Diagnosing Circumburst Environment with Multiband Gamma-Ray Burst Radio Afterglows

It has been widely recognized that gamma-ray burst (GRB) afterglows arise from interactions between GRB outflow and circumburst medium, while their evolution follows the behaviors of relativistic shock waves. Assuming the distribution of circumburst medium follows a general power-law form, that is, $n = A_{\ast} R^{-k}$, where $R$ denotes the distance from the burst, it is obvious that the value of density-distribution index $k$ can affect the behaviors of the afterglow. In this paper, we analyze the temporal and spectral behaviors of GRB radio afterglows with arbitrary $k$-values. In the radio band, a standard GRB afterglow produced by forward shock exhibits a late-time flux peak, and the relative peak fluxes as well as peak times at different frequencies show dependencies on $k$. Thus with multi-band radio peak observations, one can determine the density profile of circumburst medium by comparing the relations between peak flux/time and frequency at each observing band. Also, the effects of trans-relativistic shock waves, as well as jets in afterglows are discussed. By analyzing 31 long and 1 short GRBs with multi-band data of radio afterglows, we find that nearly half of them can be explained with uniform interstellar medium ($k=0$), $\sim 1/5$ can be constrained to exhibiting stellar wind environment ($k=2$), while less than $\sim 1/3$ samples show $0< k< 2$.