Statistical Study of Appearance Timing of H$α$ Postflare Loops: Simple Scaling Law Based on Radiative Cooling
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 appearance timing 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 appearance timing 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 ($Δt$) and the soft X-ray peak flux ($F_\mathrm{X}$). This relationship is consistent with the theoretical scaling between radiative cooling timescale ($τ_{\mathrm{rad}}$) and $F_\mathrm{X}$, where $τ_{\mathrm{rad}} \propto~F_\mathrm{X}^{-1/2}$. This statistical result indicates that the appearance timing 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 spatial scale of unresolved stellar flares can be estimated using the following scaling law: $L\propto F_\mathrm{X}^{1/3}Δt^{2/3}$. Our results are useful for interpreting secondary peaks in the H$α$ data of stellar flares and provide new method to estimate spatial scale of unresolved stellar flares.