aa r X i v : . [ a s t r o - ph . H E ] D ec A mechanism for fast radio bursts
G. E. Romero , , M. V. del Valle , ∗ and F. L. Vieyro Instituto Argentino de Radioastronom´ıa (IAR, CCT La Plata,CONICET), C.C.5, (1984) Villa Elisa, Buenos Aires, Argentina and Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata,Paseo del Bosque s/n, 1900, La Plata, Argentina
Fast radio bursts are mysterious transient sources likely located at cosmological distances. Thederived brightness temperatures exceed by many orders of magnitude the self-absorption limit of in-coherent synchrotron radiation, implying the operation of a coherent emission process. We propose aradiation mechanism for fast radio bursts where the emission arises from collisionless Bremsstrahlungin strong plasma turbulence excited by relativistic electron beams. We discuss possible astrophys-ical scenarios in which this process might operate. The emitting region is a turbulent plasma hitby a relativistic jet, where Langmuir plasma waves produce a concentration of intense electrostaticsoliton-like regions (cavitons). The resulting radiation is coherent and, under some physical condi-tions, can be polarised and have a power-law distribution in energy. We obtain radio luminositiesin agreement with the inferred values for fast radio bursts. The timescale of the radio flare in somecases can be extremely fast, of the order of 10 − s. The mechanism we present here can explainthe main features of fast radio bursts and is plausible in different astrophysical sources, such asgamma-ray bursts and some Active Galactic Nuclei. I. INTRODUCTION
Fast radio bursts (FRBs) are recently discovered tran-sient sources of unknown origin ([1], [2], [3]). Theywere detected around 1 . . δt FRB ∼ − s. Their location at high Galacticlatitudes ( | b | > ◦ ) and high dispersion measurements(DM= 375 − − ) suggest propagation throughthe intergalactic medium and high redshifts. The ob-served radio fluences and the cosmological distances im-ply a total energy realise in radio waves of about 10 ergand luminosities of ∼ erg s − .The extremely rapid variability points to relativis-tic beaming, so the linear size of the source would be δx < c Γ δt FRB , where Γ is the Lorentz factor of thesource that is moving towards the observer [4]. Thebrightness temperatures associated with such compactand bright sources are extremely high: T b > Γ − K([5], [6]). This is well above the Compton limit for in-coherent synchrotron radiation. A coherent origin of theradiation, then, seems to be unquestionable.Additional constraints on the source can be obtainedif we assume that the ultimate origin of the radi-ation is magnetic. A lower limit on the magneticfield that sets the particle flow in motion is B > Γ − (10 − s /δ t FRB ) [6]. Even for large beaming,FRBs seem to be produced by compact objects of stellarorigin such as neutron stars, magnetars or gamma-raybursts (GRBs). In fact, a number of models have beenproposed in relation to such objects: delayed collapses ofsupermassive neutron stars to black holes [7], magnetarflares [8], mergers of binary white dwarfs [9], flaring stars[10], and short GRBs [11]. The radiation mechanism forthe coherent emission is unknown. ∗ [email protected] In this article we propose that FRBs are generatedthrough coherent emission produced by a relativistic jet.Under some rather general conditions, beamed electronsinteracting with self-excited strong turbulence producecollective radiation. The emission is generated by theinteraction of the electrons with cavitons, which are theresult of beam-excited Langmuir turbulence in a plasmatraversed by the jet. This radiation mechanism, previ-ously observed in laboratory experiments, has been stud-ied in the context of intraday variability of Active Galac-tic Nuclei (AGN) jets [12].In the next section we present the basics of our modeland show that it can explain the main features of FRBs.Then, in Sect. III, we discuss a possible astrophysicalscenario where our proposed mechanism might work. InSect. IV we discuss our results, the problem of radiowave attenuation, and the sensitivity of our model tothe different physical parameters. We close with a briefsummary in Sect. V.
II. EMISSION MECHANISM
The interaction of a relativistic electron beam with atarget made out of plasma results, through plasma insta-bilities, in the generation of strong turbulence. This in-duced turbulence can be characterised as an ensemble ofsoliton-like wave packets, called cavitons ([13], [14], [15]).These cavitons result from an equilibrium between the to-tal pressure and the ponderomotive force, which causes aseparation of electrons and ions. The cavity is then filledby a strong electrostatic field E . This effect has beenverified both in the lab ([16], [17]) and through numericalsimulations ([18], [19]).Electrons passing through a caviton will radiate be-cause they are accelerated by the electrostatic field,launching a broadband electromagnetic wave packet.The acceleration of the electrons of the beam in thesoliton field results in the superposition of the radia-tion of each electron. The main contribution is pro-duced by those cavitons with perpendicular orientationwhich yields perpendicular electron acceleration. If thebeam is uniform, the out-coming emission is not coher-ent. However, when some degree of inhomogeneity ex-ists in the beam density, coherent radiation is produced.The laboratory experiments clearly show that collectiveradiation processes occur when relativistic electrons arescattered by the cavitons; the electrons then produce aBremsstrahlung-type of radiation of coherent nature [12].If a magnetic field is present, the cavitons can be aspher-ical, yielding further circular polarised emission (even inthe absence of a magnetic field some degree of polariza-tion is expected due to random fluctuations of the size ofthe cavitons in the turbulent medium).The condition for the collective radiatiation mecha-nism to operate is that the ratio of beam to plasma den-sities be no smaller than 0.01 [20]. This constraint arisesin laboratory experiments that showed that bunching inthe beam depends strongly on the ratio of beam to back-ground plasma densities [21]; a theoretical analysis forthis effect is presented in Benford & Weatherall [12]. Forthe case of a power-law density fluctuation spectrum (asexpected in several turbulent regimes), the resulting ra-diation is also a power-law. The emerging spectrum ofthe emission is broadband, extending from the plasmafrequency ω e up to a cutoff around a few eV. In addition,the radiation is also relativistically beamed. For furtherdetails on the radiation process, readers are referred toWeatherall & Benford [22] and Benford & Weatherall[12].This emission mechanism has been discussed by Ben-ford [21] in the context of intraday variability in quasars.Benford showed how the development of cavitons and co-herent emission –extensively study in the laboratory– canalso take place in astrophysical sources regardless of thedifferent scales involved. He argued that despite an astro-physical jet might have an electron energy distribution,i.e. a spread in γ e , the jet velocity is always ≈ c in theplasma frame, as in experiments, and hence the coher-ent emission is unavoidable if the approriate conditionsin the plasma and beam are satisfied. A. Physical scenario
We propose that the interaction of a leptonic relativis-tic jet with a denser plasma cloud induces strong turbu-lence within the latter; the electrons then scatter withthe cavitons producing radiation. The presence of den-sity inhomogeneities in the jet causes the emission to becoherent. This latter condition is easily fulfilled becausethere is always some level of density inhomogeneity inastrophysical fluids, given the high Reynolds number ofthe flows [e.g., 23].A sketch of the physical situation is presented in Fig-
FIG. 1. Scheme of the physical scenario considered in thiswork. Not to scale.TABLE I. Main parameters of the model.Cloud parameters Value n c : density [cm − ] 6 × T c : temperature [K] 10 R c : radius [cm] 5 × Jet parameters ValueΓ: Lorentz factor 500 n j : density [cm − ] 6 × ure 1. In this system the required condition for collec-tive emission is n j /n c ≥ .
01, with n j and n c the jet andcloud densities, respectively. The minimum radiation fre-quency is set by the requirement that it must exceed theplasma frequency: ω e = 5 . × n / Hz. For a densityof n c ∼ × cm − the plasma frequency is ∼ . D of the cavitons induced into the cloud bythe jet is ∼ λ D [22], where λ D is the Debye length ofthe plasma λ D = 6 . p T /n cm. Then, D ∼ . (cid:18) T c K (cid:19) / (cid:18) n c × cm − (cid:19) − / cm , (1)where T c is the temperature of the cloud of plasma (seeSect. IV A for further discussion, this temperature is us-sually uncertain by an order of magnitude).The impact of the jet on the denser plasma producesa shock that propagates in the direction of the jet mo-tion and heats the cloud, which violently expands and isdestroyed on timescales given by [e.g., 24]: t c ∼ R c √ ξc ∼ × (cid:18) R c × cm (cid:19) s , (2)with ξ = n c /n j , and R c the radius of the cloud. Thisyields t c >> δt FRB .The pressure exerted by the jet will also accelerate thecloud to relativistic speed on a timescale of t g ∼ ξR c c ∼ (cid:18) R c × cm (cid:19) s , (3)which is also much longer than δt FRB .The other fundamental timescale of the system is thetime in which electrons radiate coherently; this scale isdetermined by the crossing time of the cloud, and dictatesthe duration of the event. In the laboratory frame, thecrossing time is given by: t cross = R c c Γ ∼ × − (cid:18) R c × cm (cid:19) (cid:18) (cid:19) s; (4)Both dynamical timescales, t c and t g , are far longerthan the duration of the FRB given, in our model, by thecrossing time t cross . This means that the radiative phe-nomena, and not the dynamical disruption of the cloud,are relevant to the evolution of the FRB. It is worth notic-ing that the crossing time and the clump radius are con-sistent with the upper limit imposed on the linear size ofthe source by the rapid variability.Two shocks will be formed in the jet-cloud system thatmight re-accelerate particles. However, the timescaleof the coherent losses of electrons is shorter than thetimescale of acceleration even for relatively strong mag-netic fields. For instance, if the magnetic field in thecloud has a value of 10 − G, we get t acc ∼ − s.This means that in the presence of cavitons, electron re-acceleration fails and they only emit coherently. Onlyprotons can be efficiently accelerated in the cloud, buttheir emission through the pp channel is too weak to bedetectable. B. Radiated power
The radiated power per electron in the coherent regionis given by [22]: P = E σ T c π n j πD π f " (cid:18) ∆ n j n j (cid:19) .
24 ln (cid:18) cDω e (cid:19) , (5)where ∆ n j /n j is the mean squared density fluctuationin the jet, E is the value of the electric field inside thecavitons, and f is the fraction of the cloud volume filledwith cavitons. This fraction is a free parameter in ourmodel; we adopt f ∼ .
1, consistently with experimentsthat showed that f can be as high as 0 . n /n ∼ − [22]. The value of E can be estimated by the conditionthat allows the formation of the cavitons, that is theelectric energy must be greater than the thermal en-ergy, i.e. E / πn k B T c << n ; wetake n = 10 n c , as in Weatherall & Benford [22]. Weconsider the energy ratio to be 0 .
1, obtaining: E ∼ . (cid:18) n c × cm − (cid:19) / (cid:18) T c K (cid:19) / statV cm − . (6)The total power P t is calculated as the power emittedper particle times the number of relativistic electrons in-side the cloud’s volume, N e ∼ / π n j R . This resultsin P t ∼ . × erg s − × (cid:18) T c K (cid:19) (cid:18) f . (cid:19) (cid:18) n j × cm − (cid:19) (cid:18) R c × cm (cid:19) . (7)Here, we have considered n c /n j ∼ ν = 1 .
382 GHzare 0 . − . ∼ ∼ . × erg s − .For a source characterized by the parameters of Table I,our model can account for the high fluences observed. InSect. III we discuss possible astrophysical sources withthese parameters.As mentioned above, the coherent emission is broadband, extending from the plasma frequency ω e up to ν max = 2 γ c/D . For γ ∼ Γ, this yields: ν max ∼ . × (cid:18) T c K (cid:19) − / × (cid:16) n c cm − (cid:17) / (cid:18) Γ500 (cid:19) GHz . (8)Therefore ν max ∼ . III. A POSSIBLE ASTROPHYSICALSCENARIO: LONG GAMMA-RAY BURSTS
Recent bright GRBs detected with the
Fermi satelliteimply bulk Lorentz factors of Γ ∼ γ ∼
500 are obtained [e.g., 30]. GRBsinvolve, then, the fastest bulk motions known to occur inthe Universe.On the other hand, massive progenitor stars of longGRBs, such as Wolf-Rayet stars, have strong winds witha clumpy structure (e.g., [31], [32]). Once the star im-plodes, one or more clumps can be reached by the jet,because of the relative high filling factor [33]. The in-teraction of clumps and/or clouds with jets, along withstrong turbulence generation, produces different phenom-ena such as particle acceleration, non-thermal emission,etc. (see [34], [24]). We propose here that the interactionof a long GRB jet with the clumped, residual wind of itsprogenitor can lead to the coherent emission previouslydiscussed.The density of a clump in the wind of a Wolf-Rayetstar can reach very high values, similar to those expectedin the atmosphere of massive stars ( n c ∼ − cm − ,[35], [34]). As these clumps move away from the star,they expand and their density decrease to values similarto the ones adopted in our model. On the other hand,the jet density is determined by the luminosity of theGRB, and it also decreases with the distance to the cen-tral source. For typical GRB luminosities of ∼ ergs − , and interaction distances of 10 − cm (this is lo-cated near the region where the afterglow emission is pro-duced), the density can easily reach values of n j ∼ . n c ,which are necessary for the proposed mechanism to oper-ate. Since the interaction between the clump and the jetshould occur far enough to the central engine, the radiocoherent emission is not screened neither by the densestellar envelope nor the GRB prompt emission.Finally, we note that the fraction q of the jet kineticenergy of a GRB transferred into the collective emissionis q ∼ − ; this is a rather slim value of q , consider-ing the theoretical upper limits obtained in Benford &Weatherall [12], that give q ∼ − − .
5. The energeticrequirements for this implementation of our mechanismare, then, not very demanding.
IV. DISCUSSION
Coherent radiation processes have been claimed to suf-fer severe attenuation by various absorption mechanismsin the context of AGNs (e.g., [36], [20]). The bright-ness temperature can be saturated by induced Comptonscattering and/or Raman scattering ([37], [36]). How-ever, Benford & Lesch [20] argued that confrontationof induced Compton absorption with plasma experimentsuggests that there is no observed saturation effect forhigh T B . Regarding Raman scattering, the theoreticalapproach in Levinson & Blandford [36] applies only toweakly turbulent environments. An order of magnitudeestimate, however, can be obtained. Strong Raman scattering dominates other decay pro-cesses of Langmuir waves if [36]: (cid:18) n T B12 ν (cid:19) > × , (9)where n is the plasma density in units of 10 cm − , ν is the frequency of the radio emission is GHz, and T B is the brightness temperature in units of 10 K.The brightness temperature T B can be calculated asfollows [e.g., 5]: T B ∼ S ν d k B γ ν ∆ t (10) ∼ . × KΓ (cid:18) S ν Jy (cid:19) (cid:18) d Gpc (cid:19) (cid:16) ν GHz (cid:17) − (cid:18) ∆ t ms (cid:19) − . For a typical burst, T B ∼ . × K. With these val-ues condition (9) holds. However, experiments can pro-duce effective brightness temperatures in excess of 10 Kusing laser-like devices to stimulate a hot plasma [17, 20].This discrepancy between theory and experiment mightarise in the fact that the above calculation is based on theweak-turbulence limit, which is not adequate to describethe situation under consideration here. We conclude thatthere is no reason, in the absence of a comprehensive the-ory of strong Langmuir turbulence, to rule out our modelgiven the experimental results.In our model we consider that a cloud of plasma inter-acting with the relativistic jet produces the required den-sity rarefaction for the generation of cavitons; howeverregions of very high density in the jet can be formed byother processes, like internal shocks, instabilities, etc. Inthese cases the coherent emission resulting from electron-caviton scattering might also be obtained.A different scenario might be associated with the mini-jets that are produced by dissipation of magnetic energyin a larger jet [38]. For instance, in the jet of a misalignedAGN, a minijet pointing towards the observer can impactwith a cloud from the broad line region (BLR). Theseclouds have high densities, typical sizes of R c ∼ cm[39] and temperatures T c ∼ × K [24]. The jets ofAGNs have Lorentz factors Γ j ∼
10, however minijetsmight be much faster (Lorentz factors Γ mj >> Γ j , withΓ mj ∼ t cross and the total power do not differ significantly fromthose obtained in the case discussed in Sect.II.Beam decollimation might occur in astrophysical jets.Decollimation into an angle Φ will not affect the collec-tive emission until Φ > / Γ, but then emission will dropgreatly if Φ ∼ π/ ∼ − . Evenif there is a multiple interaction, it would appear as a bitlonger FRB ( t ∼ − s) with some structure, as recentlyobserved [44].In a recent FRB detected in real time, Petroff andco-workers [40] have measured circular polarisation of21 ± <
10 % (with 1 σ of significance). There areat least two ways of producing circular polarization: 1)in a jet composed by electron/positron pairs, circular po-larisation might be due to Faraday conversion ([41], [42]);2) in case that some cyclotron modes scatter with the jet,higher levels of circular polarisation are expected. Botheffects can contribute to the high degree of circular po-larisation reported by Petroff et al. [40]. The radiationmechanism we are proposing might produce linear polar-isation as well [20]. The low levels of linear polarisationdetected may be due to Faraday rotation as proposed inPetroff et al. [40].Effects of external plasma, such as propagation effects(see e.g., [43]), can be also invoked to explain the ex-istence of some spectral features claimed to be present(e.g., bright bands of a width ∼
100 MHz, [2]).The first evidence of a two-component FRB disfavorsmodels that resort to a single high energy event involvingcompact objects (e.g., [44]). Our model, on the contrary,can easily explain two components or a structured timeprofile by multiple interactions of different clumps withthe jet. It can also account for two FRBs being producedby the same repeating source, as suggested by [45]. Fur-thermore, recent investigations conclude that FRBs arisein dense star-forming regions (e.g., [46], [47]), preciselywhere massive stars with clumpy winds are expected.Another aspect to be briefly discussed is the potentialcorrelation between long GRBs and FRBs. No such a cor-relation has been observed so far, but with only around10 events detected this is not surprising at all. Noticethat the interaction might occur at some distance fromthe region where the gamma-rays are produce, hence theGRB and the FRB are not exactly simultaneous. In addi-tion, the deceleration of the jet makes the beaming angleof the radio emission larger than that of the gamma-rayflux, making the GRBs more difficult to be detected incomparison to the FRBs. Finally, not all GRBs are ex-pected to gather the adequate environment or jet condi-tions for producing the FRB phenomenon.
A. Sensitivity to model parameters
The luminosity and event duration in our modeldepend on the characteristics of the target plasma(clump)–these are properties such as clump size, density,and temperature– and the jet parameters –density andLorentz factor–. Here we discuss the sensitivity of ourresults to these parameters.The detection frequency of FRBs imposes an upperlimit to the density of the cloud in our model (as dis-cussed in Sect. II A); hence denser clumps might emit athigher frequencies, beyond the radio band, and are dis- carded. The density value we adopted is reasonable fora clump of the wind of a massive star (such as a Wolf-Rayet).In the application of our model we have adopted a den-sity ratio n j /n c = 0 .
01; the required condition for collec-tive emission is n j /n c ≥ .
01, since 0 .
01 is a lower limitto the density ratio, bigger values can be adopted. If weconsider, for example, n j /n c = 0 .
1, we obtain detectablefluxes for smaller clumps ( R c = 5 × − cm), andjets with lower Lorentz factors (Γ ∼ − cm might produce undetectable fluxes.Clumpy structures in the wind of massive stars de-velop close to the surface of the star, with typical sizesof ∼ cm. We have adopted a larger clump’s size of5 × cm. This value is justified by the fact that theclump-jet interaction in our model takes place far fromthe central source, and at such distances the clump isexpected to naturally expand. In addition, in the laterstages before the final collapse of evolved stars, massiveejections occur from the outer atmosphere, as observedin the case of Eta Carina.We have adopted a temperature of the clump of 10 K,which corresponds to the temperature of massive starwinds [48]. This is a very conservative value, since stellarwinds are known to emit even soft X-rays. This impliesthat the plasma can reach temperatures of T = 10 − K([49], [50]). Incrementing the temperature one order ofmagnitude increases approximately one order of magni-tude the resulting luminosity; in this way, by adoptinga hotter clump other parameters can be relaxed, such asthe clump’s size and/or the jet’s Lorentz factor.We conclude, then, that there is a wide range of pa-rameters over which our model is able to reproduce themain features observed so far in FRBs.
V. CONCLUSIONS
We propose a model where FRBs are the result of theinteraction of relativistic jets with plasma condensations.In this context, the jet induces strong turbulence in thecloud, exciting Langmuir waves. In the strong turbulenceregime cavitons are formed, filling a substantial part ofthe volume of the cloud. In the presence of an inhomo-geneous jet, these cavitons coherently scatter electronswith Lorentz factors of ∼
100 producing radiation upto a frequency of ∼ cD − , with D the size of thecavitons. The result is a radio flare with an effectivebrightness temperature exceeding 10 K, and a dura-tion of ∼ − s in the observer reference system. If thejet density fluctuations have a power-law distribution,the resulting radiation can have a power-law spectra asthe synchrotron radiation, but it is not restricted by thehigh brightness temperatures derived assuming incoher-ent emission. We suggest that these events can accountfor at least a fraction of the FRBs. Several possible sce-narios can accommodate this mechanism. Among them,we briefly discussed jets of long GRBs colliding with in-homogeneities of the wind of the progenitor star or thewarm component of the ISM. Other possible scenarios forthe proposed mechanism will be discussed elsewhere.Polarisation measurements [e.g., 40] can shed light onthe magnetic fields present in the clouds and the defor-mation of the cavitons. ACKNOWLEDGMENTS