Flare stars across the H R diagram

Transcription

1 doi: /mnras/stu2651 Flare stars across the H R diagram L. A. Balona South African Astronomical Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa Accepted 2014 December 11. Received 2014 November 16; in original form 2014 September 30 1 INTRODUCTION The energy release in a solar flare, typically erg, is believed to be a result of the reconnection of magnetic loops, accompanied by particle beams, chromospheric evaporation, rapid bulk flows or mass ejection, and heating of plasma confined in loops. The flare energy is emitted in a wide range of wavelengths from radio waves to gamma rays (Benz & Güdel 2010). It is possible that all flares are white light flares and that the white light continuum with a blackbody spectrum at 9000 K is the main contributor to the total radiated energy (Kretzschmar 2011). Ground-based optical observations show that stellar flares occur mostly in M dwarfs and are typically times more energetic than solar flares (Güdel & Nazé 2009). Stellar flares are qualitatively different in several respects from solar flares, but it is generally assumed that the underlying mechanism is the same as in solar flares. Flares are also seen in the RS CVn variables which are a class of detached binaries typically composed of a chromospherically active G or K star. Tidal locking in these stars ensures that the rotational period is the same as the orbital period, which ranges from a few days to as long as one month. Flares in RS CVn stars are mostly lab@saao.ac.za ABSTRACT From Kepler data, we show that the incidence of flares on stars drops by only a factor of 4 from K M dwarfs to A F stars. Allowing for visibility effects, this implies that the true relative number of flare stars does not change very much from cool dwarfs to hot A stars. The idea that flares on A stars can be attributed to a cool companion has to be rejected because it leads to flare amplitudes two orders of magnitude smaller than actually observed. We confirm that spots on flare stars are generally larger than those on non-flare stars and that flare stars rotate significantly faster than non-flare stars. Analysis of 209 flare stars observed in Kepler short-cadence mode allows accurate measurements of flare shapes and duration. We find that about one-third of the flares have a bump or slope discontinuity on the decaying branch and that flares of long duration are to be found in stars with low surface gravities. Flare energies are strongly correlated with stellar luminosity and radius. The correlation with radius leads to a rough estimate of several tens of gauss for the typical magnetic field associated with a flare. The correlation with stellar luminosity can be understood if the typical flare loop length-scales approximately as the stellar radius. We examined the flare frequency as a function of orbital phase in three eclipsing binaries in which a large number of flares are visible. There appears to be no correlation of flaring with orbital phase, which weakens the hypothesis that flares in close binaries could be a result of reconnection of field lines connecting the two stars. Key words: stars: activity stars: flare stars: magnetic field starspots. detected in radio and X-rays (Pandey & Singh 2012), though optical flares have been observed (Mathioudakis et al. 1992). It has been suggested that a magnetic reconnection in a field connecting the two stars may be the source of flaring in RS CVn systems (Simon, Linsky & Schiffer 1980; van den Oord 1988; Gunn et al. 1997). For our purposes, we define a flare star as a star in which at least one flare has been observed. Recent space observations have significantly changed our knowledge of stellar flares. These are usually termed superflares because they are typically more intense than large solar flares. Since a large solar flare has an intensity of only around 200 ppm in whole-disc white light observations (Kretzschmar 2011), the Sun would not be recognized as a flare star even in the most precise photometric observations from space. Hence there are no solar flare observations which may be directly compared to optical ground-based or space-based observations of stellar flares. Walkowicz et al. (2011) identified 373 flare stars from about cool dwarfs in the Kepler field. Maehara et al. (2012) discovered 148 solar-type stars with superflares from about stars observed over 120 d by Kepler. They find that superflares seem to occur more frequently on rapidly rotating stars, a result confirmed by Candelaresi et al. (2014). There seems to be no correlation between maximum flare energy and stellar rotation (Notsu et al. 2013). The search for superflares in solar-type stars was extended C 2015 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society

2 by Shibayama et al. (2013) who found flares on 279 G-dwarfs. They found evidence that stars with superflares tend to have extremely large starspots and that superflares occur surprisingly often (every d) on G dwarfs. Solar-type stars have convective envelopes which allows the generation of a magnetic field through the dynamo effect. Since stars earlier than F5 have very thin superficial convective zones, magnetic fields are not expected to occur. Indeed, the drop in X-ray emission in A stars (Schröder & Schmitt 2007) suggests that these stars lack a corona and support the absence of a magnetic field. It is therefore surprising that superflares have been observed in 19 out of 2000 A stars observed by Kepler (Balona 2012). Furthermore, A stars show periodic light variations consistent with rotation (Balona 2012, 2013), indicating the presence of starspots and, by implication, a magnetic field. The Kepler data consist of practically continuous photometry over a four-year period for over stars. Each data point consists of a 30-min exposure (long-cadence, LC). The study of Kepler flares have been almost entirely confined to LC observations. The 30-min exposure times prohibit the detection of short-lived flares and greatly lowers the amplitude of flares lasting for less than an hour or two. Moreover, the LC data do not permit a study of the shapes of flares which might provide clues to the mechanism which generates superflares. We searched for flares by visual inspection of a sample of stars observed in LC mode, but only for data up to Quarter 12 (about three years of continuous photometry). We were able to detect flares in 743 stars which include the 373 cool flare stars previously discovered by Walkowicz et al. (2011). Short-cadence (SC) exposures, which are of 1-min duration, are more suitable for the study of stellar flares. However, SC data are available for only 4828 stars and the time coverage is usually only a few months (mostly about two months). Characteristics of the SC data are described in Gilliland et al. (2010). We examined the light curves of stars in the Kepler field for which SC data are available and found 209 flare stars. These observations provide a unique set of flare data of micromagnitude precision and good time resolution, which is unlikely to be repeated for many years to come. Our aims are to determine the distribution of flare stars as a function of spectral type, to study the effect or rotation, to investigate possible correlations between the morphology of the flares and the stellar parameters and to determine if there is a dependence of flare frequency with orbital phase in binary systems. 2 THE DATA Prior to the launch of the Kepler spacecraft, ground-based multicolour photometry of stars in the Kepler field were observed. These data were used to obtain effective temperatures, surface gravities and radii for most stars. These values are listed in the Kepler Input Catalogue (KIC), which also gives the position of each star and the apparent magnitude in the Kepler passband. In this paper, stars are identified by their number in the KIC. A description of the KIC is given by Brown et al. (2011). The Kepler light curves are available as uncorrected simple aperture photometry (SAP) and with pre-search data conditioning (PDC) in which instrumental effects are removed. In this paper, we are interested in flares which may be mistaken as outliers in the PDC light curves. We therefore only used the raw SAP data in our search for flares. These data are publicly available on the Barbara A. Mikulski Archive for Space Telescopes (MAST, archive.stsci.edu). We searched for flares in a sample of LC light curves. These data include all stars with T eff > 6500 K as well as stars Flare stars across the H R diagram 2715 cooler than 6500 K with Kepler magnitude Kp <12.5. The survey only included data from Quarters Fig. 1 shows examples of flares observed in LC mode. Most of the 743 flare stars thus found are cool dwarfs, but flare stars as hot as early A are found. Among the 743 flare stars are 26 eclipsing binaries and 532 stars which we classified as rotational variables. Of the 4758 stars observed in SC mode, 209 stars appear to flare. A total of 3140 flares were detected in these stars. Parameters for each flare are available electronically; Table 1 is an extract from this file. The stellar parameters for flare stars observed in SC mode are listed in Table 2. Although most stars are cool dwarfs, there are 10 A-type main-sequence stars and two subdwarf B stars. For the brighter stars in Table 2, photon statistics imply a photometric precision of around 0.1 ppt (parts per thousand) per data point, while for the fainter stars it is about 1 ppt. Most of the flares detected in Figure 1. Examples of flares in stars observed in LC mode. KIC numbers are indicated. Table 1. Time of flare maximum, t max, relative to BJD , integrated flux, EW (hours), relative flux amplitude, F/F, flare duration, t (hours), and flare energy, E (erg), for flares observed in Kepler SC light curves. The complete table for 3140 flares is available electronically. KIC t max log EW log F F t log E

3 2716 L. A. Balona Table 2. Flare stars observed in SC mode. The Kepler magnitude, Kp, effective temperature, T eff (K), and gravity, log g, are from the KIC. The variability classification is as follows: ROT rotational variable, DSCT δ Scuti star, E eclipsing star, EA detached eclipsing binary, EB semi-detached eclipsing binary, SOL solar-like oscillations. The rotational period, P rot, and orbital period, P orb, are in days. KIC Kp T eff log g Type P rot P orb Notes KIC Kp T eff log g Type P rot P orb Notes ROT ROT/E ROT/E KOI DSCT ROT/E KOI E 3.17: Kepler ROT ROT/E 3.30: KOI SOL ROT ROT DSCT ROT/EA ROT/E Kepler ROT ROT/E KOI EA EA ROT/E Kepler EA ROT/E KOI ROT ROT 6.85: ROT 8.8: ROT EA ROT/EA ROT ROT ROT/SOL E Kepler ROT EA ROT SOL ROT ROT ROT 38.55: ROT/E KOI ROT EB ROT ROT ROT ROT 26.00: ROT ROT 39.00: ROT ROT/E KOI ROT ROT ROT ROT ROT ROT ROT 13.55: ROT KOI ROT ROT ROT/EA ROT ROT/EA KOI ROT/SOL ROT/SOL ROT ROT/SOL ROT ROT EA ROT/E KOI ROT ROT 16.75: ROT ROT/E KOI ROT ROT sdb ROT ROT/EA 15.10: ROT ROT ROT ROT ROT ROT ROT ROT ROT/E KOI ROT ROT ROT ROT 23.25: KOI ROT ROT/SOL KOI ROT GDOR ROT ROT ROT/E KOI ROT ROT ROT ROT ROT ROT ROT ROT/E ROT ROT/E DSCT ROT ROT ROT ROT ROT ROT/E KOI ROT ROT ROT ROT A2mF ROT ROT 0.444

4 Flare stars across the H R diagram 2717 Table 2 continued KIC Kp T eff log g Type P rot P orb Notes KIC Kp T eff log g Type P rot P orb Notes ROT ROT ROT ROT ROT/E KOI EB ROT EB SOL ROT A2p: ROT/E 45.28: KOI E ROT ROT/EA ROT ROT ROT ROT ROT ROT 1.95: ROT/EA ROT/EA ROT/E KOI ROT sdb+f/g ROT/EA ROT/EA ROT ROT 29.0: ROT/E KOI ROT ROT NGC6811-L ROT/E KOI ROT/E KOI ROT ROT ROT ROT ROT ROT ROT ROT 13.50: KOI ROT E KOI ROT ROT/EA ROT/E KOI ROT E KOI ROT ROT ROT/E 7.33: KOI ROT ROT ROT ROT/EA KOI ROT ROT ROT 4.82: ROT/E Kepler ROT ROT ROT KOI ROT/E ROT 32.97: ROT ROT/EA ROT ROT ROT ROT EB ROT 23.90: ROT KOI ROT KOI ROT ROT ROT ROT ROT ROT SOL ROT ROT/E ROT/E 35.0: Kepler-16b SC mode have peak intensities in excess of 5 ppt, but it is certainly possible that some lower amplitude flares may be a result of noise. From the nature of the light variations, we infer that nearly all these stars must have spots which causes rotational modulation. We can therefore estimate the rotation period from the highest peak in the periodogram. These periods are shown in Table 2. For many of these stars, the rotation periods have also been measured by McQuillan, Mazeh & Aigrain (2013, 2014), Reinhold, Reiners & Basri (2013) and Nielsen et al. (2013). In most cases, the agreement between our value and the published rotation period is excellent. Many stars are eclipsing binaries and some have confirmed planets. Table 2 also lists the orbital period determined from the light curve. We note that in some cases the rotation period is different from the orbital period. In Table 3, we show the integrated flux, peak intensity, total duration and energy of the strongest flare in each star. The integrated flux was measured by summing the flare intensity over the duration of the flare. Also shown is the flare frequency and number of flares detected in the star. Fig. 2 shows examples of flares observed in SC mode. It is much easier to be certain of a flare in the SC data because there are many more data points defining the flare. Also, flares of short duration may be detected which are invisible in LC data. In searching for flares in the Kepler light curve, it is important to avoid instrumental effects as far as possible. There are, indeed, 82 features in the light curves of stars which resemble flares but which occur at precisely the same time in every star. Such features are avoided since they are clearly not intrinsic to the star. Of course, the detection of a flare event is subjective since these are not repeatable. Small amplitude flares may still be artefacts it is not really possible to be certain in every case.

5 2718 L. A. Balona Table 3. Flare stars observed in SC mode. The effective temperature, T eff (K), is from the KIC. The integrated flux, EW (hours), the flare intensity, F/F,the flare duration, t (hours), and the flare energy (erg), of the most intense flare is given. The flare frequency, f (flares/day), and the number of flares, N, are also shown. KIC log EW log F F t log E f N KIC log EW log F F t log E f N KIC log EW log F F t E f N

6 Flare stars across the H R diagram 2719 Table 3 continued KIC log EW log F F t log E f N KIC log EW log F F t log E f N KIC log EW log F F t E f N Figure 2. Examples of flares in stars observed in SC mode. KIC numbers are indicated. Figure 3. Theoretical H R diagram for Kepler flare stars (large filled circles) observed in LC mode using stellar parameters in the KIC. The small dots are stars which were examined but no flares detected. 3 INCIDENCE OF FLARE STARS IN THE H R DIAGRAM Fig. 3 shows the H R diagram of all Kepler stars in which at least one flare was detected in the LC data. These are shown by large symbols. The effective temperatures and radii in the KIC were used to obtain the approximate luminosities. Also shown are stars in which no flares were detected (small symbols). Note that there are no supergiant stars among the Kepler data and very few stars earlier than A0. Therefore nothing can be inferred about the occurrence of flares on these stars. Clearly, only the most intense flares can be detected. Nearly all flares detected by Kepler have amplitudes in excess of 500 ppm. Table 4 shows the number of flare stars in each spectral type range as determined from the effective temperatures listed in the KIC. The per cent flare stars found in LC mode for K and M types are not a true reflection of the frequency of flare stars in this temperature range because we included all flare stars observed by Walkowicz et al. (2011) but omitted many non-flare stars in the same effective temperature and magnitude range. The fraction of flare stars among the cool stars observed by Walkowicz et al. (2011) is only 1.6 per cent. Flares are more easily distinguished with SC data, but fewer flare stars are found because of the smaller sample and shorter duration of the SC observations. The SC observations were not selected to detect flare stars and the relative numbers of flare stars detected in SC mode should not be biased with regard to effective temperature. The relative numbers of flare stars in different spectral classes are shown in the last column of Table 4. Fig. 4 shows the location in the H R diagram of flare and non-flare stars observed in SC mode. It is interesting to note that, while there is a decrease in relative numbers of flare stars with increasing effective temperature, the decrease is not as large as might be expected. Many factors contribute to this decline. Ground-based optical observations of cool flare stars indicate that the flux distribution of a flare closely resembles that of an A or B star (Kowalski et al. 2013). The contrast factor makes it easier to detect flares on cool stars than on hot stars. Since the stellar luminosity increases with effective temperature, flares on hot stars need to be more energetic than those on fainter, cooler stars to attain the same visibility. Both these effects conspire to a decline in flare visibility with increasing effective temperature. The observed drop of a factor of 4 in relative numbers of flare stars between K M and A F stars is therefore an upper limit and the actual drop must be considerably smaller or even non-existent. The fact that flares are seen in A stars is particularly problematic and it is natural to assume that the flare does not originate in the A star but on a cool companion. If that is the case, the flare amplitude will be very much diluted since an A star is typically about 100 times more luminous than a K or M dwarf. Therefore, if the flare originates in a cool companion, one expects a much smaller flare amplitude

7 2720 L. A. Balona Table 4. The number of flare stars, N flare found in a sample of N stars of a given spectral type. The last column is the percentage of flare stars in the given spectral type. The left-hand columns refer to LC mode and the right-hand columns to SC mode. LC SC Type T eff N N flare Per cent N N flare Per cent K+M G F A Figure 4. Theoretical H R diagram for Kepler flare stars (large filled circles) observed in SC mode using stellar parameters in the KIC. The small dots are stars which were examined but no flares detected. in A stars than in cool stars. Fig. 5 shows the average peak flare amplitude as a function of effective temperature as determined from Kepler observations. Also shown is the expected flare amplitude if the flare is attributed to a cool companion rather than the star itself. In constructing this curve, the luminosity of the presumed cool companion is taken to be log L/L = 0.74, which is the average luminosity of cool Kepler flare stars analysed by Walkowicz et al. (2011) as determined from their effective temperatures and radii in the KIC. The flare intensity in the presumed cool companion is assumed to be log F/F = 2.48, which is the average peak flare intensity of cool Kepler flare stars measured by Walkowicz et al. (2011). The curve in Fig. 5 shows that the expected flare intensity when a cool companion of the above luminosity and flare intensity is added. This shows that the expected flare intensity for A stars is about two orders of magnitude smaller than actually observed. We can conclude, without doubt, that flares on A stars cannot be attributed to a cool companion. It is possible that a flare in an A star could originate in a magnetic field spanning the A star and a companion or a disc, but this still implies that A stars must have significant magnetic fields. That A stars must have magnetic fields is, in any case, attested by the fact that a significant fraction of A stars have spots (Balona 2013). 4 CORRELATION WITH STARSPOTS It has often been suspected that flare stars have spots larger than non-flare stars (Balona 2012; Shibayama et al. 2013), but this has never been proven. From the appearance of the light curves and periodograms of the sample of over stars that we examined, a large fraction seem to have light curves similar to stars with spots. Figure 5. The average peak flare intensity derived for Kepler stars within a given effective temperature range (points) compared with the flare intensities calculated for a non-flaring star with a cool flare companion (line). One standard deviation error bars in flare intensities are shown. These data are derived from table 6 of Balona (2012). It is therefore not a surprise that the flare stars in Table 2 are nearly all rotational variables of this kind. Of course, the appearance of the light curve and periodogram is no guarantee that the cause of variability is a result of starspots. In particular, the light curves of pulsating γ Dor stars are very similar to those generally attributed to starspots. From the PDC light curve of these presumed spotted stars, we calculated the median absolute deviation which we use as a measure of the light amplitude variation. Using the median reduces the bias due to possible outliers in the Kepler LC light curves. In this way, and using the effective temperatures listed in the KIC, we can construct the amplitude distribution due to spots for stars within a given effective temperature range. This distribution can be compared with that of flare stars as shown in Fig. 6. It should be noted that the distribution continues past the amplitude limit shown in the figure. We assume that rotational modulation is confined to variables with amplitudes lower than 6 ppt for the purpose of comparing the amplitude distributions of flare and non-flare stars. The top panel of Fig. 6, which shows the amplitude distributions for all stars, certainly indicates that flare stars have larger amplitudes and hence larger starspots. The situation for individual effective temperature ranges is not as clear, but one can indeed conclude that there is a tendency for flare stars to have larger spots (larger amplitudes) than non-flare stars.

8 Flare stars across the H R diagram 2721 Figure 6. Distribution of median light variation for all stars in a particular spectral type range (solid histogram) compared to the median light variation for flare stars (dashed histogram). 5 THE EFFECT OF ROTATION Several recent studies have suggested that the rotation rate of a star plays an important role in the rate of flaring. For example, Notsu et al. (2013) find that slowly-rotating solar-type stars can still produce flares as energetic as those in more rapid rotators. However, the average flare frequency is lower in slowly-rotating solar-type stars. This trend was confirmed by Shibayama et al. (2013). We know that rapid rotation is correlated with increased chromospheric activity, which in turn implies increased magnetic activity (Noyes et al. 1984). This is consistent with the idea that superflares are a result of magnetic reconnection. For G, K and M stars, Candelaresi et al. (2014) find a quadratic increase of the flaring rate with rotation rate up to a critical rotation rate. Thereafter, the flare rate decreases linearly with increasing rotation rate. The rotation rate of a star depends on many factors, such as the age of the star. The open cluster and Mount Wilson star observations suggest that rotating stars lie primarily on two sequences. These sequences, and the fractional numbers of stars on each sequence, Figure 7. Distribution of the rotation periods for all stars in a particular spectral type range (solid histogram) compared to the rotation periods for flare stars (dashed histogram). evolve systematically with cluster age (Barnes 2003). Why these sequences exist is not entirely clear. In order to determine the role of rotation, we obtained the rotation period of 438 flare stars observed in LC mode from the catalogue of rotation periods of cool Kepler stars in McQuillan et al. (2014). These data were supplemented by the stars in Table 2. In Fig. 7,we show the distribution of rotation periods for flare stars compared to the distribution for all stars in the same spectral type range. It is quite clear from these distributions that flare stars rotate at significantly higher rates, at least for G and K M stars. There are only 48 F-type flare stars with known rotation periods, but the indication is that here, too, flare stars have generally shorter rotation periods. The mean rotation period for F stars is 6.22 d, whereas it is only 3.14 d for F-type flare stars. 6 MORPHOLOGY AND ENERGETICS OF FLARES The SC Kepler data provide a unique opportunity to study the shapes of flares in different types of stars. About half of the 3140 flares detected in Kepler SC mode have sufficient signal to noise (S/N) to enable an analysis of the shape of the flares. The majority of flares exhibit a characteristic simple rapid rise and exponential decay, but a considerable fraction (285 flares or 18 per cent) show a bump on the decay branch or else a definite change in slope (230 flares or 14 per cent). In other words, about one-third of flares have complex light decay shapes. Examples of these two types are

9 2722 L. A. Balona Figure 8. Examples of flares with a bump (left-hand panels) or with a break in decay rate (right-hand panels). KIC numbers are indicated. Figure 9. The H R diagram for SC flare stars with symbol size proportional to median duration of flares. shown in Fig. 8. We do not have enough information to understand what these flare shapes signify. We noticed that median flare duration times vary over a very wide range from about 0.10 h (e.g. KIC , , ) to about 15.0 h (e.g. KIC , , ). Such a huge range must relate to some stellar physical parameter. The only correlation we could find was with surface gravity. As Fig. 9 shows, stars with the longest flare duration tend to be giants, but the correlation is not unique. The energy stored in a magnetic field of magnitude B is approximately E = L 3 B 2 /8π,whereL is the size of the magnetically active region. If stellar flares are a result of magnetic reconnection, then the flare energy release should be closely related to the magnetic field strength and some characteristic length, L. The value of B is generally thought to be related to the depth of the convective zone and is expected to decrease with effective temperature. There is no Figure 10. The top panel shows the H R diagram of SC flare stars with symbol size proportional to the flare energy. The bottom panel shows the flare energy, (E, in erg) as a function of stellar luminosity. reason to suspect that L should increase with stellar size, in which case E is expected to decrease with effective temperature. This is the currently accepted view: stellar activity decreases as the depth of the convective zone decreases and ultimately vanishes for stars hotter than the granulation boundary. Kepler observations show that this view is not correct. In Fig. 10, we show the H R diagram of SC flare stars with symbol size proportional to flare energy. Contrary to expectations, the flare energy increases with stellar luminosity as shown in the bottom panel of the figure. The lack of low-energy flares for highluminosity stars is easily explained by observational limitations. However, the lack of high-energy flares in low-luminosity stars must be a real effect. In other words, the trend for flares to be of higher energy in more luminous stars cannot be attributed to a selection bias. This is a new and unexpected result which is not easy to understand. This could be a scaling effect, i.e. the larger the star the larger the active region, and therefore, the larger the energy that can be stored for the same magnetic field strength. Supposing that the magnetic field strength, B, is much the same for all stars, one expects the flare energy, E, to be proportional to the volume of the active region, L 3. Fig. 11 shows log E as a function of the stellar radius, log R/R. The slope has a value close to 3, suggesting that the loop length, L, is proportional to the stellar radius. In other words, the larger the star, the larger the size of the active region or loop length. Suppose that L = αr, where α is a constant of proportionality. Then we may write log E = 3log R/R + log C with C = α 3 R 3 B2 /8π. The straight line in Fig. 11 is log E = 3log R/R This leads to

10 Flare stars across the H R diagram 2723 Figure 11. The median flare energy, (E, in erg) as a function of stellar radius, R. The straight line is log E = 3log R/R , indicating that the flare energy is proportional to R 3. B 32α 3/2 G. Since the value of α cannot exceed unity by very much, this implies field strengths of the order of tens of gauss, which seems a reasonable value. We may therefore conclude that the magnetic reconnection model could account for the increase of flare energy with luminosity if the size of the active region also increases in proportion to the stellar radius. The required magnetic field strength is consistent with typical field strengths found in main-sequence stars. The fact that the observed global magnetic fields in normal A stars are typically less than 1 G (Lignières et al. 2009; Petit et al. 2011) probably implies that the magnetic fields on these stars are of complex topology with mixed polarities, as it is in the Sun. 7 ECLIPSING BINARIES It has been suggested that flares in active close binaries (RS CVn stars) could arise by magnetic reconnection in field lines joining the two stars (Simon et al. 1980; van den Oord 1988; Gunn et al. 1997). In that case, one might expect that most flare activity would occur in the region between the two stars. The visibility of this region will vary with orbital phase, so we might expect that the flare activity should vary with orbital phase. We note from Table 2 that there are several eclipsing binaries among the SC flare stars. We can look for a correlation of flare activity with orbital phase in three stars in which a large number of flares have been detected, KIC , KIC and KIC KIC (KOI-256) is a mutually eclipsing post-common envelope binary consisting of a cool white dwarf (M = M, R = R, T eff = 7100 K) and an active M3 dwarf (M = 0.51 M, R = R, T eff = 3450 K) with an orbital period of d (Muirhead et al. 2013). It is one of the most active flare stars in our sample. Fig. 12 shows the light curve and the number of flares per unit time phased with the orbital period. There are a few very strong flares occurring at what seems to be random phases. The top panel shows that there is no preferred phase. There is thus no evidence to suggest that flares occur in the space linking the two stars. KIC (FL Lyr) is a well-studied double-lined eclipsing binary system consisting of an F8V and a G8V companion (masses 1.22 and 0.96 M ) in an orbit with P = d (Popper et al. 1986). It is clearly an RS CVn system with 84 detected flares. The top panel of Fig. 13 shows quite a large variation in flare frequency with phase, but once again there is no clear dependence of flare Figure 12. Bottom panel: the light curve of KIC phased with the orbital period P = d showing the eclipse (phase 0) and many flares. Top panel: the distribution of flare frequency with orbital phase. Figure 13. Bottom panel: the light curve of KIC phased with the orbital period P = d showing the eclipse (phase 0) and many flares. Top panel: the distribution of flare frequency with orbital phase. frequency with phase. The low incidence of flares around primary minimum merely reflects the difficulty in detecting flares during the steeply descending and ascending phases. KIC consists of an M1V star (M = 0.51 M ) with a substellar companion of radius 4 earth radii in an orbit with P = d (Muirhead et al. 2012). Fig. 14 shows the light curve. The transit is too shallow to be visible in the plot. Again, the flare frequency does not seem to depend on phase. In all three systems, two stellar and one substellar, we have not been able to find any significant dependence of flare frequency with orbital phase. Although this does not disprove the idea that flares originate by magnetic recombination of field lines joining the star and its companion, it is not encouraging for this hypothesis.

11 2724 L. A. Balona Figure 14. Bottom panel: the light curve of KIC phased with the orbital period P = d showing the eclipse (phase 0) and many flares. Top panel: the distribution of flare frequency with orbital phase. 8 DISCUSSION AND CONCLUSION The most important and surprising conclusion that we can deduce from this study of white light flares in the Kepler photometry is that the relative number of flare stars is probably much the same from cool M dwarfs to hot A stars. The observed decrease from about 10 per cent for K M stars to 2.5 per cent for A F stars is largely a selection effect. There are two effects which can cause the decrease: the decreased contrast between a hot flare and a hot A F star and the dilution of relative flare intensity due to the larger luminosities of A F stars. While the superflares in solar-type stars discovered by Maehara et al. (2012) are of great interest, they can at least be broadly understood because it is easy to conceive of how a magnetic field can be generated by convection through a dynamo mechanism. Superflares on A stars are in a different category because these stars possess only very thin subsurface convective zones in which the dynamo mechanism cannot operate. The view that normal A stars have very weak global magnetic fields is supported by observations. It is therefore a great surprise that the fraction of A stars which flare is not very different from the fraction of cool flare stars. The idea that flares on A stars are not intrinsic to the star, but originate in a cool companion is easily dismissed because the expected dilution of relative flare intensity as a result of the much more luminous A star is two orders of magnitude smaller than actually observed (Fig. 5). By comparing the rotational light amplitudes of flare and nonflare stars (Fig. 6), we confirm that flare stars tend to have larger spots than non-flare stars, as previously suspected (Shibayama et al. 2013). Flare stars have distinctly higher rotation rates than non-flare stars (Fig.7). Clearly, rotation plays an important role in generating flares, but this role may be indirect. Rotation is a measure of the age of the star (Barnes 2003), and that young stars are more active (Noyes et al. 1984). However, the direct role of rotation, including differential rotation, needs to be investigated. Differential rotation could provide the necessary energy for twisting magnetic field lines and subsequent reconnection. Although the majority of flares show a sharp rise and an exponential decay, a significant proportion (about one-third) exhibit a bump on the decaying branch or else a sharp change in decay rate. The significance of this flare morphology is not clear. We also find that stars with low surface gravities tend to have flares of longer duration. This is probably related to the differences in heat transport that occurs in stars with different atmospheric densities. The flare energy is strongly correlated with stellar luminosity and also with stellar radius. The flare energy correlates with the cube of the stellar radius (Fig. 11) which can be understood if the typical loop length of a flare is roughly proportional to the stellar radius. In other words, the larger the star, the larger the active area. The resulting field strength determined from the median flare energy is a few tens of gauss, which is quite reasonable in the framework of the magnetic reconnection model. Kilogauss magnetic fields have been measured in cool dwarfs and RS CVn stars, but typical global fields for hot stars are less than 1 G. To understand flares on A F stars as due to magnetic reconnection one has to assume that the magnetic fields in these hot stars are of mixed polarity. There are three binaries in the SC Kepler data in which a large number of flares have been measured. We do not find a dependence of flare frequency with orbital phase in any of these stars. If flares in RS CVn and other close binaries are due to magnetic reconnection in field lines connecting the two stars, one might expect such a dependence because most flares would tend to occur in the space between the two stars. The observations clearly do not support this hypothesis. In order to make further progress in our understanding of stellar flares, it would be very important to obtain light curves of solar flares in white light whole-disc observations. The lack of such observations, which may perhaps be synthesized from white light images of solar flares, prevents a direct comparison of solar flare light curves with those of stellar flares. Such a comparison is essential if we are to explain stellar flares as more energetic counterparts of solar flares. ACKNOWLEDGEMENTS This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. The authors wish to thank the Kepler team for their generosity in allowing the data to be released and for their outstanding efforts which have made these results possible. All of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS Support for MAST for non-hst data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. LAB wishes to thank the South African Astronomical Observatory and the National Research Foundation for financial support. REFERENCES Balona L. 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