Appearance Timing of Hα PostflareLoops: Simple Scaling Law Based on Radiative Cooling

Postflare loops are important components of the standard flare model, and they are expected to be observed even in stellar flares. To correctly understand dynamics of stellar flares, it is important to clarify how postflare loops are observed in spatially integrated data. Recent Sun-as-a-star studies have shown that postflare loops can manifest as a secondary peak in the Hα light curve, suggesting that stellar postflare loops are detectable (Otsu+ 2024). To understand what determines the timing of such a secondary peak in the Hα light curve associated with postflare loops, we must quantitatively identify the key processes controlling the appearance timing of Hα postflare loops. Previous case studies have indicated that the appearance timing of Hα postflare loops is likely governed by radiative cooling. However, the statistical characteristics of the appearance timing of Hα postflare loops remain insufficiently investigated. In this study, we statistically investigated the appearance timing of Hα 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 Hα postflare loops (Δ t), and the soft X-ray peak flux (FX). This relationship is consistent with the theoretical scaling between radiative cooling timescale (τᵣad) and FX, where τrad ∝ FX-1/2. This result indicates that the appearance timing of Hα postflare loops relative to the soft X-ray peak time is primarily controlled by radiative cooling. Furthermore, we examined the dependence of the scaling law on flare spatial scales (L). Consequently, we demonstrated that spatial scale of unresolved stellar flares can be estimated using the following scaling law: L∝ FX1/3 Δ t2/3. Our results are useful for interpreting secondary peaks in the Hα data of stellar flares and provide a new method to estimate the spatial scale of unresolved stellar flares.