Yingjie Cai1,2,3, Yijun Hou1,2,3, Ting Li4,2,3, Ying Li5,6, Mingde Ding7,8, Dechao Song5,6, Changwen Zeng1,2, Jifeng Liu1,2,9,10
1. National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
2. School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
3. State Key Laboratory of Solar Activity and Space Weather, National Astronomical Observatories, Chinese Academy of Science, Beijing 100101, China
4. State Key Laboratory of Solar Activity and Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China
5. Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China
6. School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China
7. School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China
8. Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, China
9. Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing, China
10. New Cornerstone Science Laboratory, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China
Solar white-light flares (WLFs) are among the most energetic eruptive phenomena in the solar atmosphere. Analogous to these solar WLFs, numerous WLFs have been observed on other stars, and their energies exceed the most extreme recorded solar flares by several orders of magnitude. Such high energies of stellar WLFs are traditionally estimated from single-band photometric light curves using a blackbody approximation with a constant temperature (typically set to 10,000 K)[1]. Despite relying on simplified assumptions, bolometric energy estimates have substantially advanced systematic solar-stellar flare comparisons and the quest for a unified physical paradigm, driven by studies of flare frequency distribution and energy–duration relation. Such estimates are also crucial for assessing the impact of stellar eruptive activity on planetary habitability.
Despite its importance, accurate bolometric energy estimation for stellar WLFs from single-band photometry remains contentious due to two issues: 1) the debated radiative mechanism of WLFs (optical-thick blackbody continuum[1, 3] versus optical-thin hydrogen free-bound (Hfb) continuum[2]); 2) flare energy estimation depends on two key independent physical variables: the flare radiating area and temperature, which cannot be solved simultaneously from a single equation (the stellar flare light curve) due to the lack of spatial resolution. This fact forced pioneering stellar WLF studies to adopt simplified energy estimation methods, typically assuming either a constant flare temperature[1, 2] or a fixed radiating area[3], which inevitably introduce substantial systematic biases. The choice between different methodologies is also demonstrated to significantly affects energy estimation results[2, 3, 4].
Figure 1. Evolution of the integrated WL flux, flare radiating area, and temperature for solar WLFs. (a) and (b) Spatial distributions of WL emission enhancement signals (blue patches) at different times during an M4.2 solar WLF. (c) and (d) Similar to (a) and (b), but for an X1.0 solar WLF. (e) and (f) The temporal evolution of the integrated WL flux, flare area, and temperature for the two representative WLFs. Purple and orange vertical dashed lines indicate the peak times of GOES SXR 1–8 Å flux and its derivative, respectively. (g) and (h) The temporal evolution of the flare area and temperature for the other 68 WLFs.
To assess the physical plausibility of the existing simplified energy estimation methods of stellar WLFs, we utilize high-spatiotemporal-resolution solar observations from SDO/HMI to analyze the true evolution of source region’s radiating area and temperature of 70 solar WLFs, based on a recently released dataset of solar WLFs[4]. As shown in Figure 1, both areas and temperatures of most solar WLFs undergo significant temporal evolution, challenging the existing WLF energy estimation models assuming either a constant flare temperature or a fixed radiating area. Statistical analysis further shows that the peak WL flux is strongly correlated with the peak flare area, implying that the expansion of the radiating area plays a crucial role in the observed optical enhancement.
To better match the real evolution of solar WLFs, we propose a new energy estimation method with variable flare area and time-dependent and pixel-dependent temperature. For each flaring pixel and each time step, the effective temperature is derived from the observed HMI continuum intensity enhancement under a blackbody approximation. The instantaneous bolometric luminosity is then reconstructed from the spatially resolved temperature and area information. Ultimately, the total bolometric energy is obtained by integrating the excess luminosity over the flare duration. This approach estimates the flare energy from the observed evolution of both radiating area and temperature, rather than imposing a fixed temperature or a fixed area.
Figure 2. Comparison of energy estimates from different methods. (a) Specific intensity spectra for the blackbody model and Hfb continuum model. (b) and (c) Scatter plot and histogram comparing the bolometric energy derived from the present work and another two methods with respect to the traditional blackbody model with fixed temperature. (d) Maximum flare area versus bolometric energy estimated from these methods.
Figure 2 presents a comprehensive comparison of bolometric energy estimates of 70 solar WLFs from different methods. Compared with the classical 10,000 K fixed-temperature blackbody method[1], our dynamic method yields systematically smaller bolometric energies (with a median ratio of 0.56). This discrepancy mainly arises from the strong temperature dependence of the Stefan–Boltzmann law: even a moderate decrease in temperature can substantially reduce the inferred radiative output. Since the temperatures derived from the spatially resolved solar data are generally well below 10,000 K, the traditional assumption tends to overestimate the energy. In contrast, the variable-temperature blackbody model with a fixed area[3] and the Hfb continuum model[2] yield median ratios of 1.15 and 1.50, respectively.
Figure 3. Impact of different energy estimation methodologies on macroscopic flare scaling laws. (a) Scaling relationship between bolometric energy and GOES SXR 1–8 Å peak flux. (b) Flare frequency distribution of the bolometric energy. (c) Relationship between bolometric energy and flare duration.
The systematic energy deviations between different methods also propagate into macroscopic statistical laws involved solar/stellar flare energy. As shown in Figure 3, all methods show a positive correlation between bolometric energy and GOES soft X-ray cpeak flux, but our method gives a steeper relation. For the flare frequency distribution, the existing models yield shallow power-law indices, whereas the energies derived by our method generate a little steeper high-energy tail. Moreover, the energy–duration relation becomes shallower than that obtained from the other methods. Although these changes may appear modest, small deviations in scaling-law indices can introduce large systematic biases in statistical extrapolations, calling for a re‑examination of these established statistical results and their targeted testing or revision in future works.
For more details of this work, please refer to our publication Ref. [5].
References:
[1] Shibayama, T., Maehara, H., Notsu, S., et al. 2013, ApJS, 209, 5
[2] Simões, P. J. A., Araújo, A., Válio, A., & Fletcher, L. 2024, MNRAS, 528, 2562
[3] Heinzel, P., Falewicz, R., Bicz, K., & Preś, P. 2026, ApJL, 999, L18
[4] Cai, Y., Hou, Y., Ding, H., et al. 2026, RAA, 26, 047001
[5] Cai, Y., Hou, Y., Li, T., et al. 2026, ApJL, 1005, L28