Steep Decay Phase Shaped by the Curvature Effect. I. Flux Evolution

The curvature effect may be responsible for the steep decay phase observed in gamma-ray bursts. For testing the curvature effect with observations, the zero time point $t_0$ adopted to plot observer time and flux on a logarithmic scale should be appropriately selected. In practice, however, the true $t_0$ cannot be directly constrained from the data. Then, we move $t_0$ to a certain time in the steep decay phase, which can be easily identified. In this situation, we derive an analytical formula to describe the flux evolution of the steep decay phase. The analytical formula is read as $F_ν\propto (1+\tilde t_{\rm obs}/{\tilde t_c})^{-α}$ with $α(\tilde{t}_{\rm obs})=2+{\int_{0}^{\log (1+\tilde{t}_{\rm obs}/{\tilde{t}_c})} {β(τ)d[\log(1+τ/{\tilde{t}_c})]}}/{\log (1 + {\tilde t}_{\rm obs}/{{\tilde t}_c})}$, where $F_ν$ is the flux observed at frequency $ν$, $\tilde t_{\rm obs}$ is the observer time by setting zero time point $t_0$ at a certain time in the steep decay phase, $β$ is the spectral index estimated around $ν$, and ${\tilde t}_c$ is the decay timescale of the phase with $\tilde{t}_{\rm obs}{\geqslant}0$. We test the analytical formula with the data from numerical calculations. It is found that the analytical formula presents a well estimation about the evolution of flux shaped by the curvature effect. Our analytical formula can be used to confront the curvature effect with observations and estimate the decay timescale of the steep decay phase.