Persistent Radio Emission from Synchrotron Heating by a Repeating Fast Radio Burst Source in a Nebula

The first repeating fast radio burst (FRB), FRB 121102, was found to be associated with a spatially coincident, persistent nonthermal radio source, but the origin of the persistent emission remains unknown. In this paper, we propose that the persistent emission is produced via synchrotron-heating process by multiple bursts of FRB 121102 in a self-absorbed synchrotron nebula. As a population of bursts of the repeating FRB absorbed by the synchrotron nebula, the energy distribution of electrons in the nebula will change significantly. As a result, the spectrum of the nebula will show a hump steadily. For the persistent emission of FRB 121102, the total energy of bursts injecting into the nebula is required to be about $3.3\times10^{49}\,\unit{erg}$, the burst injection age is over $6.7\times 10^4\,\unit{yr}$, the nebula size is $\sim0.02\,\unit{pc}$, and the electron number is about $3.2\times10^{55}$. We predict that as more bursts inject, the brightness of the nebula would be brighter than the current observation, and meanwhile, the peak frequency would become higher. Due to the synchrotron absorption of the nebula, some low-frequency bursts would be absorbed, which may explain why most bursts were detected above $\sim1~\unit{GHz}$.