Probing motion of fast radio burst sources by timing strongly lensed repeaters

Given the possible repetitive nature of fast radio bursts (FRBs), their cosmological origin, and their high occurrence, detection of strongly lensed sources due to intervening galaxy lenses is possible with forthcoming radio surveys. We show that if multiple images of a repeating source are resolved with VLBI, using a method independent of lens modeling, accurate timing could reveal non-uniform motion, either physical or apparent, of the emission spot. This can probe the physical nature of FRBs and their surrounding environments, constraining scenarios including orbital motion around a stellar companion if FRBs require a compact star in a special system, and jet-medium interactions for which the location of the emission spot may randomly vary. The high timing precision possible for FRBs ($\sim {\rm ms}$) compared to the typical time delays between images in galaxy lensing ($\gtrsim 10\, {\rm days}$) enables the measurement of tiny fractional changes in the delays ($\sim 10^{-9}$), and hence the detection of time-delay variations induced by relative motions between the source, the lens, and the Earth. We show that uniform cosmic peculiar velocities only cause the delay time to drift linearly, and that the effect from the Earth's orbital motion can be accurately subtracted, thus enabling a search for non-trivial source motion. For a timing accuracy of $\sim 1\,$ms and a repetition rate (of detected bursts) $\sim 0.05$ per day of a single FRB source, non-uniform displacement $\gtrsim 0.1 - 1\,$AU of the emission spot perpendicular to the line of sight is detectable if repetitions are seen over a period of hundreds of days.Comment: 21 pages, 6 figures, 1 table. New version accepted to ApJ with abstract revised, typo corrected, and references adde

Gamma-Ray Burst Afterglows: Effects of Radiative Corrections and Nonuniformity of the Surrounding Medium

The afterglow of a gamma-ray burst (GRB) is commonly thought to be due to continuous deceleration of a relativistically expanding fireball in the surrounding medium. Assuming that the expansion of the fireball is adiabatic and that the density of the medium is a power-law function of shock radius, viz., $n_{ext}\propto R^{-k}$, we analytically study the effects of the first-order radiative correction and the nonuniformity of the medium on a GRB afterglow. We first derive a new relation among the observed time, the shock radius and the fireball's Lorentz factor: $t_\oplus=R/4(4-k)\gamma^2c$, and also derive a new relation among the comoving time, the shock radius and the fireball's Lorentz factor: $t_{co}=2R/(5-k)\gamma c$. We next study the evolution of the fireball by using the analytic solution of Blandford and McKee (1976). The radiation losses may not significantly influence this evolution. We further derive new scaling laws both between the X-ray flux and observed time and between the optical flux and observed time. We use these scaling laws to discuss the afterglows of GRB 970228 and GRB 970616, and find that if the spectral index of the electron distribution is $p=2.5$, implied from the spectra of GRBs, the X-ray afterglow of GRB970616 is well fitted by assuming $k=2$.Comment: 17 pages, no figures, Latex file, MNRAS in pres