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 ($L_ν$) and FRB rotation measure (RM), we introduce a novel diagnostic framework utilizing the intrinsic scatter of the $L_ν- |{\rm RM}|$ relation as a physical probe of nebula dynamics. We demonstrate that this scatter encodes critical information regarding the temporal evolution of the nebula radius ($R \propto t^α$). By deriving a generic scaling $L_ν\propto R^ε|\mathrm{RM}|$ and analyzing the residuals of the five confirmed FRB-PRS systems, we constrain the nebula's evolutionary index to $α|ε| = 1.5 \pm 0.7$ (1$σ$ uncertainty). This measurement provides a powerful diagnostic tool for distinguishing among different astrophysical scenarios. Its value deviates from the expectations for supernova remnants (SNRs) in the Sedov-Taylor phase ($α|ε| \sim 0.2-0.4$), reverse shocks during the free-expansion phase of SNR/interstellar medium (ISM) interactions ($α|ε| \gtrsim 3.5$), and pulsar wind nebulae (PWNe) powered by a decaying wind ($α|ε| \sim 0-0.15$). Instead, it is more consistent with forward shocks in the free-expansion phase of both SNR/ISM and PWN/SNR systems ($α|ε| \sim 2.0-2.8$), and young PWNe driven by a nearly constant wind ($α|ε| \sim 1$). These findings suggest that active repeaters are powered by dynamically young, rapidly expanding nebulae. While currently limited by the small sample size, this framework establishes a robust methodology for future population studies to constrain the physical origin of PRSs.

Figures (3)

• Figure 1: A flowchart showing the physical connections between PRS and RM. "FS", "RS" and "TS" denotes forward shock, reverse shock and termination shock, respectively. Notice that the RS in a PWN/SNR system mainly generates X-ray emission and the radio emission is produced by fossil electrons in a PWN bubble.
• Figure 2: Various shocks in a PWN/SNR/ISM nebula (panel a) and a pair of bow shocks in a binary system (panel b).
• Figure 3: The $L_\nu-|{\rm RM}|$ relation with the current five sources. Top panel: The observed data points are shown as black circles, labeled with their respective event IDs. The solid red line represents the best-fit linear model with a fixed slope of unity. The red shaded region indicates the scatter (standard deviation) of the data around the best-fit line, with $1\sigma=0.65$. Bottom panels: The left panel displays the residuals ($\Delta$) of the fit in logarithmic scale. The horizontal red dashed lines mark the $\pm 1\sigma$ boundaries. The right panel shows the distribution of these residuals, overlaid with a Gaussian profile (solid red curve) derived from the sample statistics.