Statistical Study of the Timing of the Appearance of H α Postflare Loops: Simple Scaling Law Based on Radiative Cooling

Abstract Recent Sun-as-a-star studies have shown that postflare loops can manifest as a secondary peak in the H α light curve, suggesting that stellar postflare loops are detectable. To understand what determines the timing of such a secondary peak in the H α light curve associated with postflare loops, we must quantitatively identify the key physical processes controlling the appearance of H α postflare loops. Previous case studies have indicated that the timing of the appearance of H α postflare loops is likely governed by radiative cooling. However, the statistical characteristics of the timing of H α postflare loops' appearance remain insufficiently investigated. In this study, we statistically investigated the timing of the appearance of H α postflare loops to quantify their cooling processes. As a result, we found a negative correlation between the time difference between the soft X-ray peak and the appearance of the H α postflare loops (Δ t ) and the soft X-ray peak flux ( F X ). This relationship is consistent with the theoretical scaling between radiative cooling timescale ( τ rad ) and F X , where τ rad ∝ F X − 1 / 2 . This statistical result indicates that the timing of the appearance of H α postflare loops relative to the soft X-ray peak is primarily controlled by radiative cooling. Furthermore, we examined the dependence of the scaling law on flare spatial scales ( L ). Consequently, we demonstrated that the spatial scale of unresolved stellar flares can be estimated using the following scaling law: L ∝ F X 1 / 3 Δ t 2 / 3 . Our results are useful for interpreting secondary peaks in the H α data of stellar flares and provide a new method to estimate the spatial scale of unresolved stellar flares.