# On the nature of fast radio bursts

Ya. N. Istomin
P.N. Lebedev Physical Institute, Leninsky Prospect 53, Moscow 119991, Russia
Moscow Institute Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow region, 141700, Russia
E-mail: istomin@lpi.ru
###### Abstract

Scenario of formation of fast radio bursts (FRBs) is proposed. Just like radio pulsars, sources of FRBs are magnetized neutron stars. Appearance of strong electric field in a magnetosphere of a neutron star is associated with close passage of a dense body near hot neutron star. For the repeating source FRB 121102, which has been observed in four series of bursts, the period of orbiting of the body is about 200 days. Thermal radiation from the surface of the star (temperature is of the order of ) causes evaporation and ionization of the matter of the dense body. Ionized gas (plasma) flows around the magnetosphere of the neutron star with the velocity , and creates electric potential in the polar region of the magnetosphere. Electrons from the plasma flow are accelerated toward the star, and gain Lorentz factor of . Thermal photons moving toward precipitating electrons are scattered by them, and produce gamma photons with energies of . These gamma quanta produce electron-positron pairs in collisions with thermal photons. The multiplicity, the number of born pairs per one primary electron, is about . The electron-positron plasma, produced in the polar region of magnetosphere, accumulates in a narrow layer at a bottom of a potential well formed on one side by a blocking potential , and on the other side by pressure of thermal radiation. The density of electron-positron plasma in the layer increases with time, and after short time the layer becomes a mirror for thermal radiation of the star. The thermal radiation in the polar region under the layer is accumulated during time , then the plasma layer is ejected outside. The ejection is observed as burst of radio emission formed by the flow of relativistic electron-positron plasma.

###### keywords:
radiation mechanisms: general – stars: neutron
pagerange: On the nature of fast radio burstsLABEL:lastpagepubyear: 2017

## 1 Introduction

The closest to what is discussed in this article are models of FRBs origin by the interaction of neutron stars with other bodies (planets, comets, asteroids). These are primarily direct collisions of bodies with a neutron star (Di and Dai, 2017; Dai et al., 2016) and the interaction of a relativistic pulsar wind with a companion of a neutron star (Mottez and Zarka, 2014).

However, apparently, FRB is a different class of sources of radio emission, which are neutron stars. This can be seen from analysis of observations of the source FRB 121102.

## 2 Frb 121102

The large energy density in a source most likely suggests cataclysmic event, such as an explosion, and hence destruction of the source. Indeed, no recurrent events were recorded until recently. One radio source FRB 121102, discovered in November 2012, flared up ten times within 16 days () 926 days later (May 2015). With the exception of one long time interval between consecutive flashes, also by the way about 16 days, the time interval between bursts were random from seconds to hour. The average duty cycle was . Then after 164 days (November 2015) six more bursts were recorded during 25 days () with two breaks of 6 and 18 days and with the average duty cycle of . In September 2016 (after 287 days) the source FRB 121102 flashed four more times with the average time between bursts . Finally, after 340 days fifteen new bursts were recorded with average duty cycle of in August 2017.

We see that in the temporal activity the FRB 121102, apart from the duration of the radio emission pulses, , exhibits three characteristic times: 1) is the average time interval between consecutive bursts; 2) is the duration of series of bursts and the duration of continuous breaks between bursts in the series; 3) is the average time between series of bursts. These times are very different, .

Thus, the source FRB 121102 exhibits activity during the time in the form of short bursts of few milliseconds in duration. The values of duration of radio emission, intensity and duty cycle are close to the same values observed from so-called rotating radio transients (RRATs). However, although the duty cycle of RRATs isa random value, it is a multiplier of some constant time unit. This time unit is period of rotation of a neutron star, which increases with time like that for radio pulsars. The value of time derivative of the period, , allows us to estimate the magnetic field strength on the stellar surface. It is , which is similar to radio pulsars. RRATs rotate slower than radio pulsars (the period of rotation is of the order of ). Because of this there is no permanent generation of electron- positron plasma in the neutron star magnetosphere. Although the rotation of magnetic field, frozen into the star, generates the electric field in the magnetosphere, , the electric field is not sufficient for the continuous production of electron-positron plasma. However, strong magnetic field and the electric field, induced by the rotation of magnetosphere of the neutron star, can lead to the burst of the production of electron-positron plasma under some external action. Such external action in the case of RRATs is the Galactic and extragalactic gamma rays with energies above according to mechanism of RRATs developed by Istomin and Sobyanin (2011a,b). Now a natural question arises: what will happen if magnetized neutron star rotates even more slowly () than the neutron star which is the source of RRAT? There is strong magnetic field in its magnetosphere, but there is practically absent the electric field, which is necessary for acceleration of electrons and positrons in the magnetosphere and for beginning of a cascade plasma production process.

The electric field, , can also arise in the magnetosphere when a sufficiently dense flow of charged particles moves through the magnetosphere with the velocity . Thus, we come to the conclusion that if the source of FRB is a magnetized neutron star, which rotates slowly enough, and the birth of a relativistic plasma occurs in its magnetosphere, an external action is necessary to distort the magnetosphere and to create the electric field. The presence of a neutron star as a source of FRB is indicated by short duration of the burst of radio emission. Such an effect can be a flow of a sufficiently dense plasma. Note that from one flash to another flash the dispersion measure is not constant but varies within during the time . This would correspond to the motion of electron density inhomogeneities of the size of . This size is too small either for the Galaxy, or for mentioned intergalactic medium, but reflects the presence of plasma in the immediate vicinity of the neutron star. The characteristic values of plasma flow scale and its velocity can be estimated from the following relations. First, it is . Second, we assume that the repetition time of the burst series is the orbiting period a dense body (planet, comet, asteroid) around the neutron star, . Since , the orbit of a body is strongly elongated. Therefore . Here is the gravitational constant, and is the mass of the neutron star, which we put equal to . Thus, we have

 L=1013cm,u=6⋅106cm/s. (1)

## 3 Electric field

As a result, the scenario of interaction of a dense body orbiting around the magnetized hot neutron star (for FRB 121102), looks like that: close pass of the body at the distance less than causes evaporation from the body, and dense plasma flow around the magnetosphere of neutron star perturbing magnetic field in the polar region, and generating the longitudinal electric field. For the temperature of the surface of the neutron star of the order of , the energy flux of x-ray photons is (at the stellar radius of ), which exceeds the solar luminosity by times. The evaporated plasma has the temperature on the order of the temperature of evaporation of a solid body , and its thermal velocity is equal to . Here we chose the mean atomic number of the evaporated ions to be of the order of , and is the proton mass. Thus, , and the velocity of the flow around the neutron star magnetosphere is approximately equal to . It should be noted that the temperature of the neutron star should not be too large, such that the radiation force acting on the ionized gas does not exceed the gravitation force, . Here is the Stefan-Boltzmann constant, and is the Thomson scattering cross-section. The pressure of the plasma flow destroys the magnetic field of the neutron star at distances from the center of the star larger than a certain distance determined by equality of pressures, . The value of is the average ion charge, . We obtain . The region of perturbed magnetosphere at the distance from the star, called casp, has size of the same . However, this size decreases to the polar oval of small size on the stellar surface. Electron-positron plasma fills this polar region by a cascade process described below. The magnetic field at the level can be found from that the magnetic field of the neutron star at large distances is dipole, . Here is the value of the magnetic field on the surface of NS, . Thus, for . The electric field induced in the polar region, , is equal to for . Accordingly, the arising voltage in the cusp is equal to for (). The electric field originates in the open region of the magnetosphere of the neutron star, in its polar region around the axis of a stellar magnetic moment . On the surface of the star this region is almost a circle of the radius . This polar circle is the same as in neutron stars, which are radio pulsars. Knowing the magnitude of the magnetic field , we can determine the electron density in the incoming plasma flow, for . A small fraction of electrons from the incoming stream penetrates into the polar region of the magnetosphere. Their flux is equal to . Here is the efficiency of penetration of electrons into the polar magnetosphere, .

The electric field arising in the magnetosphere at the distance penetrates deep into the magnetosphere in the polar region. Its dependence on of the height above the surface of the star can be determined by solving the Laplace equation in the region bounded by the surface of the radius with boundary conditions:

 ∂2ψ∂h2+1ρ∂∂ρ(ρ∂ψ∂ρ)=0.

Replacing the transverse derivative by , we arrive at the equation

 ∂2ψ∂h2=4rh3ψ.

The solution of this equation is , where is the McDonald function of the first order. Since , one can use asymptotic presentation of the McDonald function for large arguments. As a result, we have

 ψ(h)=ψ0(hr)3/4exp[−4(rh)1/2+4]. (2)

For the practical purpose the asymptotic presentation (2) does not differ from the exact solution expressed by the McDonald function, and we will use the expression (2) below. We see that the longitudinal electric field exists only in the upper part of the polar tube, . The field is exponentially suppressed when approaching the star,. Thus, electrons falling into the polar region at are accelerated toward the star, , and get relativistic energy equal to , i.e. their Lorentz factor becomes equal to .

## 4 Production of electron-positron plasma

Thermal photons with energies propagating from the stellar surface are scattered by relativistic electrons and produce high-energy photons with energies ,

 ϵ′=ϵ4γ21+γ2θ2+4γϵ. (3)

Here the angle is the angle of propagation of a scattered photon with respect to the velocity of the relativistic electron. Since, as can be seen from (3), is large for photons scattered in the direction of propagation of the electron, , then we use the approximation . If we neglect the electron recoil, , then the maximum energy of the scattered photon () is . For the Planck spectrum with the temperature of the maximum energy density is at the photon energy . Here is the stellar surface temperature in units of , . Thus, the characteristic value of the thermal photon energy is . The value of is equal to . This means that for small scattering angles, , the electron completely loses its energy, . At larger angles, , the energy of the scattered photon is . The scattered photon propagates toward the star where the density of thermal photons increases , and the magnetic field also grows . Therefore it is possible to create electron-positron pairs by two ways: 1) in collisions of scattered photons with thermal photons moving towards, , and 2) birth in magnetic field, when the scattered photon intersects the line of strong magnetic field at the angle , . It should be noted that in contrast to radio pulsars, energetic electrons and photons propagate toward the star, where the birth of pairs is more efficient, but not outwards. In addition, curvature photons, which play a major role in initiation of the cascade production of pairs in magnetospheres of radio pulsars, have small energies here and are incapable to produce pairs. Indeed, the energy of a curvature photon is . Here is the Compton wavelength of electron, , is the radius of curvature of magnetic field lines in the polar region, .

Let us consider the process of production of electron-positron pairs by scattered photons:

1) For the production of a pair in collision of a scattered photon with a thermal photon, , taking into account (3), we obtain condition

 θ<[(2ϵ−1/γ)2−2/γ2]1/2≃2ϵ,ϵ>ϵ1=(21/2+1)/2γ.

The inverse Compton scattering cross-section in the laboratory coordinate system, associated with the neutron star, has the form (Berestetskii, Lifshitz and Pitaevskii, 1982)

 dσ=8πr2eγ2θdθ(1+γ2θ2+4ϵγ)2[1(1+γ2θ2)2−11+γ2θ2+ (4) 14(1+γ2θ2+4γϵ1+γ2θ2+1+γ2θ21+γ2θ2+4γϵ)].

Here is the classical radius of an electron, . Integrating the cross-section (4) over the angle from to , and then averaging over the Planck spectrum, we obtain the value of the thickness gained by an electron moving toward the neutron star when it is scattered by thermal photons

 τ=∫r∗hdh′∫∞ϵ1dnphdϵdϵ∫2ϵ0dσdθdθ=4aπr2eT2R\mathchar22λ3γ(Rh−Rr∗). (5)

The coefficient is proportional to the integral of the scattering cross-section (4) over the Planck spectrum, starting from the photon energies to infinity. It depends on the electron energy and the temperature of the star , . For parameters of interest, , the value of is . The coefficient in the right hand side of (5) is large, . It means that the thickness becomes of the order of unity fairly fast, , and the electron completely loses energy, emitting a gamma quantum with the energy .

Let us now find the efficiency of production of electron-positron pairs produced by collisions of energetic photons with thermal photons . The cross-section for pair production is (Berestetskii, Lifshitz and Pitaevskii, 1982)

 σ±=πr2e2(1−v2)[(3−v4)ln1+v1−v−2v(2−v2)],

where is equal to . Again averaging over the Planck spectrum, starting from the energy , and assuming , we get the value of the thickness gained by the gamma quantum of the energy of with respect to the production of a pair,

 τe=b2πr2eT2R\mathchar22λ3γ(Rh−Rhi). (6)

The numerical coefficient for is . Here is the initial height from which the gamma photon begins to move toward the star. We see that the birth of an electron-positron pair is even more effective, times, than gamma-ray radiation by an electron. Thus, the primary electron, passing the distance through the thermal radiation, produces a pair with electron and positron energies . In turn, a born pair produces a new pair in the photon field of the star, etc. So the cascade production of electrons and positrons occurs. The minimum number of possible cascades , if we do not take into account decrease of the energy of secondary electrons ( in the expression (5)), is determined by the condition

 qK∑n=1(1−q)n=1−Rr∗.

Summing up, we obtain . Substituting characteristic values , we find . Since , the number of pairs generated by one primary electron is determined by the relation ,

2) Now we consider the single-photon production of electron-positron pairs by scattered photons in the magnetic field of the neutron star. We need to know the number of generated photons with energy . They are larger than the previously calculated density of energetic photons , which collide with thermal photons to produce pairs. Expression (3) for the energy of scattered photons determines regions of angles and energies of primary photons,

 θ<(2ϵ−4ϵγ−1−γ−2)1/2≃(2ϵ)1/2,ϵ>1/2γ(γ−2)≃1/2γ2.

As before, using the cross-section (4), integrating over the Planck spectrum, we obtain expression for the thickness ,

 τ1=4a1πr2eT2R\mathchar22λ3γ(Rh−Rr∗), (7)

where the constant in the region of parameters . Thus, the mean free path of fast electron with respect to the production of secondary photons with energy is 3.8 times smaller than the mean free path for production of energetic photons with energies .

Scattered secondary photons emitted by electrons along magnetic field lines, and propagating toward the surface of the star, begin to cross magnetic field lines at an angle due to curvature of magnetic field lines in the polar region of the magnetosphere. Thus, the angle is equal to

 β=∫hihρ−1c(h′)dh′=32[(hir)1/2−(hr)1/2].

Here the value of is equal to the initial altitude, from which the photon begins to propagate. As we will see pair production occurs below the height . Because of this and because of strong dependence of the magnetic field strength on the height, angle can be considered to be a constant, . The probability of pair production in the magnetic field per unit time is (Berestetskii, Lifshitz and Pitaevskii, 1982)

 w=33/2αc29/2\mathchar22λb|sinβ|exp{−83ϵ′b|sinβ|}Θ(ϵ′|sinβ|−2). (8)

The quantity is the magnetic field intensity in units of the critical field, , is the fine-structure constant, , is the stepwise theta function. The height , at which a pair is born with the probability , is determined by the condition

 1c∫hihw(ϵ′,h′)dh′=1.

Setting , we obtain

 (hR)3=916ϵ′b0(hir)1/2Λ, (9)

where the value of is , and

 −53lnln{31/6217/6Rα\mathchar22λ(ϵ′)−2/3b1/30(hir)1/6}.

The characteristic values of are . We see that photons with energy produce electron-positron pairs only near the stellar surface, , while energetic photons, , can produce pairs at a distance . After pairs are born, they lose their transverse momenta, emitting synchrotron photons with energy . They can not produce new pairs. Thus, the cascade single-photon production of electron-positron pairs in a strong magnetic field in our case is actually absent, in contrast to the cascade production of pairs in magnetospheres of radio pulsars.

## 5 Plasma trap

Thus, we see that primary electrons with the Lorentz factor of effectively produce electron-positron pairs by directly interacting with thermal photons. In view of the cascade character of process, most of them have multiplicity and energies of with the energy spread of . However, they do not reach the surface because the flux of thermal photons from the surface pushes them out. As a result, electrons and positrons are accelerated outward from the star,

 dγdt=σTπ2cT460\mathchar22λ3(Rh)2.

Here is the Thomson cross-section, , and we substitute for the Stefan-Boltzmann constant in energy units. Integrating and setting , we obtain

 γ=2π345r2eT4R\mathchar22λ3(1−Rh).

In the region , where there is factually no electric field (2), secondary electrons and positrons get the energy ,

 γf=2π345r2eT4R\mathchar22λ3=1.6⋅105T48R6.

Positrons freely escape outside freely, while electrons obeying the condition

 κ=0.6γ5/T48R6>1

are reflected from the electric potential (2) and become trapped. This trap is formed from one side by the electric potential created by the plasma flow in the magnetosphere, and the thermal radiation of the star itself from the other side. Electrons stop at the distance from the star defined by the condition ,

 hr=r[1+14lnκ+38ln(1+14lnκ)]−2.

After reflection electrons begin to move toward the surface of the star under the action of the trapping potential , and, if it were no interaction with thermal photons moving toward them, they would come to the stellar surface with the energy corresponding to the Lorentz factor of . However, they emit gamma quanta, which produce new electron-positron pairs, are decelerated to nonrelativistic energies and are picked up again by the radiation of the star. The force acting on electrons moving outward from the star is potential since the cross-section for scattering of thermal photons is equal to the Thomson cross-section, which does not depend on the electron energy. Corresponding potential is equal to

 ψT=mec2γfRh.

The plot of the potential is shown in Figure 1. Figure also shows the potential of the electric field (Eq. 2) and the total potential .

As a result of pair production, electrons are accumulated at the bottom of the potential well formed by the total potential . The location of the bottom, , is equal to (with logarithmic accuracy)

 hm=r[1+14lnκ+14ln(2rR) (10) −58ln(1+14lnκ+14ln(2rR))]−2

Thus, a primary electron with energy , after almost reaching the surface of the star, and losing its energy, returns back, receiving from thermal photons energy less than the original one, , then again moves toward the star and again produces quanta and pairs until it settles near the height of . The multiplicity, i. e. the number of generated electrons per one primary electron, increases as . It should be noted that the decelerating force acting on an electron when it moves toward the star is not conservative, since the scattering cross-section is inversely proportional to the electron energy, , as can be seen from expression (5). Therefore, the motion of energetic electrons toward the star is not the motion in the total potential . Their deceleration occurs faster, , but not as for the motion in the potential . But for subrelativistic electrons, , located near the bottom of the potential well, the cross-section for scattering of thermal photons is the Thomson one, and their motion is potential in the potential .

The thickness of the layer , where electrons are accumulated, is small. It is determined by the spread of electron energies near the surface of the star after production of gamma-quanta and pairs, ,

 Δh=hmΔγ2γf(hmr)1/2<

This layer creates an additional electric potential . The equation for the potential is as follows

 d2ψshdh2−4rh3ψsh=enl(h)4rh3, (11)

where is the electron column density associated with the density by the relation due to the dependence of the cross-section of the magnetic tube on the height, . Since the thickness of the electron layer is initially small, we can assume , where is the Dirac delta function. The solution of equation (11) is

 ψsh=−eNer2h3m(hhmr2)3/4⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩exp(−4r1/2h1/2+4r1/2h1/2m),hhm.

We see that the electric field of the electron layer decreases exponentially on both sides. The characteristic width of the additional potential, , does not depend on the value of the potential, and is equal to , i.e. it is equal to the width of the magnetic tube at the altitude . The total potential is shown in Figure 2 with account of the additional electric field created by electrons trapped into the minimum of the potential.

At the height electrons form a local potential well, to which positrons are attracted. Figure 3 shows the potential acting on positrons, . In the absence of the potential of the electron layer , positrons are accelerated by the thermal radiation up to energy , then get even more energy from the potential . However, if we do not take into account the interaction of positrons passing through the layer with electrons, then positrons are not trapped inside the layer. Deceleration of positrons due to bremsstrahlung radiation in the potential is not enough for them to lose energy . Here we should take into account the collective interaction of passing positrons having the Lorentz factor with electrons and positrons trapped earlier. The plasma density in the layer is equal to , their plasma frequency is . A beam of positrons with a density excites plasma oscillations in the layer, which decelerate positrons due to beam instability. The increment of the beam instability is

 νb=31/224/3(nbnpγ+3)1/3ωp,np>nb/γ+3.

Positrons are trapped into the well , if the condition is satisfied, where is the time of passage of positrons though the plasma layer, . Let us show for characteristic values of quantities that the condition is satisfied already at low densities of the plasma in the layer. We take characteristic values: . As a result we obtain the condition

 np>2⋅1016(nb1cm−3)−2cm−3,

which is obviously valid for . Thus, we see that the thin layer of width arises in the polar region, where all electrons and positrons produced in the polar magnetosphere are accumulated. Plasma is practically neutral, subrelativistic electron-positron plasma in this layer. Electrons are confined there by the locking potential (Eq. 2), while positrons are held by confined electrons.

There is slight displacement of the position of the plasma layer from the height . Indeed, the drag force of the thermal radiation of the star becomes twice larger than the drag force acting separately onto electrons and positrons. In the expression for the value of (10) it is necessary to replace by . As a result, the shift of the position of the plasma layer is equal to

 Δhmhm=ln22(hmr)1/2[1−58(hmr)1/2]≃0.1.

## 6 Burst

Electron-positron plasma constantly accumulates in the layer. The number of pairs grows linearly with time, . Accordingly, the plasma column density grows. The thickness relative to scattering of the thermal radiation of the star by the electron-positron plasma of the layer, , grows also with time, and becomes equal to unity at time . This means that the light of the star will start to reflect effectively from the plasma layer. If the reflection coefficient from the layer and the surface of the star is , then the energy density of the thermal radiation in the polar region under the plasma layer will begin to grow exponentially with the characteristic time . Effective temperature of trapped radiation will grow in time until radiation pushes the plasma layer outside the magnetosphere. This happens when the radiative force acting on an electron and a positron of the plasma layer, , at the height becomes equal to the force acting on electrons from the locking potential at the same height, . As a result, we have

 Teff=108K(11r8Rκ)1/4≃109K.

## 7 Discussion

From observations of the repeated source FRB 121102 we conclude that there exists a hierarchy of quasiperiods in its radio emission. The time is the time between consecutive bursts. Probably, because of the high power of the bursts, this is the time of accumulation of energy, which is then released during short time, considerably less than . Time is the duration of a series of bursts and also the duration of long breaks between bursts in the series. Finally, the time is the average time of recurrence of bursts. Short burst time tells us about compactness of the source of radiation, apparently, a neutron star.

It should be noted that this region of polar magnetosphere, , is a source of gamma radiation in the range of annihilation line just before the burst of radio emission during the time , when there is accumulation of electron-positron plasma in the layer. The plasma density in the layer can be estimated from the condition , when the layer becomes the mirror, . This value will be also the density of gamma quanta . Correspondingly, almost isotropic flux of gamma radiation from the layer in the annihilation line is, , and the luminosity in annihilation line is .

## 8 Acknowledgments

This work was supported by Russian Foundation for Fundamental Research, grant numbers 15-02-03063 and 16-02-00788.

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