###### Abstract

We aim to combine asteroseismology, spectroscopy, and evolutionary models to
establish a comprehensive picture of the evolution of Galactic blue supergiant
stars (BSG). To start such an investigation, we selected three BSG candidates
for our analysis: HD 42087 (PU Gem), HD 52089 ($\epsilon$ CMa) and HD 58350
($\eta$ CMa). These stars show pulsations and were suspected to be in an
evolutionary stage either preceding or succeding the red supergiant (RSG)
stage. For our analysis, we utilized the 2-min cadence TESS data to study the
photometric variability and we obtained new spectroscopic observations at the
CASLEO observatory. We used non-LTE radiative transfer models calculated with
CMFGEN to derive their stellar and wind parameters. For the fitting procedure,
we included CMFGEN models in the iterative spectral analysis pipeline XTgrid
to determine their CNO abundances. The spectral modeling was limited to
changing only the effective temperature, surface gravity, CNO abundances, and
mass-loss rates. Finally, we compared the derived metal abundances with
prediction from Geneva stellar evolution models. The frequency spectra of all
three stars show stochastic oscillations and indications of one nonradial
strange mode, $f_{r}=$ 0.09321 d-1 in HD 42087 and a rotational splitting
centred in $f_{2}=$ 0.36366 d-1 in HD 52089. We conclude that the rather short
sectoral observing windows of TESS prevent establishing a reliable mode
identification of low frequencies connected to mass-loss variabilities. The
spectral analysis confirmed gradual changes in the mass-loss rates and the
derived CNO abundances comply with the values reported in the literature. We
were able to achieve a quantitative match with stellar evolution models for
the stellar masses and luminosities. However, the spectroscopic surface
abundances turned out to be inconsistent with the theoretical predictions. The
stars show N enrichment, typical for CNO cycle processed material, but the
abundance ratios are not reflecting the associated levels of C and O
depletion. We found HD 42087 (PU Gem) to be most consistent with a pre-RSG
evolutionary stage, and HD 58350 ($\eta$ CMa) is most likely in a post-RSG
evolution. We were unable to model HD 52089 ($\epsilon$ CMa) with any evolved
BSG tracks and conclude its relatively low mass and high luminosity are most
consistent with a rejuvenated merger scenario.

###### keywords:

stars: massive; stars: supergiants; stars: winds, outflows

1 1 0 https://doi.org/ Unveiling the evolutionary state of three B supergiant
stars: PU Gem, $\epsilon$ CMa and $\eta$ CMa Unveiling the evolutionary state
of three BSG stars. Julieta P. Sánchez Arias 1*, Péter Németh 1,2, Elisson S.
G. de Almeida3, Matias A. Ruiz Diaz 4, Michaela Kraus 1 and Maximiliano
Haucke5 Julieta P. Sánchez Arias, Péter Németh, Elisson S. G. de Almeida,
Matias A. Ruiz Diaz, Michaela Kraus and Maximiliano Haucke Sánchez Arias, J.
P.; Nemeth, P.; de Almeida, E. S. G.; Ruiz Diaz, M. A.; Kraus, M. and Haucke,
M. Correspondence:sanchezasu.cas.cz

##  1 Introduction

Massive stars are one of the most important objects in the universe due to
their key role in the enrichment of interstellar medium with metals for future
star generations and the evolution of the host galaxies (Abbott, 1982).
Nevertheless, understanding their evolution is challenging due to the
significant changes they experience at different evolutionary stages,
especially during the post-Main Sequence (MS). Therefore, any inaccuracy in
the input parameters results in large uncertainties of the evolutionary models
(Martins and Palacios, 2013; Agrawal et al., 2022). Fortunately, there are
many sophisticated stellar evolutionary models calculated for massive stars
(e.g. Yusof et al., 2022; Maeder and Meynet, 2010). However, they depend on
internal parameters such as rotation, chemical mixing, and angular momentum
transport, for which no decisive observational constraints are available. In
addition, massive supergiant stars undergo mass loss via line-driven winds and
their rates are far from being firmly established, adding even more
uncertainties to the evolution of these stars.

BSGs comprise extreme transition phases, in which the stars shed huge amounts
of material into their environments. During the evolution of massive stars,
those stars with masses between 20 and 40 $M_{\odot}$ evolve back to the blue
supergiant state, after a Red Supergiant (RSG) state, either as Blue
Supergiant (BSG) or in a follow-up Wolf-Rayet phase. On the other hand,
massive stars with $<20$ $M_{\odot}$ may experience "blue loops", where the
star changes from a cool star to a hotter one before cooling again. Therefore,
hot BSGs can be found at the pre-RSG stage, burning only H in a shell or at
the post-RSG during the He-core burning. The exact reason why some massive
stars experience "blue loops" and others do not still remains unknown,
although it is known that extra mixing processes within the layers surrounding
their convective core along with mass loss events during their evolution play
an essential role (Wagle et al., 2019; Bowman et al., 2019; Langer, 2012).

In addition, BSGs can show an extremely rich spectrum of stellar oscillations.
Since these oscillations depend on the internal structure of the star, their
analysis paves the way to understanding phenomena such as the occurrence of
“blue loops” or the discernment between the different evolutionary stages in
which blue supergiant stars can exist. The observed oscillation modes in B
supergiants are driven by different excitation mechanisms, such as the
classical $\kappa$ mechanism, stochastic wave generation caused by the
presence of convective layers in the outer regions of the star or in their
envelope, tidal excitation causing the so-called Rossby modes and strange
modes. Strange modes are nonlinear instabilities that require a luminosity
over a mass ratio of $L_{\star}/M_{\star}>10^{4}L_{\odot}/M_{\odot}$ to be
excited (Glatzel, 1994). Their existence has been related to the variable mass
loss these stars experience (Kraus et al., 2015; Haucke et al., 2018; Aerts et
al., 2010).

Recent studies on stellar oscillations of individual BSGs (Saio et al., 2006;
Aerts et al., 2010; Kraus et al., 2015) improved our understanding of the
complex variability these stars show. Furthermore, considerable progress has
been made in theoretical studies on the stellar oscillations of massive stars
at different evolutionary stages focusing on specific mass ranges. For
example, Ostrowski and Daszyńska-Daszkiewicz (2015) investigated stellar
oscillations in stars at pre- and post-RSG stages for masses between 13 and 18
solar masses for different physical parameters such as metallicity and
overshooting. In Saio et al. (2013), a thorough study of stellar oscillations
is presented for a broader range of masses, revealing that the pulsation
properties in pre- and post-RSG evolutionary phases are fundamentally
different: stars in the post-RSG stage pulsate in many more modes, including
radial strange modes, than their less evolved counterparts, although these
authors did not inspect the effect of different wind efficiency on the
stability of the modes.

Despite the efforts accumulated over years in the study of the stellar
pulsations of these objects in different evolutionary stages, the correct
identification of the evolutionary stage of BSGs additionally requires a
detailed analysis of spectral observations due to the large uncertainties
these stars have in astrophysics parameters such as the mass and the radii.

Therefore, we started a comprehensive study of these objects with the aim of
gaining insights into the evolutionary state of BSG stars. Combining
information about the pulsation behaviour extracted from photometric
lightcurves with newly determined stellar parameters and precise chemical
abundances obtained from modeling of acquired spectroscopic data, we strive to
find clear evidence for either a pre- or a post-RSG state of the objects. The
current paper presents our methodology and first results on a small sample of
objects.

The paper is organized as follows: in Sect. 2 we summarize the main parameters
for these stars found in the literature and Sect. 3 describes the
spectroscopic and photometric observations employed in this work. In Sect. 4,
we present the analysis of the light curves and the frequency spectra. Sect. 5
is devoted to the spectral analysis of the selected objects. We describe the
numerical tools employed, including brand new capabilities implemented in
XTgrid (Németh et al., 2012) for this work and the results obtained. Finally,
Sect. 6 and 7 are devoted to the discussion and conclusions.

##  2 Target selection and parameters values from literature

Haucke et al. (2018) presented the most recent comprehensive analysis of the
wind properties of 19 pulsating BSGs. We focus on three objects from their
sample, for which we have obtained new spectroscopic observations. These are
the stars HD 42087 (PU Gem), HD 52089 ($\epsilon$ CMa) and HD 58350 ($\eta$
CMa). Table 1 summarizes the values for these stars derived by Haucke et al.
(2018). They serve as reference (or starting) values for our analysis. In Fig.
1 we show the position of the selected stars in an HR diagram for the
parameters in Table 1. We notice that these hot BSG stars could be either at
the immediate post-Main Sequence or at the post-Red Supergiant stage.

The three selected objects have been studied extensively in the literature,
and a compilation of literature values for their stellar and wind parameters
can be found in Haucke et al. (2018). Here we restrict to a brief overview
highlighting a variety of stellar and wind parameters.

###  2.1 HD 42087

Searle et al. (2008) derived the following parameters for this star using
spectra from October 1990: $T_{\rm eff}$ = 18000 $\pm$ 1000 K, $\log g$ = 2.5,
$\log(L_{\star}/L_{\odot})$ = 5.11 $\pm$ 0.24, $R_{\star}$ = 36.6 $R_{\odot}$,
and $v\sin i$ = 71 km s-1. They employed CMFGEN complemented with TLUSTY to
derive $T_{\rm eff}$ and $\log(L_{\star}/L_{\odot})$, $\log g$, along with the
CNO abundances being $\epsilon(C)$ = 7.76, $\epsilon(N)$ = 8.11, and
$\epsilon(C)$ = 8.80. For the wind parameters, they obtained $\dot{M}$ =
$5.0\times 10^{-7}$ $M_{\odot}$ yr-1, $\beta$ = 1.2, and $v_{\infty}$ = 650 km
s-1. Morel et al. (2004) showed that HD 42087 has a high H$\alpha$ variability
with a spectral variability index (as defined by these authors) of $\sim$91%
in this line (see Sect. 4 of Morel et al., 2004), evidencing a cyclic
behaviour with P $\sim$25 d. We also mention that the values derived for this
object in Haucke et al. (2018) were obtained with TLUSTY using an optical
spectrum covering only the H${\alpha}$ region. This spectra from 2006, showed
a P Cygni feature with a weak emission and a strong absorption component.

###  2.2 HD 52089

In Morel et al. (2008) this star was studied as a slowly rotating B-type dwarf
star. They derived $T_{\rm eff}$ = 23000 K and $\log{g}$ = 3.30 $\pm$ 0.15
using spectroscopic data from April 2005. By using the DETAIL/SURFACE code
they determined the non-LTE abundances $\epsilon(C)$ = 8.09 $\pm$ 0.12,
$\epsilon(N)$ = 7.93 $\pm$ 0.24, and $\epsilon(O)$ = 8.44 $\pm$ 0.18. Fossati
et al. (2015) derived an effective temperature and surface gravity of $T_{\rm
eff}$ = 22 500 $\pm$ 300 K and $\log g$ = 3.40 $\pm$ 0.08. They also obtained
updated values for the surface abundances by analyzing a FEROS spectra from
2011 and found $\epsilon(C)$ = 8.30 $\pm$ 0.07, $\epsilon(N)$ = 8.16 $\pm$
0.07 and $\epsilon(O)$ = 8.70 $\pm$ 0.12. Additionally, they estimated a 12.5
$M_{\odot}$ for this object.

###  2.3 HD 58350

Lefever et al. (2007) derived $T_{\rm eff}$ = 13500 K, $\log g$ = 1.75,
$R_{\star}$ = 65 $R_{\odot}$, $\log(L_{\star}/L_{\odot})$ = 5.10, $v_{\infty}$
= 250 km s-1, $\dot{M}$ = $1.4\times 10^{-7}$ $M_{\odot}$ yr-1 and $\beta$ =
2.5 for the stellar and wind parameters of this star. In Searle et al. (2008),
they derived $T_{\rm eff}$ = 15000 $\pm$ 500 K, $\log g$ = 2.13, $R_{\star}$ =
57.3 $\pm$ 2.64 $R_{\odot}$, and $\log(L_{\star}/L_{\odot})=$ 5.18 $\pm$ 0.17.
For the CNO abundances they obtained $\epsilon(C)$ = 7.78, $\epsilon(N)$ =
8.29, and $\epsilon(O)$ = 8.75.

Figure 1: Our three BSG stars in the HR diagram according to the values derived in Haucke et al. (2018). The evolutionary tracks are taken from (Ekström et al., 2012) Table 1: Stellar and wind parameters derived by Haucke et al. (2018) for our star sample. Parameter |  HD 42087 |  HD 52089 |  HD 58350
---|---|---|---
$T_{\rm eff}$ [K] |  16500 $\pm$ 1000 |  23000 $\pm$ 1000 |  15500 $\pm$ 700
$\log g$ [cgs] |  2.45 $\pm$ 0.10 |  3.00 $\pm$ 0.10 |  2.00 $\pm$ 0.10
$\log{L_{\star}}$ [$L_{\odot}$] |  5.31 $\pm$ 0.43 |  4.49 $\pm$ 0.05 |  5.18 $\pm$ 0.32
$R_{\star}$ [$R_{\odot}$] |  55 |  11 |  54
$v\sin i$ [km s-1] |  80 |  10 |  40
$\dot{M}$ [$M_{\odot}$ yr-1] |  (5.7 $\pm$ 0.5)$\times 10^{-7}$ |  (2.0 $\pm$ 0.6)$\times 10^{-8}$ |  (1.4 $\pm$ 0.2)$\times 10^{-7}$
$v_{\infty}$ [km s-1] |  700 $\pm$ 70 |  900 $\pm$ 270 |  200 $\pm$ 30
$\beta$ |  2.0 |  1.0 |  3.0

##  3 Observations

With the aim to shed light on the evolutionary state of these objects, we
analyze new spectroscopic observations and combine these results with
information extracted from their photometric light curves.

###  3.1 Spectra

The spectra we employed in this work to derive new parameters were taken on
2020 January 23 and 24 for HD 42087 and HD 58350, respectively, and 2015
February 14 for HD 52089. All of them cover the wavelength range from 4275 Å
up to 6800 Å with a signal-to-noise ratio S/N of 140, 140, and 130 for HD
42087, HD 52089, and HD 58350, respectively.

We utilized the REOSC spectrograph attached to the Jorge Sahade 2.15 m
telescope at the Complejo Astronómico El Leoncito (CASLEO), San Juan,
Argentina. The resolving power at 4500 Å and 6500 Å is $R\sim 12600$ and
$R\sim 13900$, respectively. The spectra were reduced and normalized following
standard procedures using IRAF111IRAF is distributed by the National Optical
Astronomy Observatory, which is operated by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation. routines.

In order to show the H$\alpha$ variability in our targets, we have collected
previous observations in CASLEO, depicted in Fig. 2, 3, and 4; from 2006
January 15 for HD 42087; 2013 February 5 for HD 52089; and 2006 January 15 and
2013 February 5 for HD 58350.

Figure 2: Evolution of the H$\alpha$ line profile of HD 42087 between 2006 and
2020. The emission component weakened by 2020.  Figure 3: Evolution of the
H$\alpha$ line profile of HD 52089 between 2013 and 2015.  Figure 4: Evolution
of the H$\alpha$ line profile of HD 58350 between 2006 and 2020. The emission
component has strengthened between 2006 and 2013 and got noticeably weaker by
2020.

###  3.2 Photometric light curves

We complemented the optical spectra with space photometry collected with the
Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al., 2014,
2015). We retrieved high cadence (120s) PDCSAP light curve for each object
using Astroquery (Ginsburg et al., 2019) and cleaned the light curves using
the Lightkurve package (Lightkurve Collaboration et al., 2018). For this, we
first selected only the data points with a quality flag equal to 0, meaning
they are not cosmic rays or data from bad pixels. Next, we carried out sigma
clipping to remove outliers, using a value of 6$\sigma$ for all light curves,
following a similar procedure as in Garcia et al. (2022). The resulting light
curves were transformed to display the variation of the magnitude ($\Delta m$)
from the mean magnitude:

$\Delta
m=-2.5\mathrm{log}(\mathrm{pdcsap\\_flux})+2.5\mathrm{log}(\overline{\mathrm{pdcsap\\_flux}});$
(1)

where pdcsap_flux is the individual flux at each exposure and
$\overline{\mathrm{pdcsap\\_flux}}$ is the mean flux of the entire light
curve. In this way, we obtain a normalized light curve, whose amplitude can be
expressed in units of $\mathrm{mag}$. The time axis is in units of Barycentric
Julian Date, which is the Julian Date corrected for differences in the earth’s
position with respect to the barycenter of the solar system.

##  4 Frequency analysis

We employed the Fourier Transform with Period04 (Lenz and Breger, 2005). For
each star and sector, the frequencies were searched in the interval [0;50]
$d^{-1}$, widely covering their frequency content. No frequency beyond 2
$d^{-1}$ was found for any star. The amplitude and phase were calculated using
a least square sine fit for each detected frequency. After obtaining the first
frequency, the analysis was performed on the residuals. The Fourier analysis
was stopped after obtaining 15 frequencies. Once we derived the frequencies
following this procedure, we dismissed those frequencies below 0.1 $d^{-1}$
since TESS data of a single sector do not allow us to derive periods higher
than $\sim 10$ d. Additionally, we discard those frequencies with a separation
of less than $2/T$, where $1/T$ is the Rayleigh resolution and $T$ is the time
span of the observations. The S/N ratio was computed for the derived
frequencies along with the uncoupled uncertainties in the frequencies and
amplitudes using a Monte Carlo simulation.

When available, we also analyzed the frequency content of the combined,
consecutive sectors since the longer time baseline would facilitate the
detection of longer periods from radial modes possibly connected to strange
modes. We considered the values recommended in Baran and Koen (2021) for the
S/N when dealing with individual TESS sectors and combined sectors, which
resulted in 5.037 for our individual TESS sectors, and of 5.124 and 5.194 for
the combined sectors of HD 52089 (2 sectors combined) and HD 42087 (3 sectors
combined), respectively. Nevertheless, we took into account frequencies with
lower S/N whenever they turned out to be interesting for the analysis (see
below).

Finally, when comparing frequencies from different sectors, we adopted the
separation criterion for the combined sectors.

Next, we provide the details of the light curves and the frequencies extracted
for each star.

###  4.1 HD 42087

This star was observed in 3 consecutive sectors: Sector 43, in the period 2021
September 16 to October 10; Sector 44, during 2021 October 10 to November 6;
and Sector 45, during 2021 November 6 to December 2, covering an observation
time span of 24.287 days, 24.156 days and 24.551 days, respectively. The light
curve and the amplitude spectra for the individual sectors and all sectors
combined are displayed in Fig. 5.

Figure 5: Top panel: light curve of HD 42087 corresponding to Sectors 43, 44,
and 45. Lower panels: Amplitude spectra for each sector and all sectors
combined.

The frequencies derived for sectors 43, 44, and 45 are listed in Table 2 and
the threshold considered for the frequency separation is $0.0823$ $d^{-1}$,
$0.0828$ $d^{-1}$ and $0.0815$ $d^{-1}$, respectively. For the analysis of all
sectors combined, the time span is $T$ = 72.994 $d$, therefore, the separation
criterion we employed is $2/T$ = 0.0274 $d^{-1}$. We adopted this value as a
conservative threshold when comparing frequencies from different sectors.

The pulsational content of HD 42087 lies at low frequencies below 0.8 $d^{-1}$
and the amplitudes are between 2.5 and 9.5 mmag, approximately. When analyzing
the combined sectors, we also searched for frequencies down to 0.025 $d^{-1}$
due to the extended length of the observations. This allowed us to find the
frequency $f_{r}$ and its five harmonics. Some of these harmonics appear in
the individual sectors. Due to the short time span for individual sectors,
frequencies below $\sim$0.1 $d^{-1}$ are not reliable; nevertheless, $f_{r}$
seems to appear for sectors 43 and 44 (indicated with green lines in Fig. 5).
Besides $f_{r}$ and its harmonics, five frequencies appear randomly over the
four sets of observations, which are likely related to stochastic oscillations
(see Sect. 6).

Table 2: List of frequencies, their amplitudes, and S/N ratios, found for HD 42087. Sector |  Frequency |  3$\sigma_{f}$ |  Amplitude |  3$\sigma_{A}$ |  S/N |  Id
---|---|---|---|---|---|---
|  [$d^{-1}$] |  [$d^{-1}$] |  [mmag] |  [mmag] |  |
|  0.15543 |  0.00044 |  7.7615 |  0.15 |  10.65 |  $f_{1}$
43 |  0.25729 |  0.00063 |  5.7467 |  0.14 |  8.41 |  $3*f_{r}$
|  0.47759 |  0.00064 |  4.7663 |  0.13 |  7.79 |  $5*f_{r}$
|  0.37169 |  0.00090 |  3.5347 |  0.13 |  5.45 |  $4*f_{r}$
|  0.70257 |  0.00037 |  2.9734 |  0.14 |  5.41 |  $f_{2}$
|  0.26268 |  0.00068 |  5.5317 |  0.16 |  7.68 |  $3*f_{r}$
44 |  0.15072 |  0.00082 |  5.2203 |  0.17 |  6.75 |  $f_{1}$
|  0.43536 |  0.00090 |  4.9832 |  0.19 |  7.45 |  $f_{3}$
|  0.27928 |  0.00009 |  9.2074 |  0.10 |  10.65 |  $3*f_{r}$
45 |  0.18110 |  0.00059 |  5.0188 |  0.13 |  5.57 |  $2*f_{r}$
|  0.47295 |  0.00159 |  4.0068 |  0.21 |  5.07 |  $5*f_{r}$
|  0.28022 |  0.00019 |  4.4715 |  0.12 |  8.48 |  $3*f_{r}$
All |  0.16305 |  0.00021 |  3.9728 |  0.11 |  7.05 |  $f_{1}$
|  0.49515 |  0.00022 |  3.4982 |  0.11 |  7.44 |  $f_{4}$
|  0.23874 |  0.00027 |  3.2680 |  0.11 |  6.04 |  $f_{5}$
|  0.44102 |  0.00023 |  3.2321 |  0.10 |  6.67 |  $5*f_{r}$
|  0.33531 |  0.00030 |  2.9450 |  0.11 |  5.74 |  $2*f_{1}$
|  0.19899 |  0.00029 |  2.8086 |  0.10 |  5.07 |  $2*f_{r}$
|  0.37073 |  0.00033 |  2.6635 |  0.10 |  5.29 |  $4*f_{r}$
|  0.09321 |  0.00032 |  2.4597 |  0.12 |  4.20 |  $f_{r}$

###  4.2 HD 52089

This star was observed in 4 TESS sectors: Sector 6, in the period 2018
December 15 to 2019 January 6; Sector 7, in the period 2019 January 8 to
February 1; Sector 33 from 2020 December 18 to 2021 January 13; and Sector 34
in the period 2021 January 13 to February 8. The total time spans for Sectors
6, 7, 33, and 34 are 21.771 d, 24.454 d, 25.839 d, and 24.962 d, respectively.
Fig. 6 displays the light curves and amplitude spectra for Sectors 6 and 7,
and Fig.7 displays the same information for Sectors 33 and 34. Due to the
large time gap between sectors 7 and 33, we decided to evaluate the sectors in
pairs, i.e. together with the analysis of sectors 6 and 7, we studied the
frequencies of these sectors combined, and the same with sectors 33 and 34.
For the combined sectors 6 and 7, the time span is 46.225 d, and for sectors
33 and 34 together, the time span is 50.801 d.

Figure 6: Top panel: light curve of HD 52089 corresponding to Sectors 6 and 7.
Lower panels:  Amplitude spectra for each sector and all sectors combined.
Figure 7: Top panel: light curve of HD 52089 corresponding to Sectors 33 and
34. Lower panels:  Amplitude spectra for each sector and all sectors combined.

The derived frequencies, their amplitudes and identifications are listed in
Table 4. The separation criteria we employed are 0.0918 $d^{-1}$, 0.0817
$d^{-1}$, 0.0774 $d^{-1}$, and 0.0801 $d^{-1}$ for sectors 6, 7, 33, and 34,
respectively. For sectors 6 and 7 combined and 33 and 34 combined the
separation criteria are 0.0432 $d^{-1}$ and 0.0393 $d^{-1}$, respectively and
the lowest detectable frequency for the sectors combined is $\sim$0.04
$d^{-1}$. We adopted conservative separation criteria with 0.0393 $d^{-1}$ to
compare frequencies from different sectors. We find only one significant
frequency, $f_{1}$, in Sector 6, which is also in Sector 7. In Sector 7, we
found another frequency, $f_{2}$, which stays present also in Sector 33, and
its harmonic, which appears in Sector 34. We searched for lower frequencies in
both combined light curves, but we did not find any. After the gap of $\sim$2
years in the observations, new frequencies appear randomly in the new Sectors,
which indicates they are not related to stellar oscillations due to their
short lifetime ($f_{3}$, $f_{5}$, $f_{6}$). We note that $f_{2}-f_{3}\sim
f_{4}-f_{2}$ in Sector 33, indicating the presence of a triplet centred in
$f_{2}$ with a possible rotation frequency of 0.20 $d^{-1}$. Nevertheless, the
observations’ short time span makes precise classification difficult. We did
not find any significant frequency at lower ranges in the combined sectors.

Table 3: Sector, Frequency, Amplitude, and Identification for HD 52089. Sector |  Frequency |  3$\sigma_{f}$ |  Amplitude |  3$\sigma_{A}$ |  S/N |  Id
---|---|---|---|---|---|---
|  [$d^{-1}$] |  [$d^{-1}$] |  [mmag] |  [mmag] |  |
6 |  0.24083 |  0.00115 |  0.7830 |  0.05 |  6.15 |  $f_{1}$
|  0.24316 |  0.01540 |  0.8982 |  0.47 |  6.14 |  $f_{1}$
7 |  0.38381 |  0.00192 |  0.7286 |  0.05 |  5.15 |  $f_{2}$
|  0.77252 |  0.00051 |  0.5888 |  0.02 |  4.37 |  $2*f_{2}$
|  0.23321 |  0.00072 |  0.7736 |  0.07 |  7.28 |  $f_{1}$
6 & 7 |  0.38342 |  0.02775 |  0.4930 |  0.17 |  4.79 |  $f_{2}$
|  0.77679 |  0.00073 |  0.4769 |  0.04 |  4.83 |  $2*f_{2}$
|  0.16358 |  0.00070 |  0.7716 |  0.02 |  6.868 |  $f_{3}$
33 |  0.36366 |  0.00089 |  0.5784 |  0.02 |  5.42 |  $f_{2}$
|  0.56843 |  0.00089 |  0.5083 |  0.02 |  4.72 |  $f_{4}$
|  0.42781 |  0.00107 |  0.7556 |  0.03 |  4.98 |  $f_{5}$
34 |  0.52280 |  0.00128 |  0.7113 |  0.03 |  4.74 |  $3*f_{3}$
|  0.76818 |  0.00127 |  0.7038 |  0.03 |  4.75 |  $2*f_{2}$
|  0.29483 |  0.00147 |  0.6389 |  0.03 |  4.35 |  $f_{6}$
|  0.29937 |  0.00167 |  0.6627 |  0.12 |  5.01 |  $f_{6}$
33 & 34 |  0.36220 |  0.00331 |  0.5806 |  0.34 |  4.39 |  $f_{2}$
|  0.50772 |  0.00158 |  0.4714 |  0.09 |  4.09 |  $3*f_{3}$

###  4.3 HD 58350

This star has been observed in Sector 34 in the period 2021 January 14 to
February 8, covering a total timespan of $T$ = 24.96 days. The light curve for
sector 34 is displayed in the top panel of Fig. 8 and we show the amplitude
spectra in the lower panel.

Figure 8: Top panel: light curve of HD 58350 acquired during Sector 34. Lower
panel: Amplitude spectrum for the same sector.

The resulting list of frequencies for HD 58350 is shown in Table. 4. The
separation criterion is $2/T$ = 0.0801 $d^{-1}$. The pulsational content for
this BSG lies below 0.5 $d^{-1}$. We found 2 independent frequencies with
similar amplitudes.

Table 4: Sector, Frequency, Amplitude, and Identification for HD 58350. Sector |  Frequency |  3$\sigma_{f}$ |  Amplitude |  3$\sigma_{A}$ |  S/N |  Id
---|---|---|---|---|---|---
|  [$d^{-1}$] |  [$d^{-1}$] |  [mmag] |  [mmag] |  |
34 |  0.21533 |  0.00069 |  0.4392 |  0.01 |  8.54 |  $f_{1}$
|  0.35927 |  0.00088 |  0.3339 |  0.01 |  7.07 |  $f_{2}$

##  5 Modeling tools for spectral analysis

###  5.1 The code CMFGEN

CMFGEN (Hillier and Miller, 1998) is one of the state-of-the-art 1-D non-LTE
radiative transfer codes for studying the physical and chemical properties of
massive hot stars. It has been successfully used for reproducing different
types of observables of massive stars, such as in the case of OB supergiants:
photometry, spectroscopy, and interferometry (e.g., Searle et al., 2008;
Bouret et al., 2012; de Almeida et al., 2022).

CMFGEN solves the radiative transfer and statistical equilibrium equations for
the photosphere and wind under the assumption of radiative equilibrium and
considers a spherically symmetric stationary wind. For the photospheric
region, CMFGEN solves the radiative transfer and hydrostatic equations
(describing the state of radiation and gas in the photosphere) in a self-
consistent way. Further details on this procedure can be found in Sect. 3.1 of
Bouret et al. (2013).

The wind velocity, $v(r)$, and density, $\rho(r)$, profiles are related to
each other as follows, based on the assumption of a stationary spherical
symmetric wind:

$\dot{M}=4\pi r^{2}\rho(r)v(r)f(r),$ (2)

where $r$ is the distance from the center of the star and $\dot{M}$ is the
wind mass-loss rate, which is constant at any location of the wind. Here, the
volume filling factor $f(r)$ parameterizes the inclusion of micro-clumping
(e.g., see Hillier et al., 2001), that is, inhomogeneities that are found in
the winds of massive stars (e.g., Eversberg et al., 1998; Bouret et al.,
2005). In CMFGEN, this parameterization is performed as follows:

$f(r)=f_{\infty}+(1-f_{\infty})\mathrm{e}^{-\frac{v(r)}{v_{\mathrm{initial}}}},$
(3)

where $f_{\infty}$ is the filling factor value at $r\to\infty$ and
$v_{\mathrm{initial}}$ is the onset velocity of clumping in the wind. Since
our initial guesses, in the fitting procedure (as described in Sect. 5.2), for
the stellar and wind parameters are based on the results from Haucke et al.
(2018), we set $f_{\infty}$ = 1.0 in our default models. This means that our
default CMFGEN models are calculated considering a homogeneous wind.

Furthermore, the wind’s velocity profile is parameterized in CMFGEN by the so-
called $\beta$-law approximation as follows:

$v(r)=v_{\infty}\left(1-\frac{R_{\star}}{r}\right)^{\beta},$ (4)

where $v_{\infty}$ is the wind terminal velocity and $R_{\star}$ is the
stellar radius ($r$ higher than $R_{\star}$). For OB supergiants, values of
$\beta$ are usually found to be as high as $\sim$2.0-3.0 (e.g., Martins et
al., 2015). In short, Eqs. 2 and 4 sets the relation between the most
important fundamental wind physical parameters: the wind mass-loss rate and
the terminal velocity.

Table 5:  Summary on the atoms and ionization states of our default CMFGEN models: number of energy levels, super-levels, and bound-bound transitions for each atomic specie.  Ion | Full-levels | Super-levels | b-b transitions
---|---|---|---
H i | 30 | 30 | 435
He i | 69 | 69 | 905
He ii | 30 | 30 | 435
C ii | 322 | 92 | 7742
C iii | 243 | 99 | 5528
C iv | 64 | 64 | 1446
N ii | 105 | 59 | 898
N iii | 287 | 57 | 6223
N iv | 70 | 44 | 440
N v | 49 | 41 | 519
O ii | 274 | 155 | 5880
O iii | 104 | 36 | 761
O iv | 64 | 30 | 359
O v | 56 | 32 | 314
Ne ii | 48 | 14 | 328
Ne iii | 71 | 23 | 460
Ne iv | 52 | 17 | 315
Mg ii | 44 | 36 | 348
Si ii | 53 | 27 | 278
Si iii | 90 | 51 | 640
Si iv | 66 | 66 | 1090
S iii | 78 | 39 | 520
S iv | 108 | 40 | 958
S v | 144 | 37 | 1673
Fe ii | 295 | 24 | 2135
Fe iii | 607 | 65 | 6670
Fe iv | 1000 | 100 | 37899
Fe v | 1000 | 139 | 37737
Fe vi | 1000 | 59 | 36431
Ni ii | 158 | 27 | 1668
Ni iii | 150 | 24 | 1345
Ni iv | 200 | 36 | 2337
Ni v | 183 | 46 | 1524
Ni vi | 182 | 40 | 1895

Despite its high computational cost, when compared to other non-LTE codes,
such as FASTWIND, one of the biggest advantages of CMFGEN relies on allowing
to set a complex chemical composition, with the inclusion of energy levels of
different ions from hydrogen up to nickel. For instance, Haucke et al. (2018)
studied our star sample of three BSGs employing radiative transfer models
calculated with FASTWIND considering hydrogen, helium, and silicon with an
approximation for the treatment of the line transfer of iron-group elements.
In fact, the main difference between these two codes relies on an “exact”
treatment of line-blanketing that is performed by CMFGEN (e.g., see Sander,
2017, and references therein). Table 5 summarizes the atomic species, the
number of energy levels, and bound-bound transitions that are included as
default in our models. Our models are calculated considering a robust atomic
model for studying massive hot stars, including species of hydrogen, helium,
carbon, nitrogen, oxygen, neon, magnesium, silicon, sulfur, iron, and nickel.

Finally, due to the complexity of the code and its computational cost, a very
common approach when using CMFGEN is to vary its parameters manually in order
to find an acceptable “by eye” fit to the observations (e.g., Martins et al.,
2005; Marcolino et al., 2009; de Almeida et al., 2019; Rivet et al., 2020; de
Almeida et al., 2022). However, in this paper, we implement an automatic
fitting procedure with CMFGEN to find the best-fit models for the observed
spectrum of each star, as described below.

###  5.2 Spectral analysis with XTgrid

XTgrid (Németh et al., 2012) is a steepest-descent iterative $\chi^{2}$
minimizing fit procedure to model hot star spectra. The procedure was
developed for the model atmosphere code Tlusty (Hubeny and Lanz, 1995; Lanz
and Hubeny, 2007; Hubeny and Lanz, 2017) and was previously applied to
ultraviolet and optical spectral observations of O and B-type stars (Kraus et
al., 2015), Horizontal Branch stars (Lin et al., 2023), hot subdwarfs (Lei et
al., 2023; Németh et al., 2021; Luo et al., 2020), and white dwarfs (Wang et
al., 2022; Vennes et al., 2017). It was designed to perform fully automated or
supervised spectral analyses of massive data sets. The procedure starts with
an input model and by applying successive approximations along a decreasing
global $\chi^{2}$, iteratively converges on the best solution. All models are
calculated on the fly, which is the main advantage of the procedure. XTgrid
does not require a precalculated grid and with its tailor-made models –
although at a high computational cost – it is able to address nonlinearities
in multidimensional parameter space. After the fitting procedure has converged
for a relative change less than 0.5%, parameter errors are evaluated in one
dimension, changing each parameter until the $\chi^{2}$ variation corresponds
to the 60% confidence limit. Parameter correlations are evaluated only for
effective temperature and surface gravity. If the procedure finds a better
solution during error calculations, it returns to the descent part using the
previous solution as the initial model.

Instead of using TLUSTY (plane-parallel model, no wind), we used CMFGEN since
these stars have non-negligible stellar winds with mass-loss rates in the
order of $10^{-8}-10^{-7}$ $M_{\odot}$ yr-1. Using TLUSTY, instead of CMFGEN,
would prevent us to address the wind variability and perform homogeneous
modeling for BSGs. In addition, beyond short-term photometric variability,
typical for pulsating stars and discussed in Section 4, gradual spectral
variations may occur due to inhomogeneities in the wind density structure.
These together require re-evaluating the surface and wind parameters for each
observation. Therefore, we decided to proceed with CMFGEN models and start out
from the results of (Haucke et al., 2018). We updated XTgrid to apply CMFGEN
and minimize the wind properties along with the stellar surface parameters.
Then, we performed a new analysis of the most recent CASLEO spectra to measure
the CNO abundances in each of the three stars.

The focus of our analysis was on the CNO abundances. Therefore, we kept the
abundances of all elements heavier than O at their solar values (Asplund et
al., 2009) and we adopted He abundances $n{\rm He}/n{\rm H}=$ 0.2, based on
the analysis of (Searle et al., 2008). To maintain an approximate consistency
with the results of (Haucke et al., 2018) we kept the stellar radii and
turbulent velocity fixed at the values determined in the previous analysis
(see Table 1) and we applied unclumped wind models. Adopting clumping in our
models would result in different mass-loss rates. We neglected macroturbulence
in the current analysis because it shows a degeneracy with the projected
rotation at low spectral resolutions. We note that our goal was to measure the
CNO abundances and not to make a comparison with the analysis of (Haucke et
al., 2018). Our method is not suitable for such a comparison, as we used
different model atmosphere codes, our observations were taken at different
epochs and we used a different fitting procedure.

Finally, our best-fit CMFGEN models to the CASLEO spectra of HD 42087, HD
52089, and HD 58350, are shown, respectively, in Figs. 9, 10 and 11, and the
derived parameters are listed in Table 6. The luminosity and mass in this
table were derived using $L_{\star}=4\pi R_{\star}^{2}\sigma T_{\rm eff}^{4}$
and $g=GM_{\star}/{R_{\star}}^{2}$.

Figure 9: Best-fit XTgrid/CMFGEN model for HD 42087. In each panel, the CASLEO observation is in grey, the CMFGEN model in black, and the residuals, shifted by +1.1 for clarity, is in red.  Figure 10: Best-fit XTgrid/CMFGEN model for HD 52089. In each panel, the CASLEO observation is in grey, the CMFGEN model in black, and the residuals, shifted by +1.1 for clarity, is in red.  Figure 11: Best-fit XTgrid/CMFGEN model for HD 58350. In each panel, the CASLEO observation is in grey, the CMFGEN model in black, and the residuals, shifted by +1.1 for clarity, is in red.  Table 6: Summary of the spectroscopic results from unclumped CMFGEN models using the XTgrid fitting procedure. Abundances are reported on the 12 scale: $\epsilon=\log{N_{\rm x}/N_{\rm H}}+12$. Fixed parameters are marked with an "x" prefix. Metals not listed here were included at their solar metallicity from Asplund et al. (2009). Parameter | HD 42087 | HD 52089 | HD 58350
---|---|---|---
$T_{\rm eff}$ (K) | 18400${}^{+1000}_{-\ 200}$ | 23800${}^{+3900}_{-1400}$ | 15800${}^{+100}_{-400}$
$\log{g}\ ({\rm cm\ s}^{-2})$ | 2.34${}^{+0.01}_{-0.17}$ | 3.40${}^{+0.01}_{-0.60}$ | 1.95${}^{+0.02}_{-0.03}$
$v\sin{i}\ ({\rm km\ s}^{-1})$ | 73.4 $\pm$ 8.0 | 38.4 $\pm$ 5.0 | 51.5 $\pm$ 5.0
$v_{\rm turb}\ ({\rm km\ s}^{-1})$ | x10 | x10 | x12
$\dot{M}$ ($M_{\odot}$ yr-1) | (2.3$\pm$1.0)$\times 10^{-7}$ | (1.9 $\pm$ 0.2)$\times 10^{-8}$ | (6.2 $\pm$ 2.0)$\times 10^{-8}$
$v_{\infty}\ ({\rm km\ s}^{-1})$ | x700 | x900 | x230
$\beta$ | x2 | x1 | x3
$L_{\star}$ ($L_{\odot}$) | 312700${}^{+74000}_{-13000}$ | 35000${}^{+29200}_{-\ 7500}$ | 163800${}^{+\ 4200}_{-15900}$
$M_{\star}$ ($M_{\odot}$) | 24.3 | 11.1 | 9.5
$R_{\star}$ ($R_{\odot}$) | x55 | x11 | x54
$\log{L_{\star}/M_{\star}}$ | 4.1 | 3.5 | 4.2
Mean atomic | 1.4490 | 1.5097 | 1.5095
mass (a.m.u.) |  |  |  |  |  |
Distance ($pc$) | 2470${}^{+420}_{-290}$ | 124$\pm 2$ | 608${}^{+148}_{-148}$
E($B-V$) ($mag$) | 0.4 | 0.005 | 0.03
Element |  $\epsilon$ | mass fr. |  $\epsilon$ | mass fr. |  $\epsilon$ | mass fr.
Hydrogen |  12 | 5.89$\times 10^{-1}$ |  12 | 5.52$\times 10^{-1}$ |  12 | 5.52$\times 10^{-1}$
Helium |  x11.23$\pm 0.10$ | 4.01$\times 10^{-1}$ |  x11.30$\pm 0.17$ | 4.41$\times 10^{-1}$ |  x11.31$\pm 0.12$ | 4.41$\times 10^{-1}$
Carbon |  8.31$\pm 0.08$ | 1.37$\times 10^{-3}$ |  8.19$\pm 0.15$ | 1.04$\times 10^{-3}$ |  8.07$\pm 0.08$ | 7.75$\times 10^{-4}$
Nitrogen |  8.12$\pm 0.06$ | 1.09$\times 10^{-3}$ |  7.97$\pm 0.06$ | 7.25$\times 10^{-4}$ |  8.21$\pm 0.12$ | 1.25$\times 10^{-3}$
Oxygen |  8.60$\pm 0.08$ | 3.75$\times 10^{-3}$ |  8.30$\pm 0.13$ | 1.78$\times 10^{-3}$ |  8.19$\pm 0.09$ | 1.38$\times 10^{-3}$
|  [N/C] | [N/O] |  [N/C] | [N/O] |  [N/C] | [N/O]
Abundance ratios |  0.41 | 0.38 |  0.38 | 0.53 |  0.74 | 0.88

##  6 Discussion

We used the evolutionary sequences from Ekström et al. (2012) to explore the
evolutionary stage of our star sample. The main physical ingredients of these
sequences relevant to our analysis include initial abundances of H, He, and
metals set to X = 0.720,Y = 0.266, and Z = 0.014 with chemical initial
abundances of C = 2.283$\times 10^{-3}$, N = 6.588$\times 10^{-4}$, O =
5.718$\times 10^{-3}$, in mass fraction. We considered evolutionary tracks
with differential rotation at two different rates $\Omega/\Omega_{crit}$ =
0.568, and $\Omega/\Omega_{crit}$ = 0.4, employing for the latter an
interpolation of the models with $\Omega/\Omega_{crit}$ = 0.568 and 0,
provided in Ekström et al. (2012). The mass loss recipes employed in these
sequences are those of Vink et al. (2001) for initial masses above 7
$M_{\odot}$. For initial masses above 15 $M_{\odot}$ and $\log(T_{\rm eff})$ >
3.7, de Jager et al. (1988) recipe was adopted and the correction factor for
the radiative mass-loss rate from Maeder and Meynet (2000) was implemented in
these rotating models (see, Eq. 10 from Ekström et al. (2012)). Detailed
descriptions on the microphysics and mass loss recipes employed in these
evolutionary sequencies can be found in Ekström et al. (2012) and Yusof et al.
(2022). Our selected stars with the derived $T_{\rm eff}$, $\log L_{\star}$
and mass, along with the evolutionary tracks, are depicted in Fig. 12. The
errors in this Figure correspond to an error of 10% in the radii (Haucke et
al., 2018) which in turn result in a 21% error in the mass and luminosity.

Figure 12: Left panels: HR diagram (top) and mass evolution (bottom) showing
the position of HD 42087 and evolutionary tracks with initial masses of 24,
26, and 28 $M_{\odot}$. Middle panels: The same as in the left panels for HD
52087 and initial masses of 10, 12, and 14 $M_{\odot}$. Right panels: the same
as before for HD 58350 and initial masses of 20 and 22 $M_{\odot}$. Solid and
dashed lines represent models with $\Omega/\Omega_{crit}=$ 0.568 from (Ekström
et al., 2012) and the interpolated ones for $\Omega/\Omega_{crit}=$ 0.4,
respectively.

Below, we discuss our results for each star in terms of their location in the
HR diagram, their oscillations and the mass-loss rate found and next we
compare our results on their surface abundances with previous studies.

###  6.1 HD 42087

In the left panel of Figure 12 we show the new position of HD 42087 in an HR
diagram along with evolutionary tracks corresponding to initial masses of 24,
26, and 28 $M_{\odot}$. Continuous and dashed lines indicate rotation
velocities of $\Omega/\Omega_{crit}$ = 0.568 and $\Omega/\Omega_{crit}$ = 0.4,
respectively. In the lower panels, we include a diagram showing the total mass
evolution. The derived values for $T_{\rm eff}$ and
$\log(L_{\star}/L_{\odot})$ suggest an initial mass of $\sim$26 $M_{\odot}$
and our value obtained for the current mass indicate this star is at the pre-
RSG stage, which is in agreement with the mass evolution diagram for stars
with the mentioned initial mass.

Our analysis for this object resulted in a higher $T_{\rm eff}$ and lower
$\log{g}$ than those from Searle et al. (2008) and Haucke et al. (2018). We
emphasize that Haucke et al. (2018) did not have the Si lines covered by their
spectra and in principle, our new values would be more reliable. Additionally,
with our values, HD 42087 lies in the linear relation ($\log T_{\rm
eff}-\log{g}$) found in Searle et al. (2008) for Galactic BSGs (their Fig. 6).
In Fig. 13, we show the theoretical SED using the D = $2470^{+420}_{-289}$ pc
Gaia EDR3 distance and $E(B-V)$ = 0.4 mag extinction adopted from the Stilism
maps (Capitanio et al., 2017), along with the binned IUE spectrum in black.
Our procedure uses the fit formula from Cardelli et al. (1989) based on the
extinction data from Fitzpatrick and Massa (1986, 1988) with Rv=3.1. The
theoretical SED fits the photometric measurements, including the IUE spectrum,
and the resulting SED and CMFGEN model masses agree within error bars , being
22.5 M⊙ the mass derived from the SED. However, we were unable to match the
H$\alpha$ and H$\beta$ profiles with our homogeneous wind models. Considering
the reported line variability and asymmetries by Morel et al. (2004), which is
obvious also from Fig. 2, we conclude that our steady, smooth wind model is
inadequate for the 2020 CASLEO spectrum of HD 42087.

Figure 13: SED of HD 42087. All data points were taken from the VizieR
Photometry Viewer service. The photometric data were de-reddened using
$E(B-V)$ = 0.4 mag. The green points were used to match the slope of the
passband convolved CMFGEN fluxes to the observations and the model was
normalized to the observed SED in the 2MASS/J band. The binned IUE spectrum is
included with black dots.

Nevertheless, we were able to derive a mass-loss rate of $\dot{M}$ =
$2.3\times 10^{-7}$ $M_{\odot}$ yr-1 for this star with our procedure. This
value is close to the value found by Haucke et al. (2018) of $\dot{M}$ =
$5.7\times 10^{-7}$ $M_{\odot}$ yr-1, being lower by a factor of $\sim$2.6.
From looking at their observed H$\alpha$ line profile (taken in 2006, Fig.2),
we see a stronger emission component in H$\alpha$ compared to our data (2020).
This is in line with a higher mass-loss rate (in 2006) as found by these
authors.

The frequency analysis performed for HD 42087 shows three frequencies,
$f_{1}$, $f_{2}$ and $f_{3}$ which appear randomly over the three sectors and
two more detectable in the long time span of the three consecutive sectors
($f_{4}$ and $f_{5}$). The short lifetime for these variations prevents us
from associating them with stellar pulsations but rather with stochastic
variations at the stellar surface (Krtička and Feldmeier, 2021). Additionally,
the analysis over the combined sectors allowed us to find one stable
frequency, $f_{r}$, possibly associated with stellar pulsations. The nature of
this mode is not certain. Its frequency lies in the usual range ([10, 100] d)
of strange modes, known to facilitate the mass loss in massive stars (Aerts et
al., 2010). These modes can be radial and non-radial (Saio, 2011) and they
appear trapped in the strongly inflated envelopes of highly non-adiabatic
stars, usually with $\log L_{\star}/M_{\star}>4$. The low amplitude of $f_{r}$
and the new value found for $\log L_{\star}/M_{\star}=4.1$, indicate that this
can be a non-radial strange mode. If true, it would explain the high
variability in the wind lines (Fig. 2) and why the spectrum can not be
modelled with a smooth wind approach. Nevertheless, the only way to determine
if this mode facilitates the mass loss in HD 42087 is to perform a nonlinear
stability analysis and to check whether the mode velocity on the surface can
exceed the stellar escape velocity. We did not detect any frequency
corresponding to the $\sim$25 d period observed for the variability in the
H${\alpha}$ line. We stress here that the detection of any periodicity from
the H$\alpha$ profile in the light curves is unlikely to be observed. Any
perturbation on the stellar surface produced in the large wind volume where
the H$\alpha$ line is formed may lead to new perturbations (for example due to
densities inhomogeneities in the wind), difficulting its detection.
Additionally, the H$\alpha$ profile, in most cases is composed of absorption
and emission components and the product of both components in the integrated
flux also difficult its detection. However, as shown in Krtička and Feldmeier
(2021), wind variations can cause stochastic light variations if the base
perturbations are sufficiently large. The lack of multiple independent
frequencies identified as stellar pulsations modes in this object would
indicate that this star belongs to the pre-RSG stage, considering that massive
post-RSG have in general more excited modes than pre-RSG, as shown in Saio et
al. (2013). This is in agreement with our values for $T_{\rm eff}$, $\log g$,
$M_{\star}$ (Fig. 12).

###  6.2 HD 52089

The new position of HD 52089 in the HR diagram along with evolutionary tracks
for initial masses 10, 12 and 14 $M_{\odot}$ and $\Omega/\Omega_{crit}$ =
0.568 and 0.4 (continuous and dashed lines, respectively) are depicted in Fig.
12. The HR diagram suggests initial masses between 12 and 14 M⊙ at the TAMS in
agreement with our derived value for the current mass ($\sim 11M_{\odot}$) in
this stage considering the adopted errors in the mass. However, we can not
dismiss a merger scenario for this object that would explain the measured
luminosity, which is slightly high considering an 11 $M_{\odot}$ object at the
TAMS. Such an event can lead to rejuvenation and an overluminosity of the
merger remnant due to the energy injection from the secondary during the
merging of the two components. The merger product might hence appear as if it
would have a significantly higher mass. The best known such case is the B[e]
supergiant star R4 in the Small Magellanic Cloud (Zickgraf et al., 1996; Wu et
al., 2020). If true, then HD 52089 would be a highly important object to study
merger remnants, and it would be interesting to search for possible remnants
of ejecta from the past merger event.

Additionally, this star showed an inconsistency in the $T_{\rm eff}-\log{g}$
distribution of Haucke et al. (2018), having a higher $\log{g}$ for its
temperature than other stars in their sample. Our new fit to the 2015 CASLEO
spectrum confirmed the earlier results and shows a discrepancy among the
stellar mass, luminosity, and surface gravity, given its high effective
temperature. With its close distance of D = $124\pm 2$ pc and moderate
interstellar extinction of $E(B-V)$ = $0.005$ mag, adopted from (van Leeuwen,
2007) and Stilism, respectively, we find that its spectroscopic mass is in
good agreement with the mass derived from the SED (9.8M⊙) depicted in Fig. 14.
We also notice the model from the optical fit matches the slope of the binned
IUE spectrum, but there is an offset possibly due to the low metalicity found
for this object.

Figure 14: SED of HD 52089. Same as Fig. 13, but de-reddening was done using
$E(B-V)$= 0.005 mag.

The frequency analysis for HD 52089 over the 4 TESS sectors revealed one
triplet centred in $f_{2}$ in Sector 33, corresponding to a rotational period
of 5 d ($f_{rot}$ = 0.2 d-1). The short baseline observations provided by TESS
single sectors translate into high Raylight frequency separation, preventing
making precise mode identification for short-term variabilities. In the case
of $f_{3}$, $f_{2}$ and $f_{4}$ being members of a triplet, $f_{2}$ would be a
a $\ell$ = 1 nonradial mode Saio et al. (2013). In this case, the star would
have a surface rotation velocity of $\sim$110 km s-1, considering the radius
and rotation period derived for this object. Moreover, with the value of v
$\sin i$ obtained from our analysis, we can derive the inclination of the
star, which would be $\sim$20 degree. Besides the triplet, we found $f_{2}$ in
Sectors 7 and 33, and signatures of this mode in Sectors 6 and 34 with very
low amplitude (not included in Table 4) along with its harmonic supporting the
hypothesis of $f_{2}$ being a non-radial mode due to its apparently extended
lifetime. As in the case of HD 42087, we found low-frequency signals ($f_{1}$,
$f_{5}$ and $f_{6}$) randomly excited over the observed sectors, which are
probably related to convective variabilities stochastically excited at the
stellar surface. These observations confirm the results of Burssens et al.
(2020).

Our determination of $\dot{M}$ for HD 52089, $\dot{M}$ = $1.9\times 10^{-8}$
$M_{\odot}$ yr-1, agrees very well with the one reported by Haucke et al.
(2018): $\dot{M}$ = $2.0\times 10^{-8}$ $M_{\odot}$. From comparing our
observed H$\alpha$ line profile of this star, we do not find any significant
morphological difference between the observed spectrum shown in Haucke et al.
(2018) and our data: a pure absorption H$\alpha$ line profile with flux at the
core of the line of $\sim$0.6 (normalized flux) (see Fig. 3).

###  6.3 HD 58350

The right panel of Fig. 12 shows the position of HD 58350 in the HR diagram
and the evolution of the total mass along with evolutionary tracks for 20 and
22 $M_{\odot}$ with different rotational velocities. We found an excellent
agreement between our derived values for $T_{\rm eff}$,
$\log(L_{\star}/L_{\odot})$ and its current mass for a star evolution model
with initial mass $\sim$22 M⊙ at the post- RSG stage, after losing a
considerable amount of mass during its evolution. We notice as well, that our
derived values for the $T_{\rm eff}$ and $\log g$ are in good agreement with
the linear relation found for galactic BSGs in Searle et al. (2008).

The theoretical SED fits the optical photometric measurements for HD 58350
(Fig. 15). We notice a slightly different slope for the IR photometric data,
possibly pointing towards a time-variable wind in data taken at different
dates. Considering its distance $D=$ 608 $\pm$ 148 pc (Haucke et al., 2018),
we derived an extinction $E(B-V)$ = 0.18 mag. We also found a discrepancy
between the stellar masses obtained from the SED modeling ($\sim$5.5
$M_{\odot}$) and the best-fitting CMFGEN model ($\sim$9.5 $M_{\odot}$) of
unclear origin. We have found different values for the effective temperature
(and extinction) in the literature (see Sec. 2) for this object, ranging from
13500 K to 16000 K, possibly due to a combination between the use of different
methodologies to derive it and due to stellar oscillations. This hampers a
reliable comparison between our SED model and the photometric data, possibly
leading to this mass discrepancy. Spectroscopic time-series observations
analyzed homogeneously can help to place reliable constraints on the effective
temperature.

Figure 15: SED of HD 58350. Same as Fig. 13, but de-reddening was done using
$E(B-V)$ = 0.18 mag (see main text).

For HD 58350, our derived $\dot{M}$ of $6.2\times 10^{-8}$ $M_{\odot}$ yr-1 is
lower than the one from Haucke et al. (2018) by a factor of $\sim$2.4. This
can be understood since the observed H$\alpha$ line profile reported in Haucke
et al. (2018) (taken in 2006 and 2013) shows a more intense emission component
in comparison with our data (very weak emission in H$\alpha$).

The frequency content for this star lies below 0.4 d -1. The only sector
provided by TESS at the current time allowed us to find two independent
frequencies. We were not able to find the frequency discovered in Lefever et
al. (2007), with $f$ = 0.1507 d-1, indicating that this mode is not an
opacity-driven mode. From the analysis of one single TESS sector, we can not
suggest this star is a non-radial oscillator. However, the marked H$\alpha$
line profile variation (see Fig. 4) suggests these frequencies are connected
to line-driven wind instabilities (Krtička and Feldmeier, 2021). The analysis
in (Saio et al., 2013) demonstrated that a BSG in a post-RSG state should
undergo multiple pulsations. It is therefore unfortunate that HD 58350, which
is the best candidate in our sample for a post-RSG star, was observed only in
one TESS sector. Only with multiple, and in particular consecutive TESS
observations, would it be possible to properly analyze the frequency spectrum
of this object and to confirm the theoretical predictions of Saio et al.
(2013).

###  6.4 Surface abundances

With the aim of framing these discrepancies with our derived values for the
surface abundances, we compared our results with those from different authors
and Geneva evolutionary tracks. Fig. 16 shows our measured CNO abundances for
the selected stars along with the solar abundances and the average CNO
abundances derived in (Searle et al., 2008) for their Galactic BSG sample.

In general, for all three stars, we found well-constrained CNO abundances, all
showing a slight depletion of C and O, and a mild overabundance of N when
compared to the solar mixture. This pattern agrees with the previously
reported CNO abundance profiles in (Searle et al., 2008); however, at the same
time we found higher C and slightly lower O abundances. We notice that some
systematic differences are expected from the analysis itself. The global
spectral modeling applied in XTgrid is fundamentally different from the
methodology of Searle et al. (2008) who used different diagnostic lines for
$T_{\rm eff}$, $\log{g}$ and abundance determinations. Searle et al. (2008)
noted that their CMFGEN models, with derived C abundances from the C ii 4267 Å
line, overestimated the C ii 6578 and 6582 Å lines. In contrast, our C
abundance analysis was based on the strongest C features in the CASLEO
observations (C ii 6578 and 6582 Å lines), but the global analysis is also
sensitive to variations in all other spectral lines due to a change in the C
abundance. This difference in diagnostics is applied consistently for all
surface and wind parameters in XTgrid. Furthermore, Nieva and Przybilla (2006)
showed that LTE analyses could result in discrepant abundances based on
different carbon lines, and the C ii 4267 multiplet tends to underestimate the
carbon abundance. Further possible sources of a discrepancy may be the
differences in atomic data used in the analyses and the different fit
procedures. Additionally, we note that our sample of three stars is too small
to match (Searle et al., 2008) population averages and the individual objects
in our selection may show large deviations. To uncover such systematics one
needs to process larger, homogeneously modeled datasets and multi-epoch
observations, which are beyond the scope and limitations of our current work.

Figure 16: The measured CNO abundance patterns in all three stars compared to
the solar pattern from Asplund et al. (2009). In all three stars, C and O are
depleted and N is overabundant compared to the solar mixture. The mean CNO
abundances found by Searle et al. (2008) for Galactic BSGs are also shown for
reference.

At the same time, we noticed that the total amount of metals we derived (see
Table 6) correspond to subsolar metallicities, being approximately $70\%\
Z_{\odot}$, $60\%\ Z_{\odot}$, and $50\%\ Z_{\odot}$ for HD 42087, HD 52089,
and HD 58350 respectively, possibly due to the adopted fixed value for $n{\rm
He}/n{\rm H}=0.2$. We have also adopted fixed solar abundances for all metals
beyond oxygen and did not include elements that do not contribute to the
opacity, such as Ar, K, Ti, V, Mn, and Co. All these features contribute to
reducing the metal mass fractions.

In Fig. 17 we compare the N/C and N/O abundance ratios from different star
samples with Geneva stellar evolution models for different masses. We included
the sample of A and B supergiants studied in (Searle et al., 2008) and (Georgy
et al., 2021) and the B-type main sequence (MS) stars analyzed in (Lyubimkov
et al., 2013) as a reference, along with our three stars. Three well-defined
groups can be identified in the panels of Fig. 17. B-type MS stars between 5
and 10 $M_{\odot}$ form a clear sequence in temperature and show the lowest
abundance ratios, which marks that these stars have low N abundances relative
to C and O. Below ~13000 K, cooler A-type supergiant stars show up as a
compact group, while the hot side of the observed BSG abundance ratios shows a
much larger scatter. The larger scatter can be interpreted in several ways.
Krtička et al. (2021) have shown that winds are driven mostly by C, Si, and S
for hot BSGs and iron for cooler BSGs implying a decreasing mass-loss rate for
temperatures lower than 15000 K. A lower mass-loss rate might operate in favor
of a better determination of N/C and N/C ratios. The observed compact group
could be as well a signature of the fast post-RSG evolution through the cool
BSG stage. Additionally, atmospheres of cool supergiants usually exhibit more
lines, helping to obtain precise values in their abundances. However, it is
clear that none of the measurements reaches the predicted high N/C and N/O
ratios for the post-RSG domain.

Based on the abundance ratios alone, we find that all our stars are more
consistent with a pre-RSG stage of evolution, we do not see the predicted very
high N/C and N/O ratios. This is in contrast with the spectroscopic mass of HD
58350, which suggests it is in a post-RSG stage. Saio et al. (2013) showed the
N/C and N/O ratios are increased mainly by the mass loss. However, they found
N/C and N/O abundances consistent with models at the pre-RSG stage, in
contradiction with the position in the HR diagram for Deneb and Rigel,
arriving at the same conclusion.

There is a discrepancy in the abundance ratios between evolutionary model
predictions and spectroscopic measurements. The observed ratios remain
significantly lower than predictions. Much larger samples, multifaceted
efforts, and homogeneous modeling will be necessary to address this issue
statistically.

Fig. 18 shows the correlations between the N/O and N/C ratios, which is
analogous to the distribution (Martins et al., 2015) found for O-type stars.

The offset between theoretical predictions and the measurements implies that
some systematics may exist in the C and O abundances. A systematically
underestimated C or overestimated O abundance can produce the observed offset.
It is unlikely that all the methodologies used to analyze the surface
abundances in these stars are biased in the same way, therefore, the origin of
the offset remains unclear. It may be related to stellar variability as well
as to atomic data or shortcomings in the surface abundance predictions.
Martins et al. (2015) demonstrated an anticorrelation between the N/C ratio
and $\log{g}$, the lower the gravity the larger the N/C ratio. Meanwhile, Saio
et al. (2013) concluded that recent developments in modeling of RSG (Davies et
al., 2013) make these stars more compact for a given luminosity. The
combination of the two trends acts towards decreasing the offset with respect
to the Geneva models in Fig. 18.

Figure 17: N/C ($bottom$) and N/O ($top$) abundance ratios by number from
Searle et al. (2008) and Georgy et al. (2021) compared to our measurements and
Geneva solar metallicity (Z = 0.014), $\Omega/\Omega_{crit}=$ 0.4 interpolated
evolutionary tracks for 5, 10, 20, 23 and 26 $M_{\odot}$. For reference, we
also show B-type main sequence stars from Lyubimkov et al. (2013).  Figure 18:
N/C and N/O abundance ratio correlations based on the same datasets as in Fig.
17.

##  7 Conclusions

This paper reports our first step towards a comprehensive study of BSG stars,
taking into account their photometric and spectroscopic variabilities.

The evolutionary tracks predict HD 42087 to be a pre-RSG star for our derived
values of $\log(L_{\star}/L_{\odot})$, $T_{\rm eff}$, and $M_{\star}$, in
agreement with our values for the abundance ratios. HD 52089 is most likely an
11 $M_{\odot}$ star at the TAMS. However, we noticed that the derived
luminosity is slightly high for an 11 $M_{\odot}$ star, and a binary merger
scenario is plausible. Finally, for HD 58350, the evolutionary tracks for our
derived values indicate this star is at the post-RSG. However, the derived
surface abundance ratios are compatible with those at the pre-RSG stage,
finding the same discrepancies mentioned in (Saio et al., 2013) for Deneb and
Rigel. The rather short sectoral observing windows of TESS are insufficient to
cover the low frequencies usually present in these stars. However, we were
able to detect a frequency splitting allowing us to infer a rotational period
of $\sim$5 d for HD 52089 and to find a low frequency, $f_{r}$ = 0.093
$d^{-1}$, related possibly to strange mode instabilities in HD 42089 in
agreement with its new derived value for the $\log(L_{\star}/L_{\odot})$.

We found lower values for mass-loss rates of HD 42097 and HD 58350 compared
with those derived by Haucke et al. (2018), in agreement with the detected
variability of the H$\alpha$ emission component, related to changes in the
wind mass-loss rate in these stars at these different epochs. The finding of
low-frequencies stochastic oscillations in these objects is possibly connected
with such variations.

Despite the numerous and valuable efforts to study BSG stars, there are yet
many issues to address and we highlight here the most important ones:

  * •

Large sample of BSGs needs to be studied with homogenously modeled datasets of
multi-epoch observations, with the aim of uncovering systematic deviations of
surface abundance ratios from evolutionary models. Additionally, multi-epoch
observations will allow us to place constraints to the current uncertainties
observed in the radii of these objects and to identify changes in radii caused
by radial pulsations.

  * •

To frame the current studies for the needed Mcore/Mr ratio for massive stars
to evolve back towards the blue region of the HR diagram, such as the effect
of stellar rotation, convective boundaries criteria, mixing length theories,
overshooting adopted for the stellar interior and evolutionary models, in
terms of the CNO abundances. This will allow us to untackle the observed
indetermination in the evolutionary stage of these objects with precise values
for their CNO surface abundances and, in turn, will help to set the needed
constraints to the current poorly established theoretical mass-loss recipes in
the diverse evolutionary states and mass ranges.

  * •

In addition, we emphasize that stellar pulsations play a key role in the
analysis of BSGs, not only as a test to infer their evolutionary stage as
proposed in (Saio et al., 2013), but also as a mechanism that facilitates the
mass loss in massive stars as suggested in (Kraus et al., 2015) and
theoretically confirmed in (Yadav and Glatzel, 2016), affecting, therefore,
their surface abundances. The systematic differences noticed when comparing
evolutionary tracks with surface abundance measurements for BSGs (Fig. 18)
should be discussed, considering the effect of stellar pulsations over their
evolution. Furthermore, short-term mass-loss variabilities should be
contemplated in detailed evolutionary sequences as they can act as an
additional source for the discrepancies found with evolutionary models.

From a spectral analysis software development point, we have added CMFGEN
modeling capabilities to the automatic spectral analysis procedure
XTgrid222https://xtgrid.astroserver.org/. The first results shown here
demonstrate its feasibility for processing spectra of massive stars and
deriving homogeneous parameters from diverse data. The next steps will include
improving the accuracy of spectral inference to reduce the observed
discrepancies in mass and surface abundances. Additionally, we will work on
better optimizing the calculations by recycling previously calculated models.
Future applicability for large datasets relies on the development of efficient
methods to search the parameter space either by utilizing large grids Zsargó
et al. (2020) and/or machine learning techniques.

We would like to mention as well, that we found diffuse interstellar bands
(DIB) in the spectra of HD 42087 and HD 58350. The 4428 and 6613 Å DIB bands
are clearly present in both stars. Neither of the two bands is visible in the
spectrum of the relatively nearby HD 52089. Although the two stars with DIBs
are at larger distances, and they have very different extinction values, the
DIBs show very similar strengths. The lack of DIBs in HD 52089 is likely due
to its short distance and the very low interstellar extinction in its
direction. In addition, HD 52089 is not only the hottest star in our sample,
but also the strongest ultraviolet source in the night sky (Gregorio et al.,
2002), which may be able to photodissociate DIB carriers.

“Conceptualization, J.P.S.A, P.N. and M.K; methodology, J.P.S.A. and P.N.,
E.S.G.A; software, P.N.; validation, P.N. and E.S.G.A; formal analysis, P.N.;
investigation, J.P.S.A, P.N., E.S.G.A, M.K. and M.A.R.D.; resources, M.A.R.D
and M.H.; writing—original draft preparation, J.P.S.A., P.N., M.A.R.D., M.K.
and E.S.G.A.; writing—review and editing, J.P.S.A., P.N., E.S.G.A and M.K.;
visualization, J.P.S.A, P.N., M.K, and M.A.R.D; supervision, J.P.S.A.; project
administration, J.P.S.A and P.N. All authors have read and agreed to the
published version of the manuscript."

This research received funding from the European Union’s Framework Programme
for Research and Innovation Horizon 2020 (2014-2020) under the Marie
Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical
Institute in Ondřejov is supported by the project RVO:67985815. J.P.S.A and
M.K. acknowledge financial support from the Czech Science foundation (GACR
20-00150S). E. S. G. de Almeida has been financially supported by ANID
Fondecyt postdoctoral grant folio № 3220776. P.N. acknowledges support from
the Grant Agency of the Czech Republic (GAČR 22-34467S). M.A. Ruiz Diaz
acknowledges support from CONICET (PIP 1337)

###### Acknowledgements.

We are grateful to the referees for their thoughtful reports and useful
suggestions that helped us improve the manuscript. We thank Dr. Cidale for the
inspiring and valuable discussions. Based on observations taken with the J.
Sahade Telescope at Complejo Astronómico El Leoncito (CASLEO), operated under
an agreement between the Consejo Nacional de Investigaciones Científicas y
Técnicas de la República Argentina, the Secretaría de Ciencia y Tecnología de
la Nación, and the National Universities of La Plata, Córdoba, and San Juan.
This research has used the services of www.Astroserver.org under reference
M1DE05. no References

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