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Erratum
This article is an erratum for:
[https://doi.org/10.1051/0004-6361/202348951]

Issue
A&A
Volume 703, November 2025
Article Number C4
Number of page(s) 4
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/202557625e
Published online 13 November 2025

Erratum for: A&A, 686, A239 (2024), https://doi.org/10.1051/0004-6361/202348951

1 Bug in flare energy calculation

We highlight a bug in the part of our code used for deriving flare energy. In the main paper, this was obtained by multiplying the integral under the flare profile by the ‘flare peak’ luminosity: the ‘quiescent stellar’ luminosity should have been used instead. This error caused flare energy to be underestimated, especially for the smallest flares. This affects the statistics related to flare energy in our study.

In Figure 1, we report the updated plots for Figures 13 and 14 in the original paper. The updated Figure 13 (left panel) presents the updated distribution of flare energy for simple (single-peak) and complex (part of multi-peak) flares, which we confirm differ at a high level of significance (p ≪ 0.05). Updated Figure 14 (right panel) shows the flare duration-energy correlation. We recovered steeper correlations between flare duration d and energy E: log⁡ ds=(0.439−0.004+0.005) log⁡ Es−13.2−0.2+0.1$\[\log~ d_{\mathrm{s}}=\left(0.439_{-0.004}^{+0.005}\right) ~\log~ E_{\mathrm{s}}-13.2_{-0.2}^{+0.1}\]$(1) log⁡ dc=(0.53±0.01) log⁡ Ec−(15.8±0.3)$\[\log~ d_{\mathrm{c}}=(0.53 \pm 0.01) ~\log~ E_{\mathrm{c}}-(15.8 \pm 0.3)\]$(2)

for simple (‘s’) and complex (‘c’) flares, respectively; here, duration is measured in minutes and energy in erg. The result is significantly different from Maehara et al. (2015)’s theoretical prediction, which states that d ∝ E1/3, but it might also be affected by an underestimate of flare parameter uncertainties.

Figure 2 is the updated version of Figure 15 in the original paper. The underestimation of flare energy increased the steepness of the impulse-energy relationship, which is now found to be less significant. The Spearman correlation coefficient confirms that the parameters are correlated, with p ≪ 0.05 at native cadence, and p ≃ 0.04 at binned cadence. More specifically, we found that the relationship becomes negative for flares at native cadence (3 s for CHEOPS and 20 s for TESS): log⁡ In=(−0.175±0.005) log⁡ En+(1.6±0.2)$\[\log~ \mathcal{I}_{\mathrm{n}}=(-0.175 \pm 0.005) ~\log~ E_{\mathrm{n}}+(1.6 \pm 0.2)\]$(3) log⁡ Ib=(0.07±0.01) log⁡ Eb−(6.7±0.2),$\[\log~ \mathcal{I}_{\mathrm{b}}=(0.07 \pm 0.01) ~\log~ E_{\mathrm{b}}-(6.7 \pm 0.2),\]$(4)

where ‘n’ and ‘b’ denote ‘native’ and ‘binned’ cadence, respectively, impulse is measured in s−1, and energy in erg. Qualitatively, this confirms the importance of monitoring flares with the highest cadence possible in order to constrain flare impulse.

Figure 3 presents the updated plots for the flare energy power laws for simple and complex flares (left panel) and for partially convective versus fully convective stars (right panel), as reported in the top-left and top-right panels of Figure 16 in the original paper. We do not detect any significant preference for the power law over the log-normal modelling of the observed distributions, as p > 0.05 in all cases; in the original study, we found a significant preference for the log-normal description in the only case of the complex flare energy distribution. More importantly, we retrieved a significant increase in the power-law scaling factor α: in all cases, the value 2 is now included within 3σ uncertainties, as shown in Table 1. What we found is, however, compatible with recent results that attribute high scaling factors to small inertial ranges (Aschwanden & Scholkmann 2025); we chose to explore this issue in a subsequent publication.

Finally, in Figure 4 we present the updated distribution for the trends between the detected quasi-periodic pulsation (QPP) candidate energy, the amplitude ratio as described in Section 6.7 in the original paper (left panel), and the QPP estimated period (right panel). Qualitatively, we observe similar trends to those found in the previous paper, which we chose not to interpret due the small available sample size.

thumbnail Fig. 1

Left: distribution of computed flare energies. Single-peaked and individual components of multi-peaked flares are coloured in black and red, respectively. Right: flare duration-energy correlation, with the same colour code.
Table 1

Fitted power-law coefficients α, normalised likelihood ratio R, corresponding p-value, and lower bound for the inertial range x1 for the cumulative distribution flare energy subsets.

thumbnail Fig. 2

Flare impulse as a function of flare energy for binned (blue crosses) and native (orange dots) cadence. The top and right histograms show energy and impulse distributions, respectively, with the same colour code.

2 Summary of updated results

To conclude, we here summarise the updated conclusions of our study concerning flare energy in our sample:

  • Beginning with Hawley et al. (2014)’s finding that complex flares tend to be more energetic than simple flares, we focused on complex flare components and report that these are also statistically more energetic than simple flares. This might indicate a difference in the respective triggering mechanisms, which needs to be further investigated both from observational and theoretical standpoints;

  • We found a steeper relationship between flare duration and energy than Maehara et al. (2015)’s theoretical prediction on solar-type stars. However, we cannot exclude an underestimation of flare parameter uncertainties, which might make the difference less significant. This is due to the many free parameters that need to be fitted in complex flare profiles and the large number of flares to be analysed, which challenge a systematic and homogeneous quantification of parameter uncertainties for each outburst;

  • We found a significant correlation between flare impulse and energy, both at native (3 and 20 s for CHEOPS and TESS, respectively) and 1 min binned cadence. Overall, this confirms the need to monitor flares at the highest time cadence possible, in order to better assess their morphology and potential impact on close-in exoplanets;

    thumbnail Fig. 3

    Complementary cumulative distribution functions (CCDFs) for flare energy, and their power-law and log-normal distribution fits. Left: simple and complex flares are divided. Right: partially (<M3V) and fully (>M3V) convective stars are divided.
    thumbnail Fig. 4

    QPP energy trends vs oscillation amplitude ratio (left) and period (right).

  • We failed to find a statistical preference for a log-normal modelling of the flare energy distributions compared to a power-law approach. The statistical preference was inspected for simple flares versus complex flare components, and for partially versus fully convective stars, where different dynamo processes are expected (e.g. Chabrier & Küker 2006);

  • The power-law scaling factors that we recovered for all examined subsets include the critical value 2 within 3σ uncertainties, in agreement with recent literature suggesting that observed scaling factors might be overestimated in the common case of small inertial ranges (e.g. Aschwanden & Scholkmann 2025). This highlights the need for enhancing both the precision of our surveys (to reduce the minimum flare energy that can be detected) and their duration (to increase the likelihood of detecting large and rare energy events, if they occur);

  • We confirmed tentative correlations between the energy of quasi-periodic pulsation candidates (i.e. the energy of the single-smoothed flare fitted to their profile) and their oscillation amplitude and period, as reported by Howard & MacGregor (2022). Larger samples are needed to further investigate these trends.

All in all, this update reinforces the need for large-sample statistical explorations of flare properties, such as those that will be made possible thanks to the PLATO yield.

3 Update to public code

We fixed the bug in the flare energy calculation in the code used for this study, available on GitHub1.

References

  1. Aschwanden, M. J., & Scholkmann, F. 2025, arXiv e-prints [arXiv:2505.00748] [Google Scholar]
  2. Chabrier, G., & Küker, M. 2006, A&A, 446, 1027 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  3. Hawley, S. L., Davenport, J. R. A., Kowalski, A. F., et al. 2014, ApJ, 797, 121 [Google Scholar]
  4. Howard, W. S., & MacGregor, M. A. 2022, ApJ, 926, 204 [NASA ADS] [CrossRef] [Google Scholar]
  5. Maehara, H., Shibayama, T., Notsu, Y., et al. 2015, Earth Planets Space, 67, 59 [NASA ADS] [CrossRef] [Google Scholar]

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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All Tables

Table 1

Fitted power-law coefficients α, normalised likelihood ratio R, corresponding p-value, and lower bound for the inertial range x1 for the cumulative distribution flare energy subsets.

In the text

All Figures

thumbnail Fig. 1

Left: distribution of computed flare energies. Single-peaked and individual components of multi-peaked flares are coloured in black and red, respectively. Right: flare duration-energy correlation, with the same colour code.
In the text
thumbnail Fig. 2

Flare impulse as a function of flare energy for binned (blue crosses) and native (orange dots) cadence. The top and right histograms show energy and impulse distributions, respectively, with the same colour code.
In the text
thumbnail Fig. 3

Complementary cumulative distribution functions (CCDFs) for flare energy, and their power-law and log-normal distribution fits. Left: simple and complex flares are divided. Right: partially (<M3V) and fully (>M3V) convective stars are divided.
In the text
thumbnail Fig. 4

QPP energy trends vs oscillation amplitude ratio (left) and period (right).
In the text

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