# FRB 190520B embedded in a magnetar wind nebula and supernova remnant:
luminous persistent radio source, decreasing dispersion measure and large
rotation measure

Z. Y. Zhao School of Astronomy and Space Science, Nanjing University, Nanjing
210093, China  F. Y. Wang School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China  Key Laboratory of Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093,
China F. Y. Wang<EMAIL_ADDRESS>
(Received Nov 25, 2021; Revised Nov 30, 2021; Accepted Dec 2, 2021)

###### Abstract

Recently, FRB 190520B with 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 $\sim 900$ pc cm-3, which is nearly ten times higher than
other fast radio bursts (FRBs) host galaxies. The DM decreases with the rate
$\sim 0.1$ 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 $1.8\times 10^{5}\mathrm{rad}~{}\mathrm{m}^{-2}$. 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 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 the interior magnetic field
$B_{\text{int}}\sim(2-4)\times 10^{16}$ G and the age $t_{\text{age}}\sim
14-22$ yr. The dense SN ejecta and the shocked shell contribute a large
fraction of the observed DM and RM. Our model can naturally explain the
luminous PRS, decreasing DM and extreme RM of FRB 190520B simultaneously.

Fast radio burst, magnetar, magnetar wind nebula, supernova remnant

##  1 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 ($\nu L_{\nu}\sim
10^{39}$ 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 $-0.41\pm 0.04$. 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 ($\nu
L_{\nu}\sim 10^{38}$ 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 J1834.9-0846 shows a surrounding wind nebula (Younes et al., 2016). In
this work, we focus on the PRSs associated with FRBs.

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${}_{\text{host}}\sim 900$ 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 $z\sim 0.2$ is $\sim 50-250$ 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
$-0.09\pm 0.02$ pc cm-3 day-1 (Niu et al., 2021). Under the assumption of 100%
linearly polarized intrinsically, the low limit of RM is $>1.8\times
10^{5}~{}\mathrm{rad}~{}\mathrm{m}^{-2}$ (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.

##  2 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.

###  2.1 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)

$\displaystyle L_{\mathrm{sd}}=$ $\displaystyle
L_{\mathrm{sd},i}\left(1+\frac{t}{t_{\mathrm{em}}}\right)^{-2}$
$\displaystyle\simeq\begin{cases}8.6\times
10^{45}\mathrm{erg}\mathrm{s}^{-1}P_{i,-2.5}^{-4}B_{\mathrm{dip},14}^{2}&\left(t\leqslant
t_{\mathrm{em}}\right)\\\ 8.9\times
10^{42}\mathrm{erg}\mathrm{s}^{-1}B_{\mathrm{dip},14}^{-2}t_{7}^{-2}&\left(t>t_{\mathrm{em}}\right)\end{cases},$
(1)

where $t_{\mathrm{em}}\simeq 3.2\times
10^{5}B_{\mathrm{dip},14}^{-2}P_{i,-2.5}^{2}\mathrm{~{}s}$ is the
characteristic spin-down timescale, $B_{\mathrm{dip}}$ 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)

$\mathcal{E_{\text{B,int}}}\simeq B_{\text{int}}^{2}R_{\text{ns}}^{3}/6\approx
3\times 10^{49}B_{\text{int},16}^{2}\text{~{}erg},$ (2)

is another ideal reservoir for PRS, where $B_{\text{int}}$ is the interior
magnetic field and $R_{\text{ns}}=12$ km is the neutron star radius. The
magnetic-energy-injection luminosity can be written as (Margalit & Metzger,
2018)

$L_{\mathrm{m}}=(\alpha-1)\frac{\mathcal{E_{\text{B,int}}}}{t_{0}}\left(\frac{t}{t_{0}}\right)^{-\alpha},$
(3)

where $t_{0}$ is the onset of energy injection and $\alpha>1$ 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

$Q_{\mathrm{inj}}(\gamma,t)=\begin{cases}Q_{0}(t)\left(\gamma/\gamma_{\mathrm{b}}\right)^{-p_{1}}&\quad\gamma_{\min}\leqslant\gamma\leqslant\gamma_{\mathrm{b}}\\\
Q_{0}(t)\left(\gamma/\gamma_{\mathrm{b}}\right)^{-p_{2}}&\quad\gamma_{\mathrm{b}}\leqslant\gamma\leqslant\gamma_{\max}\end{cases},$
(4)

where $Q_{0}(t)$ is the normalization factor, $\gamma_{\min}$, $\gamma_{b}$
and $\gamma_{\max}$ 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

$\int_{\gamma_{\min}}^{\gamma_{\max}}Q_{\mathrm{inj}}(\gamma,t)\gamma
m_{e}c^{2}d\gamma=\epsilon_{\mathrm{e}}L(t),$ (5)

where $\epsilon_{\mathrm{e}}$ is the electron energy fraction and $L(t)$ is
the spin-down or magnetic energy injection luminosity.

###  2.2 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)

$\rho_{\mathrm{ej}}=\frac{(3-\delta)M_{\mathrm{ej}}}{4\pi
R_{\mathrm{ej}}^{3}}\left(\frac{R}{R_{\mathrm{ej}}}\right)^{-\delta},$ (6)

where $\delta=0-1$ is widely used, and we take $\delta=1$ in this work. The
ejecta will expand freely until the Sedov–Taylor phase without the energy
injection. The initial velocity is $v_{\mathrm{ej},\mathrm{i}}\sim
10,000\mathrm{~{}km}\mathrm{~{}s}^{-1}\left(\mathcal{E}_{\mathrm{SN}}/10^{51}\mathrm{erg}\right)^{1/2}\left(M_{\mathrm{ej}}/M_{\odot}\right)^{-1/2}\mathrm{~{}cm}\mathrm{~{}s}^{-1}$.
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 $R_{\text{n}}<R_{\text{ej}}$, the nebula
radius $R_{\text{n}}$ is given by (Metzger et al., 2014)

$\frac{dR_{n}}{dt}=\sqrt{\frac{7}{6(3-\delta)}\frac{\mathcal{E}_{\mathrm{tot}}}{M_{\mathrm{ej}}}\left(\frac{R_{n}}{R_{\mathrm{ej}}}\right)^{3-\delta}}+\frac{R_{n}}{t},$
(7)

where $\mathcal{E}_{\mathrm{tot}}$ is the total injection energy. If
$R_{\text{n}}>R_{\text{ej}}$, the nebula and ejecta will move together

$\frac{\mathrm{d}R_{\mathrm{ej}}}{\mathrm{d}t}=\frac{\mathrm{d}R_{\mathrm{n}}}{\mathrm{d}t}=v_{\mathrm{ej,f}},$
(8)

where
$v_{\mathrm{ej,f}}=\sqrt{2(\mathcal{E}_{\mathrm{tot}}+\mathcal{E}_{\mathrm{SN}})/M_{\mathrm{ej}}}$
is the finial accelerated velocity. For $t<t_{0}$, the rotational energy
injection dominates, and the injection energy is
$\mathcal{E}_{\mathrm{tot}}=\mathcal{E}_{\mathrm{rot}}=\int_{0}^{t}L_{\mathrm{sd}}(t)\mathrm{d}t$.
For $t>t_{0}$, the interior magnetic energy starts to leak out into the nebula
and the total injection energy
$\mathcal{E}_{\mathrm{tot}}=\mathcal{E}_{\mathrm{rot}}+\mathcal{E}_{\mathrm{mag}}=\int_{0}^{t}(L_{\mathrm{sd}}(t)+L_{\mathrm{m}}(t))\mathrm{d}t$.
The solution of Equations (7) and (8) for the example case $B_{\text{int}}\sim
10^{16}$ G, $M_{\mathrm{ej}}=1~{}M_{\odot}$, and
$\mathcal{E}_{\mathrm{SN}}=1\times 10^{51}$ erg is shown in Figure 1. The red,
blue and green solid lines represent the case of $P_{\text{i}}=1.5$ ms,
$P_{\text{i}}=2.5$ ms and $P_{\text{i}}=5$ 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 ($\sim
10^{-2}-10^{-1}$ yr) before the magnetic flux begins to leak out. The finial
ejecta velocity is up to $\sim 20,000-60,000\mathrm{~{}km}\mathrm{~{}s}^{-1}$,
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
$B_{\mathrm{n}}=\sqrt{(6\mathcal{E_{\text{B}}}/R_{\mathrm{n}}^{3})}$, where
$\mathcal{E_{\text{B}}}$ is the magnetic energy in nebular and
$R_{\mathrm{n}}$ is the nebula radius. The magnetic energy in nebular is given
by (Murase et al., 2021)

$\frac{d\mathcal{E_{\text{B}}}}{dt}=\epsilon_{\mathrm{B}}L(t)-c_{\mathrm{B}}\frac{\dot{R}_{\mathrm{n}}}{R_{\mathrm{n}}}\mathcal{E_{\text{B}}},$
(9)

where $\epsilon_{\mathrm{B}}$ 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, 2016). The limit
$c_{\mathrm{B}}\to 0$ is a good approximation for a young source engine.

###  2.3 The Evolution of Particle Distribution

The evolution of the electron number density distribution $n_{e,\gamma}$ is
given by the continuity equation in energy space

$\frac{\partial}{\partial
t}n_{e,\gamma}+\frac{\partial}{\partial\gamma}\left(\dot{\gamma}n_{e,\gamma}\right)=\dot{Q}_{e,\gamma},$
(10)

where $\dot{Q}_{e,\gamma}$ is the injection electron number density. The
electron cooling process $\dot{\gamma}$ includes the synchrotron radiation,
synchrotron self-Compton (SSC) and adiabatic expansion

$\dot{\gamma}(\gamma,t)=\dot{\gamma}_{\mathrm{syn}}(\gamma,t)+\dot{\gamma}_{\mathrm{SSC}}(\gamma,t)+\dot{\gamma}_{\mathrm{ad}}(\gamma,t).$
(11)

The energy loss of synchrotron radiation is given by (Rybicki & Lightman,
1979)

$\dot{\gamma}_{\mathrm{syn}}(\gamma,t)=-\frac{4}{3}\frac{\sigma_{\mathrm{T}}\gamma^{2}}{m_{e}c}U_{\mathrm{B}}(t),$
(12)

where $U_{\mathrm{B}}=B_{\text{n}}^{2}/8\pi$ is the energy density of magnetic
field. The energy loss caused by SSC is (Blumenthal & Gould, 1970)

$\displaystyle\dot{\gamma}_{\mathrm{SSC}}(\gamma,t)=$
$\displaystyle-\frac{3}{4}\frac{\sigma_{\mathrm{T}}h}{m_{e}c}\frac{1}{\gamma^{2}}\int_{0}^{\infty}\nu_{\mathrm{fin}}d\nu_{\mathrm{fin}}$
(13)
$\displaystyle\times\int_{0}^{\infty}\frac{n_{\mathrm{syn}}\left(\nu_{\mathrm{ini}},t\right)}{\nu_{\mathrm{ini}}}f\left(q,\Gamma_{\epsilon}\right)\theta(1-q)\theta\left(q-1/4\gamma^{2}\right)dv_{\mathrm{ini}},$

where $\nu_{\mathrm{ini}}$ and $\nu_{\mathrm{fin}}$ are the frequencies of
initial the synchrotron radiation photons and that of scattered photons,
$\Gamma_{\epsilon}=4\gamma h\nu_{\text{ini }}/(m_{e}c^{2})$,
$q=h\nu_{\text{fin}}/(\Gamma_{\epsilon}(\gamma m_{e}c^{2}-h\nu_{\text{fin
}}))$, $f(q,\Gamma_{\epsilon})=2q\ln
q+(1+2q)(1-q)+0.5(1-q)(\Gamma_{\epsilon}q)^{2}/(1+\Gamma_{\epsilon}q)$, and
$\theta(x)$ is the step function. The seed synchrotron photon number density
$n_{\mathrm{syn}}$ is

$n_{\mathrm{syn}}(\nu,t)=\frac{L_{\nu,\mathrm{syn}}(t)}{4\pi
R_{\mathrm{n}}^{2}(t)c}\frac{1}{h\nu}\bar{U},$ (14)

where $L_{\nu,\mathrm{syn}}$ is the synchrotron radiation luminosity (see
Section 2.4), and $\bar{U}\sim 2.24$ (Atoyan & Aharonian, 1996) is used in our
calculations. The adiabatic cooling is given by

$\dot{\gamma}_{\mathrm{ad}}(\gamma,t)=-\frac{1}{3}\gamma\frac{d\ln
V_{\mathrm{n}}}{dt}=-\frac{\dot{R}_{\mathrm{n}}}{R_{\mathrm{n}}}\gamma.$ (15)

###  2.4 The synchrotron radiation of MWN

The spectral power of synchrotron radiation is

$P_{\nu}=\frac{2e^{3}B_{\mathrm{n}}}{\sqrt{3}m_{\mathrm{e}}c^{2}}F\left(\frac{\nu}{\nu_{\mathrm{c}}}\right),$
(16)

where $\nu_{\mathrm{c}}=\gamma^{2}eB/2\pi m_{\mathrm{e}}c$ is the
characteristic frequency, $F(x)=x\int_{x}^{+\infty}K_{5/3}(k)dk$ and
$K_{5/3}(k)$ is the 5/3 order modified Bessel function. The emissivity and
absorption coefficients of synchrotron radiation is

$j_{\nu}=\int\frac{n_{e,\gamma}P_{\nu}(\gamma)}{4\pi}d\gamma,$ (17)
$\alpha_{\nu}=-\int\frac{\gamma^{2}P_{\nu}(\gamma)}{8\pi
m_{\mathrm{e}}\nu^{2}}\frac{\partial}{\partial\gamma}\left(\frac{n_{e,\gamma}}{\gamma^{2}}\right)d\gamma.$
(18)

The synchrotron radiation luminosity considering the synchrotron self-
absorption (SSA) is

$L_{\nu}=4\pi^{2}R_{\mathrm{n}}^{2}\frac{j_{\nu}}{\alpha_{\nu}}\left(1-e^{-\alpha_{\nu}R_{\mathrm{n}}}\right).$
(19)

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 $\sim 0.001-0.1$
$M_{\odot}$, 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)

$\tau_{\mathrm{ff},\mathrm{ej}}=\alpha_{\nu,\mathrm{ff}}\Delta R\simeq
2.06\eta^{2}Y_{\mathrm{e},0.2}^{2}M_{\mathrm{ej},1}^{2}T_{\mathrm{ej},4}^{-3/2}\nu_{9}^{-2}v_{\mathrm{ej},9}^{-5}t_{\mathrm{yr}}^{-5},$
(20)

where $\eta$ is the ionization fraction,
$Y_{\mathrm{e},0.2}=Y_{\mathrm{e}}/0.2$ is the the electron fraction,
$M_{\mathrm{ej},1}=M_{\mathrm{ej}}/1M_{\odot}$ is the ejecta mass,
$T_{\mathrm{ej},4}=T_{\mathrm{ej}}/10^{4}$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 $M_{\text{ej}}=10$ $M_{\odot}$ and $M_{\text{ej}}=2$
$M_{\odot}$, which is shown in gray and black shaded regions in Figures 2 and
3, respectively.

##  3 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 $v_{\text{n}}\simeq
v_{\text{ej,f}}$ is a good approximation (Metzger et al., 2014; Kashiyama et
al., 2016) for $R_{\text{n}}>R_{\text{ej}}$. In our calculations, we take
$v_{\text{ej,f}}=0.1c$, 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 $\alpha=1.3$ is used in this work. The
energy fraction $\epsilon_{\mathrm{B}}=0.1$ and
$\epsilon_{\mathrm{B}}+\epsilon_{\mathrm{e}}\sim 1$ 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 $\gamma_{\min}$ and $\gamma_{\max}$
is not important as long as its value is small or large enough. The main
parameters are the interior magnetic field $B_{\text{int}}$, the source age
$t_{\text{age}}$ and the break Lorentz factor $\gamma_{\text{b}}$.

The spectral energy distribution is shown in Figure 2. The electron density
$n_{e}$ and the nebula magnetic field $B_{\text{n}}$ are from the solutions of
Equations (10) and (9). We find that parameters $B_{\text{int}}=2.76\times
10^{16}$ G, $t_{\text{age}}=14$ yr and $\gamma_{\text{b}}=5\times 10^{4}$ can
reproduce the spectrum of the PRS associated with FRB 121102 (Chatterjee et
al., 2017), and $B_{\text{int}}=3.41\times 10^{16}$ G, $t_{\text{age}}=22$ yr
and $\gamma_{\text{b}}=5\times 10^{3}$ 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 $t=t_{\text{age}}/3$, $t=t_{\text{age}}$ and
$t=3t_{\text{age}}$. The case of rotational energy injection is also plotted
as dashed line at $t=t_{\text{age}}$ for comparison, whose dipole magnetic
field is estimated under the assumption of $B_{\text{dip}}=0.1B_{\text{int}}$
(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 $\sim
0.4$ 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

$\displaystyle\mathrm{DM_{MWN}}=R_{\mathrm{n}}\cdot\int\frac{n_{e,\gamma}}{\gamma^{2}}d\gamma,$
(21) $\displaystyle\mathrm{RM_{MWN}}=\frac{e^{3}}{2\pi
m_{e}^{2}c^{4}}R_{\mathrm{n}}B_{\mathrm{n}}\cdot\int\frac{n_{e,\gamma}}{\gamma^{2}}d\gamma,$
(22)

where the electron density $n_{e}$ and the nebula magnetic field
$B_{\text{n}}$ 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).

##  4 Long-term DM evolution

FRB 190520B has been reported in a dense environment, and the estimated
$\mathrm{DM}_{\text{host}}\simeq 902^{+88}_{-128}$ 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 $-0.09\pm 0.02$ pc 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).

###  4.1 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:

$\mathrm{DM}_{\mathrm{obs}}=\mathrm{DM}_{\mathrm{MW}}+\mathrm{DM}_{\text{halo
}}+\mathrm{DM}_{\mathrm{IGM}}+\frac{\mathrm{DM}_{\text{host}}+\mathrm{DM}_{\text{source
}}}{1+\mathrm{z}},$ (23)
$\mathrm{RM}_{\mathrm{obs}}=\mathrm{RM}_{\mathrm{MW}}+\mathrm{RM}_{\text{halo
}}+\mathrm{RM}_{\mathrm{IGM}}+\frac{\mathrm{RM}_{\text{host
}}+\mathrm{RM}_{\text{source }}}{(1+z)^{2}}.$ (24)

Using the IllustrisTNG simulation, Zhang et al. (2020) found that the DM
contributed by FRB 190520B-like host galaxies at $z\sim 0.2$ is $\sim 50-250$
pc cm-3. Therefore, the DM from the source of FRB 190520B can be inferred to
be $\sim 524-940$ pc cm-3. Following our previous work (Zhao et al., 2021),
the DM from the local environment of FRBs is given by

$\mathrm{DM}_{\text{source}}=\mathrm{DM}_{\mathrm{MWN}}+\mathrm{DM}_{\mathrm{unsh},\mathrm{ej}}+\mathrm{DM}_{\mathrm{sh},\mathrm{ej}}+\mathrm{DM}_{\mathrm{sh},\mathrm{ISM}}+\mathrm{DM}_{\mathrm{unsh},\mathrm{ISM}},$
(25)

where $\mathrm{DM}_{\mathrm{MWN}}$, $\mathrm{DM}_{\mathrm{unsh},\mathrm{ej}}$,
$\mathrm{DM}_{\mathrm{sh},\mathrm{ej}}$,
$\mathrm{DM}_{\mathrm{sh},\mathrm{ISM}}$ and
$\mathrm{DM}_{\mathrm{unsh},\mathrm{ISM}}$ are the contributions from the MWN,
unshocked ejecta, shocked ejecta, shocked ISM and unshocked ISM, respectively.
Usually, $\mathrm{DM}_{\mathrm{unsh},\mathrm{ISM}}$ 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

$\mathrm{RM}_{\text{source}}=\mathrm{RM}_{\mathrm{MWN}}+\mathrm{RM}_{\mathrm{sh},\mathrm{ej}}+\mathrm{RM}_{\mathrm{sh},\mathrm{ISM}}.$
(26)

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 $\mathcal{E_{\text{SN}}}\sim
1\times 10^{51}$ erg, the power-law index of outer ejecta $n=10$, ionization
fractions of unshocked ejecta $\eta=0.1$, the wind velocity of progenitors
$v_{\text{w}}=10$ km s-1 and $\epsilon_{\mathrm{B}}=0.1$. The solid and dashed
lines represent the case of $M_{\text{ej}}=10$ $M_{\odot}$ and
$M_{\text{ej}}=2$ $M_{\odot}$, respectively. The blue, red and green lines
represent different progenitors’ mass-loss rate. The orange shading is the
range of estimated $\mathrm{DM}_{\text{source }}$. We can see that only the
case of $\dot{M}=10^{-4}$ $M_{\odot}$ yr-1 and the source age
$t_{\text{age}}=10-30$ yr can provide the large enough DM. The constraint on
the source age is consistent with that derived from PRS in the previous
section.

###  4.2 Fitting Results

We assume that the variations of DM is only from
$\mathrm{DM}_{\text{source}}$. Thus, we can define the unchangeable
$\mathrm{DM}_{\text{other}}=\mathrm{DM}_{\text{obs}}-\mathrm{DM}_{\text{source}}$
for facilitate fitting. From the estimation of $\mathrm{DM}_{\text{source}}$
above, we can get $\mathrm{DM}_{\text{other}}\sim 270-686$ pc cm-3. The Markov
Chain Monte Carlo (MCMC) method performed by Python package
emcee111emcee.readthedocs.io (Foreman-Mackey et al., 2013) is used to estimate
the parameters $\mathrm{DM}_{\text{other}}$ and the age of the source
$t_{\text{age}}$. The $\chi^{2}$ for the observed DMs is

$\chi_{\mathrm{DM}}^{2}=\sum_{i=1}^{n}\frac{\left(\mathrm{DM}_{\mathrm{SNR},i}-\mathrm{DM}_{\mathrm{obs},i}\right)^{2}}{\sigma^{2}},$
(27)

where $\mathrm{DM}_{\mathrm{SNR}}$ is the DM from SNR given by our model, and
$\mathrm{DM}_{\mathrm{obs}}$ and $\sigma$ is the observed DM and uncertainties
in the frame of observers (Niu et al., 2021). The likelihood is

$\mathcal{L}\propto\exp\left[-\left(\chi_{\mathrm{DM}}^{2}\right)/2\right].$
(28)

The posterior corner plots obtained from fitting the models of two typical
ejecta mass models ($M_{\text{ej}}=10$ $M_{\odot}$ and $M_{\text{ej}}=2$
$M_{\odot}$) to the deta are shown in Figure 4. Our best-fit parameters are
shown in blue solid lines, and the parameters with 1-$\sigma$ ranges are shown
in dashed lines. For $M_{\text{ej}}=10$ $M_{\odot}$, we find
$\mathrm{DM}_{\text{other}}=464\pm 15$ pc cm-3
($\mathrm{DM}_{\text{SNR}}=746\pm 15$ pc cm-3) and the source age is $21.9\pm
0.5$ yr. For $M_{\text{ej}}=2$ $M_{\odot}$, we find
$\mathrm{DM}_{\text{other}}=642\pm 12$ pc cm-3
($\mathrm{DM}_{\text{SNR}}=568\pm 12$ pc cm-3) and the source age is $16.6\pm
0.4$ 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 $M_{\text{ej}}=10$ $M_{\odot}$ and $M_{\text{ej}}=2$
$M_{\odot}$, respectively. The shaded regions represent the SNR is opaque to
radio signals of $\nu\sim$ 1 GHz. If we assume that the typical SN ejecta mass
is $2M_{\odot}<M_{\text{ej}}<10M_{\odot}$, the source age of FRB 190520B can
be estimated as $16-22$ yr. In the same way, we have
$\mathrm{DM}_{\text{other}}=449-654$ pc cm-3
($\mathrm{DM}_{\text{SNR}}=556-761$ pc cm-3).

When DM${}_{\text{source}}$ is taken into consideration, the
DM${}_{\text{host}}$ 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 $\mathrm{DM}_{\text{other}}\gg\mathrm{DM}_{\text{SNR}}$. Finally,
we have $\mathrm{DM}\sim\mathrm{DM}_{\text{other}}$ for $t\sim 100-1000$ 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${}_{\mathrm{MWN}}\sim
10^{4}-10^{5}$ rad m-2 at $t_{\text{age}}=14$ 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 $\sim
10^{7}-10^{8}\mathrm{rad}~{}\mathrm{m}^{-2}$).

##  5 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 $B_{\text{int}}=2.76\times 10^{16}$ G, the source age
$t_{\text{age}}=14$ yr for FRB 121102, and $B_{\text{int}}=3.41\times 10^{16}$
G, $t_{\text{age}}=22$ yr for FRB 190520B.

  * •

FRB 190520B is embedded in a dense SNR whose DM contribution is $\sim 746\pm
15$ pc cm-3 and $\sim 568\pm 12$ pc cm-3 for $M_{\text{ej}}=10$ $M_{\odot}$
and $M_{\text{ej}}=2$ $M_{\odot}$, 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 $-0.09\pm
0.02$ 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 $\sim
10^{7}-10^{8}~{}\mathrm{rad}~{}\mathrm{m}^{-2}$). 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.

## acknowledgements

We thank the anonymous referee for helpful comments. We acknowledge Yuan-Pei
Yang, Ling-Jun Wang, Fan Xu, Long Li, Abudushataer Kuerban, Younes George and
Kohta Murase for helpful discussions, Jin-Jun Geng, Zhao Zhang and Qiao-Chu Li
for the help of numerical calculations. This work was supported by the
National Natural Science Foundation of China (grant No. U1831207), and the
Fundamental Research Funds for the Central Universities (No. 0201-14380045).

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Figure 1: The nebula velocity derived from Equations (7) and (8) for
$B_{\text{int}}\sim 10^{16}$ G, $M_{\mathrm{ej}}=1~{}M_{\odot}$, and
$\mathcal{E}_{\mathrm{SN}}=1\times 10^{51}$ erg. The red, blue and green solid
lines represent the case of $P_{\text{i}}=1.5$ ms, $P_{\text{i}}=2.5$ ms and
$P_{\text{i}}=5$ 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 ($\sim
10^{-2}-10^{-1}$ yr) before the magnetic flux begins to leak out. The finial
ejecta velocity is up to $\sim 20,000-60,000\mathrm{~{}km}\mathrm{~{}s}^{-1}$,
which is consistent with the observations of SN Ib/Ic. Figure 2: The spectral
energy distribution for the PRS associated with FRB 121102 (top panel) and FRB
190520B (middle panel). The main parameters are $B_{\text{int}}=2.76\times
10^{16}$ G, $t_{\text{age}}=14$ yr and $\gamma_{\text{b}}=5\times 10^{4}$ for
FRB 121102 , and $B_{\text{int}}=3.41\times 10^{16}$ G, $t_{\text{age}}=22$ yr
and $\gamma_{\text{b}}=5\times 10^{3}$ for FRB 190520B. The red, blue and
green solid lines represent the spectra observed epoch at
$t=t_{\text{age}}/3$, $t=t_{\text{age}}$ and $t=3t_{\text{age}}$. The case of
rotational energy injection is also plotted in dashed line at
$t=t_{\text{age}}$ for comparison, whose dipole magnetic field is estimated as
$B_{\text{dip}}=0.1B_{\text{int}}$ and initial spin period $P_{i}=5$ ms is
taken. The black circles are the observations values from Chatterjee et al.
(2017) for FRB 121102 and Niu et al. (2021) for FRB 190520B. The light curves
of PRS (bottom panel) 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). The SNR will be optically thick for 3 yr and 1.5 yr for
$M_{\text{ej}}=10$ $M_{\odot}$ and $M_{\text{ej}}=2$ $M_{\odot}$, which is
shown in gray and black shaded regions. Figure 3: DM (left panel) and RM
(right panel) contributed by SNR based on Equations (24) and (45)-(48) of Zhao
et al. (2021). We adopt the typical parameters of SNRs: the explosion energy
$E_{\text{SN}}\sim 1\times 10^{51}$ erg, the power-law index of outer ejecta
$n=10$, ionization fractions of unshocked ejecta $\eta=0.1$, the wind velocity
of progenitors $v_{\text{w}}=10$ km s-1 and $\epsilon_{\mathrm{B}}=0.1$. The
solid and dashed lines represent the case of $M_{\text{ej}}=10$ $M_{\odot}$
and $M_{\text{ej}}=2$ $M_{\odot}$, respectively. The blue, red and green lines
represent different progenitors’ mass-loss rates. The orange shading is the
range of estimated $\mathrm{DM}_{\text{source }}$. We can see that only the
case of $\dot{M}=10^{-4}$ $M_{\odot}$ yr-1 and the source age
$t_{\text{age}}=10-30$ yr can provide the large enough DM required by
observations. The SNR will be optically thick for 3 yr and 1.5 yr for
$M_{\text{ej}}=10$ $M_{\odot}$ and $M_{\text{ej}}=2$ $M_{\odot}$, which is
shown in gray and black shaded regions. Figure 4: Posteriors parameters of DM
model fit to FRB 190520B for the case of $M_{\text{ej}}=10$ $M_{\odot}$ (left
panel) and $M_{\text{ej}}=2$ $M_{\odot}$ (right panel) performed by MCMC
method. The posterior probability of the parameters of
$\mathrm{DM}_{\text{other}}=\mathrm{DM}_{\text{obs}}-\mathrm{DM}_{\text{source}}$
and $t_{\text{age}}$ are plotted in the histograms, and the 1-$\sigma$ range
is shown in the dashed vertical lines. The medians are shown in blue lines.
The contours indicate the parameter space with 1-$\sigma$, 2-$\sigma$ and
3-$\sigma$ range. If we assume that the typical SN ejecta mass is
$2M_{\odot}<M_{\text{ej}}<10M_{\odot}$, the source age of FRB 190520B can be
estimated as $16-22$ yr. In the same way, we have
$\mathrm{DM}_{\text{other}}=449-654$ pc cm-3
($\mathrm{DM}_{\text{SNR}}=556-761$ pc cm-3). Figure 5: Samples of
DM${}_{\text{SNR}}$ for $M_{\text{ej}}=10$ $M_{\odot}$ (left panel) and
$M_{\text{ej}}=2$ $M_{\odot}$ (right panel) from the MCMC method (orange
curves). Black squres and circles are the DM${}_{\text{obs}}$ 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 $\nu\sim$ 1 GHz.