Fast Radio Bursts as Crustal Dynamical Events Induced by Magnetic Field Evolution in Young Magnetars

J.E.Horvath ,P.H.R.S.Moraes,3 ,M.G.B.de Avellar ,and L.S.Rocha

1 Universidade de S?o Paulo(USP),Instituto de Astronomia,Geofísica e Ciências Atmosféricas,Rua do Mat?o 1226,Cidade Universitária,05508-090 S?o Paulo,SP,Brazil;foton@iag.usp.br

2 Universidade Federal de S?o Paulo(UNIFESP),Instituto de Ciências Ambientais,Químicas e Farmacêuticas,Rua S?o Nicolau 210,09913-030 Diadema,SP,Brazil

3 Universidade Federal do ABC (UFABC),Centro de Ciências Naturais e Humanas (CCNH),Avenida dos Estados 5001,09210-580,Santo André,SP,Brazil

Abstract We revisit in this work a model for repeating Fast Radio Bursts based of the release of energy provoked by the magnetic field dynamics affecting a magnetar’s crust.We address the basics of such a model by solving the propagation of the perturbation approximately,and quantify the energetics and the radiation by bunches of charges in the so-called charge starved region in the magnetosphere.The(almost)simultaneous emission of newly detected X-rays from SGR 1935+2154 is tentatively associated with a reconnection behind the propagation.The strength of f-mode gravitational radiation excited by the event is quantified,and more detailed studies of the nonlinear(spiky)soliton solutions are suggested.

Key words:stars:magnetars–relativistic processes–radiation mechanisms:non-thermal

1.Introduction

The last decade has witnessed the emergence of the study of a class of transients which keep the community very active in search of answers.As a relatively recently observed phenomenon,Fast Radio Bursts(FRBs)are bright pulses of emission at radio frequencies which last for～ms or even less (Katz 2018;Popov et al.2018;Petroff et al.2019).Their emission has been detected in the interval 400 MHz–8 GHz,considered as“typical,” with at least one case in which slightly delayed X-ray bursts coincide with the radio spikes (Mereghetti et al.2020).

The first ever detected FRB accepted as so,later named FRB 010724,was discovered by Lorimer et al.(2007)in surveys of radio pulsars (Beskin 2018).For years,such a discovery remained the only known signal of its kind.Strong support for a short-duration transient with characteristics similar to the burst reported in Lorimer et al.(2007)only became available in 2013,when Thornton et al.(2013) reported the observation of four high-dispersion measure pulses with the Parkes Telescope facility (Keith et al.2010).

Over the years,more and more FRBs have been detected,not only by Parkes Telescope,but also by Arecibo Observatory(Spitler et al.2014),Australian Square Kilometre Array Pathfinder (Bannister et al.2017),Canadian Hydrogen Intensity Mapping Experiment (CHIME/FRB Collaboration 2019a) and others.The high values of the dispersion measure of FRBs mentioned above strongly indicate that they have extragalactic or cosmological origin,a conjecture which is indeed confirmed by most FRB observations.It became clear later that FRBs are quite luminous outbursts,with luminosity L～1043erg s-1if arising from extragalactic sources,and much consideration and interest have been given to them,among other reasons due to the amazing mechanism behind the progenitor systems of such extreme events that must be operating.

In 2016 the first repeating FRB was observed,FRB 121102(Spitler et al.2016).Possibly,repeating FRBs come from an entirely different source class or progenitor system compared to the non-repeating population,if the latter indeed exists.It is not known whether all sources repeat,because the repeating times may be long,thus the need for some so-called “cataclysmic”models of the non-repeating sources is not yet compelling.Some models able to explain repeating FRBs,such as FRB 121102 and also the later reported FRB 180814.J0422+73(CHIME/FRB Collaboration 2019b),are:relativistic beams accelerated by impulsive magnetohydrodynamic driven mechanism,which interact with clouds at the center of star-forming dwarf galaxies (Vieyro et al.2017);soft-gamma repeaters(Wang &Yu 2017);starquakes of a magnetar (Wang et al.2018);mass transfer in a magnetic white dwarf and neutron star(NS) with strong bipolar magnetic field binary systems (Gu et al.2016),highly magnetized pulsars traveling through asteroid belts (Dai et al.2016) and binary NSs not far away from merging (Totani 2013;Wang et al.2016).Several other proposals can be seen,for instance,in Wadiasingh et al.(2020),Dai et al.(2016),Wadiasingh &Timokhin (2019),Gupta &Saini(2018),Michilli et al.(2018),Levin et al.(2020),Ioka&Zhang (2020),Kashihama &Murase (2017),a state that is heavily reminiscent of the situation of gamma-ray bursts(GRBs) in the decade before the 1990s (Nemiroff 1994) and suggests that more than one event is actually contributing to the FRB phenomenon.The issue of repeating versus non-repeating sources is actively under discussion.Given the recent detections and localizations of the non-repeating FRB 190824(Bannister et al.2019) and FRB 190523 (Ravi et al.2019),Gourdji et al.(2020) placed constraints on the magnetic field strength of the putative-emitting NS.In order to explain FRB 150418(Keane et al.2016),Zhang invoked a merging NS–NS binary producing an undetected short GRB and a supramassive NS,which subsequently collapses into a black hole(Zhang 2016b).Binary black holes were also considered,in spite of the general thought that electromagnetic signals should not emerge from these events (Liu et al.2016;Zhang 2016a).

If FRBs are indeed related to NSs and/or black holes,it is natural to consider them in current and future gravitational wave (GW) searches.Callister et al.(2016) demonstrated that Advanced Laser Interferometer Gravitational Wave Observatory(LIGO)and Virgo observations can severely constrain the validity of FRB binary coalescence models.FRBs can also provide an unprecedented tool for observational cosmology.In Caleb et al.(2019),for instance,FRBs were used for tracing the He II reionization epoch from simulations of their dispersion measures.Finally,in Wei et al.(2018) it was proposed that upgraded standard sirens can be constructed from the joint measurements of luminosity distances derived from GWs and dispersion measures derived from FRBs.Such an upgrade has been shown to be more effective for constraining cosmological parameters.These considerations are especially important for non-repeating,catastrophic event models of FRBs.

On the other hand,an important report on repeating FRBs associated with a Galactic magnetar source (Bochenek et al.2020) SGR 1935+2154 has confirmed that NSs with high magnetic fields are involved in the production of FRBs.The magnetar has been associated with the supernova remnant SNR G57.2+0.8 with age in excess of～104yr (Zhou et al.2020),although this may be misleading if the object inserted energy into the remnant at birth(Allen&Horvath 2004),because this makes the remnants look much older that they really are.However,the recent work of Zhou et al.(2020) did not find evidence in favor of a large energy injection.Independently of this last consideration,it is clear that a successful model for the production of FRBs in magnetars is needed.

We address here a model in which FRBs originate from NS crustal events induced by magnetic field evolution in young magnetars.As a supporting fact,it is worth mentioning that in Wang et al.(2018),the burst rates of FRB 121102 were shown to be consistent with a kind of seismicity rate.However,the scenario proposed in Wang et al.(2018)consists of a magnetar with a solid crust in which the stellar shape changes from oblate to spherical,which induces stresses in the crust,yielding a starquake.

We argue below that it is the behavior of the magnetic field,suffering from instabilities in the first few centuries after birth,that may produce (i) the free-energy source;(ii) the right timescales and (iii) the perturbations at the crust’s plates that suitably shake the field lines,sending Alfvén waves/solitons into the magnetosphere,resulting in the production of curvature radiation that may be the origin of the detected FRBs with the required frequency.The production of associated X-ray bursts and the possibility of detection of f-modes from the crust induced by these events are also briefly addressed.

2.Magnetic Field Evolution in Young Magnetars

The group of magnetar NSs,in which activity is supposed to arise from the behavior of magnetic fields,includes Soft-Gamma Repeaters and Anomalous X-Ray Pulsars (AXPs).A very high value of the magnetic field B has been inferred for them,in spite that in a few cases (notably AXPs) a much smaller field seems to be present(see Turolla&Esposito 2013 and references therein).Since a pure poloidal field is known to be unstable,it is suspected that a toroidal component acting as a reservoir of free-energy is also present.

A relevant detailed investigation for this picture has been performed by Gourgouliatos et al.(2016).They have shown that in a plausible scenario of energy equipartition between global-scale poloidal and toroidal magnetic field components,magnetic instabilities transfer energy to non-axisymmetric,subkm-sized features,in which local field strength can greatly exceed that of the global-scale field.Such intense small-scale magnetic features were demonstrated to induce high-energy bursts through local crust yielding.Essentially,it was shown that the observed diversity in magnetar behavior can be explained well with mixed poloidal-toroidal fields of comparable energies.

Neutron starquakes were considered as a model for gammaray bursts long ago (Pacini &Ruderman 1974;Epstein 1988;Blaes et al.1989).In such models,elastic energy is released in a crustquake,exciting oscillations of the surface magnetic field.The induced electric field accelerates high energy particles,which in turn radiate gamma-rays.The later discovery of the extragalactic character of the sources diminished the interest in quake models,now revived by the identification of SGR 1935+2154.Instead of the release of elastic energy,our scenario suggests the magnetic field to be the cause of crust breaking,and the propagation of Alfvén waves (ordinary or solitonic)along the field lines radiating the observed radio photons.

We start by the sort of basic energetic calculation that motivates the whole picture.Consider a solid NS crust in which a magnetic field of strength B evolves on sub-km-sized patches with length l.The magnetic energy is

where we have scaled the quantities as B15≡(B/1015G) and l4≡(l/104cm)(Gourgouliatos&Esposito 2018).Note that for extragalactic FRBs with energy～1040erg,if the energy ratio between radio and X-rays is～10-5,the energy reaches the giant flare level.This could be accommodated by changing B and l in the above equation.

On the other hand,the crust magnetoelastic energy is

with σ-3≡(σ/10-3) being the dimensionless stress modulus(Thompson&Duncan 1995).Detailed calculations have shown(Horowitz &Kadau 2009;Hoffman &Heyl 2012) that the crust cracks at a critical value ofso that from Equations (1) and (2) we see that the critical field enough to crack the crust is

Since the local field exceeds the global value by an order of magnitude (Gourgouliatos et al.2016),it is the magnetic field that drives the energetics.The instabilities will induce a“l(fā)ifting” of the cracked crust material of size l,typically displaced a distance Δl ?100σ-3l4cm.A more accurate calculation (Lander et al.2015) has employed a cracking condition based on the local von Mises criterion,i.e.,

and also attempted to predict where exactly the crust should crack.We refer to the work of Lander et al.(2015)and keep the simplest estimates in the following.

3.Transmission of Energy into the Magnetosphere:Linear and Nonlinear Regime Solutions

The Alfvén equation derived in the Appendix(Equation A10) has been studied before and,at least in its simplest form,admits the propagation of solutions for the perturbation ξ of relevance for the FRB problem.A brief description of this kind of propagating mode is in order.First of all,the amplitude of the Alfvén wave is estimated to be B/B15～10-4,therefore a linearization seems justified.The physical picture is that an electric current is carried along the field lines by electrons and positrons moving in opposite directions.Bunches of particles with approximately zero electric charge are formed,and along the propagation the velocity must speed up to compensate the decreasing density,as required to comply with the current density needed to sustain the wave.When the plasma density in the magnetosphere falls below a critical value,the propagation can no longer be supported,and a large electric field develops to compensate via the displacement current the decrease of the current density.This electric field boosts electrons and positrons to high Lorentz factors in opposite directions along the field lines.This is when the synchro-curvature radiations of the bunches,“summed up”in the quantity ξ,are produced,as discussed,for example,by Kumar et al.(2017).Therefore we must solve the propagation of ξ to calculate the emerging radiation.

We shall not treat the behavior in the crust because in our picture of magnetoquakes the instability of the field would provoke a lifting of a crust plate with a speed vpin which the field lines are frozen.Therefore,we shall focus on the magnetosphere propagation z ＞0.Recalling the assumptions made when we wrote down Equation(A10),in the region z ＞0 evidently μ=0,and we can also neglect the density ρ compared to the B2term (by the very nature of a magnetosphere,Blaes et al.1989).We shall further assumecosαto be constant,independent of z to allow an analytical solution.(We have checked that an expected dependence on z of a dipolar B field does not change the picture much.The latter case can be solved and the solutions happen to be linear combinations of the first-order Bessel functions H1and Y1,with a slightly decreasing amplitude for large z.)The equation to be solved in this simplified case is just

Once the Green function (having two contributions G0and G1) is found,the solution for ξ reads

Imposing the appropriate initial conditions and transforming to the Fourier spacethe transformed amplitude g(k,t) is the solution of the equationthat isEnforcing the two sets of initial conditions

which is easily integrated using the symmetry k →-k to yield

with signx being the signum function of the argument.Now the convolution of Equation(8)with the perturbation having ξ0=0 makes the G0(z,t) unnecessary,and we are left with the propagating solution

This is a very simple solution for the perturbation ξ traveling along the field lines,but a particularly suitable one because a sudden,unitary perturbation is physically invoked as the origin of the propagating ξ as stated above.The charges traveling inside ξ could,in principle,radiate photons by synchrotron and curvature mechanisms as explained above.The general case,dubbed synchro-curvature radiation embedding both effects is,however,reduced to the curvature contribution in our case because high magnetic fields are known to cool very efficiently,reducing the radiative and conductive opacities κradand κcond(Istomin&Sobyanin 2007).For a single electron the curvature radiation is (Xiao et al.2021)

where Rcis the curvature radius,ωc(3/2)(γ3c/Rc) a characteristic curvature frequency and K5/3is the modified Bessel function of second kind.The contribution to the total radiated power of the charges in the perturbation is essentially the coherent contribution of the charges traveling inside the perturbation ξ.

The radiation spectrum can be evaluated using Equation(14)and the solution Equation (13).On very general grounds,it is quite possible to understand the main features.It is well-known that the Bessel function K5/3admits an integral representationcoshαt dt,and this form immediately shows that the main contribution is given for low values of the dummy argument μ ≈1.In fact,it is enough to use for evaluating the approximate form Kα→(Γ(α)/2)(2/μ)αvalid for0The result is

where the coherence signature,N2is the result of evaluating the so-called phase stacking integral at the relevant frequencies(Xiao et al.2021).Since the perturbation has a size of～10 m and the charge bunches are much smaller,many of the latter may be considered to contribute,although Equation(15)refers to only one bunch.The size of the region in which coherence can be maintained scales as 1/γ,and the condition of charge starvation has been generally considered as important for the radiation(Blaes et al.1989;Chen et al.2020).A more detailed treatment shows additional spectral features not treated in this approximate expression (Yang &Zhang 2018).

An application of this formula to our problem yields a curvature radiation power below ωcof

We must point out that the general propagation of waves in the magnetosphere is not restricted to the Alfvén linear waves of the type just discussed.The ultra-relativistic plasma actually supports a variety of propagating modes which have to be studied case-by-case(Treumann&Baumjohann 1997),without linearizing the equations.We have not performed such a study here,however,we suggest that a class of these nonlinear solutions,dubbed spike solitons,may be of particular interest in this problem.Their rapid phase rotation has been associated with concrete observations in the solar wind,and they have been shown to exist in ultra-relativistic plasmas,of which the magnetospheres constitute a prime example.

Two important conditions of these solutions is that their amplitude is large at infinity,and their velocity approaches the light value vA→c.Spiky solitons have been pictured as sharp density “humps” with a length Λ,which travel along the field lines and which produce curvature and additional radiation(called “transition radiation” in Guinzburg &Tsytovich 1979)when electrons or positrons from the ambient environment enter the hump region,a feature that is shared by the linear solutions discussed above.If we denote by v the relative velocity of Neelectrons or positrons that contribute to the transition radiation,its total power has been estimated as(Sakai&Kawata 1980)

These estimates are quite crude,and a complete,explicit solution of the spiky solitons would be needed to properly evaluate the emission(for example,the actual length Λ and the velocity vA).A definite radiation mechanism for the solitons to power the FRBs remains to be identified.The total release of magnetic energy ultimately exciting the solitary wave is given in Equation (2),and even this may be an extreme limit.At vA～c,the soliton would travel several radii in ≤1 ms and could be the cause of the pulses.

4.Gravitational Wave from f-modes of the Crust

As suggested above,the very recent detection of FRB activity from the Galactic magnetar SGR 1935+2154(CHIME/FRB Collaboration 2020) opens the possibility of a future test of this class of models using GWs.The scenario of magnetic instabilities cracking the crust suggests the excitation of crustal modes,akin to ripples in a pond.These are known as f-modes in astroseismology works.On very general grounds(Thorne 1987;Ho et al.2020) one can write down the amplitude of an oscillation decaying on a timescale τgwasUsing the general definition of the GW luminosity dEgw/dt for an impulsive source at a distance d,an integration in time yields the burst energy (Ho et al.2020).

There are ongoing searches for GW sources from transient signals (Klimenko et al.2016;Abbott et al.2020) associated,for instance,with supernovae(Lundquist et al.2019;Iess et al.2020;Skliris et al.2020;Szczepanczyk et al.2021)and FRBs.Recently,a GW transient candidate,S191110af,was detected,consisting of a signal at 1.78 kHz and lasting for 0.104 s(Chatterjee 2019).It was pointed out that the frequency and duration of such an event are consistent with the fundamental stellar oscillation mode (f-mode,for short) of an NS with～1.25 M⊙and～13.3 km (Kaplan et al.2019).It can be seen,for instance,in Andersson &Comer (2001),van Eysden&Melatos (2008),Keer &Jones (2015) and Sidery et al.(2010) that f-mode oscillations can be triggered by transient events internal to the NS,such as magnetic field reorganization.

Note that the importance and interest that have been given to the f-mode associated GWs lies in the fact that the mode frequency depends on the dynamical timescale of NSs,so that it is a probe of NS macroscopic features,such as density,mass and radius (Andersson &Kokkotas 1996;Andersson &Comer 2001;Kokkotas et al.2001).

Considering a stellar oscillation with frequency νgwthat damps on a timescale τgw,the GW amplitude from such an oscillation is (Echeverria 1989;Finn 1992)

The peak luminosity h0can be determined from the integration of the GW luminosity of a source at distance d (Owen 2010),

Such an integration over 0 ＜t ＜∞naturally gives the total GW energy emitted,Egw.Solving the integration for h0yields

These GW modes must be excited by some mechanism,which in our case,as in Sieniawska&Bejger(2019),de Freitas Pacheco(1998),is the magnetoquake energy release.However,the excitation of the Alfvén waves depends on the exact fracture induced by the magnetic field,the state of the crust and other details.As a result,the energy effectively going into this propagation can only be calculated in a detailed simulation,and even then many uncertainties would remain.Given that Equation (2) is an upper limit,if it is totally converted into fmode excitation,we obtain h0～3×10-24at 10 kpc as an upper limit to the GW emission,in total agreement with other values reported in the literature.

Studies of the f-modes have shown frequencies fgwbetween 1.25 and 2 kHz and a variety of damping times.However,it has been shown that the product ωgwτgwcan be considered only a function of the compactness M/R (Chirenti et al.2015).Therefore,a measurement of the frequency and damping time would be consistent with a variety of equations of state with a given fixed compactness.If the maximum released energy,Equation(2),is effectively employed to excite the f-modes,the estimate h0≈3–4×10-24for the FRBs of SGR 1935+2154 at 10 kpc may be modified by observing that recent works(Mereghetti et al.2020;Zhou et al.2020) obtained a lower distance to the source.The design goals of Advanced LIGO(LIGO Scientific Collaboration 2015) would be still insufficient,but the projected Einstein Telescope could reach this figure (Huerta &Gair 2010) although the detection of a closer source in a monitoring campaign and a careful treatment of the data around the time of the burst could improve the situation.On the other hand,h0would be within the reach of a project such as the Einstein Telescope within the next decade(Maggiore et al.2020).The detection of GWs associated with FRBs would be important to support or discard magnetar models in its various versions,even good upper limits would be useful for this task.

5.X-Ray Emission

Although the previous evidence in favor of high-energy emission in coincidence with FRBs was scarce,the detection of a burst of hard X-rays by the INTEGRAL mission(Mereghetti et al.2020) from SGR 1935+2154 has added an important piece of evidence that must be addressed when discussing the generation of FRBs by magnetars.The presence of X-rays slightly delayed (by a few ms) from the FRB has been interpreted as evidence in favor of synchrotron maser emission,but there are other mechanism(s) which can also be invoked.

One could imagine that dissipation of the Alfvén waves could be enough to power the observed X-ray spikes.However,a detailed study by Chen et al.(2020)suggests that,in spite that the plasma arranges itself to propagate beyond the chargestarved region,its dissipation falls short of power to explain the X-ray emission.Therefore,the X-rays should come from another process.One of these proposals (Yuan et al.2020)simulated the behavior of Alfvén waves converted to plasmoids at high z in the magnetosphere.The magnetic field lines,broken by the plasmoid ejection process,reconnect behind the plasmoid ejection region and extract energy from the magnetosphere (i.e.,a reservoir much bigger than the release in Equation (2)).As a general,quasi-dimensional estimate,the energy release by the reconnection process can be written as(see,for instance,Asai et al.2002) dE/dt|rec=(B2/4π)viA,where viis the inflow velocity entering the reconnection zone and A～L2is the area of the reconnection happening up in the magnetosphere.The two well-studied mechanisms suggested for the physical picture of the reconnection,the“slow”Sweet–Parker (Parker 1957;Sweet 1958) and “fast” Petschek (1964)mechanisms,have been found to be adequately described by vi∝B1+α,with α=0.5 and α=1 for the Sweet–Parker and Petschek proposals.The energy release estimate is

mainly in X-rays and accelerated electrons.Detailed simulations (Werner &Uzdensky 2017) suggest that～1/3 of the energy contributes to accelerate particles and～2/3 is dissipated in hard X-ray photons,with a spectrum which is not particularly important here.This figure must be compared with the inferred 1.4×1039erg obtained by Mereghetti et al.(2020) with INTEGRAL or the slightly higher value reported by Tavani et al.(2020) with AGILE,and shows that the propagation of the Alfvén perturbation can trigger a suitable magnetospheric release,logically delayed from the FRB as observed.

6.Discussion

We have considered in this work a scenario for the production of FRBs and related issues.The recent confirmation of the Galactic magnetar SGR 1935+2154 source at～10 kpc as the origin of FRB pulses leads us to believe that at least one type of source has been identified,and therefore models of FRB generation from them need to be re-examined.Starting with the idea that magnetic fields do not achieve definitive configurations for～centuries(Gourgouliatos&Esposito 2018),we have considered their dominant role in crust dynamics when their intensity is high enough (e.g.,Equation (3)).It is important to remark that the sources may not be restricted to the“magnetar”class.Empirical evidence for a link between young pulsars and FRBs has been presented by Nimmo et al.(2021),still requiring high magnetic fields but also finding a range of timescales and luminosities.These authors then associate the production of FRBs with young pulsars in general.For the magnetoquake scenario to work,however,it would be enough that a local field reaches B～1015G,not necessarily the largescale field thought to be harbored by magnetars.In fact,and independently of our analysis,a patch～100 m in size possessing higher multipole intensities in this ballpark is currently considered and found some support in direct observations using the NICER data (Bilous et al.2019).A local cracking of the crust by this local field would produce,in principle,FRBs with some variation in timescales and luminosities.This is why a closer look to the arrival directions of FRBs from known young pulsars would be potentially revealing.

The Alfvén wave bunches are a simple,but not the only,mechanism invoked to produce the FRBs.A discussion by Wang (2020) addressed these possibilities and concluded that all of them have some difficulty for their identification with the main emission of FRBs,specifically,the short duration predicted for the coherent curvature radiation.Within our simple picture we are not able to address this issue,which involves the detailed geometry of the field lines (simplified by our treatment) and damping effects.Cooper &Wijers (2021),on the other hand,address positively the coherent curvature as a “universal” origin for a large range of luminosities,and suggest a scale of ≤107cm,low inside the magnetosphere for its origination.On the other hand,Belobodorov (2021) argues for the “choking” of the FRBs if the emission originates at distances ?1010cm.The issue of locating the charge-starvation region closer to the end of the magnetosphere becomes critical in this sense,because it may or may not allow the observed radio emission at all.

Next we examined the propagation of a perturbation ξ excited by plate “blowout” by the magnetic field,by solving the simplest Green function in the magnetosphere z ＞0.The curvature radiation as the most likely mechanism for the production of the radio pulse was shown to be consistent with the expected features started by the event.We have also pointed out that“spiky solitons”(not treated in detail here)may be involved in the problem,although their detailed physics is even more complicated and has not been clarified in this context.The detection of diffuse radiation of a plerion-type(Resmi et al.2020) could also hold clues about the overall nature of the radiation mechanism and its diffusion in the magnetospheric region,although not necessarily related to the pulse events (in particular,the transition radiation through the propagating hump is found to be too weak to contribute).The detection of a change in the spindown (Younes et al.2020)after the FRBs in SGR 1935+2154 is another relevant clue,possibly interpreted via a topology change in the magnetic field,which is quite expected in our model.Reasons for a closely associated X-ray burst from the source (Mereghetti et al.2020) can be also accommodated (Section VI),but it is premature to claim a convincing explanation.Finally,an estimation of the gravitational signal expected from the crust fmodes was found to yield weak amplitudes,but likely crucial to establish the reality of the whole picture once advanced facilities could be operated.Meanwhile,the characterization of magnetars as sources of FRBs will continue,as will the discussion about the source(s) clarified by continuing observations and related statistical considerations (Katz 2016).

Acknowledgments

P.H.R.S.M.would like to thank CAPES for financial support.J.E.H.has been supported by the CNPq Agency(Brazil) and the FAPESP foundation (S?o Paulo,Brazil).

Appendix Review of the Transmission of Energy into the Magnetosphere:Basic Equations of the Physical Picture

The starting point for the calculations of the transmission of the released energy into the magnetosphere is the derivation of a suitable wave equation,by linearizing the motion and continuity equations about a static equilibrium.After a standard procedure,the result is (Blaes et al.1989)

In Equations(A1)and(A2),ρ is the density profile,obtained from integrating the hydrostatic equilibrium equation,ξ is the displacement of an element of material from its equilibrium position,c is the speed of light,j is the current density,is the Newtonian gravitational acceleration and p is the pressure due to degenerate electrons.

The components of the perturbed elastic stress tensor are

in which κ is the bulk modulus and the ions in the solid crust are arranged in a Coulomb lattice whose shear modulus is given by μ.When writing Equation (A3),terms which arise if there is a static elastic stress are neglected.

The crust is effectively a perfect electrical conductor of seismic frequencies.The perturbed electric field can be written in terms of ξ as

Let us also write the perturbed Maxwell equations

In order to derive a simpler wave equation,Blaes et al.(1989) considered a vertically propagating shear wave polarized in such a way that ξ ∝z×B.In this case,?.ξ=δρ=δp=0 and Equations (A1)–(A3),after standard vector calculus manipulations,now read

being the effective shear modulus and density,respectively,and cos

Up to now,the wave equation applies to the crust and beyond.However,instead of studying the transmission across the crust (Blaes et al.1989),we shall simply assume that a fraction of the total released energy,Equation (2),makes its way into the magnetosphere being transmitted by the sudden motion at the base of the line,in the form of an impulsive perturbation.That is,we disturbed the field lines by assuming a sudden “blowout” of the local crust material patch with the field lines frozen in it in the form f(z)δ(t-t0) and seek the response of Equation (5) in Section 3.This kind of picture differs from the seismological approach in which the cracking of the crust is assumed.Here we rather envisage the magnetic instability lifting the base of the lines and seek to solve the Alfvénic wave propagation and its radiated power.