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Astronomy & Astrophysics manuscript no. paper (cid:13)cESO 2017 January 6, 2017 Pulsar magnetospheric convulsions induced by an external magnetic field Fan Zhang12 1 Gravitational Wave and Cosmology Laboratory, Department of Astronomy, Beijing Normal University, Beijing 100875, China e-mail: [email protected] 2 Department of Physics and Astronomy, West Virginia University, PO Box 6315, Morgantown, WV 26506, USA Received July 6, 2016; accepted 7 1 ABSTRACT 0 2 The canonical pulsar magnetosphere contains a bubble of closed magnetic field lines that is separated from the open lines by current sheets, and different branches of such sheets intersect at a critical line on the light cylinder (LC). The n LC is located far away from the neutron star, and the pulsar’s intrinsic magnetic field at that location is much weaker a thanthecommonlyquotednumbersapplicabletothestarsurface.Themagneticfieldsurroundingsupermassiveblack J holes that reside in galactic nuclei is of comparable or greater strength. Therefore, when the pulsar travels inside such 5 regions, a non-negligible Lorentz force is experienced by the current sheets, which tends to pull them apart at the critical line. As breakage occurs, instabilities ensue that burst the bubble, allowing closed field lines to snap open and ] releaselargeamountsofelectromagneticenergy,sufficienttopowerfastradiobursts(FRBs).Thisprocessisnecessarily E associatedwithanenvironmentofastrongmagneticfieldandthusmightexplainthelargerotationmeasuresrecorded H for the FRBs. We sketch a portrait of the process and examine its compatibility with several other salient features of . the FRBs. h p Key words. radiation mechanisms: general - plasmas - (stars:) pulsars: general - o r t1. Introduction of 1011−1013G). Having this in mind, much attention has s a beenpaidtomagnetars(see,e.g.,Lyubarsky(2014)),which [Fast radio bursts (FRBs) are transient radio events with possess a stronger magnetic field (up to 1015G), for which millisecond durations, and they are typically associated relatively minor magnetospheric events would already suf- 1 with large dispersion measures (DMs). Sources of cosmo- fice energetically. v 9logical and galactic origins have both been proposed (see This thinking underestimates the flexibility of the mag- 0the review article Katz (2016a) and references therein), netospheres, however, and the possibility remains that 2with the latter challenged by more recent observations and catastrophic global reconfigurations of a normal pulsar’s 1theoretical arguments (Masui et al. (2015); Katz (2016b); magnetosphere can occur (under the influence of external 0Luan&Goldreich(2014)),thereforeweassumeextragalac- factors), but without altering the star itself, which then re- .tic sources to be responsible for FRBs here. stores the magnetosphere to its usual state. Such a global 1 0 The enormous distance scales that come with this as- event produces a large emission region that it takes radio 7sumption imply that a large amount of energy needs to be signalsmillisecondstotraverse(seethebeginningofSec.7.1 1available,1038−1040ergs,tobemoreprecise(Katz(2016a), below), thus providing a more natural explanation for the :assuming isotropic emission). In addition, the short pulse duration of FRBs, as opposed to having to amalgamate a v iduration requires a compact source region. Neutron stars large number of short-duration events, such as giant pulses X(NSs)representnaturalcandidatesthatsimultaneouslysat- of young pulsars (see, e.g., Kulkarni et al. (2015) for more risfy these two requirements. The gravitational energy of discussions). In this paper, we explore this alternative. a an NS can obviously serve as a deep energy well, but to To begin with, we note that a large rotation measure draw from it, significant changes (such as a collapse, see has been recorded for an FRB (Masui et al. (2015)), sug- Falcke & Rezzolla (2014)) probably have to occur on the gesting that a strong magnetic field exists near the source. star itself, which would be difficult to repair. This contra- On the other hand, we know that the region surrounding a dictstherecentobservationthatFRBsrepeat(Spitleretal. supermassive black hole (SMBH) is pervaded by a strong (2016); Scholz et al. (2016)), which excludes the possibility magnetic field, which is a vital ingredient in jet launching that disruptive processes cause irreversible damages to the mechanisms such as the Blandford-Znajek process (Bland- source. An alternative energy well is the electromagnetic ford & Znajek (1977)). Therefore, it is interesting to see if (EM) well residing in the NS magnetosphere. This well is thisfieldcanactasthedestabilizingexternalinfluencethat shallower, however, and the FRB energy estimate consti- triggers magnetospheric reconfigurations. tutes a substantial portion of the total EM energy in the The field strength is sufficient to play this role. The magnetosphere of a normal pulsar (magnetic field strength keyistoconcentrateonastructurallyvitalplaceforpulsar Article number, page 1 of 12 magnetospherescalledthelightcylinder(LC).Ifthetypical The underlying assumption for their treatment is that the pulsarrotationperiodis1second,thenwithingeometrized plasmaparticles’contributiontothetotalstress-energyten- unitstheangularvelocityofthepulsarrotationcanbecom- sor of the system is subdominant to that of the EM field, puted as Ω = (2π/1)(106/3×1010) ≈ 2×10−4R−1. This so that the particles’ inertia is negligible, and they can ex- ∗ impliesthataparticlethatcorotateswiththeNSwillhave perience no forces, or else will be infinitely accelerated. totravelatthespeedoflightifitislocatedat∼5×109cm GLP also asserted that the spacetime metric outside of away from the rotation axes of the star. Such a place is the NS (r >1 in our units) takes the form of termed the LC. At such a large distance from the star, the star-generatedmagnetic fieldwouldhave droppedto 10−11 (cid:18) C(cid:19) (cid:18) C(cid:19)−2 ds2 = − 1− dt2+ 1− dr2 ofitsstrengthnearthestar,assumingadipolarfieldoutto r r tshiveeLbCla.cIknhcoonletriasstli,mthiteedmabgynethtiecEfiedlddinclgotsoentofiealdsuspterremngatsh- +r2(cid:34)dθ2+sin2θ(cid:18)dφ− CIΩdt(cid:19)2(cid:35) , (1) to 6×104M−1/2G (Dermer et al. (2008); Palenzuela et al. r3 8 (2010)) where M is the mass of the supermassive black 8 holedividedby108M .Fromactivegalacticnucleijetcon- where C is a dimensionless compactness parameter with a (cid:12) siderations, Blandford & Znajek (1977) estimated that the typical value of 1/2, and I is the dimensionless moment fieldstrengthmustexceed100G,whileforourquieterMilky of inertia, with a value of 2/5 for a uniform density sphere. Way, the field strength close to the SMBH Sgr A* is about ThesequantitiesaredependentontheNSequationofstate, 20G (Melia (2013)). These values are not negligible com- and we take the aforementioned example values for con- pared to pulsar’s intrinsic field at the LC, and can exert creteness. The quantity Ω, on the other hand, is the an- strong influences on the magnetospheric dynamics. gular frequency of the NS rotation, which we retain as a We begin a rough sketch of this influence in Sect. 2 by freeparameterintheexpressionsbelow.Inaddition,asthe computing the current density along current sheets (CSs) essentialingredientsweneedalreadyappearforalignedro- thatencloseclosed(returntothestar)magneticfieldlines. tators (rotation and magnetic axes of the NS are aligned), Even though they are of vital importance (they determine this is the case we examine. the force-free regions through boundary conditions), a de- A most salient feature of the pulsar magnetosphere is tailed description of the CS dynamical evolution is gen- that a bundle of closed field lines without accompanying erally lacking. Nevertheless, we can evaluate (in Sect. 3) currents exists in a bubble, separated from the open field the Lorentz force that they experience when immersed in lines outside, along which currents do flow (see Fig. 1 (a)). the background magnetic field near an SMBH, and show In order for such regions of distinct characters to coexist, a that force discontinuities would dismember the bubble en- compressedlayerofhighcurrentdensitycalledtheCSneeds closure they provide, with the magnetosphere experiencing to be present to separate them (across which the magnetic a catastrophic transition as a result. We then estimate the field is allowed be discontinuous), and the integral version overall energyreleased from such a violent event in Sect. 4, of the Maxwell equations (see Sect. 6 in Gralla & Jacobson and describe in Sect. 5 the process through which bubbles (2014)) dictates that the CSs need to be tangential to the regrow, thereby completing a full repeatable dynamical cy- magnetic field lines. cle (noting that FRBs have been observed to repeat). We Quantitatively, we note that the Faraday tensor within finally examine some potential complications to our com- the stationary axisymmetric force-free magnetosphere is putations in Sect. 6, and evaluate the conformity of the given by GLP, Eq. 6, in the exterior calculus notation as present proposal to salient features of the FRBs in Sect.7, rI(ψ) (cid:16) (cid:17) introducing some new analysis, for instance, on the tem- F = dr∧dθ+dψ∧ dφ−Ωdt . (2) poral pulse profile. The results show good agreement with π(2r−1)sinθ FRB observations, especially with several previously over- The quantity 2πψ(r,θ) is the polar magnetic flux through looked features of the signals. Finally, we conclude with an anysurfaceboundedbythetoroidalcurveofconstantrand outlook for future work in Sect. 8. θ(asgivenbytheargumentsofthefunctionψ),andI isthe The formulae in this paper are in geometrized units polarcurrentthroughthatsamesurface.Thebubbleregion where c = G = (cid:15) = 1, unless stated otherwise. With 0 corresponds to high values ψ > ψ ≈ 1.23µΩ, where µ is this choice, only one fundamental unit, that of length, is 0 the dipole moment of the magnetic field close to the star. required, and we take it to be the radius R of the NS. In ∗ In this region, we have I(ψ) = 0, which is quite different otherwords,alengthof1R inourformulaecorrespondsto roughly 106cm in cgs units.∗The index notation adopted in from the outside region with ψ <ψ0, where (GLP, Eq. 22, + sign for the northern hemisphere) this paper is that the beginning part of the Latin alphabet denotes four dimensional spacetime quantities, while the (cid:32) ψ 1(cid:18) ψ (cid:19)3(cid:33) middle part of the Latin alphabet denotes spatial compo- I(ψ)=±2πΩψ 2− − . (3) nents. ψ 5 ψ 0 0 Although the closed field lines are of secondary impor- 2. Pulsar magnetosphere in isolation tance to pulsars in isolation because they are unrelated to Before examining the dynamical evolution brought about the regular pulsed radiations, they nevertheless serve as a by the background magnetic field of the galactic center, we reservoir holding on to energy that is available to be re- first enumerate some of the important structural features leased in a sudden outburst, should the bubble walls ex- of a pulsar magnetosphere in isolation. We take the outline perience any catastrophic failure. A general rule for the of Goldreich & Julian (1969), and in particular the quanti- closed field lines is that they must reside within the LC, tativerefinementfromGrallaetal.(2016)(GLPforshort). never venture outside. The reason being that beyond the Article number, page 2 of 12 Fan Zhang : Pulsar magnetospheric convulsions induced by an external magnetic field 1.0 cf. Gralla & Jacobson (2014), Eq. 66) F Fab = 2(|B|2−|E|2) ab I2 = +|dψ|2|dφ−Ωdt|2, (4) 0.5 r(2r−1)π2sin2θ andnotethat|dψ|2 <0asitisspace-likeandthatdφ−Ωdt (2) changesfrombeingspace-liketotime-likewhenwecrossthe (1) L LC outward bound. Therefore when I = 0, as in the case R 0.0 z (cid:144) Y of the closed field lines, the region beyond the LC is an electrically dominated forbidden zone, while for open field lines with non-vanishing I, this region can remain magnet- ically dominated. It is then not surprising that the tip of (cid:45)0.5 the bubble enclosing those close field lines is on the LC, as depicted in Fig. 1(a). From Eq. (2), we can also obtain the explicit form of the magnetic field using Ba = (cid:15)bcdaF τ /2, where (cid:15)abcd is bc d (cid:45)1.0 the 4D Levi-Civita tensor, and τa is the time-like one form 0.0 0.5 1.0 1.5 2.0 orthogonal to constant t slices of spacetime. The result is (a) Ρ RL (upper index spatial vector in the basis of {∂φ,∂r,∂θ}) 1.0 √ (cid:26) (cid:27) cscθ 2r−1 cscθ r ∂ψ ∂ψ B = √ I(ψ), , − . (5) (cid:144) 2r5/2 π 2r−1 ∂θ ∂r Weseethatthetoroidalmagneticcomponentsimmediately 0.5 inside and outside of the CS have values of B √ * 2µΩ2csc2θ C B(1)φ =0, B(2)φ ≈± √ . (6) r3/2 2r−1 L * R 0.0 z (cid:144) Therefore, by the usual boundary condition across the CS (nˆ istheoutwardnormaltotheCS,whichispurelypoloidal as a result of axisymmetry) (cid:45)0.5 (cid:16) (cid:17) (cid:15)ijknˆ B(2)−B(1) =σi, (7) j k we have that there is a return current (with surface density σ) flowing inward toward the star along the singu- (cid:45)1.0 lar separatrix (separating open and closed lines) CS in 0.0 0.5 1.0 1.5 2.0 (b) the poloidal direction. In particular, its distribution is Ρ R L reflection-symmetric against the equatorial plane, or in Fig.1.(a):Schematicdepictionofapoloidalsliceofthemagne- other words, the currents along the top and bottom red tosphereofanisolatedpulsar,reproducedfromGLPFig.1.The arches in Fig. 1(a) are both flowing to the left. vertical dashed lines mark the(cid:144)location of the LC (ρ/R = 1, L withρandzbeingthecylindricalcoordinatesmeasuringthedis- tancetoandalongtherotationaxis,andtheLCisatρ=R ). 3. Destablizing Lorentz force L The red curves represent the CSs that demarcate the different magnetic domains (1) (inside the shaded bubble containing We now place the pulsar in the galactic center, but still far closed field lines) and (2) (open field line region). (b): A depic- away from the innermost stable circular orbit, so that the tionofthesplitmonopolesolution,withaCSontheequatorial general relativistic effects from the SMBH are negligible. plane, facilitating the change of field line direction across it. The speed of the NS, on the other hand, is approximated (cid:112) bytheKeplerianexpressionv ∼ GM/r,whichfororbital radiusr ∼0.005pc(estimatedsizeofthemagnetizedregion, LC, the particles that are stuck on the field lines satisfy- see Sect. 7 for details) and SMBH mass M ∼ 4×106M (cid:12) ing force-free conditions will have to move superluminally, givesv ∼10−2c(Lorentzfactorγ−1∼10−5),andthusspe- which is impossible (Goldreich & Julian (1969)). Alterna- cialrelativisticeffectsarenotimportanteither.Inaddition, tively, magnetic dominance is lost for such field lines be- the electric field strength when we move into the comoving yond the LC (Gralla & Jacobson (2014)). That these two frameoftheNSiscorrespondinglyweak.Forthepurposeof statements are equivalent can be seen simply by writing grasping the basics of the proposed FRB mechanism then, down the flat spacetime expression for the Lorentz force it suffices to consider a stationary pulsar immersed in a q(E + v × B), and observe that in an electrically domi- static magnetic-only background EM field. nated region, we would need |v| > 1 in order to achieve a As is clear from Fig. 1, the CSs form the skeleton of vanishing force. To see what distiunguishes the closed and themagnetosphere(ithasbeenobservedthatforce-freeso- openfieldlinesintermsofpenetratingtheLC,wecompute lutions are in general numerous and the CS configuration from Eq. (2) the invariant (with notation |B|2 ≡ B Ba, is the deciding factor in selecting particular solutions out a Article number, page 3 of 12 cannot in fact exist. This is because the three branches of (cid:159) (cid:159) (cid:159) CSsintersectatacriticalpointY(seeFig.1(a)),wherethe current flows (with a non-vanishing density σ ∝ µΩ2/R2, L seeEqs.(6)and(7))dictatethatthedifferentbranchesare pulled by Lorentz forces in different directions when the non-negligible(neartheYpoint)backgroundmagneticfield is orientated as in Fig. 2(a). In other words, the forces will tear the three branches apart at the seams. While a sim- (cid:159) (cid:159) (cid:159) ple deformation may bring the three branches to intersect B Y tangentially(Fig.2(b)),thecurrentsinthetopandbottom sheetsthatflowbacktothestararenecessarilyopposite,so that the forces would still not align at the critical point. A proper force alignment requires more drastic deformations, wherethetopandbottombranchescollapsetogetheratthe (cid:159) (cid:159) (cid:159) criticalpoint,asshowninFig.2(c).Theforcedirectionswill align at Y, but the volume of the bubble will be squeezed, (a) and the energy stored in the closed field lines compacted. This implies that significant magnetic pressure (∝ energy (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) density) would have built up inside the bubble that prevented this configuration from being achieved in the first place, especially since the Lorentz force acting on the top separatrix CS also tries to expand instead of compress the bubble. In short, tearing is unavoidable 1. (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) B Y B Y Theattentionisalsodirectedtothefactthattheregion surrounding the Y point is not perfectly force-free; Mestel & Shibata (1994) predicted that dissipative zones neces- (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) sarily exist, which means that magnetic reconnections are allowed there. Therefore, failure in the form of rapidly re- (b) (c) curringforcedreconnections,andthesubsequentdisorderly Fig. 2. Force distribution near the critical point. The back- current flows (i.e., charged particle motion), would develop groundmagneticfieldpointsoutofthepaper,thedottedlinede- intheCSsattheYpoint,whichisalsocharacterizedbyhigh notestheLC,andtheblackarrowsindicatetheforcedirections. particlespeedsandthusahighReynoldsnumber.Thetur- (a) The isolated magnetospheric configuration. (b) Slightly ad- bulent flows would then be convected along the CSs to re- justedsothatthethreebranchesofthecurrentsheetsintersect gionsfartherawayfromtheYpoint,introducingdissipation tangentially,buttheforcesdonotsynchronize.(c)Heavilymod- and resistivity at these places as well. Resistive magneto- ified so that the three branches have matching force directions, hydrodynamic instabilities such as the rippling and tearing but the volume of the bubble is curtailed near the LC. modes (Furth et al. (1963)) would then be able to develop. The configuration with a sudden jump of Bφ across a current layer (see Eq. (6)) is a stable equilibrium only when of large families of possible candidates (Goldreich & Julian the conductivity of the CS is infinite, and once a resistiv- (1969); Yang et al. (2015))), therefore we concentrate on ity η is introduced, instabilities grow on a timescale much their reaction to our placing the pulsar in the galactic nu- greater than that of Alfvén (∼ R √ρ /B(2)φ, where ρ is L 0 0 clei.Specifically,theywouldexperienceaLorentzforcefrom the plasma density), but much shorter than that of resis- theexternalbackgroundfield,providedthatsuchfieldshave tive diffusion (∼R2/η), destroying the structural integrity not been cancelled by a slight adjustment of the magneto- L of the CSs and thus removing the segregation between dis- sphericcurrents.Becausetheexternalfieldstrengthiscom- tinct magnetic domains (the magnetic field is no longer al- parableto(oreasilyordersofmagnitudestrongerthan)the lowed to be discontinuous in the absence of CSs, therefore star-generated field near the LC, it is clear that small per- thetwodomainsmustassimilate).Inotherwords,thebub- turbativealterationstothemagnetosphericcurrentsarein- ble containing the closed field lines that were punctured sufficient to shield the background field in that region. The at the Y point would burst open, allowing these lines to resultisthattheCSsaremovedbytheLorentzforce,result- spring out into open field lines (to match smoothly with ing in a loss of stationarity for the entire magnetosphere, the open field configuration outside), along which currents withtheCSsservingasmovingboundariestotheforce-free and Alfvén waves can travel, possibly as the null solutions regions. of Brennan et al. (2013) (see also Zhang et al. (2015) for The result of this loss of stationarity is episodic magne- their stability). When these plasma winds and waves prop- tospheric reconfigurations. Specifically, in the regions close agate along the originally open field lines near the poles of totheNS,theCSsinastationarysolution,ifitexists,would isolatedpulsars,theenergytheycarryeventuallyturnsinto besimilartotheircounterpartsintheisolatedcasewithtwo theregularpulsarradioemissions.Itisthereforenotunrea- inflowing currents (computed in Sect. 2) above and below the equatorial plane with similar fluxes (obeying approx- 1 Beyond tearing, that force acting on the top separatrix sheet imate reflection symmetry) because these regions are still even tries to take it across the LC, which would incite further dominated by the magnetic field of the NS itself. However, instability,astheclosedfieldlinesabuttingitandtangentialto neartheLC,theCSconfigurationasdepictedinFig.1isno itwillhavetoaccompanyitonthisjourney(ourdiscussiondoes longer stable, which means that such stationary solutions not rely on this also happening though). Article number, page 4 of 12 Fan Zhang : Pulsar magnetospheric convulsions induced by an external magnetic field sonable to expect that the same conversion process could for the split-monopole and channelsomeofthelargeamountsofenergyinitiallystored in the closed field lines into a bright, although short-lived, ρl=1 =− µ2 (cid:104)(6r+1)cos2θ+10r−1(cid:105)× radio burst, or FRB. E 100(2r−1)2r9 ×(cid:104)(cid:0)1−5r3(cid:1)2Ω2cos2θ−(cid:0)1−5r3(cid:1)2Ω2+25(1−2r)r3(cid:105), 4. Bubble-free magnetosphere (12) The now fully open magnetosphere resembles a split- for the dipole (setting I = 0 as we are interested in the monopole (see Fig. 1(b) and Michel (1974); Gralla & Ja- region inside the bubble). In addition, the magnetic field cobson (2014)), which is a valid solution for force-free elec- strengths in the two cases are trodynamics given by (q being the monopole charge) q (cid:113) F =qsinθdθ∧[dφ−Ωd(t−r)], (8) Bl=0 = √ r(2r−1)Ω2sin2θ+2, (13) 2r2 and has long been used as a simpler surrogate for the true and isolatedmagnetosphereinregionsfarfromtheNS(see,e.g., Bparnenelnsan(a)&anGdra(lbla) (o2f0F14ig).).1Ashvoiwsusalthcaotmtphaeritsrounebiseotlwaeteend Bl=1 = µ√sinθ(cid:113)r(cid:0)8rcot2θ+2r−1(cid:1). (14) 2r4 solutionasymptotestoamonopole-likefielddistributionat large r, which is expected because monopole-like radial (in Take a pulsar with a magnetic field strength of ∼ thepoloidaldirections)fieldlinesisimposedasaboundary B ×1011Gatthestarsurfacewherethedipoledominates 11 condition in GLP. (wenotethatthevariableB isonlythecoefficientinfront 11 The split-monopole is a poor approximate for the iso- of the power 1011, and does not include the power term it- latedmagnetospherewhenweareclosertotheNS,however, self, so that for a pulsar with a field strength of 1012G at as the current inside the star is expected to favor a dipolar theNSsurface,wehaveB =10,andthefieldstrengthat 11 near-zone field. However, the field lines of a dipole, given the LC for this pulsar would be 1×B =10G), then from 11 by Eq. (14) we can work out µ. Since the dipole dominance is taken over by a split-monopole at the LC (at the typical µ ψ = sin2θ, (9) valueofR =5×103R forapulsarofaone-secondperiod), r L ∗ themonopolefieldstrengthfromEq.(13)mustbecommen- are all closed, so that energy cannot be transported out suratetothedipolefieldvalueatthatlocation,fromwhich asPoyntingorwindfluxesmovingoutalongthefieldlines. we can determine q. Because a dipole magnetic field drops Therefore,toaccountforthefactthatwedoobserveregular off as 1/r3 as opposed to the monopole’s 1/r2, the bubble pulsaremissions,fieldlinesnearthepolarregionswillhave configurationwillhavetocontainamuchstrongermagnetic to be opened up into monopole-like lines by the currents field (as compared to the post-burst monopole) in the near in the magnetosphere. In the regions closer to the equato- zone to match the same asymptotic field strengths. This is rial plane, on the other hand, the influence of the dipole is in essence the energy reservoir. Substituting the aforemen- restrictedtowithintheLC,allowingthemonopoletodom- tionedvaluesintoEqs.(11)and(12),wecanthenintegrate inate outside. The bubble is then essentially an island of the resulting ∆ρ ≡ ρl=1 − ρl=0 over the bubble region E E E dipolar dominance in this tug of war, trapping the dipole- to yield a total released energy of ≈4×1039B2 ergs. This 11 likeclosedfieldlineswithin,untilreleasedbythecollapseof resultappearstobeingoodagreementwiththeimplieden- the separatrix CSs, at which time the same split-monopole ergyof1038−1040ergsfortheFRBs(Katz(2016a)),noting thatdominatedoutsideoftheLCbecomesdominantevery- thattheseobservation-impliednumbersmaybeslightover- where. A turbulent current region close to the star surface estimates,however,dependingonwhichfractionoftheDMs (generated during the collapse of the CSs) is then respon- is appropriated into the intergalactic medium (see Sect. 7 sible for keeping the influence of the dipole temporarily at below). In which case we would have additional room for a bay (allowing for a temporary alteration of the boundary much lower radiation efficiency (especially since for a typi- condition near the star). cal pulsar B ∼10). 11 Given this general scenario, we can then estimate the energy budget available to be released from the bubble by 5. Recharging the bubble comparing the energy originally stored there in the form of the dipole solution (9), with the same region filled with Although the split monopole is a valid description of the monopole fields (8) after the bursting. It is straightforward magnetosphereimmediatelyafterthebubble’sbursting,the to compute the energy density, or the ·tt component of the toroidalcurrentswithintheNShavenotbeenremoved,and stress-energy tensor theywilltrytoimposeadipolarboundaryconditiononthe 1 magnetosphere (such boundary conditions supported the Tab =FacFbc− 4gabFcdFcd, (10) original dipolar closed field line region in the first place) and restore the closed field line bubble. Immediately af- which is ter the bursting, their efforts are hindered by a turbulent current layer near the NS surface that temporarily shields q2 (cid:104) ρl=0 = 50(2r−1)r3 the dipolar boundary condition from the magnetosphere. E 50(2r−1)2r6 However, the high dissipation within the turbulence would +(cid:0)5(cid:0)5r(cid:0)8r2−4r+1(cid:1)−8(cid:1)r3+4(cid:1)Ω2sin2θ(cid:105), (11) eventuallyvanquishsuchcurrents,andthedipolarinfluence wouldbegintopushoutwardagainbyinjectingenergyinto Article number, page 5 of 12 a nascent bubble, causing it to grow gradually in size. The in Sect. 3. When its Y point once again breaches a cer- detailed process most likely resembles the one outlined in tain threshold radius and reaches close to the LC, however Contopoulos (2007), that is, magnetic reconnection across (i.e.,whenthelostenergyhasbeenreplenishedbytheNS), a resistive equatorial CS (see the red line in Fig. 1(b)), but the intrinsic pulsar magnetic field becomes subdominant applied in a different context inside of the LC: the resid- in strength to the external field and will not be able to ual resistivity from the bursting episode inside of the split- neutralize the latter through its own perturbations. Conse- monopole’s equatorial CS facilitates reconnection of open quently,Lorentz-force-inducedinstabilitiessetinonceagain fieldlinesacrosstheequatorialplane,intoclosedlines.The to burst the newly grown bubble, and the cycle repeats it- process begins near the NS where the toroidal current in- self. When the environmental field strength is denoted as side the star bends the field lines into dipolar shapes to be B G, the stability threshold is approximately located near e reconnected, and then marches outward. Let v and η de- (B ×1011/B )1/3R ,wheretheintrinsicandexternalmag- 11 e ∗ note the material velocity and the magnetic diffusivity in neticfieldsareofcomparablestrengths.AstheYpointwill the equatorial CS, then the evolution of the closed mag- not push beyond the LC, we need this threshold radius to netic field lines along the resistive CS is governed by the beinsideoftheLCforanyinstabilitytooccuratall,which induction equation (Contopoulos (2007) Eq. 13) givesusacriterion(anadditionalcriterionistobegivenin Eq. (22) below) dBi =(cid:15)ijk∇ (cid:15) lm(v −η∇ )B , (15) dt j k l l m B ≥(ΩR )3B ×1011G, (16) where the first term denotes the advection of the closed e ∗ 11 field lines by the particle flow in the CS and the second term a diffusion that has a characteristic timescale of meaning that the pulsar has to be sufficiently close to the T =R h/η,wherehisthehalfthicknessoftheCS.Con- ch L SMBH such that B is large enough to trigger our FRB topoulos (2007) integrated and plotted (in their Fig. 2) the e mechanism. We further note that the short bursts we pro- outwardmarchingofclosedfieldlinesaccordingtoEq.(15) pose are not induced by sudden changes in the environ- for a toy model. Although that toy model was designed to mental magnetic field, but rather are a violent constituent illustratetheprinciplesforadifferentprocessoccurringout- episode in an intrinsic dynamical cycle for the pulsar mag- side of the LC, it turns out to be more directly analogous netosphere immersed in a steady external field. In other to our bubble growth process, and we refer to that paper words, the episodic short-duration FRBs is due to the in- for more details (in particular, their Fig. 2 provides a good trinsic cyclic dynamics of an out-of-equilibrium nonlinear visualizationforagrowingbubble).Wenote,however,that system,anddoesnotrequireanexternaltriggerwithasim- themagneticdiffusivityηisproportionaltoelectricresistiv- ilartemporalvariationprofile,somewhatlikethecrashesin ity, which means that lower turbulence-induced resistivity business cycles that do not need to be induced by wars or in the outer regions (which had more time to settle down natural disasters. before the new bubble reaches them) would prevent effec- tive reconnection of the field lines and thus slow down the growthofthenewbubble,causingprotractedperiodsofin- activity and thus low FRB duty cycles. The overall length ofthebubble’sgrowthperiodisotherwisestochastic,asthe 6. Complications turbulentpost-burstingenvironmentinjectsvariabilityinto bothη andv.Nevertheless,thediffusiontimescaleT pro- 6.1. Tidal forces ch vides a crude estimate for the FRB recurrence interval and thusitsrepeatrate.Wefirstnotethatη =1/(µ σ ),where We have placed the pulsars close to the SMBH, which 0 0 µ isthevacuumpermeability,andσ istheplasmaconduc- means that tidal forces might become significant. In this 0 0 tivity, which we approximate by the Coulomb collision for- section, we briefly estimate their strengths as compared to mulaσ =n q2/(m ν ),withn beingtheelectronnumber the Lorentz forces to obtain a rough idea of when tidal ef- 0 e e e c e density,q andm theelectronchargeandmass,andv the fects have to be taken into account in our computations. e e c collision frequency. For ne, we can take Goldreich & Julian To evaluate the tidal forces, we note that the tide- (1969)Eq.9,whichgivesne =7×10−2BzΩ/(2π),andnote induced relative acceleration ∆a between two freely falling that the regrowth process in the outer regions consumes observersthatarespatiallyseparatedbyavectorξ isgiven the most time, so that the relevant Bz ≈ B11 ∼ O(10)G by (see Nichols et al. (2011)) for a typical pulsar (we also adopt a typical 1s pulsar rotation period). Substituting all these numbers, we obtain T ≈ 10−2h/ν . Recalling that FRB 121102 has been ob- ∆aj =−Ej ξk, (17) ch c k served to repeat on 10min intervals (Spitler et al. (2016)), we obtain ν ≈2×10−5h s−1, which translates into an ef- e fective thermal motion temperature of ∼ 1.4×104/h2/3K, where E represents the tidal tensor. For a Kerr black hole broadlyinlinewithtemperaturestypicallyfoundnearpul- of mass M and dimensionless spin a, the tidal tensor as sars (e.g., Page et al. (1996) and Pavlov et al. (2001) found measuredbylocallynonrotatingBoyer-Lindquistobservers a surface temperature of 106K for the Vela pulsar). is given by (see Zhang et al. (2012)) During the growing phase, while the bubble remains small and hidden inside regions of strong intrinsic pulsar magnetic field, the external field can be shielded by per- −Qe12−+ηη µQm 0  turbations to the intrinsic field, and in this way, the bub- Eij = µQm Qe11+−2ηη 0  , (18) ble is protected against the destabilizing effects described 0 0 Q e Article number, page 6 of 12 Fan Zhang : Pulsar magnetospheric convulsions induced by an external magnetic field closed magnetic field lines at the LC are loaded with elec- trons and positrons traveling close to the speed of light, we Λ1Ξ(cid:158)Θ(cid:61)Π2 arrive at a Lorentz-force-induced acceleration on the order 105 Λ2Ξ(cid:158)Θ(cid:61)Π2 of 1019cm s−2. A close examination of Fig. 3 then shows (cid:144) thatthetidaleffectscanbesafelyignoredallthewaydown Λ3Ξ(cid:158)Θ(cid:61)Π2 2 100 (cid:144) to the event horizon. The reason for this is of course that (cid:45)ms Λ1Ξ(cid:158)Θ(cid:61)0(cid:144) the black hole is very large, so that although the resulting c Λ2Ξ(cid:158)Θ(cid:61)0 gravitational field is strong, it varies slowly on a very long 0(cid:144).1 a length scale. (cid:68) Λ3Ξ(cid:158)Θ(cid:61)0 10(cid:45)4 6.2. Pulsar wind 10(cid:45)7 In Sect. 2 we have sketched an analytical outline for the 10(cid:45)6 10(cid:45)5 10(cid:45)4 0.001 0.01 structure of pulsar magnetospheres in regions extending as r pc far out as the LC. In real astronomical surroundings, relativistic winds typically dominate in regions farther out, Fig. 3. Relative accelerations corresponding to the three prin- formingaterminationshockfrontatastand-offdistanceR cipal directions of the tidal tensor(cid:144)(the blue curves are covered s where the wind pressure balances that of the surrounding by the red ones). Both axes are in log scale. medium.Thisstand-offdistanceforabowshockisexpected to satisfy (see, e.g., Gaensler & Slane (2006)) whereby L Qe = Mr(r2−Σ33a2cos2θ), 4πRS2c =ρ0vP2 , (21) Macosθ(3r2−a2cos2θ) whereListhespin-downluminosityandv thespeedofthe Q = , P m Σ3 pulsar, while ρ is the density of the interstellar medium. √ 0 ∆a2sin2θ 3 η Substituting some typical values for field pulsars listed in η = , µ= , Gaensler & Slane (2006), which are L ≈ 1033ergs s−1, (r2+a2)2 1−η v ≈ 5×107cm s−1, and ρ ≈ 1.67×10−25g cm−3 (hy- P 0 Σ=r2+a2cos2θ, ∆=r2−2Mr+a2, (19) drogen with a number density of 0.1 per cm3), we obtain R ≈2.5×1015cm,whichisfargreaterthantheLCradius and(r,θ)areBoyer-Lindquistcoordinates.Torepresentthe S of ≈ 5×109cm. Although both ρ and v should statisti- directionaldependenceofthetidalforcesinamoreexplicit 0 P callybegreaterforpulsarsinthegalacticcenterthantheir and economic manner (as a result of frame dragging, the fieldcounterparts,thevastgapinorderofmagnitudesfrom maximumtidaleffectdoesnotnecessarilymanifestitselfin our estimate suggests that R would still most likely to be theradialdirection),wecancomputetheeigenvaluesofthe S ofsufficientsizetoenclosetheclosedfieldlineregion,form- matrix (18), which are ingineffectacocoonthatisolatesthisregionfromtheout- (cid:115) sideenvironment,atleastintermsofmattercontents.Itis λ =−Qe − (cid:18)3Qe(cid:19)2(cid:18)1+η(cid:19)2+µ2Q2 , thereforeimportanttoassesswhetheraneffectiveshielding 1 2 2 1−η m from the magnetic field near the SMBH is also thus estab- (cid:115) lished (in addition to the interstellar particles being blown λ =−Qe + (cid:18)3Qe(cid:19)2(cid:18)1+η(cid:19)2+µ2Q2 , away), which may shut down our FRB mechanism. 2 2 2 1−η m Our conclusion is in the negative, since in our strong field scenario, the blowing away of interstellar charged par- λ =Q . (20) 3 e ticles by the wind does not translate into an expulsion of These eigenvalues correspond to the tidal field strength in the environmental magnetic field. To see this, we note that thethreeprincipal(eigenvector)directionsofthetidalten- the magnetic pressure from a Be = 20G galactic center sor. magnetic field (we note that the pulsar wind literature is Using Sgr A* as a concrete example (4.9 million so- typically concerned with much weaker fields, e.g., the Crab lar masses and a spin of 0.996) for the SMBH and |ξ| ≈ Nebula has an estimated field strength of 300µG (Trim- 5×109cm (distance from the star to the LC) as a generous ble (1982)), while 3C 58 carries 80µG (Green & Scheuer upper limit for the length scale inside the LC, we can com- (1992)))isaround107MeV cm−3,whichwhenequatedwith pute the tidally induced relative acceleration in the three the left-hand side of Eq. (21), which represents the pulsar principaldirectionsasafunctionofthedistancer fromthe wind pressure, yields the stand-off radius against the mag- pulsar to the SMBH. We plot the results for two angles netic pressure at ≈ 1010cm. Comparing this with the LC θ = π/2 and 0 (i.e., when the pulsar is on the equatorial radius of ≈5×109cm for a pulsar of period 1s, we see that planeandinthepolarregionsoftheSMBH)inFig.3,forr a slight increase in the period and/or decrease in the spin- ranging from the black hole event horizon to about 0.01pc, down luminosity and/or increase in the B will allow the e which has been used in the literature as a rough bound for environmental magnetic field to penetrate the closed field the galactic center region (see references in Sect. 7.4). We lineregionandtriggertheinstabilitiesdescribedinSect.3. cancomparethistidalaccelerationwiththatresultingfrom The pulsar wind consideration thus sets a (an astronomi- the Lorentz force, using B = 20G as an example (applica- callyachievable)lowerboundonB thathastobesatisfied e ble for our Milky Way, see Sect. 1), and noting that the in addition to Eq. (16), so that now we have an overall Article number, page 7 of 12 threshold of (cid:43) (cid:43) 0 (cid:43) (cid:40) (cid:114) (cid:41) (cid:43) 2L B ≥max (ΩR )3B ×1011, Ω G. (22) e ∗ 11 c (cid:45)1 (cid:43) y (cid:76) (cid:43) sit (cid:43) n e (cid:45)2 7. Observables D (cid:43) x u (cid:43) (cid:43) We have proposed a candidate mechanism for the FRBs, Fl (cid:45)3 n (cid:72) which relies on the interaction between the pulsar magne- l (cid:43) (cid:43) tospheres and a strong background magnetic field within (cid:45)4 galactic nuclei. This scenario needs to comply with the ob- (cid:43)(cid:43) servedpropertiesoftheFRBs.Somehavealreadybeendis- (cid:45)5 cussed; we consider the rest in turn. (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 (a) ln∆t(cid:43)R (cid:42) 7.1. Intrinsic pulse duration and temporal profile 0 (cid:43) (cid:43) (cid:43)(cid:43) (cid:43) A distinguishing feature of the present model is that the (cid:43)(cid:43) (cid:72) (cid:76) (cid:43) release of energy locked up in the bubble is a global event, (cid:45)1 whose observation signatures are intrinsically of millisec- y (cid:76) (cid:43)(cid:43) onddurations.Specifically,thereleasedenergystreamsout sit (cid:43)(cid:43) en (cid:45)2 (cid:43)(cid:43) along sprung-open magnetic field lines in an orderly fash- D (cid:43) itohna,tspooitnhtattowonalrydtEhaorstehswouilrlcceonlotcraibtuiotnestoalothneg sfiigelndallsinwees nFlux (cid:45)3 (cid:72) (cid:43)(cid:43)(cid:43) (cid:43) observe. Furthermore, radially along such lines, only those l regions of sufficiently high energy density (i.e., close to the (cid:45)4 (cid:43)(cid:43) star) will contribute fluxes that register above the back- (cid:43) groundnoise.Whenweassumethatsuchregionsextendto (cid:45)5 ∼10−100R∗,itwilltake∼0.3−3millisecondsforlightsto (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 2 traverse (Aflvén waves in force-free plasma also travel with (b) ln∆t(cid:43)R a group velocity equaling the speed of light). This is in (cid:42) contrast to other pulsar events such as giant pulses, which Fig.4.Logofflux(inarbitraryunits)againstthelogoftimeto lastmicroseconds.Thiswouldhelpexplaintheratherbroad peak-amplitude,withareferencestraightline(black)−4ln(δt+ (cid:72) (cid:76) FRB pulses, especially since evidence of multipath scatter- R∗)+A. The inset indicates the location of the data points on the overall signal profile (with space between data points filled ing is sometimes lacking (Spitler et al. (2016)). inwithstraightlines).Thehorizontallineintheinsetisthezero We can even predict more than just the overall pulse flux.(a)Burst11oftherepeatingFRB121102.(b)TheLorimer width. The energy originally stored farther away from the burst FRB 010724. star (e.g., at point B in Fig. 1(b) as compared to point C) would have a head-start in terms of streaming along the field lines pointed toward Earth and will arrive at our ra- index −4 power law would diverge there), so that given dio telescopes slightly earlier. Because the density of the a fixed temporal sampling interval, only few data points released energy is lower at B than it is at C, we expect to would land on the leading edge. Therefore we should look see an increase in the FRB signal at its leading edge, be- forthestrongestsignalsforwhichmoreofthepre-peakseg- ginning a few hundred milliseconds before the peak time, ments rise above the background noise. The cleanest and butonlybecomingstrongenoughtoriseabovenoisesmuch simplest (single-peaked) signal with a high signal-to-noise later.Semi-quantitatively,thereleasedenergyisroughlythe ratio and several data points on the leading edge is burst difference between a dipole (ρl=1 ∼ r−6) and a monopole E 11 reported in Spitler et al. (2016). We display it in Fig. 4 (ρl=0 ∼ r−4). On the other hand, the transverse area sub- E (a), and observe that the early part of the leading edge tending the same solid angle is ∝ r2, which means that appears to agree with a power law with index −4, but the the energy density along the radial direction is ∝ r2∆ρ . E partclosertothepeakdeviatesfromit.Thismaybedueto Near the peak of the signal (arriving at t ) where energy p otherpulse-smearingeffectsexperiencedduringsignalprop- is released from close to the star, the monopolar term is agation, or if intrinsic, possibly indicates the existence of subdominant and can be dropped. We have then that the complicateddynamicssuchasturbulencesveryclosetothe log of the flux is approximately star that prevent an efficient transportation of the released −4ln(δt+R)+A, (23) energy.Anotherstrongburstthathadmultipledatapoints on the leading edge of its pulse profile is the Lorimer burst where we have used r −R ≈ t −t = δt, with t being (Lorimer et al. (2007)), which, as seen from Fig. 4(b), dis- P P thearrivaltime forthepeakof the signalandR the radius plays a similar morphology. For prudence, we caution that from which the peak radiation comes, which we set to R≈ several points at the very beginning of the pulses are not R = 106cm or 0.03ms in temporal units. We can then fit above the larger noise peaks during quiescent periods, and ∗ the logarithm of the frequency-summed burst profiles to that the precise slope of the fitting lines will be influenced expression (23) by varying the overall amplitude A. by DC offsets in the flux (i.e., the uncertainty in where Because 0.03 ms is a small offset, the pulse profile is true zero is). There is thus the need for more dedicated expected to be rather steep near the peak (without it, the high temporal resolution searches. Article number, page 8 of 12 Fan Zhang : Pulsar magnetospheric convulsions induced by an external magnetic field 7.2. Rotation measure Wealsoassumethatthislinetraversesalargeportionof the SMR, so that the RM measurement for FRB 110523 A close examination of FRB 110523 yields a rotation mea- is not a chance underestimate of the average FRB value. sure (RM) of ∼ −186 rad m−2, far in excess of that to be – For the Milky Way, ∼ 0.01 − 0.1pc (scales to other expected from the intergalactic medium and the signal’s galaxies linearly with SMBH mass) is where the mat- journey through the Milky Way, which when combined, ter is captured by Sgr A* and begins infalling (Melia onlygivesanRMthatisanorderofmagnitudelower(Ma- (2013, 1994); Quataert (2002)), which means that an sui et al. (2015)). The present proposal places the source accreting-material-supported magnetic field extending pulsar within a strongly magnetized region (SMR) in the out to around LSMR ∼ 0.005pc should be a safe guess galactic nucleus, and is therefore endowed with the ability (the headroom is larger for heavier SMBHs). to account for the dominant remainder. The expression for computing the RM is From these values, the n comes out at 0.04/B cm−3. e 11 For a reference scale, we can compute the n of the pul- RM (cid:90) n B dl e =0.812 e (cid:107) , (24) sar magnetosphere near its LC (where the magnetic field radm−2 cm−3µGpc strength is similar), which is given by Goldreich & Ju- lian (1969) Eq. 9 (not strictly valid at the LC, but the where B is the magnetic field along the line of sight, and (cid:107) Lorentz factor drops sharply as we move inward from it, n is the electron density. Ideally, we would like to know e so that this is still an approximate for nearby locations) as all three terms on the right-hand side and see if we can 7×10−2B /P ∼0.07B cm−3,whereB ∼B isthemag- reproduce the observed RM. However, we know very little z 11 z 11 netic field strength along the rotation axis in Gauss, and about the status of the interstellar medium in the close P ∼ 1s is the pulsar period. One major difference exists vicinityofaSMBH,andthusn isuncertain.Asasurrogate e between the SMR and the pulsar magnetosphere, however. test, then, we attempt to paint a general picture for n e The positively charged particles in the SMR are protons within the present proposal instead, and see if its inferred (with number density n = n ) instead of the less massive value from the RM is at least broadly consistent with this p e positrons. Therefore, to ensure self-consistency, we need to picture. checkthattheenergydensityoftheprotonsisstillsubdom- We first note that the SMR is expected to be relatively inanttothatoftheEMfieldintheSMR.Asimplecompu- clean outside of the accretion disk (we are statistically un- tationshowsthattheratiobetweenthem(protonoverfield) likely to be viewing the host galaxy disk exactly edge-on, is ∼2×10−3γB−3 , where γ is the average Lorentz factor therefore the FRB signal path to Earth traverses mostly 11 oftheprotons.Therefore,withaplasmatemperaturelower non-diskpartsoftheSMR,regardlessofwhetherthesource than∼1015B3 K,ourn estimateisindeedconsistentwith pulsar is itself embedded in the disk), as compared to far- 11 e the force-free assumption. ther out in the galactic nucleus. This is because charged particles need a source of resistivity to cross the magnetic field lines and therefore are prevented from entering our 7.3. Dspersion measure and the scattering tail region of strong magnetization (except for those in the accretion disk near the equatorial plane where mechanisms A galactic-center location for FRB sources has been in- for strong Ohmicdissipation exist, seeBlandford& Znajek voked by Pen & Connor (2015) to account for the large (1977)).Magneticreconnectionmayalsogenerateheatand DMs. The high electron density within the nucleus of the buoyancy in infalling gas, pushing it out (Igumenshchev & host galaxy is made responsible for the majority of the Narayan(2002)).We therefore expectmagnetic dominance DM, and the remainder places the sources at extragalac- (i.e., matter stress-energy density is negligible as compared tic but non-cosmological distances of hundreds of millions to that of the EM field), and thus a force-free environment of parsecs. Our scenario is somewhat different, as we place within the SMR. the pulsars close to the SMBH in a relatively clean SMR, in the eye of any scattering screen of a toroidal topology More specifically, as the magnetic field lines thread (see the inset of Fig. 5). Therefore, unless we view the host through the accretion disk, the electric field component E (cid:107) galaxy approximately edge-on, the radio signal would not along those lines accelerate charged particles and lift them havepassedthroughthecentralcoreofthescreens,andthe out of the disk. The charges are subsequently distributed intergalactic medium would still have contributed a signifi- along the magnetic lines in such a way that the lines be- cant fraction to the total DM. (The local SMR contributes come equipotential, shutting down further particle extrac- a negligible amount to the DM. For example, with FRB tion and establishing a state of quasi-equilibrium. This en- forces E Bi = 0, which is a necessary condition for force- 110523, the SMR contribution to DM can be estimated as i ∼ RM/(0.812B ) ∼ 10−4/B cm−3pc (note B is in µG), free electrodynamics. The charged particle density is then (cid:107) 11 (cid:107) the minimum value sufficient to short out E . out of a total of ∼ 600cm−3pc. The SMR is also not sig- (cid:107) We now turn to computing the RM-inferred n (after nificant in its contribution to the scattering tail, as it is e excluding the intergalactic and Milky Way contributions), physically too small, thus all locations within are too close and see if it is indeed consistent with a force-free environ- to the source). In fact, the scattering tail may offer a way ment. We start with the following ingredients: to distinguish which FRBs are viewed through a screen. Some FRBs (e.g., FRB 110220) show clear exponential – The magnetic field strength in the SMR is determined tails typical of multipath broadening, and Luan & Goldre- by the requirement that it exerts significant influence on ich (2014) also showed that the intergalactic medium does the pulsar magnetosphere at its LC, therefore we set its not appear sufficient to achieve this amount of scattering. value to B G. Therefore, screens closer to the sources need to be involved 11 – We take the geometric factor from projecting the mag- (the detection of scintillation in FRB 110523 further sup- neticfieldontoourlineofsighttobeontheorderofunity. ports this scenario). On the other hand, some signals, in- Article number, page 9 of 12 5 does appear to display a general trend. With better data (cid:43)(cid:43) I II quality and quantity, and a much more refined method for extractingthescatteringcontributiontothetrailingedge,it 4 h dt may become possible to separate the intergalactic medium Wi (cid:43)(cid:43) contribution to the DM from that of the screen, with the ad 3 (cid:43) (cid:43)(cid:43)(cid:43)(cid:43)(cid:43) former appearing as a stochastic spread in the horizontal e H (cid:43)(cid:43) direction around a smooth monotonic curve representing dth 2 (cid:144) (cid:43)(cid:43) the latter (assuming the existence of a universal temporal Wi (cid:43)(cid:43) (cid:43)(cid:43)(cid:43)(cid:43) scattering versus DM relationship for screens across galax- ail (cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43) ies). T 1 (cid:43)(cid:43)(cid:43) 7.4. Event rate 0 400 600 800 1000 1200 TheFRBshaveahighimpliedaggregateeventrateofupto DME cm(cid:45)3pc 7×104 per day at above 1.5Jy ms fluence across all sky directions (Law et al. (2015)). There are two ways to achieve Fig. 5. DM against tail-to-head ratio. The tail (head) section ahighrate:alargenumberofindependentsources,orlarge width is the distance from the end (beginning) of the pulse (cid:144) number of repeat events from each individual source. The to its peak. The red crosses correspond to DM values without the Milky Way contribution (i.e., DM ), and the gray shadow bubble-bursting model falls within the second category. If E crosses provide the original overall DM for reference. The two 500 Mpc is to be used as a conservative detectability hori- bluecrossescorrespondtothetwopulseprofilesplottedinFig.4 zon for the FRBs (the DM is contaminated by the source’s and replaces two red crosses. The inset shows the two lines of local environment, therefore we take predictions from Pen sight. No points are obscured by the inset. & Connor (2015) as a lower bound), then there would be millionsofgalaxieswithinavolumeofthatradius,assuming a∼1011 totalgalaxypopulationintheobservableuniverse cludingthoseplottedinFig.4,donotdisplaythesignature (Gottetal.(2005)).Furthermore,assumingthatthereisan ofascatteringtailasclearly.Onepossibilityisthatinsome SMBH at the center of each, we reach the upper FRB rate cases, our line of sight (line I in the inset of Fig. 5) threads limit with an average recurrence rate of once per month through the scattering screens in the galactic center, while (lower than the upperlimitof 3.2 perday set by Law et al. in other cases, we peek a more direct view (line II). If this (2015)) if there are ∼ 0.5 pulsars in the strongly magne- is true, then repeat bursts should retain their classifica- tizedcentralregion.Thenumberofnormalpulsarscloseto tion under the dichotomy, as they originate from the same our Sgr A* was estimated by Pfahl & Loeb (2004) to be source,anexpectationthatappearstohavebeenborneout around 100−1000 within about 0.02pc from it, and with a (Spitler et al. (2016); Scholz et al. (2016)). naivevolumetricaveraging,thenumberofpulsarsexpected Moreover, we should expect statistically that signals within a SMR of LSMR ∼0.005pc is 1.6−16, which covers with larger DMs would exhibit more pronounced scatter- our required population comfortably. Wecaution,however, ingsignatures,manifestingthemselvesasafattertailwhen that there appears to be a“missing-pulsar problem”(Mac- compared to the width of the leading edge (lack of such an quart et al. (2010); Dexter & O’Leary (2014)). Within the asymmetryisquotedinSpitleretal.(2016)asasignofthe present model, however, it is relatively easy to accommo- missing scattering tail). In other words, we normalize the date any deficiencies in the galactic center pulsar number tailbytheintrinsicwidthofthesignal,insteadofassigning by noting that the size of LSMR can be increased further all of the extendedness of the trailing half or the whole of with more massive SMBHs, and that the burst repeat fre- the profile as being due to scattering, which may lead to quency as well as the detectability horizon also have some significant overestimates (a simple plot of DM versus pulse room to expand into. width shows no discernible correlation). Our investigation is of course further complicated by the fact that we cannot remove the intergalactic component from the overall DM, 7.5. Correlated observations therefore a correlation will be loose at best, but a scatter plot of DM vs width ratios could still demonstrate a bias The observables considered so far have been somewhat cir- providedthatthescreencontributiontotheDMisnotcom- cumstantial in terms of placing the sources of the FRBs pletelyoverwhelmed.WemakesuchaplotinFig.52,which in galactic centers. In this section, we briefly consider the possibility and difficulties in using correlated observations 2 The data are taken from Thornton et al. (2013) for FRB to carry out more reliable source localizations. 110220, FRB 110627, FRB 110703, FRB 120127; from Lorimer Although for our proposed FRB mechanism to operate, et al. (2007) for FRB 010724; from Burke-Spolaor & Bannister thepulsarscanbeonstableorbitsaroundtheSMBHanddo (2014) for FRB 011025; from Champion et al. (2016) for FRB not need to be falling directly into it, occasions may never- 090625, FRB 130626, FRB 130628, FRB 130729; Ravi et al. thelessarisewhenthelatterscenariobecomesrealized,such (2015) for FRB 131104; Spitler et al. (2016) for FRB 121102; aswhenorbitalparametersarealteredbyperturbationsby Petroff et al. (2015) for FRB 140514; Masui et al. (2015) for other stars. This may present us with a distinguishing test FRB 110523. We chose the single-peak profiles, with the high- forthemechanism,namelythattheobservationoftheneu- est signal-to-noise ratio when data from multiple bursts were tron star’s tidal disruption by the SMBH (for example, if available. In particular, this excludes FRB 121002 from Thorn- ton (2013), which has the largest DM. The temporal sampling the disruption occurs before the NS reaches the innermost rateforthisdouble-peakedsignalisrelativelylow,butthefitted stable circular orbit, or ISCO, of the SMBH, the gravita- profile does not appear to show a scattering tail. tional wave signal detectable by space-based detectors will Article number, page 10 of 12