aa r X i v : . [ h e p - ph ] N ov Nisho-3-2014
Fast Radio Bursts from Axion Stars
Aiichi Iwazaki
International Economics and Politics, Nishogakusha University,6-16 3-bantyo Chiyoda Tokyo 102-8336, Japan. (Dated: Dec. 25, 2014)Axions are one of the most promising candidates of dark matter. The axions have been shownto form miniclusters with masses ∼ − M ⊙ and to become dominant component of dark matter.Some of the axion miniclusters condense to form axion stars. We have recently shown a possibleorigin of fast radio bursts ( FRBs ) by assuming that the axion stars are main component of halos:FRBs arise from the collisions between the axion stars and neutron stars. It is remarkable that themasses of the axion stars obtained by the comparison of the theoretical and observational event ratesare coincident with the mass ∼ − M ⊙ . In this paper, we describe our model of FRBs in detail.We derive the approximate solutions of the axion stars with large radii and constraint their massesfor the approximation to be valid. The FRBs are emitted by electrons in atmospheres of neutronstars. By calculating the optical depth of the atmospheres, we show that they are transparent for theradiations with the frequency given by the axion mass m a such as m a / π ≃ . m a / − eV).Although the radiations are linearly polarized when they are emitted, they are shown to be circularlypolarized after they pass magnetospheres of neutron stars. We also show that the FRBs are notbroadband and their frequencies have finite bandwidths owing to the thermal fluctuations of theelectrons in the atmospheres of the neutron stars. The presence of the finite bandwidths is adistinctive feature of our model and can be tested observationally. Furthermore, we show thatsimilar FRBs may arise when the axion stars collide with magnetic white dwarfs B ∼ G. Thedistinctive feature is that the durations of the bursts are of the order of 0 . PACS numbers: 98.70.-f, 98.70.Dk, 14.80.Va, 11.27.+dAxion, Neutron Star, Fast Radio Burst
I. INTRODUCTION
Fast Radio Bursts have recently been discovered[1–3] at around 1 . ∼ GeV/s is produced at the radiofrequencies. The event rate of the burst is estimated to be ∼ − per year in a galaxy. Furthermore, no gamma or Xray bursts associated with the bursts have been detected. Follow up observations[4] of FRBs do not find any signalsfrom the direction of the FRB. To find progenitors of the bursts, several models[5] have been proposed. They ascribeFRBs to traditional sources such as neutron star-neutron star mergers, magnetors, black holes, et al..Our model[6, 7] ascribes FRBs to axions[8], which are one of most promising candidates of dark matter. A prominentfeature of axions is that they are converted to radiations under strong magnetic fields. The axions form axion starsknown as oscillaton[9] made of axions bounded gravitationally. The axion stars are condensed objects of axionminiclusters[10], which have been shown to be produced after the QCD phase transition and to form the dominantcomponent of dark matter in the Universe. Furthermore, the axion miniclusters have been shown to form the axionstars by gravitationally losing their kinetic energies[10, 11]. Thus, the axion stars are the dominant component ofdark matter. We have recently proposed a progenitor of the FRBs that the FRBs arise from the collisions betweenthe axion stars and neutron stars. All of the properties ( duration, event rate and total radiation energy ) of theFRBs observed can be naturally explained in our model.In the paper, we present the details of our models. First of all, we derive the solutions of the axion stars by expandingthe axion field and the gravitational fields in terms of eigen modes cos( nωt ) with n integers, where the eigen value ω can be determined by solving axion field equation representing gravitationally bounded axions. It is given such that ω = m a − k / m a where k / m a ( ≪ m a ) denotes a gravitational binding energy of an axion bounded by the axionstar; k ≃ GM a m a where G and M a are the gravitational constant and the mass of the axion star, respectively. Then,we find that the mass M a of the axion star is given in terms of the radius R a = 1 /k of the axion stars such that M a = 1 / ( Gm a R a ). The value of the mass M a can be obtained by the comparison of the theoretical and observationaleven rate of FRBs in our production mechanism of the FRBs. It is remarkable that a mass M a ∼ − M ⊙ obtainedin such a way is coincident with the mass of the axion miniclusters previously obtained. The radus R a is given by R a ∼ km, which is much larger than neutron stars.The field configurations of the solutions are not static but oscillating and localized with radius R a = 1 /k ; a ( t, ~x ) ∝ cos( ωt ) exp( − k | ~x | ) ≃ cos( m a t ) exp( − k | ~x | ). We show that the oscillating electric fields ~E a ( t, ~x ) ∝ a ( t, ~x ) ~B ( ~x ) aregenerated on the axion stars under external magnetic fields ~B ( ~x ). Thus, when the axion stars collide with neutronstars with strong magnetic fields, the electric fields ~E a ( t, ~x ) are generated, which make electrons in atmospheres[12]of neutron stars coherently oscillate. Thus, the electrons emit coherent radiations with the frequency given by theaxion mass. Since the electrons are much dense in the atmospheres, the large amount of radiations with the frequency m a / π ≃ . m a / − eV) can be produced in the collisions. The total amount of the energy of the radiations isgiven by 10 − M ⊙ (10km / km) ∼ GeV, where the radii of the neutron stars and the axion stars are supposedto be 10km and 10 km, respectively. This is our production mechanism of FRBs.We also discuss the optical properties of the atmospheres of neutron stars. The geometrical depth of the hy-drogen atmospheres of old neutron stars is of the order of 0 . m a / π ) ≃ . m a / − eV). They can also pass through magnetospheres of neutron stars. After they pass the magne-tospheres, the radiations are circularly polarized owing to the absorption of the radiations with either right or lefthanded polarization. The circular polarization of a FRB has recently been observed[4].Although the radiations produced by our mechanism are monochromatic having the frequency given by ω = m a / π ,the observed radiations have bandwidths at least wider than the range 1 . ∼ . ω th ( ω ± ω th ) of FRBs arise from thermal fluctuations of electrons. Thus, thebandwidths are narrow. The presence of such narrow bandwidths owing to the thermal fluctuations is a distinctivefeature of our model and can be tested observationally.The relative velocities v c in the collisions are given by p GM ns /R ns , which is of the order of 10 km/s; M ns ( R ns ) denote mass ( radius ) of neutron stars. The fact explains the duration of the FRBs being of the order of ∼ p − GM ns /R ns ) and cosmolgical redshifts z . Owing to theeffects, the actual frequencies observed at the earth are given by ( m a / π ) × (1 − GM ns /R ns ) / (1 + z ) which is lessthan ( m a / π ) ≃ . m a / − eV).It is interesting to see that similar radio bursts are emitted when the axion stars collide with white dwarfs withvery strong magnetic fields ∼ G. In particular, the duration of the bursts is of the order of 0 . p GM wd /R wd ∼ × km/s, where M wd = 0 . M ⊙ and R wd = 10 km denote the typicalmass and radius of the white dwarfs, respectively. Furthermore, the bandwidths of the radiations are wider thanthose of radiations emitted by neutron stars. This is because the magnetic fields of the white dwarfs are much weakerthan those of neutron stars. Thus, we can distinguish them from the radiations emitted from neutron stars. Theproduction rate of the bursts is larger than the one of the FRBs observed, if the number of the white dwarfs withstrong magnetic fields B ≥ G is larger than 10 in a galaxy. Furthermore, their luminocities are much larger thanthose of the FRBs observed. On the other hand, the number of typical white dwarfs with magnetic fields ∼ Gis much larger than the one of the white dwarfs with B ≥ G. However, it is difficult to observe the radiationsfrom the white dwarfs with small magnetic fields ≤ G because the total amount of the radiation energies is is notsufficientlly large to be observable.In the next section (II), we derive approximate solutions of axion stars with small masses coupled with gravity. Wecan see that the axion field oscillates with the frequency m a / π . In the section (III), we show that electric fields areproduced on axion stars under external magnetic fields. They are parallel to the magnetic fields and oscillate withfrequency m a / π . In the section (IV), we determine masses of axion stars by comparison of theoretical with observedrates of FRBs. The masses are found to be coincident with those estimated previously as the masses of the axionminiclusters. In the section (V), we describe how the radiations are emitted from the collisions. Especially, we showthat they are emitted from atmospheres of neutron stars. We find that once the axion stars touch the atmospheres,they lose their energies by emitting radiations. In the section (VI), we show by calculating optical depth that theatmospheres are transparent for the radiations. We also discuss that the radiations are circularly polarized afterthey pass magnetospheres of neutron stars. In the section (VII), we discuss how the thermal fluctuations of electronsemitting FRBs give rise to narrow bandwidths of the radiations. In the section (VIII), we discuss that similar RFBsarise in the collisions between axion stars and magnetic white dwarfs. We find that their durations are of the orderof 0 . II. AXION STARS
First we would like to make a brief review of axions and axion stars. Axions described by a real scalar field a areNambu-Goldstone boson associated with Pecci-Quinn global U(1) symmetry[13]. The symmetry was introduced tocure strong CP problems in QCD. After the breakdown of the Pecci-Quinn symmetry at the period of much highertemperature than 1GeV, axions are thermally produced as massless particles in the early Universe. They are howeveronly minor components of dark matter. Since the axions interact with instanton density in QCD, the potential term − f a m a cos( a/f a ) develops owing to instanton effects at the temperature below 1GeV; f a denotes the decay constantof the axions. Thus, the axion field oscillates around the minimum a = 0 of the potential. But, the initial value ofthe field a at the temperature 1GeV is unknown. It can take a different value in a region from those in other regionscausally disconnected. Thus, there are many regions causally disconnected at the epoch around the temperature 1GeV,in each of which the axion field takes a different initial value; energy density is also different. With the expansion ofthe Universe, the regions with different energy densities are causally connected. Thus, there arises spatial fluctuationsof the axion energy density. The nonlinear effects of the axion potential cause the fluctuations with over densities insome regions grow to form axion miniclusters[10] at the period of equal axion-matter radiation energy density in theregions. Their masses have been estimated to be of the order of 10 − M ⊙ . Furthermore, these miniclusters condenseto form axion stars with gravitationally losing their kinetic energies[11]. Therefore, masses of axion stars are expectedto be of the order of 10 − M ⊙ .Now we explain the classical solutions of the axion stars obtained in previous papers[11, 14–16]. The solutionsare found by solving classical equations of axion field a ( ~x, t ) coupled with gravity. In particular we would like toobtain spherical symmetric solutions of axion stars with much smaller masses than the critical mass[16] M max ofthe axion stars; axion stars are stable when their masses are smaller than the critical mass M max ∼ . m /m a ≃ . × − M ⊙ (10 − eV /m a ). The mass is obtained only for free axion field without self-interaction. The gravity ofthe axion stars with much small masses is much weak so that we may take the space-time metrics given by ds = (1 + h t ) dt − (1 + h r ) dr − r ( dθ + sin θdφ ) (1)with both h t and h r ≪
1. It is easy to derive the equations of motion of the axion field and gravity,(1 − h t ) ∂ a = ∂ h t − ∂ h r ∂ a + (1 − h r )( ∂ r + 2 r ∂ r ) a + ∂ r h t − ∂ r h r ∂ r a − m a a∂ r h t = h r r + 4 πGr (cid:16) ( ∂ r a ) − m a a + ( ∂ a ) (cid:17) (2) ∂ r h r = − h r r + 4 πGr (cid:16) ( ∂ r a ) + m a a + ( ∂ a ) (cid:17) with the gravitational constant G , where we assume the axion potential such that V a = − f a m a cos( a/f a ) ≃ − f a m a + m a a / a/f a ≪
1. As we will see later, the assumption holds for the axion stars with small masses, e.g. 10 − M ⊙ .It is well known that there are no static solutions of real scalar fields coupled with gravity. The fields and themetrics oscillate with time. We expand the axion field and the metric such that a ( t, r ) = X n =0 , , ,,, a n ( r ) cos((2 n + 1) ωt ) = a ( r ) cos( ωt ) + a ( r ) cos(3 ωt ) , , ,h t,r ( t, r ) = X n =0 , , ,,, h nt,r ( r ) cos(2 nωt ) = h t,r ( r ) + h t,r ( r ) cos(2 ωt ) , , , (3)with a ( r ) ≡ a ( r ). Then, only by taking the terms proportional to cos ( ωt ) = 1 and cos( ωt ) in the equations (2), weobtain − ω (1 − h t + h r − h t a ( r ) = − ω h r − h t a ( r ) + (1 − h r )( ∂ r + 2 ∂ r r ) a ( r ) − m a a ( r ) (4) −
12 ( ∂ r h r − ∂ r h t + ∂ r h r − ∂ r h t ∂ r a ( r ) ∂ r h t = h r r + 2 πGr (cid:16) ( ∂ r a ( r )) − m a a ( r ) + ω a ( r ) (cid:17) (5) ∂ r h r = − h r r + 2 πGr (cid:16) ( ∂ r a ( r )) + m a a ( r ) + ω a ( r ) (cid:17) (6) ∂ r h t = h r r + 2 πGr (cid:16) ( ∂ r a ( r )) − m a a ( r ) − ω a ( r ) (cid:17) (7) ∂ r h r = − h r r + 2 πGr (cid:16) ( ∂ r a ( r )) + m a a ( r ) − ω a ( r ) (cid:17) . (8)We note that when the gravitational effects vanish i.e. G →
0, there is a solution a = ˜ a cos( m a t ) with ω = m a aswell as h , t,r = 0. Since we consider axion stars with small masses, ω is almost equal to m a ; m a − ω ≪ m a . That is,the binding energies m a − ω are much smaller than m a . Furthermore, as we will see later, the radius R a of the axionstars with small masses is very large; R a ≫ m − a . Thus, the term ∂ r a ( r ) is much smaller than the term m a a ( r ), i.e.( ∂ r a ( r )) ≪ ( m a a ( r )) . We may approximate the above equations in the following,( m a − ω ) a ( r ) + m a h t a ( r ) = ( ∂ r + 2 ∂ r r ) a ( r ) (9) ∂ r h t ≃ h r r (10) ∂ r h r ≃ − h r r + 4 πGrm a a ( r ) (11) ∂ r h t ≃ h r r − πGrm a a ( r ) (12)(13)with h r ≪ h t,r ≪ h r ≪ h t ≪
1, since h r ∼ O (cid:0) Gr (( ∂ r a ) + ( m a − ω ) a (cid:1) and the other metrics h ∼ O (cid:0) Gr m a a (cid:1) .Therefore, we obtain the equation of the axion field, − k m a a ( r ) = − m a ( ∂ r + 2 ∂ r r ) a ( r ) + m a φa ( r ) (14)with k ≡ m a − ω , where “gravitational potential” φ ≡ h t / ∂ r + 2 ∂ r r ) φ = 2 πGm a a ( r ) . (15)Obviously, k / m a represents a binding energy of the axion bounded to an axion star whose mass M a is given by M a = R d x (( ∂ a ) +( ∂ r a ) + m a a ) / ≃ R d xm a a ( t, r ) = R d xm a a ( r ) / ω ≃ m a .The equation (14) can be rewritten in the limit r → ∞ as − k m a a ( r ) = − m a ( ∂ r + 2 ∂ r r ) a ( r ) − Gm a M a r a ( r ) . (16)A solution in eq(16) is given by a ( r ) = ˜ a exp( − kr ) with k = Gm a M a . Thus, we find that the radius R a = k − =( Gm a M a ) − of the axion star is much larger than m − a for small mass M a . We can confirm numerically that thesolution in eq(16) represents approximate solutions of the equations (14) and (15). In this way we approximatelyobtain spherical symmetric solutions, a ( ~x, t ) = a f a exp( − rR a ) cos( m a t ) , (17)with r = | ~x | . The solutions represent boson stars made of the axions bounded gravitationally, named as axion stars.The solutions are valid for the axion stars with small masses M a ≪ − M ⊙ . The radius R a of the axion stars isnumerically given in terms of the mass M a by R a = m m a M a ≃
260 km (cid:16) − eV m a (cid:17) (cid:16) − M ⊙ M a (cid:17) , (18)with the Planck mass m pl . The coefficient a can be obtained by using the relations M a ≃ R d xm a a ( r ) / πm a a f a R a / m a ≃ × − eV × (10 GeV /f a ), where the average is taken in time, a ≃ . × − (cid:16) km R a (cid:17) − eV m a . (19)Thus, the condition a/f a ≪ M a ∼ − M ⊙ . We have simply usedthe mass 10 − M ⊙ for reference. But, the mass is the one we determine by the comparison of the theoretical andobservational event rates of FRBs, as we show below. The mass is much smaller than the critical mass M max ∼ − M ⊙ . Thus, the solutions represent stable axion stars. In this way the solutions along with the parameters R a and a can be approximately obtained. Obviously, the axion stars are composed of axions with much small momenta ∼ /R a .Here we should make a comment on the quartic term − ( m a /f a ) a /
24 of the potential V a = − f a m a cos( a/f a ) ≃− f a m a + m a a / − ( m a /f a ) a /
24. We have neglected the term in the above discussion. Although the term is muchsmaller than the mass term m a a /
2, the term gives a contribution − m a a ( r ) / (12 f a ) in the eq(9) which is comparableto the term ( ω − m a ) a ( r ) ≃ R − a a ( r ) when R a ≃ a . That is, the axion stars becomeunstable when the term is comparable to the term ( ω − m a ) a ( r ). Thus, our solutions are only stable for the axionstars with larger radii than 130km ( smaller masses than 0 . × − M ⊙ .) This indicates that the critical massbecomes much smaller than M max when the quartic term is taken into account. We do not yet know real critical masswhen we take account of the full potential V a of the axions. But the axion stars at least with masses smaller than0 . × − M ⊙ are stable since the approximation of neglecting the quartic term is valid. ( Much small critical masses ∼ − M ⊙ have been previously pointed out[17] using the procedure in the previous work[18]. But the procedure isonly valid for free fields coupled with gravity. Especially, it is not applicable for the real scalar fields with nonlinearinteractions such as the axions. The approximations such as < ˆ a > = < ˆ a >< ˆ a > is used in the reference forobtaining the axion stars where ˆ a represents axion field operator and the state | > does a state with number of axionsfixed, i.e. eigenstates of the axion number operator. On the other hand, our classical approximation is to assumethat the state | > represents a coherent state of axions. Thus, we have < ˆ a > = < ˆ a > . The critical mass shown inthe previous paper is of the order of 10 − M ⊙ , while as we have shown perturbatively, the critical masses are of theorder of 10 − M ⊙ . The difference comes from the use of the different approximations. We point out that the criticalmasses we obtain are of the same order of the magnitude as the ones shown in the paper[7]. Anyway, more rigoroustreatments are needed to see the precise values of the critical mass. )In our analysis, the radius 130km of the axion stars with the mass 0 . × − M ⊙ was roughly derived only as aguide for a critical radius. We may use the radius 10 km as a reference of the stable axion stars in the discussionbelow. III. AXION STARS IN MAGNETIC FIELDS
We proceed to discuss electric field ~E a generated on the axion stars under magnetic field ~B . It is well-known thatthe axion couples with both electric ~E and magnetic fields ~B in the following, L aEB = kα a ( ~x, t ) ~E · ~Bf a π + ~E − ~B α ≃ / k depends on axion models; typically it isof the order of one. Hereafter we set k = 1. From the Lagrangian, we derive the Gauss law, ~∂ · ~E = − α~∂ ( a ~B ) /f a π .Thus, the electric field generated on the axion stars under the magnetic field ~B is given by ~E a ( r, t ) = − α a ( ~x, t ) ~Bf a π = − α a exp( − r/R a ) cos( m a t ) ~B ( ~r ) π (21) ≃ . × ( = 2 × eV / cm ) cos( m a t ) (cid:16) km R a (cid:17) − eV m a B G . (22)We find that the electric field is very strong at neutron stars with magnetic fields ∼ G, while it is much weak atthe sun with magnetic field ∼ ~E a is parallel to the magnetic field ~B and oscillates coherentlyover the whole of the axion stars. When the axion stars are in magnetized ionized gases, the field induces coherentlyoscillating electric currents with large length scale R a of the axion stars. Thus the large amount of dipole radiationscan be emitted. Especially, electrons in the atmospheres of neutron stars oscillate and emit coherent dipole radiationswith the frequency m a / π . We should mention that the motions of charged particles accelerated by the electric field ~E a are not affected by the magnetic field ~B since ~B is parallel to ~E . Thus, they can emit dipole radiations.Here we make a comment that the electric fields in eq(21) are the ones generated at the rest frame of the axionstars. When the axion stars collide with neutron stars, the magnetic field ~b = ~v c × ~E a with v c ≪ ~v c represents a relative velocity between the axion stars and the neutron stars.Since v c ∼ .
1, the magnetic field ~b is much smaller than ~B . Thus, the effect can be neglected. Similarly, we canneglect the effects of the rotations of the neutron stars, whose velocities are much smaller than the relative velocities v c .As will be shown later, all the energy of a part of the axion star touching the atmospheres of neutron stars isreleased into the radiations, since the atmospheres of neutron stars are composed of highly dense electrons andions. The electric fields of the axion stars induce oscillating electric currents which produce the radiations. Namely,the neutron stars make the axion stars evaporate into the radiations. The frequency of the radiations is given by m a / π ≃ . × (10 − eV /m a )GHz at the rest frame of the axion stars. Therefore, when the axion stars collide withneutron stars, the large amount of the radiations is produced within a short period R a /v c being of the order of milliseconds; v c is of the order of 10 km/s, see later. These radiations can escape the atmospheres and magnetosphere ofneutron stars, because they are optically thin for the radiations as we show below. Thus, it is reasonable to identifythe radiations as the FRBs observed. IV. EVENT RATE OF FAST RADIO BURSTS
We calculate the rate of the collisions between axion stars and neutron stars in a galaxy. The collisions generateFRBs so that the rate is the event rate of the FRBs. By the comparison of theoretical with observed rate of thebursts, we can determine the mass of the axion stars. We assume that halo of a galaxy is composed of the axionstars whose velocities v relative to neutron stars is supposed to be 3 × km/s. Since the local density of the halo issupposed to be 0 . × − g cm − , the number density n a of the axion stars is given by n a = 0 . × − g cm − /M a .The event rate R burst can be obtained in the following, R burst = n a × N ns × Sv × , (23)where N ns represents the number of neutron stars in a galaxy; it is supposed to be 10 . The cross section S for thecollision is given by S = π ( R a + R ns ) (cid:0) GM ns /v ( R a + R ns ) (cid:1) ≃ . π ( R a + R ns ) GM ⊙ /v where R ns (= 10km )denotes the radius of neutron star with mass M ns = 1 . M ⊙ . It follows that the observed event rate is given by R burst = 0 . × − g cm − M a × × . π (10km + R a ) GM ⊙ − × ∼ − (cid:16) − M ⊙ M a (cid:17) (cid:16) − eV m a (cid:17) − M ⊙ M a . (24)Therefore, we can determine the masses M a of the axion stars by the comparison of R burst in eq(24) with the observedevent rate ∼ − per year in a galaxy. We obtain M a ∼ − M ⊙ when m a = 10 − eV. The parameters used abovestill involves large ambiguities. Furthermore, the rate becomes larger than that in eq(24) when we take into accountthe cosmological evolution of the Universe. Thus, the observed rate only constrains the masses of the axion stars ina range such that M a = 10 − M ⊙ ∼ − M ⊙ . It is remarkable that the mass M a ∼ − M ⊙ of the axion starsobtained is coincident with the masses of axion miniclusters[19] estimated previously.Using the formula eq(18) we find the radius ∼ km of the axion stars with the mass ∼ − M ⊙ , which is largerthan those of neutron stars. Then, when the collisions take place, the neutron stars pass through the insides of theaxion stars. As we will show later, when the axion stars touch the atmospheres of the neutron stars, the large amountof the radiations is emitted instantaneously so that the parts of the stars touching them lose their energies. Theamount of the radiation energy released in the collision is given by 10 − M ⊙ (10km / km) ∼ GeV. Thus, ourproduction mechanism of FRBs can explain the observed energies of the FRBs.
V. RADIATIONS FROM AXION STARS IN ATMOSPHERES OF NEUTRON STARS
We estimate how large amount of energies the axion stars emit as radiations in the collisions with neutron stars.In particular, we show that they rapidly lose their energies in the atmospheres[12] of neutron stars. We consider oldneutron stars which are dominant components of neutron stars in the Universe. Their temperatures ( magnetic fields) are assumed to be of the order of 10 K ( 10 G ). We also assume that the neutron stars have hydrogen atmospheres.First, we show how an electron emit radiations in the electric fields of the axion stars. The electric field ~E a onthe axion stars generated under magnetic fields makes an electron oscillate according to the equation of motion ~ ˙ p = ( − e ) ~E a + ( − e ) ~v × ~B + m e ~g with electron mass m e , where ~p and ~v denote momentum and velocity of the electron,respectively and ~g does surface gravity of neutron stars; | ~g | = M ns G/R ns . We note that the electric field ~E a is parallelto the magnetic field ~B . Thus, the direction of the oscillation is parallel to ~B . The magnetic field does not affect theoscillation. Similarly, the gravitational forces does not affect it since they are much weaker than the electric fields.Then, the equation of motion of the electron parallel to ~E a is given by ˙ p = − eE a . Since the electric fieldsoscillate such as E a ∝ cos( m a t ), the electron oscillates with the frequency m a / π so that it emits a dipole radiation.The amplitude of the oscillator is given by eαa B/ ( m a m e π ) ≃ . λ ∼ − eV /m a ) of the radiations. Thus, the emission rate of the radiation energy produced by a single electronwith the mass m e is given by˙ w ≡ e ˙ p m e = 2 e ( eαa B/π ) m e ≃ . × − GeV/s (cid:16) km R a (cid:17) (cid:16) − eV m a (cid:17) (cid:16) B G (cid:17) . (25)Electrons coherently oscillate in the volume λ , in which there exist a number of the electrons with their number N e = n e λ where n e denotes the number density of electrons. Then, the total emission rate ˙ W from the electrongas is given such that ˙ W = ˙ w ( n e λ ) = 2( n e λ ) ˙ p / (3 m e ). On the other hand, if the depth d of the atmosphere ofneutron stars is less than the wave length of the radiations, the number of electrons coherently oscillating is givenby n e dλ . Actually, the depth d of the hydrogen atmosphere with temperature of the order of 10 K is about 0 . ∼ − eV /m a ).We make a comment that the thermal effects of electron gas under consideration do not disturbe the oscillation bythe electric field. Since the temperatures of the atmospheres are supposed to be 10 K, the thermal energy ∼ p / m e = ( eE ) / m e m a ∼ eV( B/ G) with m a = 10 − eV. Although the oscillation is never disturbed in the thermal bath, the frequency of the radiationsrecieves the effect of the thermal fluctuations so that the radiations have finite bandwidth.We also make a comment about the depth d of atmospheres of neutron stars. We suppose that the densitydistribution ρ ( r ) is given by ρ ( r ) = ρ exp( − r/d ) with the depth d = k B T /mg ( m denotes average mass of theatoms composing the atmospheres, T does temperature of the atmospheres and k B does Boltzmann constant. ) Thedistribution may be obtained by solving the equation of the dynamical valance ∂ r P ( r ) = − ρ ( r ) g between pressure P and surface gravity g ≡ GM/R with the use of the equation of state P ( r ) = n ( r ) k B T of ideal gas. Here T denotesthe constant temperature and M ( R ) and n denote mass ( radius ) of the star and number density ( n = ρ/m )of atoms composing the atmosphere. For example, d ∼ T = 300K, g e = 9 . and m = 28GeV in thecase of the earth, while d ∼ . T = 10 K, g n = 10 × g e and m = 1GeV in the case of hydrogen atmosphereof neutron stars. Although the estimation is very rough, we can grip on the depth of the atmosphere of the neutronstars, which is given by 0 . K. We can see that the number density ofelectrons n e ( r ) = n exp( − r/ . r from the bottom of the atmospheres. Theradiations emitted in the atmospheres can pass through the atmospheres without absorption because the atmosphereis transparent for the radiations with transverse polarizations, as shown in the next section.We proceed to show that the axion stars rapidly evaporate into the radiations when they touch the atmospheres ofneutron stars. We assume that the atmospheres are composed of fully ionized hydrogen gas with temperature of theorder of 10 K, whose depth d is about ∼ . n e ( r ) = n exp( − r/ . n e ( r = 0) = n at the bottom is much larger than 10 /cm . In the paper we take n = 10 /cm .It approximately corresponds to the density 1g/cm . We consider the radiations arising from a region with volume dλ ∼ in the atmospheres. The emission rate ˙ W of the radiations from the region is given by,˙ W ∼ − ( dλ n e ) GeV/s (cid:16) B G (cid:17) ∼ GeV/s (cid:16) n e cm − (cid:17) (cid:16) km R a (cid:17) (cid:16) − eV m a (cid:17) (cid:16) B G (cid:17) , (26)where we have taken, for instance, the number density n e = 10 / cm of electrons in the region with the density ρ ∼ − g/cm , which is located roughly at the height r = 0 . n e , ˙ W becomes larger. ) On theother hand, the energy of the axion stars contained in the volume dλ = 10cm is given by 10 − M ⊙ / (4 πR a / ∼ GeV. This energy is smaller than the energy of the radiations ˙ W × − s ≃ GeV emitted within a time0 . /v e ∼ − s in which the axion stars pass the depth d = 0 . v e ofthe axion stars when they collide with the neutron stars, is given by v e = p G (1 . M ⊙ ) /R ns ≃ × − ≃ × km/s.Therefore, we find that the whole energy of the region with the volume λ d in the axion stars is transformed into theradiation energy when the region pass through the atmospheres.The purpose using the specific values n e = 10 cm − or the depth d = 0 . / cm in the atmosphere and that strong magnetic fields ≥ G is present. These assumptions are generallyacceptable. Thus, our production mechanism of the FRBs is fairly promising.
VI. TRANSPARENCY OF NEUTRON STAR ATMOSPHERE
The radiations produced in the atmospheres can pass through them and arrive at the earth. They are neverabsorbed within the atmospheres. We will show that the atmospheres are transparent for the radiations, even if n e ≃ cm − when the strong magnetic field stronger than B = 10 G is present. We assume that the temperaturesof the atmospheres are of the order of 10 K and that electron density is given by n e ( r ) = n exp( − r/ . . C ǫ ( r ) = n e ( r )( ω + ǫ ω c ) + ν ǫ ( r ) πe ν ǫ ( r ) m e (27)with ω = m a / π , ω c = eB/m e and ω p = eB/m p , where m p denotes proton mass and ν ǫ is given by ν ǫ ( r ) = 2 e ω m e + 4 n e ( r ) e Λ ǫ ( T, B, ω )3 T r πm e T (28)where we used ω/T ≪ T = 10 K and m a = 10 − eV. The parameter ǫ = 0 , ± denotes three types ofpolarizations; circular polarizations ǫ = ± ( polarized transverse to ~B ) and longitudinal polarization ǫ = 0 ( polarizedlongitudinal to ~B ). The explicit formula of Λ ǫ ( T, B, ω ) is given byΛ ǫ ( T, B, ω ) = 34 ∞ X n = −∞ Z ∞ Q ǫ ( n, T, B, ω, y ) dy (29)where Q ǫ ( n, T, B, ω, y ) = yA ǫn ( T, B, ω, y ) p θy + y (cid:0) y + θ + p θy + y (cid:1) | n | (cid:0) sinh( b/ (cid:1) | n | (30) A n ( T, B, ω, y ) = x n K ( x n ) y + b/ , A ± n ( T, B, ω, y ) = ω ( ω ∓ ω p ) ( y + θ + | n | p θy + y ) K ( x n )1 + 2 θy + y (31) b = 13 . B G 10 K T , x n = | ω/T − nb | p .
25 + y/b, θ = 1 + exp( − b )1 + exp( − b ) ≃ x ≃ | ω/T | p .
25 + y/b and x n =0 = | nb | p .
25 + y/b since ω/T ≃ − . K and K represent modified Besselfunctions.Here we note that the contributions of the sum over large integer n are very small because there are dampingfactors such as 1 / (2 | n | sinh( b/ | n | ) ≃ / (800) | n | and integrands of y have the factor exp( −| n |√ yb ) for large y . Theintegration R ∞ dy exp( −| n |√ yb ) gives a damping factor n − for large n . Thus, the main contribution comes from theintegral of R ∞ Q ǫ ( n = 0 , T, B, ω, y ) dy . We should also note the presence of the small factor ω / ( ω ± ω p ) ≃ ω /ω p ≃ − (10 G / B) in A ± n . The factor comes from the finiteness of proton mass; the recoil effect of the proton owing tothe absorption of the radiations. Therefore, the absorption coefficient can be approximately rewritten by C ± ( r ) ≃ n e ( r ) ω c πe ν ± ( r ) m e (33)where ν ± ( r ) ≃ n e ( r ) e Λ ± ( T, B, ω )3 T r πm e T ≃ n e ( r ) e R ∞ Q ± ( n = 0 , T, B, ω ) T r πm e T (34)with θ ≃ Q ± ( n = 0 , T, B, ω ) ≃ ω ω p yK ( | ω/T | p .
25 + y/b )(1 + y ) . (35)On the other hand, the absorption coefficient C ( r ) for longitudinally polarized radiations is given by C ( r ) ≃ n e ( r ) ω c πe ν ( r ) m e (36)where ν ( r ) ≃ n e ( r ) e Λ ( T, B, ω )3 T r πm e T ≃ n e ( r ) e R ∞ Q ( n = 0 , T, B, ω ) T r πm e T (37)with Q ( n = 0 , T, B, ω ) ≃ y | ω/T | p .
25 + y/b K ( | ω/T | p .
25 + y/b )(1 + y )( y + b/ . (38)Using these formulae, we can see the optical depth τ ± ( r c ) = R ∞ r = r c dr ′ C ± ( r ′ ) < r c in which thenumber density n e ( r c ) is equal to 10 /cm . Therefore, we find that the atmospheres are transparent for the radiationswith the circular polarizations. The transparency comes from the fact that the frequency ω = m a / π is much smallerthan the cyclotron frequencies ω c and ω p under the strong magnetic fields B = 10 G. Physically, the electric fields ofthe radiations hardly make electrons move transversely to the direction of the magnetic fields B . Thus, they cannotbe absorbed. On the other hand, we can easily see that the atmospheres are opaque ( R ∞ r = r c dr ′ C ( r ′ ) ≫ B . The charged particles are distributed to screen the electric field. Thenumber density of electrons ( positrons ) in the magnetospheres is given by the Goldreich-Julian density ≃ Ω B/ π ∼ cm − (cid:0) Ω / (2 π/ s) (cid:1)(cid:0) B ( r ) / G (cid:1) with angular velocity Ω of neutron stars. These electrons ( positrons ) absorb right( left ) handed circularly polarized radiations when the cyclotron frequency ω = eB ( r ) /m e becomes equal to m a / π ,respectively. Since B ( r ) decreases such that B ( r ) ∝ /r , the absorption arises around the location at the height r ab ∼ km above the surface of neutron stars. It implies that the absorption coefficient C ± ( r ab ) is much large fora type of circularly polarized radiations compared with the one for the other type of circularly polarized radiations,for instance, C + ( r ab ) ≫ C − ( r ab ). The spatial distribution of the electrons is different from the distribution of the0positrons. Therefore, the radiations passing through the magnetospheres are circularly polarized. Such a polarizationhas been observed[4] in FRB 140514.We would like to point out that the atmospheres may evapolate instantaneously when the radiations pass them. Thisis because even if only a fraction of the radiation energies is dissipated in the atmospheres, the energy is sufficientlylarge to make the atmospheres evapolate. For example, a fraction e.g. 10 − of the radiation energies 10 GeV, (10 − × GeV = 10 GeV ) gives a large energy 10 / ( N = 10 ) = 1GeV to each nucleon in the atmospheres; N ∼ / cm × . cm) = 10 . Then, it apparently seems that our production mechanism of the FRBs doesnot work. But we should note that the FRBs are also produced in envelopes present just below the atmospheres.The envelopes are more dense ( 10 / cm ∼ / cm ) in electron number density and deeper ( ∼ cm ) thanthe atmospheres. Thus, even if the atmospheres instantaneously evapolate, the radio bursts with sufficiently largeenergies as observed are produced in the envelopes of neutron stars. VII. NARROW BANDWIDTH
It apparently seems that the radiations are monochromatic, that is, their frequencies are given by the axion mass.On the other hand, the FRBs have been observed with the frequencies in the range of 1 . ∼ . K ≃
10 eV. Thus, we take account of thermal effects on the oscillations. Thekinetic energies ǫ k of the oscillations are given by p / m e = ( eE ) / m e m a ∼ eV( B/ G) with m a = 10 − eV.The energy is equal to the potential energy m e ω x e / ω = m a / π ; x e represents the amplitude of electrons. When the thermal fluctuations are added to the harmonic oscillations, theelectron motion may be described by the following Langevin equation, m e ¨ x e = − m e ω x e + η (39)where the thermal fluctuation is represented by η . Since we consider only the effect on the harmonic oscillation x e = x cos( ωt ), we take only the term of η = η cos( ω ′ t ). Then, the frequency ω ′ = ω + ω th ( ω ≫ ω th ) of electronscan be derived from the Langevin equation such that m e ( ω ′ − ω ) x ≃ m e ωω th x = η ; ω th = η / (2 m e ωx ). Thus,the fluctuations ω th in the frequency is obtained by taking average of the thermal fluctuation η with an appropriateGaussian distribution, < ω > = 14 m e ω x < η > = ω < y > x = T m e x = ω T ǫ k , (40)with η ≡ m e ω y . The Gaussian distribution is assumed to be given by exp( − m e ω y / T ).Thus, the thermal fluctuations in the frequencies of electrons are given by ω ± ω th = ω (1 ± ω th ω ) = m a π (cid:16) ± r T ǫ k (cid:17) ≃ m a π (cid:16) ± r × eV (cid:17) ≃ m a π (1 ± . , (41)where we take values T = 10 , B = 10 G and m a = 10 − eV. The thermal fluctuations of the electrons cause thefinite but narrow bandwidth of the FRBs. We should note that the fluctuation ω th /ω depends on the temperature T ,magnetic field B and axion mass m a , ω th ω = r T ǫ k ∼ . m a − eV 10 GB r T . (42)The equation is used for the estimation of the bandwidths of the radio bursts from the collision between axion starsand white dwarfs, which is discussed in next section.We should mention that the observed radiations receive several redshifts. The frequency of the electric fields inducedon the axion stars under the magnetic fields is equal to ω = m a / π . Since the axion stars collide with the neutronstars at the relative velocity p GM ns /R ns , the frequencies ω ns of oscillating electrons induced by the electric fieldsis given by ω ns = ω p − GM ns /R ns at the rest frame of the neutron stars. Thus, the radiations with the frequency1 ω ns are emitted by the electrons at the rest frame of the neutron stars. The radiations receive gravitational redshiftswhen we observe them far from the neutron stars. The frequency ω ′ is given by ω ′ = ω ns p − GM ns /R ns . Finally,the frequency of the radiations observed at the earth is given by ω ob = ω ′ / (1 + z ) = ω (1 − GM ns /R ns ) / (1 + z ) whenthe neutron stars are located at the places with redshift z . VIII. COLLISIONS WITH MAGNETIC WHITE DWARFS
Up to now, we have considered that FRBs arise from the collisions between axion stars and neutron stars. Similarly,FRBs may arise from the collisions between axion stars and magnetic white dwarfs[21]. Some of the magnetic whitedwarfs have strong magnetic fields such as 10 G. They have dense hydrogen atmospheres with temperatures of orderof 10 K and depths of the order of 10 cm. ( We can easily estimate the depth by taking account of the physicalparameters, the surface gravity g wd ≃ × g e and the temperature T = 10 K of the white dwarfs. Thus, the densitydistribution is given by n = n exp( − r/ cm). ) They have dense free electrons similar to the case of neutron stars.Thus, by the collisions with axion stars, radiation bursts similar to the observed FRBs are produced. It turns out thatthe duration of the bursts is of the order of 0 . p GM wd /R wd ≃ M wd and radius R wd of the white dwarfs are typically given by0 . M ⊙ and 10 km, respectively. Thus it approximately takes 0 . B ∼ G havewider bandwidths than those of the radiations from the neutron stars. Since the temperatures of the white dwarftsare equal to or less than 10 K and the magnetic field is equal to 10 G, we find using eq(42) that the fluctuations ω th /ω is three times larger than those of the radiations from the neutron stars with T = 10 K and B = 10 G.When we observe them at earth, they receive gravitational and cosmological red shifts. The effect of the gravitationalred shift is, however, very small; p − GM wd /R wd ≃
1. Hence, the frequencies observed at the earth are given by( ω ± ω th )(1 + z ) − , which are larger than the frequencies of the radiations from neutron stars located at the placeswith the redshift z .We should mention that the radio bursts from white dwarfs with B = 10 G are more energetic than those of theobserved FRBs. This is because the whole energies of the axion stars colliding with such white dwarfs are transformedinto radiations. The energies are of the order of 10 − M ⊙ ≃ erg. As we have shown, the axion stars are muchsmaller than the white dwarfs so that the whole of axion stars collide with the white dwarfs.Actually, the emission rate of the radiation energy produced by a single electron in the atmospheres is given by˙ w ≡ e ˙ p m e ≃ . × − GeV/s (cid:16) km R a (cid:17) (cid:16) − eV m a (cid:17) (cid:16) B G (cid:17) , (43)where the radiations are also dipole ones.The electrons in a volume λ ( λ = 2 π/m a ≃ − eV /m a ) denotes the wave length of the radiations )coherently emit the radiations. Thus, the emission rate of the coherent radiations in the volume is˙ w ( n e λ ) ≃ GeV/s (cid:16) B G (cid:17) (cid:16) n e cm (cid:17) (44)where n e denotes number density of electrons in the atmospheres. On the other hand, the axion stars have theenergies 10 − M ⊙ ( λ/R a ) ≃ GeV in the volume λ . Thus, when the axion stars pass the atmospheres in a period0 . B = 10 G and T = 10 K, we find that the optical depth τ ± ( r c ) = R ∞ r = r c dr ′ C ± ( r ′ ) < r = r c in which n ( r c ) = 10 cm − .We should make a comment that the typical white dwarfs have magnetic fields B ∼ G much smaller than 10 G.It leads to the numerical parameters ω /ω p ≃ − , b ≃ .
134 and θ ≃ /b . Then, we find that the atmospheres arenot transparent when the radiations are produced in the deep inside of the atmosphere with the electron density suchas n e = 10 cm − . However, when they are produced at the depth with n e = 10 cm − , they can pass through theatmospheres. But the emission rate in the volume λ is much small,˙ w ( n e λ ) ≃ GeV/s (cid:16) B G (cid:17) (cid:16) n e cm (cid:17) . (45)2Thus, the axion stars emit radiations with their energies ˙ w ( n e λ ) ( R a /λ ) ∼ GeV/s. Hence, the collisionsbetween the white dwarfs with B ∼ G and the axion stars does not produce the radiations with enough luminositiesto be observed at the earth when they arise in extragalactic origins.If the number of white dwarfs in a typical galaxy is of the order of 10 , only a small fraction 10 − ∼ − of thewhite dwarfs would be those with strong magnetic fields ≥ G and hydrogen atmospheres. Then, the production rate R burst of the FRBs emitted in the collisions with such magnetic white dwarfs is found such that R burst ∼ (10 − /year ∼ − /year) in a galaxy. The values are obtained by using the formula in eq(24) with the use of R wd = 10 km. Therate is ten times larger than or equal to the rate of the FRBs actually observed. In other words, if the typical numberof such magnetic white dwarfs in a galaxy is of the order of 10 , the rate of the bursts is approximately equal to therate of the FRBs observed. Thus, the FRBs associated with the white dwarfs can be observed with their frequenciesin a range 2GHz ∼ ≥ G in a galaxy is not known and the estimationof the number is difficult. Although we know the presence of such white dwarfs, the number of them could be veryfew. Thus, the event rate of the FRBs associated with the white dwarfs[21] could be much small so that the FRBsare undetectable.
IX. SUMMARY AND DISCUSSIONS
We have shown the details of a possible production mechanism of FRBs; FRBs arise from the collisions betweenaxion stars and neutron stars. We have found that the masses and radii of the axion stars are given by M a ∼ − M ⊙ and R a ∼ km, respectively. The axion stars are rapidly converted into radiations under strong magnetic fieldsof the neutron stars. The radiations are emitted in the atmospheres of the neutron stars. We have shown that theatmospheres are transparent for the radiations. The transparency comes from the presence of strong magnetic fields B ≥ G and the low frequencies ∼ ∼ ms ) and amount of the energies ( 10 erg ) of the bursts.It apparently seems that the radiations is monochromatic with the frequency given by the axion mass. But,the observed frequencies have finite bandwidths. We have shown that the bandwidths are caused by the thermalfluctuations of electrons emitting the radiations.In the actual collisions the tidal forces of the neutron stars distort the formation of the axion stars. When the axionstars are close to the neutron stars, the gravitational forces of the neutron stars are stronger than those of axion starsbinding themselves. Then, the axions freely fall to the neutron stars. But the coherence of the axions is kept becausethe number density of the axions in the volume m − a is quite large.Our mechanism predicts that there are no radiations with any frequencies after the bursts. This is consistentwith the results of follow-up observations[4]. It also predicts that FRBs contain circular polarizations. The circularpolarizations arise owing to the absorption of either right or left handed polarized radiations in the magnetospheres.Circular polarizations have recently been observed[4] in a FRB.Similar radio bursts may arise when the axion stars collide with magnetic white dwarfs. We have found that theduration of the bursts is of the order of 0 . ≥ G. Although the number of such white dwarfs in a galaxy is unknown, the production rate of the bursts issufficienlly large for them to be detectable if their number is larger than 10 in a galaxy.If the our production mechanism of FRBs is true, we can reach a significant conclusion that the axions are thedominant component of dark matter and their mass is about 10 − eV, which is in the window allowed by observationaland cosmological constraints[22].The author expresses thanks to Prof. J. Arafune for useful comments and discussions. [1] D. R. Lorimer, M. Bailes, M. A. McLaughlin, D. J. Narkevic, F. Crawford, Science, 318 (2007) 777.E. F. Keane, D. F. Ludovici, R. P. Eatough, et al., MNRAS, 401 (2010) 1057.[2] D. Thornton, B. Stappers, M. Bailes, et al. Science, 341 (2013) 53.[3] L. G. Spitler, J. M. Cordes, J. W. T. Hessels, et al., ApJ, 790 (2014) 101.[4] E. Petroff, et al., arXiv:1412.0342.V. Ravi, R. M. Shannon and A. Jameson, arXiv:1412.1599.3