On the magnetoionic environments of fast radio bursts

The average magnetic field strength (i.e. the absolute value) along the line of sight (LOS) may be derived by combining the measurements of RM and DM of radio pulses (e.g. Manchester, 1972, 1974; Han et al., 2006; Noutsos et al., 2008; Han et al., 2018)):

where RM is in units of $\rm rad\,m^{-2}$ and DM is the dispersion measure in units of $\rm pc\,cm^{-3}$. For Galactic pulsars (data are quoted from Manchester et al. 2005), the contribution to RM from the plasma associated with the source is negligible, so that $\langle B_{\parallel}\rangle$ can be used to map the magnetic field structure of the Galaxy (Han & Qiao, 1994). The RM term $\rm RM_{iono}$ originated from the Earth’s ionosphere (Sotomayor-Beltran et al., 2013) is $0.5-3\rm\,rad\,m^{-2}$. The RMs for most Galactic pulsars are larger than $\rm 10\,rad\,m^{-2}$. For pulsars that have $\rm DM\sim 100pccm^{-3}$, the variability of the ionospheric RM only slightly affects $\langle B_{\parallel}\rangle$ and can, hence be neglected. The $\langle B_{\parallel}\rangle$ of Galactic pulsars are calculated with Equation (1) and plotted in Figure 1.

We consider whether Galactic magnetars can host a more extreme magneto-ionic environment than pulsars. According to theory, a young magnetar may power an MWN, which may be more magnetized than a regular pulsar wind nebula. In an expanding MWN, the DM and RM would generally decrease with time (Metzger et al., 2017; Yang & Zhang, 2017; Piro & Gaensler, 2018; Margalit & Metzger, 2018; Metzger et al., 2019), suggesting that one may observe a trend of decreasing $\langle B_{\parallel}\rangle$ with time or surface magnetic field strength. Among over twenty observed magnetars, there are six sources that emitted polarized radio waves to allow $\langle B_{\parallel}\rangle$ to be measured (The CHIME/FRB Collaboration et al. 2020; Lower et al. 2020, also see Kaspi & Beloborodov 2017 for a review).

In Figure 1(a) and (b), we plot $\langle B_{\parallel}\rangle$ as a function of magnetar surface magnetic field $B_{\rm s}$ and characteristic age for these six radio magnetars. The characteristic age is close to real age under the assumptions that the initial spin period of the pulsar is much less than its current period and that pulsar spindown is dominated by dipolar magnetic braking. The surface magnetic field configuration is likely more complicated but the exact strength of the surface field is difficult to infer from observations. For simplicity, we use $B_{\rm S}$ as a proxy of the magnetar surface magnetic field and study how $\langle B_{\parallel}\rangle$ depends on it. For comparison, we also select some pulsars that are within 0.5 kpc from the magnetars. The distance between a pulsar and a magnetar is calculated by combining their radial distances inferred from DM measurements and their transverse distance measured based on their celestial coordinates. The distance to Swift J1818.0–1607 is estimated to be 4.8 kpc (according to the YMW16 electron density model, Yao et al. 2017) or $8.1\pm 1.6$ kpc (according to NE2001 electron density model, Cordes & Lazio 2002). Another source, SGR 1935+2154, has an unknown distance, but one may assume that the SGR is physically related to a supernova remanent SNR G57.2+0.8, which has a distance of $9.0\pm 2.5$ kpc (Zhong et al., 2020; Zhou et al., 2020).

Figure 1(a) shows that except the magnetar PSR J1745–2900, whose abnormally large $\langle B_{\parallel}\rangle$ may be related to its proximity with the Galactic super-massive black hole, the other five magnetars have $\langle B_{\parallel}\rangle$ consistent with the pulsar $\langle B_{\parallel}\rangle$ distribution (two within $1\sigma$, one slightly outside the $1\sigma$ region). Indeed some regular pulsars have larger $\langle B_{\parallel}\rangle$ values than magnetars. Compared with normal pulsars within 5 kpc, the five magnetars also do not have systematically higher $\langle B_{\parallel}\rangle$ values. Figure 1(b) plots $\langle B_{\parallel}\rangle$ against pulsar age. One can see that PSR J1745–2900 is not the youngest but has the highest $\langle B_{\parallel}\rangle$, further suggesting that its abnormal $\langle B_{\parallel}\rangle$ is related to its special environment. For the five magnetars, there is also no trend of decreasing $\langle B_{\parallel}\rangle$ with characteristic age.

In view of the abnormally high $\langle B_{\parallel}\rangle$ of the magnetar PSR J1745–2900, we also investigate the relationship between $\langle B_{\parallel}\rangle$ and the distance of pulsars from the Galactic Centre. The results are shown in Figure 1(c). The $\langle B_{\parallel}\rangle$ values for the magnetar PSR J1745–2900 and two pulsars close to it (PSR J1746–2849 and PSR J1746–2856) have $\langle B_{\parallel}\rangle$ significantly exceeding those of Galactic pulsars, further suggesting that it is the special location near the Galactic centre that caused the abnormally high $\langle B_{\parallel}\rangle$. The Galactic Centre super-massive black hole likely provides the strong local magnetic field. The magnetar PSR J1745–2900 indeed shows a larger $\langle B_{\parallel}\rangle$ than the pulsars. This may suggest that the magnetar may contribute to an additional $\langle B_{\parallel}\rangle$ value. However, due to the large uncertainty of the distance measurements from the Galactic Centre (noticing the large error bars of the red dots), this conclusion cannot be firmly drawn. It is interesting to note that at distances beyond 1 kpc, $\langle B_{\parallel}\rangle$ is essentially independent of distance from the Galactic Centre.

The $\langle B_{\parallel}\rangle$ distribution of Galactic pulsars can be fitted with a log normal function, as shown in Figure 3. The mean value of log$\langle B_{\parallel}\rangle$ is $\mu_{\rm pulsar}=-0.002$ with a standard deviation of $\sigma_{\rm pulsar}=0.40$ for the log normal distribution. Excluding the Galactic Centre magnetar PSR J1745–2900, the other five magnetar sources have a mean $\langle B_{\parallel}\rangle$ of $1.84\,\rm\mu G$, which is statistically consistent with the pulsar $\langle B_{\parallel}\rangle$ distribution.

For a source at cosmological distances, the observed total $\rm RM_{obs}$ consists of contributions from several different terms,

where $\rm RM_{Gal}$ is the Galactic component, $\rm RM_{IGM}$ is contributed from the intergalactic medium (IGM), and $\rm RM_{HG,sr}$ is contributed by the host galaxy and the FRB source as measured in the Observer’s frame. The true value of the latter in the rest frame of the host galaxy is

where $z$ is the redshift.

In order to determine $\rm RM_{Gal}$, we identify the RM of the known NRAO VLA Sky Survey (NVSS) sources (Taylor et al., 2009). The closest sources are $\lesssim 2^{\circ}$ away from the position of FRBs. For the FRB sources located in the survey blind regions, we identify the RM based on their closest pulsars (Han et al., 2018). Alternatively, a simulation result of Galactic RM is given by Oppermann, et al. (2015), even though the simulation results are based on some inputs and assumptions. The strength of intergalactic magnetic fields is much lower. A safe upper limit is $\sim 10^{-9}$ G (Dai et al., 2002; Ando & Kusenko, 2010; Dermer et al., 2011; Arlen et al., 2014), which gives $\rm RM_{IGM}\lesssim 1\,\rm rad\,m^{-2}$, so that it can usually be neglected. After subtracting the contributions from $\rm RM_{iono}$ and $\rm RM_{Gal}$, one can finally derive $\rm RM_{HG,sr}$ and $\rm RM^{Loc}_{HG,sr}$.

The observed DM also consists of several terms:

where $\rm DM_{Gal}$ can be derived from the Galactic electron density models. However, the two well-known Galactic electron density models NE2001 and YWM16, give quite different results sometimes (Cordes & Lazio, 2002; Yao et al., 2017). In the following discussion, we consider the electron density models, separately. The DM associated with the Milky Way halo is adopted as $\rm DM_{halo}=30\pm 15\,pc\,cm^{-3}$ according to Dolag et al. (2015). The average value of the IGM component is (Deng & Zhang, 2014; Zhang, 2018b)

where the free electron number per baryon in the universe is $\chi(z)\approx 7/8$ and the fraction of baryons $f_{\rm IGM}\sim 0.83$. For a distant cosmological FRB, the DM is mainly contributed by IGM rather than Milky Way or the host galaxy, in contrast to the RM that has a significant contribution from the host. The $\Lambda$CDM cosmological parameters are taken as $\Omega_{m}=0.315\pm 0.007,\Omega_{b}h^{2}=0.02237\pm 0.00015$, and $H_{0}=67.36\rm\pm 0.54\,\rm km\,s^{-1}Mpc^{-1}$ (Planck Collaboration et al., 2018). The remaining term in Eq.(4) is $\rm DM_{HG,sr}$ in the observer frame. The intrinsic value in the source frame is related to it by

In practice, $\rm DM_{HG,sr}$ is difficult to derive from the observed $\rm DM_{obs}$, even if the redshift of the FRB is precisely known because there is a large scatter of $\rm DM_{IGM}$ around Eq.(5) due to the large scale structure fluctuation (McQuinn, 2014). Several FRB sources have measured redshifts, so that their $\rm DM_{IGM}$ and uncertainties can be derived (McQuinn, 2014). As a result, we can calculate $\rm DM_{HG,sr}$ and its uncertainty $\rm\delta DM_{HG,sr}$ directly. For the sources without precise redshift measurements, we adopt the opposite approach, to consider the distribution of $\rm DM^{Loc}_{HG,sr}$ based on host galaxy models. Following a generic constraint by Li et al. (2020b), we assume that $\rm DM^{Loc}_{HG,sr}=85\pm 35\rm\,pc\,cm^{-3}$, which is consistent with the results of the average DM contribution from the host galaxy in the local frame e.g., Xu & Han (2015) and Luo et al. (2018).

The average magnetic field for the host galaxy along the LOS can then be derived as

where for the redshift we either adopted the spectroscopic value if measured, or estimated using Eq.(6) with error properly introduced.

Based on the FRB catalog (http://frbcat.org/, Petroff et al. 2016), in Table 1 we list all the FRBs with both RM and DM measured. According to Eq.(7), we estimate $\langle B_{\parallel}\rangle$ for these FRBs. The results are listed in Tables 1 and 2, where $\rm DM_{Gal}$ is derived using the NE2001 and YMW16 models, respectively. Several sources have $\rm DM_{HG,sr}$ smaller than $\rm DM_{IGM}$ and $\rm\delta DM_{IGM}$, which differ from FRB 121102 whose $\rm DM_{HG,sr}$ and $\rm DM_{IGM}$ are comparable. We show the lower limits for these sources. The distribution of $\langle B_{\parallel}\rangle$ is also presented in Figure 3, which can be also fitted with a log normal function when FRB 121102 is excluded. The mean value derived by the two Galactic electron density models are $\mu_{\rm FRB}=0.25,\,\sigma_{\rm FRB}=0.78$ (NE2001) and $\mu_{\rm FRB}=0.24,\,\sigma_{\rm FRB}=1.00$ (YMW16), respectively. These values are slightly higher but not inconsistent with the distributions of both Galactic pulsars and magnetars.