The Environment of FRB 121102 and Possible Relation to SGR/PSR J1745-2900

Variations of the dispersion (DM) and rotation (RM) measures of FRB 121102 indicate magnetic fields $\sim$ 317 mG in the dispersing plasma. The electron density may be $\sim 10^4\,$cm$^{-3}$. The observed time scales $\sim 1$ year constrain the size of the plasma cloud. Increasing DM excludes simple models involving an expanding supernova remnant, and the non-zero RM excludes spherical symmetry. The varying DM and RM may be attributable to the motion of plasma into or out of the line of sight to or changing electron density within slower-moving plasma. The extraordinarily large RM of FRB 121102 implies an environment, and possibly also a formation process and source, qualitatively different from those of other FRB. The comparable and comparably varying RM of SGR/PSR J1745$-$2900 suggests it as a FRB candidate. An appendix discusses the age of FRB 121102 in the context of a "Copernican Principle".

\[firstpage\]

radio continuum, transients: fast radio bursts

Introduction
============

@WWY recently reviewed the magnetoionic environments of Fast Radio Bursts (FRB). The dispersion measures (DM) and rotation measures (RM) of most FRBs have shown no detectable changes and are consistent with paths through the intergalactic medium and the interstellar media of our Galaxy and a host galaxy (whose properties are necessarily very uncertain). There is one striking exception, FRB 121102, whose RM has extraordinary values $\sim 10^5$/m$^2$, decreased by $\approx 30\%$ during about three years of observations, and whose DM increased by about 2.5 pc/cm$^3$ [@H20]. From these data it is possible to estimate the magnetic field and electron density in the dense, strongly magnetized, region in which the RM and the varying part of the DM are produced. No other FRB has such a high RM, but the other phenomenology of FRB 121102 is not extraordinary among FRB.

The magnetoionic environment of SGR/PSR J1745$-$2900 may be similar to that of FRB 121102; its RM is also large and rapidly varying, although no change in its DM has been reported. If these two objects are related, SGR/PSR J1745$-$2900 would be a candidate FRB source.

Magnetic Fields
===============

The mean parallel component of magnetic field along the line of sight is related to the RM and DM: $$\label{B}
    \langle B_\parallel \rangle_{n_e} = 1.23 {\text{RM} \over \text{DM}}
    \mu\text{G},$$ where the RM is in units of radians/m$^2$, the DM is in units of pc/cm$^{3}$ and the subscript $n_e$ indicates this is the electron density-weighted average of the parallel (to the line of sight) component of magnetic field. In a homogeneous medium $\langle B_\parallel \rangle_{n_e}$ equals a single value $B_\parallel$.

The fact that the RM is an integral of the product $B_\parallel n_e$ implies that it is dominated by the densest and most strongly magnetized regions, even if they make only a small contribution to the DM; the RM contains two powers of parameters that are expected to be correlated, while the DM is linear. Most of the DM is likely produced in long paths in low $B$ interstellar and intergalactic plasmas that contribute negligibly to the RM. In order to determine the magnitude of $B_\parallel$ it is necessary to estimate the DM of the regions that dominate the RM. Only then can Eq. \[B\] be used to estimate $\langle B_\parallel \rangle_{n_e}$. Unfortunately, for most FRB it is not possible to separate any contribution to the DM of a dense and strongly magnetized region from those of the interstellar (in both galaxies) and intergalactic media.

FRB 121102 is an exception. Not only is its RM extraordinarily large and rapidly varying, but it is the only FRB whose DM is known to change [@H20; @O20], increasing by about 2.5 pc/cm$^3$ in two years of accurate measurements. Characteristic time scales of variation of contributions to the DM are $T \sim L/v$, where $L$ is a length scale and $v$ a velocity. For interstellar media $T \sim \text{100 pc}/(\text{30 km/s}) \sim 3 \times
10^6\,$y, while for the intergalactic medium $T \sim 1/h \sim 10^{10}\,$y, where $h$ is the Hubble constant; the corresponding $d\text{DM}/dt \sim
\text{DM}/T$ are unobservably small ($\ll 10^{-3}\,$pc/cm$^3$ during the period of observation). The observed change in DM must be attributed to a compact dense region that is also the source of most of the (also varying) RM.

It is then possible to estimate the magnitude of $B_\parallel$ in this region. In the simplest possible model the dense region is homogeneous, is the source of the entire RM, and its RM and DM vary in proportion. If correct, this would also permit estimating this region's contribution to the DM. A possible physical realization would be the intrusion into, or the withdrawal from, the line of sight of a homogeneous (in density and magnetic field) wedge of plasma, or a change in its ionization state. Then Eq. \[B\] becomes [@K18] $$\label{DB}
    \left|\langle B_\parallel \rangle_{n_e}\right| = 1.23{|\Delta
    \text{RM}| \over |\Delta \text{DM}|}\,\mu\text{G}.$$

The observations of FRB 121102 by @H20 fall into three groups: I, comprising bursts 5 and 6 during MJD +1-757-540-5213; II, comprising bursts 7–15 during MJD 58216–58348; and III, comprising bursts 16–20 during MJD +1-757-540-5213. Between groups I and II $\Delta \text{RM} \approx -15000$ m$^{-2}$ and $\Delta \text{DM} \approx 1.1$ pc/cm$^3$; between groups II and III $\Delta \text{RM} \approx -4000$ m$^{-2}$ and $\Delta \text{DM} \approx
1.6$ pc/cm$^{-3}$. Groups I and II are separated by about 160 days while groups II and III are separated by about 450 days.

Eq. \[DB\] indicates $|B_\parallel| \sim 17$ mG for the transition between groups I and II and $|B_\parallel| \sim 3$ mG for the transition between groups II and III. Within this model, these estimates of the magnetic field do not depend on the times over which the transitions occurred. These fields are three or more orders of magnitude greater than typical estimated interstellar fields, and indicate that FRB 121102 is behind or immersed in an extraordinary region.

Assuming that the entire DM of the dense region, DM$_{dr}$, varies when its RM varies, $$\label{DMdr}
    \text{DM}_{dr} \sim {|\text{RM}| \over |\Delta \text{RM}|}
    |\Delta \text{DM}| \sim 10\,\text{pc/cm}^3.$$ This is only a rough estimate, but is consistent with the attribution of most of the DM to interstellar and intergalactic media, even though nearly all the RM must be attributed to the dense region.

Numerical Estimates
===================

If the moving plasma that is the source of the changing DM and RM has a speed (that might be the phase speed of an ionization or recombination front rather than a material speed) $v$, then a change in properties over a time $\Delta t$ corresponds to a length scale $v \Delta t$. The characteristic electron density $$\label{ne}
    n_e \sim {\Delta \text{DM} \over v \Delta t}.$$

If the magnetic stress is in equipartition with the gas pressure $P = 2 n_e
k_B T$ (assuming equal ion and electron temperatures and a statistically isotropic field), there is a characteristic temperature $T$: $$\label{P}
    2 n_e k_B T = P = 3{B_\parallel^2 \over 8 \pi}.$$ Combining Eqs. \[DB\], \[ne\] and \[P\], $$\label{T}
    k_B T = 2 \times 10^{-13} {|\Delta \text{RM}|^2 \over |\Delta
    \text{DM}|^3} {\Delta t \over \text{1 s}} v_7\ \text{eV},$$ where RM and DM are in their usual units (radians/m$^2$ and pc/cm$^3$) and $v_7 \equiv v/(10^7\text{cm/s})$.

Alternatively, the magnetic stress could be equated to the dynamic pressure $n_e \mu v^2$, where $\mu$ is the molecular weight per electron, taken as $2 \times 10^{-24}\,$g: $$\label{Q}
    n_e \mu v^2 = 3{B_\parallel^2 \over 8 \pi}.$$ Combining Eqs. \[DB\], \[ne\] and \[Q\], $$\label{v}
    v = 3 \times 10^{-8} {|\Delta\text{RM}|^2 \over |\Delta\text{DM}|^3}
    {\Delta t \over \text{1 s}}\ \text{cm/s}.$$

The inferred values of the parameters are shown in the Table. The assumptions, particularly those of equipartition, are uncertain, but the inferred orders of magnitude of $k_B T$ and $v$ may be plausible in regions of high energy density.

  Epoch                               I$\to$II                   II$\to$III
  -------------------------- --------------------------- ---------------------------
  $\Delta$RM (rad/m$^2$)              $-15000$                     $-4000$
  $\Delta$DM (pc/cm$^3$)               $+1.1$                      $+1.6$
  $|B_\parallel|$ (mG)                   17                           3
  $\Delta t$ (s)                  $1.5 \times 10^7$            $4 \times 10^7$
  $\ell = v \Delta t$ (cm)    $1.5 \times 10^{14}\,v_7$     $4 \times 10^7\,v_7$
  $n_e$ (cm$^{-3}$)           $2 \times 10^4\,v_7^{-1}$   $1 \times 10^4\,v_7^{-1}$
  $L$ (cm)                          $10^{15} v_7$               $10^{16} v_7$
  $k_B T$ (eV)                        500 $v_7$                   30 $v_7$
  $v_7$                                   7                          0.5

  : \[table1\] The parameters of the plasma responsible for changes in DM and RM of FRB 121102 between Epochs I (bursts 5 and 6) and II (bursts 7-15) and between Epochs II and III (bursts 16–20) of @H20, estimated from Eqs. \[DB\] and \[ne\]. $k_B T$ is estimated from Eq. \[T\], assuming magnetic equipartition with the gas pressure with $v$ as a parameter, and $v$ is estimated from Eq. \[v\], assuming magnetic equipartition with dynamic pressure.

An additional constraint may be imposed on $n_e$ by the observation that the dispersion index $\alpha = -1.997 \pm 0.015$ [@S16]. Most of the DM is produced in low density interstellar and intergalactic plasma whose contribution to any deviation from $\alpha = -2$ is infinitesimal. The portion of the DM attributable to the dense magnetized region (Eq. \[DMdr\]) is $\sim 2\%$ of the total DM, so in the dense region the emprical bound is $\Delta \alpha_{dr} \sim (-\alpha - 2)/0.02 \lesssim 0.6$, implying [@K14; @K15] $$n_e \approx {4 \over 3} {\omega^2 m_e \over 4 \pi e^2} \Delta
    \alpha_{dr} \lesssim 2 \times 10^{10}\ \text{cm}^{-3}.$$ This is not restrictive.

The electron density inferred for FRB 121102 implies very short ($\sim
10^9\,$s, unless the temperature is very high) recombination times. This may suggest very young ages, but might also be explained by an intense ionizing ultraviolet flux.

Implications
============

FRB 121102 is behind or immersed in an extraordinarily dense and strongly magnetized region, plausibly associated with the FRB source or with the event that made it. The inferred $n_e$ and $|\langle B_\parallel
\rangle_{n_e}|$ are much greater than those found in other interstellar plasmas. The data are consistent with any location of this region on the line of sight between the FRB source and the observer, but it is natural to associate it with the immediate environment of the source of FRB 121102. Combining Eqs. \[DMdr\] and \[ne\] suggests a length scale $$\label{L}
    L \sim {\text{DM}_{dr} \over n_e} \sim {|\text{RM}| \over
    |\Delta\text{RM}|} v \Delta t \sim 10^{15} v_7\ \text{cm}.$$

The identification of FRB 200428 with SGR 1935$+$2154, associated with the supernova remnant (SNR) G57.2$+$0.8 [@KSGR18], suggests by analogy that the dense region on the line of sight to FRB 121102 may be a very young, dense and strongly magnetized remnant of the event that made its source. However, the magnetoionic environment of SGR 1935$+$2154 is very different, with a modest RM plausibly attributed to the interstellar medium.

This explanation of the RM and DM of FRB 121102 is not entirely satisfactory. The expansion of a SNR generally produces a decreasing DM [@MM18; @PG18], in contradiction to the observed [@H20; @O20] increase, although increasing ionization could increase DM. More complex models [@ZZWW] may have more complex behavior. A spherically symmetric remnant cannot have a magnetic field at all, but a remnant might be approximately symmetric on large scales while being asymmetric (turbulent or otherwise structured) on smaller scales.

The changing DM and RM of FRB 121102 could be attributed to either a change in the magnetic field and density of a nebular filament on the line of sight or to the motion of such a filament across the line of sight, as suggested for PSR J1745$-$2900 by @P18. The observed reduction in |RM|, combined with an increase in DM, implies the movement into the line of sight of comparatively dense plasma whose $B_\parallel$ is opposite to that dominant along the total line of sight, or an increase in the integrated column density of such an opposed region.

It is natural to compare the inferred parameters with those of the best-studied SNR, the Crab Nebula. Its magnetic field [@BK90; @RGB12] is less than that inferred for FRB 121102. However, direct comparison is inappropriate because the magnetic field of the Crab nebula is inferred for the low density synchrotron-radiating volume that must be a negligible contributor to both DM and RM; it is not possible to measure the field in the high density filaments directly. Further, both the Crab Nebula and its central neutron star differ from those that produce FRB 200428, and very plausibly other FRB. FRB 200428/SGR 1935$+$2154 has a spin rate about 100 times less than that of the Crab pulsar and a magnetic moment about 60 times greater. The age of FRB 121102 may be much less than that of the Crab Nebula and its pulsar; it must be at least eight years, but is otherwise empirically unconstrained.

Discussion
==========

It is remarkable that the RM and inferred $\langle B_\parallel
\rangle_{n_e}$ of FRB 121102 are orders of magnitude greater than those of any other FRB [@WWY][^2]. If the sources of the DM are qualitatively similar in FRB 121102 and other FRB, as would be expected if they are gradually expanding and dissipating SNR with a continuous distribution of parameters (such as age, electron density and magnetic field), a smooth and continuous distribution of RM would be expected, perhaps a power law. This is inconsistent with the extraordinary and unique RM and inferred $\langle B_\parallel \rangle_{n_e}$ of FRB 121102; its environment differs qualitatively from those of other FRB, and its nature and origin may also differ.

The source of FRB 200428 shows an analogously anomalous distribution of burst strengths, with FRB 200428 orders of magnitude more intense than subsequent bursts from this source. This may be explained by the collimation of radiation emitted by relativistic charges into a narrow beam that may jitter or wander in direction [@K20]. No such explanation is apparent for the distribution of RM among FRB sources.

@H20 compared the observed properties of FRB 121102 to the SNR models of @MM18 and @PG18, and suggested birthdates in the range 2000–2010. The rapid decay of RM (30% in three years), if interpreted as exponential decay, would also suggest a characteristic age of about 10 years. No trend in the rate and strengths of outbursts of FRB 121102, that might be expected to decay as a young source ages, has been qualitatively evident in eight years of observation, but insufficient data exist to bound any trends quantitatively. The *increase* in DM argues against a rapidly dissipating (and therefore very young) SNR. This increase, and the fact that FRB 121102 has been observed over a time scale $\gtrsim \tau
\equiv |\text{RM}|/|d\text{RM}/dt|$ without evident decay, suggests that its environment is fluctuating chaotically rather than systematically evolving and that its age and life expectancy may be $\gg \tau$. As discussed in the Appendix, characteristic time scales of variation are only lower bounds on actual ages.

If a repeating FRB had a measured period and period derivative, these would constrain its age, although the possibility of a neutron star born recently with a long spindown age could not be excluded—spindown age is only an upper bound on the actual age. Because FRB are episodic emitters, a birth date could only be established statistically from its absence in surveys prior to some date; CHIME/FRB might make this possible.

The only other astronomical objects with such extraordinarily large (and rapidly varying) RM are PSR J1745$-$2900, the similarity of whose properties to those of FRB was suggested by @P18, and the Galactic center black hole Sgr A$^*$. This PSR has a projected separation from Sgr A$^*$ of only 0.1pc, so the line of sight to PSR J1745$-$2900 may pass through a region filled with plasma accreting onto the massive black hole. It is unsurprising that this region has unique properties, and @K19 suggested that FRB are emitted by jets produced by intermediate mass black holes. That cannot explain FRB 200428, known to be emitted by a SGR (and that has an unremarkable RM, plausibly attributed to interstellar plasma), but is consistent with the hypothesis that there are at least two, qualitatively different, kinds of objects that produce FRB whose phenomenologies are not obviously distinct.

It is difficult to explain the rapid variation of the RM of PSR J1745$-$2900 as the result of a path through an accretion flow onto Sgr A$^*$. A characteristic time scale $\sim 6\,$y (10% change in 7 months) [@MSH18] at a projected distance from Sgr A$^*$ of 0.1 pc would suggest a velocity $\sim 1.5 \times 10^9\,$cm/s, about 50 times the escape or orbital velocity of the $4 \times 10^6 M_\odot$ black hole in Sgr A$^*$. For a smaller and more plausible $v$ the scale $L$ (Eq. \[L\]) of the birefringent plasma is $\ll 0.1\,$pc. Alternative explanations include propagation through a very fast and unslowed supernova shell and varying refraction, in which the high velocity is not a material velocity but a displacement velocity of the propagation path as a result of refraction by more slowly moving material. Application of this hypothesis to FRB 121102 would not invalidate the inference of extraordinary $n_e$ and $\langle
B_\parallel \rangle_{n_e}$, but would emphasize that $v$ is uncertain.

Only upper limits to |$\Delta$DM| of PSR 1745$-$2900 exist, so it is not possible to estimate $\langle B_\parallel \rangle_{n_e}$. However, these upper limits are comparable to the measured |$\Delta$DM| of FRB 121102 and the $\Delta\,$RM are of the same order of magnitude, so $|\langle B_\parallel
\rangle_{n_e}| \sim 10\,$mG is at least plausible.

If varying refraction explains the rapid variation of the RM of FRB 121102, we cannot be on a caustic. The converging rays of a caustic, while having the same optical path (foci occur at extrema of the optical paths), would traverse paths of differing $B_\parallel$, averaging out their linear polarization.

The large and rapidly varying RM of SGR/PSR J1745$-$2900 may be unrelated to its proximity to Sgr A$^*$ (a separation of $0.1\,\text{pc} \gg v \Delta t$ for likely $v$), but may rather be associated with the environment of the SGR/PSR on scales $\ll 0.1\,$pc. PSR J1745$-$2900 and FRB 121102 may be members of a distinct class of object characterized by the behavior of their RM. If so, then the FRB activity of FRB 121102 suggests that PSR J1745$-$2900 might also be a source of FRB, a speculation supported by the discovery that another SGR/PSR with similar parameters (1935$+$2154) made FRB 200428.

Appendix {#appendix .unnumbered}
========

It has been suggested [@G93] on the basis of a "Copernican Principle" (of much older origin) that if a phenomenon has been observed for a period $t$, the probability that its lifetime is $\gtrsim T$ is ${\cal O}(t/T)$. This has been used to argue that the expected lifetime of humanity (estimated to be $\sim 300,000$ years old, depending on the definition of when our ancestors became human) is unlikely to be more than a few million years, that the expected lifetime of the present American constitutional government, established in 1788, is unlikely to be more than a few thousand years, <span>*etc.*</span>

This argument depends on an assumed uniform prior probability of discovery through the lifetime of the phenomenon. This can be shown to be wrong, at least in some applications. For example, the rapid variation of some AGN might suggest ages of no more than a few years, but we are confident, on the basis of the physics of black hole formation and accretion and observation of their accompanying double radio sources, that their ages are $10^6$–$10^{10}$ years. The variation of the RM of PSR J1745$-$2900 suggests an age $\approx 6\,$y, but the absence of a neutrino signal in Kamiokande and other neutrino observatories from its formation implies an age $\ge 37\,$y, and the properties of SNR G57.2$+$0.8 in which it is embedded indicate an age of thousands or tens of thousands of years [@KSGR18]. Hence, while the rapid variation of the RM of FRB 121102 may suggest on physical grounds a very young SNR, its age cannot be estimated by a statistical argument.

Data Availability {#data-availability .unnumbered}
=================

This theoretical study did not obtain any new data.

<span>99</span> Bietenholz, M. F. & Kronberg, P. P. 1990 357, L13. Gott, J. R. 1993  363, 315. Hilmarsson, G. H., Michilli, D., Spitler, L. G. <span> *et al.*</span> 2020 arXiv:2009.12135. Katz, J. I. 2014  788, 34. Katz, J. I. 2015 arXiv:1505.06220. Katz, J. I. 2018 Prog. Nucl. Part. Phys. 103, 1 arXiv:1804.09092. Katz, J. I. 2019  487, 491 arXiv:1811.10755. Katz, J. I. 2020  499, 2319 arXiv:2006.03468. Kothes, R., Sun, X., Gaensler, B. & Reich, W. 2018  852, 54. Margalit, B. & Metzger, B. D. 2018  868, L4. Michilli, D., Seymour, A. & Hessels, J. W. T. 2018  553, 182. Oostrum, L. C., Maan, Y., van Leeuwen, J. <span>*et al.*</span> 2020  635, 61. Pearlman, A. B., Majid, W. A., Prince, T. A. <span>*et al.*</span> 2018  866, 160. Piro, A. L. & Gaensler, B. M. 2018  861, 150. Reynolds, S. P., Gaensler, B. M. & Bocchino, F. 2012 Sp. Sci. Rev. 166, 231. Scholz, P., Spitler, L. G., Hessels, J. W. T. <span>*et al.*</span> 2016  833, 177. Wang, W.-Y., Zhang, B., Chen, X. & Xu, R. 2020 499, 355. Zhao, Z. Y., Zhang, G. Q., Wang, Y. Y. & Wang, F. Y. 2020 arXiv:2010.10702.

\[lastpage\]

[^1]: E-mail josemurray@example.com

[^2]: @WWY use Eq. \[B\] rather than Eq. \[DB\] to infer $\langle B_\parallel \rangle_{n_e}$, leading to an underestimate if most of the RM but little of the DM are contributed by a dense strongly magnetized region. For FRB 121102, most of whose RM must be produced in such a region (because the evidence of other FRB shows that interstellar and intergalactic plasma contribute much less RM), Eq. \[B\] may underestimate $\langle B_\parallel \rangle_{n_e}$ by a factor $\sim
    (\Delta \text{RM}/\text{RM})(\text{DM}/\Delta \text{DM}) \sim 50$. The error is much less if most of the RM is produced in low density Galactic and host galactic interstellar media, as it may be for other FRB.

{'source': 'arXiv_out/2011'}