Are Persistent Emission Luminosity and Rotation Measure of Fast Radio Bursts Related?

Fast radio bursts (FRBs) are extragalactic radio transients with millisecond durations, large dispersion measures (DMs), and extremely high brightness temperatures (e.g., Lorimer et al. 2007; Thornton et al. 2013; Chatterjee et al. 2017; Bannister et al. 2019; CHIME/FRB Collaboration et al. 2019a, 2019b; Prochaska et al. 2019; Ravi et al. 2019; Fonseca et al. 2020; Marcote et al. 2020). The physical origin of FRBs is still unknown. Among all the published FRBs (http://frbcat.org, Petroff et al. 2016), only the first repeating FRB source, FRB 121102, has a persistent radio counterpart and a large, evolving Faraday rotation measure (RM; Chatterjee et al. 2017; Michilli et al. 2018).

FRB 121102 was first discovered with the Arecibo telescope (Spitler et al. 2014). Its repeating behavior was further confirmed and studied with other radio telescopes all over the world, including Karl G. Jansky Very Large Array (VLA), Green Bank Telescope, the Five-hundred-meter Aperture Spherical Telescope, etc. (e.g., Chatterjee et al. 2017; Zhang et al. 2018; Li et al. 2019a). Thanks to the precise localizations and multiwavelength follow-up observations, the host galaxy of FRB 121102 was identified as a dwarf galaxy at redshift z = 0.193 (Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017), and a persistent radio counterpart with luminosity of νL_ν ∼ 10³⁹ erg s⁻¹ at a few gigahertz was discovered to be coincident with FRB 121102 spatially. The persistent emission has a nonthermal spectrum that deviates from a single power-law spectrum from 1 to 26 GHz (Chatterjee et al. 2017). The RM of FRB 121102 has a very large absolute value, i.e., $| \mathrm{RM}| \sim {10}^{5}\,\mathrm{rad}\,{{\rm{m}}}^{-2}$, which decreased by 10% during seven months (Michilli et al. 2018). Such a large RM implies that the corresponding magnetic field is orders of magnitude stronger than that in the interstellar medium, and the variation RM might be related to the change of the magnetic field configuration (e.g., Zhang 2018) or strength (e.g., Metzger et al. 2019) along the line of sight. The host galaxy of another repeater, FRB 180916.J0158+65 (abbreviated as FRB 180916 as follows), was reported recently by Marcote et al. (2020), which is a nearby massive spiral galaxy at z = 0.0337. There is no coincident persistent emission above 3σ of 18 μJy at 1.6 GHz for this source, which places an upper limit on the persistent source luminosity νL_ν < 7.6 × 10³⁵ erg s⁻¹. This upper limit is at least three orders of magnitude lower than that of FRB 121102, which gives one of the strongest constraints on the persistent emission luminosities of FRBs.

The persistent emission of FRB 121102 could be explained by the radiation from a nebula surrounding an FRB source, e.g., a supernova remnant (SNR) or a pulsar wind nebulae (PWN; e.g., Murase et al. 2016; Yang et al. 2016; Metzger et al. 2017; Margalit & Metzger 2018). Alternatively, it could be associated with a supermassive black hole (e.g., Michilli et al. 2018; Zhang 2018). Yang et al. (2016) and Li et al. (2020) studied the synchrotron-heating process by an FRB source in a self-absorbed synchrotron nebula and found that the observed persistent emission associated with FRB 121102 could be generated via multi-injection of bursts. Dai et al. (2017) and Yang & Dai (2019) suggested that the persistent emission could be generated via an ultra-relativistic PWN sweeping up its ambient medium. Wang & Lai (2020) studied the multiwavelength afterglow emission from the nebula powered by a repeating or nonrepeating FRB central engine, and they found that a brighter nebula emission associated with a repeating source would have a larger RM.

In this work, we consider the possibility that the persistent emission and the RM of an FRB source originate from the same region. In this scenario, we derive a simple relation between the persistent emission luminosity and RM, and we find that FRB 121102 falls into the relation. If this applies to all FRBs, our result implies that most FRBs, which have $\left|\mathrm{RM}\right|\lesssim {\rm{a}}\,\mathrm{few}\times 100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$ (e.g., Petroff et al. 2017; Bhandari et al. 2018; Caleb et al. 2018; Bannister et al. 2019; Osłowski et al. 2019), would not have detectable persistent emission with the current radio telescopes. This paper is organized as follows. In Section 2, we set up the theoretical framework for the relation between persistent emission and RM of an FRB source. We test the relation with some FRBs with the measurements of persistent emission and RM in Section 3. The results are summarized and discussed in Section 4.

Let us consider an FRB propagating in a plasma with number density of nonrelativistic electrons n_e, magnetic field strength B, and scale length R. The RM in this region is given by

$$\begin{eqnarray}&&\left|\mathrm{RM}\right|=\displaystyle \frac{{e}^{3}{\xi }_{B}B}{2\pi {m}_{e}^{2}{c}^{4}}{n}_{e}R,\end{eqnarray} \tag{ 1 }$$

where the parameter ξ_B is defined as ${\xi }_{B}=\left\langle {B}_{\parallel }\right\rangle /\left\langle B\right\rangle$, B_∥ is the line-of-sight magnetic field, and the $\left\langle \,\right\rangle$ sign denotes the average value. For a random magnetic field, one has $\left\langle {B}_{\parallel }\right\rangle =0$ and hence, ξ_B = 0. Notice that in Equation (1), we abbreviate $\left\langle B\right\rangle$ to B.

On the other hand, the persistent emission with a continuous nonthermal spectrum is generally explained by synchrotron radiation from relativistic electrons. We assume that the number density of relativistic electrons in this region is ${\zeta }_{e}{n}_{e}$, where ζ_e is the ratio between the relativistic and nonrelativistic electron numbers.⁵ The radiation power and the characteristic synchrotron frequency from a randomly oriented electron with Lorentz factor γ ≫ 1 in a magnetic field B are $P=(4/3){\sigma }_{{\rm{T}}}c{\gamma }^{2}{B}^{2}/8\pi$ and $\nu ={\gamma }^{2}{eB}/2\pi {m}_{e}c$, respectively. Thus, the spectral radiation power is given by ${P}_{\nu }\simeq P/\nu ={m}_{e}{c}^{2}{\sigma }_{{\rm{T}}}B/3e$, which is independent of γ. Let the total number of relativistic electrons be ${N}_{e}=4\pi {R}^{3}{\zeta }_{e}{n}_{e}/3$. The maximum specific luminosity is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\nu ,\max } & = & {N}_{e}{P}_{\nu }=\displaystyle \frac{64{\pi }^{3}}{27}{m}_{e}{c}^{2}\displaystyle \frac{{\zeta }_{e}}{{\xi }_{B}}{R}^{2}\left|\mathrm{RM}\right|\\ & \simeq & (5.7\times {10}^{29}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}){\zeta }_{e}{\xi }_{B}^{-1}\\ & & \times \,\left(\displaystyle \frac{\left|\mathrm{RM}\right|}{{10}^{5}\,\mathrm{rad}\,{{\rm{m}}}^{-2}}\right){\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 2 }$$

where Equation (1) has been used. One can see that there is a simple linear relation between $| \mathrm{RM}|$ and ${L}_{\nu ,\max }$, with dependences on the size of the persistent emission regions and the parameters ζ_e and ξ_B. The observed peak flux for an FRB source at the distance D is

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\nu ,\max } & = & \displaystyle \frac{{L}_{\nu ,\max }}{4\pi {D}^{2}}=\displaystyle \frac{16{\pi }^{2}}{27}{m}_{e}{c}^{2}\displaystyle \frac{{\zeta }_{e}}{{\xi }_{B}}\displaystyle \frac{{R}^{2}}{{D}^{2}}\left|\mathrm{RM}\right|\\ & \simeq & 480\,\mu \mathrm{Jy}{\zeta }_{e}{\xi }_{B}^{-1}\left(\displaystyle \frac{\left|\mathrm{RM}\right|}{{10}^{5}\,\mathrm{rad}\,{{\rm{m}}}^{-2}}\right)\\ & & \times \,{\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{2}{\left(\displaystyle \frac{D}{1\mathrm{Gpc}}\right)}^{-2}.\end{array}\end{eqnarray} \tag{ 3 }$$

For FRB 121102, the peak flux of the persistent emission is ∼200 μJy (Chatterjee et al. 2017), the RM is ∼10⁵ rad m⁻² (Michilli et al. 2018), and the persistent source has a projected size constrained to be ≲0.7 pc (Marcote et al. 2017). The flux of the persistent emission source has a variation with a timescale of ${\rm{\Delta }}{t}_{\mathrm{per}}\sim 10\,\mathrm{day}$ (see Figure 2 of Chatterjee et al. 2017), which further constrains the emission region to $R\sim c{\rm{\Delta }}{t}_{\mathrm{per}}\simeq 0.01\,\mathrm{pc}$. These numbers for FRB 121102 match Equation (3) very well given that ξ_B ∼ ζ_e ∼ 1 is satisfied. This result might imply that the large RM of FRB 121102 is physically associated with its large persistent emission luminosity. The condition ξ_B ∼ 1 requires that the magnetic field is coherent in large scale, which is consistent with the large RM observation. The condition ζ_e ∼ 1 requires that the number of relativistic and nonrelativistic electrons are of the same order. According to Equation (3), the variation of the persistent emission in the timescale of Δt_per ∼ 10 day would result in an RM variation. Michilli et al. (2018) reported that the RM of FRB 121102 decreases by ∼10% within seven months. Such a long-term evolution of the observed RM might be due to the change of the magnetic field configuration so that ξ_B is a function of t (e.g., Zhang 2018).

The above discussion assumes that both the magnetic field B and the electron number density n_e are uniform in a region with scale length R. If the magnetic field B and electron number density n_e satisfy a power-law distribution with radius from the source, the results should be of the same order of magnitude or somewhat lower compared with the uniform case presented in Equation (3). A detailed discussion is presented in the Appendix.

For a source with the brightness temperature T_B and scale length R, the observed flux is ${F}_{\nu }=(2{{kT}}_{{\rm{B}}}/{\lambda }^{2})({R}^{2}/{D}^{2})$. According to Equation (3), one has

$$\begin{eqnarray}\begin{array}{rcl}\left|\mathrm{RM}\right| & = & \displaystyle \frac{27}{8{\pi }^{2}}\displaystyle \frac{{\xi }_{B}{{kT}}_{{\rm{B}}}}{{\zeta }_{e}{\lambda }^{2}{m}_{e}{c}^{2}}\simeq (0.6\times {10}^{5}\,\mathrm{rad}\,{{\rm{m}}}^{-2})\\ & & \times \,{\zeta }_{e}^{-1}{\xi }_{B}\left(\displaystyle \frac{{T}_{{\rm{B}}}}{{10}^{12}\,{\rm{K}}}\right){\left(\displaystyle \frac{\nu }{10\mathrm{GHz}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 4 }$$

where we have normalized the frequency to 10 GHz, since the FRB 121102 persistent source has a broad spectrum that extends beyond 10 GHz. In general, for an incoherent stationary source, the maximum brightness temperature is ${T}_{{\rm{B}},\max }\sim ({10}^{12}\mbox{--}{10}^{13})\,{\rm{K}}$ due to the constraint of the inverse Compton (IC) catastrophe (Kellermann & Pauliny-Toth 1969). According to Equation (4), the observed large RM from FRB 121102 demands a T_b close to this limit. This provides direct evidence that the persistent emission associated with FRB 121102 originates from a strong compact radio source.

One should check the absorption effect in the emission region, including the Razin effect and free–free absorption. The synchrotron radiation of relativistic particles is subject to the plasma propagation effects. If the radiation frequency satisfies ω < γω_p, where γ is the electron Lorentz factor, and ${\omega }_{p}=\sqrt{4\pi {e}^{2}{n}_{e}/{m}_{e}}$ is the plasma frequency, the synchrotron spectrum would be cut off due to the suppression of beaming, which is called the Razin effect (e.g., Rybicki & Lightman 1979). Using $\nu ={\gamma }^{2}{eB}/2\pi {m}_{e}c$ and ω > γω_p, the condition for plasma transparency to the persistent emission is

$$\begin{eqnarray}&&{n}_{e}\lt \displaystyle \frac{B\nu }{2{ec}}.\end{eqnarray} \tag{ 5 }$$

According to Equations (2) and (5), the transparency condition for the Razin effect can be written as

$$\begin{eqnarray}&&\begin{array}{l}{n}_{e}\lt 3\times {10}^{4}\,{\mathrm{cm}}^{-3}{\zeta }_{e}^{-1/2}{\left(\displaystyle \frac{{L}_{\nu ,\max }}{{10}^{29}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{1/2}\\ \quad \times \,{\left(\displaystyle \frac{\nu }{10\mathrm{GHz}}\right)}^{1/2}{\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{-3/2}\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{l}\simeq \,6\times {10}^{4}\,{\mathrm{cm}}^{-3}{\xi }_{B}^{-1/2}{\left(\displaystyle \frac{| \mathrm{RM}| }{{10}^{5}\mathrm{rad}{{\rm{m}}}^{-2}}\right)}^{1/2}\\ \quad \times \,{\left(\displaystyle \frac{\nu }{10\mathrm{GHz}}\right)}^{1/2}{\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ 7 }$$

Thus, for FRB 121102, one needs $n\lt ({10}^{4}\mbox{--}{10}^{5})\,{\mathrm{cm}}^{-3}$. On the other hand, we can also define the DM in the persistent emission source region as ${\mathrm{DM}}_{\mathrm{src}}={n}_{e}R$. The above condition can be then also written as

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{DM}}_{\mathrm{src}}\lesssim 655\,\mathrm{pc}\,{\mathrm{cm}}^{-3}{\xi }_{B}^{-1/2}\\ \quad \times \,{\left(\displaystyle \frac{\nu }{10\mathrm{GHz}}\right)}^{1/2}{\left(\displaystyle \frac{| \mathrm{RM}| }{{10}^{5}\mathrm{rad}{{\rm{m}}}^{-2}}\right)}^{1/2}{\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{1/2}.\end{array}\end{eqnarray} \tag{ 8 }$$

Therefore, through measuring the RM, the persistent emission frequency ν, and the variability timescale of the persistent emission Δt_per ∼ R/c, the upper limit DM of the emission region can be derived, which can be compared against the host galaxy DM constrained from the data. We note that recent observations suggest that the excess DMs of several localized FRBs are generally consistent with being mostly due to the intergalactic medium (IGM) contribution (Bannister et al. 2019; Ravi et al. 2019; Marcote et al. 2020), suggesting that Equation (8) is readily satisfied. Therefore, the Razin effect is not important unless the studied FRB has an abnormally large host DM.

Next, we consider the free–free absorption from the emission region. The free–free optical depth is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}={\alpha }_{\mathrm{ff}}R=0.018{T}^{-3/2}{Z}^{2}{n}_{e}{n}_{i}{\nu }^{-2}\bar{{g}_{\mathrm{ff}}}R,\end{eqnarray} \tag{ 9 }$$

where T is the thermal gas temperature, n_e and n_i are the number densities of electrons and ions, respectively, and $\bar{{g}_{\mathrm{ff}}}\sim 1$ is the Gaunt factor. Here n_e = n_i and Z = 1 are assumed for the emission region. The transparency condition requires τ_ff < 1, which corresponds to

$$\begin{eqnarray}&&{n}_{e}\lt 4\times {10}^{5}\,{\mathrm{cm}}^{-3}{\left(\displaystyle \frac{T}{{10}^{4}{\rm{K}}}\right)}^{3/4}\left(\displaystyle \frac{\nu }{10\,\mathrm{GHz}}\right){\left(\displaystyle \frac{R}{0.01\mathrm{pc}}\right)}^{-1/2}.\end{eqnarray} \tag{ 10 }$$

Combining the constraints from both Razin effect and free–free absorption, the transparency condition for the persistent emission of FRB 121102 requires ${n}_{e}\lesssim ({10}^{4}\mbox{--}{10}^{5})\,{\mathrm{cm}}^{-3}$. If the above conditions are not satisfied in some special environments (e.g., those FRBs with abnormally large source DM), the persistent emission luminosity would be much lower than that given by Equation (2) due to the absorption effects.

In Table 1, we list 11 FRBs from the FRB catalog of Petroff et al. (2016) with the measured RM and the measured values (or upper limits) of the persistent emission flux. Among them, only FRB 121102 has the measured values of both. For some FRBs, the persistent emission flux was constrained in a wide frequency range. We take the minimum value of the upper limits of the persistent emission flux density, because the predicted maximum flux density given by Equation (3) is independent of frequency. FRB 121102, FRB 180916, FRB 180924, and FRB 181112 have precise localizations (Chatterjee et al. 2017; Tendulkar et al. 2017; Bannister et al. 2019; Prochaska et al. 2019; Marcote et al. 2020), so the redshifts in Table 1 are the directly measured values. For other FRBs, due to the lack of precise localization, we estimate their redshifts and luminosity distances via the extragalactic DM, i.e., ${\mathrm{DM}}_{{\rm{E}}}={\mathrm{DM}}_{\mathrm{obs}}-{\mathrm{DM}}_{\mathrm{MW}}={\mathrm{DM}}_{\mathrm{IGM}}+{\mathrm{DM}}_{\mathrm{HG}}$, where DM_obs is the observed total DM, and DM_MW, DM_IGM, and DM_HG are the DMs contributed by Milky Way, IGM, and the FRB host galaxy, respectively. In this work, the DMs contributed by the Milky Way are taken from⁶ the FRB catalog (Petroff et al. 2016), and we assume that the local DM contributed by FRB host galaxies is ${\mathrm{DM}}_{\mathrm{HG},\mathrm{loc}}=100\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ (e.g., Xu & Han 2015; Yang et al. 2017; Luo et al. 2018; Li et al. 2019b). The extragalactic DM is given by (Deng & Zhang 2014)

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{DM}}_{{\rm{E}}}(z) & = & \displaystyle \frac{3{{cH}}_{0}{{\rm{\Omega }}}_{b}{f}_{\mathrm{IGM}}}{8\pi {{Gm}}_{p}}{\displaystyle \int }_{0}^{z}\displaystyle \frac{\chi (z)(1+z){dz}}{{\left[{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right]}^{1/2}}\\ & & +\,\displaystyle \frac{{\mathrm{DM}}_{\mathrm{HG},\mathrm{loc}}}{1+z},\end{array}\end{eqnarray} \tag{ 11 }$$

where the fraction of baryons in the IGM is f_IGM ∼ 0.83 (Fukugita et al. 1998; Shull et al. 2012), and the free electron number per baryon in the universe is χ(z) ≃ 7/8. The ΛCDM cosmological parameters are taken as Ω_m = 0.315 ± 0.007, ${{\rm{\Omega }}}_{b}{h}^{2}=0.02237\pm 0.00015$, and ${H}_{0}=67.36\pm 0.54\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ (Planck Collaboration et al. 2018). For an FRB with redshift inferred from DM_E, the corresponding redshift error is given by σ_IGM due to the IGM density fluctuation (McQuinn 2014). For FRB 160102, since its redshift is out of the range given by McQuinn (2014), we assume ${\sigma }_{\mathrm{IGM}}=350\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$.

Table 1.  FRB Sample with Measurements (or Upper Limits) of RM and Persistent Emission Flux

FRB Name	DMobsa	DMMWb	zc	dLd	RMe	Fν, RMf	Fνg	νh	Lνi	References
	(pc cm−3)	(pc cm−3)		(Gpc)	(rad m−2)	(μJy)	(μJy)	(GHz)	(1029 erg s−1 Hz−1)
FRB 121102	557	188	0.19273	0.98	1.4 × 105	698	180	1.7	2.1	1,2,3,4
FRB 180916	348.76	200	0.0337	0.15	−114.6	24.4	<18	1.6	<0.0048	5,6
FRB 180924	361.42	40.5	0.3214	1.74	14	0.022	<20	6.5	<0.72	7
FRB 181112	589.27	102	0.47550	2.76	10.9	0.0068	<21	6.5	<1.91	8
FRB 110523	623.3	43.52	${0.58}_{-0.21}^{+0.21}$	${3.5}_{-1.4}^{+1.6}$	−186.1	0.073	<40	0.8	$\lt {5.8}_{-3.7}^{+6.6}$	9
FRB 150215	1105.6	427.2	${0.69}_{-0.22}^{+0.22}$	${4.3}_{-1.6}^{+1.7}$	1.5	0.00039	<6.48	10.1	$\lt {1.4}_{-0.9}^{+1.4}$	10
FRB 150418	776.2	188.5	${0.59}_{-0.21}^{+0.21}$	${3.6}_{-1.4}^{+1.6}$	36	0.013	<70	1.4	$\lt {11}_{-7}^{+12}$	11
FRB 150807	266.5	36.9	${0.17}_{-0.11}^{+0.10}$	${0.85}_{-0.57}^{+0.57}$	12	0.08	<240	5.5	$\lt {2.1}_{-1.8}^{+3.7}$	12
FRB 160102	2596.1	13	${3.04}_{-0.48}^{+0.51}$	${26.4}_{-4.9}^{+5.4}$	−220.6	0.0015	<30	5.9	$\lt {249}_{-84}^{+112}$	13,14
FRB 180309	263.42	44.69	${0.16}_{-0.10}^{+0.10}$	${0.79}_{-0.51}^{+0.57}$	<150	<1.2	<105	2.1	$\lt {0.8}_{-0.7}^{+1.5}$	15
FRB 191108	588.1	52	${0.53}_{-0.20}^{+0.20}$	${3.1}_{-1.3}^{+1.5}$	474	0.24	<213	1.4	$\lt {24}_{-16}^{+29}$	16

Notes.

^aThe observed DMs of FRBs. ^bThe DM contribution from Milky Way, which is from the FRB catalog (Petroff et al. 2016). ^cThe measured/inferred redshifts of FRBs. For FRB 121102, FRB 180916, FRB 180924, and FRB 181112, their redshifts are from the redshift measurements of their host galaxies (Chatterjee et al. 2017; Tendulkar et al. 2017; Bannister et al. 2019; Prochaska et al. 2019; Marcote et al. 2020). For other FRBs, their redshifts are inferred by the extragalactic DMs, i.e., Equation (11). ^dThe luminosity distance inferred by redshift. ^eThe observed RMs of FRBs. ^fThe predicted flux density given by RM, i.e., Equation (3). ^gThe observed flux density of the persistent emission. For the FRBs without persistent emission detected, the upper limits correspond to the 3σ flux density limits. ^hThe frequency at which the persistent emission is measured. ⁱThe persistent emission luminosity inferred by the observed flux density and luminosity distance.

References. (1) Spitler et al. (2014), (2) Tendulkar et al. (2017), (3) Marcote et al. (2017), (4) Michilli et al. (2018), (5) CHIME/FRB Collaboration et al. (2019c), (6) Marcote et al. (2020), (7) Bannister et al. (2019), (8) Prochaska et al. (2019), (9) Masui et al. (2015), (10) Petroff et al. (2017), (11) Keane et al. (2016), (12) Ravi et al. (2016), (13) Bhandari et al. (2018), (14) Caleb et al. (2018), (15) Osłowski et al. (2019), (16) Connor et al. (2020).

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In Figure 1, we plot the relation between the specific luminosity of the persistent emission and RM. The FRB data are taken from Table 1. According to Equation (2), the red solid line corresponds to the predicted relation for ζ_e(R/0.01 pc)² = 1, and the red dotted line corresponds to the predicted relation for ζ_e(R/0.01 pc)² = 0.1. For FRB 121102 with $| \mathrm{RM}| \sim {10}^{5}\,\mathrm{rad}\,{{\rm{m}}}^{-2}$, the observed flux is close to the predicted value for ζ_e(R/0.01 pc)² ∼ (0.1–1). For FRB 180916, the VLA observations show that there is no coincident persistent emission above a 3σ rms noise level of 18 μJy per beam at 1.6 GHz (Marcote et al. 2020). Such an upper limit is close to the predicted flux density given by Equation (3) for ζ_e(R/0.01 pc)² ∼ 1. For other FRBs, the upper limits of the observed flux densities are significantly higher than the predicted persistent emission flux density.

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So far, both persistent radio emission and a large RM value are discovered only in FRB 121102. Although the persistent emission is found to be spatially coincident with the repeating bursts, it does not show a direct physical connection with the FRB 121102 bursting source (Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017). In this work, we find a linear positive relation between the specific luminosity and RM (Equation (2)). Such a relation is indeed satisfied for FRB 121102, given that the following conditions are satisfied:

• 1.  
  The persistent emission of FRB 121102 and its large RM originate from the same region.
• 2.  
  The magnetic field that contributes to RM and the persistent emission is coherent in large scale (i.e., ξ_B ∼ 1).
• 3.  
  The ratio between relativistic and nonrelativistic electrons in the emission region, ζ_e, is of the order of unity.
• 4.  
  The Razin effect and free–free absorption are not significant, which requires n_e ≲ (10⁴–10⁵) cm⁻³ in the emission region for FRB 121102 and Equations (8) and (10) in general.

If these conditions are satisfied for all other FRBs, we would expect that most FRBs with $\left|\mathrm{RM}\right|\lesssim {\rm{a}}\,\mathrm{few}\times 100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$ (e.g., Petroff et al. 2017; Bhandari et al. 2018; Caleb et al. 2018; Bannister et al. 2019; Osłowski et al. 2019) would not have a detectable persistent emission counterpart. This seems to be consistent with the current observations. Considering that most FRBs have small RMs with $| \mathrm{RM}| \lesssim {\rm{a}}\,\mathrm{few}\times 100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$, the expected luminosity of the persistent emission is ${L}_{\nu ,\mathrm{RM}}\sim {10}^{27}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}$ for ζ_e(R/0.01 pc)² ∼ 1. For a radio telescope with 3σ limiting fluxes of a few × 10 μJy, the observable distance for the persistent emission satisfies d_L ≲ 100 Mpc or z ≲ 0.03. Thus, our model can be tested via the observations of nearby FRBs. A deviation of the prediction Equation (2) would suggest that at least one of the above conditions is not satisfied. For example, a bright persistent emission source with relatively small RM would suggest that the magnetic field in the persistent emission region is mostly random. At last, as shown in Figure 1, some FRBs (e.g., FRB 180916 and even FRB 110523) have an upper limit not too far away from the predicted zone (enclosed by the red lines). We suggest that observers may spend more observing times on target trying to make a positive detection of the persistent emission from these sources. A detection or a more stringent upper limit can help greatly to confirm or constrain the model proposed here.

The large-scale magnetic field requirement offers insight into the FRB mechanism. A large-scale magnetic field has been discovered near supermassive black holes or active galactic nuclei (Eatough et al. 2013; Michilli et al. 2018). It was also hypothesized to exist in shocked nebula (e.g., SNR, PWN and etc.) surrounding a magnetized neutron star (e.g., Margalit & Metzger 2018; Metzger et al. 2019). For the latter scenario, the synchrotron maser FRB mechanism requires that the magnetic fields lie in the plane of the shock. Such a field configuration needs to be destroyed to produce a radially ordered B field in the region where RM is generated.

As shown in the Appendix, for more general setups, e.g., ${n}_{e}\propto {r}^{-\alpha }$ and $B\propto {r}^{-\beta }$, for a given $\left|\mathrm{RM}\right|$, the predicted flux of persistent emission is of the same order or slightly lower than that given by Equation (3) for the uniform case. The observations of the persistent emission and RM of FRB 121102 imply that α + β < 1, which is close to the uniform distribution assumption. FRB 180916 (Marcote et al. 2020), on the other hand, has a persistent emission flux upper limit very close to the predicted ${L}_{\nu ,\max }\mbox{--}\mathrm{RM}$ relation, suggesting that either ζ_e < 1, or a smaller emission region (R < 0.01 pc), or a stratified medium with α + β > 1.

Finally, different from DM measurements that require transients, RM measurements could be made for persistent sources as long as they are polarized. According to our analysis, the persistent emission region for FRB 121102 carries an ordered B field, so that its emission should be linearly polarized. We suggest a direct measurement of RM of the persistent emission of FRB 121102 to test our prediction.

We thank the anonymous referee for helpful comments and suggestions, and Qiancheng Liu and Xiaohui Sun for helpful discussions. This research has made use of the FRB catalog (http://frbcat.org, Petroff et al. 2016).

In this appendix, we perform a more general treatment on the relation between the persistent emission specific flux and the RM of FRBs. We assume that at the radius r₀ < r < R from the FRB source, the electron number density follows ${n}_{e}={n}_{e,0}{(r/{r}_{0})}^{-\alpha }$ and the magnetic field follows $B={B}_{0}{(r/{r}_{0})}^{-\beta }$. Then the RM is given by

$$\begin{eqnarray}&&\begin{array}{l}\left|\mathrm{RM}\right|=\displaystyle \frac{{e}^{3}{\xi }_{B}}{2\pi {m}_{e}^{2}{c}^{4}}{\displaystyle \int }_{{r}_{0}}^{R}B(r){n}_{e}(r){dr}\\ =\,\left\{\begin{array}{ll}\tfrac{{e}^{3}{\xi }_{B}}{2\pi {m}_{e}^{2}{c}^{4}}{B}_{0}{n}_{e,0}R{\left(\tfrac{R}{{r}_{0}}\right)}^{-(\alpha +\beta )}, & \mathrm{for}\,\alpha +\beta \lt 1,\\ \tfrac{{e}^{3}{\xi }_{B}}{2\pi {m}_{e}^{2}{c}^{4}}{B}_{0}{n}_{e,0}{r}_{0}, & \mathrm{for}\,\alpha +\beta \gt 1.\end{array}\right.\end{array}\end{eqnarray} \tag{ A1 }$$

In the radius range $r\sim r+{dr}$, the radiation power of a single electron is ${P}_{\nu }(r)={m}_{e}{c}^{2}{\sigma }_{{\rm{T}}}B(r)/3e$, and the number of electrons is $4\pi {r}^{2}{\zeta }_{e}{n}_{e}(r){dr}$. The observed peak flux at the distance D from the source is

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\nu ,\max } & = & \displaystyle \frac{1}{4\pi {D}^{2}}{\displaystyle \int }_{{r}_{0}}^{R}{P}_{\nu }(r)4\pi {r}^{2}{\zeta }_{e}{n}_{e}(r){dr}\\ & = & \displaystyle \frac{{m}_{e}{c}^{2}{\sigma }_{{\rm{T}}}{\zeta }_{e}{B}_{0}{n}_{e,0}}{3{{eD}}^{2}}\\ & & \times \,\left\{\begin{array}{ll}\tfrac{{R}^{3}}{3-(\alpha +\beta )}{\left(\tfrac{R}{{r}_{0}}\right)}^{-(\alpha +\beta )}, & \mathrm{for}\,\alpha +\beta \lt 3,\\ \tfrac{{r}_{0}^{3}}{3-(\alpha +\beta )}, & \mathrm{for}\,\alpha +\beta \gt 3.\end{array}\right.\end{array}\end{eqnarray} \tag{ A2 }$$

According to Equations (A1) and (A3), one finally has

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\nu ,\max } & = & \displaystyle \frac{16{\pi }^{2}}{9(3-\alpha -\beta )}{m}_{e}{c}^{2}\displaystyle \frac{{\zeta }_{e}}{{\xi }_{B}}\displaystyle \frac{{R}^{2}}{{D}^{2}}\left|\mathrm{RM}\right|\\ & & \times \,\left\{\begin{array}{ll}1, & \mathrm{for}\ \alpha +\beta \lt 1,\\ {\left(\tfrac{R}{{r}_{0}}\right)}^{1-(\alpha +\beta )}, & \mathrm{for}\ 1\lt \alpha +\beta \lt 3,\\ {\left(\tfrac{R}{{r}_{0}}\right)}^{-2}, & \mathrm{for}\ \alpha +\beta \gt 3.\end{array}\right.\end{array}\end{eqnarray} \tag{ A3 }$$

This result is consistent with the uniform case with α = β = 0. For any value of α + β, one always has

$$\begin{eqnarray}&&{F}_{\nu ,\max }\leqslant \displaystyle \frac{16{\pi }^{2}}{9(3-\alpha -\beta )}{m}_{e}{c}^{2}\displaystyle \frac{{\zeta }_{e}}{{\xi }_{B}}\displaystyle \frac{{R}^{2}}{{D}^{2}}\left|\mathrm{RM}\right|.\end{eqnarray} \tag{ A4 }$$

The equal sign corresponds to the case with α + β < 1.