Radius-to-frequency Mapping and FRB Frequency Drifts

An important concept is the observer time—a time measured from the arrival of the first emitted signal (see, e.g., models of gamma-ray bursts; Piran 2004). In our case, both the relativistic motion and the curved trajectory strongly affect the relation between the coordinate time, t, and the observer time, t_ob. To separate effects of rotation from the propagation, we first consider stationary magnetospheres.

Let us assume that at time t = 0, an emission front is launched from radius, r₀, propagating with velocity βc along the local magnetic field; Figure 1. Thus, we assume that the whole of the magnetosphere starts to produce emission instantaneously. The trigger could be, e.g., an onset of magnetospheric reconnection event (Lyutikov 2006, 2015). A reconnection event that encompasses the whole region near the surface of the neutron star will be seen at some distance away as a coherent large-scale event. If only a patch of the magnetosphere produces an emission, the light curves will be truncated accordingly. Given that we already have a number of model parameters, we did not explore the finite size of the emission regions in the r–θ–ϕ space.

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Let the observer be located at an angle θ_ob, measured from the instantaneous direction of the magnetic dipole. Angle θ_ob defines a field line with a tangent (the magnetic field) along the LOS at the radius r₀. That field line can be defined by the angle θ₀ of the magnetic foot point. At time t the photons emitted from the surface at t = 0 propagated a distance ct. It is assumed that at each point emission is produced along the LOS (solid points and arrows in Figure 1). We assume that at given location r_em the emission front produces coherent emission at a frequency $\omega ({r}_{\mathrm{em}})$. (We neglect the fact that in a dipolar magnetosphere the strength of the magnetic field at given radius varies by a factor of 2 depending on the magnetic latitude.) As the emission front propagates in the magnetosphere, different magnetic field lines contribute to the observed emission. At time t emission point is located at distance r_em and the polar angle θ_em. Emission from the point r_em arrives at the observer at time t_ob that depends on (i) the emission time, (ii) the velocity of the emission front, and (iii) the geometry of field lines.

For a field line parametrized by polar angle θ₀ at r₀, a distance along the field line from θ₀ to θ > θ₀ is

$$\begin{eqnarray}&&{\rm{\Delta }}s=\displaystyle \frac{{r}_{0}}{{\sin }^{2}{\theta }_{0}}{\int }_{{\theta }_{0}}^{\theta }\sqrt{1+3{\cos }^{2}\theta }\sin \theta d\theta .\end{eqnarray} \tag{ 3 }$$

Emitting particles move according to

$$\begin{eqnarray}&&{\rm{\Delta }}s=\beta t,\end{eqnarray} \tag{ 4 }$$

where t is the coordinate time.

The points θ_em in the dipolar magnetosphere that have magnetic field along the LOS satisfy

$$\begin{eqnarray}&&\cos 2{\theta }_{\mathrm{em}}=\displaystyle \frac{1}{6}\left(\sqrt{2}\cos {\theta }_{\mathrm{ob}}\sqrt{\cos \left(2{\theta }_{\mathrm{ob}}\right)+17}+\cos \left(2{\theta }_{\mathrm{ob}}\right)-1\right).\,\end{eqnarray} \tag{ 5 }$$

As Figure 1 demonstrates, the observer time is given by (speed of light is set to unity)

$$\begin{eqnarray}&&{t}_{\mathrm{ob}}=t-({r}_{\mathrm{em}}-{r}_{0})\cos \left({\theta }_{\mathrm{ob}}-{\theta }_{\mathrm{em}}\right)\end{eqnarray} \tag{ 6 }$$

with r_em(t) given by the requirement that particles propagating along the curved field with velocity β emit along the local magnetic field. For nearly straight field lines and highly relativistic velocity, $\beta \approx 1-1/(2{{\rm{\Gamma }}}^{2})$, the effects of field line curvature dominate for θ_ob − θ_em ≥ 1/Γ. (In the calculations below, the time is normalized to unites r₀/c, where r₀ is some initial radius, not necessarily the neutron star radius.)

We then implement the following procedure (see Figure 1):

• 1.  
  Given is the observer angle θ_ob (with respect to the magnetic dipole).
• 2.  
  Find the polar angle of the footprint ${\theta }_{0}^{(0)}$ by solving Equation (5) and setting ${\theta }_{\mathrm{em}}={\theta }_{0}^{(0)}$. (Superscript ⁽⁰⁾ indicates the moment t = 0).
• 3.  
  After time t the emission front moved along the field lines according to Equations (3) and (4), where θ₀(t) is a parameter for the field line emitting at time t (at t = 0 we have ${\theta }_{0}(0)={\theta }_{0}^{(0)}$).
• 4.  
  For t ≥ 0, using Equations (3)–(5) with θ = θ_em, find the polar angle of the foot point θ₀, Equation (5), where magnetic field is along the LOS at time t.
• 5.  
  Using Equation (4) find θ₀—the polar angle of the field line that produces emission at time t.
• 6.  
  Given time t and the location of the emission point, we can calculate the observer time; see Equation (6).
• 7.  
  We then find dependence of r_em versus t_ob.
• 8.  
  Assuming some ω(r_em), we find the radius-to-frequency mapping ω(t_ob).

Thus, we take into account relativistic transformations and curvature of the field lines in calculating the relations between the observer time versus the coordinate time. We implicitly assume that emitted frequency is the function of the emission radius, ω(r_em), but given our uncertainty about the emission properties we do not specify a particulate dependence ω(r_em). We plot curves for generic profiles $\omega \propto {r}_{\mathrm{em}}^{-1}$ and $\omega \propto {r}_{\mathrm{em}}^{-3}$; the last scaling is expected if the emission is linearly related to the local magnetic field.

The velocity of the emitting front has a complicated effect on the overall duration of the observed pulse and a range of emitted frequencies. For subrelativistic velocities ${\rm{\Delta }}{r}_{\mathrm{em}}\sim \beta {\rm{\Delta }}t$ and ${t}_{\mathrm{ob}}\sim {\rm{\Delta }}t$. As $\beta \to 1$, the observed duration shortens, t_ob ≪ Δt. But for sufficiently high velocity, β ≈ 1 with θ_em − θ_ob ≥ 1/Γ, this relativistic LOS contraction become unimportant, as the observed duration is determined by the curvature of field lines.

In Figure 2, we implement the procedure described above showing r_em(t_ob) for the extreme relativistic limit of β = 1. This figure demonstrates that the effects of magnetic field line curvature can dominate over the relativistic effects along the LOS.

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For assumed scaling ω(r_em) $\omega \propto {r}_{\mathrm{em}}^{-1}$ and $\omega \propto {r}_{\mathrm{em}}^{-3}$, the corresponding curves ω(t_ob) are given in Figure 3. The model generally reproduces approximate linear drifts rates (see, e.g., Josephy et al. 2019, their Figure 6), regardless of the particular power-law dependence ω(r_em). We consider this as a major success of the model.

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We assume next that the star is rotating with spin frequency Ω. The magnetic polar angle of the LOS at time t in the pulsar frame is then

$$\begin{eqnarray}&&\cos {\theta }_{\mathrm{ob}}=\cos \alpha \cos {\theta }_{\mathrm{ob}}^{(0)}+\sin \alpha \sin {\theta }_{\mathrm{ob}}^{(0)}\cos ({\rm{\Delta }}\phi +t{\rm{\Omega }}),\end{eqnarray} \tag{ 7 }$$

where α is the inclination angle between rotational axis and magnetic moment; ${\theta }_{\mathrm{ob}}^{(0)}$ is the observer angle in the plane comprising vectors of Ω, μ and the LOS; and Δϕ is the azimuthal angle of the observer with respect to the Ω–μ plane when the injection starts: this is the phase at t = 0. (For example, if emission starts when the LOS is in the Ω–μ plane, at that moment ${\theta }_{\mathrm{ob}}=\alpha -{\theta }_{\mathrm{ob}}^{(0)}$.)

We then implement the procedure outlined in Section 1, with the following modifications; see Figure 4:

• 1.  
  Given are the ${\theta }_{\mathrm{ob}}^{(0)}$, α, β, Ω, and Δϕ.
• 2.  
  For t ≥ 0, implement procedure of Section 1 with time-dependent θ_ob given by Equation (7).

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2.3.1. Frequency Drifts in Rotating Magnetosphere

In the rotating magnetospheres, the observed frequency drifts are generally more complicated, as the LOS samples larger part of the magnetosphere. A key limitation in the approach is that we assume that the whole surface r = 0 produced as emission front—thus, different parts of the emission front maybe casually disconnected—under certain circumstances this leads to unphysical results (e.g., upward frequency drifts).

In Figure 5, we plot emission radius as function of observer time for different pulsar spins. It is clear that for a given intrinsic burst duration larger Ω produce emission that is seen for a longer observer time. This is due to the fact that the LOS samples larger angular range and correspondingly larger differences in the LOS advances of the emitting region.

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In Figures 6 and 7, we show the corresponding frequency drifts for selected parameters. As is clear from the plots, the evolution of the peak frequency can be more complicated in the rotation magnetospheres, as the LOS samples large variations of plasma parameters. (The dimensionless spin Ω = π/2 in Figure 6 is relatively high; smaller Ω produce more linear scalings).

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Importantly, depending on the trigger phase Δϕ the same object will produce different r_em(t_ob) curves; see Figure 8. This explains why in the Repeaters FRB 121102 different burst have different drifts (Hessels et al. 2019). The fact that different parts of the magnetar magnetosphere can become active also explains the lack of periodicity in repeating FRBs.

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2.3.2. Prediction: Polarization Swings

Polarization behavior of FBRs is, arguably, the most confusing overall (Masui et al. 2015; Petroff et al. 2015; Caleb et al. 2019; Petroff et al. 2019). We are not interested here in the propagation effects (e.g., sometimes huge and sometimes not rotation measure). There is a clear repeated detection of linear polarization. Importantly, FRBs have thus far not shown large polarization angle swings (Petroff et al. 2019).

The present model does not address the origin of polarization, as it would depend on the particular coherent emission mechanism. On basic grounds, polarization is likely to be determined by the local magnetic field within the magnetosphere. The model then does predict polarization angle swings. In the rotating vector model (RVM; Radhakrishnan & Cooke 1969) polarization swings reflect a local direction of the magnetic field at the emission point. In our notations, the position angle of polarization χ is given by

$$\begin{eqnarray}&&\tan \chi =\displaystyle \frac{\sin {\theta }_{\mathrm{ob}}^{(0)}\sin ({\rm{\Omega }}t+{\rm{\Delta }}\phi )}{\cos \alpha \sin {\theta }_{\mathrm{ob}}^{(0)}\cos ({\rm{\Omega }}t+{\rm{\Delta }}\phi )-\sin \alpha \cos {\theta }_{\mathrm{ob}}^{(0)}}.\end{eqnarray} \tag{ 8 }$$

Generally, we do expect polarization swings through the pulse; see Figure 9. Qualitatively, the fastest rate of change of the position angle occurs when the LOS passes close to the magnetic axis; this requires $\alpha \approx {\theta }_{\mathrm{ob}}^{(0)}$ (so that the denominator comes close to zero). This is the case for rotationally powered pulsars. If emission is generated far from the magnetic axis, the expected PA swings are smaller. Thus, we do predict that PA swings will be observed within the pulses, but with values smaller than the ones seen in radio pulsars.

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