Rotation Measure Evolution of the Repeating Fast Radio Burst Source FRB 121102

Data from the 305 m William E. Gordon Telescope at the Arecibo Observatory were acquired by using the C-band receiver at an observing frequency between 4.1 and 4.9 GHz, composed of two orthogonal linear-polarization feeds. The Puerto Rican Ultimate Pulsar Processing Instrument (PUPPI) backend recorded dual-polarization data every 10.24 μs in 512 frequency channels, each coherently dedispersed to DM = 557 pc cm⁻³ to reduce intrachannel dispersive smearing to < 2 μs. The time and frequency resolution were reduced to 81.92 μs and 12.5 MHz, respectively, before searching for bursts. We used PRESTO¹³ (Ransom 2011) to create 200 dedispersed time series between 461 and 661 pc cm⁻³, which were searched by single_pulse_search.py with boxcar filters ranging from 81.92 μs to 24.576 ms. A large fraction of detections due to noise and radio frequency interference (RFI) were excluded by using dedicated software¹⁴ (Michilli et al. 2018a). A "waterfall" plot of signal intensity as a function of time and frequency was produced and visually inspected for the rest of the detections. The DSPSR package¹⁵ (van Straten & Bailes 2011) was used to create PSRCHIVE¹⁶ (Hotan et al. 2004) files containing the full-resolution data recorded by PUPPI.

For each observation, the differential gain of the two receiver feeds was corrected for by analyzing with PSRCHIVE utilities a one-minute noise injection performed with a diode. Additional calibrations, such as observations of a calibration source, were not performed and residual artifacts in the measured polarization fraction may be present. RM values were calculated by using a technique called RM synthesis (Burn 1966; Brentjens & de Bruyn 2005), which reconstructs the Faraday dispersion function (FDF) of a source through a Fourier transform, in the implementation included in the RM-tools package.¹⁷ We cleaned the resulting FDF by using a deconvolution algorithm (Heald 2009). RM uncertainties are estimated from the width of the peak in the FDF, rescaled by the maximum signal-to-noise ratio. The FDF of bursts detected on MJDs 58222 and 58712 (bursts 8, 19, and 20) shows signs of a poor polarization calibration, namely, symmetric RM peaks around the origin. This is likely caused by a delay calibration issue, and while the RM measurements are still valid, the resulting linear-polarization fractions should be considered lower limits. Conversely, the lack of an RM = 0 rad m⁻² peak for the bursts with ∼100% linear-polarization fractions indicates that cross-coupling between the X and Y polarizations does not have a significant effect on our results. For these bursts, we also find no evidence for significant circular polarization varying with radio frequency, which can occur if there are cross-coupling and poor calibration. Polarization angle (PA) curves were calculated by derotating the data with PSRCHIVE at the RM value obtained for each burst.

We have used the Effelsberg 100 m radio telescope to observe FRB 121102 at 4–8 GHz using the S45 mm receiver with a roughly two-week cadence for 2–3 hr each session from late 2017 to early 2020, totaling 115 hr. The S45 mm receiver has dual linear-polarization feeds.

The data were recorded with full Stokes information using two ROACH2 backends with each one capturing 2 GHz of the band. The channel bandwidth is 0.976562 MHz across 4096 channels, with a 131 μs sampling rate. The recorded data were in a Distributed Acquisition and Data Analysis (DADA) format.¹⁸ Before processing, Stokes I was extracted from the data into a SIGPROC filterbank¹⁹ format in order to perform the initial burst searching.

Observations on 2018 October 22 encountered a receiver issue, forcing us to use the S60 mm receiver instead. The S60 mm receiver has 500 MHz of bandwidth from 4.6 to 5.1 GHz, a 0.976562 MHz channel bandwidth across 512 channels, and an 82 μs sampling rate. The data were recorded as SIGPROC filterbanks.

The data were searched for single pulses using the PRESTO software package. We used rfifind to identify RFI in the data over two-second intervals and to make an RFI mask that was applied to the data during searching. We used PRESTO to create dedispersed time series of the data from 0–1000 pc cm⁻³ in steps of 2 pc cm⁻³, which were searched for single pulses using single_pulse_search.py to convolve the time series with boxcar filters of varying widths to optimize the signal to noise of a burst. A predetermined list of boxcar widths from PRESTO was used, where the widths are multiples of the data sampling time. We searched for burst widths up to 19.6 ms and applied a signal-to-noise threshold of 7. DM–time and frequency–time plots of candidates were visually inspected to search for bursts.

For further RFI mitigation, we calculated the modulation index of the candidates. The modulation index assesses a candidate's fractional variations across the frequency channels in order to discriminate between narrowband RFI and an actual broadband signal (Spitler et al. 2012). We applied this thresholding following Hilmarsson et al. (2020).

If a burst was detected, we performed polarization calibration in order to obtain the RM, PA, and degree of polarization of the burst. We used the psrfits_utils package²⁰ to create a psrfits²¹ file containing the burst and used PSRCHIVE to calibrate the data by first dedispersing the burst data using pam, then pac to polarization-calibrate those data with noise diode observations. RM was obtained using RM synthesis, described in Section 2.1.

FRB 121102 was observed with the VLA as part of a monitoring project (VLA/17B-283) from 2017 November to 2018 January. Ten 1 hr observations were conducted at 2–4 GHz using the phased-array pulsar mode. The VLA has dual circular polarization feeds. Data were recorded with full Stokes information with 8096 × 0.25 MHz channels and 1024 μs time samples. Each observation had ≈ 30 minutes on source. Data were dedispersed at 150 trial DMs from 400–700 pc cm⁻³, and the resulting time series were searched for pulses using the PRESTO single_pulse_search.py.

Polarization calibration was done using the 10 Hz injected noise calibrator signal. After polarization calibration, the RMs were obtained using RM synthesis, as described in Section 2.1.

Previous surveys of FRB 121102 at frequencies between 4 and 8 GHz reported rates based on fewer observed hours (Spitler et al. 2018) and anomalously high burst rates (Gajjar et al. 2018). Spitler et al. (2018) detected three bursts from observing at 4.6–5.1 GHz for 22 hr consisting of 10 observing epochs spanning five months using the Effelsberg telescope. Gajjar et al. (2018) detected 21 bursts in a single six-hour observation, observing at 4–8 GHz at the Green Bank Telescope. Furthermore, Zhang et al. (2018) re-searched the data from Gajjar et al. (2018) using a convolutional neural network and detected an additional 72 bursts within the data.

Our Effelsberg survey spans over two years of observing FRB 121102 for 2–6 hr at a time at 4–8 GHz with a two-week cadence, amounting to 115 hr of observations. Included here are 10 hr of observations presented in Caleb et al. (2020). We can therefore report a robust, long-term average burst rate of FRB 121102 in this frequency range of ${0.21}_{-0.18}^{+0.49}$ bursts/day (1σ error) above a fluence of 0.04 (w/ms)^1/2 Jy ms for a burst width of w ms. We list the details of the surveys discussed here in Table 2.

A caveat to our observed burst rate is the suspected periodic activity of FRB 121102 (Rajwade et al. 2020). Roughly 40% of our Effelsberg observations were performed during the suspected inactivity of FRB 121102, which if true would affect the observed burst rate. No bursts were detected during this state of inactivity. Including only observations while FRB 121102 is active, the average burst rate becomes ${0.35}_{-0.29}^{+0.80}$ bursts/day above a fluence of 0.04 (w/ms)^1/2 Jy ms.

The observed burst rates of FRB 121102 also seem to be frequency dependent, with the rate being lower at higher frequencies. At 1.4 GHz the FRB 121102 burst rate has been observed to be 8 ± 3 bursts/day above a fluence of 0.08 Jy ms for 1 ms burst widths (Cruces et al. 2021).

We plot the dynamic spectra, polarization profile, and PAs of our detected bursts in Figure 1. The PA is equal to RMλ² + PA_ref, where λ is the observing wavelength, and PA_ref is a reference angle at a specific frequency (central observing frequency in our case). The PAs are flat across each burst, as has been seen previously from FRB 121102 (Gajjar et al. 2018; Michilli et al. 2018b). We do not discuss PA changes over time, as we did not observe an absolute calibrator for polarization. In the absence of an absolute calibrator, we cannot compare PAs across multiple telescopes, and a more detailed discussion is outside the scope of this work.

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The bursts are mostly ∼100% linearly polarized. Bursts from FRB 121102 have been consistently ∼100% linearly polarized since its first polarization measurement in late 2016 (Michilli et al. 2018b), which suggests a stability in its emission process. The Arecibo bursts at MJD 58222, 58247, and 58712 (bursts 8, 15, 19, and 20) are not fully linearly polarized, which is uncharacteristic for FRB 121102, and can be attributed to polarization calibration issues (see Section 2.1).

The circular polarization fraction is on average 4% for all the bursts and 17% by summing the absolute Stokes V values. The Stokes V on-pulse and off-pulse standard deviations are the same for the majority of the bursts, indicating that the circular polarization is consistent with noise; therefore, the circular polarization fractions quoted above are upper limits. For the VLA and Effelsberg bursts (number 5, 6, and 10) the on and off-pulse statistics differ, where the on-pulse standard deviation is larger by a factor of 2, and we obtain circular polarization fractions for Stokes V and absolute Stokes V of 5%/5%, 2%/4%, and 3%/11% for bursts 5, 6, and 10, respectively. The 100% linear polarization and the lack of circular polarization for the majority of bursts indicate that little to no Faraday rotation conversion, where linear polarization is converted to circular in a magneto-ionic environment (Gruzinov & Levin 2019; Vedantham & Ravi 2019), occurs at our observing frequencies.

The VLA burst on MJD 58075 (burst 6) exhibits a triple-component profile. The second and third components exhibit a downward drift in frequency, a feature predominantly observed from repeating FRBs (e.g., Hessels et al. 2019; The CHIME/FRB Collaboration et al. 2020). An apparent upward drift in frequency between the first two components can be seen, and the first component has a different PA from the other two. While the temporal spacing between the components is not large, the difference in PAs between the first component and the other two might suggest that these are in fact two separate bursts.

The Effelsberg burst at MJD 58228 (burst 10) was only detected between 4–5.2 GHz of the 4–8 GHz bandwidth. We were affected by strong edge effects in the bandpass, resulting in an uneven frequency response across the bandwidth. Thus, we are uncertain whether the burst frequency envelope is inherent to the burst or due to the bandpass.

The RMs we obtained from our bursts are listed in Table 1. We plot the RMs over time in Figure 2. The observed RM of FRB 121102 has dropped by 34% over 2.6 yr from ∼ 10⁵ rad m⁻² to ∼ 6.7 × 10⁴ rad m⁻². As Figure 2 shows, the drop in RM has not been steady over time. From MJD 57757 to MJD 58215 (bursts 1–7), the RM decreased rapidly to ∼7 × 10⁴ rad m⁻² and has declined only slightly (∼5000 rad m⁻²) since then. An FDF plot of the bursts reported here can be seen in Figure 3.

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Within a 32 day time span, the observed RM of FRB 121102 exhibited significant short-timescale variations (bursts 7–15). At epochs separated by a week, the RM increased by ∼1000 rad m⁻² (bursts 7–8). For three epochs during the following week, the RM remained stable between bursts 8–10, before increasing again by ∼1000 rad m⁻² a week later (bursts 11–12). During three epochs in the following two weeks, the RM was observed to drop rapidly by a total of ∼4500 rad m⁻² (bursts 12–15). This short-timescale behavior can be seen in the inset of Figure 2. No RM measurement is available between MJDs 58247 and 58677 (430 days), but the RMs are consistent with each other at these dates (bursts 15–16). Another drop in RM of ∼2000 rad m⁻² can be seen between bursts 18 and 19, separated by 28 days.

Only minor changes in DM have been observed during the observed RM evolution of FRB 121102. While the RM decreased significantly, the DM has increased by ∼4 pc cm⁻³, from 559.7 ± 0.1 pc cm⁻³ (Michilli et al. 2018b) up to 563.3 pc cm⁻³ from the aforementioned linear interpolation of the L-band burst DMs used in this work.

Michilli et al. (2018b) constrained the average magnetic field along the line of sight (LoS) in the region where Faraday rotation occurs, 〈B_∥〉, between 0.6 and 2.4 mG using their measured FRB 121102 RM in the source frame of RM_src ∼ 1.4 × 10⁵ rad m⁻² and the estimated host DM contribution of DM_host 70–270 pc cm⁻³ (Tendulkar et al. 2017). From a measured DM and RM, 〈B_∥〉 can be calculated, ignoring sign reversals, as

$$\begin{eqnarray}&&\langle {B}_{\parallel }\rangle =1.23\ {\mathrm{RM}}_{\mathrm{src}}/{\mathrm{DM}}_{\mathrm{host}}\,\mu {\rm{G}}.\end{eqnarray} \tag{ 1 }$$

The most recent DM and RM values in our sample yield 〈B_∥〉 = 0.4–1.6 mG. This is a lower limit as the DM in the Faraday-rotating region could be much lower.

For the constant-density ISM model variety, the number density of the ISM, n, is a free parameter, and for the progenitor wind model varieties, the wind mass-loading parameter, K, is a free parameter. For all model varieties, we also set t_age and Σ as free parameters for supernova ejecta masses of 10 and 2 M_⊙. In all cases, the supernova explosion energy is 10⁵¹ erg.

We estimate the posterior of n, K, t_age, and Σ for each model variety (Piro & Gaensler 2018, Equations (26) and (57), and the Appendix) using the measured RM values of FRB 121102 (Table 1). Our initial guesses are the median values of n (1 cm⁻³) and K (10¹³ g cm⁻¹) from Piro & Gaensler (2018), t_age = 5 yr, and Σ = 10³ rad m⁻² (roughly 1% of the observed RM magnitude). Our results are listed in Table 3.

Table 3.  Model Parameters of Piro & Gaensler (2018) for Each Scenario

Model	E (erg)	M (M⊙)	n (cm−3)	${\mathrm{log}}_{10}(K)$ (g cm−2)	tage (yr)	${\mathrm{log}}_{10}({\rm{\Sigma }}$) (rad m−2)
Const. ISM	1051	10	${1.7}_{-0.1}^{+0.1}$	⋯	${1.4}_{-0.2}^{+0.2}$	${3.9}_{-0.1}^{+0.1}$
	1051	2	${2.5}_{-0.1}^{+0.1}$	⋯	${1.4}_{-0.2}^{+0.2}$	${3.8}_{-0.1}^{+0.1}$
Wind	1051	10	⋯	${15.3}_{-0.1}^{+0.1}$	${7.8}_{-1.1}^{+0.9}$	${3.9}_{-0.1}^{+0.1}$
	1051	2	⋯	${15.6}_{-0.1}^{+0.1}$	${6.4}_{-0.7}^{+0.6}$	${3.9}_{-0.1}^{+0.1}$
Wind + SNR	1051	10	⋯	${11.7}_{-0.1}^{+0.1}$	${8.3}_{-1.2}^{+1.0}$	${3.9}_{-0.1}^{+0.1}$
	1051	2	⋯	${11.9}_{-0.1}^{+0.1}$	${8.3}_{-1.1}^{+1.0}$	${3.9}_{-0.1}^{+0.1}$

Note. From left to right: model scenario, supernova explosion energy (E), supernova ejecta mass (M), number density of surrounding uniform ISM (n), wind mass-loading parameter (K), age of bursting source (t_age), and the factor that takes into account additional errors and astrophysical variance in RM, Σ. The parameters n, K, t_age, and Σ were obtained in this work (Section 4.1). Uncertainties are 1σ.

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For the constant ISM model, we obtain a t_age of 1.4 yr at the time of the first RM detection. For the wind and wind plus SNR evolution models, we obtain t_age between ∼ 6–8 yr. The range of RM from our results (1σ error) for each model and mass is plotted as a function of time in Figure 4 and overplotted with the observed RM values of FRB 121102.

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We also plot the local DM versus RM for the models in Piro & Gaensler (2018) in Figure 5, showing how the DM changes as RM decreases over time. The estimated source-frame local DM (up to 270 pc cm⁻³; Tendulkar et al. 2017) and the source-frame RM values of FRB 121102 are overplotted in the figure.

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In this model, we set t_age, Σ, and the power-law parameter, α, as free parameters. The free magnetic energy of the magnetar, ${E}_{{B}_{* }}$, the time since the onset of the magnetar's activity, t₀, and the expansion velocity of the nebula, v_n, differ between model varieties "A, B, and C." We estimate the posterior of α, t_age, and Σ. The initial guesses are the parameters of models A, B, and C and t_age in Margalit & Metzger (2018), and as before, Σ = 10³ rad m⁻². Our results are listed in Table 4.

Table 4.  Model Parameters of Margalit & Metzger (2018)

Model	${E}_{{B}_{* }}$ (erg)	t0 (yr)	vn (cm s−1)	αa	tage (yr)a	αb	tage (yr)b	${\mathrm{log}}_{10}({\rm{\Sigma }})$ (rad m−2)
A	5 × 1050	0.2	3 × 108	1.3	12.4	${1.6}_{-0.4}^{+0.3}$	${16.8}_{-0.6}^{+2.0}$	${3.9}_{-0.1}^{+0.1}$
B	5 × 1050	0.6	108	1.3	37.8	${1.1}_{-0.1}^{+0.1}$	${16.2}_{-2.6}^{+2.0}$	${3.9}_{-0.1}^{+0.1}$
C	4.9 × 1051	0.2	9 × 108	1.83	13.1	${1.6}_{-0.3}^{+0.3}$	${15.3}_{-0.3}^{+0.8}$	${3.9}_{-0.1}^{+0.1}$

Notes. From left to right: model, free magnetic energy of the magnetar (${E}_{{B}_{* }}$), onset of the magnetar's active period (t₀), radial velocity of expanding nebula (v_n), power-law parameter (α), and age of bursting source (t_age) used in Margalit & Metzger (2018), and α, t_age, and the factor that takes into account additional errors and astrophysical variance in RM, Σ, obtained in this work (Section 4.2). Uncertainties are 1σ.

^aIn Margalit & Metzger (2018). ^bThis work.

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A similar t_age of ∼ 15–17 yr was obtained for all models. Our obtained α values lie in the range of 1.1–1.6 and are consistent with the values in Margalit & Metzger (2018). The resulting RM range (1σ error), overplotted with observed FRB 121102 RM values is plotted in Figure 6.

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The GC magnetar PSR J1745−2900 has exhibited similar behavior to FRB 121102 regarding changes in RM. Since its first RM measurements of −67,000 rad m⁻² (Eatough et al. 2013), it showed some variations in RM of a few hundred rad m⁻² per year for a few years until its RM suddenly exhibited a steep drop in absolute magnitude (Desvignes et al. 2018). This drop in RM is similar to FRB 121102, albeit not as intense, as PSR J1745−2900 had a drop of 5% in RM over the course of a year while the RM of FRB 121102 has dropped by an average of 15% yr⁻¹ over roughly two years. Both PSR J1745−2900 and FRB 121102 exhibit short-term variations in their observed RMs. Although somewhat similar, the magnitude of the FRB 121102 variations is greater. Desvignes et al. (2018) also report a constant DM and attribute the RM evolution to the changing LoS toward the moving magnetar where either the projected magnetic field or the GC free electron content varies.

Desvignes et al. (2018) use the measured proper motion of PSR J1745−2900 to estimate the characteristic size of magneto-ionic fluctuations to be ∼2 astronomical units (au). Assuming the bursts from FRB 121102 originate from the magnetosphere of a neutron star with a speed of ∼100 km s⁻¹, the source moves a distance of 20 au per year. The observations of PSR J1745−2900 show that spatial variations on the scale of a few to tens of astronomical units are possible in the vicinity of a massive black hole. If the host of FRB 121102 also harbors a massive black hole, the variations seen in the RM of FRB 121102 could be caused by the changing medium in its accretion disk. The velocity of the medium could be much higher than in the Galactic center, contributing to the observed fluctuation.

Piro & Gaensler (2018) model the temporal evolution of both RM and DM of an expanding SNR. They consider three cases of evolutionary environments, which we expand upon below.

The first evolutionary case is an SNR that expands into an ISM of constant density. The shocked, ionized regions of the supernova ejecta and ISM, as well as ionized material from the pulsar wind nebula close to the SNR center, provide sufficient free electrons to disperse an FRB. The Faraday rotation arises from the magnetic fields generated by the forward and reverse shocks during the SNR expansion. The SNR dominates both the DM and RM contributions at early times until the ISM takes over on a timescale of ∼ 10²–10³ yr. The free parameters in this model are the number density of the uniform ISM, n, and the supernova ejecta mass, M. The energy of the explosion is kept constant as E = 10⁵¹ erg for all cases.

The second case is where the stellar wind of the massive progenitor affects the circumstellar environment. The magnetized wind provides another source of the magnetic field as well as altering the DM evolution. The DM is much higher initially compared to the previous scenario due to the high density of the wind adjacent to the SN, but the DM decreases more rapidly because of the wind's decreasing density. The wind environment can produce an ordered magnetic field, which is swept up by the SNR. This is the focal point of RM generation in this scenario as opposed to the shock generation of magnetic fields in the previous scenario. The RM also drops rapidly due to the steep decline with time of the wind's density and magnetic field. Here, the free parameters are the ejecta mass, M, and the wind mass-loading parameter, K, which is a function of the mass-loss rate, $\dot{M}$, and wind velocity, v_w, and is given in units of g cm⁻¹.

The third is a mixture of the first two scenarios: an SNR expands into an ISM affected by a constant-velocity wind, with M and K as free parameters. For all three cases, they assume supernova ejecta masses of 10 M_⊙ (red supergiant progenitor) and 2 M_⊙ (stripped-envelope supernova).

Margalit & Metzger (2018) consider a magnetar surrounded by a magnetar nebula. Flares and winds from the magnetar inject particles and magnetic energy into the nebula that is, in turn, responsible for the large observed RM. Their model is a one-zone magnetar nebula model, where they assume a spherical, freely expanding nebula with a constant radial velocity, v_n. The free magnetic energy of the magnetar, ${E}_{{B}_{* }}$, is released into the nebula at a rate following a power law in time, $\dot{E}\propto {t}^{-\alpha }$ (Margalit & Metzger 2018, Equation (4)), where α ≳ 1. The Faraday rotation occurs in nonrelativistic electrons ejected earlier in the nebula's history and cooled from radiation and adiabatic expansion.

In this model, the RM can be approximated as (Margalit & Metzger 2018, Equation (19), values normalized to 1 are omitted for clarity)

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{RM}}_{5} & \approx & 6\ {\left(\displaystyle \frac{{E}_{{B}_{* }}}{{10}^{50}\mathrm{erg}}\right)}^{3/2}{\left(\displaystyle \frac{{v}_{n}}{{10}^{17}\mathrm{cm}{{\rm{s}}}^{-1}}\right)}^{-7/2}\\ & \times & {\left(\alpha -1\right)}^{3/2}{t}_{0}^{(\alpha -1)/2}{t}^{-(6+\alpha )/2}\,\mathrm{rad}\,{{\rm{m}}}^{-2},\end{array}\end{eqnarray} \tag{ A1 }$$

where RM₅ ≡ RM/10⁵ rad m⁻², ${E}_{{B}_{* }}$ is in erg, v_n in cm s⁻¹, t is seconds since the supernova explosion, and t₀ is the time in seconds since the onset of the active period of the magnetar's energy release into the nebula. We obtain t_age from Equation (A1) by replacing t with ${t}_{\mathrm{age}}+t^{\prime}$, where $t^{\prime}$ is the time elapsed in seconds of each RM measurement since the first one.

For completeness, the estimated DM contribution from the Faraday-rotating medium is given by

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{DM} & \sim & 3\times {10}^{18}\ \left(\displaystyle \frac{{E}_{{B}_{* }}}{{10}^{50}}\right){\left(\displaystyle \frac{{v}_{n}}{{10}^{8}}\right)}^{-2}\\ & \times & (\alpha -1){t}^{-2}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}.\end{array}\end{eqnarray} \tag{ A2 }$$

In their analysis, Margalit & Metzger (2018) consider three variations of their model with each having its own set of values for ${E}_{{B}_{* }}$, t₀, v_n, and α. They call these variations "models A, B, and C", and we keep the same notation to avoid confusion. Margalit & Metzger (2018) use models A, B, and C to estimate the t_age of FRB 121102 from Equation (A1) using the RM measurements from Michilli et al. (2018b) and Gajjar et al. (2018). Their choice of parameters and their results are shown in Table 4.