Emission Properties of Periodic Fast Radio Bursts from the Motion of Magnetars: Testing Dynamical Models

Recent observations of the periodic Fast Radio Burst source 180916.J0158+65 (FRB 180916) find small linear polarization position angle swings during and between bursts, with a burst activity window that becomes both narrower and earlier at higher frequencies. Although the observed chromatic activity window disfavors models of periodicity in FRB 180916 driven by the occultation of a neutron star by the optically-thick wind from a stellar companion, the connection to theories where periodicity arises from the motion of a bursting magnetar remains unclear. In this paper, we show how altitude-dependent radio emission from magnetar curvature radiation, with bursts emitted from regions which are asymmetric with respect to the magnetic dipole axis, can lead to burst activity windows and polarization consistent with the recent observations. In particular, the fact that bursts arrive systematically earlier at higher frequencies disfavors theories where the FRB periodicity arises from forced precession of a magnetar by a companion or fallback disk, but is consistent with theories where periodicity originates from a slowly-rotating or freely-precessing magnetar. Several observational tests are proposed to verify/differentiate between the remaining theories, and pin-down which theory explains the periodicity in FRB 180916.


INTRODUCTION
Fast radio bursts (FRBs) are short (∼ s to ∼ms duration) radio bursts with unknown origins. A decade after the first discovery (Lorimer et al. 2007), massive progress has been made in understanding the nature of FRBs. The discovery repeating FRBs (e.g. Spitler et al. 2014Spitler et al. , 2016CHIME/FRB Collaboration et al. 2019a;Fonseca et al. 2020) suggests at least a portion of the FRB population has a non-catastrophic origin. The first localization of an FRB source (Chatterjee et al. 2017) confirmed their cosmological origin. The detection of extremely bright radio bursts from a galactic magnetar (CHIME/FRB Collaboration et al. 2020b; Bochenek et al. 2020) places magnetars as a most promising progenitor for FRBs.
Repeating FRBs offer great opportunities for follow-up observations, as well as studying the burst properties with time and frequency. FRB 180916.J0158+65 (hereafter FRB 180916) is the closest localized extragalactic FRB source ( = 0.0337; Marcote et al. 2020), and also the most active repeating source detected within the frequency band 400-800 MHz by the Canadian Hydrogen Intensity Mapping Experiment (CHIME, CHIME/FRB Collaboration et al. 2019b). Remarkably, a periodicity in the activity of FRB 180916 has recently been found. Bursts detected at 400-800 MHz occur within a 5-day window that repeats every 16.3 days (CHIME/FRB Collaboration et al. 2020a). Multiple models have been proposed to explain the periodicity: the orbital motion of the FRB source (neutron star or magnetar) around a windy companion (Dai et al. 2016;Lyutikov, Barkov, & Giannios 2020;Ioka & Zhang 2020), precession of the burst emitting object itself (e.g. Zanazzi & Lai 2020;Levin et al. 2020;, or ultralong rotational periods of the bursting object (Beniamini et al. 2020).
Multi-wavelength follow-ups of this source provide constraints on the FRB progenitor and the local environment, and also introduce additional puzzles. Imaging from the Hubble Space Telescope shows that the location of FRB 180916 is offset by ∼ 250 pc from the nearest knot of star formation in the L & Z host galaxy (Tendulkar et al. 2020) , while young magnetars are expected to born in a star forming region. For bursts detected in the -band, the change of linear polarization position angle (PA) is constrained to be 10 • -20 • across the bursts of the same phase , and 50 • across the -band active phases (Pastor-Marazuela et al. 2020). No PA swing is observed, as would generically be expected in the magnetospheric origin models (as oppose to the diverse polarization angle swing detected in FRB 180301, Luo et al. 2020). Most strikingly, the active phase is observed to be chromatic, with the activity window being both narrower and earlier at higher frequencies (Pastor-Marazuela et al. 2020;Pleunis et al. 2020). This disfavours models explaining the periodic activity with the eclipse of a companion wind, as these theories predict a narrower activity window at lower frequencies Ioka & Zhang 2020).
Recently, a period of ∼160 days was detected in the repeating FRB 121102 (Rajwade et al. 2020;Cruces et al. 2021). Similar to FRB 180916, FRB 121102 also has small PA swings during and between bursts (Michilli et al. 2018), but data on the frequency-dependence of the activity window has yet to be gathered. Clearly, the polarization emission and frequency-dependence expected from different models of periodic FRBs is becoming highly topical.
The goal of this work is to show how the recent constraints on the PA variation, as well as the dependence on the FRB activity window with frequency, fit into theories which argue the periodicity of FRB 180916 originates from the motion of a Neutron Star (NS) or magnetar. Section 2 justifies our phenomenological emission model for NSs emitting FRBs. Section 3 discusses the three dynamical theories which have been put forth to explain the periodicity of FRB 180916, which we test in this work: a NS with a long rotation period, a NS undergoing free precession, and a NS undergoing forced precession. Section 4 presents the main results of this work, and discusses which dynamical models are favored to explain the periodicity of FRB 180916. Section 5 discusses future observations which can potentially distinguish between the remaining theories. Section 6 summarizes our work.

EMISSION MODEL FOR FRBS
To make testable predictions for periodic FRB emission, we construct a phenominological model of radio emission from a rotating (and possibly precessing) NS, motivated by models for radio pulsar emission (e.g. Ruderman & Sutherland 1975;Rankin 1993). Figure 1 displays the geometry of the model. A NS with rotation axisˆand dipole moment unit vectorˆis viewed by an observer, withˆbeing the unit vector pointing in the direction of the Line of Sight (LOS) of the observer. FRBs are emitted in a directionˆwith frequency , and an observer sees the FRB emission whenˆlies within the cone centered onˆwith opening angle (orange region of  The emission model assumed in this paper is significantly more complex than the model in Zanazzi & Lai (2020), which assumedˆ=ˆ. In this section, we justify our model assumptions about the FRB emission region, assuming the coherent radio emission arises due to curvature radiation from charged particles travelling along the NS magnetic field lines. Magnetar curvature radiation has already been invoked by many models to explain the coherent radio emission characteristic of FRBs (e.g. Katz 2014;Cordes & Wasserman 2016;Kumar et al. 2017;Lu & Kumar 2018;Yang & Zhang 2018;Lu et al. 2020;) Section 2.1 discusses our expectations for how the emission directionˆchanges with FRB frequency , while Section 2.2 discusses the linear polarization of the emission.

FRB Emission Direction
In the study of pulsar emission, it is widely believed that different frequencies are produced at different altitudes within the pulsar magnetosphere (e.g. Manchester & Taylor 1977). It has been shown the spread of field lines at higher altitude could account for the widening of pulse profiles at lower  frequencies (e.g. Cordes 1978;Phillips 1992). This kind of "radius-to-frequency mapping" has also been used to explain the downward drifting pattern seen in some repeating FRBs (Wang et al. 2019;Lyutikov 2020). Here, we demonstrate this same effect in the context of curvature radiation, and show how it will lead to the magnetic polar angle ob and cone opening angle of the observed FRB emission to decrease with an increase in frequency .
For simplicity, we assume the magnetic field exterior to the NS is a dipole in vacuum, and ignore how plasma affects the magnetic field (e.g. Goldreich & Julian 1969; Tchekhovskoy N S Emission Region Figure 3. Cartoon representation of offset magnetic field geometry, which can give rise to FRB emission from curvature radiation consistent with our model (Fig. 1). The strong exposed magnetic fields on one side of the NS lead to burst emission occuring in "cones" around the emission axisˆ, with both the magnetic polar angle ob (angle betweenˆandˆ) and opening angle increasing with decreasing frequency .  (1) where em is the magnetic polar angle of the field line, is the distance to the center of the NS, 0 is the NS radius, while 1 ≤ < ∞ is a constant which varies for different field lines. Emission at em will be observed by a distant observer at a different magnetic polar angle ob , which is related to em by cos ob = 1 + 3 cos 2 em √ 10 + 6 cos 2 em . ( The trajectory of a number of different magnetic field lines, as well as the relation between em and ob , are displayed in the top panel of Figure 2. The characteristic frequency of emission from curvature radiation is (e.g. Ruderman & Sutherland 1975): where is the Lorentz factor, while is the curvature radius.
The upper panel of Figure 2 shows the emission heights of different frequency bands, displayed in green (1.36 − 2 GHz), orange (400 − 800 MHz), and blue (110 − 180 MHz). Clearly for curvature radiation, high (low) frequency radiation originates at low (high) altitudes. The bottom panel displays a histogram for number of bursts with a given ob value, assuming an equal number of bursts are emitted per unit length of the field line, with the field line values spaced linearly between 55 ≤ ≤ 80. From this, we see both the mean and spread of ob values for a given frequency band increases with decreasing frequency. Within this simple model, a dipole situated at the center of a NS will lead to FRB emission symmetric about the NS dipole momentˆ, rather than localized on a small "patch" at a location displaced fromˆ. One simple modification to the magnetic field geometry which can lead to asymmetric emission about theˆaxis is displayed in Figure 3: a dipole displaced from the NS center. The strong magnetic fields exposed on one side of the NS (which are buried on the other side in this cartoon) can lead to emission "cones" similar to those assumed in our model (Fig. 1), with opening angles comparable to the width of the histogram widths displayed in the bottom panel of Figure 2. There is growing observational evidence many NSs may have similar magnetic field geometries as displayed in Figure 3. An offset dipole has been invoked to explain X-ray emission from the mode-switching pulsar PSR B0943+10 (Storch et al. 2014). Recently, X-ray observations from the NICER mission found the hot spots on the surface of isolated pulsars to be far from antipodal, implying a highly complex magnetic field far from the classic assumption of a centered dipole (Riley et al. 2019;Bilous et al. 2019). For this work, rather than attempting to construct a complex magnetic field which can lead to FRB emission consistent with our model, we simply leave ob and as free parameters, with the general expectation both ob and should increase with decreasing .

FRB Polarization
To model the polarization of FRB emission, we use the rotating vector model (Radhakrishnan & Cooke 1969), where the position angle PA of the linear polarization is given by where Ψ is the rotational phase (Ψ = 0 corresponds to when has its closest approach toˆ), while the other quantities Notice that in Equation 5, a change from to can be compensated by a change of 0 to 0 3 / 3 to emit in the same frequency. Therefore, the emission height in Figure 2 can be scaled with a change of 0 . are displayed in Figure 1. Although the magnetic field close to the NS may be complex, Lu et al. (2019) argued the propagation of a FRB across a plasma-filled NS magnetosphere causes the electric field of the burst to "freeze" in a direction perpendicular to the magnetic field when the plasma density is sufficiently low (and the magnetic field is approximately dipolar), implying equation (6) should be an adequate model for the linear polarization from FRBs. Other works have also used PA measurements constrain the free-precession of NSs (Weisberg et al. 2010). We neglect how additional propagation effects can cause the polarization to differ from the rotating vector model (e.g. Wang et al. 2010;Beskin & Philippov 2012).

DYNAMICS OF PERIODIC FRB MODELS
Models which ascribe the periodicity of FRB 180916 from the motion of a magnetar fall into three different categories. The simplest dynamical model postulated the 16.3 day periodicity was the rotation period of the magnetar (Beniamini et al. 2020), which implies this magnetar must have a rotation period much longer than those typically observed (∼0.1-10 s, e.g. Kaspi & Beloborodov 2017). Models which assumed more typical magnetar spin periods argued the periodicity of FRB 180916 came from either free (or "inertial") precession of the NS (Zanazzi & Lai 2020;Levin et al. 2020; Sob'yanin 2020), or forced precession of the NS from a companion  or a fallback disk (Tong et al. 2020). In this section, we show how each scenario predicts different motion of a bursting magnetar with respect to a distant observer. Figure 4 summarizes how the motion of the NS spin axisˆ, dipole moment axisˆ, emisison direction axisˆ, and observer LOS axisˆdiffer between the three different dynamical theories. We defer a discussion of the physics which lead to these three different classes of magnetar motions to the references above.
The dynamics of these different motions are more conveniently analyzed in either a Cartesian coordinate system anchored into and co-rotating with the NS {ˆ,ˆ,ˆ} (body frame), or stationary with respect to a distant observer {ˆ,ˆ,ˆ} (inertial frame). The time evolution of a vector ( ) in the body d /d | or inertial d /d | frames are related via d /d | + × = d /d | , where =ˆ, with the spin frequency of the NS.

Magnetar undergoing Free Precession
The middle panel of Figure 4 displays our dynamical model for a freely-precessing NS. We work in the body frame, where the Cartesian coordinates {ˆ,ˆ,ˆ} define the (effective) principal axis of the biaxial NS, withˆprecessing aroundâ ccording tô ( ) = sin cosˆ+ sin sinˆ+ cosˆ, (8) where ( ) = Ω prec + (0) is the precession phase, with Ω prec the precession frequency (see Zanazzi & Lai 2015 for details). The vectorˆlies in the direction of the projection ofˆonto the plane perpendicular toˆ.
When the precession frequency is much smaller than the rotational frequency (Ω prec ),ˆremains approximately constant asˆrotates aroundˆover the rotational period of the NS (in the body frame). Henceˆ( ) is described by an equation similar to (7), except in a frame whereˆ≠ˆ: where ( ) = + (0) is the spin phase.

Magnetar undergoing Forced Precession
The right panel of Figure 4 displays our model for forced precession, either by a companion  or a fallback disk (Tong et al. 2020). We work in an inertial reference frame {ˆ,ˆ,ˆ}, with the orbital angular momentum axis of the companion or disk definingˆ, withˆlying in the direction of the projection ofˆonto the plane perpendicular toˆ. Here, the magnetic longitude is defined as the angle between the planes spanned by the vector pairs {ˆ,ˆ} and {ˆ,ˆ}. All other quantities are the same as those illustrated in Figure 1.

APPLICATION OF PERIODIC EMISSION MODEL TO FRB 180916
Now that we have developed a phenomenological emission model for periodic FRBs due to the motion of NSs, we apply these results to the recent multi-wavelength and polarization measurements of FRB 180916. For a dynamical model to remain a plausible explanation for the periodicity of FRB 180916, it must explain the following three features of the FRB 180916 emission (Pleunis et al. 2020;Pastor-Marazuela et al. 2020): 1. The linear polarization angle PA varies by less than ΔPA 10 • -20 • for single bursts, and ΔPA 40 • between bursts.
2. The bust activity window (region in phase busts seen after folding over 16.3 day periodicity) widens with a decrease in frequency.
3. The burst activity windows in different frequency bands have phase centers which are offset from one another, or busts at one frequency are systematically delayed with respect to burst at another frequency. We will call this effect activity window phase drift with frequency.
The first constraint (with ΔPA) lied at odds with the original predictions of the Zanazzi & Lai (2020) free precession model. This is because the emission region was assumed to emit in the direction of the NS dipole moment (ˆ=ˆ, see Fig. 1). Because FRBs occur whenˆ≈ˆ=ˆ, the rotational phase Ψ 1 during an FRB, implying equation (6) reduces to tan PA sin sin( − ) Ψ.
Sinceˆ≈ˆimplies ≈ (see Fig. 1), the denominator of equation (13) becomes large, and hence the model of Zanazzi & Lai (2020) predicted large variations in PA during individual FRB bursts, as well as modulation of the PA variation between bursts as the NS precessed. However, becausê andˆcan have significant differences in orientation (see §2.1), it is possible for PA variations to be sufficiently small to be consistent with the polarization measurements from FRB 180916.
In this section, we calculate the variations in PA, as well as how the burst activity window depends on frequency , with different dynamical models (Fig. 4). We create maps of the range in PA variation, and estimate the activity window of different models, by first fixing parameters which are constant for a given dynamical model (see Fig. 1), and in particular specifyingˆand for different frequency bands. Notice here, we assume the shift of emission region in the observed polar angle ob while the center of the emission region has the same magnetic longitude . A burst is considered observed at a specific -band ifˆandˆlie within an angle of each other, with no emission if this condition is not met. The Phase = ( + Φ)/(2 ) for the slow magnetar model, Phase = ( + Φ)/(2 ) for the free/forced precession model (with Φ a constant picked so the 400-800 MHz -band is centered at Phase = 0.5), and Spin Phase = /(2 ) are then cycled through their possible parameter values ( , ∈ [0, 2 ], middle panels of Fig. 5) to calculate the range of PA values with (spin or precession) Phase (top panels of Fig. 5). The number of bursts over all spin phases are then binned by precessional phase to construct burst activity window profiles (bottom panels of Fig. 5).
The top panels of Figure 5 display the PA variation with either spin (left panels) or precession (middle & right panels) phase. From this, we see all three models are able to have PA variations consistent with observations, as long as the emission region directionˆhas a significant offset from the dipole axisˆ( obs sufficiently large). The spin phase has similar variations as the PA with spin or precession phase, with differences due to the slight differences in geometry between the two angles (see eq. [6]).
The bottom panels of Figure 5 display the activity window over different -bands. The magnetar with a slow rotation period, as well as the magnetar undergoing free precession, can accommodate an activity window which widens at lower values, as well as activity window phase drifts betweenbands. The widening of the activity window is primarily due to the increase of at lower -bands, while the phase drift is primarly due to the higher magnetic polar angle obs values at lower -bands (see §2.1 for discussion). The roughly monochromatic bust rate with spin phase for the slow magnetar model is because the event rate is assumed to be uniform over the NS spin phase: dropping this assumption can lead to activity windows which are not monochromatic. We conclude that the dynamical model of a magnetar with a slow rotation period, as well as a magnetar undergoing free precession, are capable of causing periodic emission consistent with observations of FRB 180916.
The bottom right panel of Figure 5 displays the activity windows for the model describing a magnetar undergoing forced precession. Although these parameters clearly show a widening activity window with a lower -band, the histogram also displays no activity window phase shift with frequency. The lack of activity window phase shift is not unique to these particular model parameters, but is rather a general feature of the forced precession model. Consider a magnetar undergoing forced precession, with LOS directionˆ= sinˆ+cosˆ. Becauseˆ·ˆ FRB 180916 disfavors the dynamical model of a magnetar undergoing forced precession.

PROPOSED OBSERVATIONAL TESTS
The previous section showed the dynamical model of a magnetar with a slow rotation period, and a magnetar undergoing free precession, could explain the polarization and frequency-dependent activity window of FRB 180916, while the magnetar undergoing forced precession was disfavored. In this section, we describe how further polarization and activity window measurements can refine constraints on model parameters, and additional observations can favor or rule-out the remaining dynamical theories to explain periodic FRBs.
If additional periodic FRB sources are detected, the PA swings and activity windows are expected to change from source to source. Within the context of our phenomenological emission model, the phase-drift of the FRB activity window with frequency results from a change of emission region. This phase-drift can be in either direction, depending on the direction of rotation/precession, as well as the orientation of the emission regions with respect to each-other. Event rates of FRBs can also differ significantly across frequency due to different magnetic field geometries. However, both the slowly-rotating and freely-precessing magnetars require the emission regions to be asymmetric with respect to the NS dipole axis. In contrast, it is difficult for the induced precession model to have an activity window which drifts asymmetrically in phase with frequency.
All dynamical models allow a small PA change with precession or spin phase, while larger PA changes like in the case of FRB 180301 (Luo et al. 2020) is also allowed. The PA variations of both models differ between frequency bands.
L & Z However, for a slowly-rotating magnetar, there is a constant PA angle at each (spin) phase, while for the freely-precessing magnetar, a range of PA values are expected at each (precession) phase (Fig. 5, upper panels). Moreover, for a fixed precession phase, only certain spin phases will lead to an observed burst (Fig. 5, middle panels). More accurate PA measurements from high S/N bursts would be able to distinguish between these two models (or rule out both).
The precession model also requires fixed emission region, and a small timescale periodicity due to the rotation period of the magnetar. This short periodicity may be concealed by various timing noises common in young magnetars. However, if a short period is found, the precession model predicts a change of the short periodicity duty cycle with the NS precession phase. For instance, in Figure 5 central middle panel, the spin duty cycle appears smaller at the edge of each active (precession) phase. Moreover, the active window in the spin phase is changing against frequency with this setup of asymmetric emission against magnetic pole.
To get the same activity windows, the free-precession model requires a smaller emission region than the slowlyrotating magnetar model. This is because for each spin phase of the slowly-rotating magnetar, the LOS points at a specific location on the NS surface, while for each precession phase of the freely-precessing magnetar, the LOS rotates around the NS rotation axis. Distinguishing between these two models would constrain the FRB emission region altitude and angular size.
Recently, Katz (2020) discussed how detecting a change in the FRB period can further differentiate between dynamical theories. We note the freely-precessing magnetar model expects a much larger period change than the slowly-rotating magnetar model, since the spin-down timescale is much shorter in the former model due to the faster rotation frequency (e.g. Zanazzi & Lai 2020).

SUMMARY AND CONCLUSIONS
In this paper, we construct a phenomenological model of Fast Radio Burst (FRB) emission from rotating magnetars to test periodic FRB theories, in light of recent measurements of small polarization Position Angle (PA) swings during and between bursts, as well as the narrowing and phase shift of burst activity windows with frequency. Our model assumes burst are emitted from a region anchored into the magnetar and offset from the dipole axis, with region size and dipole offset angle increasing with a decrease in frequency, due to the altitude dependence of curvature radiation on frequency ( §2.1). The model PA values are given by the rotating vector model ( §2.2).
Using this model, we constrain three separate dynamical models which have been invoked to explain the 16.3 day periodicity of FRB 180916: a magnetar with a slow rotation period ( §3.1), a magnetar undergoing free precession ( §3.2), and a magnetar undergoing forced precession due to an external body ( §3.3). We find the slowly-rotating and freely-precessing magnetar models can produce PA swings and frequency-dependent activity windows consistent with recent observations, but the magnetar undergoing forced precession is disfavored as a dynamical model, due to its inability to produce a frequency-dependent phase drift of the burst activity window ( §4). Future observations are necessary to distinguish between the remaining theories, and understand what is causing the periodicity of FRB 180916 ( §6)