Photometric properties and stellar parameters of the rapidly rotating magnetic early-B star HD 345439

The multicolor photometric measurements of HD 345439 were performed using the Nanshan One-meter Wide-field Telescope (NOWT; Bai et al. 2020; Shen et al. 2021) of the Xinjiang Astronomical Observatory (XAO), Chinese Academy of Sciences (CAS). NOWT is an alt-azimuth mount reflector telescope at the Nanshan in Xinjiang. NOWT is equipped with a 4096$\times$4136-pixel CCD and a Johnson-Cousins $UBVRI$ filter system. The CCD provides a field of view (FoV) of $1.3\times 1.3$ degree², and the per-pixel resolution is 1.28^′′. The photometric observations were performed over a period of 12 nights from October 13 to November 12, 2020, and 2457 frames of HD 345439 were obtained. During the observations, when a single exposure of one band was finished, the filter wheel rotated to the next band. The FoV of the CCD readout was set as $42.67\times 42.67$ arcmin² to ensure the efficiency of the observations, and the scan rate mode was set to medium. The observations were supplemented with a series of 10 bias and 3 flat-field exposures per filter every night to calibrate the CCD instrumental signature. The observing log is summarized in Table 1.

The data reductions, including bias subtraction, flat-fielding, and illumination correction, were carried out using the CCDPROC task of the Image Reduction and Analysis Facility (IRAF¹¹1IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.) software. Part of one observation image is shown in Fig. 1, where HD 345439, the comparison star, and the check star are indicated. Differential photometry was carried out by using the standard aperture photometric package of the IRAF APPHOT task. The photometric error distribution is shown in Fig. 2. Although the photometric error in the $U$ band is considerably higher than that in the other bands, the overall multicolor photometry process is stable.

The archival photometric data of HD 345439 employed in this work were obtained from the All-Sky Automated Survey for Supernovae (ASAS-SN; Kochanek et al. 2017; Jayasinghe et al. 2018; Pawlak et al. 2019), the Super Wide Angle Search for Planets (SuperWASP; Butters et al. 2010), and the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015). The ASAS-SN was the first project to routinely survey the entire visible sky in the V band, reaching a depth of roughly 17 mag. As of the mid-2017, ASAS-SN consists of two observing stations, namely the Haleakala Observatory (Hawaii) and the Cerro Tololo International Observatory (CTIO, Chile) sites. By the end of 2017, ASAS-SN expanded to comprise five observing stations with the addition of a second unit at the CTIO and two more units at the McDonald Observatory (Texas) and the South African Astrophysical Observatory (SAAO, Sutherland, South Africa). Two observational datasets were obtained from the ASAS-SN Photometry Database²²2https://asas-sn.osu.edu/variables using different sources. In the first observational dataset, ASAS-SN observed HD 345439 using the APOGEE input catalog; 349 photometric data points were obtained from February 2015 to April 2018 using bc and bd cameras. In the second observational dataset, the observations were conducted using the ASAS-SN DR9 source and the bc camera and consisted of 192 epochs from February 2015 to October 2018. SuperWASP is an extra-solar planet detection program employing identical robotic telescopes. The telescopes are equipped with a 2048$\times$2048 thinned CCD with a pixel size of 13.5 $\mu$m. SuperWASP carried out observations of HD 345439 over 67 nights in 2007 and provided 979 points of photometric observations in a broadband filter (bandpass from 400 to 700 nm). TESS is a NASA Astrophysics Explorer mission. The satellite is equipped with four identical wide-field cameras, which together can monitor the sky. TESS conducted photometric observations of HD 345439 in Sector 41 with a 2-min cadence mode, providing the most complete light curve of the star. The photometric data of TESS was reduced using the lightkurve package (hereafter denoted as LK)³³3http://docs.lightkurve.org/.

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST; Cui et al. 2012) is a reflecting Schmidt telescope located at the Xinglong station of the National Astronomical Observatories, Chinese Academy of Sciences. The mean aperture of the LAMOST is 4.3 m, and the corresponding FoV is 5 degree in diameter (Luo et al. 2015). A Low-resolution spectrum of HD 345439 was obtained in the LAMSOT Low-Resolution Survey Data Release 7 (LAMOST-LRS DR7)⁴⁴4http://dr7.lamost.org/.

To fit the He I line profiles, Eikenberry et al. (2014) hypothesized that the rotational velocity $v$sin$i$ has a value of $270\pm 20$ km s^-1. Wisniewski et al. (2015) derived a rotational period of 0.7701 day from the KELT, SuperWASP, and ASAS surveys. In this work, we determine the rotational period of HD 345439 by analyzing the new archival data and new photometric observations. The Lomb-Scargle method (Lomb 1976; Scargle 1982) provided by Astropy (Astropy Collaboration et al. 2013, 2018) was used to detect the periodic signals in the unevenly spaced SuperWASP and ASAS-SN observations. The PERIOD04 software (Lenz & Breger 2005) was used to detect the periodic signature in both the TESS and NOWT photometric observations. The validity of the frequencies that detected in the TESS and NOWT photometric observations has followed the example of Breger et al. (1993). The period-power spectra and corresponding light curves of the archival data are shown in Fig. 3.

The Lomb–Scargle periodogram of ASAS-SN using the ASAS-SN DR9 source shows that the rotational period exhibits a distinct signature at $0.7701\pm 0.0002$ day. The value of $0.7702\pm 0.0001$ day was derived from the ASAS-SN observations using the APOGEE source with the bc camera. A consistent periodic signal with a value of $0.77017\pm 0.00013$ day appears also in the ASAS-SN measurements using the bd camera. The SuperWASP photometric data also reveals a rotational period of $0.77\pm 0.0042$ day. The TESS observations indicate a rotational period of around 0.77044 $\pm$ 0.00009 day, and this result is statistically consistent with previous results. We summarized the rotational periods obtained from the different datasets in Table 2.

We extract the rotational frequencies from the NOWT photometric data in each band, which are denoted as F0 in Table 3. The corresponding rotational periods of each band are listed in the fourth column of the table. The photometry data of each band exhibits a clear signature at around 0.7691 $\pm$ 0.0025 day; however, the signal-to-noise ratio (SNR) in the B band is low. The consistency of all observations indicates that the rotational period of HD 345439 is $0.7699\pm 0.0014$ day, which is consistent with the value of $0.7701\pm 0.0001$ derived by Wisniewski et al. (2015) and the value of $0.77018\pm 0.00002$ day derived by Hubrig et al. (2017a).

We utilized the PERIOD04 software (Lenz & Breger 2005) to analyze the TESS and NOWT photometric data for investigating the pulsating behavior of HD 345439. The Nyquist frequency of the TESS observations is $f\rm_{N}=360\rm\;d^{-1}$; thus, the frequencies in the range of $0<f<360\rm\;d^{-1}$ were removed from our analysis. We calculate the resolution frequency $f\rm_{res}=1/T\,$(T represents the duration of the observations) and use this information to distinguish between two contiguous frequencies. If the difference between two contiguous frequencies is less than the $f\rm_{res}$, we consider them to be unresolved. The rectified light curve is fitted using the following formula:

where $m\rm_{0}$ is the zero-point, $A_{i}$ is the amplitude, $f_{i}$ is the frequency, and $\phi_{i}$ is the corresponding phase. The multifrequency least-square fit of the light curve for all detected significant frequencies is then performed using Eq.1 to obtain the solution for all frequencies. The following search is conducted with the residual data that is obtained by subtracting the above solution from the rectification data. We adopt SNR $\geq 4$, which is the criterion suggested by Breger et al. (1993), to detect the significant frequencies.

We identify the significant frequencies in the multicolor NOWT photometry data; the Fourier amplitude spectrum of each band is shown in Fig. 4. All significant frequencies are obtained according to the $f\rm_{res}=0.09\rm\;d^{-1}$ and SNR$\geq 4$ restrictions, which are listed in Table3. We find an independent frequency in the $U$ band. However, this independent frequency is not detected in the other bands, which indicates that it may be an instrumental artifact. Regarding the $B$, $V$, $R$, and $I$ bands, only the harmonics of the fundamental rotational frequency are detected.

The Fourier amplitude spectra of the TESS photometric data are shown in Fig. 5. In addition to the rotational frequency, 39 significant frequencies are detected from the TESS photometric observations. Among these, 15 significant frequencies were removed after taking into account the $f\rm_{res}=0.04\rm\;d^{-1}$ restriction. The remaining frequencies are listed in Table 4; the corresponding SNR values, amplitudes, identifications, and periods are provided in columns 3–6, respectively. All the resolved significant frequencies are identified as the harmonics of the rotational frequency, and no evidence of pulsating behavior is found in HD 345439.

The light curves of all new archival observations are shown in Fig. 3. We calculate the median magnitude in bins of 0.02-phase width and overlay these median values on the initial phase diagram to aid in the visual interpretation of the SuperWASP observations. The light curves show an analogical variation with a feature of the S-wave, and the maximum amplitude$\;\sim 0.1\;$mag. The multicolor NOWT photometry is shown in Fig. 6. The profiles and amplitudes of the light curves are approximately consistent with the archival observations.

The shapes of the light curves of all photometric observations show a typical double S-wave feature. The light curve obtained from the TESS observations exhibits a pronounced asymmetry between the primary and secondary eclipses. The same asymmetric components are detected in the multicolor observations. An unequal spacing between the primary and secondary eclipses is visible in each band, and the secondary eclipse is delayed with respect to the primary one by a phase of $\sim$ 0.4. The presence of these unequal spacings constitutes a strict constraint for the shape and density profile of the magnetosphere clouds as well as a restriction on the magnetic obliquity angle $\beta$ and the rotational inclination angle $i$. On the other hand, the multicolor NOWT photometry light curves show a different trend in the phase 0.1 to 0.2, compared with that of other photometric observations, and the amplitude variation decreases with the increase in the central wavelength of the filter. This feature region is marked between two vertical red dashed lines in Fig. 6. The same trend does not appear in the other archival observations. Regarding the ASAS-SN and SuperWASP observations, it is the low time resolution that causes the absence of this amplitude variation. The bandpass of the TESS detector ranges from 600 to 1000 nm, and the central wavelength is 786.5 nm (Ricker et al. 2015). However, the variability of the multicolor NOWT photometry at the 0.15 phase is considerably reduced in the $I$ band; therefore, the amplitude variation is unobservable for the TESS detector. The amplitude variation of the multicolor light curves suggests that the material in the magnetic clouds may condense into larger particles, thereby causing an increase in the short-wavelength opacity.

Being the archetype of helium-rich stars with large-scale magnetic fields, $\sigma$ Ori E was conformed to have hard X-ray flares (e.g., Groote & Schmitt 2004; Sanz-Forcada et al. 2004). Townsend & Owocki (2005, hereafter referred to as TO05) speculated that the X-ray flares are associated with the centrifugal breakout and are universal in rapidly rotating magnetic early B star. Subsequently, ud-Doula et al. (2006, hereafter referred to as UD06) demonstrated the feasibility of centrifugal breakout hypothesis via numerical magnetohydrodynamic (MHD) simulations.

In this study, we explore the existence of centrifugal breakout of HD 345439 using the TESS photometric data. We use the functions in the LK package to prewhiten the TESS light curve for eliminating the instrumental variations, and the result is shown in the upper panel of Fig. 7. After subtracting the light curve with the periodic signal, the residual magnitudes are shown in the bottom panel of Fig. 7. As estimated by TO05 and UD06 works, the entire centrifugal breakout event would occur on a timescale of days. However, apart from the variation caused by the magnetosphere, the smoothed light curve shown in the bottom panel of Fig. 7 is relatively devoid of features. Indeed, the residual magnitudes do not vary significantly. Additionally, no systematic variations are observed at the extremum of the light curve. It can thus be inferred that no centrifugal breakout episodes occurred during the TESS photometric observations.

The luminosity is derived by using the bolometric correction ($BC$), and the extinction coefficient$\;A_{V}$ is estimated from the intrinsic color$\;(B-V)\rm_{0}$. The distance modulus$\;(DM)\;$is evaluated using the distance from Gaia DR2. The Gaia DR2 distances are corrected using a weak distance prior that varies smoothly as a function of Galactic longitude and latitude according to a Galaxy model (Bailer-Jones et al. 2018). The intrinsic color$\;(B-V)\rm_{0}$ is determined from the smooth interpolation of the grids provided by Lanz & Hubeny (2007). We use the relationship of$\;R(V)\equiv A(V)/E_{(B-V)}=3.1$ (Cardelli et al. 1989) to derive the extinction coefficient$\;A_{V}$. The solar values, including $BC_{V,\odot}=-0.19$ mag, $M_{bol,\odot}=4.60$ mag, $M_{V,\odot}=4.83$ mag, and $V_{\odot}=-26.74$ mag were taken from a previous work (Torres 2010). The absolute visual magnitude$\;(M_{V}=V-A_{V}-\rm DM$), the bolometric magnitude$\;(M_{bol}=M_{V}+BC)$, and the luminosity [$log(L/L_{\odot})=0.4\times(M_{bol,\odot}-M_{bol})$] are compiled in Table 5.

We download the grids of ATLAS9 model atmospheres (Castelli & Kurucz 2003) from the ATLAS websites ⁵⁵5https://wwwuser.oats.inaf.it/castelli/grids.html. The grids include the effective temperature ($T\rm_{eff}$) spanning a range of 15 000$\;\sim\;$35 000 K with a step of 1000 K, the log$\,{g}$ in the range of 2.50$\;\sim\;$5.00 dex with a step of 0.50 dex, and the metal abundance in the range of -4.00$\;\sim\;$5.00 dex. The SPECTRUM program (Gray & Corbally 1994) was adopted to synthesize the spectral templates. The unbroadened infinite-resolution spectra were transformed into low resolution ($R\sim 1800$)  spectra to be comparable with the LAMOST-LRS. To ensure the accuracy of analysis, we removed the spectral region with wavelength below 390 nm, which is affected by instrumental ripples, and truncated the spectrum at 700 nm. For determining the stellar atmospheric parameters, we matched the LAMOST-LRS spectrum with templates using a methodology similar to LSP3. The LSP3 adopts a cross-correlation algorithm to determine stellar radial velocity and uses the weighted means of parameters of the best-matching templates and values yielded by $\chi^{2}$ minimization (Xiang et al. 2015). The $\chi^{2}$ values calculated from the observing spectrum and matching template spectra are defined as:

where $O_{i}$ is flux densities of the observing spectrum of the $i$th, $T_{i}$ is flux densities of the template spectra of the $i$th. N is the total pixel number used to calculate the $\chi^{2}$ values, and $\sigma_{i}$ is the error of flux density of the observing spectrum of the $i$th pixel. As shown in Fig. 8, the best fitting result reproduces with the effective temperature at $22\pm 1$ kK, log$\,g=4.00\pm 0.22$ and $[Fe/H]=-0.50$ dex.

A grid of evolution tracks with a mass ranging from 4 M_⊙ to 10 M_⊙ with a step of 0.02 M_⊙ was generated using the Modules for Experiments in Stellar Astrophysics (MESA) (Paxton et al. 2011, 2013, 2015, 2018, 2019). The rotational effect of the star having $v/v\rm_{crit}$ = 0.75, a restriction which is imposed by photometric observations, was used in the grid. The luminosity, mass, and age of HD 345439 were estimated from its location on the Hertzsprung$-$Russell Diagram (HRD) with the evolutionary tracks. The positions of HD 345439 on the Kiel-Diagram (KD) and HRD are shown in Fig.9. The mass, luminosity, and age were derived as $\;M=8.00\,_{-0.51}^{+0.50}\rm\,M_{\odot}$, log$(L/L_{\odot})=3.82\pm 0.1$ dex, and $\tau\rm_{age}=26.79\,_{-6.79}^{+7.03}$ Myr, respectively. According to the Stefan–Boltzmann law, the radius was estimated to be $5.59\pm 0.06\rm\,R_{\odot}$.

Following a previous report (Nakajima 1985), the TO05 work developed an RRM model to interpret the interaction of the magnetosphere with other phenomena in hot stars and indicated that magnetospheric clouds are the primary factor responsible for the variation in the light curve of rapidly rotating magnetic stars.

By applying the light-curve synthesis demonstrated by the UD06 work⁶⁶6http://www.astro.wisc.edu/ townsend/static.php?ref=rrm-movies, we obtained the shapes of light curves based on the RRM model. Since the light-curve synthesis does not provide the absolute flux, we used the normalized flux converted to the corresponding $\Delta$mag. To generate the theoretical light curves, we used a grid with the rotational factor ($v/v\rm_{crit}$) in the range from 0.00 to 0.99 with a step of 0.25, the magnetic obliquity angle $\beta$ in the range of 0 ^∘$\sim$90 ^∘ with a step of 15 ^∘, and the rotational inclination angle $i$ in the range of 0 ^∘$\sim$90 ^∘ with a step of 15 ^∘. For the accuracy of the analysis, we used the light curves obtained from the TESS observations and the multicolor NOWT photometry. The $\chi^{2}$-minimum method was used to estimate the difference between the RRM model and the observations. Fig. 10 shows the comparison between the photometry light curves and the light curves synthesized using the RRM model. The best fit (black line) indicates that we can restrict the magnetic obliquity angle $\beta$ and the rotational inclination angle $i$ to the values satisfying the approximate relation $\beta$ + $i$ $\approx$ 105 ^∘ ($i=$75 ^∘ and $\beta$=30 ^∘) Based on the work by Shultz et al. (2020), we also adopt the assumptions that $i$=60 ^∘ and $\beta$=45 ^∘. However, there are several differences between the simulated result and photometric observations. The unequal eclipses may be caused by the mismatch parameters with the magnetic obliquity angle $\beta$ and the rotational inclination angle $i$ (Townsend et al. 2005). Our result is more consistent with that report by Hubrig et al. (2017a).

Although we are able to reproduce the light curve for HD 345439, there are evident discrepancies between the theoretical and observational light curves. In contrast to the synthesized light curves, asymmetrical depths of eclipses are detected in the observational light curves. Townsend et al. (2005) pointed out that this asymmetry in the eclipses may be caused by an offset between the center of the dipole magnetic fields and the rotation axis. Furthermore, in the phase range of $0.3\sim 0.7$, the observational and synthesized light curves show opposite trends. One hypothesis that could explain this discrepancy is that the magnetosphere clouds of HD 345439 have more complex geometries, as was also reported by Carciofi et al. (2013). In the $0.3\sim 0.7$ phase, the magnetospheric column density is affected by other small scale magnetic fields, which results in larger density values than those estimated by the RRM model. The TO05 work presented the morphology of magnetic clouds with pure poloidal magnetic fields. Nevertheless, the MHD simulations show that in order to maintain stable global scale magnetic fields on long timescale, the ratio of the energies of toroidal and poloidal components of the magnetic fields must have a specific value (Braithwaite 2009; Duez et al. 2010). On the other hand, the opposite trend may be caused by chemical anomalies and starspots, which are from microscopic chemical transport processes. It is thus necessary to determine the surface chemical spot structure using Doppler imaging (DI) to further restrict the light curve of HD 345439 (e.g., Kochukhov et al. 2022).

No evidence of centrifugal breakout was found in the TESS photometry. Several factors could explain the failure of the RRM model in detecting the centrifugal breakout. One of the reasons is that no relevant scheduling of the TESS run was suitable for searching for the flare of the centrifugal breakout. Another factor could be that the mass of the magnetic clouds does not reach the crucial value necessary to generate the centrifugal breakout. The TO05 work predicted that the crucial mass was dependent strongly on the magnetic strength, stellar radius, mass-loss rate, and escape velocity. HD 345439 has a more extreme state than $\sigma$ Ori E; thus, its breakout timescale is larger than the latter with a low incidence of breakout episode.

Although the centrifugal breakout is a natural result in MHD for rapidly rotating magnetic stars (ud-Doula et al. 2006), no centrifugal breakout event was detected for HD 345439. A uniform situation occurred in $\sigma$ Ori E, which indicates the low incidence of events (Townsend et al. 2013). In addition to the centrifugal breakout, the ongoing reductions in the magnetospheric column density might indicate the mass leakage of the magnetosphere. Owocki & Cranmer (2018) presented diffusion-plus-drift magnetospheres (DDM) model to interpret the mass leakage of magnetic hot stars. In this scenario, plasma diffuses in both directions and escapes from the star via drift. The mass leakage process in the DDM model is more gradual than in the RRM model, and in particular the centrifugal breakout does not occur in the DDM model. Shultz et al. (2020) analyzed the H$\alpha$ line profiles of the magnetic B-type stars, and pointed out that the H$\alpha$ emission and strength are independent of the effective temperature, luminosity, and mass-loss rate; these results support the hypothesis of the occurrence of centrifugal breakout events. However, Shultz et al. (2020) also reported that centrifugal breakouts must be a continuous process, and the H$\alpha$ emission is in contradiction with the model predictions. The detection of the centrifugal breakout using the TESS data presented in this work supports the results of Shultz et al. (2020).

We obtained the stellar parameters of HD 345439 from the LAMOST-LRS spectrum. The basic parameters were also measured by previous works: Guo et al. (2021) derived $T\rm_{eff}=22\pm 1.5$ kK, log$\,g=3.67\pm 0.25$, and $[Fe/H]=-0.35$ dex with the Stellar LAbel Machine (SLAM). The SLAM (Zhang et al. 2020) is a data-driven method based on support vector regression (SVR) which is a robust nonlinear regression technique and often used to build the mapping from stellar spectra to stellar labels in previous works (e.g., Bu & Pan 2015; Lu & Li 2015). Shultz et al. (2020) determined $T\rm_{eff}=23\pm 2$ kK, log$\,g=4.29\pm 0.19$, and log$(L/L_{\odot})=4.0\pm 0.3$ dex from the Gaia photometric observations and high-resolution NLTE atmosphere spectra. Regarding the LAMOST-LRS spectrum, it should be noted that the lowest value of the surface gravity was obtained from the NLTE atmosphere using the SLAM, while the highest value was from the LTE atmosphere. A similar discrepancy occurred for other hot stars (e.g., Shultz et al. 2021). As one of the main reasons, we adopted a different atmospheric model, which indicated the different line list, model atoms, and opacity will be used to generate final template spectra. Secondly, there are differences in the method of how to derive the stellar parameters of the HD 345439. Different grids were used between our study and Guo et al. (2021) to increase the difference in the surface gravity. The surface gravity and effective temperature derived in the present work are consistent with the results of Shultz et al. (2020).

The locations of the HD 345439 derived from the NLTE and LTE atmospheres are marked in Fig. 9. The mass, radius, and age were also derived using the evolution tracks in the HRD, and the final results are listed in Table 5. Being a magnetic He-rich star, HD 345439 is located in the colony of other main-sequence (MS) magnetic He-rich stars on the HRD and KD. HD 345439 is around the terminal-age main sequence (TAMS), which suggests that HD 345439 is an old B-type star. HD 345439 has a larger surface gravity than that inferred from its luminosity derived from the NLTE or LTE atmosphere. The other magnetic He-strong early-type stars are also marked in the HRD and KD. The locations of the He-strong stars in the HRD and KD are also characterized by systematic discrepancies. We also mark the other magnetic He-strong early-type stars on the HRD and KD. The locations of He-strong stars between the HRD and KD also have systematic discrepancies.

The surface gravity of magnetic early-type stars is usually inferred from the pressure broadening of some specific H and He lines (e.g., Cidale et al. 2007; Sikora et al. 2015). Strong magnetic fields contribute to the magnetic pressure via the Lorentz force, which results in the flux of the spectral lines changing by several percent. Additionally, the magnetic clouds reduce this flux by approximately 2% (Valyavin et al. 2004; Vallverdú et al. 2014; Shultz et al. 2019a, 2020). The flux variations result in a larger value of the surface gravity inferred from the template, and it can be seen that 0.2 dex variations in the surface gravity result in a mass uncertainty as high as $\sim 20\%$. On the other hand, $\tau\rm_{age}$ of HD 345439 has $\sim 30\%$ uncertainty caused by the uncertainty in both log$(L/L_{\odot})$ and $T\rm_{eff}$. Although the number of magnetic stars are growing, only a dozen rapidly rotating magnetic stars have been confirmed at the moment. It is important to obtain a detailed characterization of all known samples for understanding the evolution of rapidly rotating magnetic stars. Thus, more future works are required to perform further photometric observations of rapidly rotating magnetic stars and search for more samples for this type of star (e.g., Bagnulo et al. 2006; Hubrig et al. 2013).