Axionic Boson Stars in Magnetized
Conducting Media and Monochromatic Radiations

Aiichi Iwazaki Department of Physics, Nishogakusha University, Shonan Ohi Chiba 277, Japan.
June 28, 1998
Abstract

Axions have been argued to form coherent axionic boson stars as well as incoherent axion gas. These are ones of most plausible candidates of dark matters. Since the axionic boson stars generate oscillating electric fields in an external magnetic field, they induce oscillating currents in magnetized conducting media, which result in emitting radiations. We show that colliding with a magnetic white dwarf, an axionic boson star can emit a monochromatic radiation with a frequency given by a mass of the axion.

preprint: Nisho-98/3

The axion is one of most plausible candidates of dark matters in the Universe. It is the pseudo Nambu-Goldstone boson associated with Peccei-Quinn U(1) symmetry (1), which was introduced to solve naturally the strong CP problem. The axions are produced mainly by the decay of axion strings (2); (3)which arise when the symmetry is broken spontaneously, or coherent oscillations (2); (3)of axion field in the early Universe; the coherent oscillations arise when the axion gains mass through QCD instanton effects. These axions form incoherent axion gas as an axion dark matter in the present Universe. Thus it is quite significant to confirm their existence. Several ways of detecting these axions have been proposed.

In addition to these incoherent axions, the existence of coherent axionic boson stars (4); (5)has been argued (6); (7). It have been shown numerically (7)that axion clumps are formed around the period of 111 GeV owing to both the nonlinearity of an axion potential and the inhomogeneity of coherent axion oscillations on the scale beyond the horizon. Their masses are about 10-12Msuperscript1012subscript𝑀direct-product10^{-12}M_{\odot} or less. Then, these clumps contract gravitationally to axion miniclusters (8)after separating out from the cosmological expansion. They are incoherent axions localized contrary to axion gas distributed uniformly. Furthermore, depending on their energy densities, some of these miniclusters may contract gravitationally to more compact coherent boson stars (9); (7); (10). Eventually we expect that in the present Universe, there exist the coherent axionic boson stars as well as the incoherent axion miniclusters and axion gas as axion dark matters. It has been estimated (11)that both of the axionic boson stars and the miniclusters may constitute a substantial fraction of the total mass of the axion dark matters. An observational implication of the axion miniclusters has been discussed (11).

In this letter we point out that colliding with the axionic boson stars, strongly magnetized stars can emit detectable amount of electromagnetic radiations with a frequency given by the axion mass. As we have shown in the previous paper (5), the coherent axionic boson stars ( we call them axion stars in this paper ) oscillate and generate oscillating electric fields in external magnetic fields. These electric fields induce oscillating electric currents in the conducting media. Thus we expect emission of electromagnetic radiations in the magnetized conducting media, when they collide with the axion star. Since most of them are absorbed inside the conducting media, observable emission is made at the surface of the media. Although the electric field itself is quite weak, the amount of the energy of the radiation is very large due to astrophysically large surface area of the media, e.g. white dwarfs, neutron stars. Consequently detectable monochromatic radiations are expected from the media. Because the strength of the electric fields is proportional to the strength of the magnetic field, the phenomena are revealed especially in strongly magnetized media such as neutron stars, white dwarfs e.t.c.. We show that when the white dwarf with magnetic field 106superscript10610^{6} Gauss collides with the axion star, the luminosity of the radiations is approximately 1021superscript102110^{21} erg/s (M/10-14M)4(m/10-5eV)5(σ/1020/s)superscript𝑀superscript1014subscript𝑀direct-product4superscript𝑚superscript105eV5𝜎superscript1020𝑠(M/10^{-14}M_{\odot})^{4}(m/10^{-5}\mbox{eV})^{5}(\sigma/10^{20}/s) where we have assumed that the radius of the white dwarf is 109superscript10910^{9} cm and that the conductivity σ𝜎\sigma is about 1020/ssuperscript1020𝑠10^{20}/s around the surface of the core with degenerate electrons in the white dwarf. M(M)𝑀subscript𝑀direct-productM\,(M_{\odot}) is the mass of the axion star ( the sun ) and m𝑚m is the mass of the axion.

Let us first explain briefly our solutions of the axion stars (5). The stars are coherent axionic boson stars with masses estimated to be typically 10-12MΩah2superscript1012subscript𝑀direct-productsubscriptΩ𝑎superscript210^{-12}M_{\odot}\Omega_{a}h^{2} (7); (9); ΩasubscriptΩ𝑎\Omega_{a} is the ratio of the axion energy density to the critical density in the Universe and hh is Hubble constant in the unit of 100100100 km s -11{}^{-1} Mpc -11{}^{-1} . The solutions of the stars have been obtained by solving a field equation of the axion a𝑎a coupled with gravity. Since the axion stars are expected to have small masses, 10-12M10-14Msimilar-tosuperscript1012subscript𝑀direct-productsuperscript1014subscript𝑀direct-product10^{-12}M_{\odot}\sim 10^{-14}M_{\odot} for 0.01Ωah210.01subscriptΩ𝑎superscript210.01\leq\Omega_{a}h^{2}\leq 1 , we have solved a free field equation of the field a𝑎a along with Einstein equations in a weak gravitational limit. Namely nonlinearity of axion potential has been neglected; we found that the nonlinearity appears when masses of the axion stars are bigger than 10-9Msuperscript109subscript𝑀direct-product10^{-9}M_{\odot} . It has turned out that our numerical solutions of the axion star with radius Rasubscript𝑅𝑎R_{a} may be approximated by the explicit formula,

a=fPQa0sin(mt)exp(-r/Ra),𝑎subscript𝑓𝑃𝑄subscript𝑎0𝑚𝑡𝑟subscript𝑅𝑎a=f_{PQ}a_{0}\sin(mt)\exp(-r/R_{a})\quad, (1)

where t𝑡t ( r𝑟r ) is time ( radial ) coordinate and fPQsubscript𝑓𝑃𝑄f_{PQ} is the decay constant of the axion whose value is constrained from cosmological and astrophysical considerations (3)such as 1010superscript101010^{10} GeV <fPQ<absentsubscript𝑓𝑃𝑄absent<f_{PQ}< 1012superscript101210^{12} GeV. a0subscript𝑎0a_{0} is a dimensionless numerical constant representing an amplitude of the axion field.

We note that the solutions oscillate with the single frequency given by the axion mass. In general, solutions of the axion stars with larger masses have many modes with various frequencies (4). This is a distinctive feature of the real scalar field a𝑎a , compared with that of boson stars (10)composed of complex scalar fields.

The radius Rasubscript𝑅𝑎R_{a} in the solutions is a free parameter and is related with the mass M𝑀M of the star,

M=6.4mpl2m2Ra,𝑀6.4superscriptsubscript𝑚𝑝𝑙2superscript𝑚2subscript𝑅𝑎M=6.4\,\frac{m_{pl}^{2}}{m^{2}R_{a}}\quad, (2)

with Planck mass mplsubscript𝑚𝑝𝑙m_{pl} . Numerically, for example, Ra=1.6×108m5-2cmsubscript𝑅𝑎1.6superscript108superscriptsubscript𝑚52cmR_{a}=1.6\times 10^{8}m_{5}^{-2}\mbox{cm} for M=10-12M𝑀superscript1012subscript𝑀direct-productM=10^{-12}M_{\odot} , Ra=1.6×1010m5-2cmsubscript𝑅𝑎1.6superscript1010superscriptsubscript𝑚52cmR_{a}=1.6\times 10^{10}m_{5}^{-2}\mbox{cm} for M=10-14M𝑀superscript1014subscript𝑀direct-productM=10^{-14}M_{\odot} , e.t.c. with m5m/10-5eVsubscript𝑚5𝑚superscript105eVm_{5}\equiv m/10^{-5}\mbox{eV} . A similar formula has been obtained in the case of the complex scalar boson stars. We have also found an explicit relation (5)between the radius and the dimensionless amplitude a0subscript𝑎0a_{0} in eq( 1),

a0=1.73×10-8(108cm)2R210-5eVm.subscript𝑎01.73superscript108superscriptsuperscript108cm2superscript𝑅2superscript105eV𝑚a_{0}=1.73\times 10^{-8}\frac{(10^{8}\mbox{cm})^{2}}{R^{2}}\,\frac{10^{-5}% \mbox{eV}}{m}\quad. (3)

Thus it turns out that our solutions are characterized by one free parameter, e.g. the mass of the axion star. These explicit formulae are used for the evaluation of the amount of radiations emitted by the axion stars in the magnetized conducting media.

We proceed to show how the coherent axion field of these axion stars generates an electric field in an external magnetic field B𝐵\vec{B} . The point is that the axion couples with the electromagnetic fields in the following way,

Laγγ=cαaEB/fPQπsubscript𝐿𝑎𝛾𝛾𝑐𝛼𝑎𝐸𝐵subscript𝑓𝑃𝑄𝜋L_{a\gamma\gamma}=c\alpha a\vec{E}\cdot\vec{B}/f_{PQ}\pi (4)

with α=1/137𝛼1137\alpha=1/137 , where E𝐸\vec{E} is electric field. The value of c𝑐c depends on the axion models (12); (13); typically it is the order of one. It is easy to see how the Gauss law is modified by this interaction,

E=-cα(aB)/fPQπ+“matter”𝐸𝑐𝛼𝑎𝐵subscript𝑓𝑃𝑄𝜋“matter”\vec{\partial}\vec{E}=-c\alpha\vec{\partial}(a\vec{B})/f_{PQ}\pi+\mbox{``% matter''} (5)

where the last term “matter” denotes an electric charge formed by ordinal matters. The first term in the right hand side represents electric charge, ρa=-cα(aB)/fPQπsubscript𝜌𝑎𝑐𝛼𝑎𝐵subscript𝑓𝑃𝑄𝜋\rho_{a}=-c\alpha\vec{\partial}(a\vec{B})/f_{PQ}\pi , formed by the axion under the magnetic field. Thus we find (14)that the axion star possesses an electric field, Ea=-cαaB/fPQπsubscript𝐸𝑎𝑐𝛼𝑎𝐵subscript𝑓𝑃𝑄𝜋\vec{E_{a}}=-c\alpha a\vec{B}/f_{PQ}\pi associated with this electric charge. Note that the strength of the field is quite small owing to smallness of the factor a/fPQ𝑎subscript𝑓𝑃𝑄a/f_{PQ} ; for instance, the factor is 10-8similar-toabsentsuperscript108\sim 10^{-8} for an axion star with mass 10-12Msimilar-toabsentsuperscript1012subscript𝑀direct-product\sim 10^{-12}M_{\odot} .

As is shown, the field a𝑎a oscillates in time, so that the electric field Easubscript𝐸𝑎\vec{E_{a}} also oscillates. Thus electric currents induced by this field oscillate with the same frequency as that of the field and consequently electromagnetic radiations are emitted.

Here we wish to point out that there are two types of the currents induced. One is the current, Jm=σEasubscript𝐽𝑚𝜎subscript𝐸𝑎J_{m}=\sigma E_{a} , carried by ordinary charged particles, e.g. electrons, protons etc, which is induced by the electric field Easubscript𝐸𝑎\vec{E_{a}} in a magnetized conducting medium with a conductivity σ𝜎\sigma . The other one is the current, Jasubscript𝐽𝑎J_{a} , constituted by the axion field itself; since electric charge, ρa=-cα(aB)/fPQπsubscript𝜌𝑎𝑐𝛼𝑎𝐵subscript𝑓𝑃𝑄𝜋\rho_{a}=-c\alpha\vec{\partial}(a\vec{B})/f_{PQ}\pi , in eq( 5) oscillates in time, an oscillating current, Ja=-cαtaB/fPQπsubscript𝐽𝑎𝑐𝛼subscript𝑡𝑎𝐵subscript𝑓𝑃𝑄𝜋\vec{J_{a}}=-c\alpha\partial_{t}a\vec{B}/f_{PQ}\pi arises (14)associated with this charge owing to the conservation of the current, Ja-tρa=0subscript𝐽𝑎subscript𝑡subscript𝜌𝑎0\vec{\partial}\vec{J_{a}}-\partial_{t}\rho_{a}=0 . Hence this type of the current is present even without conducting medium as far as the magnetic field is present.

It is important to know which is dominant one in astrophysical circumstances. The ratio of these two currents is given by Jm/Ja=σ/msubscript𝐽𝑚subscript𝐽𝑎𝜎𝑚J_{m}/J_{a}=\sigma/m . Since the axion mass is constrained approximately such as 1010/s<m<1012/ssuperscript1010s𝑚superscript1012s10^{10}/\mbox{s}<m<10^{12}/\mbox{s} , corresponding to the above constraint on fPQsubscript𝑓𝑃𝑄f_{PQ} , the current Jmsubscript𝐽𝑚J_{m} is dominant in the medium with larger conductivity than m𝑚m . For instance, conductivities σ𝜎\sigma inside of the white dwarf or the neutron star are much larger than m𝑚m . This is mainly because electrons density is much larger than that of normal metals whose σ𝜎\sigma is the order of 1017superscript101710^{17} /s with room temperature. Thus Jmsubscript𝐽𝑚J_{m} dominates over Jasubscript𝐽𝑎J_{a} inside of these stars with magnetic fields. On the other hand, the current Jasubscript𝐽𝑎J_{a} is dominant in the medium with smaller conductivity σmmuch-less-than𝜎𝑚\sigma\ll m . For instance, the conductivity around the surface of the core with degenerate electrons in the white dwarfs must be much smaller than m𝑚m . This is because electron density at the surface must vanishes by the definition of the surface of the white dwarf. In a realistic situation there is the atmosphere of H or He with nonvanishing conductivity above the core of the white dwarf. The boundary between the atmosphere and the core is obscure; the atmosphere near the core involves a fraction of carbons which compose the outer core and electrons are not fully degenerate in the outer core. Hence actual conductivity increases with the depth in the atmosphere, starting at the value of 00 , but the behavior of the value is not well understood in the region near the boundary. Apart from the region, however, it reaches the large value expected in the degenerate crystallized core whose properties are well understood. As a result an average value of the conductivity over a region including the envelope and the outer core can be smaller than the mass of the axion. In such a case the current Jasubscript𝐽𝑎J_{a} dominates over Jmsubscript𝐽𝑚J_{m} . Therefore we should take an appropriate current dominantly generating electromagnetic radiations, depending on whether or not the conductivity σ𝜎\sigma is larger than m𝑚m .

Anyway we found that the axion star generates the oscillating electric field Easubscript𝐸𝑎E_{a} under the external magnetic field and induces the oscillating currents Jmsubscript𝐽𝑚J_{m} and Jasubscript𝐽𝑎J_{a} in the conducting media. These oscillating currents generate the monochromatic radiations with the frequency given by the mass of the axion. Thus if the luminosity of the radiation is sufficiently large, we can determine the mass of the axion by the observation of the radiation. Thus it is important to estimate how large the luminosities in various media are. Here we take only white dwarfs as such media, which are estimated to dominate the halo of our galaxy and to cause the gravitational microlensing (15). Thus we expect that collisions between the white dwarfs and the axion stars occur so frequently in our galaxy that these radiations can be observed. In addition, we assume that the mass of the axion star is the order of 10-14Msuperscript1014subscript𝑀direct-product10^{-14}M_{\odot} corresponding to Ωah20.01similar-tosubscriptΩ𝑎superscript20.01\Omega_{a}h^{2}\sim 0.01 : The recent observations indicate much smaller values of Ωh2Ωsuperscript2\Omega h^{2} than 111 ; ΩΩ\Omega is the ratio of the energy density to the critical density.

Before evaluating the luminosity we need to take into account that almost of all radiations generated in this mechanism are absorbed inside of the white dwarfs owing to the large conductivity; there are so many free electrons to absorb the radiations. Only radiations emitted around the surface of the stars can escape the white dwarfs. Thus we need to know electromagnetic properties of the surface of the white dwarfs, in order to estimate the luminosity of the radiations emitted at the surface. ( Most of the energies of the radiations are dissipated in the degenerate cores of the white dwarfs. Hence the white dwarfs are heated and their temperatures increase. Among others, sufficiently cooled dark white dwarfs become bright again (16)with such a collision.)

In the white dwarfs there are H or He atmospheres above the surface of the core with degenerate electrons. Opacities of the atmospheres of relatively hot white dwarfs whose effective temperatures are much larger than 104superscript10410^{4} K, are known both observationally and theoretically. But knowledge of opacities of the atmospheres of the cool white dwarfs is very poor. In particular our concerns are dark white dwarfs with very low effective temperature. Such white dwarfs are expected to cause the gravitational microlensing (17). Their population is estimated to be very large; their total mass is about the half of the halo mass in our galaxy. Thus these very faint white dwarfs are dominant over ones having been observed. Such white dwarfs have been cooled enough to enter a Debye cooling regime where speed of the cooling is very fast; most of them in the halo must be very dim. So we do not have any observational evidence about their atmospheres.

Now, noting these circumstances we evaluate the luminosity of the radiations from the surface of the magnetized conducting media. Since we only consider white dwarfs and the axion stars with small masses 10-14Msimilar-toabsentsuperscript1014subscript𝑀direct-product\sim 10^{-14}M_{\odot} , the typical radius 109similar-toabsentsuperscript109\sim 10^{9} cm of the white dwarf is smaller than those of the axion stars; their radii are 1010similar-toabsentsuperscript1010\sim 10^{10} cm. Thus the white dwarfs can emit radiations when they are inside of the axion star. We denote a depth of a region from the surface by d𝑑d , in which radiations are emitted and can escape from the white dwarf. We also denote an average conductivity in the region by σ𝜎\sigma . These values are not well known so that we take them as free parameters. Noting that only radiations from a semi-sphere facing observers can arrive at them, we calculate electromagnetic gauge potentials Aisubscript𝐴𝑖A_{i} of the radiations with an appropriate gauge condition,

Aisubscript𝐴𝑖\displaystyle A_{i} =\displaystyle= 1R0surfaceJm(t-R0+xn)d3x1subscript𝑅0subscriptsurfacesubscript𝐽𝑚𝑡subscript𝑅0𝑥𝑛superscript𝑑3𝑥\displaystyle\frac{1}{R_{0}}\int_{\mbox{surface}}J_{m}(t-R_{0}+\vec{x}\cdot% \vec{n})\,d^{3}x (6)
=\displaystyle= cασa0BiπR0surfacesinm(t-R0+xn)d3x𝑐𝛼𝜎subscript𝑎0subscript𝐵𝑖𝜋subscript𝑅0subscriptsurface𝑚𝑡subscript𝑅0𝑥𝑛superscript𝑑3𝑥\displaystyle\frac{c\alpha\sigma a_{0}B_{i}}{\pi R_{0}}\int_{\mbox{surface}}% \sin m(t-R_{0}+\vec{x}\cdot\vec{n})\,d^{3}x (7)
=\displaystyle= 2cασa0BiRR0m2(mdcosm(t-R0)-2cosm(t-R0+R-d/2)sin(md/2))2𝑐𝛼𝜎subscript𝑎0subscript𝐵𝑖𝑅subscript𝑅0superscript𝑚2𝑚𝑑𝑚𝑡subscript𝑅02𝑚𝑡subscript𝑅0𝑅𝑑2𝑚𝑑2\displaystyle\frac{2c\alpha\sigma a_{0}B_{i}R}{R_{0}m^{2}}\,(md\cos m(t-R_{0})% -2\cos m(t-R_{0}+R-d/2)\sin(md/2)) (8)

where we have integrated it over the region around the surface with the depth dRmuch-less-than𝑑𝑅d\ll R . R0subscript𝑅0R_{0} is the distance between the observer and the white dwarf. Here we have used the current Jm=σEasubscript𝐽𝑚𝜎subscript𝐸𝑎J_{m}=\sigma E_{a} with the field a𝑎a in the approximate formula eq( 1) with setting exp(-r/Ra)=1𝑟subscript𝑅𝑎1\exp{(-r/R_{a})}=1 ; the white dwarf is involved fully in the axion star so that r/Ra1much-less-than𝑟subscript𝑅𝑎1r/R_{a}\ll 1 . On the other hand, the current Jasubscript𝐽𝑎J_{a} should be used for σm𝜎𝑚\sigma\leq m , in which case σ𝜎\sigma should be replaced with m𝑚m in the above formula. Using the gauge potentials, we evaluate the luminosity of the radiations,

L=83(σm)2c2a02B2R2K2,𝐿83superscript𝜎𝑚2superscript𝑐2superscriptsubscript𝑎02superscript𝐵2superscript𝑅2superscript𝐾2L=\frac{8}{3}(\frac{\sigma}{m})^{2}\,c^{2}\,a_{0}^{2}\,B^{2}\,R^{2}\,K^{2}\quad, (9)

with R𝑅R being radius of the white dwarf, where we have taken an average both in time and the direction of the magnetic field. K2superscript𝐾2K^{2} is given such that

K2superscript𝐾2\displaystyle K^{2} =\displaystyle= (m2d2+4sin2(md/2)-4mdcos(mR)sin(md/2))/2superscript𝑚2superscript𝑑24superscript2𝑚𝑑24𝑚𝑑𝑚𝑅𝑚𝑑22\displaystyle(m^{2}d^{2}+4\sin^{2}(md/2)-4md\cos(mR)\sin(md/2))/2 (10)
\displaystyle\cong m2d2/2for md1superscript𝑚2superscript𝑑22for much-greater-than𝑚𝑑1\displaystyle m^{2}d^{2}/2\quad\mbox{for $md\gg 1$} (11)
\displaystyle\cong m2d2(1-cos(mR))for md1.superscript𝑚2superscript𝑑21𝑚𝑅for much-less-than𝑚𝑑1\displaystyle m^{2}d^{2}(1-\cos(mR))\quad\mbox{for $md\ll 1$}\quad. (12)

In both limit K2superscript𝐾2K^{2} is proportional to m2d2superscript𝑚2superscript𝑑2m^{2}d^{2} . Thus it turns out that L𝐿L is proportional to σ2d2superscript𝜎2superscript𝑑2\sigma^{2}d^{2} for mσ𝑚𝜎m\leq\sigma , or to m2d2superscript𝑚2superscript𝑑2m^{2}d^{2} for σm𝜎𝑚\sigma\leq m ; as is shown soon later, the factors are large. We should note that the luminosity is proportional to the surface area R2superscript𝑅2R^{2} of the white dwarf. Thus the quantity is enhanced even if a luminosity per unit area in the surface is quite weak. This is the point we wish to stress. Generally, phenomena caused by the axion are too faint to be detected owing to small factor of m/fPQ𝑚subscript𝑓𝑃𝑄m/f_{PQ} . But in our case we have a large factor R2/m2superscript𝑅2superscript𝑚2R^{2}/m^{2} of the order of 1018superscript101810^{18} . We also have a large factor, σ2d2superscript𝜎2superscript𝑑2\sigma^{2}d^{2} or m2d2superscript𝑚2superscript𝑑2m^{2}d^{2} , depending on whether or not the mass is larger than the conductivity of the medium.

Since it is difficult to estimate accurately the depth d𝑑d of a region in which the radiation emitted can escape from the white dwarf, we assume for convenience that the depth is given by the penetration depth of the electromagnetic radiation. Then the depth is given in terms of the conductivity. In the case that the frequency m𝑚m of the radiation is much larger than the conductivity σ𝜎\sigma , the depth d𝑑d is given by d=1/2πσ𝑑12𝜋𝜎d=1/2\pi\sigma . On the other hand, the depth is given by d=1/2πσm𝑑12𝜋𝜎𝑚d=\sqrt{1/2\pi\sigma m} for mσmuch-less-than𝑚𝜎m\ll\sigma . Hence it follows that the factor σ2d2superscript𝜎2superscript𝑑2\sigma^{2}d^{2} of the enhancement is the order of σ/m𝜎𝑚\sigma/m for mσmuch-less-than𝑚𝜎m\ll\sigma and the factor m2d2superscript𝑚2superscript𝑑2m^{2}d^{2} is the order of m2/σ2superscript𝑚2superscript𝜎2m^{2}/\sigma^{2} for σmmuch-less-than𝜎𝑚\sigma\ll m . They are much larger than 111 .

Then with the assumption we find that the luminosity is numerically given by

L𝐿\displaystyle L similar-to\displaystyle\sim 1021erg/sB62R92σ20m55M144c2for mσsuperscript1021erg/ssuperscriptsubscript𝐵62superscriptsubscript𝑅92subscript𝜎20superscriptsubscript𝑚55superscriptsubscript𝑀144superscript𝑐2for much-less-than𝑚𝜎\displaystyle 10^{21}\mbox{erg/s}\,B_{6}^{2}\,R_{9}^{2}\,\sigma_{20}\,m_{5}^{5% }\,M_{14}^{4}\,c^{2}\quad\mbox{for $m\ll\sigma$} (13)
similar-to\displaystyle\sim 1021erg/sB62R92m58M144c2/σ52for mσsuperscript1021erg/ssuperscriptsubscript𝐵62superscriptsubscript𝑅92superscriptsubscript𝑚58superscriptsubscript𝑀144superscript𝑐2superscriptsubscript𝜎52for much-greater-than𝑚𝜎\displaystyle 10^{21}\mbox{erg/s}\,B_{6}^{2}\,R_{9}^{2}\,m_{5}^{8}\,M_{14}^{4}% \,c^{2}/\sigma_{5}^{2}\quad\mbox{for $m\gg\sigma$} (14)

where

B6=B106G,R9=R109cm,σ20=σ1020/s,σ5=σ105/s,M14=M10-14M.formulae-sequencesubscript𝐵6𝐵superscript106Gformulae-sequencesubscript𝑅9𝑅superscript109cmformulae-sequencesubscript𝜎20𝜎superscript1020sformulae-sequencesubscript𝜎5𝜎superscript105ssubscript𝑀14𝑀superscript1014subscript𝑀direct-productB_{6}=\frac{B}{10^{6}\mbox{G}},\quad R_{9}=\frac{R}{10^{9}\mbox{cm}},\quad% \sigma_{20}=\frac{\sigma}{10^{20}/\mbox{s}},\quad\sigma_{5}=\frac{\sigma}{10^{% 5}/\mbox{s}},\quad M_{14}=\frac{M}{10^{-14}M_{\odot}}\quad. (15)

We have used the penetration depth d=1/2πσ𝑑12𝜋𝜎d=1/2\pi\sigma for mσmuch-greater-than𝑚𝜎m\gg\sigma , and d=1/2πσm𝑑12𝜋𝜎𝑚d=\sqrt{1/2\pi\sigma m} for mσmuch-less-than𝑚𝜎m\ll\sigma , respectively. In both case we have simply assumed that values of both dielectric constant and magnetic permeability are the same as those of the vacuum. Numerically, d105similar-to𝑑superscript105d\sim 10^{5} cm /σ5absentsubscript𝜎5/\sigma_{5} for mσmuch-less-than𝑚𝜎m\ll\sigma and d10-5similar-to𝑑superscript105d\sim 10^{-5} cm (σ20m5)-0.5superscriptsubscript𝜎20subscript𝑚50.5(\sigma_{20}m_{5})^{-0.5} for mσmuch-less-than𝑚𝜎m\ll\sigma .

L𝐿L in eq( 13) is the luminosity of the radiation generated by the ordinary current Jmsubscript𝐽𝑚J_{m} dominant in the medium with σmmuch-greater-than𝜎𝑚\sigma\gg m . This is the case that the atmosphere of the white dwarf has a vanishing conductivity and the core with degenerate electrons has much large conductivity, e.g. 1020/ssuperscript1020s10^{20}/\mbox{s} , even at the surface of the core. This is the case of the sharp boundary of the white dwarfs. On the other hand, L𝐿L in eq( 14) is the luminosity of the radiation generated by the current Jasubscript𝐽𝑎J_{a} dominant in the medium with σmmuch-less-than𝜎𝑚\sigma\ll m . This is the more realistic case that a small conductivity of the atmosphere increases with the depth and reaches a large value at the core. Then, the average value of the conductivity in the region with the depth d𝑑d may be much smaller than m𝑚m ; for instance, σ105/ssimilar-to𝜎superscript105s\sigma\sim 10^{5}/\mbox{s} .

We would like to mention that the result in eq( 14) heavily depends on the mass of the axion. If the mass is given by 10-4superscript10410^{-4} eV, the luminosity becomes large such as L1029similar-to𝐿superscript1029L\sim 10^{29} erg/s. Similarly, the above results depend on the unknown parameters, mass M𝑀M of the axion star, magnetic field B𝐵B , conductivity σ𝜎\sigma , or depth d𝑑d of the white dwarf. Hence it is difficult to obtain the accurate value of the luminosity. But it is interesting that there is a possible range of the parameters for us to be able to observe the radiations.

The radiations are emitted during the collision between the axion star and the white dwarf. Thus the emission continues approximately for 2×1032superscript1032\times 10^{3} sec which is taken for one passing the other, when the relative velocity is 10-3csuperscript103𝑐10^{-3}c ; the velocity is supposed to be typical one of matters composing the halo. On the other hand if the axion star is trapped gravitationally by the white dwarf, the emission goes on longer.

In the above evaluation we have assumed implicitly that the axion star colliding with the white dwarf does not received any effects from it. Since the mass of the axion star is much smaller than typical mass 0.5×Msimilar-toabsent0.5subscript𝑀direct-product\sim 0.5\times M_{\odot} of the white dwarf, the form of the coherent axionic boson star may be fairly deformed by the gravitational effect of the white dwarf. Especially the density of a part of the axion star occupied by the white dwarf will increase. Then the amplitude a0subscript𝑎0a_{0} around the part becomes so large that the luminosity is enhanced. These effects of the deformation and the dissipation of the energy will result in the axion star being trapped by the white dwarf.

Finally we point out that the rate of the collision between the axion star and the white dwarf in our galaxy is large to be detectable, if the half of the halo is composed of the axion stars and the other half is composed of the white dwarfs. Suppose that the distribution of the halo with total mass 4×1011Msimilar-toabsent4superscript1011subscript𝑀direct-product\sim 4\times 10^{11}M_{\odot} is given such that its density (r2+3Rc2)/(r2+Rc2)2proportional-toabsentsuperscript𝑟23superscriptsubscript𝑅𝑐2superscriptsuperscript𝑟2superscriptsubscript𝑅𝑐22\propto(r^{2}+3R_{c}^{2})/(r^{2}+R_{c}^{2})^{2} with Rc=28subscript𝑅𝑐2similar-to8R_{c}=2\sim 8 kpc where r𝑟r denotes a radial coordinate with the origin of the center of the galaxy. Then it is straightforward to evaluate the rate of the collisions in a solid angle ω𝜔\omega per year,

0.5/year×1M1431m54ω5×50.5year1superscriptsubscript𝑀1431superscriptsubscript𝑚54𝜔superscript5superscript50.5/\mbox{year}\times\frac{1}{M_{14}^{3}}\,\frac{1}{m_{5}^{4}}\,\frac{\omega}{% 5^{\circ}\times 5^{\circ}} (16)

where relative velocity between the axion star and the white dwarf is assumed to be 10-3csuperscript103𝑐10^{-3}c . We have taken into account the fact that the earth is located at about 888 kpc from the center of our galaxy, simply by counting the number of the collisions arising in the region from 888 kpc to 505050 kpc. The rate is not necessarily large enough, but not so small for us not to be able to detect the radiation.

In summary, we have shown that the electromagnetic radiations are emitted with the collision between the white dwarf and the axion star. The radiations are monochromatic with the frequency given by the mass of the axion. Although the luminosity heavily depends on the unknown parameters, e.g. mass of the axion, there is a range of the parameters with which the luminosity is sufficiently large to be observable. Therefore, the detection of the radiations makes us determine the value of the mass. Both monochromatic radiations discussed in this paper and thermal radiations caused by heating up dark white dwarfs with the collision discussed in a previous paper (16)can be used for the detection of the coherent axionic dark matters.

The author would like to express his thank for useful discussions and comments to Professors M. Kawasaki and R. Nishi, and also for the hospitality in Tanashi KEK. This research is supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports No. 10640284

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