Angular dependence of coherent radio emission from magnetars with multipolar magnetic fields

ABSTRACT

1 INTRODUCTION

Highly magnetized neutron stars (NSs), in short magnetars (Duncan & Thompson 1992), have magnetic field strengths above the Schwinger limit |$B_{\rm cr}\equiv m_{\rm e}^2c^3/(\hbar e)\sim 4.4\times 10^{13}$| G and slow spin periods of about seconds, and emit recurrent X-ray bursts, which are short in duration but extremely energetic events (Kaspi & Beloborodov 2017; Enoto, Kisaka & Shibata 2019). Magnetic field strengths of magnetar sources are generally inferred assuming that the NS loses its energy through the magnetic dipole radiation. Namely, the magnetic field estimates using their spin periods and spin-down rates are the indicators of their dipole field strengths. However, magnetars likely possess multipolar field structures, which could play important roles in producing energetic X-ray bursts. For example, the inferred magnetic field strength of SGR 0418+5729 is only 6 × 10¹² G (Rea et al. 2013). However, Güver, Göǧüş & Özel (2011) investigated the persistent X-ray spectrum of this source with a physically motivated spectral model and determined its surface magnetic field averaged over the spin cycle as 10¹⁴ G (see also Mondal (2021) for the similar result independently obtained based on the modelling of pulsar spin down). They concluded that the stronger fields reside in multipolar components, and likely give rise to energetic bursts. By performing a spin phase-resolved spectral investigation of the same data set, Tiengo et al. (2013) found a variable spectral absorption feature in a fraction of the spin cycle. As interpreted as a proton cyclotron feature, the corresponding magnetic field strength was found to be above 2 × 10¹⁴ G, implying the presence of multipolar magnetic topology in a magnetar. Furthermore, although not a magnetar, recent modelling of NICER observations of thermal X-ray pulses from a radio pulsar PSR J0030+0451 has shown a strong preference for high-temperature surface regions created by non-dipolar magnetic field configurations (Bilous et al. 2019; Riley et al. 2019; Kalapotharakos et al. 2021). Recent advanced numerical simulations by Igoshev et al. (2021) also indicate the presence of complicated magnetic field structure of magnetars.

With magnetic field strengths above B_cr, magnetars have the necessary energy budget and ambient conditions to operate the coherent mechanisms of fast radio bursts (FRBs; Lorimer et al. 2007; Thornton et al. 2013), short-duration (typically a few milliseconds), intense (peak flux densities up to about 100 Jy), coherent (brightness temperatures of ∼10³⁵ K) signals observed in radio bands (GHz), even from Gpc distances. Therefore, the currently popular models to elucidate FRBs generally invoke NSs with magnetar-type field strengths (see e.g. Beloborodov 2020). A Galactic magnetar SGR J1935+2154 has entered into a burst active episode on 2020 April 27, emitting hundreds of energetic X-ray bursts in a day (Palmer 2020). During this intense burst activity phase, a bright radio burst from this magnetar was detected independently with two radio facilities: CHIME (CHIME/FRB Collaboration 2020) and STARE2 (Bochenek et al. 2020b). The detection of FRBs (Bochenek et al. 2020a; CHIME/FRB Collaboration 2020) associated with the X-ray bursts (Mereghetti et al. 2020; Li et al. 2021a; Ridnaia et al. 2021; Tavani et al. 2021) from the Galactic magnetar SGR J1935+2154 establishes that at least some FRBs originate from magnetars.

Recent observations of repeating source FRB 20121102A revealed that the distribution of burst waiting times is bimodal, with peaks at ∼3 ms and ∼70 s (Hewitt et al. 2021; Li et al. 2021b). The existence of a burst population with such a short waiting time may favour the magnetospheric origin of bursts (Li et al. 2021b). Among these, there are models for FRBs in which some sort of magnetic disturbance near the NS, such as stellar quake-driven Alfvén waves (Katz 2016; Ghisellini 2017; Kumar, Lu & Bhattacharya 2017; Yang & Zhang 2018; Lu, Kumar & Zhang 2020), trigger coherent curvature radiation along open magnetic field lines in the dipolar magnetosphere. Meanwhile, as evidenced by observations, the NS population generally could possess multipolar field components, which can affect emission processes occurring near the star and may result in the modified angular characteristics of coherent radio emission from magnetars. For instance, Yang & Zhang (2021) recently showed schematically that the diversity in the burst occurrence rate of FRBs might have to do with the potential existence of multipolar field geometry near the star in the conjuncture of the Galactic FRB-like bursts from SGR 1935+2154 (see also Wang et al. 2022). Since the presence of such multipolar fields might significantly affect the curvature of the magnetospheric field lines in the emission zone, it must be taken into account when modelling and interpreting the observed emission.

In this work, we show with simple but realistic multipolar magnetic field configurations how angular characteristics for observing transient coherent radio emission (including FRBs and transient pulsed radio emission) from magnetars are modified assuming that they could arise from the curvature emission along the open multipolar magnetic fields. In particular, we consider the ‘quadrudipolar’ field configuration in which the quadrupolar component is superimposed on to the dipolar component with an arbitrary field strength ratio and inclination angle between two components (Barnard & Arons 1982, hereafter BA82). In general, a quadrudipolar field configuration yields two open field line regions at each pole. The open field line region in the Northern hemisphere (or ‘polar cap’) has a nearly circular shape, whereas that in the Southern hemisphere exhibits a thin annular shape when dipolar and quadrupolar moments are aligned, and it could be much more complicated when they are misaligned (BA82; see also Gralla, Lupsasca & Philippov 2017 for detailed numerical simulation of an aligned quadrupole). As a first step towards understanding the general field configuration, we consider the radio emission from the Northern¹ polar cap, which has a larger area at the stellar surface than the Southern one and presumably a higher ability to generate observable radio emission (Gralla et al. 2017; Lockhart et al. 2019). Although our ultimate goal is to study transient radio emissions from magnetars using the force-free field, which could approximate the magnetic field of the plasma-filled magnetosphere, in this paper we concentrate on the qualitative investigation of angular characteristics of radio beams using a vacuum magnetic field only.

This paper is organized as follows. In Section 2, we present the quadrudipolar field configurations that we use. In Section 3, we discuss the implications of quadrudipole for FRBs, and in turn for transient radio pulses from magnetars. Finally, we conclude by summarizing our findings and prospects in Section 4.

2 MODIFICATION OF MAGNETIC GEOMETRY BY QUADRUPOLE FIELDS

We consider the emission geometry from an NS by solving the viewing geometry in an inclined dipolar and quadrupolar magnetic field following BA82. We will use the vacuum field as an approximation to the magnetic field of the plasma-filled magnetosphere, where the magnetospheric structure is stationary in the corotating frame. Important model parameters are the surface field strength ratio between quadrupole and dipole f_Q ≡ B_Q/B_D and the inclination angle between dipole and quadrupole moments i_QD.

2.1 Field geometry

Figure 1.

Polar flux tubes for different inclinations with f_Q ≡ B_Q/B_D = 10 (left) and i_QD = 80° (right). The assumed parameter for the NS is x_LC ≡ R_LC/R_* = 2 × 10⁴ (i.e. P ∼ 4.2 s), where R_* = 10⁶ cm. The expected geometry significantly deviates except for the aligned quadrupole and dipole case (i_QD = 0°) in the left-hand panel, which is almost identical to a pure dipole one. The case of f_Q = 10 and i_QD = 80° (reddish curves) is identical between the left- and right-hand panels for reference.

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2.2 Beam characteristics

Here, we discuss the angular characteristics of radio emission. Fig. 2 shows the polar angle of the last open field lines, θ_B, and the angular width between them, W, as a function of radius r from the NS for different field geometries. There are two effects of including the quadrupolar component: significant deviations of (i) the polar angle of the emission and (ii) the width of the flux tube (i.e. the radio beam) from those in a pure dipolar field geometry.

Figure 2.

Top: colatitude of the last open field lines as a function of distance from the star for a pure dipolar geometry (dashed lines) and for quadrudipolar geometries with i_QD = 40° (left) and i_QD = 80° (right) when varying f_Q ≡ B_Q/B_D from 4 to 400 (solid lines). In the extreme limit of f_Q → 0, the quadrudipole asymptotically approaches the pure dipole. Bottom: the beamwidth W, defined by the angular extent of the last open field lines, as a function of radius. The assumed parameters for the NS here are the same as in Fig. 1.

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First, the polar angle of the quadrudipolar field has a stronger dependence on r than that of a dipole. For a pure dipolar case, the polar angle monotonically increases with r, and thus there is always a single solution of θ_B(r) = θ_obs for r where θ_obs is the observer viewing angle measured from the North Pole. For a quadrudipolar case, on the other hand, each field line has more than one solution at which the local field line is directed towards the observer, satisfying θ_B(r_i) = θ_obs (see top panels of Fig. 2). Specifically, a field line passing through the equatorial point at θ = π/2 (θ = −π/2) could have at most two (three) solutions. Since field lines at locations corresponding to such solutions have different curvatures, this may lead to complex temporal and spectral radio pulse profiles.

Second, multipolar fields generally have much narrower opening angles at the surface than dipole fields (BA82). However, this does not necessarily mean that the chance probability of seeing radio emission from the multipolar fields is lower than that from pure dipole. In the case of quadrudipolar fields, the angular extent between the polar flux tubes near the point where a sign of curvature changes r/R_* ∼ f_Q could be even larger than the dipolar one for high inclinations i_QD ≳ 50° (see the bottom right-hand panel of Fig. 2). Moreover, as a polar angle of quadrupolar field lines is a strong function of radius it could have a larger coverage of the sky below a given radius than a purely dipolar one.

Special relativistic effects, such as the aberration of photon emission direction and photon traveltime delay, could be in principle important near the light cylinder, where the structure of the last open field lines is modified (e.g. Yadigaroglu 1997). However, it does not significantly change the structure of the inner magnetosphere (at radii below |$f_\mathrm{ Q} \, R_\ast \sim R_{\rm LC}/100$| for P = 1 s magnetars), where the quadrupole plays important roles.

3 IMPLICATIONS FOR TRANSIENT RADIO EMISSIONS FROM MAGNETARS

Now we turn to discuss the implications of multipolar fields for the coherent radio emission from magnetars, including FRBs and transient radio pulsations from galactic magnetars (Camilo et al. 2006, 2007), in the framework of curvature emission from multipolar field geometries.⁴ The curvature emission frequency is estimated as ν∝Γ³/R_c, where Γ is the Lorentz factor of electrons in bunch and R_c is the curvature radius. Although the demand that the emitted frequency be in the radio band (∼GHz) constrains Γ and R_c, a full determination of them requires self-consistent modelling of particle generation and acceleration (e.g. Ghisellini 2017; Kumar et al. 2017; Yang & Zhang 2018; Kumar & Bošnjak 2020; Lu et al. 2020), which remains unclear (see e.g. Lyubarsky 2021 for criticisms). Instead, we consider only the geometrical effects of magnetic fields on the emission direction assuming that coherent radio emission is successfully generated at some radius.

The observed emission is not isotropic, but it is beamed within some fraction of the solid angle 4π. In case of quadrudipolar field structure, the maximal time-scale for the observer’s line of sight (LOS) to sweep the beam is typically |$T_{\rm beam}=PW/(2\pi)\sim 110 {\ \rm ms}\ (P/4{\, \rm s})(W/{\cal O}\,(10{\, \rm deg}))$|⁠, where W, of course, depends on r, f_Q, and i_QD (see the bottom panels of Fig. 2).

The trigger time-scale is also important in determining the observed duration of coherent emission t_obs. FRBs are believed to be generated by a series of instantaneous and stochastic triggers such as occasional starquakes. Namely, FRBs would be observed only when the magnetic field lines (or the beam), along which instantaneous generation and acceleration of charged particles take place, cross the observer’s LOS. Meanwhile, the radio pulsations from magnetars, although highly variable, could be almost steadily triggered during the active period of magnetars. Therefore, the observed duration of FRBs is controlled by the intrinsic trigger time-scale t_obs = t_int/(1 + z) (with z being the source redshift), whereas the observed pulse width of radio pulsation from magnetars is determined by the geometric beam size t_obs = T_beam when neglecting the effects of the potential substructure inside the beam and the impact parameter of the LOS with respect to the beam centre.

3.1 Fast radio bursts

In the presence of a quadrudipolar field, a polar angle of field line is a strong function of r. This means that even a small variation in emission altitude could lead to the large variation in the observed emission angle. This would naturally give rise to the highly sporadic burst rate of repeating FRBs.

In most FRBs, the time–frequency structure exhibits a downward drifting pattern, i.e. the subpulses arriving later have lower frequencies with some exceptions (e.g. Pleunis et al. 2021). Theoretical models for frequency drift patterns are already complex even in a pure dipolar case (e.g. Wang et al. 2019b, 2020; Lyutikov 2020). Including the quadrupole gives further freedom in generating more complex frequency drift patterns, which may explain the observed diversity of frequency drift. Also, the inclusion of a multipolar field should alter the radius-to-frequency mapping that assumes a pure dipole (e.g. Lyutikov 2020; Wang et al. 2022) due to the non-monotonic behaviour of the field line curvature.

3.2 Transient radio pulsation from magnetars

To date, coherent radio pulsations have been detected from six⁶ (including SGR 1935+2154, the source of the Galactic FRB) of the 30 currently known magnetars (Olausen & Kaspi 2014).⁷ All of the six sources are transients, and the radio emission occurred preferentially when they exhibited bursting/flaring activities in X-rays. Although the origins of transient radio pulses of magnetars remain unclear, it has also been proposed that they may be generated in the closed magnetic field region of a dipolar configuration (Wadiasingh & Timokhin 2019; Wang et al. 2019a). Such models invoking acceleration along closed field lines are partly motivated by the fact that the angular size of open field lines in a pure dipole tends to be extremely small for magnetars rotating slowly (beam size of pure dipole scales with ∝P^−1/2) while that of closed field lines could be arbitrarily large. Another motivation comes from the fact that the relatively hard spectra of magnetar radio pulsation extending up to a few tens of GHz are better explained by the emission from closed field lines that are more curved than open field lines.

Alternatively, our results imply that magnetar radio pulsations may arise from the open field region modified by multipolar components, as suggested by early polarization observations (Kramer et al. 2007), for the following reasons. First, as shown in Section 2.2, open field lines of a quadrudipole with high obliquity could have a wider beam size at a given radius and cover a larger fraction of the sky below a given radius than those of a pure dipole do (the latter point is critical since it significantly increases the effective beam size). Indeed, the duty cycle of radio pulsation from a magnetar XTE J1810−197 with a spin period of 5.5 s (Ibrahim et al. 2004) is ∼5 per cent (translated in beam size of W ≳ 18°) at 2–8 GHz range (Camilo et al. 2016; Eie et al. 2021), which is comparable to the duty cycle of ordinary millisecond pulsars (Maciesiak, Gil & Ribeiro 2011). In this respect, the generation of radio emission in the open field region with multipolar configuration could also be a viable option to have a wide enough beamwidth. Secondly, the open field lines modified by a multipolar field component naturally give rise to the hard spectra of magnetar radio pulsations since their curvature could be effectively smaller than that of a pure dipole if Γ is constant along the field lines. Finally, the magnetar radio pulsation is known to be highly variable in time, which may imply that the physical condition Γ and R_c could change dramatically from one pulse to another. For the same reason, we predict that the highly complicated beam structure would result in a significant dispersion in the peak position of the radio pulse profile.

4 CONCLUSION

In this work, the magnetic field geometry of an inclined dipolar and quadrupolar magnetic field is considered based on an analytic model by BA82 in conjuncture with coherent radio emission from magnetars. While we do not attempt to explain the phenomena at the emission level, we demonstrate that a consideration of a multipolar component could give considerable freedom in interpreting/modelling observations of a family of coherent emissions from magnetars, FRBs, and transient radio pulsations.

We analytically demonstrate that the curvature of the open field lines can vary significantly depending on both the ratio of quadrupole to dipole field strength and their inclination angle at the NS surface. This means that there are multiple points along each magnetic field line that coincide with the observer’s line of sight, and may explain the complex spectral and temporal structure of the observed FRBs. This also implies that even a small variation in emission altitude could result in a large variation in the observed emission angle, leading to a highly sporadic burst rate of repeating FRBs. It is also found that in quadrudipole, the radio beam can take a wider angular range and the beamwidth can be wider than that in pure dipole. This may explain why the pulse width of the transient radio pulsation from magnetars is as large as that of ordinary radio pulsars.

The fact that the magnetar SGR 1935+2154, which is the source of Galactic FRB 200428, has now turned into a radio pulsar (Zhu et al. 2020) makes it interesting to continue monitoring it with high-sensitivity radio telescopes. Although testing whether non-dipolar field components are involved only from radio observations may be difficult, it is important for analysis of radio emissions from magnetars to keep in mind that the magnetic field topology may well deviate from the simple dipole model.

ACKNOWLEDGEMENTS

We thank Shota Kisaka for useful comments and discussion. We also thank the anonymous referee for their careful reading of the manuscript and suggestions. SY was supported by the advanced ERC grant TReX. KYE acknowledges support from TÜBİTAK with grant number 118F028.

DATA AVAILABILITY

This is a theoretical paper that does not involve any new data. The model data presented in this article are all reproducible.

Footnotes

REFERENCES