FRB 190520B Embedded in a Magnetar Wind Nebula and Supernova Remnant: A Luminous Persistent Radio Source, Decreasing Dispersion Measure, and Large Rotation Measure

Recently, FRB 190520B, which has the largest extragalactic dispersion measure (DM), was discovered by the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The DM excess over the intergalactic medium and Galactic contributions is estimated as ∼900 pc cm−3, which is nearly ten times higher than that of other fast-radio-burst (FRB) host galaxies. The DM decreases with the rate ∼0.1 pc cm−3 per day. It is the second FRB associated with a compact persistent radio source (PRS). The rotation measure (RM) is found to be larger than 1.8 × 105rad m−2. In this Letter, we argue that FRB 190520B is powered by a young magentar formed by core collapse of massive stars, embedded in a composite of a magnetar wind nebula (MWN) and supernova remnant (SNR). The energy injection of the magnetar drives the MWN and SN ejecta to evolve together and the PRS is generated by the synchrotron radiation of the MWN. The magnetar has an interior magnetic field B int ∼ (2–4) × 1016 G and an age t age ∼ 14–22 yr. The dense SN ejecta and the shocked shell contribute a large fraction of the observed DM and RM. Our model can naturally and simultaneously explain the luminous PRS, decreasing DM, and extreme RM of FRB 190520B.


INTRODUCTION
Fast radio bursts (FRBs) are mysterious radio transients with millisecond-duration (Lorimer et al. 2007), whose physical origins are still unknown though they were first reported more than a decade ago (Katz 2018;Cordes & Chatterjee 2019;Zhang 2020;Xiao et al. 2021;Petroff et al. 2021).The large dispersion measures (DMs) of them well above the contribution from the Milky Way imply they may originate at cosmological distances.Some FRBs show repeating bursts and other seem to be one-off events.Many models have been proposed to interpret the origins of FRBs (see Platts et al. 2019 for a recent review).Among those models, the ones relational to magnetars are promising because of the detection of FRB 200428 from a Galactic magnetar (Bochenek et al. 2020;CHIME/FRB Collaboration et al. 2020a).
Recently, the repeating FRB 190520B with the largest extragalactic DM till now was discovered by the Five-hundred-meter Aperture Spherical radio Telescope (FAST) (Niu et al. 2021).It locates in a dwarf galaxy with a high star formation rate, and it is associated with a compact luminous (νL ν ∼ 10 39 erg s −1 ) persistent radio source (PRS), which is too luminous to come from the star-formation activity of the host galaxy (Law et al. 2021).From the observations of Karl G. Jansky Very Large Array (VLA), the power-law spectrum index of the compact PRS has been found to be −0.41 ± 0.04.This is the second PRS associated with FRBs after FRB 121102 (Chatterjee et al. 2017).The similarity between the two PRSs indicates that they have similar physical origin.Interestingly, a similar PRS is found associated with Type I superluminous supernova (SLSN) PTF10hgi, but is less luminous (νL ν ∼ 10 38 erg s −1 ) than that of FRBs (Eftekhari et al. 2019).From the comparative study between the wide-band spectrum of PTF10hgi and FRB 121102 (Mondal et al. 2020), they found the PRS is most probably originating from a pulsar/magnetar wind nebula (PWN/MWN).It has been found that magnetar Swift J1834.9-0846shows a surrounding wind nebula (Younes et al. 2016).In this work, we focus on the PRSs associated with FRBs.

Zhao & Wang
When a pulsar-driven relativistic wind interacts with the surrounding medium, the luminous PWN generates.For rapidly rotating pulsars, the rotational energy is the main reservoir for powering the wind nebula, which has been well-studied for Galactic PWNe (Tanaka & Takahara 2010).Some FRBs' energy injection (Li et al. 2020) and rotational energy injection (Kashiyama & Murase 2017;Dai et al. 2017;Yang & Dai 2019;Wang & Lai 2020) models have been proposed to explain the PRS associated with FRB 121102.However, for a decades-old magnetar, the rotational energy is less significant than the interior magnetic energy.The case of the magnetic energy injection was proposed, and it is successfully explaining the PRS's luminosity and large rotation measure (RM, Michilli et al. 2018;Hilmarsson et al. 2021) of FRB 121102 (Margalit & Metzger 2018).
From the estimation of Niu et al. (2021), the DM of host galaxy is DM host ∼ 900 pc cm −3 , which is nearly ten times higher than other FRBs' host galaxies.However, using the state-of-the-art IllustrisTNG simulation, Zhang et al. (2020) showed that the DM contributed by FRB 190520B-like host galaxies at z ∼ 0.2 is ∼ 50 − 250 pc cm −3 .Different from the increasing DM of FRB 121102 (Hessels et al. 2019;Josephy et al. 2019;Oostrum et al. 2020;Li et al. 2021) and the nearly unchangeable DM of FRB 180916 (CHIME/FRB Collaboration et al. 2020b;Pastor-Marazuela et al. 2021;Nimmo et al. 2021), the DM of FRB 190520B decreases with the rate −0.09 ± 0.02 pc cm −3 day −1 (Niu et al. 2021).Under the assumption of 100% linearly polarized intrinsically, the low limit of RM is > 1.8 × 10 5 rad m −2 (Niu et al. 2021), which is larger than that of FRB 121102.The large DM and RM together with decreasing DM may come from the expanding shocked shell of supernova remnant (SNR, see Yang & Zhang 2017;Piro & Gaensler 2018;Zhao et al. 2021;Katz 2021a).Katz (2021a) rejected the possibility that the large host DM of FRB 190520B is contributed by interstellar cloud, and proposed that the excess of host DM attributes to a young SNR.It has also been claimed that the luminous PRS correlates with the large RM, if RM mostly arises from the persistent emission region (Yang et al. 2020).
In this letter, we propose that the magnetar associated with FRB 190520B is embedded in the composite of MWN and SNR.The magnetar is formed by core-collapse of massive star.Due to the energy injection of the young magnetar, the wind nebula and the SN ejecta will evolve together.The observed PRS is produced by the synchrotron radiation of the nebula.Our numerical calculations are based on the spectral evolution model of Galactic PWNe (Tanaka & Takahara 2010, 2013) with magnetic-energy injection (Margalit & Metzger 2018).The dense SNR ejecta and the shocked shell contribute considerable DM and RM.With the expansion of SNR, the DM will decrease rapidly, similar to the observed trend.Our model can simultaneously explain the luminous PRS, decreasing DM and extreme RM of FRB 190520B.
This letter is organized as follows.In Section 2, the synchrotron spectral evolution model from MWN is shown.We present our numerical results of the PRS energy spectrum in Section 3. The long-term DM evolution model to explain that of FRB 190520B is shown in Section 4. Finally, summary is given in Section 5.

THE COMPACT PERSISTENT RADIO SOURCE
The compact PRS associated with FRBs is from the synchrotron radiation from the MWN powered by the young magnetar in our model.Rotational or magnetic energy together with the particle is injected into the nebula, and the electron will undergo radiation or adiabatic cooling.In this section, we will introduce the cases of rotational and magnetic energy injection, and give the explanation of the radio spectra of PRSs associated with FRB 190520B and FRB 121102.

Energy injection
The case of rotational energy injection is well studied for wide-band spectrum of the Crab Nebula (Tanaka & Takahara 2010).The spin-down luminosity can be estimated as (Dai & Lu 1998;Zhang & Mészáros 2001;Murase et al. 2015) where t em 3.2 × 10 5 B −2 dip,14 P 2 i,−2.5 s is the characteristic spin-down timescale, B dip is the dipole magnetic and P i is the initial spin period.The model of magnetic energy injection has been proposed to explain the exceptionally high RM and PRS associated with FRB 121102 (Margalit & Metzger 2018).The interior magnetic energy (Katz 1982) is another ideal reservoir for PRS, where B int is the interior magnetic field and R ns = 12 km is the neutron star radius.The magnetic-energy-injection luminosity can be written as (Margalit & Metzger 2018) where t 0 is the onset of energy injection and α > 1 is the power-law index.
The injected electron-positron pairs will be accelerated to relativistic energy by the termination shock before entering the nebula.Similar to Galactic PWNe (Tanaka & Takahara 2010, 2013), the injection particles spectrum is described as a broken power-law form where Q 0 (t) is the normalization factor, γ min , γ b and γ max is the minimum, break, and maximum Lorentz factors.p 1 and p 2 are the injection spectrum indices for low and high energy particles, respectively.The normalization factor is determined by where e is the electron energy fraction and L(t) is the spin-down or magnetic energy injection luminosity.

Dynamics and the nebular magnetic fields evolution
The inner density profile of the ejecta can be described as a smooth or flat power-law (Chevalier & Soker 1989;Kasen & Bildsten 2010) where δ = 0−1 is widely used, and we take δ = 1 in this work.The ejecta will expand freely until the Sedov-Taylor phase without the energy injection.The initial velocity is v ej,i ∼ 10, 000 km s −1 E SN /10 51 erg When a newborn millisecond magnetar exists, the nebula and ejecta radius will evolve together because the injected energy will significantly accelerate the ejecta via magnetized wind.For R n < R ej , the nebula radius R n is given by (Metzger et al. 2014) where E tot is the total injection energy.If R n > R ej , the nebula and ejecta will move together where v ej,f = 2(E tot + E SN )/M ej is the finial accelerated velocity.For t < t 0 , the rotational energy injection dominates, and the injection energy is For t > t 0 , the interior magnetic energy starts to leak out into the nebula and the total injection energy The solution of Equations ( 7) and (8) for the example case B int ∼ 10 16 G, M ej = 1 M , and E SN = 1 × 10 51 erg is shown in Figure 1.The red, blue and green solid lines represent the case of P i = 1.5 ms, P i = 2.5 ms and P i = 5 ms, respectively.The initial ejecta velocity is shown as black lines.The onset of the magnetic energy injection t 0 = 0.2 yr and 0.6 yr from the benchmark model of Margalit & Metzger (2018) is shown in cyan solid and dashed lines, respectively.We can see that ejecta is accelerated significantly in a short time (∼ 10 −2 − 10 −1 yr) before the magnetic flux begins to leak out.The finial ejecta velocity is up to ∼ 20, 000 − 60, 000 km s −1 , which is well consistent with the observations of SN Ib/Ic (Kawabata et al. 2002;Rho et al. 2021).
The evolution of nebular magnetic fields is , where E B is the magnetic energy in nebular and R n is the nebula radius.The magnetic energy in nebular is given by (Murase et al. 2021) where B is the magnetic energy fraction.In this work, we do not consider the magnetic energy loss caused by adiabatic expansion, which has been used in Galactic PWNe (Tanaka & Takahara 2010, 2013) and high-energy emission of pulsar-powered PWNe (Murase et al. 2015(Murase et al. , 2016)).The limit c B → 0 is a good approximation for a young source engine.

The Evolution of Particle Distribution
The evolution of the electron number density distribution n e,γ is given by the continuity equation in energy space where Qe,γ is the injection electron number density.The electron cooling process γ includes the synchrotron radiation, synchrotron self-Compton (SSC) and adiabatic expansion γ(γ, t) = γsyn (γ, t) + γSSC (γ, t) + γad (γ, t). (11) The energy loss of synchrotron radiation is given by (Rybicki & Lightman 1979) γsyn (γ, t) where U B = B 2 n /8π is the energy density of magnetic field.The energy loss caused by SSC is (Blumenthal & Gould 1970) γSSC (γ, t) = − 3 4 where ν ini and ν fin are the frequencies of initial the synchrotron radiation photons and that of scattered photons, Γ = 4γhν ini /(m e c 2 ), q = hν fin /(Γ (γm e c 2 − hν fin )), f (q, Γ ) = 2q ln q + (1 + 2q)(1 − q) + 0.5(1 − q)(Γ q) 2 /(1 + Γ q), and θ(x) is the step function.The seed synchrotron photon number density n syn is where L ν,syn is the synchrotron radiation luminosity (see Section 2.4), and Ū ∼ 2.24 (Atoyan & Aharonian 1996) is used in our calculations.The adiabatic cooling is given by γad (γ, t) 2.4.The synchrotron radiation of MWN The spectral power of synchrotron radiation is where ν c = γ 2 eB/2πm e c is the characteristic frequency, F(x) = x +∞ x K 5/3 (k)dk and K 5/3 (k) is the 5/3 order modified Bessel function.The emissivity and absorption coefficients of synchrotron radiation is The synchrotron radiation luminosity considering the synchrotron self-absorption (SSA) is In addition to SSA, free-free absorption due to the ejecta is also important for radio signals from a young magnetar.From the study of DM and RM evolution of FRB 121102 (Zhao et al. 2021), the associated magnetar is in a clean environment, which means that the magnetar is born in the merger of two compact stars.For the merger channel, the ejecta mass is ∼ 0.001 − 0.1 M , whose free-free absorption process is not obvious.However, for SNe channel, free-free absorption due to the ejecta can not be neglected.The free-free optical depth of the ejecta is (Wang et al. 2020;Zhao et al. 2021) where η is the ionization fraction, Y e,0.2 = Y e /0.2 is the the electron fraction, M ej,1 = M ej /1M is the ejecta mass, T ej,4 = T ej /10 4 K is the ejecta temperature.Due to the free-free absorption of electrons, the SNR will be optically thick for 3 yr and 1.5 yr for M ej = 10 M and M ej = 2 M , which is shown in gray and black shaded regions in Figures 2 and 3, respectively.

NUMERICAL RESULTS
From the dynamics equations of MWN, we know that the nebula/ejecta velocity is mainly accelerated by the rotational energy injection and is almost constant for the time of our interest.The assumption of v n v ej,f is a good approximation (Metzger et al. 2014;Kashiyama et al. 2016) for R n > R ej .In our calculations, we take v ej,f = 0.1c, which is the mean ejecta velocity of SN Ib/Ic (Soderberg et al. 2012) and compact binary mergers.Following Margalit & Metzger (2018), t 0 = 0.2 yr and α = 1.3 is used in this work.The energy fraction B = 0.1 and B + e ∼ 1 is used.The injection spectrum index p 1 = 1.3 and p 2 = 2.5 is taken from Law et al. (2019); Mondal et al. (2020).The exact value of γ min and γ max is not important as long as its value is small or large enough.The main parameters are the interior magnetic field B int , the source age t age and the break Lorentz factor γ b .
The spectral energy distribution is shown in Figure 2. The electron density n e and the nebula magnetic field B n are from the solutions of Equations ( 10) and ( 9).We find that parameters B int = 2.76 × 10 16 G, t age = 14 yr and γ b = 5 × 10 4 can reproduce the spectrum of the PRS associated with FRB 121102 (Chatterjee et al. 2017), and B int = 3.41 × 10 16 G, t age = 22 yr and γ b = 5 × 10 3 for that of FRB 190520B (Niu et al. 2021).The source age of FRB 121102 we guessed as is to be roughly consistent with previous study (Yang & Dai 2019;Margalit & Metzger 2018;Zhao et al. 2021).For FRB 190520B, the source age is given by the estimated from the DM evolution (see Section 4).The red, blue and green solid lines represent the observed epoch at t = t age /3, t = t age and t = 3t age .The case of rotational energy injection is also plotted as dashed line at t = t age for comparison, whose dipole magnetic field is estimated under the assumption of B dip = 0.1B int (Levin et al. 2020) and initial spin period P i = 5 ms is taken.The light-curves at 1 GHz, 3 GHz, and 5.5 GHz (blue, red, and green lines, respectively) for FRB 121102 and FRB 190520B (solid and dashed lines, respectively) are shown in the bottom panel in Figure 2. The size of MWN we obtained is ∼ 0.4 pc for FRB 121102, which satisfies the constraints < 0.7 pc given by very long baseline interferometry (VLBI, Marcote et al. 2017).
The DM and RM from the relativistic electrons in MWN are where the electron density n e and the nebula magnetic field B n are from the solutions of Equations ( 10) and ( 9).We obtain that the DM from the MWN is < 1 − 10 pc cm −3 and RM is < 10 4 − 10 5 rad m −2 .The contributions from the MWN are negligible compared with the SNR (see Section 4).

LONG-TERM DM EVOLUTION
FRB 190520B has been reported in a dense environment, and the estimated DM host 902 +88 −128 pc cm −3 (Niu et al. 2021) is nearly ten times higher than other FRB host galaxies.The DM of FRB 190520B systemically decreases with the rate −0.09±0.02pc cm −3 day −1 together with some irregular variations (Niu et al. 2021).In our model, the long-term DM variation is from the expanding SNR (Yang & Zhang 2017;Piro & Gaensler 2018;Zhao et al. 2021;Katz 2021a) and the random variations may be caused by turbulent motions of filament (Katz 2021b).

The DM from the local environment
For cosmological FRBs, the observed DM or RM contains the contributions of the Milky Way (MW), the Milky Way halo, the intergalactic medium (IGM), the host galaxy and the local environment of FRBs: Using the IllustrisTNG simulation, Zhang et al. (2020) found that the DM contributed by FRB 190520B-like host galaxies at z ∼ 0.2 is ∼ 50 − 250 pc cm −3 .Therefore, the DM from the source of FRB 190520B can be inferred to be ∼ 524 − 940 pc cm −3 .Following our previous work (Zhao et al. 2021), the DM from the local environment of FRBs is given by DM source = DM MWN + DM unsh,ej + DM sh,ej + DM sh,ISM + DM unsh,ISM , where DM MWN , DM unsh,ej , DM sh,ej , DM sh,ISM and DM unsh,ISM are the contributions from the MWN, unshocked ejecta, shocked ejecta, shocked ISM and unshocked ISM, respectively.Usually, DM unsh,ISM is negligible because of the low ionization fraction.The unshocked region is not magnetized, so the RM from the source only contributed by three parts The contributions from the MWN are also negligible compared with the SNR.The total DMs and RMs from the SNR are shown in Figure 3.Our calculations are based on Equations ( 24) and ( 45)-( 48) of Zhao et al. (2021).We adopt the typical parameters of SNRs: the explosion energy E SN ∼ 1 × 10 51 erg, the power-law index of outer ejecta n = 10, ionization fractions of unshocked ejecta η = 0.1, the wind velocity of progenitors v w 10 km s −1 and B = 0.1.The solid and dashed lines represent the case of M ej = 10 M and M ej = 2 M , respectively.The blue, red and green lines represent different progenitors' mass-loss rate.
The orange shading is the range of estimated DM source .We can see that only the case of Ṁ = 10 −4 M yr −1 and the source age t age = 10 − 30 yr can provide the large enough DM.The constraint on the source age is consistent with that derived from PRS in the previous section.

Fitting Results
We assume that the variations of DM is only from DM source .Thus, we can define the unchangeable DM other = DM obs −DM source for facilitate fitting.From the estimation of DM source above, we can get DM other ∼ 270 − 686 pc cm −3 .The Markov Chain Monte Carlo (MCMC) method performed by Python package emcee 1 (Foreman-Mackey et al. 2013) is used to estimate the parameters DM other and the age of the source t age .The χ 2 for the observed DMs is where DM SNR is the DM from SNR given by our model, and DM obs and σ is the observed DM and uncertainties in the frame of observers (Niu et al. 2021).The likelihood is The posterior corner plots obtained from fitting the models of two typical ejecta mass models (M ej = 10 M and M ej = 2 M ) to the deta are shown in Figure 4. Our best-fit parameters are shown in blue solid lines, and the parameters with 1-σ ranges are shown in dashed lines.For M ej = 10 M , we find DM other = 464 ± 15 pc cm −3 (DM SNR = 746 ± 15 pc cm −3 ) and the source age is 21.9 ± 0.5 yr.For M ej = 2 M , we find DM other = 642 ± 12 pc cm −3 (DM SNR = 568 ± 12 pc cm −3 ) and the source age is 16.6 ± 0.4 yr.The DM evolution after the SN explosion is plotted in Figure 5. Due to the free-free absorption of electrons, the SNR will be optically thick for 3 yr and 1.5 yr for M ej = 10 M and M ej = 2 M , respectively.The shaded regions represent the SNR is opaque to radio signals of ν ∼ 1 GHz.If we assume that the typical SN ejecta mass is 2M < M ej < 10M , the source age of FRB 190520B can be estimated as 16 − 22 yr.In the same way, we have DM other = 449 − 654 pc cm −3 (DM SNR = 556 − 761 pc cm −3 ).
When DM source is taken into consideration, the DM host of FRB 190520B is not special, which is consistent with that derived from the IllustrisTNG simulation (Zhang et al. 2020).The DM decline will continue for another few decades, and then DM will trend to be stable when DM other DM SNR .Finally, we have DM ∼ DM other for t ∼ 100 − 1000 yr.The unchangeable DM has been reported for FRBs (e.g., FRB 180916, see CHIME/FRB Collaboration et al. 2020b;Pastor-Marazuela et al. 2021;Nimmo et al. 2021).For FRB 121102, the estimated RM MWN ∼ 10 4 − 10 5 rad m −2 at t age = 14 yr is consistent with the study of DM and RM evolution (Zhao et al. 2021, their estimated age starts on 2012).For FRB 190520B, the large RM is from the young SNR (RM ∼ 10 7 − 10 8 rad m −2 ).

SUMMARY
In this letter, we argue that the magnetar associated with FRB 190520B is embedded in the 'composite' of MWN and SNR.Due to the energy injection of the young magnetar (16-22 yr), the wind nebula and the SN ejecta will evolve together.The observed PRS is from the synchrotron radiation of the nebula.The dense SNR ejecta and the shocked shell contribute the observed DM and RM.Our model can simultaneously explain the luminous PRS, decreasing DM and extreme RM of FRB 190520B.Our conclusions are summarized as follows: • The compact PRSs associated with FRBs are from the synchrotron radiation of the MWNe.From the observed luminosities and spectra, we find the interior magnetic field B int = 2.76 × 10 16 G, the source age t age = 14 yr for FRB 121102, and B int = 3.41 × 10 16 G, t age = 22 yr for FRB 190520B.
• FRB 190520B is embedded in a dense SNR whose DM contribution is ∼ 746 ± 15 pc cm −3 and ∼ 568 ± 12 pc cm −3 for M ej = 10 M and M ej = 2 M , respectively.Considering the DM from the SNR, the DM from the interstellar medium of FRB 190520B host galaxy is not special any more, which is consistent with that derived form the state-of-the-art IllustrisTNG simulation (Zhang et al. 2020).The DM decay rate −0.09 ± 0.02 pc cm −3 d −1 can be well understood in the context of a SNR with the age of 16 − 22 yr, well in the range required by the PRS.The decline will continue for another few decades, and then DM will trend to be stable.
• For FRB 190520B, the large RM is from the young SNR (RM ∼ 10 7 − 10 8 rad m −2 ).The RM attributed to MWN is < 10 4 − 10 5 rad m −2 in our model, which is much lower than the lower limit given by Niu et al. (2021) and the contributions from SNR.
10 2 10 1 10 0 10 1 10 2 t (yr) 10 8 10 9 10 10 v n (cm s 1 ) P i = 1.5 ms P i = 2.5 ms P i = 5 ms v ej, i (M ej = 1 M , SN = 1 × 10 51 erg) Figure 1.The nebula velocity derived from Equations ( 7) and ( 8) for B int ∼ 10 16 G, M ej = 1 M , and E SN = 1 × 10 51 erg.The red, blue and green solid lines represent the case of P i = 1.5 ms, P i = 2.5 ms and P i = 5 ms, respectively.The initial ejecta velocity is shown in black lines.The onset of the magnetic energy injection t 0 = 0.2 yr and 0.6 yr is shown in cyan solid and dashed line, respectively.We can see that ejecta is accelerated significantly in a relatively short time (∼ 10 −2 − 10 −1 yr) before the magnetic flux begins to leak out.The finial ejecta velocity is up to ∼ 20, 000 − 60, 000 km s −1 , which is consistent with the observations of SN Ib/Ic.

Figure 3 .Figure 4 .
Figure 3. DM (left panel) and RM (right panel) contributed by SNR based on Equations (24) and (45)-(48) ofZhao et al. (2021).We adopt the typical parameters of SNRs: the explosion energy E SN ∼ 1 × 10 51 erg, the power-law index of outer ejecta n = 10, ionization fractions of unshocked ejecta η = 0.1, the wind velocity of progenitors v w = 10 km s −1 and B = 0.1.The solid and dashed lines represent the case of M ej = 10 M and M ej = 2 M , respectively.The blue, red and green lines represent different progenitors' mass-loss rates.The orange shading is the range of estimated DM source .We can see that only the case of Ṁ = 10 −4 M yr −1 and the source age t age = 10 − 30 yr can provide the large enough DM required by observations.The SNR will be optically thick for 3 yr and 1.5 yr for M ej = 10 M and M ej = 2 M , which is shown in gray and black shaded regions.

Figure 5 .
Figure 5.Samples of DM SNR for M ej = 10 M (left panel) and M ej = 2 M (right panel) from the MCMC method (orange curves).Black squres and circles are the DM obs of FRB 190520B from Niu et al. (2021).Black lines represent the best-fit values given by the MCMC method.The shaded regions represent the SNR is opaque to radio signals of ν ∼ 1 GHz.