New Approach to Superflare Energy Determination

TL;DR

The paper tackles large biases in stellar flare energy estimates arising from assuming a constant flare temperature in single-band data. It introduces a framework with a fixed emitting area and a time-evolving flare temperature, enhanced by empirically estimated peak temperatures from a semi-empirical grid. Through analysis of a TESS superflare (TIC 325178532) and a large sample of 50{,}320 flares from 268 stars, the study shows that energy estimates can shift by up to an order of magnitude depending on the temperature treatment, with peak temperatures clustering around $11{,}000$ K. This approach improves the physical realism of flare energetics and has important implications for exoplanet space weather studies and cross-star flare comparisons, while highlighting the need for multi-band data to further constrain temperatures and emission measures.

Abstract

We present a new method for estimating the total energy radiated by stellar flares in broad-band continua, which assumes a constant emitting area but incorporates a time-dependent temperature evolution. This physically motivated approach offers an alternative to the commonly used method that assumes a fixed flare temperature about of 10 000\,K and variable area. By allowing the temperature to vary over time while keeping the emitting area constant, our method captures more realistic flare behaviour. This time-dependent treatment of the flare temperature is supported by numerous solar observations, numerical simulations, and multiwavelength studies of active stars. We demonstrate that using peak flare temperatures estimated from a semi-empirical model grid, rather than assuming an ad-hoc flare temperature value, improves the accuracy of total energy estimates. Although the most precise results still require a multi-band photometry/spectroscopy or independently constrained flare temperatures, our method offers a practical and scalable solution for single-band observations. It is particularly well suited for main-sequence stars of spectral types K4 and later with known effective temperatures. Finally we discuss how the flare continuum behaves under varying chromospheric conditions. Our method improves flare energy estimates by incorporating a physically relevant time-dependent temperature evolution and empirically derived peak temperatures, rather than assuming a constant 10 000\,K value. This modification reduces systematic errors that can reach factors up to ten as compared to previous estimates. We proved this on a sample of 50,320 TESS flares.

Figures (4)

• Figure 1: The relationship between the flare energy estimated using our modified method IIa and the assumed peak flare temperature is illustrated for a representative event. The red dots show the energy values corresponding to different temperatures, while the black error bars indicate the associated uncertainties.
• Figure 2: Time evolution of the flare temperature for a superflare TIC 325178532 for three different methods. Note an asymptotic temperature decrease towards stellar $T_{\rm eff}$ for methods IIa and IIb.
• Figure 3: Comparison of three methods for estimating the bolometric energy of flares. The panels show the energy estimated with Method I versus Method IIa (left), Method I versus Method IIb (middle), and Method IIa versus Method IIb (right). The green shaded areas indicate the uncertainty of the polynomial fits (linear for the left panel, quadratic for the middle and right). The dashed red lines represent a one-to-one correspondence. The orange square in the middle panel marks the energy range between $1\times 10^{33} - 2\times10^{33}\,$erg at which the quadratic fit intersects the one-to-one line. Note that the points are plotted as non-transparent to ensure visibility in the final scale; consequently, significant overlap in the panels may affect the visual impression of point density. While the distribution may visually appear balanced, the underlying data confirms that 92% of cases yield lower energies for Method IIa compared to Method I.
• Figure 4: Distribution of the peak temperatures of 50$\,$320 flares on 268 stars, from Pietras22 and Bicz_2025, estimated using our semi-empirical grid.