Scatter in the Relation between Persistent Radio Source Luminosity and Fast Radio Burst Rotation Measure: A Window into Circum-burst Environments

Abstract

The association of persistent radio sources (PRSs) with repeating fast radio bursts (FRBs) offers unique insights into their circum-burst environments. Building upon the physical link between PRS luminosity ($L_\nu$) and FRB rotation measure (RM), we introduce a novel diagnostic framework utilizing the intrinsic scatter of the $L_\nu - |{\rm RM}|$ relation as a physical probe of nebula dynamics. We show that this scatter encodes critical information regarding the temporal evolution of the nebula radius ($R \propto t^\alpha$). By deriving a generic scaling $L_\nu \propto R^\epsilon |{\rm RM}|$, we apply this framework to the current sample of five confirmed FRB-PRS systems as a proof-of-concept. Our preliminary analysis yields a combined evolutionary index of $\alpha|\epsilon| = 1.5 \pm 0.8$. This measurement illustrates the potential of our methodology to distinguish among diverse astrophysical scenarios, such as supernova remnants (SNRs) in different evolutionary stages, pulsar wind nebulae (PWNe) driven by constant wind or varying winds, and binary bow-shock systems. While the current conclusions are limited by the small sample size, this work establishes a robust statistical methodology for future population studies. As next-generation radio surveys localize a larger population of active repeaters, this framework will serve as a powerful tool to statistically constrain the physical origin and dynamical life cycle of FRB circum-burst environments.