Simultaneous Detection of Optical Flares of the Magnetically Active M Dwarf Wolf 359

We present detections of stellar flares of Wolf\,359, an M6.5 dwarf in the solar neighborhood (2.41~pc) known to be prone to flares due to surface magnetic activity. The observations were carried out from 2020 April 23 to 29 with a 1-m and a 0.5-m telescope separated by nearly 300~km in Xinjiang, China. In 27~hr of photometric monitoring, a total of 13 optical flares were detected, each with a total energy of $\gtrsim 5 \times 10^{29}$~erg. The measured event rate of about once every two hours is consistent with those reported previously in radio, X-ray and optical wavelengths for this star. One such flare, detected by both telescopes on 26 April, was an energetic event with a released energy of nearly $10^{33}$~erg. The two-telescope lightcurves of this major event sampled at different cadences and exposure timings enabled us to better estimate the intrinsic flare profile, which reached a peak of up to 1.6 times the stellar quiescent brightness, that otherwise would have been underestimated in the observed flare amplitudes of about $0.4$ and $0.8$, respectively, with single telescopes alone. The compromise between fast sampling so as to resolve a flare profile versus a longer integration time for higher photometric signal-to-noise provides a useful guidance in the experimental design of future flare observations.


INTRODUCTION
Solar flares are commonly observed surface phenomena, attributable to acceleration of plasma during magnetic reconnection near sunspots and active regions, that lead to sudden brightening observed in radio, optical, to X-ray wavelengths. While the detailed heating mechanism, i.e., how the magnetic energy is converted to gas kinetic energy is still unclear (Benz & Güdel 2010), it is believed that conductive and radiative processes are involved in the cooling phase. The solar flares are classified by the peak flux in soft X-rays, with the most powerful being class X peaking at > 0.1 erg s −1 cm −2 . A typical solar flare releases 10 29 -10 32 erg. Rare, major flares, which release a total energy more than ∼ 10 33 erg, are linked to coronal mass ejection events which influence space weather and pose potential hazards to terrestrial environments.
Other stars, notably late-type dwarfs, being largely convective are even more predisposed to flare activity and encompass a larger range of energy output, particularly the young ones with fast rotation rates (Feinstein et al. 2020). Based on the K2 lightcurves of G-to M-type dwarfs, Lin et al. (2019) conclude that later type stars have higher flare occurrence frequencies but generally with less energetic output. M-dwarf flares with much shorter durations have been detected in millimeter wavelengths (MacGregor et al. 2020). Later spectral types (brown dwarfs) have also shown surface activity (e.g., Rutledge et al. 2000).
Our target, Wolf 359 (GJ 406; CN Leo), at a heliocentric distance of 2.41 pc (Gaia Collaboration et al. 2021), is a known eruptive-type red subdwarf (Kesseli et al. 2019). Previously, six flare events were detected during 12.8 hours of monitoring in radio frequencies, equivalent to 47 events per 100 hours (Nelson et al. 1979). Extreme ultraviolet flare events have been reported to occur at least daily (Audard et al. 2000), and major X-ray flares have also been detected in this star (Liefke et al. 2010). Using ground-based and Kepler/K2 observations combining long-and short-cadence lightcurves of Wolf 359, Lin et al. (2021) derived a flare occurrence rate of once per day for events with a total flare energies > 10 31 erg, and ten times per year for superflares with released energies > 10 33 erg. Such an activity level is considered high even among known flaring M dwarfs.
Magnetic reconnection may not be limited to surface extrusion as in the case of the Sun. The field lines may be linked to some intricate spin-orbit magnetospheric interaction with close-in companion stars, as in RS Canum Venaticorum or BY Draconi type variables. It is known that newly born stars may anchor their magnetic field to circumstellar disks (Feigelson & Montmerle 1999), with which the entwined field lines are susceptible to reconnection and result in outbursts (10 36 erg) or in extended flaring loops (Hayashi et al. 1996;Shibata & Magara 2011). Superflare events of red dwarfs are suspected to have similar interactions with orbiting giant exoplanets (e.g., Klocová et al. 2017), though there is so far no definite supporting evidence. Wolf 359 is a fast rotator (Guinan & Engle 2018, 2.72 d) and is known to host at least two exoplanets (Tuomi et al. 2019). One of these, Wolf 359c (radius 0.1272 R Jupiter ) is hot and close in (0.018 au, orbital period 2.88 d) suggestive of a possible spin-orbit tidal lock. In the flare star AD Leo, a periodicity of 2.23 d was inferred and attributed to stellar rotation (Hunt-Walker et al. 2012). However, Lin et al. (2021) found no flare timing in synchronization with the planetary orbital phase in Wolf 359.
Here we report on an optical monitoring campaign of Wolf 359. The star was monitored for one week in 2020 April simultaneously with two telescopes. In addition to reaffirmation of the flare rate, with 13 events detected in 27 observing hours, this paper focuses on one major flare observed by both telescopes, affording the possibility to derive the underlined flare profile, whose amplitude would have been underestimated by any single lightcurve as a result of finite exposure time. While a flare is quantified by the total released energy (essentially scaled with the amplitude multiplied by the duration), usually an event is recognized mainly by a brightness spike. Our study indicates how intrinsically moderate events could escape detection, and provides guidelines for proper sampling specific to certain profiles in the experimental design.

CCD Imaging and Lightcurve Extraction
The observations reported here were carried out from 2020 April 23 to 29 simultaneously by the Nanshan One-meter Wide-field Telescope (NOWT) in Xinjiang, and one of the TAOS telescopes, except for the night of April 24 for which the TAOS site was weathered out. The TAOS telescopes, each of f/2 50 cm, used to be installed at Lulin Observatory to catch chance stellar occultation events by transneptunian objects (e.g., Alcock et al. 2003;Zhang et al. 2013). Two of the original four TAOS telescopes were relocated in the spring of 2020 to Qitai Station in Xinjiang, some 300 km from Nanshan, also operated by Xinjiang Astronomical Observatory.
The NOWT was equipped with an E2V back-illuminated CCD203-82 camera, with 12-micron pixels, spanning 1. 13/pixel on the sky. For the data reported here, the NOWT observed Wolf 359 in the R band for the first five nights, and in the V band for the rest two nights. The exposure time was 18 s, with a dead time of approximately 12 s between exposures, amounting to a cadence of ∼ 30 s.
The TAOS telescope was equipped with a Spectral Instrument 800 camera with 13.5-micron pixels and a plate scale of 2. 78/pixel. A custom-made filter was used which has a flat-response in 500-700 nm approximately comparable to an SDSS r filter. For the observing campaign of Wolf 359, the exposure time was 45 s, with a dead time of ∼ 0.5 s.
Wolf 359 has a high proper motion due to its proximity, with µ α = −3866.338±0.081 mas yr −1 and µ δ = −2699.215± 0.069 mas yr −1 (Gaia Collaboration et al. 2021). Figure 1 displays its position in four epochs, three recorded by the Digital Sky Survey in years 1953, 1988 and 1995, whereas the last one was taken in 2020 reported in this work.
No attempt was made to synchronize the shutter openings of the two telescopes. The different cadences, hence sampling functions, of the two telescopes observing the same flare event in turn provide the possibility to derive the underlined flare profile, which would not have been possible otherwise with a single data set. Images were processed by the standard procedure of bias, dark, and flat-field corrections. Aperture photometry was then performed using SExtractor (Bertin & Arnouts 1996 with an adaptive aperture to measure the brightness of Wolf 359 with an aperture size of 4-8 pixels for the NOWT images and of 8-9 pixels for the TAOS images. Figure 2 shows an illustrative image taken by NOWT and by TAOS reported here with the photometric aperture marked. No standard star was observed and the brightness is referenced to that of the stellar quiescent state. In every case, two reference stars near Wolf 359 in the same image frame were also measured to assess any variations due to the sky. Figure 3 exhibits the NOWT lightcurves obtained during the campaign with the individual flare events marked. The major event detected on 2020 April 26 by both NOWT and TAOS telescopes is presented separately in Figure 4. While the reference stars remained steady in brightness, Wolf 359 experienced a brightening of ∼ 0.65 mag detected by NOWT, and ∼ 0.38 mag detected by TAOS. Table 1 summaries the photometric measurements used to plot the lightcurves in Figure 3. Columns 1, 2, and 3 list, respectively the calendar date, telescope, and the Heliocentric Julian Date (HJD) of the observation (middle of an exposure). The remaining columns are magnitude and associated error of Wolf 359, and of the two reference stars. b The variable m1 is the magnitude of reference star 1.
c The variable m2 is the magnitude of reference star 2.
Note- Table 1 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.
In a total of 27 data hours, 13 flare events were identified visually in the lightcurves, in accord with the (non)variation of the reference stars at the same time. The flare parameters for each event such as the peak amplitude, the rising and decay time scales were derived, from which the integrated total energy was computed. Figure 3. The NOWT lightcurves of Wolf 359 (in black) and of the two reference stars (in gray). Each flare event detected is marked by a vertical red line. The observations from April 23 to 27 were taken in the R band, whereas on April 28 and 29 they were in V . The sudden flux drop on April 27 around HJD 2458967.2 was due to weather conditions, manifest also in the reference lightcurves. The major event on 2020 April 26 detected also with TAOS was analyzed separately and not shown here.

Flare Properties
A flare profile is parameterized with (1) the epoch and amplitude of the observed peak in the lightcurve, and relative to the peak, (2) the rising time scale, and (3) the decline time scale. Usually (2), signifying the energizing process is relatively fast, whereas (3), relevant to the cooling mechanism drops off slower, typified by an exponential or a power-law decay.
First, the observed lightcurve in unit of flux (count) f (t) is subtracted of then divided by the detrended quiescent stellar count (a linear fit to the flux away from an event) within the spectral filter f 0 . To compute the released energy from such a normalized lightcurve, ∆f /f 0 = (f (t) − f 0 )/f 0 ), we follow the equivalent duration method described by Gershberg (1972). The flare is approximated by a typical black-body of effective temperature T flare of 9000 K (Mochnacki & Zirin 1980), thereby having a bolometric luminosity of where σ is the Stefan-Boltzmann constant, and A flare is the flare area, related to the observed flare luminosity as Here B λ (T ) is the Planck function and R λ is the spectral response function for which only the filter response is considered. The observed photospheric luminosity of a star of radius R * is then The ratio of L flare to L * is the amplitude, and the area of the flare becomes with the flare luminosity being computed as, Finally the total flare energy is estimated from the bolometric luminosity of the star multiplied by the equivalent duration.
where α is the constant accounting for the correction for the black body assumption and the filter response. Taking the stellar temperature as 2900K (Fuhrmeister et al. 2005), α of 0.11 for the standard Johnson-Cousins R filter, and 0.05 for the V filter, Equation 1, adopting log L * /L = −2.95 ± 0.05 (Pavlenko et al. 2006), leads to the derivation of the flare energy, E flare released in an event. Note that this method assumes the flare temperature to be constant throughout the event, and the behaviour of the flare is the same in all spectral bands. Table 2 summarizes the parameters of the 13 events including the superflare detected simultaneously by two telescopes. For all events reported here, the rising time is less than ∼ 30 s, i.e., shorter than the cadence of each of the telescopes, so was not derived. This rising/heating time scale is contrasted to those of several minutes among solar-type flares (Yan et al. 2021). In our analysis, the lightcurve then takes a straight line from the stellar quiescent state, i.e., one data point prior to the peak. The date/time refers to the middle of the exposure within which a flare occurred. The duration of an event is estimated from one data point prior to the peak to where the lightcurve falls below the uncertainty in ∆f /f 0 , typically 0.002 for NOWT and 0.01 for TAOS. The determination of duration time therefore is somewhat subjective, but serves to gauge the relative time length of an event. The events had energies ranging from ∼ 3 × 10 29 erg to ∼ 3 × 10 31 erg, lasting for a couple of minutes to over 20 minutes. Our campaign was not long enough to catch more powerful, presumably rarer events.
We note that the term "superflare" is applicable to solar events, but not well defined for stars, whether it refers to total released energy (in unit of energy) or relative to stellar photospheric luminosity (in unit of power). A superflare of solar-type stars releases 10 33 -10 38 erg (Schaefer et al. 2000), which given a typical duration of ∼ 30 min (Yan et al. 2021) amounts to a ratio to stellar luminosity 10 −4 L L flare 10 1 L . An M dwarf flare, on the other hand, gives out a total energy, 10 31 -10 34 erg, with the fast rotators liberating more, up to 10 35 erg (Lin et al. 2021). For the events reported here, the most energetic one has E ∼ 3 × 10 31 erg within ∼ 25 min, hence with L * = 1.1 × 10 −3 L for the star, leading to L flare /L * ≈ 0.5%. The two-telescope event is powerful, having the rising and exponential decay time scales both less than about 30 s, with a peak amplitude comparable to the stellar flux, we refer this major event as a superflare.

THE FLARE RATE OF WOLF 359
The flare rate we detected, 13 flares in 27 hours, equivalent to about 48 events per 100 hours, is consistent with literature results (e.g., Nelson et al. 1979, 47/100 in radio wavelengths). In terms of energetics, Figure 5 presents  Table 2. Fitted with a power-law, log ν ∝ β log E, where ν is the flare frequency distribution with energy greater than E (Hunt-Walker et al. 2012), the index β = −0.87 ± 0.19 if the data at the lowest-energy end, incomplete in our experiment, are excluded. Even with our limited number of events, spanning two orders of magnitude in energy, the distribution seems more complex than a single power-law, a conclusion that has been drawn for this star and for active M dwarfs in general (Lin et al. 2021). By and large, Wolf 359 produces a flare as powerful as 10 30 erg approximately once every three hours. While a fast rotating M dwarf like Wolf 359 tends to produce frequent and powerful flares (Lin et al. 2019), it is not known whether the boosted magnetic activity of our target is linked to its own rapid spin alone or to the close-in planet with an enlarged emission volume. A more energetic flare conceivably would take a longer time to dissipate. Because a flare profile may be complex (e.g., more than a simple exponential or power-law decay, or multiple flares in the lightcurve segment), we correlate the duration of a flare with energy, exhibited also in Figure 5. A log-log linear fit gives t flare ∝ E 0.49±0.08 flare . Maehara et al. (2015) related the magnetic reconnection time scale to the Alvén time and derived analytically for solar-type stars, t flare ∝ E 1/3 flare , to be comparable to their observed slope of 0.39 ± 0.03. Such a duration-energy relation is also observed in the active dM 4e dwarf GJ 1243 , of a slope about 0.4, reading from their Figure 10), and in other M dwarfs (Raetz et al. 2020). Our slope is marginally steeper but, given the incompleteness for weaker events, should be overestimated. If so, this indicates a similar energizing mechanism in M dwarfs as in solar-type stars.

INTRINSIC FLARE PROFILE DERIVED FROM TWO-TELESCOPE LIGHTCURVES
The simultaneous detection of the same event with different sampling functions allows us to break the parameter degeneracy and thereby to infer the underlined flare profile. By varying profile parameters, the lightcurve that would have resulted for a given sampling function of a telescope is compared with what was actually observed. The profile that yields the least deviation in the chi-squared sense in the two lightcurves, is considered the "best" solution, closer to the "truth" than any individual observed lightcurve alone.
The normalized lightcurves by the two telescopes are displayed in Figure 6. Note that NOWT detects a relative amplitude of ∼ 0.8, whereas TAOS detects an amplitude of ∼ 0.4. There is a caveat that, because of the different filters used, f 0 differs for each telescope. Conceivably, the star itself is red in color, but the flare should be relatively blue, but here we assume the same f 0 to normalize each lightcurve. The peak of the flare was sampled differently: TAOS detected it at HJD of 2458966.133611, whereas NOWT detected it at HJD of 2458966.1336424, with an offset of about 2.7 s. Figure 6. As in Figure 4 for the superflare observed in 2020 April 26, but now each lightcurve has been detrended and rescaled relative to the quiescent stellar brightness, in flux/count unit instead of magnitude. The horizontal segments mark the sampling function for NOWT (in red) and for TAOS (in blue), with the filled parts for exposure and open (or a dark bar for TAOS) for dead time intervals.
The sampling effects on an intrinsically continuous function include the integration time (averaging the signal), dead time (no signal), and the lapse, i.e., the offset time between the peak relative to the sampling window. It is this offset and finite sampling time that smear off the peak of the flare and distort the shape of a lightcurve, accounting for different flare statistics between the long-cadence versus short-cadence K2 lightcurves (e.g., Raetz et al. 2020;Lin et al. 2021). A grid of event parameters were used to compute the simulated lightcurves with steps of 0.01 in the peak amplitude and 1 s in decay time scale, chosen as the appropriate step parameters with extensive simulations. For the decay portion, different models were exercised: a single exponential function (A e −t/τ1 ), a double-exponential function (A e −t/τ1 + B e −t/τ2 ), and a power-law fall-off (A t −γ ), where A and B are amplitudes, t is time, τ 1 and τ 2 are correspondent exponential time constants, and γ is the power-law exponent index. In each case, the computed lightcurve according to a specific sampling function was compared with the actual observed one ("observed" minus "computed", "O − C") to evaluate the chi-squared value (sum of (O − C)/O). Figure 7 presents the best-fit results, whose parameters are summarized in Table 3. The double-exponential model, adding one more degree of freedom in the fitting, gives an over-all better account than the single-exponential function of the fading part of the lightcurves, judged by the residual χ 2 . This is consistent with the time-resolved flares of another eruptive red dwarf, GJ 1243 , and supports the notion of possibly more than one cooling mechanism (radiative and conductive, Benz & Güdel 2010).
For the power-law model, we present two cases, one with an instantaneous impulse rise, and the other with a finitetime rise. The latter is more realistic, but our data could not constrain the rising time scale. Therefore an exponential rise of 16 s is adopted here as an example, for which a peak of 1.63 times of the stellar brightness is required to fit the data, whereas for an instantaneous rise, the peak would have been 2.92 then falling off faster (with a slightly larger value of γ). In general a power-law decay requires a higher peak than an exponential model, leading to an elevated total flare energy. In our data, the impulse plus the rapid decay portion of the lightcurve spans no more than a few data points. This means that a higher time resolution is needed than reported here in order to distinguish one cooling function from another.

IMPLICATION FOR FLARE OBSERVATIONS
A flare event, detected either visually or by an algorithm (e.g., to recognize in a lightcurve an abrupt rise followed by a few data points above quiescence) is characterized by the amplitude and duration, from which the total energy is derive. The fact that the true superflare event reaches to at least 1.6 times of stellar brightness, while the observed lightcurves peak, respectively, at 0.8 and 0.4 times, as demonstrated in this work, manifests how the sampling function affects the amplitude of an observed flare. The experimental design to detect sporadic stellar flares hence pertains to an integration time as short as possible so as to resolve the flare profile given the kind of flare events targeted for detection, while commensurate with sufficient signal-to-noise. Figure 8 plots how cadence affects the detected amplitude of the superflare event reported here, for which the peak of 1.6 times of stellar brightness would be degraded quickly; e.g., with a 30-s integration the detected peak drops to less than 70%. This applies only to the specific event detected on 2020 April 26 for Wolf 359, but serves to demonstrate vividly the essence of fast sampling. The lesson is while the total energy of a flare can be reasonably estimated with a single data set, different samplings of a flare is necessary to derive the true profile in order to distinguish the heating and cooling processes. One improvement of our experiment, other than with larger telescopes to afford faster cadences, is to measure the same event at different passbands, or better yet with spectroscopy, thereby diagnosing the temperature variation during the flare.
Stellar flare activity may be elevated if the field lines have an external source to anchor to, be it a circumstellar disk, a companion, or an exoplanet, increasing the magnetic filling factors hence the emitting volume than by surface starspot pairs or coronal loops (Benz & Güdel 2010). In stars like Wolf 359 there may well be a combination of solartype surface flares plus inflated star-planet events. Long-term high cadenced monitoring observations are called for to derive any possible rotation or orbital periodicity.
As in the case of Kepler/K2, the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) provides data as useful for stellar flare research as in the primary science in exoplanets, particularly if complemented with groundbased observations of high sampling rates, (e.g., Howard et al. 2020). TESS are monitoring some bright stars of three different cadences: 20 s, 2 min, and 10 min. Figure 9 illustrates how these three sampling rates would have detected the April 26 event. Here a peak amplitude of 1.6 times of the stellar brightness is adopted (c.f., Figure 7(d)) with a zero phase lag, i.e., with the peak coinciding with the start of the sampling window, versus with a 0.5 phase lag. One sees that for this particular flare only the shortest (20 s) cadence, similar to the data reported here, can resolve the profile, with a phase-dependent amplitude of 0.6 or 0.8, respectively, but neither the 2 min (for selected targets) nor the 10 min (for the whole frame) cadence can.
In summary, our photometric monitoring of the red dwarf Wolf 359 in 2020 April detected, in 27 data hours, 13 flare events with released energy in the range ≈ 3 × 10 29 -3 × 10 31 erg, consistent with the flare occurrence rate for this star reported previously in the literature. For any single-telescope data, the peak and energy are underestimated as the result of sampling by finite integration time with a phase lapse. A major flare was detected simultaneously by two telescopes on 2020 April 26, for which the underlying flare profile is estimated. The profile parameters are model dependent, but the "true" flare amplitude might reach as high as 1.6 times of the quiescent stellar flux, whereas the two telescopes detected a peak level of 0.8 and 0.4, respectively, with the total released energy nearly four times as large. Figure 9. The computed lightcurves of the intrinsic flare profile with a peak of 1.6 times of the stellar brightness (Figure 7(d)) sampled at a cadence of 20 s (in blue), 2 min (in orange), and 20 min (in green) (a) with a zero phase lag, or (b) with a 0.5 phase lag.