THE SOFTENING PHENOMENON DUE TO THE CURVATURE EFFECT: IN THE CASE OF AN EXTREMELY SHORT INTRINSIC EMISSION

Both the light curve and spectral evolution of the radiation from a relativistic fireball with extremely short duration are studied in order to examine the curvature effect for different forms of the radiation spectrum. Assuming a δ function emission we get formulae that get rid of the impacts from the intrinsic emission duration, applicable to any forms of the spectrum. It shows that the same form of the 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 ∝ t –1. When the emission is early enough, the t 2 f ν(t) form as a function of time will possess exactly the same form that the intrinsic spectrum as a function of frequency has. Assuming f ν ∝ ν–β t –α one finds α = 2 + β, which holds for any intrinsic spectral forms. This relation will be broken down and α>2 + β or α 2 + β will hold at much later time when the angle between the moving direction of the emission area and the line of sight is large. An intrinsic spectrum in the form of the Band function is employed to display the light curve and spectral evolution. Caused by the shifting of the Band function spectrum, a temporal steep decay phase and a spectral softening appear simultaneously. The softening phenomenon will appear at different frequencies. It occurs earlier for higher frequencies and later for lower frequencies. The terminating softening time t s,max depends on the observation frequency, following t s,max ∝ ν–1. This model predicts that the softening duration would be linearly correlated with t s,max; the observed βmin and βmax are determined by the low and high-energy indices of the Band function; both βmin and βmax are independent of the observation frequency.