Zg. Dai et T. Lu, GAMMA-RAY BURST AFTERGLOWS - EFFECTS OF RADIATIVE-CORRECTIONS AND NONUNIFORMITY OF THE SURROUNDING MEDIUM, Monthly Notices of the Royal Astronomical Society, 298(1), 1998, pp. 87-92
The afterglow of a gamma-ray burst (GRB) is commonly thought to be the
result of continuous deceleration of a relativistically expanding fir
eball in the surrounding medium. Assuming that the expansion of the fi
reball is adiabatic and that the density of the medium is a power-law
function of shock radius, i.e. n(ext) proportional to R-k, we study th
e effects of the first-order radiative correction and the non-uniformi
ty of the medium on a GRB afterglow analytically. We first derive a ne
w relation among the observed time, the shock radius and the Lorentz f
actor of the fireball: t(+) = R/4(4 - k)gamma(2)c, and also derive a n
ew relation among the comoving time, the shock radius and the Lorentz
factor of the fireball: t(co) = 2R/(5 - k)gamma c. We next study the e
volution of the fireball by using the analytic solution of Blandford &
McKee. The radiation losses may not significantly influence this evol
ution. We further derive new scaling laws both between the X-ray flux
and observed time and between the optical flux and observed time. We u
se these scaling laws to discuss the afterglows of GRB 970228 and GRB
970616, and find that if the spectral index of the electron distributi
on is p = 2.5, implied from the spectra of GRBs, the X-ray afterglow o
f GRB 970616 is well fitted by assuming k = 2.