}$. This method assumes independent knowledge of the
orbital eccentricity and that the variations in the timing differences can be
represented by a linear relationship with the slope corresponding to the time
derivative of the argument of periastron. This method is only valid when
considering a small fraction, typically less than 1%, of the total apsidal
motion period, $U$. Otherwise, the nonlinear component of the $T_{2}-T_{1}$
variations becomes relevant and the analysis requires a different approach.
When this is the case, one can use the equations given by Giménez & Bastero
(1995), which are complete up to $\mathcal{O}(e^{5})$. Linearizing the
variations permits computing the argument of periastron, $\omega$,
corresponding to each observed value of $T_{2}-T_{1}$ with the adopted orbital
eccentricity and inclination, as derived from the light curve analysis. This
is based on the relation between the phase of the secondary eclipse and the
value of $e\cos\omega$. Potential ambiguities in the argument of periastron
can be resolved with the value of $e\sin\omega$ resulting from the light curve
analysis. Using this approach, a linear fit to the variation of $\omega$ with
time yields the determination of the apsidal motion rate with no restriction
in terms of the coverage of the period, $U$.

In both approaches, the basic observational information is the measurement of
$T_{2}-T_{1}$. We provide in Table 1 the full list of the $T_{2}-T_{1}$ TESS
values for all systems in Table LABEL:tab:sample, and we discuss each
individual system in Appendix A. In Table 2 we summarize the results of the
apsidal motion determination where the orbital eccentricity, $e$, and the
apsidal motion rate, $\dot{\omega}$ expressed in deg/cycle, are provided. The
anomalistic period, $P_{a}$, can be easily computed from the sidereal period
in Table LABEL:tab:sample, using the relation
$P_{a}=P_{s}/(1-\dot{\omega}_{\rm obs}/360)$. The letter code in the second
column denotes the adopted methodology: A for those based on the linear
variation of $T_{2}-T_{1}$, and B for those using the argument of periastron,
including the comparison of values derived from the light curve analysis. In
addition, we use L to denote values adopted from the literature, generally
with method B, or using variable $\omega$ in the analysis of light curves
spanning a long time base. In this case, we checked that the adopted solutions
predict values for the relative position of the secondary eclipse in agreement
with the new TESS measurements. Obviously, B and L determinations correspond
to faster apsidal motion rates and A to slower ones, including those with a
higher fractional contribution of the relativistic term. Finally, those
systems where the presence of a third body has been proposed, either from the
light curve analysis or from the variations of the individual eclipse timings,
are marked with an asterisk.

Table 2 does not include all systems in Table LABEL:tab:sample because some
were discarded due to various reasons. Firstly, let us note that we defined a
precision criterion to accept a system for further analysis. We only kept
systems with a relative precision better than 25% in the classical term of the
apsidal motion determination, for which we derive the internal stellar
structure constants. This corresponds to a maximum $\sigma_{\log k_{2}}$ of
0.11, necessary for a constraining comparison with theoretical models. After
our analysis, we found a small fraction of the eclipsing binaries in Table
LABEL:tab:sample to yield no significant apsidal motion detections. For two
systems, BP Vul and AI Phe, we were not able to measure apsidal motion in
spite of the precise TESS observations. We encountered a similar situation
with MU Cas and V1022 Cas, which resulted in apsidal motion rate
determinations having uncertainties above our acceptance threshold because of
the narrow time span covered by precise timings. In the case of BF Dra, we
found disagreement between the different methodologies to determine the
apsidal motion and could thus be affected by large systematic effects.
Moreover, the potential presence of a perturbing third body as well as the
highly-evolved nature of the component stars, close to the termination-age
main-sequence, call for an individual study of this system, including the
analysis of a new light curve. Finally, EW Ori does not meet the relative
precision limit in the classical term because of the high relative GR
contribution ($\sim$80%). This system could have been included in Paper I but
we did not have an apsidal motion rate determination at the time. However, we
briefly discuss the comparison with GR and the updated post-Newtonian
parameters including this eclipsing binary in Appendix A.

##  4 The theoretical stellar models

We use the Modules for Experiments in Stellar Astrophysics package (MESA;
Paxton et al., 2011, 2013, 2015) version r-7385, and adopted the solar-
calibrated value of the mixing length parameter to be 1.84 (Torres et al.,
2015). The equation relating the temperature gradients was solved using the
Henyey option, and for the localization of the boundary of the convection zone
we adopted the Schwarzschild criterion. For the opacities we use the mixture
of Asplund et al. (2009), for which $Z_{\odot}=0.0134$. These opacities were
paired with a linear enrichment law given by $Y_{p}=0.249$ (Planck
Collaboration et al., 2016) and a slope $\Delta Y/\Delta Z=1.67$, where
$Y_{p}$ is primordial helium mass fraction. Microscopic diffusion was
considered (see details below) for all evolutionary tracks that were computed
starting from the pre-main sequence (PMS) stage. For each component of our
sample of DLEBs we computed grids for the observed stellar mass exploring
ranges of values of the overshooting parameter $f_{\rm ov}$ (see below), the
mixing-length parameter $\alpha_{\rm MLT}$, and the metallicity $Z$. To find
the solution providing the best fit to both stellar components we adopted a
$\chi^{2}$ statistic and considered $f_{\rm ov}$ and $\alpha_{\rm MLT}$ as the
optimization parameters. We also adopted the initial metal abundance $Z$ to be
the same for the two components.

We calculate evolutionary tracks for the exact component masses and other
parameters. Thus, our methodology does not involve computing grids for a wide
variety of parameters and avoids interpolation in masses, metallicities,
mixing-length, or core overshooting, which could lead to systematic effects.
In our calculations we allowed the optimized ages for both components to
differ by up to 5%, as long as the radii, effective temperatures and masses
were predicted to be within their respective observational error bars. This
flexibility is justified by the expected uncertainties in the input physics of
the evolutionary tracks, for instance, opacities, equations of state, mass
loss, etc., and also in the observational parameters. In some cases this
procedure was improved with some additional computations. Starting from
previously converged models, we increased the resolution in the input physics
($\alpha_{\rm MLT}$, $f_{\rm ov}$ and $Z$) to refine the fitting process.

A fraction of the systems in our sample have high-mass components and
therefore core overshooting can have a significant effect in the computation
of evolutionary model grids. Convective core overshooting is related to the
increase of the stellar core beyond the boundary defined by the Schwarzschild
criterion. Stellar models computed taking into account this extra-mixing lead
to longer main-sequence lifetimes and a higher degree of mass concentration
toward the center, with direct impact on the comparison between theoretical
and observational rates of apsidal motion. Claret & Torres (2019, and
references therein) selected 50 well-measured detached DLEBs to calibrate the
dependency of $f_{\rm ov}$ on stellar mass. In such a formulation, the extra-
mixture is modeled as a diffusive process (Freytag et al., 1996; Herwig et
al., 1997) with a diffusion coefficient at a radial distance $r$ from the
boundary given by $D(r)=D_{\rm o}\exp(-2r/f_{\rm ov}H_{p})$, where $D_{\rm o}$
is the coefficient inside the boundary and $H_{p}$ the pressure scale height.
As a result of the comparison between theoretical models and observational
data from these systems, it has been shown that $f_{\rm ov}$ increases sharply
up to a mass on the order of 2.0 M⊙ and is practically constant up to 4.43 M⊙,
the upper limit in mass of the observational sample adopted by Claret & Torres
(2019). There are also some evidences of a dependence of core overshooting on
stellar mass from a theoretical point of view (see, for example, the
appendices in Claret & Torres, 2017; Jermyn et al., 2018). Such kind of
calibrations can only be performed with highly evolved stars where the effects
of core overshooting are more evident. Unfortunately, this is not the case in
our observational sample with only two systems with moderately evolved
components (V453 Cyg and V478 Cyg). For all stars more massive than 10 M⊙, we
have extended the range of values of $f_{\rm ov}$ to be explored with our
search methodology.

In all our computations we have included microscopic diffusion (8 elements:
H1, He3, He4, C12, N14, O16, Ne20, and Mg24). For those stars showing
convective envelopes, we adopted the standard mixing-length formalism (Böhm-
Vitense, 1958). For the most massive stars in our sample, we have adopted the
Vink et al. (2001) formulation for mass loss assuming a multiplicative scale
factor $\eta=0.1$ (not to be confused with $\eta$ of the Radau equation; see
below). On the other hand, rotational-mixing was not included in the
calculations. For a more complete description of our modeling framework, see
Claret & Torres (2019) and references therein.

We also ran a consistency check of our model comparisons with observational
data (masses, Teff, $\log g$, $Z$) between our adopted MESA models and those
computed with the GRANADA code (Claret, 2004). This was done for a subsample
of systems (DI Her, V1143 Cyg, V1647 Sgr, IQ Per, $\zeta$ Phe). Convective
core overshooting in the GRANADA code is treated as a step-function,
characterized by the parameter $\alpha_{\rm ov}$. A similar check-up procedure
was already carried out by Claret & Torres (2016, 2017) for a larger sample of
DLEBs. From this cross-comparison the relationship $\alpha_{\rm ov}/f_{\rm
ov}\approx 11.36\pm 0.22$ was derived. With this procedure we check the
consistency between the log k2 computed with MESA and with the GRANADA codes.
A very good agreement has been found between the k2 computed with the two
codes.

To calculate the theoretical values of log k2 to be compared with the
corresponding empirically-determined values, we applied the methodology
described above to all systems in Table LABEL:tab:sample. The theoretical
internal structure constants were computed for each component through the
integration of the differential equations of Radau of order $j$, namely

${a{d\eta_{j}(a)\over
da}}+{6\rho(a)\over\overline{\rho}(a)}{(\eta_{j}\\!+\\!1)}+{\eta_{j}(\eta_{j}\\!-\\!1)}={j(j+1}),\,j=2,3,4,$
(1)

where

$\eta\equiv{{a}\over{\epsilon_{j}}}{d\epsilon_{j}\over{da}}$ (2)

and $a$ is the mean radius of the equipotential, $\epsilon_{j}$ is a measure
of the deviation from sphericity (the tesseral harmonics), $\rho(a)$ is the
mass density at the distance a from the center, and $\overline{\rho}(a)$ is
the mean mass density within an equipotential of radius a. The boundary
conditions are: $\eta_{j}(0)$ =$j-2$, and
$\left(d\eta_{j}\over{da}\right)_{\eta=0}$ =
$-{3(j-1)\over{j+1}}{dD\over{da}}$, where $D={\rho/{\overline{\rho}}}$.

The theoretical internal structure constant kj, for each component $j$, is
given by

$k_{j}={{j+1-\eta_{j}(R)}\over{2\left(j+\eta_{j}(R)\right)}},$ (3)

where $\eta_{j}(R)$ are the values of $\eta_{j}$ at the surface of the star.
Finally, the theoretical value of k2 to be compared with observations is given
by the weighted average over the model predictions for the two components:

${\overline{k}_{2,{\rm theo}}}={{c_{21}{k_{21,{\rm theo}}}+{c_{22}{k_{22,{\rm
theo}}}}}\over{c_{2,1}+c_{2,2}}}.$ (4)

The parameters c2i, for the case of aligned rotation axes, are computed
following the equations,

${c_{2i}}\\!=\\!\left[\\!\left({\Omega_{i}\over{\Omega_{K}}}\right)^{\\!\\!2}\\!\\!\left(1+{m_{3-i}\over{m_{i}}}\right)\\!f(e)+{15\,m_{3-i}\over{m_{i}}}\,g(e)\right]\\!\\!{\left({R_{i}\over{A}}\right)^{\\!\\!5}}$
(5)

where $A$ is the semi-major axis of the orbit, while $f$ and $g$ are auxiliary
functions of the orbital eccentricity given by

${f(e)}={(1-e^{2})^{-2}}$ (6)

and

${g(e)}={(8+12e^{2}+e^{4})f(e)^{2.5}\over{8}}.$ (7)

Additionally, ${\Omega_{i}/{\Omega_{K}}}$ is the ratio between the rotational
angular velocity of component $i$ and the average orbital angular velocity,
$m_{i}$ are the stellar masses, $R_{i}$ are the stellar radii and the
eccentricity is denoted by the symbol $e$. As indicated by Claret & Giménez
(1993), the contribution of the terms $k_{n}$ (for $n>2$) are negligible in
the present context.

On the other hand, using the quasi-spherical approximation, Claret (1999)
found that the influence of rotation on internal structure depends on the
distortion of the configuration following the expression $\delta\log k_{2}$
$\approx-0.87\Lambda_{s}+0.004$, where $\Lambda_{s}=2v^{2}/(3gR)$, with $v$
being the rotational velocity, $g$ the local gravity and $R$ the stellar
radius. Such correction was computed at the surface of each star in our
sample, using the observed rotational velocities given in Table
LABEL:tab:sample, and applied to the corresponding theoretical models. We note
that this correction does not take into account the effects of rotational-
mixing on the mass concentration. It is also important to point out some
uncertainty in relation to the observed rotational velocities regarding
whether or not they are representative of the stellar interior. Most of the
systems in our observational sample are not very evolved, from which it
follows that the angular velocity gradients in their interiors should not be
high. This circumstance reduces the uncertainties although it does not remove
them completely.

The effects of stellar compressibility and dynamic tides were also taken into
account according to the computations by Claret & Willems (2002). We could
verify that in all systems in their Table 2, no significant effect is present
except in the most massive and close systems, EM Car and Y Cyg, for which we
have applied the calculated correction (2% and 1%, respectively).

As an example of our procedure, we present in Fig. 1 the fitting process in
the case of AI Phe. The best input physics for the corresponding models and
for each component (subindices 1 and 2) is given by $\alpha_{\rm MLT,1}=2.00$,
$\alpha_{\rm MLT,2}=2.10$ and $f_{\rm ov1}=f_{\rm ov2}=0$. In Fig. 2 we show
the resulting models in the $\log g$ vs. $\log{\rm age}$ diagram. The
observational error bars are very small and were shown in different color for
the sake of clarity. This solution, as explained above, is obtained on the
basis of the minimum $\chi^{2}$ of the model grid search and imply the log k2i
values for the system. In the case of AI Phe, we find $\log k_{2,1}=-1.58\pm
0.30$ and $\log k_{2,2}=-2.45\pm 0.03$. Although the masses of both components
are similar, the large difference between apsidal motion constants is due to
the advanced evolutionary stage of the system. While the secondary is still in
the main sequence, the primary is much more evolved and is located in the
giant branch, where $k_{2}$ varies very rapidly, and hence the large error
bar. As we do not yet have a reliable determination of the apsidal motion rate
for AI Phe, we have performed Monte Carlo simulations (see section 5.1 for
more details), assuming that the two components are aligned with the orbital
spin (see Fig. 3). The asymmetry of the simulations reflect the differences
between the uncertainty in $\log k_{2}$ for each of the components. The most
probable value of $\dot{\omega}_{\rm theo}\approx 2.24_{-0.50}^{+1.12}\times
10^{-4}$ degrees/cycle, should be compared with the observational value when
available, but the slow predicted rate is consistent with the lack of a
measurable value (see Appendix A). We have selected AI Phe to illustrate our
methodology, in spite of not having an observed value of the apsidal motion
rate, as a tribute to the late Prof. J. Andersen. This was one of his favorite
DLEBs, frequently used as a test case for stellar structure and evolution.

Figure 1: HR diagram for AI Phe. The models were calculated adopting
$Z=0.011$. The solid line indicates the primary component while the dashed one
represents the secondary. Figure 2: $\log g$ and respective error bars as a
function of time for AI Phe. The two vertical lines indicate the time interval
corresponding to the individual ages for each component. The horizontal lines
represent the error bars in log g. Line coding as in Fig. 1.  Figure 3: Monte
Carlo simulations ($10^{6}$ realizations) for the theoretical apsidal motion
of AI Phe.

##  5 The apsidal motion test of stellar structure

The weighted values of the model-predicted $\log k_{2}$ given by Eq. (4) are
presented in column 6 of Table 2, together with the corresponding predicted
apsidal motion rates. The adopted methodology ensures that both components
have the same age (within the previously mentioned 5% tolerance limit), and
that the observed masses, radii and effective temperatures, are reproduced by
the respective models within their errors.

The comparison of the theoretical $\dot{\omega}$ with the observed values is
shown in Fig. 4. As can be seen, the agreement between observed and predicted
values is excellent. The only system with marginal agreement, at the limit of
the adopted uncertainties, is EM Car, having the highest mass components of
our sample. Such good agreement includes the cases where a third body has been
identified, as shown in the bottom panel of Figure 4 (see 5.2).

Figure 4: Comparison between $\dot{\omega}_{\rm theo}$ and $\dot{\omega}_{\rm
obs}$. The top panel shows a comparison between the theoretical predictions
and the observed values of the apsidal motion rates for all systems in Table 3
while the bottom panel shows the same comparison but only for systems with a
third body.  Figure 5: The top panel depicts a comparison between the
theoretical predictions and the observed values of the apsidal motion constant
log k2 for all systems in Table 2. The bottom panel shows the same comparison
but without plotting the low mass binary V530 Ori, which has a rather extreme
value, so that the bulk of the sample can be better visualized.

For the test of stellar models, we have to use the values of $k_{2}$. Their
observational value is a function of $\dot{\omega}$ and the $c_{2j}$
coefficients given in Eq. (5):

${\overline{k}_{2,{\rm obs}}}={1\over{(c_{2,1}+c_{2,2})}}{\dot{\omega}_{\rm
Newt}\over{360}}.$ (8)

This refers only to the classical, or Newtonian, term. The total observed
apsidal motion rate in Table 2 contains the contribution of this classical
term plus a second additive term corresponding to the relativistic or GR
contribution. The GR term is independent of the tidal/rotational distortions
and can be computed using the equation given by Levi-Civita (1937) and
rewritten by Giménez (1985) as a function of observable parameters. This
equation is, in degrees per cycle:

${\dot{\omega}_{GR}}={5.45\times
10^{-7}}\left({m_{1}+m_{2}\over{P}}\right)^{2/3}{1\over{(1-e^{2})}},$ (9)

where the apsidal motion rate corresponding to the relativistic term is
expressed in degrees per cycle with masses in solar units and the orbital
period in days. The classical term in Eq. (8), used to calculate the observed
$\log k_{2}$, is then simply calculated as

${\dot{\omega}_{\rm Newt}}={\dot{\omega}_{\rm obs}}-{\dot{\omega}_{\rm GR}}.$
(10)

It should be noted that we have assumed the rotational axes of both components
to be aligned with the orbital spin in all cases except DI Her, for which the
expression by Shakura (1985) was used together with the observed tilt angles
by Albrecht et al. (2009). This is further discussed in Sect. 5.1, together
with the suspected, albeit not corrected, cases of V1143 Cyg and EP Cru.
Furthermore, all systems for which a third body has been identified through
eclipse timing variations or from the measurement of third light are
identified by an asterisk and included in Table 2 with no specific correction.
They are shown in Fig. 4, at the bottom, and are discussed in Sect. 5.2.

Figure 6: Differences between $\log\overline{k}_{\rm 2obs}$ and
$\log\overline{k}_{\rm 2theo}$ as a function of the relative radius of the
primary. The mean differential for systems with $r_{1}<0.18$ is $\delta\log
k_{2}=-0.002\pm 0.012$ and for those with $r_{1}>0.18$ is $\delta\log
k_{2}=-0.010\pm 0.008$, but no significant trend is detected over a constant
model.

The resulting values of $\log k_{2}$ derived from the observed apsidal motion
rates are given in Table 2 and are compared with the theoretical values in
Fig. 5. The agreement is very good, with the values of $\log k_{2}$,
theoretical and observational, agreeing within their uncertainties for all
systems. In spite of the general good agreement, a tendency for slightly more
concentrated stars (for instance, lower $\log k_{2}$) than predicted by the
models seems to be present around $\log k_{2}=-2.2$, including EM Car.
However, we emphasize that all systems agree within the corresponding
uncertainties. This effect is not observed in the comparison of
$\dot{\omega}$, shown in Fig. 4, except for the case of EM Car. A detailed
inspection of the observed minus computed values shows that it only affects
those binaries with large relative radii, above 0.18, and with massive
components, the most extreme case being indeed EM Car (Fig. 6). The deviation
might therefore be linked to the observational values used for Eq. (5), given
in Table LABEL:tab:sample. This could be explained through multiple factors,
and given that the deviation is insignificant, we did not attempt to study it
in further detail in the present work.

###  5.1 Tilted rotational axes

When the rotational axes of the component stars are tilted with respect to the
orbital spin, Eq. (5) is no longer valid (Company et al., 1988) and the
alternative formulation given by Shakura (1985) is needed. The total Newtonian
term of the apsidal motion rate is given by:

$\dot{\omega}_{\rm Newt}=\dot{\omega}_{\rm tidal,1}+\dot{\omega}_{\rm
tidal,2}+\dot{\omega}_{\rm rot,1}\phi_{1}+\dot{\omega}_{\rm
rot,2}\phi_{2}\leavevmode\nobreak\ ,$ (11)

where,

$\dot{\omega}_{{\rm tidal,}j}=\left[{15m_{3-j}\over{m_{j}}}k_{2j}\thinspace
g(e)\right]{\left({R_{j}\over{A}}\right)^{5}}\leavevmode\nobreak\ ,$ (12)
$\dot{\omega}_{{\rm
rot,}j}=\left[\left({\Omega_{j}\over{\Omega_{K}}}\right)^{2}\left(1+{m_{3-j}\over{m_{j}}}\right)f(e)\thinspace
k_{2j}\right]{\left({R_{j}\over{A}}\right)^{5}}\leavevmode\nobreak\ ,$ (13)
$\displaystyle\phi_{j}=-{1\over{\sin^{2}i}}\left[\cos\alpha_{j}(\cos\alpha_{j}-\cos\beta_{j}\cos
i)\right]$ $\displaystyle-{1\over{2}}\left[(1-5\cos^{2}\alpha_{j})\right].$
(14)

The angle $i$ is the inclination of the orbital plane, $\alpha_{j}$ are the
angles between the rotation axes and the normal to the orbital plane, and
$\beta_{j}$ are the angles between the rotation axes and the line of sight,
while the angles $\lambda_{j}$ are the projection of the spin axes and the
orbital axis on the plane of the sky for star $j$. All other parameters are
the same as in Eq. (5). The angles $\alpha_{j}$ and $\beta_{j}$ cannot be
directly measured although they are related by the equation,

$\cos\alpha_{j}=\cos\beta_{j}\cos i+\sin\beta_{j}\sin i\cos\lambda_{j},$ (15)

where the angles $\lambda_{j}$ can be directly measured using the Rossiter-
McLaughlin effect (Rossiter, 1924; McLaughlin, 1924).

The Monte Carlo simulations of apsidal motion rates were performed from
realizations of the masses, radii, $k_{2}$, orbital parameters and
$\lambda_{j}$ angles considering Gaussian distributions for the error bars.
Also, we assumed that $\beta_{j}$ are distributed randomly, for instance,
${\rm Prob}(\beta_{j})d\beta_{j}=\sin\beta_{j}d\beta_{j}$. In other words,
$\cos\beta_{j}$ has a uniform distribution. Finally, using Eq. (15), the
values of $\cos\alpha_{j}$ can be determined. Angles $\beta_{j}$ implying
equatorial rotational velocities exceeding the breakup velocities were
discarded.

DI Her was puzzling for long time due to the flagrant disagreement between
observed and predicted apsidal motion rates (Guinan & Maloney, 1985; Claret,
1998), but Albrecht et al. (2009) confirmed, using the Rossiter-McLaughlin
effect, that the rotational axes of the components stars are actually tilted.
The authors measured the angles $\lambda_{1}=+72^{\circ}\pm 4^{\circ}$ and
$\lambda_{2}=-84^{\circ}\pm 8^{\circ}$ for the primary and the secondary
components, respectively. Adopting the values of $\log k_{2}$ derived from the
stellar models described in Sect. 4 ($\log k_{2,1}=-2.146\pm 0.050$, $\log
k_{2,2}=-2.171\pm 0.050$), the histogram in Fig. 7 is generated by simulation
of the angles not directly measured. The comparison with the observed apsidal
motion rate, confirms the solution of the old discrepancy.

Figure 7: Monte Carlo simulations ($10^{6}$ realizations) for the apsidal
motion of DI Her. The two vertical lines indicate the observational error
bars.

Two other eclipsing binaries in our sample, V1143 Cyg and EP Cru, have been
suspected of having tilted rotational axes. For V1143 Cyg, Albrecht et al.
(2007) observed that the axes of both components are aligned with the orbital
spin within uncertainties. To check the effect in the apsidal motion
computations we have carried out Monte Carlo simulations for both cases,
aligned and misaligned, using their values of $\lambda_{1}=0.3^{\circ}\pm
1.5^{\circ}$ and $\lambda_{2}=-1.2^{\circ}\pm 1.6^{\circ}$. The theoretical
internal structure constants derived from our models, namely log
$k_{2,1}=-2.19\pm 0.05$ and $\log k_{2,2}=-2.29\pm 0.05$, were used to
generate the histograms shown in Fig. 8. Both simulations yield very similar
results, as expected from the very small angles $\lambda_{1,2}$ and their
uncertainties. Although the agreement with the observational value is
marginally better in the case of misaligned axes, they both agree within
errors and we have adopted the aligned solution in Table 2.

Figure 8: Monte Carlo simulations ($10^{6}$ realizations) for the total
apsidal motion of V1143 Cyg. In the upper panel, we assumed misaligment while
in the bottom panel we simulate the case of alignment. The two vertical lines
indicate the observational error bars.

We have also explored the effect of inclined rotation axes in EP Cru, with
direct measurement of the angles $\lambda_{1}=-1.8^{\circ}\pm 1.6^{\circ}$ and
$\left|\lambda_{2}\right|<17^{\circ}$ (Albrecht et al., 2013). For the
simulation, we adopted the internal structure constants derived from our
models as $\log k_{2,1}=-2.351\pm 0.015$ and $\log k_{2,2}=-2.348\pm 0.015$,
corrected for a significant rotation-induced internal density concentration.
The corresponding histograms, representing the total predicted apsidal motion,
are shown in Fig. 9 (upper panel: not aligned; bottom: aligned). The solutions
based on aligned rotation axes show better agreement with the observed
$\dot{\omega}_{obs}$ and we have adopted such aligned configuration for Table
2.

Figure 9: Monte Carlo simulations ($10^{6}$ realizations) for the total
apsidal motion of EP Cru. In the upper panel, we assumed misalignment while in
the bottom panel we simulate the case of alignment. The two vertical lines
indicate the observational error bars.

###  5.2 Systems with a third body

We have found that some systems with physically bound third bodies actually
perform well in the comparison between the observed and predicted internal
structure constants. To identify possible dynamical effects in the observed
values, we refer to Fig. 4, with the comparison of observed and predicted
apsidal motion rates, where systems with a potentially perturbing third body
are denoted with a red symbol. The good agreement, within the uncertainties,
is evident and no systematic trend is detected. In fact, slightly faster rates
should have been observed, compared to those predicted without considering
their companions, if an additional term to the total apsidal motion rates is
noticeable (see, for instance, Martynov, 1973).

For some DLEBs in the sample, the properties of the third body are quite well
measured and have determinations of key parameters such as the orbital period
and the mass function. This is the case of CW Cep (Wolf et al., 2006), V539
Ara (Wolf & Zejda, 2005) and $\zeta$ Phe (Zasche & Wolf, 2007). For these
systems we calculated the additional term in the apsidal motion rate using the
approximation of Martynov (1973) for coplanar orbits. The resulting
corrections were found to be very small compared with the total apsidal motion
rate. In the case of V539 Ara, that shows the largest contribution, it is on
the order of $1.5\times 10^{-6}$ deg/cycle, totally undetectable with errors
in the observed rate of $0.6\times 10^{-3}$ deg/cycle. Their impact thus
should be negligible, as observed.

###  5.3 Convective core overshooting

As a result of the grid search for the best models fitting the observed
properties of the component stars in our sample, we also derive their values
for $f_{\rm ov}$. Unfortunately, our sample is dominated by quite unevolved
systems that do not provide strong constraints to the core overshooting
parameter. We tried, nevertheless, to check the consistency of the results
with the semi-empirical calibration obtained by Claret & Torres (2019) for
much more evolved binary systems.

Figure 10: The solid line is an approximate representation of the derived
$f_{\rm ov}$ as a function of stellar mass obtained by Claret & Torres (2019)
while the symbols represent the values of $f_{\rm ov}$ resulting from the
comparison between the theoretical models and the observed absolute dimensions
performed in the present study. The error bar displayed represents the typical
uncertainties for unevolved or mildly evolved stars. For the more massive
stars, beyond the calibration of $f_{ov}$ by Claret & Torres (2019), we find
an average of the best values from the same methodology to be around 0.025.
The only slightly evolved massive system is V453 Cyg, which is discussed in
section 5.3, and shows a better fit of the observed parameters with
$f_{ov}=0.03$.

We show in Fig. 10 the values of $f_{\rm ov}$ resulting from the comparison
between the theoretical models and the observed physical parameters, together
with the relationship between mass and core overshooting obtained by Claret &
Torres (2019). The observed correlation is quite suggestive, especially when
considering that most of our systems have components well within the main
sequence. We assume a conservative formal error for $f_{\rm ov}$ of 0.005. The
most massive stars in our sample cannot be included in such a comparison
because they are above the upper limit considered by Claret & Torres (2019),
$\approx$5 M⊙.

In order to extend the analysis of the variation of $f_{\rm ov}$ to higher
masses, we need DLEBs with accurate dimensions like those in our sample but,
unfortunately, most are quite unevolved systems. At such early evolutionary
stages it becomes difficult to distinguish the impact of different amounts
core overshooting both on the physical parameters and also on log k2. The only
potential case may be the high-mass system V453 Cyg, whose moderately evolved
stage could allow exploring the effect of convective overshooting. Our
methodology to model the DLEB observables, described in Sect. 4 and applied to
V453 Cyg, favored $f_{\rm ov}=0.03$, clearly above the highest value given by
Claret & Torres (2019) for less massive stars. Comparing the corresponding
apsidal motion rate with the observed value, $f_{\rm ov}=0.03$ is also favored
with respect to adopting 0.02. This is illustrated in Figure 11, which shows
log k2 for models with different amounts of core overshooting ($f_{\rm
ov}=0.02$ – dashed line; $f_{\rm ov}=0.03$ – solid line) for the most massive
and evolved component of V453 Cyg. We can see that the effect of core
overshooting in $\log k_{2}$ is only noticeable beyond the middle of the main
sequence. At less evolved stages (including the PMS phase), the differences in
the model predictions when changing the overshooting parameter are well below
the observational errors. The differences though become easily observable as
we consider evolutionary stages approaching the giant branch phase.

The results for V453 Cyg seem to indicate that apsidal motion can help to put
constraints on the convective cover overshooting parameter using suitable
DLEBs with eccentric orbits. This was already explored by Guinan et al. (2000)
and revisited by Claret (2003) using the evolved high-mass system V380 Cyg.
The question of a dependence of core overshooting with mass using DLEB data
has been discussed previously without reaching conclusive results (Ribas et
al., 2000a; Claret, 2007; Tkachenko et al., 2020; Rosu et al., 2020). The
tentative increase of $f_{\rm ov}$ with stellar mass that we find in the
present work would imply a more pronounced dependence than the relationship
found by Claret & Torres (2019) for stars more massive than $\approx 5$ M⊙. In
the same sense, recent theoretical studies carried out by Martinet et al.
(2021) also find a need for larger convective cores at higher masses (see also
Scott et al., 2021). However, the indications found are still not sufficiently
well established. Further detailed analyses of suitable massive DLEBs should
provide the necessary observational evidence to establish a possible
overshooting-mass dependence in this interval of masses.

Figure 11: $\log g-\log k_{2}$ diagram for the primary of V453 Cyg and
adopting $Z=0.0134$. The solid line indicates a model with $f_{\rm ov}=0.03$
while the dashed one represents a model computed with $f_{\rm ov}=0.02$. The
two arrows indicate the observed value of $\log g$. The resulting difference
in $log\overline{\rm k}_{\rm 2,theo}$ is $\approx-0.01$.

##  6 Conclusions

We have studied 34 DLEBs with eccentric orbits employing TESS data that we use
to determine eclipse timings. This allowed us to determine the apsidal motion
rates for 27 DLEBs with sufficient precision to compare their internal
structure constants with those predicted by state-of-the-art theoretical
models. The precision of the TESS eclipse timings, as well as the expanded
span of time covered when compared with archival values, have allowed us to
significantly improve previous apsidal motion rate determinations and detect
their presence in some cases for the first time.

The sample was selected in such a way that the dominant term in the apsidal
motion is the classical, or Newtonian, contribution. In paper I we used a
complementary sample to demonstrate that the relativistic term calculated
using GR shows excellent agreement with the observations. Therefore, in the
present paper, where we study the internal structure of stars, we subtracted
out the relativistic term to the measured apsidal motion rate to ultimately
estimate empirical values for the internal concentration parameters of the
component stars, namely $\log k_{2}$. A comparison of our theoretical and
observational results reveals excellent agreement.

The case of EM Car deserves further attention. In spite of a reasonable
agreement between theory and observation, its high mass indicates that some
additional effects could be present, including the relative proximity of the
components to their Roche limit. For more detailed conclusions, additional
observations and improved physical parameters are needed. EM Car could also
help to eventually understand the small deviations, in the same sense,
observed in our comparison for massive stars with large relative radii.

A rather unexpected result is the good agreement found for systems with an
identified third body. The impact of the companion in the dynamical behavior
of the close binary should have been observed as systematic differences in the
observed vs. computed diagram. This is not the case, and probably indicates
that the companions are less massive than the components of the close system,
and/or orbit them at a large distance. For those cases with good enough
observational constraints on the properties of the third bodies, the
negligible contribution of the gravitational effects on the apsidal motion,
assuming coplanar orbits, was confirmed.

The systems in our sample are not especially suitable, owing to their
relatively unevolved nature, to provide constraints on the best value of the
convective core overshooting parameter, $f_{\rm ov}$. Nevertheless, the values
resulting from our analysis are clearly compatible with the relation found by
Claret & Torres (2019). The particular case of V453 Cyg, a high-mass system
with moderately evolved components, albeit still in the main sequence,
indicates a larger value of $f_{\rm ov}$ than that for lower-mass stars. The
analysis of a larger sample of massive DLEBs will be needed to further
constrain the increasing value of the overshooting parameter with mass, as
suggested in the past in several works.

Table 1: $T_{2}-T_{1}$ computed from TESS lightcurves.

System & $T_{0}$ | $N$ |  $T_{2}-T_{1}$ [d] | $dN$
---|---|---|---
EM Car | 0 | $1.723492\pm 0.000045$ | 0
2458572.9908 | 1 | $1.723193\pm 0.000101$ | 0
| 2 | $1.723907\pm 0.000082$ | 0
| 4 | $1.724324\pm 0.000080$ | 0
| 5 | $1.723740\pm 0.000054$ | 0
| 6 | $1.724186\pm 0.000064$ | 0
| 8 | $1.723906\pm 0.000098$ | 0
| 9 | $1.723282\pm 0.000064$ | 0
| 10 | $1.722915\pm 0.000061$ | 0
| 12 | $1.724228\pm 0.000082$ | 0
| 13 | $1.722725\pm 0.000064$ | 0
| 14 | $1.724169\pm 0.000084$ | 0
Y Cyg | 0 | $1.743685\pm 0.000058$ | 0
2458713.0054 | 1 | $1.743724\pm 0.000035$ | 0
| 2 | $1.743574\pm 0.000034$ | -1
| 5 | $1.743077\pm 0.000032$ | 0
| 6 | $1.742991\pm 0.000025$ | 0
| 7 | $1.742894\pm 0.000027$ | -1
111The BJD value below each system name defines the origin epoch of the
orbital cycle count ($N$). This table is presented in its entirety in the
online version of the article.

Table 2: Observed and theoretical apsidal motion rates

System | Method** | $e$ | $\dot{\omega}_{\rm theo}$ | $\dot{\omega}_{\rm obs}$ | $\log\overline{k}_{\rm 2,theo}$ | $\log\overline{k}_{\rm 2,obs}$ |
---|---|---|---|---|---|---|---
EM Car | B | 0.0120$\pm$0.0005 | 0.090$\pm$0.007 | 0.080$\pm$0.002 |  $-$2.258$\pm$0.026 |  $-$2.311$\pm$0.030 |
Y Cyg | L | 0.14508$\pm$0.00029 | 0.06244$\pm$0.00031 | 0.06178$\pm$0.00003 |  $-$1.943$\pm$0.012 |  $-$1.948$\pm$0.020 |
V478 Cyg | L | 0.0158$\pm$0.0007 | 0.1062$\pm$0.0086 | 0.1047$\pm$0.0010 |  $-$2.239$\pm$0.036 |  $-$2.245$\pm$0.017 |
V578 Mon | L | 0.07755$\pm$0.00026 | 0.0690$\pm$0.0053 | 0.07089$\pm$0.00021 |  $-$2.002$\pm$0.037 |  $-$1.989$\pm$0.013 |
V453 Cyg | B | 0.0250$\pm$0.0014 | 0.0453$\pm$0.0046 | 0.0431$\pm$0.0015 |  $-$2.380$\pm$0.050 |  $-$2.402$\pm$0.019 |
CW Cep* | B | 0.0305$\pm$0.0009 | 0.0591$\pm$0.0017 | 0.0583$\pm$0.0004 |  $-$2.065$\pm$0.011 |  $-$2.071$\pm$0.010 |
QX Car | L | 0.278$\pm$0.003 | 0.0132$\pm$0.0011 | 0.01222$\pm$0.00022 |  $-$2.112$\pm$0.037 |  $-$2.149$\pm$0.033 |
V539 Ara* | B | 0.0530$\pm$0.0010 | 0.0195$\pm$0.0014 | 0.0196$\pm$0.0006 |  $-$2.309$\pm$0.012 |  $-$2.306$\pm$0.034 |
DI Her | A | 0.489$\pm$0.003 | 0.00046$\pm$0.00009 | 0.00044$\pm$0.00002 |  $-$2.135$\pm$0.062 |  $-$2.157$\pm$0.032 |
EP Cru | A | 0.1874$\pm$0.0005 | 0.00229$\pm$0.00011 | 0.00237$\pm$0.00015 |  $-$2.350$\pm$0.013 |  $-$2.331$\pm$0.043 |
V760 Sco | B | 0.0265$\pm$0.0010 | 0.0450$\pm$0.0051 | 0.0434$\pm$0.0005 |  $-$2.180$\pm$0.046 |  $-$2.196$\pm$0.032 |
GG Lup | B | 0.155$\pm$0.005 | 0.0168$\pm$0.0014 | 0.0173$\pm$0.0003 |  $-$2.197$\pm$0.040 |  $-$2.208$\pm$0.023 |
$\zeta$ Phe* | B | 0.0116$\pm$0.0024 | 0.0326$\pm$0.0019 | 0.0328$\pm$0.0006 |  $-$2.287$\pm$0.026 |  $-$2.284$\pm$0.016 |
IQ Per | B | 0.0662$\pm$0.0005 | 0.01526$\pm$0.00088 | 0.0150$\pm$0.0005 |  $-$2.303$\pm$0.025 |  $-$2.311$\pm$0.023 |
PV Cas | B | 0.0325$\pm$0.0005 | 0.0224$\pm$0.0012 | 0.0212$\pm$0.0002 |  $-$2.379$\pm$0.022 |  $-$2.404$\pm$0.015 |
V364 Lac | A | 0.2873$\pm$0.0014 | 0.00180$\pm$0.00005 | 0.00181$\pm$0.00006 |  $-$2.611$\pm$0.015 |  $-$2.609$\pm$0.021 |
SW CMa | A | 0.3180$\pm$0.0005 | 0.00067$\pm$0.00002 | 0.00069$\pm$0.00005 |  $-$2.581$\pm$0.015 |  $-$2.565$\pm$0.067 |
PT Vel | B | 0.112$\pm$0.003 | 0.01231$\pm$0.00049 | 0.0125$\pm$0.0006 |  $-$2.452$\pm$0.012 |  $-$2.445$\pm$0.031 |
V1647 Sgr* | B | 0.4130$\pm$0.0005 | 0.00543$\pm$0.00036 | 0.00554$\pm$0.00005 |  $-$2.384$\pm$0.037 |  $-$2.373$\pm$0.019 |
AI Hya | B | 0.234$\pm$0.002 | 0.00177$\pm$0.00015 | 0.00191$\pm$0.00005 |  $-$2.682$\pm$0.015 |  $-$2.642$\pm$0.048 |
VV Pyx* | A | 0.0956$\pm$0.0009 | 0.00138$\pm$0.00004 | 0.00132$\pm$0.00005 |  $-$2.488$\pm$0.010 |  $-$2.517$\pm$0.031 |
EK Cep | A | 0.109$\pm$0.003 | 0.00089$\pm$0.00002 | 0.00088$\pm$0.00004 |  $-$2.167$\pm$0.014 |  $-$2.177$\pm$0.041 |
VV Crv* | B | 0.0852$\pm$0.0010 | 0.01177$\pm$0.00068 | 0.0109$\pm$0.0012 |  $-$2.686$\pm$0.031 |  $-$2.721$\pm$0.031 |
IM Per* | B | 0.0491$\pm$0.0010 | 0.01520$\pm$0.00050 | 0.0146$\pm$0.0004 |  $-$2.615$\pm$0.012 |  $-$2.633$\pm$0.017 |
V1143 Cyg | A | 0.5386$\pm$0.0012 | 0.00082$\pm$0.00003 | 0.00080$\pm$0.00004 |  $-$2.237$\pm$0.035 |  $-$2.259$\pm$0.045 |
IT Cas | A | 0.089$\pm$0.002 | 0.00112$\pm$0.00005 | 0.00114$\pm$0.00010 |  $-$2.449$\pm$0.014 |  $-$2.436$\pm$0.067 |
V530 Ori | A | 0.0862$\pm$0.0010 | 0.00081$\pm$0.00006 | 0.00086$\pm$0.00005 |  $-$0.796$\pm$0.042 |  $-$0.763$\pm$0.043 |

The values of $\dot{\omega}$ are expressed in deg/cycle.

* Binary systems with a suggested third body.

** Adopted methodology to compute $\dot{\omega}_{\rm obs}$: A, from linear
variation of $T_{2}-T_{1}$; B, from variations of the angle of periastron; L,
adopted from the literature.

###### Acknowledgements.

This paper is dedicated to the memory of J. Andersen, a very good friend and
colleague, that initiated a systematic approach to the accurate measurement of
masses and radii in DLEBs, and motivated us to continue his path. We thank an
anonymous referee for the insightful comments and suggestions. The Spanish MEC
(ESP2017-87676-C5-2-R, PID2019-107061GB-C64, and PID2019-109522GB-C52) is
gratefully acknowledged for its support during the development of this work.
A.C. also acknowledges financial support from the State Agency for Research of
the Spanish MCIU through the “Center of Excellence Severo Ochoa” award for the
Instituto de Astrofísica de Andalucía (SEV-2017-0709). D.B., I.R., and J.C.M.
acknowledge support from the Spanish Ministry of Science and Innovation and
the European Regional Development Fund through grant PGC2018-098153-B-C33,
from the Generalitat de Catalunya/CERCA programme, and from the Agència de
Gestió d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya.
This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. This research has made use
of the SIMBAD database, operated at the CDS, Strasbourg, France, and of NASA’s
Astrophysics Data System Abstract Service.

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##  Appendix A Apsidal motion studies

  * •

EM Car

Andersen & Clausen (1989) made a complete study of EM Car obtaining an
eccentricity $e=0.0120\pm 0.0005$ and an apsidal motion rate
$\dot{\omega}=0.081\pm 0.010$ deg/cycle. This value has a relatively large
error bar because of the unfavorable configuration with $\omega\approx 0$ deg,
which affects the use of a linear fit to the differential timings. The light
curve solution by Andersen & Clausen (1989) yields an argument of periastron
of $\omega=350\pm 5$ deg, and the TESS measurements give a phase for the
secondary eclipse of 0.50484$\pm$0.00010 which corresponds, adopting also
$e=0.0120\pm 0.0005$, to a value of $\omega=308\pm 2$ deg. We used the TESS
light curve to exclude the other possible value with positive $esin\omega$. By
using the two $\omega$ determinations at different epochs we obtain an apsidal
motion rate $\dot{\omega}=0.080\pm 0.002$ deg/cycle, considering all
uncertainties. This value is in excellent agreement, but significantly more
precise, than that given by Andersen & Clausen (1989) thanks to the large
time-span increase, which is now covering nearly a complete apsidal motion
period. Furthermore, Mayer (2004) measured $T_{2}-T_{1}=1.683\pm 0.003$ days
at HJD 2452280, between the two light curve solutions. This is in excellent
agreement with the predicted value from our apsidal motion solution
$T_{2}-T_{1}=1.6828\pm 0.0005$ days.

  * •

Y Cyg

The physical parameters of Y Cyg were given in the comprehensive study of
Harmanec et al. (2014), who analyzed light curve changes with time, using a
variable $\omega$, and obtained an orbital eccentricity $e=0.1451\pm 0.0003$
together with a precise apsidal motion rate of $\dot{\omega}=0.06178\pm
0.00004$ deg/cycle. The authors also studied the changing position of the
times of eclipse, obtaining a less precise $e=0.1448\pm 0.0012$, using old
photographic timings. An earlier study with the same methodology but
restricted to the best photoelectric measurements was carried out by Holmgren
et al. (1995), determining $e=0.1458\pm 0.0007$. Our TESS data, given in Table
1, indicate a clear apsidal motion variation in the observed $T_{2}-T_{1}$,
with a slope of $-0.000138\pm 0.000009$ d/cycle and a value of
$T_{2}-T_{1}=1.74357\pm 0.00003$ days at BJD 2458718.9981. The apsidal motion
rate cannot be accurately determined using these data alone due to the very
narrow time interval, but a value of $\dot{\omega}=0.062\pm 0.004$, with
$\omega=27.8\pm 0.3$ deg, is obtained assuming 0.1451 for the eccentricity.
Nevertheless, it confirms the solution by (Harmanec et al. 2014), based on the
global analysis of all photometric data with variable longitude of the
periastron, that we adopt for our discussion.

  * •

V478 Cyg

The absolute dimensions of V478 Cyg were recently measured by Pavlovski et al.
(2018) using their own radial velocity curves and a new analysis of the light
curve of Sezer et al. (1983). Unfortunately the light curve could not
establish a precise value of the orbital eccentricity and $e=0.021\pm 0.005$,
as derived from the radial velocity curve, was fixed for the analysis. The
analysis of the TESS light curve using the Wilson-Devinney code (Wilson &
Devinney 1971) resulted in a best fit with $e=0.016$. This is in agreement
with the apsidal motion variation produced by the eclipse timings presented by
Wolf et al. (2006) yielding $e=0.0158\pm 0.0007$ and $\dot{\omega}=0.1047\pm
0.0010$ deg/cycle. TESS measurements give an average $T_{2}-T_{1}$ value of
$1.4120\pm 0.0005$ days, in excellent agreement with the expected value,
$T_{2}-T_{1}=1.4122\pm 0.0005$ days, using the solution by Wolf et al. (2006).
The comparison of our TESS data with the $T_{2}-T_{1}$ extracted from Table 1
of Wolf et al. (2006), with timings within less than 10 orbital cycles,
confirms the apsidal motion rate determination by Wolf et al. (2006). However,
using the larger and less precise values of the eccentricity $e=0.019\pm
0.002$ by Mossakovskaya & Khaliullin (1996) or $e=0.021\pm 0.005$ by Pavlovski
et al. (2018) produces much poorer fits to the data and predictions that are
in contradiction with the TESS light curve. We therefore adopted the solution
given by Wolf et al. (2006) for Table 2.

  * •

V578 Mon

The most recent and complete determination of the physical parameters of V578
Mon was carried out by Garcia et al. (2014) adopting an earlier apsidal motion
determination (Garcia et al. 2011) of $\dot{\omega}=0.07089\pm 0.00021$
deg/cycle with $e=0.07755\pm 0.00026$. Their global analysis of the light
curve variations with $\dot{\omega}$ as a free parameter over one complete
apsidal motion period allowed such precise results. The TESS measurements of
the $T_{2}-T_{1}$ differences, obtained in two epochs, yield a less precise
determination $\dot{\omega}=0.0697\pm 0.0010$ deg/cycle, assuming the
eccentricity of Garcia et al. (2011). The individual light curve solutions in
table 3 by Garcia et al. (2011) give an apsidal motion rate
$\dot{\omega}=0.0718\pm 0.0012$ deg/cycle, and including the arguments of
periastron derived from the TESS $T_{2}-T_{1}$ values, yield a rate of
$\dot{\omega}=0.0701\pm 0.0006$ deg/cycle. All values agree within their
uncertainties and we adopted for our discussion $\dot{\omega}=0.07089\pm
0.00021$ deg/cycle, as derived from the light curve analysis with variable
omega.

  * •

V453 Cyg

A detailed analysis of the TESS light curve of V453 Cyg has been recently
carried out by Southworth et al. (2020) in their quest to use $\beta$ Cep
pulsations as a tracer of the physical processes that govern the evolution of
massive stars. The authors determined the orbital eccentricity to be
$e=0.0250\pm 0.0014$ with the argument of periastron at $\omega=152.5\pm 5.1$
deg, but did not analyze the apsidal motion variations. This was done in an
earlier paper (Southworth et al. 2004) with less precise data, providing an
apsidal motion rate $\dot{\omega}=0.0577\pm 0.0016$ deg/cycle, with an orbital
eccentricity $e=0.022\pm 0.003$. Nevertheless, this solution predicts an
argument of periastron at the time of the TESS observations that is not
compatible with the TESS light curve analyzed by Southworth et al. (2020). We
have used the TESS measurements in Table 1, giving an average eclipse timing
difference $T_{2}-T_{1}=1.8896\pm 0.00015$ days, and the timings by Wachmann
(1973) and Cohen (1971), using only photoelectric measurements separated by
less than 10 orbital cycles. We computed the corresponding arguments of
periastron, assuming $e=0.0250\pm 0.0014$, and a linear fit yielded
$\dot{\omega}=0.0431\pm 0.0015$ deg/cycle, considering the uncertainty of the
eccentricity.

  * •

CW Cep

The light curve of this eclipsing binary system has been recently studied by
Lee et al. (2021) using TESS measurements. An eccentricity $e=0.0305\pm
0.0009$, with an argument of periastron $\omega=212.7\pm 3.2$ deg were
obtained from the analysis. The authors do not discuss the apsidal motion
rate, well known from previous studies, but focus on the analysis of the
detected $\beta$ Cep pulsations and confirm the presence of a significant
third light. The physical parameters of the components are well determined
using the radial velocity amplitudes measured by Johnston et al. (2019). The
apsidal motion of CW Cep was studied most recently by Wolf et al. (2006), who
clearly observed the effect of the third body through the light-time effect,
with a period of $P_{3}=38.5\pm 1.5$ years. After correction for the third-
body effect in the eclipse timings, the authors could determine an orbital
eccentricity $e=0.0297\pm 0.0005$ and an apsidal motion rate
$\dot{\omega}=0.0582\pm 0.0005$ deg/cycle. With the TESS measurements of
$T_{2}-T_{1}$ in different sectors, apsidal motion is already evident and we
determine a slope of $(2.8\pm 0.2)\times 10^{-5}$ days/cycle over a time span
of 70 orbital cycles. We have added the $T_{2}-T_{1}$ values retrieved from
the literature to compute the corresponding argument of periastron with the
orbital eccentricity given by Lee et al. (2021). We have restricted eclipse
timings to those obtained by means of photoelectric measurements and
considered pairs within less than 10 orbital cycles. A weighted linear fit
yields $\dot{\omega}=0.0583\pm 0.0004$ deg/cycle, in excellent agreement with
previous determinations.

  * •

QX Car

Precise absolute dimensions of QX Car were obtained by Andersen et al. (1983).
The authors also studied apsidal motion, fixing the eccentricity to the result
of the light curve analysis, $e=0.278\pm 0.003$, and obtained
$\dot{\omega}=0.01248\pm 0.00015$ deg/cycle. A revised analysis by Giménez et
al. (1986), with the same adopted eccentricity, yielded
$\dot{\omega}=0.01222\pm 0.00022$ deg/cycle considering more realistic
uncertainties. TESS measurements give an average value of
$T_{2}-T_{1}=1.4853\pm 0.0002$ days. Adopting the eccentricity by Andersen et
al. (1983), this time difference yields an argument of periastron of
$\omega=163.1\pm 2.5$ deg. The apsidal solution by Giménez et al. (1986)
predicts a more precise value of $\omega=164.6\pm 0.8$ deg at the time of the
TESS observations, in good agreement with the value obtained from the
$T_{2}-T_{1}$ measurements, and we have adopted it for our discussion.

  * •

V539 Ara

Accurate physical parameters of this system were obtained by Clausen (1996),
who also detected apsidal motion with $\dot{\omega}=0.021\pm 0.002$ deg/cycle
but showing significant variations between different epochs that suggested the
presence of a perturbing third body. No third light was reported. The light
curve provided a value $e=0.053\pm 0.001$. The analysis by Wolf et al. (2005)
confirmed the presence of a third body by means of the light-time effect, with
a period $P_{3}=42.3\pm 0.8$ years. An apsidal motion rate
$\dot{\omega}=0.0193\pm 0.0010$ deg/cycle was obtained together with an
orbital eccentricity $e=0.0548\pm 0.0015$. TESS data provide a value
$T_{2}-T_{1}=1.4873\pm 0.0005$ days and, adopting the eccentricity given by
Clausen (1996), the argument of periastron comes out to be $\omega=204.9\pm
2.2$ deg. When relating it with the result of the light curve analysis in
Clausen (1996), $\omega$ = 125.1$\pm$1.0 deg, an apsidal motion rate
$\dot{\omega}=0.0196\pm 0.0006$ deg/cycle can be derived, in agreement with
all previous results.

  * •

DI Her

This system has historically played a key role as a paradigmatic case for
relativistic apsidal motion tests. Early apsidal motion rate measurements
showed a conspicuous and disturbing disagreement with theoretical predictions.
This was resolved by the work of Albrecht et al. (2009), who found a large
misalignment of the spin axes of the component stars from measuring the
Rossiter-McLaughlin effect. The most recent apsidal motion determination for
DI Her, by Claret et al. (2010), yielded a value $\dot{\omega}=0.00042\pm
0.00012$ deg/cycle, in good agreement with the predicted value when the
observed rotational axes misalignment is considered. Precise values of
$T_{2}-T_{1}$ can be retrieved from the TESS measurements, as given in Table
1. Using values from the literature, and restricting to photoelectric
measurements only, the increase of the observed separation in $T_{2}-T_{1}$
gives a slope of $(9.3\pm 0.5)\times 10^{-6}$ days/cycle using a weighted
least-squares minimization. Assuming an orbital eccentricity $e=0.489\pm
0.003$, as given by Torres et al. (2010), the observed apsidal motion rate of
DI Her is found to be $\dot{\omega}=0.000435\pm 0.000023$ deg/cycle, in good
agreement but significantly more precise than previous studies.

  * •

EP Cru

A first light curve of this eclipsing binary was obtained by Clausen et al.
(2007), while Albrecht et al. (2013) determined precise physical parameters
with the aim of studying circularization and synchronization timescales as
well as a possible misalignment of the rotational axes. With improved absolute
dimensions, Albrecht et al. (2013) could predict the presence of apsidal
motion but could not measure it. They also pointed out the rotational
velocities of the component stars, much larger than their synchronized values.
We have used the precise value $T_{2}-T_{1}=6.8140\pm 0.0002$ days given by
Clausen et al. (2007) and compared it with the values obtained from the TESS
measurements. A weighted mean of the time differences in Table 1 yields
$T_{2}-T_{1}=6.8255\pm 0.0005$ some 1000 orbital cycles later. This is an
increase rate of $(1.16\pm 0.05)\times 10^{-5}$ days/cycle. Assuming the
orbital eccentricity given by Albrecht et al. (2013), $e=0.1874\pm 0.0005$, we
obtain $\dot{\omega}=0.00237\pm 0.00015$ deg/cycle, which becomes the first
detection of apsidal motion in EP Cru.

  * •

V760 Sco

This system was studied by Andersen et al. (1985), who obtained precise
physical parameters and measured an apsidal motion $\dot{\omega}=0.0430\pm
0.0017$ deg/cycle with an orbital eccentricity $e=0.0265\pm 0.0010$. Wolf
(2000) re-discussed the apsidal motion parameters with new eclipse timings and
obtained $\dot{\omega}=0.0443\pm 0.0004$ deg/cycle, with $e=0.0270\pm 0.0005$.
Nevertheless, the new observations were not sufficiently precise for an
improved determination of the eccentricity. From the TESS measurements, we
obtain a value $T_{2}-T_{1}=0.8925\pm 0.0005$ days, which is in good agreement
with the predicted value by the solution of Andersen et al. (1985) of
$T_{2}-T_{1}=0.8919\pm 0.0025$ days, but not with the value predicted from
Wolf (2000) of $T_{2}-T_{1}=0.8946\pm 0.0004$ days. Considering all the
possible $T_{2}-T_{1}$ values retrieved from Andersen et al. (1985), computed
with individual timings within less than 10 orbital cycles, and adopting their
value of the eccentricity, we obtain the corresponding arguments of periastron
at different epochs. We adopted for the argument of periastron corresponding
to the TESS value of $T_{2}-T1$, a negative value of $e\sin\omega$. This
conclusion was achieved by fitting the light curve fixing the binary
parameters to those reported in Andersen et al. (1985). A weighted linear
least-squares fit yields an apsidal motion rate of $\dot{\omega}=0.0434\pm
0.0005$ deg/cycle, which includes the uncertainty in the orbital eccentricity.

  * •

MU Cas

A complete photometric and spectroscopic study of MU Cas was carried out by
Lacy et al. (2004). The system shows a well-defined orbital eccentricity of
$e=0.1930\pm 0.0003$, and the phase of the secondary eclipse is observed at
$0.61914\pm 0.00015$. The argument of periastron obtained from the light curve
analysis is $\omega=13.4\pm 0.4$ deg, but the small time span of eclipses
available did not allow the authors to claim the detection of apsidal motion.
Moreover, the argument of periastron close to 0 deg, which makes it difficult
to observe changes in timing differences, and, also, apsidal motion is
expected to be very slow given the observed relative radii. The measurements
of $T_{2}-T_{1}$ from TESS give a phase of the secondary eclipse of
$0.61881\pm 0.00003$, yielding an poorly-significant determination of the
apsidal motion rate $\dot{\omega}=0.0010\pm 0.0005$ deg/cycle. To obtain this
value we adopted the orbital eccentricity from Lacy et al. (2004). The
alternative method, that is, using the measured times of eclipse in Table 1 of
Lacy et al. (2005) to calculate $T_{2}-T_{1}$ differences, provides
$\dot{\omega}=0.0013\pm 0.0003$ deg/cycle. The alternative method of comparing
the argument of periastron in table 4 of Lacy et al. (2004) and that derived
from TESS phase of the secondary eclipse ($\omega=14.00\pm 0.05$ deg) yields
$\dot{\omega}=0.0009\pm 0.0006$ deg/cycle. The low statistical significance of
the measurement, although consistent using different techniques, does not
allow the use of MU Cas to test internal stellar structure. We derive a log k2
value with a large uncertainty of 0.14 when using the observed apsidal motion
rate and after subtracting the relativistic contribution.

  * •

GG Lup

This system was studied by Andersen et al. (1993), who obtained precise
absolute dimensions as well as a good determination of the apsidal motion rate
of $\dot{\omega}=0.0181\pm 0.0006$ deg/cycles and an orbital eccentricity
$e=0.150\pm 0.005$. Wolf et al. (2005) revisited the apsidal motion
determination and obtained results in good agreement with those of Andersen et
al. (1993), with an orbital eccentricity $e=0.1545\pm 0.0010$. The most recent
study by Budding et al. (2015) confirmed the latter value of the eccentricity
with an apsidal motion rate $\dot{\omega}=0.0172\pm 0.0003$ deg/cycle. TESS
observations show the expected slow increase of the eclipse timing differences
over the short time span covered, and yield an average value
$T_{2}-T_{1}=0.7561\pm 0.0002$ days. This is not compatible with predictions
using the eccentricity by Andersen et al. (1993). We have computed the
argument of periastron corresponding to the $T_{2}-T_{1}$ values available
from individual timings separated by less than 10 orbital cycles and adopting
an orbital eccentricity $e=0.155\pm 0.005$, as indicated by Wolf et al.
(2005). A linear least-squares fit of the variation of $\omega$ with time
yields $\dot{\omega}=0.0173\pm 0.0003$ deg/cycle, in excellent agreement with
the study by Budding et al. (2015).

  * •

$\zeta$ Phe

This bright eclipsing binary system has been the subject of a recent study
using TESS data by Southworth (2020a). The early photometric analyses by
Clausen & Grønbech (1976) already showed a significant third light
contribution and the spectroscopic analysis by Andersen et al. (1983) allowed
accurate absolute parameters to be determined. Apsidal motion was reported by
Giménez et al. (1986) using the individual eclipse timings available and
obtained $\dot{\omega}=0.0373\pm 0.0055$ deg/cycle with an eccentricity
$e=0.0113$ given by the photometric light curve analysis but not taking into
account light-time effects. Zasche & Wolf (2007) combined astrometric
measurements with light-time modeling to study the orbit of the third body and
derived an apsidal motion rate of $\dot{\omega}=0.028\pm 0.001$ deg/cycle with
an eccentricity of $e=0.0107\pm 0.0020$ for the close orbit as well as a
period of $P_{3}=221$ years and an eccentricity of $e=0.366$ for the wide
orbit of the outer third companion. The light curve analysis of Southworth
(2020a) provides precise absolute dimensions for $\zeta$ Phe, and confirms the
presence of third light. The orbital eccentricity was found to be $e=0.0116\pm
0.0024$, in excellent agreement with previous studies. The argument of
periastron is $\omega=307\pm 12$ deg. The comparison of this value with the
light curve of Clausen & Grønbech (1976), almost 10 000 cycles before,
indicates an apsidal motion rate of $\dot{\omega}=0.031\pm 0.003$ deg/cycle.
Using the argument of periastron computed from the $T_{2}-T_{1}$ values
retrieved from Clausen & Grønbech (1976), and those from all the TESS
observations, using $e=0.0116\pm 0.0024$, we derive $\dot{\omega}=0.0328\pm
0.0006$ deg/cycle, which is in agreement with but more precise than all the
other results.

  * •

IQ Per

This eccentric binary was studied by Lacy & Frueh (1985), who provided
absolute dimensions after the preliminary values by Hall et al. (1970) and
derived an apsidal motion rate of $\dot{\omega}=0.012\pm 0.003$ deg/cycle for
an orbital eccentricity $e=0.075\pm 0.006$, derived from the radial velocity
curve. The apsidal motion rate was revised by Değirmenci (1997) to
$\dot{\omega}=0.0141\pm 0.0008$ deg/cycle, using an eccentricity $e=0.076\pm
0.004$, and by Wolf et al. (2006), obtaining $\dot{\omega}=0.0138\pm 0.0003$
deg/cycle, using $e=0.0763\pm 0.0008$. TESS measurements yield a separation
between primary and secondary eclipses of $T_{2}-T_{1}=0.79827\pm 0.00011$
days that does not agree with the previously mentioned apsidal motion
solutions. Using the most recent determination, by Wolf et al. (2006), the
argument of periastron would be $\omega=178\pm 2$ deg at the TESS epoch and
the predicted time separation is $T_{2}-T_{1}=0.78755\pm 0.00007$ days, a
significant difference of $0.01072\pm 0.00012$ days. To solve this, we
performed a fit of the high-precision TESS light curve and obtained
$e=0.0662\pm 0.0005$ and $\omega=182\pm 3$ deg, with the physical parameters
being in agreement with those found by Lacy & Frueh (1985). Precise
measurements of the time of the secondary eclipse are difficult to obtain and
only a few are available in the literature. Adopting the eccentricity derived
from the TESS light curve analysis, we calculated the argument of periastron
corresponding to the observed values of $T_{2}-T_{1}$ in the literature,
restricting to photoelectric data and with primaries and secondaries within 10
orbital cycles. A linear fit yielded an apsidal motion rate
$\dot{\omega}=0.0150\pm 0.0005$ deg/cycle, which we adopted for Table 2.

  * •

PV Cas

Precise absolute dimensions of PV Cas are based on the studies by Popper
(1987) and Barembaum & Etzel (1995). An apsidal motion $\dot{\omega}=0.0189\pm
0.0015$ deg/cycle was measured by Giménez & Margrave (1982) assuming an
eccentricity value $e=0.0322\pm 0.0005$. The most recent subsequent study by
Švaříček et al. (2008) gives an apsidal motion rate $\dot{\omega}=0.01896\pm
0.00014$ deg/cycle with an orbital eccentricity $e=0.03248\pm 0.00014$. The
TESS timing data alone indicate the presence of apsidal motion by comparing
the measurements of $T_{2}-T_{1}$ in the two available sectors, separated by
110 orbital cycles. Adopting an eccentricity $e=0.032$, the observed variation
yields $\dot{\omega}=0.019\pm 0.005$ deg/cycle but with poor precision. The
more precise apsidal motion solution by Švaříček et al. (2008) predicts an
argument of periastron of $\omega=21.7\pm 0.6$ deg at the time of the TESS
observations, equivalent to an eclipse separation $T_{2}-T_{1}=0.9090\pm
0.0004$ days, which differs from the TESS measurements by $0.0036\pm 0.0005$
days. This disagreement prompted us to reanalyze the apsidal motion
determination on the basis of the observed $T_{2}-T_{1}$ values using all
photoelectric data available in the literature, with individual timings
separated by no more than 10 orbital cycles. Adopting an orbital eccentricity
of $e=0.0325\pm 0.0005$ as given by Švaříček et al. (2008) but with a larger,
more realistic, uncertainty, the corresponding argument of periastron is
calculated for each value of $T_{2}-T_{1}$. The value of $\omega$ for TESS,
close to 360 deg, was checked with an analysis of the full light curve. A
linear fit yielded $\dot{\omega}=0.0212\pm 0.0002$ deg/cycle, which we adopt
for Table 2. Concerning the argument of periastron at the time of the
observations by Ibanoglu (1974), we did not include the value of $\omega$ =
180$\pm$8 deg from their light curve analysis due to the uncertain
determination of $e\sin\omega$. Nevertheless, our solution predicts a value in
excellent agreement of $\omega=175.8\pm 1.64$ deg.

  * •

V364 Lac

A detailed spectroscopic and photometric study of this A-type eclipsing binary
was performed by Torres et al. (1999). The authors could detect the presence
of apsidal motion in V364 Lac at a rate of $\dot{\omega}=0.00258\pm 0.00033$
deg/cycle from the combined study of the radial velocity curve and eclipse
timings, with an orbital eccentricity $e=0.2873\pm 0.0014$. More recently,
Bulut (2013) adopted the same value of the eccentricity and obtained
$\dot{\omega}=0.00207\pm 0.00034$ deg/cycle. The TESS data indicate a position
of the secondary eclipse with respect to the primary of
$T_{2}-T_{1}=3.73870\pm 0.00005$ days, which yields an argument of periastron
$\omega=87.43\pm 0.02$ deg when adopting the eccentricity of Torres et al.
(1999). Comparing with the value in table 3 of Torres et al. (1999), an
apsidal motion rate $\dot{\omega}=0.00181\pm 0.00006$ deg/cycle is derived,
where the uncertainty in the eccentricity is considered.

  * •

SW CMa

Lacy (1997) first determined the absolute dimensions of this relatively
evolved DLEB and indicated the possible presence of apsidal motion. Clausen et
al. (2008) obtained a better-covered light curve and measured a rate
$\dot{\omega}=0.00067\pm 0.00021$ deg/cycle. Finally, Torres et al. (2012)
reanalyzed the light curve of Clausen et al. (2008), together with newly
obtained radial velocities, and obtained precise physical parameters. SW CMa
was observed by TESS in two sectors separated by 60 orbital cycles. The
variation in $T_{2}-T_{1}$ over this time interval is too small to allow
determining a apsidal motion rate value. The average timing difference is
$T_{2}-T_{1}=3.1141\pm 0.0002$ days. The apsidal motion parameters of Clausen
et al. (2008) predict an argument of periastron $\omega=164.32\pm 0.30$ deg
while, adopting the eccentricity from the same authors with an uncertainty
$\pm$0.0005, the TESS $T_{2}-T_{1}$ yields $\omega=164.34\pm 0.30$ deg. The
excellent agreement shows that the apsidal motion rate given by Clausen et al.
(2008) is accurate. However, the uncertainty could be reduced thanks to the
wider time span. A comparison of the argument of periastron derived from the
TESS measurements with the light curve analyses by Lacy (1997) and Clausen et
al. (2008) does not allow to improve the determination due to the very slow
apsidal motion rate and the proximity of $\omega$ to 180 deg. We analyzed the
light curve obtained by TESS and obtained a best-fitting $e=0.3180\pm 0.0005$.
A fit using the observed values of $T_{2}-T_{1}$ with TESS, together with
those provided by Clausen et al. (2008), yields an apsidal motion rate
$\dot{\omega}=0.00069\pm 0.00005$ deg/cycle considering all the uncertainties
involved, which improves on the precision of the previous determination.

  * •

PT Vel

Bakıs et al. (2008) studied this system and derived the physical properties of
their components as well as the apsidal motion rate. Together with the orbital
eccentricity of $e=0.127\pm 0.006$, the authors obtained
$\dot{\omega}=0.0099\pm 0.0003$ deg/cycle. Nevertheless, these results are
based on the light curves obtained by the ground-based All-Sky Automated
Survey (ASAS) over a decade, with poor coverage of the secondary eclipse. We
have analyzed the TESS light curve and obtained a much better fit with a lower
eccentricity $e=0.112\pm 0.003$, but with the same relative radii and orbital
inclination, thus with no change in the physical parameters. Adopting the
eccentricity derived from the TESS light curve, the $T_{2}-T_{1}$ measurements
already show the presence of apsidal motion. To increase the time span, we
have compared the average eclipse time difference of the TESS measurements,
$T_{2}-T_{1}=1.00687\pm 0.00012$ days, with the values obtained from the ASAS
survey given by Kim et al. (2018). A linear fit to the corresponding values of
$\omega$ yields an apsidal motion rate of $\dot{\omega}=0.0125\pm 0.0006$
deg/cycle. The significant difference of the eccentricity, determined from the
TESS light curve or the ASAS analysis, that we could not reproduce, remains
unexplained.

  * •

V1647 Sgr

The absolute dimensions of V1647 Sgr were obtained by Andersen et al. (1985),
who could also measure an apsidal motion rate of $\dot{\omega}=0.00546\pm
0.00006$ deg/cycle with an orbital eccentricity $e=0.4130\pm 0.0005$. The
authors also confirmed that the visual companion of V1647 Sgr is physically
bound. Wolf (2000) re-analyzed the apsidal motion and obtained a rate
$\dot{\omega}=0.00609\pm 0.00012$ deg/cycle with an eccentricity $e=0.4142\pm
0.0011$. The TESS data provide an average position of the secondary eclipse
with respect to the primary of $T_{2}-T_{1}=1.0880\pm 0.0003$ days. The
argument of periastron calculated for the TESS measurements and those obtained
from the individual timings in table 4 of Andersen et al. (1985) show a well-
defined variation of $\omega$. A linear fit yields an apsidal motion rate of
$\dot{\omega}=0.00554\pm 0.00005$ deg/cycle, considering the value and
uncertainty in $e$ from Andersen et al. (1985).

  * •

AI Hya

First absolute dimensions for this system were published by Popper & Guinan
(1998) on the basis of their own radial velocity curve and the well-covered
light curve by Joergensen & Gronbech (1978). An orbital eccentricity
$e=0.230\pm 0.002$ was derived from the light curve, and the existence of
pulsations in the secondary component was reported but no apsidal motion. AI
Hya has recently been studied by Lee et al. (2020), who analyzed the light
curve observed by TESS to characterize the $\delta$ Scuti-type pulsations. The
authors also analyzed the displacement of the argument of periastron and
derived a low-significance apsidal motion rate using individual times of
minimum of $\dot{\omega}=0.0017\pm 0.0007$ deg/cycle with an orbital
eccentricity $e=0.241\pm 0.083$. The more precise value derived from their own
light curve analysis, $e=0.234\pm 0.002$, was not used. We have computed the
$T_{2}-T_{1}$ values corresponding to individual eclipse timings within less
than 10 orbital cycles from table 1 of Lee et al. (2020), including the TESS
measurements in our Table 2, and determined the argument of periastron for
each of them, adopting $e=0.234\pm 0.002$. A linear fit yields
$\dot{\omega}=0.00191\pm 0.00005$ deg/cycle.

  * •

VV Pyx

This system has kept the name of VV Pyx in spite of its re-naming as V596 Pup
in the 78th name-list of variable stars (Kazarovets et al. 2006). A detailed
photometric and spectroscopic study of the system by Andersen & Vaz (1984)
yielded precise absolute dimensions and an apsidal motion rate
$\dot{\omega}=0.00142\pm 0.00045$ deg/cycle was obtained with the orbital
eccentricity derived from the light curve analysis, $e=0.0956\pm 0.0009$. A
significant contribution from third light was measured and attributed to the
close visual companion of the eclipsing binary but no effect in the measured
apsidal motion was measured. TESS data alone indicate a small variation of the
$T_{2}-T_{1}$ values with time, corresponding to an apsidal motion rate of
$\dot{\omega}=0.00138\pm 0.00012$ deg/cycle. Using also the two additional
$T_{2}-T_{1}$ values derived from table 5 in Andersen & Vaz (1984), we
obtained a final rate $\dot{\omega}=0.00132\pm 0.00005$ deg/cycle, using
$e=0.0956\pm 0.0009$.

  * •

EK Cep

Popper (1987) was the first to identify the pre-main sequence nature of the
secondary component in EK Cep. Radial velocities were measured by Tomkin
(1983) and the light curve was analyzed by Hill & Ebbighausen (1984). An
orbital eccentricity $e=0.109\pm 0.003$ is derived from the radial velocity
curve and adopted for the photometric studies. Giménez & Margrave (1985)
obtained an apsidal motion rate $\dot{\omega}=0.00107\pm 0.00032$ deg/cycle
using eclipse timings. TESS measurements provide a value
$T_{2}-T_{1}=2.3988\pm 0.0002$ days, and we have used all available values of
$T_{2}-T_{1}$ from the literature with precision better than 0.001 days, which
are challenging because of the shallow secondary eclipse. With these values,
we obtained a slope of $(-3.61\pm 0.08)\times 10^{-6}$ days/cycle, from which
an apsidal motion rate of $\dot{\omega}=0.00088\pm 0.00004$ deg/cycle is
derived, adopting $e=0.109\pm 0.003$.

  * •

VV Crv

Fekel et al. (2013) studied this bright system combining spectroscopic and
photometric observations, and obtained the physical parameters given in Table
1. Conspicuous third light was observed, which complicated the light curve
analysis. Moreover, the long duration and shallow depth of the eclipses make
their timing difficult, and no apsidal motion had been reported. From the TESS
measurements we obtain an average eclipse difference $T_{2}-T_{1}=1.559\pm
0.004$ days. The corresponding argument of periastron, adopting an
eccentricity $e=0.0852\pm 0.0010$ given by Fekel et al. (2013), is
$\omega=265.5\pm 1.5$ deg with its uncertainty dominated by that of the
eclipse timings. Given the observed dispersion in the values of $T_{2}-T_{1}$,
we performed an analysis of the TESS light curve with the same orbital
eccentricity, and obtained an argument of periastron $\omega=266.5\pm 0.5$
deg, in agreement with the previous value but more precise. The argument of
periastron given by Fekel et al. (2013) in their table 7, from the light and
velocity curve combined analysis, is $\omega=257.7\pm 0.2$ deg, thus
indicating for the first time the presence of apsidal motion in VV Crv. The
change in omega, from the light curve fits, after 807 orbital cycles, gives an
apsidal motion rate $\dot{\omega}=0.0109\pm 0.0012$ deg/cycle. This result
should be nevertheless considered preliminary due to the observed dispersion
in the eclipse timings, and difficulties with the photometric analysis, due to
the presence of third light.

  * •

IM Per

Absolute properties of IM Per were obtained by Lacy et al. (2015) and revised
recently by Lee et al. (2020) using the TESS light curve. A simultaneous
radial velocity and timing analysis carried out by Lacy et al. (2015) gave an
apsidal motion rate of $\dot{\omega}=0.0147\pm 0.0008$ deg/cycle, while a
value $\dot{\omega}=0.0155\pm 0.0008$ deg/cycle was obtained by the same
authors from fitting the light curve with variable $\omega$. In both cases,
the orbital eccentricity was fixed to $e=0.047\pm 0.003$. The TESS light curve
solution by Lee et al. (2020) gives a compatible but slightly larger value at
$e=0.0491\pm 0.0010$. Using all the $T_{2}-T_{1}$ measurements available from
table 1 of Lacy et al. (2015) plus the value from TESS $T_{2}-T_{1}=1.09484\pm
0.00014$ days, we calculated the corresponding argument of periastron adopting
this eccentricity. A linear fit to the variation of $\omega$ with time then
yields $\dot{\omega}=0.0146\pm 0.0004$ deg/cycle. Considering only the timings
used by Lacy et al. (2015), a rate of 0.0149$\pm$0.0005 deg/cycle is obtained.

  * •

BP Vul

Lacy et al. (2003) obtained precise absolute dimensions for BP Vul, an
eccentricity $e=0.0355\pm 0.0027$, and the secondary eclipse at phase
$0.47968\pm 0.00021$. The authors also analyzed the apsidal motion rate but
was found to be negative and the potential presence of a perturbing third body
was argued to explain this anomaly, but the timespan covered was too narrow
for a conclusion. Csizmadia et al. (2009) revised the situation with the
analysis of an earlier, unpublished, light curve showing a positive apsidal
motion with a rate of 1 deg/year but with only one new secondary eclipse
timing. Moreover, their analysis of individual timings requires a light-time
effect that is not quantified. On the other hand, TESS measurements give a
phase for the secondary eclipse of $0.47989\pm 0.00010$ days. Given that this
value is not significantly different from the light curve by Lacy et al.
(2003) no claim of apsidal motion detection can be made at this time.

  * •

V1022 Cas

A recent analysis of this eclipsing system by Southworth (2020b), using TESS
data, provides precise physical parameters in good agreement with those
published by Lester et al. (2019) on the basis of a combined astrometric and
spectroscopic study. The lack of precise eclipse timings and the small time
separation with respect to the TESS measurements makes the detection of
apsidal motion very difficult. We have used the spectroscopic solution of
Lester et al. (2019), $e$ and $\omega$, in their table 4 to obtain the
position of the secondary eclipse at phase $0.66565\pm 0.00009$. Comparing
this value with the TESS measurements from the light curve solution by
Southworth (2020b), 452 orbital cycles later, gives a phase displacement of
$-0.00026\pm 0.00011$, equivalent to an apsidal motion rate
$\dot{\omega}=0.00032\pm 0.00015$ deg/cycle. When subtracting the relativistic
term, the classical term comes out to be $\dot{\omega}=0.00007\pm 0.00015$
deg/cycle, which cannot be used in the present paper.

  * •

PV Pup

Despite the presence of intrinsic variability of unknown origin, the light
curve obtained by Vaz & Andersen (1984) provides well-defined photometric
elements, and their combination with radial velocity measurements yield
precise physical properties of the component stars. The orbital eccentricity
was found to be $e=0.0503\pm 0.0011$ but no apsidal motion was reported. PV
Pup belongs to the stellar system ADS 6348 but is most probably an optical
pair. TESS measurements of the separation between primary and secondary
eclipses present a large dispersion, higher than the individual errors,
probably due to the observed intrinsic variability. The average values of
$T_{2}-T_{1}$ for two sectors, separated by 445 orbital cycles, allowed us to
calculate the corresponding argument of periastron with the eccentricity given
by Vaz & Andersen (1984), and thus estimate an apsidal motion rate of
$\dot{\omega}=0.0087\pm 0.0010$ deg/cycle. Given the noise in the eclipse
timings, also noticed by Vaz & Andersen (1984), we tried to analyze the light
curve of TESS with $e=0.0503$. A significant fraction of third light was
observed and the intrinsic variability did not allow to achieve a reasonable
fit. Leaving only $\omega$ as a free parameter, the best fit indicated, when
compared with the value of the light curve analysis by Vaz & Andersen (1984),
an apsidal motion rate of $\dot{\omega}=0.0072\pm 0.0005$ deg/cycle.
Therefore, the preliminary result with only the TESS timings cannot be
confirmed and we call for further monitoring of PV Pup before establishing its
apsidal motion rate.

  * •

BF Dra

Apsidal motion was determined by Wolf et al. (2010) to be
$\dot{\omega}=0.00056\pm 0.00008$ deg/cycle but leaving free the eccentricity
to converge to $e=0.3898$ despite the poor coverage of the apsidal motion
variations. Masses and radii for the components of BF Dra were obtained by
Lacy et al. (2012), who also analyzed the apsidal motion by fitting the
individual eclipse timings and obtained $\dot{\omega}=0.00049\pm 0.00008$
deg/cycle with an eccentricity $e=0.3865\pm 0.0005$. BF Dra was observed in 12
TESS sectors, spanning a total of 352 days, and the large number of precise
values of $T_{2}-T_{1}$ allowed us to determine its apsidal motion rate
independently of any previous measurement. A linear fit to the TESS values of
$T_{2}-T_{1}$ in Table 1 yields a slope of the variation of $(-2.19\pm
0.14)\times 10^{-6}$ days/cycle. Adopting the eccentricity obtained by Lacy et
al. (2012), this gives an apsidal motion of $\dot{\omega}=0.00042\pm 0.00003$
deg/cycle. Because of the difference with previous results, we calculated
separately the linear periods for primary and secondary eclipses and obtained
a difference $\Delta P=(-2.26\pm 0.04)\times 10^{-5}$ days/cycle, but with
systematics present in the residuals of both primary and secondary TESS
timings. We interpret these as potentially due to the light travel-time effect
induced by a third body orbiting the system. Lacy et al. (2012) did indeed
identify the presence of third light in the light curve of BF Dra. The effect
of a third body, however, should not hamper the apsidal motion determination
using the $T_{2}-T_{1}$ values. From the archival times of minimum we could
retrieve five precise $T_{2}-T_{1}$ measurements, considering only individual
eclipse timings within 10 orbital cycles. When adding these to the TESS
$T_{2}-T_{1}$ measurements the slope becomes $(-2.35\pm 0.05)\times 10^{-5}$
days/cycle and $\dot{\omega}=0.00045\pm 0.00002$ deg/cycle, compatible with
the values resulting from the TESS measurements alone and from Lacy et al.
(2012). We detected some remaining systematic residuals in the TESS
observations that remain unexplained. An alternative determination method is
the use of the value of $\omega$ from the light curve analysis, rather than
the times of eclipse. Comparing the argument of periastron given by Lacy et
al. (2012) in their table 7 with that computed from the average TESS
$T_{2}-T_{1}$ measurements, 430 orbital cycles later, a small but significant
increase in $\omega$ of $0.16\pm 0.03$ deg is observed. This yields an apsidal
motion rate of $\dot{\omega}=0.00037\pm 0.00007$ deg/cycle, which is in
marginal agreement with the value described before. More observations are
needed to obtain a definitive apsidal motion rate as well as analyzing the
impact of the third body. In any case, the large fractional relativistic term
(68%) of BF Dra implies a classical term with an uncertainty of $\sim$30% thus
rendering this system unsuitable for the comparison with theoretical stellar
models.

  * •

V1143 Cyg

Andersen et al. (1987) obtained absolute dimensions for V1143 Cyg, which were
more recently revised by Lester et al. (2019) with new observations. Apsidal
motion has been studied by Khaliullin (1983), Giménez & Margrave (1985), Burns
et al. (1996), Dariush et al. (2005) and, finally, by Wolf et al. (2010). The
latter authors gave $\dot{\omega}=0.00072\pm 0.00008$ deg/cycle with an
eccentricity $e=0.535\pm 0.004$. A fit to the TESS timing data alone yields a
faster rate at $\dot{\omega}=0.00088\pm 0.00005$ deg/cycle. A wider time span
can be obtained by using the individual timings from Wolf et al. (2010), with
their original errors when available or otherwise adopting an uncertainty of
$\pm$0.005 days, that we judged realistic. The residuals of the fit to all
timing differences show a significantly larger dispersion than the
uncertainties of the archival measurements, suggesting an underestimation of
the errors. However, we assumed that they are globally unbiased and obtained a
slope of the $T_{2}-T_{1}$ variation of $(2.4\pm 0.1)\times 10^{-5}$
days/cycle. Adopting the orbital eccentricity given by Lester et al. (2019),
$e=0.5386\pm 0.0004$, we obtain an apsidal motion for V1143 Cyg of
$\dot{\omega}=0.00080\pm 0.00004$ deg/cycle.

  * •

IT Cas

Precise physical properties of the components of IT Cas were obtained by Lacy
(1997). From their light and radial velocity curves the authors obtained an
orbital eccentricity $e=0.085\pm 0.004$ and a longitude of periastron of
$\omega=332.6\pm 1.4$ deg, but apsidal motion could not be determined due to
the short time span covered with photoelectric measurements. Moreover, the
same authors obtained different values of the eccentricity from the radial
velocity, the light curve and the ephemeris solution. Kozyreva & Zakharov
(2001) used their own measurements together with those by Lacy (1997) to
obtain $e=0.089\pm 0.002$ with an apsidal motion rate $\dot{\omega}=0.0012\pm
0.0003$ deg/cycle, comparing the values of $\omega$ from the different light
curve solutions. TESS timings confirm the values of Kozyreva & Zakharov
(2001). We have combined the TESS eclipse timing differences with those
derived from the most precise individual timings listed in table 1 of Kozyreva
& Zakharov (2001), only photoelectric and within 10 orbital cycles. The fit
provides a slope of $(2.01\pm 0.05)\times 10^{-6}$ days/cycle. Adopting
$e=0.089\pm 0.002$, yields an apsidal motion rate $\dot{\omega}=0.00114\pm
0.00010$ deg/cycle.

  * •

AI Phe

The light curve obtained by Hrivnak & Milone (1984) was analyzed by Andersen
et al. (1988) together with their own radial velocity curve. Maxted et al.
(2020) analyzed the light curve provided by TESS and improved the physical
parameters using the radial velocity measurements by Hełminiak et al. (2009).
TESS measurements indicate a precise position of the secondary eclipse at
phase $0.457865\pm 0.000005$ and an argument of periastron $\omega=110.34\pm
0.11$ deg. Unfortunately, accurate eclipse timings are difficult to obtain due
to the long orbital period and the duration of the eclipses. Kirkby-Kent et
al. (2016) showed significant variations in the sidereal period, probably due
to the presence of a third body, and determined the phase of the secondary
eclipse at $0.4584\pm 0.0015$, which does not differ significantly from the
TESS measurement. On the other hand, the light curve solution by Andersen et
al. (1988) gives a value of $\omega=109.6\pm 1.0$ deg, which is again very
similar to the value from the TESS light curve. More eclipse timings are
certainly needed, as well as a complete dynamical study to evaluate the impact
of the potential third body in the measurement of apsidal motion.

  * •

EW Ori

This DLEB was analyzed by Clausen et al. (2010), who obtained the physical
parameters of its components and estimated an apsidal motion rate
$\dot{\omega}=0.00042\pm 0.00010$ deg/cycle with an orbital eccentricity
$e=0.0758\pm 0.0020$ derived from the light curve. We obtained precise
$T_{2}-T_{1}$ values from the TESS light curve. In addition, we retrieved four
$T_{2}-T_{1}$ values from the literature (Wolf et al. 1997; Clausen et al.
2010) from the evaluation of the secondary phase displacement and using only
precise individual timings separated by less than 10 orbital cycles. A linear
fit to the $T_{2}-T_{1}$ variation with time yields a slope of $(1.48\pm
0.11)\times 10^{-6}$ days/cycle. Taking the orbital eccentricity given by
Clausen et al. (2010) produces an apsidal motion rate of
$\dot{\omega}=0.00033\pm 0.00002$ deg/cycle. Unfortunately, this system cannot
be used to compare with stellar structure models. EW Ori´s highly relativistic
periastron precession is dominant (79%) and the determination of the classical
term carries an uncertainty that is above our threshold. However, this system
could be added to our sample in paper I. Following the methodology there, we
determine $\dot{\omega}_{\rm GR}=0.00026\pm 0.00002$ deg/cycle after
subtracting the classical term calculated from the models, as described in
section 4. It can be shown that the measured relativistic rate is in very good
agreement with the other systems plotted in figures 10 and 11 of paper I. A
recalculation of the corresponding post-Newtonian parameters, now including EW
Ori, yields $A=1.000\pm 0.011$ and $B=0.000\pm 0.051$.

  * •

V530 Ori

A complete analysis of this G+M system was carried out by Torres et al.
(2014), who provided the physical parameters of its components. The shallow
total secondary eclipse makes it difficult to study variations in phase.
Torres et al. (2014) determined the orbital eccentricity to be $e=0.0862\pm
0.0010$ with $\omega=130.08\pm 0.14$ deg. Apsidal motion was detected by the
authors but could not be determined with sufficient significance. From their
table 1, only two precise $T_{2}-T_{1}$ values could be retrieved with
individual timings within 10 orbital cycles. Using those with the new TESS
data, we determine an apsidal motion rate $\dot{\omega}=0.00086\pm 0.00005$
deg/cycle, with the orbital eccentricity of $e=0.0862\pm 0.0010$.
Nevertheless, additional observations are needed to confirm the apsidal motion
rate, given the short span of time available and the difficulty to obtain
precise timings.