Superflares on Solar-type Stars from the First Year Observation of TESS

TESS was launched on 2018 April 18, and carries four identical cameras. During its first year of observations, TESS has scanned the southern hemisphere of the sky and obtained data products for 13 segments (sector 1–sector 13). Each segment covers about 27 days. In this work, we adopt the presearch data conditioned (PDC) light curves to avoid the instrumental systematics.

First, the selection criteria of solar-type stars and Sun-like stars should be clarified. Solar-type stars are selected according to following criteria: (1) the surface effective temperature satisfies $5100\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\lt 6000\,{\rm{K}}$ and (2) the surface gravity in log scale is log g > 4.0 (Schaefer et al. 2000; Maehara et al. 2012). Those solar-type stars with $5600\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\lt 6000\,{\rm{K}}$ and stellar periodicity >10 days are considered as Sun-like stars. In total, 26,034 targets meet the requirements based on the latest TIC v8 (Stassun et al. 2019), which is expected to be the last version of the TIC. Then, we examine these solar-type stars with the Hipparcos-2 catalog (van Leeuwen 2007) to exclude some confirmed binary stars. In total, 145 stars are excluded from the data set.

In contrast to Kepler, of which the pixel scale is about 4'', TESS has a larger pixel scale of 21''. 90% of the energy is ensquared by 4 × 4 pixels, in other words, it is encircled by a radius of 42'' (Ricker et al. 2015). Therefore, it is possible that in one pixel from TESS, the primary object is contaminated by other stars. Therefore, we use Gaia-DR2 (Gaia Collaboration et al. 2018) to search for stars within a 42'' radius near the primary solar-type star. Within a 21'' radius, we find 155 stars containing other brighter stars, which are excluded from the data set. The reasons for this are: (1) flux from these brighter stars may significantly affect the light curves of the main target; and (2) in just one pixel scale (21''), we cannot separate these brighter stars apart from main targets. Next, 2849 solar-type stars, of which the nearby stars show a surface effective temperature within 3000 K–4000 K, are flagged as possessing M dwarf candidates. The reason why they are not excluded from the data set is shown in Appendix A.

The PDC light curves of the solar-type stars are used to search for superflares. There are several selection methods in the literature. For example, clean light curves with only flare candidates can be acquired by fitting the quiescent variability (e.g., Walkowicz et al. 2011; Wu et al. 2015; Yang & Liu 2019). In addition, flare candidates can also be obtained by calculating the distribution of brightness changes between consecutive data points (e.g., Maehara et al. 2012; Shibayama et al. 2013; Maehara et al. 2015). In this paper, we choose the latter method to detect superflares. According to Maehara et al. (2015), brightness changes between two pairs of consecutive points ΔF² are defined as:

$$\begin{eqnarray}&&{\rm{\Delta }}{F}^{2}\left({t}_{i,n}\right)=s\left({F}_{i}-{F}_{i-n-1}\right)\left({F}_{i+1}-{F}_{i-n}\right),\end{eqnarray} \tag{ 1 }$$

where F_i and t_i represent the flux and the time of the ith data point, n is an integer number, and s = ±1. s = 1 when both $\left({F}_{i}-{F}_{i-n-1}\right)\gt 0$ and $\left({F}_{i+1}-{F}_{i-n}\right)\gt 0$, otherwise s = −1. ΔF² during a superflare event will become much larger than the quiescent situation. We set n = 2 to detect flare candidates with rise times larger than 4 minutes (Maehara et al. 2015). Besides, we also set n = 5 and obtained another group of ΔF² in case of missing flares with longer rise times. A data point is recognized as a flare candidate when its ΔF² is at least three times larger than the value at the top 1% of the ΔF² distribution. We then located the peak data point for each flare candidate. To obtain a complete flare event, we use the data from 0.05 to 0.01 days before the peak and from 0.05 to 0.10 days after the peak in the case of n = 2, and use the data from 0.15 to 0.03 days before the peak and from 0.15 to 0.25 days after the peak in the case of n = 5. A quadratic function, F_q(t), is adopted to remove the long-term stellar variability. The start time and end time of each flare candidate are the first and last point when F(t) − F_q(t) is three times larger than the photometric error. We only reserve a flare candidate when (1) there are at least three consecutive points during the spike event, and (2) the decay time is longer than the rise time.

Then, we check each flare candidate by using pixel-level data to exclude false positives such as eclipses, random flux jumps, and cosmic rays. First, we remove the candidates occurring simultaneously on different stars. Second, the pixel-level light curves during the flare event should match the optimal aperture given by TESS. Third, visual inspection is applied to ensure the flare-like shape and obvious flux increases in pixel-level light curves as suggested by Wu et al. (2015). We present the example of a true flare event in Figure 1. The light curve of TIC121011020 in sector 4 perfectly shows a standard case of a flare-shaped curve, with rapid rise and slow decay. Figure 2 shows a case with pixel-level fluctuations under the noise level. We exclude it from superflares since it is more likely to be a very weak flare or just noise. After excluding all the false positives, we finally find 1216 superflares occurring on 400 solar-type stars.

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The periodicity of the solar-type stars are estimated through the Lomb–Scargle method (Lomb 1976; Scargle 1982), which is suitable for unevenly sampled data (VanderPlas 2018). We set the false alarm probability to 10⁻⁴ to search the stellar period (e.g., Cui et al. 2019).

Figure 3 presents the period distributions of the solar-type stars, superflares and flare stars. Note that we may miss some slowly rotating solar-type stars with P > 10 days because of the limited observing span. Most of the TESS targets were observed only in one sector, i.e., about 27 days, as shown in Figure 4. Therefore, periods longer than ∼14 days are not reliable for these targets. In total, we obtain stellar periods for 3827 slowly rotating (P > 10 days) solar-type stars and 22,207 fast rotating (P < 10 days) solar-type stars.

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Following the method of Wu et al. (2015), the stellar luminosity can be estimated with the Stefan–Boltzmann law

$$\begin{eqnarray}&&{L}_{* }=4\pi {R}_{* }^{2}{\sigma }_{\mathrm{sb}}{T}_{* }^{4},\end{eqnarray} \tag{ 2 }$$

where R_* and T_* are the stellar radius and effective temperature given by TIC v8, and σ_sb is the Stefan–Boltzmann constant. The flare energy thus can be calculated by integrating the fluxes within the flare event

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}=\int {L}_{* }{F}_{\mathrm{flare}}(t){dt},\end{eqnarray} \tag{ 3 }$$

where ${F}_{\mathrm{flare}}(t)$ is the normalized flux above the fitted quadratic function (see Section 2.2)

$$\begin{eqnarray}&&{F}_{\mathrm{flare}}(t)=F(t)-{F}_{{\rm{q}}}(t).\end{eqnarray} \tag{ 4 }$$

The observation mode of TESS, unlike Kepler, causes various observing spans for different targets. It is not suitable for calculating the occurrence frequency of superflares directly using the unequal observing spans. We therefore improve the method suggested by Maehara et al. (2012). First of all, we subdivided all the solar-type stars into different sets based on how many sectors the star was observed in. For example, Set-1 means that the stars were observed in only one sector. Similarly, Set-13 covers the stars observed in all 13 sectors. The count of the solar-type stars in each Set-n can be found in Figure 4 and Table 4.

Table 4.  Numbers of Solar-type Stars, Superflares, and Flare Stars of Each Set-n

Set-n	$5100\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\lt 5600\,{\rm{K}}$						$5600\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\lt 6000\,{\rm{K}}$							Total
	P < 10 days			P > 10 days			P < 10 days			P > 10 days
	Nstar	Nflare	Nfstar	Nstar	Nflare	Nfstar	Nstar	Nflare	Nfstar	Nstar	Nflare	Nfstar	Nstar	Nflare	Nfstar
1	7023	249	128	1204	0	0	8798	113	67	1018	5	2	18043	367	197
2	1465	233	67	406	0	0	1894	113	36	335	0	0	4100	346	103
3	326	34	14	88	0	0	471	39	14	80	1	1	965	74	29
4	92	16	3	24	0	0	123	11	3	14	0	0	253	27	6
5	73	0	0	19	0	0	109	0	0	15	0	0	216	0	0
6	54	18	2	14	0	0	68	18	3	21	0	0	157	36	5
7	39	65	5	16	0	0	63	3	1	13	0	0	131	68	6
8	37	1	1	15	0	0	48	5	2	15	0	0	115	6	3
9	80	31	5	29	0	0	90	1	1	25	0	0	224	32	6
10	62	25	2	35	0	0	73	3	2	28	0	0	198	28	4
11	91	20	6	44	4	1	115	10	3	35	0	0	285	34	10
12	182	29	5	77	1	1	213	23	8	59	1	1	531	54	15
13	173	38	5	76	0	0	193	106	11	74	0	0	516	144	16
Total	9697	759	243	2047	5	2	12258	445	151	1732	7	4	25734	1216	400

Note. The definition of Set-n is in Section 3.1. Basically, according to the stellar surface temperature and rotation period, we classify solar-type stars into four categories for each Set-n. N_star, N_flare, and N_fstar represent the numbers of solar-type stars, flares, and flare stars, respectively. The total numbers are also listed in the last three columns and the last row.

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For each Set-n, the superflare frequency distribution as a function of flare energy is defined as

$$\begin{eqnarray}&&{f}_{n}=\displaystyle \frac{{N}_{\mathrm{flares}}}{{N}_{\mathrm{stars}}\cdot {\tau }_{n}\cdot {\rm{\Delta }}{E}_{\mathrm{flare}}},\end{eqnarray} \tag{ 5 }$$

where N_flares and N_stars are the numbers of superflares and solar-type stars in Set-n. ΔE_flare represents the bin width of flare energy. τ_n can be calculated as

$$\begin{eqnarray}&&{\tau }_{n}=n\times 23.4\,\mathrm{days}.\end{eqnarray} \tag{ 6 }$$

As the observing time is affected by satellite orbit, TESS may not fully and effectively observe for 27 days in each sector. Here, we calculated the mean value of continuous observation length of each sector, which is 23.4 days. The final occurrence frequency of superflares for all the solar-type stars can be calculated as

$$\begin{eqnarray}&&f=\displaystyle \frac{\sum {f}_{n}}{13}.\end{eqnarray} \tag{ 7 }$$

Figure 5(a) illustrates the distribution of flare peak amplitudes, i.e., the peak flux of F_flare(t) in Equation (4). Panel (b) presents the frequency distributions as a function of flare energy for all the solar-type stars (solid line), and slowly (dashed line) or rapidly (dotted line) rotating solar-type stars. After comparing the dotted and dashed lines, it is evident that rapidly rotating stars are much more active than slowly rotating stars. According to the relation between stellar periodicity and age (Skumanich 1972; Barnes 2003), this result therefore indicates that younger solar-type stars generate superflares more frequently.

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A power-law model is used to fit the frequency distributions

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dE}}\propto {E}^{-\gamma }.\end{eqnarray} \tag{ 8 }$$

The power-law index γ is fitted by using the same linear regression method as in Tu & Wang (2018). For all solar-type stars, γ is 2.16 ± 0.10, which is consistent with γ ∼ 2.2 from Shibayama et al. (2013) within a 1σ interval, but higher than γ ∼ 1.5 from Maehara et al. (2015). From Figure 5(c), it is obvious that hotter stars (with 5600 K ≤ T_eff < 6000 K) have a lower frequency than cooler stars (with $5100\,{\rm{K}}\,\leqslant {T}_{\mathrm{eff}}\lt 5600\,{\rm{K}}$). The above results are basically similar to those found from Kepler data (Maehara et al. 2012; Shibayama et al. 2013; Maehara et al. 2015).

Figure 5(d) compares the flare frequency derived in this work and by Maehara et al. (2015), which used the 1 minute short-cadence data of Kepler and therefore detected more flares in the low-energy region (<2 × 10³⁴ erg). In the high-energy region (>2 × 10³⁴ erg), our superflare frequencies are higher than theirs. We speculate that this result is caused by the higher proportion of young stars in our data set, as we have discussed above (panel (b)). Since young stars rotate more rapidly and are more active, they are more likely to generate superflares. In total, 25,734 solar-type stars are selected, among which 21,955 stars have periods of less than 10 days. The proportion of young stars is 85% in our work, compared to 32% in the work by Maehara et al. (2015). One may refer to Appendix B for more discussions about the potential changes caused by the number fraction of rapidly and slowly rotating stars.

The superflare frequency of Sun-like stars (dotted line) is also presented in Figure 5(d). For Sun-like stars (with $5100\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\lt 6000\,{\rm{K}}$ and P > 10 days), we obtain a higher frequency of superflares (>10³⁵ erg) than that from Kepler (Maehara et al. 2012; Shibayama et al. 2013; Maehara et al. 2015). However, there are only seven such events, which are insufficient to robustly make a statistical conclusion. We look forward to more observations from TESS to extend the sample of Sun-like stars.

An apparent negative correlation between flare frequency and stellar period has been found (Maehara et al. 2012; Notsu et al. 2013; Maehara et al. 2015). In Figure 6(b), in the range of the stellar period over a few days, it is clear that the flare frequency decreases with an increase of the rotation period. A similar result is found by Notsu et al. (2013) and Notsu et al. (2019). Moreover, stellar age has a positive correlation with the stellar period (Skumanich 1972; Barnes 2003). This result indicates that rapidly rotating stars, or young stars, are more likely to generate superflares.

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Notsu et al. (2019) found that the upper limit of flare energy in each period bin has a continuous decreasing trend with the rotation period. From Figure 6(a), compared with the previous result (Figure 12(b) of Notsu et al. 2019), this trend is not obvious for the whole range of periods. However, superflares within the period range over a few days (the tail part) clearly show the decreasing trend. We separate superflares into two parts according to their surface effective temperatures, which are shown in Figures 6(c) and (d) respectively. Superflares with a temperature 5600 K ≤ T_eff < 6000 K perhaps show a slightly more clear decreasing trend for the whole period range, compared with panel (a). A similar decreasing trend can also be found in these two panels with period range over a few days (the tail part). As period is the key factor, we look forward to other methods precisely determining the periods of these solar-type stars (Appendix B). Meanwhile, slow-rotation stars (P ∼ 25 days) need to be searched for specially, in order to statistically and strongly confirm this trend.

Given that solar-type stars are grouped in different Set-n because of unequal observing spans, the flare frequency for an individual star is described by

$$\begin{eqnarray}&&{f}_{* }=\displaystyle \frac{{N}_{* \mathrm{flares}}}{{\tau }_{* }},\end{eqnarray} \tag{ 9 }$$

where τ_* is the continuous observation length of each flare star, and N_*flares denotes the number of flares from an individual star.

Tables 5 and 6 list the top 20 stars sorted by flare frequency f_* and the number of flares N_*flares, respectively. TIC43472154 is the most active star with an impressive flare frequency. If it is not coincidentally observed at an extraordinary active period, TIC43472154 can generate 233.17 flares per year. Besides, this star is not accompanied by any fainter stars or M dwarfs, according to the cross-match results from Gaia and Hipparcos data. This occurrence rate is much higher than that of KIC10422252, which is 41.6 flares per year (Shibayama et al. 2013). Figure 7 plots the light curve of TIC43472154. Superflares are marked with downward arrows.

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Another target, TIC364588501, was observed by all sectors of TESS and exhibits the largest number of flares (63). Additionally, like TIC43472154, this star is also a single star, according to the cross-match results from Gaia and Hipparcos data. TIC364588501 became more active than usual in the last 15 days of sector 5, and generated 15 superflares (Figure 8).

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The most energetic superflare comes from TIC93277807, releasing 1.77 × 10³⁷ erg in around 1.5 hr. This value consists with the inference that energy releasing through a stellar flare is saturated at ∼2 × 10³⁷ erg (Wu et al. 2015).

Hot Jupiters are considered as one of the essential factors that can produce superflares on host stars (e.g., Rubenstein & Schaefer 2000; Ip et al. 2004). According to these studies, it is not possible for the Sun to generate superflares. However, a lot of statistical works based on Kepler data did not detect hot Jupiters around flare stars (Maehara et al. 2012; Shibayama et al. 2013; Maehara et al. 2015).

We made a cross-check between our 400 flare stars with the Exoplanet Follow-up Observing Program for TESS (ExoFOP–TESS³ ). Cross-matching results are listed in Table 7. There are four stars (TIC25078924, 44797824, 257605131, and 373844472) that have planet candidates. These four stars are unlikely to affect our statistical results. We thus did not exclude them in the analysis. Besides, there is only one star (TIC410214986) that possesses a confirmed hot planet (Newton et al. 2019). This is an interesting planet, and we list its parameters in Table 7, even if its hosting star (TIC410214986) is flagged as a binary system from the Hipparcos-2 catalog.

Compared with other stars, there is nothing special about the superflares of these four targets. Therefore, our results suggest that the planet–star interactions are unlikely to be a general mechanism for producing superflares.

Impacts of stellar activities toward their hosted planets have been studied comprehensively (e.g., Segura et al. 2010; Airapetian et al. 2016; Atri 2017; Lingam & Loeb 2017). Some reviews have briefly introduced how superflares would affect their hosted planets (e.g., Riley et al. 2018; Airapetian et al. 2019; Linsky 2019). Planets around flare stars may not only help us to understand planet–star interactions in generating flares, but also importantly extend our knowledge about how superflares will affect the space weather of related planets. This will definitely improve our understanding about the relation between the Sun and the Earth as well. Many more space- and ground-based missions will focus on searching for habitable planets. However their habitability is just simply decided by habitable zones. From a more foresighted aspect, it is also important and necessary to estimate habitability in detail by examining the impacts of their hosting stars' activities.

Maehara et al. (2015) proposed that the energy and duration of superflares are connected through a power-law function, which was also proved by Tu & Wang (2018) using the statistical testing method. By applying magnetic reconnection theory, which has been widely accepted as the mechanism of solar flares, the correlation between the energy (E) and duration (T_duration) of superflares can be denoted as

$$\begin{eqnarray}&&{T}_{\mathrm{duration}}\propto {E}^{1/3}.\end{eqnarray} \tag{ 10 }$$

Benefiting from the 1 minute short-cadence data from Kepler, their calculations of the duration and energy of superflares are estimated more accurately than when using long-cadence data. The value of β is found to be 0.39 ± 0.03 (Maehara et al. 2015).

The 2 minute cadence of TESS also makes it possible to acquire accurate flare duration and energy. We use the superflares detected in this paper and obtain β = 0.42 ± 0.01, which is a little larger than the result from Kepler. Figure 9 presents the strong correlation between the duration and energy of superflares in our data set.

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Maehara et al. (2015) concluded that the same physical mechanism may be shared by both solar flares and superflares of solar-type stars, according to their similarity. Later, Namekata et al. (2017) used solar white-light flares, and got β = 0.38 ± 0.06, which is remarkably consistent with the result β = 0.39 ± 0.03 of Maehara et al. (2015). This similarity may support the idea that stellar superflares and solar flares are both generated by magnetic reconnections. But stellar superflares and solar white-light flares cannot be fitted by just one single power-law function, which indicates their potential difference. By using data from Galaxy Evolution Explorer missions, Brasseur et al. (2019) found that short-duration and near-ultraviolet flares do not show a strong correlation between energy and duration. Here, the observational values of β derived from Kepler and TESS are both larger than the theoretical prediction β = 1/3. Since this value is totally derived by the theory, which is successfully applied to solar flares, it may not precisely illustrate superflares on solar-type stars.

We consider the magnetic field strength of the Alfvén velocity (${v}_{A}=B/\sqrt{4\pi \rho }$) as a variable, and assume the preflare coronal density as a constant. The scaling power-law relation of Equation (10) can be expressed as

$$\begin{eqnarray}&&{T}_{\mathrm{duration}}\propto {E}^{1/3}{B}^{-5/3},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{T}_{\mathrm{duration}}\propto {E}^{-1/2}{L}^{5/2},\end{eqnarray} \tag{ 12 }$$

where B is the magnetic field strength and L is the flaring length scale. For a more detailed introduction of these two relations, one may refer to Namekata et al. (2017). Using the coefficients obtained by Namekata et al. (2017), we also plot these two scaling laws in Figure 9(b). Meanwhile, solar white-light flares (Namekata et al. 2017) and superflares of solar-type stars found by using Kepler short-cadence data (Maehara et al. 2015) are also presented in this figure.

From Figure 9, the majority of superflares found in this work (blue circles) have a magnetic field strength around 60–200 G and length scales near 10¹⁰–10¹¹ cm. The magnetic strength of superflares in this work is lower than that of superflares found by using Kepler short-cadence data (red squares). As we discuss in Section 2 and Appendix A, our method of selecting superflares based on pixel-level data and the bandpass of TESS tends to exclude weak flares with a faint flare signal, which means that the superflares in this work have a higher energy compared to those of Kepler (from Figure 10(c)). Apparently, superflares have a higher magnetic strength and a larger flaring length than solar white-light flares do. Superflares with lower energies tend to be generated from magnetic fields with a higher strength and a smaller flaring length scale. In contrast, those superflares with higher energies tend to be from the weaker magnetic fields and larger flaring length scales. In the future, the observation of stellar magnetic strength and the improvement of photometric imaging for precisely measuring flaring length scales will definitely advance our understanding of superflares. Also, by including more parameters of the superflares under consideration, a close-to-reality physical pattern will describe superflares in detail.

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