Spots and flares in hot main sequence stars

About 22000 Kepler stars and nearly 60000 TESS stars from sectors 1-24 have been classified according to variability type. A large proportion of stars of all spectral types appear to have periods consistent with the expected rotation periods. A previous analysis of A and late B stars strongly suggests that these stars are indeed rotational variables. In this paper we have accumulated sufficient data to show that rotational modulation is present even among the early B stars. A search for flares in TESS A and B stars resulted in the detection of 110 flares in 68 stars. The flare energies exceed those of typical K and M dwarfs by at least two orders of magnitude. These results, together with severe difficulties of current models to explain stellar pulsations in A and B stars, suggest a need for revision of our current understanding of the outer layers of stars with radiative envelopes.


INTRODUCTION
High-precision space photometry of upper main sequence stars show periodic or quasi-periodic variations with periods consistent with the expected rotational periods of these stars (Balona 2013(Balona , 2016(Balona , 2017(Balona , 2019. High-resolution spectroscopic time series of Vega (A0V) indicates the presence of a spotted stellar surface (Böhm et al. 2015), providing independent confirmation of the photometric results.
In addition, early results from the Kepler mission (Borucki et al. 2010) indicated the presence of flares associated with some A and late-B stars (Balona 2012). Further studies (Balona 2013(Balona , 2015Balona et al. 2016b) seem to indicate that around 2.5 percent of A stars flare with energies in the range 10 35 -10 36 erg. Pedersen et al. (2017) have argued that the flares are likely a result of cool flare stars in the same aperture or binary companions. Direct evidence of possible X-ray flares in A stars have been reported by Schmitt et al. (1994), Robrade & Schmitt (2010), while a flare on a B star has also been reported (Yanagida et al. 2004(Yanagida et al. , 2007. Rotational modulation and flares in A and B stars implies the presence of surface magnetic fields, contrary to the long-held view that it is not possible for stars with radiative envelopes to host magnetic fields. The Ap and Bp stars have strong global magnetic fields, but these are explained as being of fossil origin (Mestel 1967). Photometric studies referenced above indicate that rotational modulation is present in as many as 40 percent of stars on the upper main sequence, most of which are not known Ap/Bp stars. ⋆ E-mail: lab@saao.ac.za First results from the Kepler space mission on pulsations in main-sequence A stars (Grigahcène et al. 2010) already indicated a serious problem. It turns out that nearly all δ Scuti stars have multiple low frequency pulsations which cannot be explained by current models (Balona 2014(Balona , 2018. A further surprise was the confirmation that many late-B stars pulsate with high frequencies (Maia variables, Balona et al. 2015Balona et al. , 2016aBalona & Ozuyar 2020b). These are difficult to explain in terms of incorrect opacities alone (Daszyńska-Daszkiewicz et al. 2017). Perhaps of even more significance is the fact that less than half of the stars in the δ Sct instability strip pulsate. Also it seems that the γ Dor variables may be just a subset of the δ Sct stars (Balona 2018). None of these findings are explained by current pulsation models.
New ideas regarding the outer layers of stars in radiative envelopes have recently emerged. It has been suggested, for example, that generation of magnetic fields by dynamo action may occur in sub-surface convective zones in A and B stars (Cantiello et al. 2009;. At the surface they give rise to bright starspots. Also, it has been suggested that differential rotation may act to provide dynamo-generated magnetic fields in radiative zones (Spruit 1999(Spruit , 2002Maeder & Meynet 2004).
In this paper we report on further evidence for rotational modulation among TESS A and B stars, indicating that starspots are common among all B stars, including the early-type B stars. We also report on a survey for flares in TESS stars on the upper main sequence. We argue that current ideas regarding the outer layers of stars in radiative equilibrium need to be revised.

DATA AND METHODOLOGY
The data used here are the full four-year light curves from Kepler and sectors 1-24 of TESS data. In both cases the light curves are obtained using pre-search data conditioning (PDC) which corrects for time-correlated instrumental signatures in the light curves (Jenkins et al. 2016). All stars with effective temperatures T eff > 6000 K brighter than magnitude 12.5 were selected for the analysis of rotational modulation. This results in 5643 stars from Kepler and over 50000 stars from TESS. In the search for flares, the uncorrected light curves were used and only stars with T eff > 7500 K were selected.
Visual inspection of the light curves and the Lomb-Scargle periodograms (Scargle 1982) of Kepler and TESS stars were used to assign variability types whenever appropriate. The variability classification follows that of the General Catalogue of Variable Stars (GCVS, Samus et al. 2017). The only recognized class of rotational variable among the A and B stars are the chemically peculiar α CVn and SX Ari classes. A new ROT class has been added to describe any star in which the variability is suspected to be due to rotation and not known to be Ap or Bp. Aided by suitable software, visual classification of over 100 stars an hour is possible. In this way, several thousand stars with T eff > 6000 K have been assigned the ROT type.

STELLAR PARAMETERS
The most commonly used test for rotational modulation is comparison of the rotation rate derived from the photometric frequency, νROT, with that derived from the projected rotational velocity, v sin i. To derive the equatorial rotational velocity, v, from νROT requires an estimate of the stellar radius, R. This can be done if we know the effective temperature, T eff and luminosity, L/L⊙.
The most precise method of deriving T eff is by modelling absorption line profiles from medium-or high-resolution spectroscopy. For A and B stars, this involves fitting the Hβ and/or Hα line profiles using a suitable model atmosphere. The resulting standard deviation in T eff ranges from about 100 K for A stars to about 1000 K for early B stars. Spectroscopic estimates of T eff exist for about 25 percent of the sample considered here.
The next best method is the use of narrow-band photometry. This involves measuring the strength of the Hβ line (the Strömgren β index) usually in conjunction with uvby narrow-band photometry. The value of T eff is obtained either by direct comparison with synthetic photometry derived from model atmospheres or by using stars with known T eff (Moon & Dworetsky 1985;Gray 1991;Napiwotzki et al. 1993;Smalley & Dworetsky 1993;Balona 1994). Estimates of T eff from Sloan ugriz (Brown et al. 2011) are of this type and are available for most of the Kepler and TESS stars. However, they cannot be used for stars earlier than A0 because they lack u-band measurements. Without the u band, it is impossible to distinguish between A and B stars of the same colour. Estimates of T eff using narrowand intermediate-band photometry are available for about 55 percent of stars.
If neither spectroscopy or narrow-band photometry is Table 1. Number of ROT stars within the given T eff range, N ROT . Also shown is the fraction of ROT stars, f ROT within the range. N vsini is the number of stars used to construct the v sin i vs v diagrams ( Fig. 1 left panel) and N dist is the number of stars used in obtaining the v sin i distribution ( Fig. 1 right panel). available, wide-band photometry can be used to estimate T eff provided that the reddening is known. This method is used in 7 percent of our sample. Finally, if nothing else is available, a crude estimate of T eff can be derived from the spectral classification together with suitable calibration such as the calibration of Pecaut & Mamajek (2013). This method was used for 18 percent of the stars. The stellar luminosity is best estimated from Gaia DR2 parallaxes (Gaia Collaboration et al. 2016Collaboration et al. , 2018 in conjunction with reddening estimated using a three-dimensional map by Gontcharov (2017) and the bolometric correction calibration by Pecaut & Mamajek (2013). From the error in the Gaia DR2 parallax, the typical standard deviation in log(L/L⊙) is estimated to be about 0.05 dex, allowing for standard deviations of 0.01 mag in the apparent magnitude, 0.10 mag in visual extinction and 0.02 mag in the bolometric correction in addition to the parallax error. Table 1 lists the number of stars classified as ROT variables in the given range of effective temperature. Also shown is the percentage of main sequence stars for which the ROT classification was assigned. Note that Be stars were excluded from the sample. While the light variations in Be stars can be interpreted as rotational modulation, the light amplitude is typically an order of magnitude larger than for non-Be stars (Balona & Ozuyar 2020a). It is suggested that the cause of the variability are co-rotating clouds which obscure a larger fraction of the photosphere than starspots. Because of the large amplitude, Be stars are disproportionately represented among the ROT stars. Since Be stars are rapid rotators, their inclusion leads to an over-estimate of the proportion of ROT stars with rapid rotation. Most Be stars are of early B type and this leads to a severe distortion of the velocity distribution for stars with T eff > 18000 K.

RESULTS
The photometric period, obtained from Kepler, K2 and TESS light curves together with the stellar radius is used to estimate the equatorial rotational velocity, v. If the variability is rotational modulation, there should be a relationship between v and the projected rotational velocity, v sin i. Since most stars will be observed roughly equator-on, one expects that most data points in the v-v sin i diagram will lie on or just below the straight line defining sin i = 1. There will be a diminishing scatter of points below the line due to stars with lower angles of inclination, i. Due to unavoidable errors, some points are to be expected above the v sin i = 1 line.
Projected rotational velocities are not available for every star for which v has been estimated. In the left panel of Fig. 1, the v-v sin i diagram is shown for all stars for which measurements are available. It is clear that the expected distribution of points is present from the F stars to the early B stars. This justifies the original assumption that the periodic light variation is due to rotation.
As expected, nearly all stars with v > 60 km s −1 lie on or below the sin i = 1 line. For low rotation rates, ever increasing observational precision is required to determine whether or not v sin i < v. Often, v sin i values are truncated at some positive number corresponding to the resolution limit of the instrument. An important factor is that it becomes increasingly difficult to distinguish between binarity and rotation at low frequencies. For example, amplitude variability, which is a typical attribute of rotational modulation, is not so easily detected. Thus one may expect significant contamination from binaries at low rotation rates. These factors are probably responsible for the increased scatter in this region.
A more rigorous test can be made by considering the rotational velocity distribution for main-sequence stars within a limited T eff range. The rotational velocity distribution is the relative number of stars at a given rotational velocity. This is an important quantity which provides information on the physics of stellar rotation. The test involves the comparison of the v distribution with the v sin i distribution. These two distributions should be similar, though not identical due to variation of the inclination of the rotation axis. Close agreement is expected because most stars would be viewed equator-on. This is a far more rigorous test because it involves not just comparison of v and v sin i for the same star, but also tests whether the detailed distribution of v corresponds closely to that of v sin i.
It might be thought that one could derive v from v sin i by deconvolution assuming random orientation of the axis of rotation. In this way one could compare the photometric and spectroscopic v directly. The reason why this has not been done is that the stars from which the photometric v are derived cannot have a random axis of rotation. Clearly, if a star is nearly pole-on, no rotational modulation will be detected. This is, of course, not true for v sin i since rotational broadening can be made for any inclination angle. Therefore the distribution of v from deconvolution of v sin i cannot match the photometric v distribution. For the same reason, testing the distribution of i is not possible.
Obtaining the distribution requires sufficient numbers of stars within the chosen range of v or v sin i in order to be statistically meaningful. The number of stars for which photometric v measurements are available is sufficiently large for this purpose. If the corresponding values of v sin i are restricted to the same stars for which v is available, the numbers would be too small. Fortunately, it is not necessary to impose this restriction because it is reasonable to assume that the v sin i distribution will be the same for any set of main-sequence stars within the chosen range of T eff . In other words, one may select any set of main sequence stars with known v sin i within the required effective temperature range.
In the right panel of Fig. 1, the v and v sin i distributions for stars in six temperature ranges are shown. There is good agreement for all temperature ranges, reinforcing the results derived from the v-v sin i diagrams.

TESS EARLY-TYPE FLARE STARS
In our catalog of nearly 60000 stars classified for variability, there are 14495 stars with T eff > 7500 K. Flares are difficult to detect in eclipsing binaries and other types of variable with high amplitude. Excluding these stars results in 6072 A and 1616 B stars. This sample was searched for flares by visual inspection.
The 68 stars in Table 2  The former is a sdB star and the latter is an SX Ari variable (Bp star). Also excluded is the δ Sct flare star TIC 439399707 (Balona et al. 2019b) and the Be X-ray source TIC 207176480 (Balona & Ozuyar 2020a).
The number of TESS A-type stars which appear to flare constitute about 1.1 percent of the sample of A stars which were examined, which is less than half of the 2.5 percent flare incidence among Kepler A stars reported by Balona (2013) and Balona (2015). This can be understood given the fact that the long-cadence Kepler data span 4 years, while the TESS data mostly span a few months and always less than one year. There are 61 Kepler A stars known to flare (Balona 2015). The additional 68 flare stars reported here do not include any of the Kepler stars, bringing the total of flaring A stars to 129.
Whereas the Kepler pixel size is 4 arcsec, the TESS pixel size is 21 arcsec. This means that the probability of a flare originating in a star other than the A star is much larger for TESS. The fields of all 68 stars were examined and in each case the A star is by far the dominant optical source. Any cool star in the aperture would need to be of comparable brightness to the A star for the flare to be detected. A cool dwarf within the same aperture would be among the nearest stars and would have long ago been catalogued. Therefore the source of the flare would need to be an exotic faint object, a companion of the A star or the A star itself.
The flare energy was estimated by integrating the light curve under the flare. In cool dwarfs and the Sun, most of the flare energy is radiated in the UV. The UV contribution in Kepler and TESS light curves is negligible, and therefore the estimated flare energy is likely to be an underestimate. As can be seen from Table 2, the typical flare energy is 10 35 -10 36 erg, which is about 10-1000 times more energetic than flares in cool dwarfs. The most energetic flare in cool dwarfs detected in a survey of TESS stars by Maximilian et al.
(2020) has an energy of 10 34.7 erg. It is difficult to understand how a unique event like this can be repeated in over 1 percent of A stars. Multiple flares in the same star, all with energies exceeding the highest ever seen, also would seem to rule out a cool companion. Table 2. Flare stars detected from visual inspection of TESS light curves. The TIC and HD numbers are followed by the assigned variability class. This is followed by the effective temperature (K) and luminosity. The time of peak flare intensity, tmax, is relative to BJD 2458000. The flare energy, E (erg), is followed by the relative peak flare intensity, ∆F F . The number of flares, N Fl , is followed by a reference to X-ray detection and the spectral type.

CONCLUSION
It is demonstrated that the periodic light variations seen in about 40 percent of A and B stars is consistent with rotational modulation. The expected relationship between the estimated equatorial rotation velocity, v, and the projected rotation velocity, v sin i, is confirmed for B stars in three effective temperature ranges. Furthermore, the detailed distribution of v matches the distribution of v sin i even for the hottest B stars. Balona & Ozuyar (2020a) proposed a model for Be stars in which the energy released by flares and magnetic recon- nection, in conjunction with rapid rotation, ejects gas which is trapped in two diametrically opposite locations defined by the intersection of the geographic and magnetic equators. This gas eventually dissipates into the circumstellar disc. This model appears to explain most of the known characteristics of Be stars. In this analysis Be were excluded, even though they appear to show rotational modulation (Balona & Ozuyar 2020a). It turns out that the rotational light amplitudes are an order of magnitude larger than in non-Be stars. Rotational modulation in Be stars is, after all, detectable from the ground which is not true of most A and B stars. The large amplitude in Be stars may possibly be a result of circumstellar clouds rather than starspots (Balona & Ozuyar 2020a). Due to their rapid rotation, inclusion of Be stars will distort the true velocity distribution. A full discussion of activity and rotation in Be stars will be presented elsewhere.
The nature of the presumed starspots responsible for rotational modulation is not known. The idea of magnetic field generation in subsurface convective zones first postulated by Cantiello et al. (2009);Cantiello & Braithwaite (2019) seems to be quite promising. However, one would have expected that starspots should occur only within certain effective temperature ranges depending on the ionization species.
According to Cantiello & Braithwaite (2019), the largest effects are caused by a convective layer driven by second helium ionization. The amplitude of surface magnetic fields and their associated photometric variability are expected to decrease with increasing stellar mass and surface temperature, so that magnetic spots and their observational effects should be much harder to detect in late B-type stars. This is clearly not the case, since the fraction of late B stars showing rotational modulation is about the same as in A stars (Table 1). In fact, the fraction stays about the same at 30-40 percent for all stars in the upper main sequence.
Another problem is that sub-surface convection predicts the creation of bright spots. We know that spots on the Sun are dark, and this seems to be true of solar-type stars as well. Since rotational modulation is present for the full range of main sequence stars, there must be a transition between dark spots and bright spots around early F or late A. As a result, one might expect a decrease in the numbers of stars with rotational modulation in this spectral type range. This does not seem to be the case unless the transition is very sharp.
An alternative mechanism proposed many years ago involves the interaction between magnetic fields, convective flows and differential rotation. A dynamo cycle operating on differential rotation in stellar radiative interiors was described by Spruit (1999Spruit ( , 2002 and Maeder & Meynet (2004) (see also Braithwaite & Spruit 2017). In this theory, a magnetic instability in the toroidal field wound up by differential rotation replaces the role of convection in closing the field amplification loop in conventional dynamo theory. It is possible that completely stable radiative envelopes do not exist and that turbulence generated by differential rotation may lead to surface magnetic fields capable of forming conventional dark spots.
Examination of TESS A and B stars has led to the detection of 68 new early-type flare stars, doubling the total number. These include some Ap and Am stars, with some stars being X-ray sources. If starspots are deemed to be present, in A and B stars, then there should be no barrier to accepting that flares may be generated by magnetic reconnection, as in the Sun and cool stars. Indeed, it then becomes necessary to provide reasons why flares should not be generated in A and B stars.
The apparent magnitudes of the A stars are 4-10 mag with a median 8.2 mag. A cool foreground dwarf of comparable brightness, or even significantly fainter, will be one of the nearest stars and well documented. Flares originating in a cool foreground star can be excluded. One can also exclude foreground F or G giants because the combined colour would not be that of an A star (in any case the stars all have A or B spectral classifications).
A cool K or M binary companion can also be excluded because all 110 detected flares have energies considerably larger than that of the largest flare ever seen in a cool dwarf. There are two remaining possibilities: the flare arises in magnetic reconnection involving the A star and a close companion, or solely on the A star itself. Either way it means that a significant magnetic field must be present on the A star. We therefore return to the original problem regarding the presence of magnetic fields in radiative envelopes.
For further progress it will be important to design observations which might lead to resolving the problem of whether the spots are bright or dark. Further high-resolution spectroscopy of A or B stars, as performed by Böhm et al. (2015) on Vega would be important to place limits on the size and distribution of the spots. It would also be important to obtain time-series spectroscopy on A/B flare stars to determine possible interacting companions.