The softening phenomenon in early X-ray afterglows as a consequence of the curvature effect: in the case of extremely short intrinsic emission

The curvature effect is explored in the case of extremely short intrinsic emission. Assuming a $\delta$ function emission we get formulas that get rid of the impacts from the intrinsic emission duration, which are applicable to any forms of continuum. The formulas predict that the same form of spectrum could be observed at different times, with the peak energy of the spectrum shifting from higher energy bands to lower bands following $E_{peak}\propto t^{-1}$. When the emission is early enough the light curve in the form $f_{\nu }(t)t^{2}$ will possess exactly the intrinsic spectral form, for which the temporal power law index and the spectral power law index will be related by $\alpha =2+\beta $. The analysis shows that there do exist a temporal steep decay phase and a spectral softening which occur simultaneously, and both are caused by the shifting of the Band function spectrum. According to the analysis, we predict that the softening due to the curvature effect will appear at different frequencies; it occurs earlier for higher frequencies and later for lower frequencies; the maximum spectral index time follows the $t_{b,max}\propto \nu^{-1}$ law. We also find: the softening duration would be linearly correlated with the maximum spectral index time; the observed $\beta_{min}$ and $\beta_{max}$ are determined by the low and high energy indexes of the observed Band function spectrum. We propose to check the curvature effect with the $\log f_{\nu}(t)t^{3}$ vs. $log t$ curve which would be in agreement with the $\log \nu f_{\nu}$ vs. $log \nu$ curve. Applying this to GRB 060614 shows that the peak energy of its observed spectrum is expected to pass through the observation band at $\sim $175 s.