On the existence of ‘Maia variables’

Abstract

There are different classes of pulsating stars in the H-R diagram. While many of those classes are undisputed, some remain a mystery such as the objects historically called ‘Maia variables’. Whereas the presence of such a class was suggested seven decades ago, no pulsational driving mechanism is known that could excite short-period oscillations in these late B to early A-type stars. Alternative hypotheses that would render the reports of variability of those stars erroneous have been proposed such as incorrect effective temperatures, binarity or rapid rotation, but no certain conclusions have been reached yet. Therefore, the existence of these variables as a homogeneous class of pulsating star is still under discussion. Meanwhile, many new candidates of these variables have been claimed especially by using photometric observations of space telescopes. In this study, we examined 31 objects that are alleged members of this hypothetical group and carried out detailed spectroscopic and photometric analyses to test the proposed hypotheses for their cause of variability. The $T_\textrm{eff}$, $\log g$, $v \sin i$, and chemical abundances of the targets were determined and the TESS photometric data were examined. As a result, we found that most of these targets are located inside the $\delta$ Scuti, $\beta$ Cephei, or SPB star instability strips, a few show evidence for binarity and others for rapid rotation. We give arguments that none of the apparently rapid pulsations in our targets is caused by a star outside any known instability strip. By extrapolation, we argue that most stars proposed as pulsators outside well-established instability domains are misclassified. Hence there is no sufficient evidence justifying the existence of a class of pulsating stars formerly known as the ‘Maia variables’.

1. Introduction

Perhaps the most efficient and accurate way to investigate stellar interior structure is asteroseismology which uses the eigenfrequencies of stellar oscillations to probe the interiors of stars. Using asteroseismology phenomena such as internal rotation, angular momentum, and convective core overshooting have been successfully examined (e.g., Christophe et al. 2018; Townsend, Goldstein, & Zweibel 2018; Walczak et al. 2019; Kurtz 2022).

The loci of several different classes of pulsating stars can be delineated in the Hertzsprung-Russell (H-R) diagram. These loci are called ‘instability strips’ and comprise groups of different evolutionary statuses. Among those on the main sequence, six types of pulsating stars are commonly known. Moving from hotter to cooler temperatures, these variables are the $\beta$ Cephei stars, the slowly pulsating B-type (SPB) stars, the $\delta$ Scuti stars, the rapidly oscillating Ap (roAp) stars, the $\gamma$ Doradus stars, and solar-like oscillators. The $\beta$ Cephei stars are objects of spectral types B0 – B3 showing low-order pressure (p) and gravity (g) modes with frequencies $\gtrsim$ 3.5 d $^{-1}$ (Stankov & Handler 2005) On the other hand, the SPB variables are B3 – B9 stars which exhibit high-order g-mode oscillations with frequencies $\lesssim$ 3.5 d $^{-1}$ (Aerts, Christensen-Dalsgaard, & Kurtz 2010). The $\delta$ Scuti stars exhibit high-frequencies ( $\gtrsim$ 5 d $^{-1}$ ) similar to $\beta$ Cephei variables, however, their spectral types range from A2 to F5 (Chang et al. 2013; Uytterhoeven et al. 2011). The roAp stars comprise similar spectral types, but are chemically peculiar due to the presence of strong magnetic fields and pulsate in high-order pressure modes with even higher frequencies ( $\gtrsim$ 50 d $^{-1}$ ) (Holdsworth et al. 2024). The $\gamma$ Doradus stars are low-frequency ( $\lesssim$ 5 d $^{-1}$ ) g mode pulsators with spectral types of F2 – F6 (Uytterhoeven et al. 2011). Solar-like oscillators are of even later spectral type on the main sequence and show stochastic high-order p mode oscillations with frequencies around 300 d $^{-1}$ (e.g., Jackiewicz 2021). All these pulsators have their own instability strips where theoretical models can explain the observed types of oscillations. Consequently, almost the whole main sequence is populated with pulsating stars with spectral types ranging from early B to late G, but with a marked gap between spectral types B9 and A2. The reason for this gap is that both B-type variable star classes show pulsations triggered by the $\kappa$ mechanism operating in the partial ionization zone of iron group elements (Dziembowski, Moskalik, & Pamyatnykh 1993) but the $\delta$ Scuti stars are triggered by the $\kappa$ and/or the turbulent mechanism acting in the partial ionization zones of helium (Baker & Kippenhahn 1962; Antoci et al. 2014) and hydrogen (e.g., Cunha et al. 2013). In the gap between those groups of pulsators at late B/early A spectral type, none of the currently known pulsation mechanisms are able to drive pulsations in models, consistent with observations. In particular, no pulsations with intrinsic frequencies in excess of 3 d $^{-1}$ should exist in stars with spectral types between B3 and A2.

However, a group of high-frequency pulsating stars in this gap and in the cool part of the SPB instability strip has been proposed by several authors (e.g., Struve 1955; Lehmann et al. 1995). These variables seem to have high frequencies like $\beta$ Cephei and $\delta$ Scuti stars but they are too cool to be a $\beta$ Cephei and too hot to be a $\delta$ Scuti pulsator. Hence they appear to populate the gap where pressure mode pulsations should not exist. These variables have historically been known as the ‘Maia stars’ following the apparent prototype of these variables, Maia in the Pleiades cluster. Maia was first thought to be a high-frequency pulsator (Struve 1955), but this claim was refuted several times (e.g., McNamara 1985, and references therein) and in the end it was shown that Maia is actually a rotational variable (HgMn star, White et al. 2017) without any detectable pulsational variability. However, recent studies, especially those based on space-based data, revived interest in those objects (Balona et al. 2016; Balona & Ozuyar 2020; Balona 2023).

Furthermore, examples of objects with slower apparently pulsational variations have been reported in the gap between the main sequence instability strips (e.g., Aerts & Kolenberg 2005; Degroote et al. 2009; Mowlavi et al. 2013; Sharma et al. 2022), which are not to be confused with the short period variables mentioned above. Nevertheless, the hypotheses to explain all these variable stars are essentially the same. We call the first one the ‘rapid rotation hypothesis’. This idea would explain the apparent pulsators as rapidly rotating SPB or $\beta$ Cephei stars seen equator-on. Because of this fast rotation, the star would have an ellipsoidal-like shape, and its effective temperature ( $T_\textrm{eff}$ ) would be lower near the equator and higher near the poles. This would have several consequences. First, the observationally determined $T_\textrm{eff}$ would be different from the overall value due to projection effects, modifying the star’s position in the HR Diagram. Second, the rapid rotation would shift the observed pulsational mode frequencies (Townsend 2005; Salmon et al. 2014; Szewczuk & Daszyńska-Daszkiewicz 2017; Mowlavi et al. 2016) to higher values. Third, Rossbywaves (r modes) could be excited (Szewczuk & Daszyńska-Daszkiewicz 2017; Saio et al. 2018). Fourth, the non-uniform temperature distribution in the star could facilitate pulsational excitation near the stellar poles or the equator (cf. Kurtz et al. 2023). Fifth, rapid rotation would extend the SPB instability strip, also towards lower temperatures (Townsend 2005). Another hypothesis to explain pulsation outside of any known instability strip is associated with binarity (hereinafter the ‘binary hypothesis’). The pulsating stars could be components of binary systems consisting of a non-pulsating late B-type star, which dominates the spectrum, and a $\delta$ Scuti pulsator. Given that most B-type stars have companions, and often more than one (Moe & Di Stefano 2017) such a scenario is not unlikely. Therefore, apparent p-mode pulsations would be associated with a star too hot to have them excited. The third hypothesis involves simple observational uncertainties that lead to incorrect $T_\textrm{eff}$ values (hereinafter the ‘wrong $T_\textrm{eff}$ hypothesis’). Most of the variables under question were observed by the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2014). The $T_\textrm{eff}$ of these B type systems given in the TESS input catalogue (Brown et al. 2011; Stassun et al. 2019) may not always be reliable. These $T_\textrm{eff}$ parameters are in most cases photometric estimates based on a system lacking a U-band filter, and that carry some uncertainties regarding dereddening for stars located close to the Galactic plane (Stassun et al. 2019). Therefore, the $T_\textrm{eff}$ values for hot stars from the TIC catalogue have to be treated with caution. Finally, regarding TESS data, a ‘contamination hypothesis’ must be considered. Due to the large (21^′) size of TESS’s CCD pixels on the sky, the light of nearby stars may fall into the photometric aperture, hence, for the purpose of example, an unrelated $\delta$ Scuti star may introduce short-period pulsational variations into the light curve of a star outside of any pulsational instability strip.

Balona et al. (2016), Balona & Ozuyar (2020) tested some of these hypotheses, but without an adequate amount of good quality spectroscopic data. In a recent study Balona (2023) even suggested that Maia variables are an extension of $\delta$ Scuti stars and there is an interplay between available pulsational driving mechanisms, but this conclusion was partly based on inhomogeneous literature data. To improve on this situation one needs to study these variable stars with medium to high-resolution spectroscopic data to determine some important parameters (e.g. $T_\textrm{eff}$ , $v \sin i$ ) accurately.

In a recent study, Kurtz et al. (2023) analyzed an apparently genuine, very rapidly rotating p-mode pulsator of spectral type A0 (HD 42477), which is located between the $\beta$ Cephei and $\delta$ Scuti instability strips, by taking into account the incorrect $T_\textrm{eff}$ , contamination, and binarity hypotheses. They ruled out or placed strong constraints against all these hypotheses and concluded that p-mode pulsations can be present in the mentioned region of the H-R diagram. Their excitation would be explained by coupling with several g and r modes or by the classical $\kappa-\gamma$ mechanism operating in the Heii ionization zone in the stellar equatorial regions, hence is understood based on current astrophysical knowledge.

Recapitulating, there is one well-documented case of pressure mode pulsation between the instability strips of the SPB and $\delta$ Scuti stars. In addition, there are claims of hundreds of such stars based on space photometry data, but without proper assessments of whether the fast rotation, the binary, the contamination, or the wrong $T_\textrm{eff}$ hypotheses would apply. Given the sheer number of these objects, is it really possible that most of them were claimed by misfortune?

In this study, we present detailed analyses of some candidate variables in the gap between the main sequence instability strips to test the proposed hypotheses and to establish whether a class of ‘Maia variables’^a is justified. We use TESS photometric data and medium and/or high-resolution spectra and the paper is organized as follows. In Section 2, we introduce the target selection and observations. In Section 3, the spectroscopic analyses including radial velocity measurements, determination of atmospheric parameters, and chemical abundances are presented. A contamination analysis on TESS data is introduced in Section 4. Discussions and conclusions are given in Sections 5, and 6, respectively.

2. Target selection and observations

Our variable star candidates were selected from the list of Balona&Ozuyar (2020). These authors defined priority classes considering the methods used to estimate $T_\textrm{eff}$ values. The highest priority was given for spectroscopic $T_\textrm{eff}$ while the lowest priority was defined for $T_\textrm{eff}$ estimated from the spectral types. In between, there are $T_\textrm{eff}$ estimations from some photometric colours. In this study, we preferentially chose stars of lower priority, as this adds more value to our spectral analyses for accurate $T_\textrm{eff}$ determinations.

We reanalyzed the TESS Simple Aperture Photometry (SAP) 2-min cadence light curves of our selected candidates to confirm the presence of pressure mode oscillations in these variables. We made use of the Period04 (Lenz & Breger 2005) package, which utilizes simultaneous multi-frequency sine-wave fitting with input parameter optimization. After removing the sine-wave fits from the original data, the residuals were then re-analyzed in the same manner to detect further periodicities. The signals were extracted until we reached a Signal-to-Noise ratio (SNR) of 4.5 (Baran & Koen 2021). In addition, the initial positions of the selected candidates in the H-R diagram were checked using the luminosity (L) values calculated from the Gaia parallaxes (Gaia Collaboration 2022), interstellar reddening, E(B-V), calculated from the interstellar extinction map (Amôres & Lépine 2005), the bolometric corrections (Flower 1996) and the $T_\textrm{eff}$ values given by Balona & Ozuyar (2020) (see, Fig. 1). As a result, 31 candidates were selected for this study; they are listed in Table 1.

Figure 1. The initial positions of the selected candidates in the H-R diagram as derived by Balona&Ozuyar (2020). The theoretical instability strips of the $\beta$ Cephei and SPB stars were taken from Pamyatnykh (1999). The domain of the $\delta$ Scuti stars was taken from Murphy et al. (2019).

Table 1. The list of the targets. The number of the taken spectra from FIES and HERMES and their signal-to-noise ratios (SNR) are given in the last two columns, respectively. These spectra were taken between 2020 and 2021.

For these candidates, we carried out spectroscopic surveys. Some spectra were taken from the Fibre-fed Échelle Spectrograph (FIES, Telting et al. 2014) that is a cross-dispersed high-resolution échelle spectrograph attached to the 2.56-m Nordic Optical Telescope of the Roque de los Muchachos Observatory (ORM, La Palma, Spain). It has three different resolution options; low ( $R=25\,000$ ), medium ( $R=46\,000$ ) and high ( $R=67\,000$ ). Considering the $T_\textrm{eff}$ values and the V-band magnitudes of the target stars, the low-resolution mode was used during the observations to achieve a higher signal-to-noise ratio (SNR). In addition, this resolving power is good enough for spectral analysis of systems having medium to high projected rotational velocity ( $v \sin i$ $\gtrsim$ 30 km s $^{-1}$ ). We also gathered some spectra from the High Efficiency and Resolution Mercator Échelle Spectrograph (HERMES, Raskin et al. 2011) which is a high-resolution ( $R=85\,000$ ) fibre-fed échelle spectrograph attached to the 1.2-m Mercator telescope at the Roque de los Muchachos Observatory (ORM, La Palma, Spain). Some stars have spectra from both spectrometers, while some have only from one. For each target, we tried to obtain at least two spectra taken at different times to check for possible binary-induced radial velocity shifts. The information about the spectroscopic surveys is given in Table 1.

3. Spectroscopic analysis

3.1 Radial velocity measurements

To discover possible variability of the targets induced by orbital motion in a binary system, we used spectra taken at different epochs. There are at least a few hours (mostly a day) or a maximum of around two months between two consecutive spectra of the same star for the same spectrometer.

Table 2. Atmospheric parameters of the stars determined from the hydrogen Balmer lines and the updated parameters. An asterisk represents a fixed $\log g$ value for TIC 352781227 that we could not determine because of rapid rotation and low SNR. Since some stars are not suitable for further analysis because of their large rotational velocities and low SNR values of their spectra, only some of the parameters could be updated (see, Section 3.3). $E(B-V)$ was determined from the sodium lines and the final L parameters are given. The Gaia RUWE parameters are also listed in the last column.

The radial velocities of the systems were measured using the cross-correlation technique. The IRAF^b (Tody 1986) FXCOR task was used in the investigation. As templates, some synthetic spectra were generated with different $T_\textrm{eff}$ values of 8 000, 10 000, and 15 000 K.These atmospheric models were generated using local thermodynamic equilibrium (LTE) ATLAS9 models (Kurucz 1993) and the SYNTHE code (Kurucz & Avrett 1981) Each spectrum of the individual stars was analyzed. However, for most of the stars, no trace of another component and no significant difference between the radial velocity measurements were found within error bars. There are two systems, TIC 202567458 and TIC 251196433, whose line profile shapes differ from those of single stars. However, because of high rotation and low SNR values, we could not detect double peaks in cross-correlation analyses of these systems. The line profile shapes of the two atypical systems could be caused by pulsations as well. We classified these systems as suspected binaries. Table A1 gives a list of the radial velocity measurements.

Figure 2. Upper panels: Theoretical spectral fits (red dashed lines) to the observed spectra of target stars. Lower panel: Residuals.

3.2 Atmospheric parameters

For the analysis of stars having $T_\textrm{eff}$ lower than $\sim$ 15 000 K LTE model atmospheres are convenient (Przybilla et al. 2011). Therefore, at the beginning of our study, to estimate the $T_\textrm{eff}$ , surface gravity ( $\log g$ ), and $v \sin i$ parameters we used LTE ATLAS9 models (Kurucz 1993) and SYNTHE code (Kurucz & Avrett 1981) to generate theoretical spectra. The $v \sin i$ values were determined by the profile fitting method using unblended metal or helium (He) lines (Gray 2008). The $T_\textrm{eff}$ and $\log g$ parameters were derived utilizing the hydrogen lines. During the analysis, the minimization method was used to obtain the atmospheric parameters (Catanzaro, Leone, & Dall 2004). Since most stars in our list seem to have moderate to high rotational velocity, rather than taking into account the resolving power of the spectrometers we preferred the spectra which have the highest SNR, if a star was observed by both spectrometers.

According to initially derived $T_\textrm{eff}$ parameters from the LTE models, some stars were found hotter than 15 000 K. For these stars, the same analysis was carried out by using the non-LTE Tlusty BSTAR2006 grids (Lanz & Hubeny 2007). These non-LTE models were generated using the Synspec (Hubeny & Lanz 2011) program. As a result of LTE and non-LTE analyses, the estimated parameters are listed in Table 2. The uncertainties of the determined parameters were estimated by the 1- $\sigma$ tolerance in the goodness-of-fit parameter. The consistency between the theoretical and observed spectra is shown in Fig. 2 for some samples.

3.3 Chemical abundances

After we determined the parameters given in Table 2, in the next step we singled out stars that are suitable for chemical abundance analysis. The stars having $v \sin i$ lower than $\sim$ 200 km s $^{-1}$ , and spectra with a good SNR were chosen for that purpose. During this examination, previously derived atmospheric parameters were taken as input and they were updated utilizing the excitation and ionization equilibrium as described by Kahraman Aliçavuş et al. (2016).

In the analysis, the Kurucz line list^c was used for the line identification and we used the LTE ATLAS9 models. The spectral synthesis method was considered in the analysis. As a result of the investigation, updated atmospheric parameters and chemical abundances were derived. The list of the updated parameters is given in Table 2, while the element abundances are presented in Table A2. The distribution of the final $T_\textrm{eff}$ , $\log g$ , and $v \sin i$ parameters for all targets are shown in Fig. 3. In addition, the Fe abundance distribution of stars is shown in Fig. 4.

Figure 3. Distributions of the $T_\textrm{eff}$ , $\log g$ and $v \sin i$ parameters.

4. Examination of contamination

To complement our spectroscopic analyses, we used data from TESS. However, the TESS pixel scale is 21”, which may induce contamination of the photometric signal from stars located nearby in the sky. To characterize the extent of the contamination in the TESS data of our candidates, we worked directly with target pixel files (TPFs).

For any star in which the contamination percentage was greater than 1%, we queried Gaia DR3 for all the stars brighter than a $G_\textrm{mag}$ of 17 within a 200” radius of the TESS source. Any targets that had a star within the SPOC photometric aperture that had $\Delta G_\textrm{mag}$ < 5 were thoroughly investigated through the use of custom apertures for photometry. We checked to see whether we recover the same periodicities for the Maia stars from the SPOC light curve and the light curve calculated using the custom aperture, which will allow us to have confidence in our identification of the pulsation modes.

As part of this analysis, 22 of the 31 stars were found to have contamination ratios greater than 1%. Several individual stars were selected for individual discussion in Appendix A if one of the following criteria was true: (i) There are changes in the number or location of significant frequencies when the periodogram of the custom light curve is calculated, (ii) there exist potentially contaminating stars that are within the SPOC photometric aperture and have a $\Delta G_\textrm{mag} \leq 3$ compared to the target of interest, or (iii) there is a star with $G_\textrm{mag} \leq 14$ whose signal falls into the same pixel as the star of interest. Some of the custom-generated light curves were corrected with cotrending basis vectors to remove long-term trends. The stars that are not discussed in Appendix A can be considered to have intrinsic variability that is appropriately reflected in the SPOC-generated light curves.

Figure 4. The differences between the obtained and solar (Asplund et al. 2009) Fe abundances for each star. The x-axis represents the TIC numbers.

Note that there is a significant tradeoff between the custom light curves and the light curves generated by the SPOC pipeline – while the former will isolate signals to the star of interest, the white noise level will be somewhat higher because the detrending and photometry for this light curve are more rudimentary than the tools SPOC uses. This may mean that some of the lower S/N peaks in the periodogram for the SPOC light curve are washed out by the higher white noise level. Cases where the white noise level of the custom light curve’s periodogram is comparable to the height of the peak in the periodogram of the SPOC light curve, are noted below.

Additionally, in some cases, we used the TESS-Localize package (Higgins & Bell 2022) to identify the most likely source of some of the frequency peaks in the periodogram. All calls to TESS-Localize were used with the principal_components parameter set to 2, in order to de-trend the light curve generated from the TPF.

In Fig. 5, a plot of the TESS pixel files of TIC 64774351, TIC 352781227, TIC 467219052, and TIC 450085523 is presented. Custom apertures were created containing just the target stars; for some sources, multiple apertures were created to verify the results, as discussed in in Appendix A. The periodograms for the custom apertures are shown in blue in the same figure. According to periodograms and the discussion given in in Appendix A, for TIC 352781227, TIC 467219052, and TIC 450085523 we do not find any significant contamination that would cause frequencies higher than 3 d $^{-1}$ . However, for TIC 64774351 the result of contamination analysis could not give a reliable result for high-frequencies seen in the original periodogram.

Figure 5. A plot of the TESS target pixel files of 4 of the 7 stars discussed in Appendix A. These stars were the ones for which custom light curves were made. The top panel shows the target pixel files, with the star of interest shown in green (and indicated in the title) and the potentially contaminating stars shown in tan. The middle panel shows the light curve created using a custom aperture (shown in blue in the top panel); the bottom panel shows the light curve generated via the SPOC pipeline. Note that the y-axis scales differ between the middle and bottom panels as a result of the fact that the custom aperture imperfectly captures the flux from the star of interest. This, however, provides the advantage of allowing us to highlight the peak locations.

Figure 6. Each panel contains the target pixel file for a star that is so heavily contaminated that we are unable to determine where the true source of the frequency peaks in the periodogram lies. As in Fig. 5, the green points represent the source of interest, while the tan points represent potentially contaminating stars.

In Fig. 6, the heavily contaminated stars are shown. For TIC 255996252 according to our investigation, we found that the target of interest is the original source of high-frequencies (for detail see Appendix A). However, for other systems, we were unable to distinguish whether or not the high-frequency peaks come from our targets.

5. Discussion

We now proceed to the overall discussion of the variability of our sample of target stars. Each individual star is considered in more detail in Appendix B. We present the positions of our targets in the theoretical H-R Diagram in Fig. 7. The frequency spectra of all systems are shown in Fig. A1 in order from the hottest star to the coldest star.

The positions of the candidates in this diagram take into account the final atmospheric parameters, the structure of frequency spectra, and the $v \sin i$ values. To plot the final H-R diagram the L parameters were recalculated taking into account the final $T_\textrm{eff}$ values. Additionally, the E(B-V) parameters were re-determined from the interstellar sodium lines as described in the study of Kahraman Aliçavuş et al. (2016) (cf. Table 2).

5.1 Incorrect $T_\textrm{eff}$ hypothesis

In stark contrast to Fig. 1 most of the target stars now lie in known pulsational instability regions, suggesting that most literature $T_\textrm{eff}$ values used to select them are inaccurate. We compared our spectral $T_\textrm{eff}$ values with the $T_\textrm{eff}$ of the TIC and Balona & Ozuyar (2020) in Fig. 8. As can be seen from the figure there are significant differences between the $T_\textrm{eff}$ values. This result is an expected conclusion for the TIC $T_\textrm{eff}$ values for hot stars because of the uncertain reddening corrections for stars in the Galactic plane. A similar explanation may apply to the $T_\textrm{eff}$ values used by Balona &Ozuyar (2020), as those were derived mostly from photometric colours. According to this comparison, we conclude that for most targets previously derived $T_\textrm{eff}$ values show significant discrepancies from the spectral ones. Even so, in Fig. 7 there are still some stars located outside of any known pulsational instability strip.

Figure 7. The final positions of the selected Maia candidates in the H-R diagram. The theoretical instability strips of $\beta$ Cephei, SPB and $\delta$ Scuti stars were taken the same as in Fig. 1. A detailed explanation of the brief expressions of frequencies in the figure can be found in Section 5. Some targets are not included in this figure, since they were excluded from the Maia candidate list according to the results of our contamination analysis.

Figure 8. Comparison of the $T_\textrm{eff}$ of the TIC (Stassun et al. 2019) (upper panel) and Balona & Ozuyar (2020) (lower panel) with the spectral $T_\textrm{eff}$ determined in this study. The x-axis shows the spectral $T_\textrm{eff}$ values while the y-axis illustrates the $T_\textrm{eff}$ differences between the spectral and TIC (upper panel) and Balona & Ozuyar (2020) (lower panel).

At this point it is important to stress that the instability strip boundaries depicted in Figs. 1 and 7 are not immutable. The theoretical boundaries ( $\beta$ Cep stars, SPB stars) depend on the input physics used, as discussed by e.g., Aerts & Tkachenko 2023), and the observational boundaries for the $\delta$ Scuti stars indicate where pulsations are less likely (Murphy et al. 2019). Nevertheless, these Figures give an idea in which stars pulsations could be difficult to explain.

5.2 Binary hypothesis

According to the binary hypothesis, it was suggested that late B/early A type pressure mode pulsators could be binary systems including a B-type star with a $\delta$ Scuti companion. Hence, the system seems to be an oscillating star but it is too cool to be a $\beta$ Cephei and too hot to be a $\delta$ Scuti star. As a result of our investigation of the spectral line profiles and radial velocities of the stars in our sample, we did not reveal any obvious binary systems. Only two systems were classified as suspected binary systems because of their line profile changes, which however could be well caused by pulsations. Additionally, our method is not sensitive to binary systems with long orbital periods (more than a few months) given the temporal sampling of our spectra. Also, many stars in our sample show high rotational velocity (>100 km s $^{-1}$ ) and some spectra have low SNR, which affects the radial velocity measurements and the detection of a possible binary component in the systems. Consequently, our spectroscopic investigation of possible binarity remained mostly inconclusive.

Another possibility to detect binaries would be to search for phase modulations of the pulsational signals (Murphy et al. 2014). Unfortunately, we were not able to find any evidence thereof, presumably owing to the small amplitudes of the pulsations detected and the relatively short duration of the individual TESS light curves.

We also examined the Gaia Renormalized Unit Weight Error (RUWE Lindegren et al. 2021). RUWE is a parameter that shows the degeneration in the astrometric parameters determined with Gaia (Gaia Collaboration et al. 2021) High values of RUWE (higher than 1.4) indicate that the system of interest could be a binary (Lindegren et al. 2018, 2021). Stassun & Torres (2021) also point out that any RUWE value above 1.0 may already be indicative of binarity. Therefore, we collected the RUWE parameters of our targets from Gaia DR3 (Gaia Collaboration et al. 2021) which are given in Table 2. As can be seen from the table, there are nine systems that have RUWE > 1.4. The largest values, 11.59 and 20.59, relate to TIC 36557487, and TIC 367351021, respectively. One suspected binary system from our radial velocity analysis (TIC 251196433) has a RUWE value of 2.01. The systems with high RUWE values (>1.4) are likely astrometric binaries, even if none of our targets is in the list of non-single stars according to Gaia DR3. Catalogues of double and/or multiple stars were also searched (e.g. Balega et al. 2012; Eggleton & Tokovinin 2008; Fabricius et al. 2002; Mason et al. 2001; 199, 1997; Dommanget & Nys 1995). Indeed some of our targets were found in these catalogs and are classified as double or multiple systems. TIC 36557487, TIC 64774351, and TIC 354793407 are members of visual double systems.

5.3 Rapid rotation hypothesis

As discussed in the introduction, rapid rotation could (a) modify a star’s position in the H-R Diagram, (b) shift the observed pulsation frequencies, (c) excite Rossby waves, and (d) provide means of $\kappa-\gamma$ -driven excitation of p and g modes. To evaluate our candidate stars with respect to rotation we first took into account the stars which show frequencies between 1–10 d $^{-1}$ . There are four stars that show frequencies in that range (TIC 251196433, TIC 352781227, TIC 377099704, TIC 467219052) that are shown with black dots in Fig. 7 and two of them exhibit high rotation; TIC 251196433 and TIC 352781227. TIC 251196433 is positioned slightly below the instability strip, situated between SPB and $\beta$ Cephei stars. This fast-rotating star ( $\gtrsim$ 255 km s $^{-1}$ ) could either be a hybrid pulsator or an SPB star displaying shifted pulsation frequencies. TIC 352781227 is also a fast-rotating star ( $\gtrsim$ 270 km s $^{-1}$ ) and placed between the $\delta$ Scuti and SPB instability strip. For this star, high rotational velocity could cause low frequencies to be shifted towards high frequencies and since we likely see this star close to equator-on, our $T_\textrm{eff}$ determination would be too low. Therefore, this variable is likely an SPB star. A more detailed discussion on this object is also given in Appendix B.

The other stars showing frequencies between 1–10 d $^{-1}$ have low to moderate $v \sin i$ . One (TIC 467219052) is simply a somewhat evolved $\delta$ Scuti star, while the other is located in the SPB domain, very close to the $\beta$ Cephei territory (TIC 377099704). These stars are hence well placed in known instability strips. Both show both low and high frequencies, these stars could be SPB $-$ $\beta$ Cephei and a $\delta$ Scuti $-$ $\gamma$ Doradus hybrids.

There is a group of stars that are distinguished by two close frequencies dominant in the Fourier spectra (TIC 26369887, TIC 36557487, TIC 312637783, TIC 374038418, TIC 434340780). As depicted in Fig. 7 most of these stars exhibit rapid rotation with some falling between the SPB and $\delta$ Scuti domains. Because of high rotational velocity, the rapidly rotating ones could appear cooler than their actual temperature, leading to a shift in their pulsation frequencies towards higher values. We therefore classify them as cool SPB stars. Further discussions on these targets are provided in Appendix B.

When examining other stars exhibiting oscillations in a wide frequency range (up to 50 d $^{-1}$ ), we noticed that most of them are located inside the $\delta$ Scuti instability strip. However, six of them are placed beyond the hot border of $\delta$ Scuti instability strip. Among these, TIC 20534584 lies between the $\delta$ Scuti and SPB instability strips. Notably, it possesses a moderate to high $v \sin i$ (150 km s $^{-1}$ ), and its relatively high RUWE parameter (2.26) suggests a potential binary nature with a hotter companion, placing it beyond the hot border of $\delta$ Scuti instability strip. There are also four stars (TIC 83803744, TIC 192780911, TIC 236785664, TIC 354793407) placed inside the SPB domain and showing frequencies up to 40 d $^{-1}$ . While two of them have low $v \sin i$ values, TIC 83803744 exhibits a low RUWE parameter, and TIC 354793407 has a RUWE of 1.81. Regarding TIC 83803744 its high frequencies lack a clear explanation. However, it is classified as HgMn star (see Appendix B). As HgMn stars are often found in binaries (e.g., Gerbaldi, Floquet, & Hauck 1985), which suggests a binary nature. The remaining two systems (TIC 192780911 and TIC 236785664) exhibit high rotational velocities with high values of the RUWE parameter indicative of a possible binary nature. TIC 192780911 displays a weak signal at around 3.9 $^{-1}$ , possibly indicating an SPB-type pulsation modified by the rapid rotation. TIC 236785664 shows both low and high frequencies typically of $\delta$ Scuti stars, yet it’s $T_\textrm{eff}$ exceeds the typical range for $\delta$ Scuti stars. Finally, TIC 202567458, a rapid rotator, manifests a pulsation spectrum comprising apparent g-mode pulsations, their linear combinations, and a stochastic component implying the presence of Internal Gravity Waves (IGWs, e.g., see Ratnasingam et al. 2023; Bowman et al. 2019). The higher frequency signals are compatible with combination frequencies.

5.4 Variability classification of the target stars

We summarize the discussions above and the results on the individual stars presented in Appendix B in Table 3, respectively. Out of the 31 stars considered in this paper and previously claimed as variables outside of any known instability strip, the largest group (14 objects) are normal $\delta$ Scuti stars that simply had overestimated $T_\textrm{eff}$ . Six stars are gravity mode pulsators with combination frequencies that protrude into the pressure mode domain, but these higher frequencies are not due to independent pulsation modes (cf. Kurtz et al. 2015). Among the hottest stars, we found one star with gravity modes and Internal Gravity waves, two $\beta$ Cephei pulsators, and two $\beta$ Cephei/SPB ‘hybrids’.

Table 3. Variability classification of our target stars.

Regarding the remaining six objects, the TESS light curve of one target was revealed to be contaminated by a background ellipsoidal variable. Two stars were identified as SPB stars with a very fast rotation that presumably modifies the pulsation frequencies in the observer’s reference frame to values that are comparable with those of pressure modes. That leaves three stars, two of which are in the SPB star instability strip and one is in between the SPB and the $\delta$ Scuti star domains. Two of these stars are rotational variables with additional frequencies in the $\delta$ Scuti range. For both of those, there is some indirect evidence of binarity, one TIC 354793407 = HD 21699 has a RUWE of 1.81, and the other TIC 83803744 = HD 1009 is a HgMn star which in most cases are binaries. Consequently, the short-period signals in these star’s light curves might well arise from a $\delta$ Scuti companion. The last object is a rapidly rotating Be star with a somewhat high RUWE and several frequencies typical for $\delta$ Scuti stars. This latter object, TIC 236785664 = HD 175511, deserves some further investigation.

6. Conclusions

In this study, a detailed analysis of 31 stars that have been claimed as late B/early A type short period pulsators is presented. In this part of the H-R diagram no such pulsations are expected, so these objects need to be explained. There are four different hypotheses that are to be considered: incorrect $T_\textrm{eff}$ , binarity, rapid rotation, and possible contamination of the light curves. To this end, a detailed spectroscopic investigation was carried out for our candidate variables. For all stars accurate stellar parameters such as $T_\textrm{eff}$ , $\log g$ , $v \sin i$ were determined. Furthermore, individual element abundances were derived for all stars where this was possible. In addition, a contamination analysis was carried out to be sure that the determined high pulsation frequencies arise from the stars of interest.

There are several stars in our sample whose frequency spectra appear phenomenologically similar, hence we analysed them in groups. The first group shows a wide frequency range extending to $\sim$ 40 d $^{-1}$ . Another group exhibits two dominant frequencies close to each other and the final group possesses frequencies between 1–10 d $^{-1}$ . Taking into account these groups, the positions of the stars in the H-R diagram, $v \sin i$ values, the radial velocity measurements, and the RUWE parameter we discussed the proposed hypotheses of the existence of the so-called ‘Maia variables’ located on the main-sequence in between the instability strips of the SPB and $\delta$ Scuti stars. First, we conclude that incorrect $T_\textrm{eff}$ values by themselves can not explain all of these stars even if for some systems there are significant differences between the spectroscopic and photometric $T_\textrm{eff}$ values (see Fig. 8). The binary hypothesis alone is also insufficient. When we examined the high rotation hypothesis by considering the systems’ positions in the H-R diagram and the structure of frequency spectra. In this way, we identified some stars whose frequencies were likely shifted towards higher values due to rapid rotation.

However, taking into account a combination of all hypotheses, most of the 31 stars we studied can be explained easily. 27 are members of established variability classes, ranging from Internal Gravity Wave pulsators, $\beta$ Cephei stars, SPB stars to $\delta$ Scuti and $\gamma$ Dor stars. The variability of one object is due to a contaminating neighbouring star. There are three stars that are not that easy to explain. Two of them are hot rotational variables which also exhibit short-period pulsations, but for both of them, we found indirect evidence for binarity. The last star is a rapidly rotating Be star in the SPB instability strip that also shows multiperiodic short-period pulsations, but also, in this case, the Gaia RUWE parameter is suspicious in terms of binarity.

Extrapolating our results to the hundreds of main sequence pulsators of B/A spectral type outside of any known instability region claimed recently (Balona 2023), we expect that most if not all of these objects can be explained in different ways as well. From today’s point of view, there is thus no need to claim the existence of an unexplained class of variable stars in this domain of the H-R diagram, let alone call them ‘Maia variables’ as Maia itself is not a pulsating star (White et al. 2017).