Kosuke Namekata1, Takahito Sakaue1, Kyoko Watanabe2, Ayumi Asai3,4, Hiroyuki Maehara5, Yuta Notsu1, Shota Notsu1, Satoshi Honda6, Takako T. Ishii3, Kai Ikuta1, Daisaku Nogami1, Kazunari Shibata3,4
1. Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
2. National Defense Academy of Japan, 1-10-20 Hashirimizu, Yokosuka, 239-8686, Japan
3. Kwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto 607-8471, Japan
4. Unit of Synergetic Studies for Space, Kyoto University, Yamashina, Kyoto 607-8471, Japan
5. Okayama Astrophysical Observatory, National Astronomical Observatory of Japan, Asaguchi, Okayama, 719-0232, Japan
6. Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo, Sayo-cho, Sayo-gun, Hyogo, 679-5313, Japan
Recently, many “superflares” were discovered on solar-type stars (G-type main sequence stars) whose energies (1033-36 erg) are 10–10,000 times stronger than those of maximum solar flares (~1032 erg)1. Interestingly, these superflares are discovered as “white-light flares”, whose emission mechanism is not yet understood. A statistical study2 found a correlation between their energies (E) and durations (τ): τ∝E0.39, similar to those of solar hard/soft X-ray flares: τ∝E0.2-0.33. Moreover, the observed relations can be explained by a theoretical scaling law (τ∝E1/3) derived based on the magnetic reconnection theory. This similarity is thought to indicate that superflares can also occur through the magnetic reconnection.
A direct comparison between solar flares and stellar superflares on the white-light range is expected to give a clear evidence about the energy release mechanism of superflares. However, solar white-light flares are rare events due to low contrasts and short durations, so the statistical properties are not enough investigated. Only recently, HMI onboard SDO has provided a huge amount of data of white-light flares, which enables us to research the statistical properties. We therefore carried out a statistical research on 50 solar white-light flares with the SDO/HMI and examined the E–τ relation.
Figure 1| Left: light curves of a solar flare on 23th October 2012 observed by GOES, RHESSI, and HMI (white light). Right: temporal evolution of the pre-flare-subtracted images observed by HMI continuum. The black components show white-light emissions above 1σ level and the black lines show the RHESSI contours of 30%, 50%, 70%, and 90% of the maximum emission.
Figure 1 shows one of the white-light flares analyzed in this work after the pre-flare images were subtracted. Since previous studies showed clear spatial and temporal correlations between white-light and hard X-ray emission, we identified the white-light emissions inside the region with strong HXR emissions observed by the RHESSI. The radiated energies and decay times were calculated from the light curves of white-light flares.
Figure 2| Comparison between the flare energy and duration. The red filled squares show solar white-light flares analyzed in this paper and the open squares show superflares on solar-type stars obtained from Kepler 1 minutes cadence data. The data of superflares are taken from Ref. 2.
Figure 2 is the result of comparison of energy and duration of solar and stellar white-light flares. The E–τ relation on solar white-light flares (τ∝E0.38) is quite similar to that on stellar superflares (τ∝E0.39). However, the durations of stellar superflares are one order of magnitude shorter than those expected from solar white-light flares. Based on the magnetic reconnection theory, we present the following two physical interpretations for the discrepancy.
Figure 3| Two physical understandings of the observed distribution. Left: The purple open circles are solar flares whose durations are those of HXR flares. Right: Theoretical E–τ relations for the magnetic field strength B = 60 and 200 Gs are overlaid on the observed E–τ relation in this study.
First one is that the decay time can be elongated by the cooling effect, especially in solar white-light flares. There are some observations reporting that solar white-light flares have relatively long decay components (a few minutes). Assuming that the observed decay time does correspond to the cooling time (tcool), we can simply understand the observed distribution as follows: in solar flares, since the reconnection time (trec) is less than tcool, the decay time is elongated by the cooling effect. On the other hand, in superflares, tcool is negligible compared to trec, therefore the decay time cannot be affected by cooling effect. In fact, the decay times of solar hard X-ray flares are much shorter than those of white-light flares (Figure 3), which may support the above scenario.
The second interpretation is that the distribution can be understood by stronger coronal magnetic field of superflares. In detail, we derived a theoretical scaling law of energy and duration of flares with the dependence of coronal magnetic field strength B as τ∝E1/3B-5/3. This means that the stronger the coronal magnetic field is, the shorter the decay time becomes. On the basis of the scaling law, the observed discrepancy can be explained by the strong B of superflares, and the observed superflares are expected to have 2-4 times stronger magnetic field strength than solar flares (see, Figure 3). In this case, the scaling law can be useful because we can estimate the magnetic field strength and loop length of unresolved stellar flares from photometric observations. For validation, measurements of stellar magnetic field are necessary in future.
For more details of this work, please refer to our publication Ref 3.
 Maehara, H., Shibayama, T., Notsu, S., et al. 2012, Nature, 485, 478
 Maehara, H., Shibayama, T., Notsu, Y., et al. 2015, Earth, Planets, and Space, 67, 59
 Namekata, K. et al., 2017, ApJ, in press, https://arxiv.org/abs/1710.11325