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\preprint{Nisho-98/2} \title{Radiations from Oscillating Axionic Boson Stars
in an External Magnetic Field} 
\author{Aiichi Iwazaki}
\address{Department of Physics, Nishogakusha University, Shonan Ohi Chiba
  277,\ Japan.} \date{March 29, 1998} \maketitle
\begin{abstract}
We solve numerically a field equation of axions coupled with gravity 
and show solutions representing oscillating axionic boson stars 
with small mass $\sim 10^{-12}M_{\odot}$ and large radius $\sim 10^{8}$ cm.
We present explicitly a relation between the mass and the radius of 
the boson stars with such a small mass.
These axionic boson stars are shown to possess oscillating electric currents
in an external magnetic field and to radiate electromagnetic fields with 
a frequency given by mass of the axions. 
We estimate roughly the luminosity of the radiation in a strong magnetic field 
$10^{12}$G of neutron stars,  it is $\sim 10^{34}$ erg/s with
the axion mass $10^{-5}$eV. 
\end{abstract}
%\pacs{73.61.-r,73.20.Dx,73.40.Hm,73.40.Gk}
%\pacs{14.80.Mz, 98.80.Cq, 95.30.+d, 05.30.Jp, 98.70.-f \\Axion, 
%Boson Star, Dark Matter, Radiation 
%\hspace*{3cm}}
\vskip2pc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The axion is 
a Goldstone boson associated with 
Peccei-Quinn symmetry\cite{PQ} and is one of most plausible candidates of 
the dark matter in the Universe. 
The axions are produced \cite{kim,text} in early Universe 
mainly by the decay of the axion strings or 
coherent oscillation of the axion field. 
These axions form incoherent axion gas in the present Universe. 
It has been recently argued\cite{kolb} 
that not only the incoherent axion gas but also coherent axionic 
boson stars\cite{re} are produced 
by the coherent oscillation of the axion field.
Namely, inhomogeneity of the coherent oscillation on the scale beyond 
the horizon 
generates axion clumps as well as axion domain walls. The domain walls decay 
soon after their productions and leave a primordial magnetic field\cite{iwaza} 
as well as incoherent axions. 
On the other hand, the axion clumps contract gravitationally 
in later stage of 
the Universe to the axionic boson stars.
We call them simply as axion stars.  

In this letter we present numerical solutions \cite{real}
of the axion star by solving 
a field equation of the axion together with Einstein equations. Our concern
is confirming the existence of the axion stars with quite small masses 
and finding the explicit relation
between the radius ( or the typical value of the axion field ) 
and the mass of the star. This is because 
the mass of the axion star produced by the mechanism mentioned above 
has been estimated to be order of 
$10^{-12}M_{\odot}$ by Kolb and Tkachev\cite{kolb}.
Furthermore, using these solutions, 
we show that the axion stars in an external magnetic field 
can emit electromagnetic radiations with 
a frequency given by the mass of the axion.    
The luminosity of the radiations is roughly $10^{34}$ erg/s 
in the magnetic field $\sim 10^{12}$ G of neutron stars.
  
Now we show solutions of the axion stars 
by analyzing the field equations,

\begin{eqnarray}
\label{a}
\ddot{a}&=&\frac{(\dot{h_t}-\dot{h_r})\dot{a}}{2}+a''
+(\frac{2}{r}+\frac{h_t'-h_r'}{2})a'-m^2a\quad,\\
\label{h_t}
h_t'&=&\frac{h_r}{r}+4\pi Gr(a'^2-m^2a^2+\dot{a}^2)\quad,\\
\label{h_r}
h_r'&=&-\frac{h_r}{r}+4\pi Gr(a'^2+m^2a^2+\dot{a}^2)\quad,
\end{eqnarray}
where we have assumed gravity being small, i.e. $h_{t,r}\ll 1$ 
so that the metric is such that 
$ds^2=(1+h_t)dt^2-(1+h_r)dr^2-r^2(d\theta^2+\sin^2\theta d\phi^2)$;
$r,\theta$, and $\phi$ denote the polar coordinates.
The first equation is the equation of the axion field $a$ 
coupled only with gravity.
The second and the third equations are Einstein equations.  
A dot ( dash ) indicates a derivative in time $t$ ( $r$ ). $G$ ( $m$ ) 
is the graviational constant ( the mass of the axion ).
We have neglected the potential term of the axion
because an amplitude of the field $a$ is assumed to be sufficiently small for 
the nonlinearity of $a$ not to arise. This implies that the mass of the axion 
star is small enough. Actually the masses we are concerned with are 
such as $\sim 10^{-12}M_{\odot}$. We need to 
impose a boundary condition such as 
$h_r(r=0)=0$ for the regularity of the space-time.

Changing the scales such that $\tau=mt$, $x=mr$ and $b=a/m$, we rewrite 
the equations as follows,


\begin{eqnarray}
\label{bd}
\ddot{b}&=&\dot{V}\dot{b}+b''+(\frac{2}{x}+V')b'-b\quad,\\
\label{V}
V'&\equiv&\frac{h_t'-h_r'}{2}=\epsilon
(\frac{\int_0^xdxx^2(b'^2+\dot{b}^2+b^2)}{x^2}
-xb^2)
\end{eqnarray}
with $\epsilon=4\pi Gm^2$,
where we have expressed $V'$  
in terms of the field $b$, solving eq(\ref{h_t}) and eq(\ref{h_r});
here a dot ( dash ) denote a derivative in $\tau$ ( $x$ ).  
We understand that if the gravitational effect is neglected ( $\epsilon=0$ ),
the equation of $b$ is reduced to a Klein-Gordon equation.
Thus the frequency $\omega$ of the field $b$ is modified from the value
of the free field, $1$ ( $m$ in the physical unit ), only by a 
small quantity proportional to the gravitational effect, 
$\epsilon$; $\omega=1-o(\epsilon)$. 
We look for such a solution \cite{real} that 

\begin{equation}
\label{bB}
b=A_0 B(x)\sin\omega\tau+o(\epsilon)\sin3\omega\tau\quad, 
\end{equation}
where $B(x)$ represents coherent axions 
bounded gravitationally with its spatial extension representing 
the radius of the axion star; $B(x)$ is normalized 
such as $B(x=0)=1$. Later we find that 
$A_0$ is a free parameter determining
a mass or a radius of the axion star.  
The second term is a small correction of the order of
$\epsilon$.
Inserting the formula eq(\ref{bB}) into eq(\ref{bd}) and eq(\ref{V}) and 
taking account of the gravitational effects only with the order of $\epsilon$, 
we find that    

\begin{equation}
\label{B}
k^2B=B''+(\frac{2}{x}+\epsilon A_0^2(T+\frac{3U'}{4}))B'+
\frac{\epsilon A_0^2(U+v)B}{2}
\end{equation}
with 
\begin{equation}
T\equiv\frac{\int_0^xz^2B^2dz}{x^2}\quad \mbox{and}\quad
U\equiv\int_0^xdy(\frac{\int_0^yz^2B'^2dz}{y^2}-yB^2)\quad,
\end{equation}
where $k^2$ ( $=1-\omega^2$ ) is a binding energy of axions. We have imposed a 
boundary condition for the consistency such that 
$V(x=0)=h_t(x=0)/2=\epsilon A_0^2v\omega \sin2\omega\tau$ ; this is 
the definition of constant $v$ in the above formula.

We can see that the parameter $\epsilon A_0^2$ can take an arbitrary value
and that it represents the gravitational effect of this system.
Namely the mass of the axionic boson star is determined by choosing 
a value of the parameter. Note that the normalization of $B$ has been 
fixed in eq(\ref{B}) although the equation is a linear in $B$. 


We may take the value of $v$ without loss of generality such that 
$v=-U(x=\infty)$. Then the inverse $k^{-1}$ of the binding energy 
is turned out to represent
a radius of the axion star; $B$ decays exponentially 
such as $\exp(-kx)$ for $x\to\infty$.  
It turns out from eq(\ref{B}) that 
the choice of small values of $\epsilon A_0^2$ lead to solutions representing 
the axion stars with small mass and large radius $k^{-1}$.  




Before solving eq(\ref{B}) numerically, it is interesting to rewrite the 
equation as following. That is, we rewrite the equation by 
taking only dominant terms of a ``potential'', $V_b$, in eq(\ref{B}),

\begin{equation}
\label{Vb}
V_b=\epsilon A_0^2((T+\frac{3U'}{4})B'+\frac{(U+v)B}{2})
\end{equation}
in the limit of 
the large length scale; setting $x=\lambda y$, we take a dominant term  
as $\lambda \to \infty$. 
This corresponds to thinking the axion stars 
with their spatial extension being large. Note that our concerns are 
such axion stars with small mass and 
with large radius.


Then, since the dominant term in the limit is 
the last term in eq(\ref{Vb}), 
$U+v\sim \int_x^{\infty}dxxB^2$,
we obtain the following equation,

\begin{equation}
\label{BB}
\bar{B}=\bar{B}''+\frac{2\bar{B}'}{z}+
\frac{\bar{B}\int_z^{\infty}dyy\bar{B}^2}{2}\quad,
\end{equation}
where we have scaled the variables such that 
$B^2=k^2\bar{B}^2/\epsilon A_0^2$ and $x=k^{-1}z$;
a dash denotes a derivative in $z$.
This equation is much simpler than eq(\ref{B}), where we need to find each 
eigenvalue of $k$ for each value of $\epsilon A_0^2$ given, in order to obtain 
solutions of the axion stars with various masses. On the other hand,
we need only to find an appropriate value of 
$\bar{B}(z=0)=k^2/\epsilon A_0^2$ in order to obtain such solutions.
A relevant solution we need to find is the solution without any nodes.
Obviously, the solution is characterized by 
one free parameter, $k^2$ or $\epsilon A_0^2$, which is related to the mass 
of the axion star. Namely the choice of a value of the mass determines 
uniquely the properties of the axion star,
e.g. radius of the star, distribution of 
axion field $a$, e.t.c.. 

Although the equation (\ref{BB}) 
governs the axion star only with the large radius, 
we have confirmed by solving numerically original equation (\ref{B}) that 
the stars of our concern,
whose masses are at most $10^{-12}M_{\odot}$, can be controlled 
by the equation (\ref{BB}). Thus we may 
determine the explicit relation between the mass 
and the radius of the axion star, by finding 
the above solution of eq(\ref{BB}),
   
\begin{equation}
\label{MR}
M=6.4\,\frac{m_{pl}^2}{m^2R}\quad,
\end{equation}
where $m_{pl}$ is Planck mass.
Numerically we can see that for example, 
$R=1.6\times10^5m_5^{-2}\mbox{cm}$ 
for $M=10^{-9}M_{\odot}$, 
$R=1.6\times10^8m_5^{-2}\mbox{cm}$ for $M=10^{-12}M_{\odot}$,
e.t.c. with $m_5\equiv m/10^{-5}\mbox{eV}$.
We have also numerically confirmed that the axion field $a$ of the 
solution can be 
practically parameterized such that $a=fa_0\exp(-r/R)$ where $f$ is the decay 
constant of the axion whose value is constrained from cosmological
and astrophysical considerations\cite{text}; 
$10^{10}$ GeV $\le$ $f$ $\le$ $10^{12}$ GeV. This implies a constraint 
on the mass of the axion; $10^{-5}\mbox{eV}<m<10^{-3}\mbox{eV}$.
Amplitude $a_0$ ( $=mA_0/f$ ) is found to be expressed 
in terms of the radius $R$,

\begin{equation}
\label{aR}
a_0=1.73\times 10^{-8}\frac{(10^8\mbox{cm})^2}{R^2}\,
\frac{10^{-5}\mbox{eV}}{m}\quad.
\end{equation}  

In this way we have found solutions representing the axion stars with their 
mass $M$ being typically $10^{-12}M_{\odot}$ 
and with the radius $R$ being $10^{8}$cm.
The binding energy is turned out to be extremely small so that 
the frequency $\omega$ of the field $a$ is given by $m$. 


Next, we wish to show that 
the axion stars can emit electromagnetic radiations
with energy $m$ in an external magnetic field. The reason is that 
the coherent axion field gain an oscillating current in the magnetic field
owing to the following interaction,


\begin{equation}
   L_{a\gamma\gamma}=\frac{c\alpha a\vec{E}\cdot\vec{B}}{f\pi}
\label{EB}
\end{equation}
with $\alpha=1/137$,
where $\vec{E}$ and $\vec{B}$ denote electric and 
magnetic field, respectively. 
The value of $c$ depends on the axion models\cite{DFSZ,hadron};
typically it is the order of one.  


It is easy to see from this interaction that the coherent axion
field may have electric charge density 
$-c\alpha\vec{\partial}a\cdot\vec{B}/f\pi$ 
in the magnetic field $\vec{B}$\cite{Si}. 
We assume that the field $\vec{B}=(0,0,B)$ is spatially uniform and 
that the geometry of the axion field $a$  
representing the boson stars is spherical. Then we understand easily that 
this axion field has a charge distribution such that 
it has negative charges on a hemisphere 
( $z>0$ ) and positive charges on the other side of the sphere ( $z<0$ ). 
Net charge is zero. Thus, the star possesses the electric field, $E$,
parallel to the magnetic field associated with 
the charge distribution. As we have shown before, the field $a$
representing the axion star oscillates with frequency being approximately
given by $m$. Therefore, the charge distribution oscillates with the frequency
and so an electric current associated with the charge oscillates similarly.
This fact induces emission of the electromagnetic fields. 

The radiation from the axion stars can be understood in another way.
Since a mixing angle\cite{mix} of an axion and a photon generated 
by the interaction eq(\ref{EB}) 
is the order of $c\alpha B/fm\pi$ in the magnetic field $B$, 
equations of motion of gauge potentials $A_i$ can be derived such as 
$\partial^2 A_i=\partial^2 c\alpha B_ia/fm\pi \sim c\alpha B_im a/f\pi$. 
Hence, the oscillation of this current,
$c\alpha B_im a/f\pi$, produces the radiation as 
dictated above. 


It is easy to evaluate the luminosity $L$ of the radiation by assuming 
the configuration of the field $a$ such that $a=fa_0\exp(-r/R)$ whose
form we have confirmed numerically. 
Noting the wave length ( $=m^{-1}$ ) of the radiation is 
much smaller than the radius $R$ of the axion star, we obtain

\begin{equation}
\label{L}
L=\frac{64B^2a_0^2c^2\alpha^2}{3m^4R^2}\quad,
\end{equation} 
where we have assumed 
that whole of the axion star is involved in the external magnetic field.
On the other hand, when the length scale of the magnetized objects 
like neutron stars is smaller than the axion stars, $R$ in the above 
formula eq(\ref{L}) should be understood to denote the radius 
of the magnetized objects. Especially what we are concerned with is 
the collision of a neutron star with the axion star
whose radius is $10^8\mbox{cm}\sim10^{10}\mbox{cm}$. This radius is 
$10^2\sim10^4$ times 
larger than the radius of the neutron star.   
Thus $R$ in eq(\ref{L}) denotes the radius of the neutron star in such an 
example. 

We understand that since the luminosity $L$ is proportional to $B^2$, 
it is larger as the axion star is exposed to a stronger magnetic field.
Furthermore, as $L$ depends on $R^{-2}$, it is larger as the radius of 
the axion star or the magnetized object is smaller. Hence we understand 
that the neutron star among astrophysical objects yields large luminosity 
when it collides with the axion star. We estimate the luminosity of the case, 

\begin{equation}
L=2.7\times 10^{4}c^2\,\mbox{erg/s}\,\frac{B^2}{(10^{12}\mbox{G})^2}\,
\frac{M^4}{(10^{-12}M_{\odot})^4}\,
\frac{(10^6\mbox{cm})^2}{R^2}\quad,
\end{equation}
where we have used the above formulae eq(\ref{MR}) and eq(\ref{aR}) for 
expressing $a_0$ in terms of the mass $M$ of the axion star.
We note that this luminosity is that of the monochromatic radiation with the 
frequency, $m/2\pi=2.4\times 10^{9}$Hz$\,(m/10^{-5}\mbox{eV})$. 
This is much weaker than a corresponding luminosity of synchrotron radiations 
with the frequency from the neutron stars.
This emission continues until the axion star passes through the neutron stars.
It takes 
$10^8$cm $\times(10^{-12}M_{\odot}/M) /(3\times10^7$cm/s) $\sim10\,\mbox{sec}$ 
$(10^{-12}M_{\odot}/M)$,
assuming that the velocity of the axion star is 
$3\times10^7$cm/s, which is the typical 
velocity of matters composing the halo in our galaxy.
Thus it is impossible to observe the radiation. 

Until now we do not take into account 
the influence of the medium of the astrophysical object such 
as the neutron star.
When the medium is electric conducting, we need to consider 
its effects on the radiations or the electric field associated with the axion
star.
Here we would like to mention that the electric field possessed 
by the axion star in an external magnetic field induces 
an electric current in 
an astrophysical medium. This current also oscillates and emits a radiation.
Since the strength of the current is proportional to 
an electric conductivity, $\sigma$, 
of the medium, the medium with large conductivity such as the crust of 
the neutron star leads to strong radiations. 
In the case, the luminosity is approximately given by

\begin{equation}
L_{\sigma}=(\frac{\sigma}{m})^2L
\end{equation} 
where the conductivity should be taken 
as an conductivity averaged over the volume 
of the medium. For instance, in the case of the neutron star  
the conductivity of its crust has been estimated\cite{con}
theoretically such as $\sigma=$O($10^{24}$/s). When we adopt this value
for $\sigma$, the luminosity $L_{\sigma}$ is given by  
$\sim10^{34}$ erg/s. Although this is a rough estimation,
we can expect that the neutron star emit sufficiently strong radiations
to be detectable when it collides with the axion star. The detail analysis
is in progress.



%It is practically useful to rewrite the luminosity as 
%an intensity of the radiation received at the earth,
 

%\begin{equation}
%I=5.6\times 10^{-7}c^2\,\mbox{Jy}\,\frac{B^2}{(10^{12}\mbox{G})^2}\,
%\frac{M^4}{(10^{-12}M_{\odot})^4}\,
%\frac{m^3}{(10^{-5}\mbox{eV})^3}\,\frac{(1\mbox{kpc})^2}{r^2}
%\end{equation}
%where we have assumed that the neutron star or the white dwarf 
%is located at the distance $r$ from the earth; 
%$1$ Jy $=10^{-23}$ erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$.
%This intensity is much smaller than 
%those of synchrotron radiations from pulsars.


Finally we point out that the monochromatic radiation 
under the discussion should be  
detected for confirmation of the phenomena associated with the axion
just after the discovery of rebrightness of dark neutron stars
or white dwarfs caused by the axion star. The rebrightness of these stars 
has been discussed in the previous paper\cite{iwa} 
to be caused by energy dissipation 
of the axion star when it collide with them. 
These two phenomena, monochromatic radiations and 
thermal radiations owing to the energy dissipation, are distinctive in 
the collision of the axion star with these strongly magnetized stars.  


In conclusion, we have shown the solutions of the oscillating 
axionic boson stars with small masses. We have made explicit 
the relation between the mass and the radius of these axion stars.
They have also been shown to gain two types of oscillating electric currents 
in the magnetized conducting medium; currents made of axions themselves
and currents made of electrons in the medium.
Both of them emit monochromatic radiations with the frequency 
given by the mass of the axion. Having estimated the intensity of the 
radiations, we found that the radiations
by the currents of electrons are strong enough to be detectable.
But, since the duration of the emission is much short, 
it seems difficult to observe the radiations. 
  




%condense and form topological objects\cite{kim,text}, i.e. 
%strings and domain walls, 
%although they decay below the temperature of QCD phase transition. 
%After their decay, however, they have been 
%shown to leave a magnetic field\cite{iwa} 
%as well as cold axion gas as observable effects in 
%the present Universe; the field is a candidate of a primordial
%magnetic field leading to galactic magnetic fields observed 
%in the present Universe.
 

%In addition to these topological objects,  
%the existence of axion miniclusters has been argued\cite{hogan,kolb}.
%It have been shown numerically\cite{kolb} that in the early Universe, 
%axion clumps are formed around the period of 
%$1$ GeV owing to both the nonlinearity of an axion potential and 
%the inhomogeneity of coherent axion oscillations
%on the scale beyond the horizon. These clumps are called axitons 
%since they are similar to solitons in 
%a sense that its energy is localized. Then,
%the axitons contract gravitationally to axion miniclusters\cite{kolb2}
%after separating out from the cosmological expansion.
%They are compact bosonic objects contrary to 
%rather uniformly distributed axion gas 
%generated by the axion strings or the coherent axion oscillations.
%Furthermore, depending on their energy densities, 
%some of these miniclusters may contract gravitationally 
%to coherent boson stars\cite{Tk,kolb,re}.
%Eventually we expect that in the present Universe, 
%there exist the axion miniclusters and the axion boson stars
%as well as the incoherent axion gas as dark matter candidates. 
%It has been estimated\cite{fem} that a fairly amount of the fraction of 
%the axion dark matter comprises these  
%axion clumps. An observational implication
%of the axion miniclusters has been discussed\cite{fem}.


%In this letter we wish to point out an intriguing 
%observable effect yielded 
%by these coherent axionic boson stars. Namely, they 
%release their energies in magnetized conducting media so much that
%resultant radiations are observable. 
%The coherent axion stars are shown to generate electric fields 
%in external magnetic fields. These electric fields induce electric currents
%in the conducting media and loose their energies owing to existence of 
%resistances. Consequently radiations are expected from the media heated. 
%Because the strength 
%of the electric fields is proportional to the strength of the magnetic field,
%the phenomena are distinctive especially
%in strongly magnetized media such as neutron stars, white dwarfs e.t.c..
%We show that the amount of the energy released in neutron stars with
%magnetic field $\sim 10^{12}$ Gauss is approximately
%$10^{33}(M/10^{-13}M_{\odot})^4$ erg/s.
%where $M\, ( M_{\odot} )$ is the 
%mass of the axion star ( the sun ) and $m$ is the mass of the axion.    
 

%Let us we explain how the coherent axion field generates an electric field.
%The point is that the axion couples with the electric magnetic fields
%in the following way,

%\begin{equation}
%   L_{a\gamma\gamma}=ca\vec{E}\cdot\vec{B}/f
%\label{EB}
%\end{equation}
%with $f$ being the axion decay constant, where $a$ is axion field and, 
%$\vec{E}$ and $\vec{B}$ are electric and magnetic fields respectively. 
%The value of $c$ depends on the axion models;
%typically it is the order of one for DFSZ axion models\cite{DFSZ} 
%or the order of $\alpha/2\pi=1/(137\times 2\pi)\sim 10^{-3}$ 
%for hadronic axion models\cite{hadron}. 
%The value of $f$ is constrained from cosmological 
%and astrophysical considerations\cite{text};
%$10^{10}$GeV $< f <$ $10^{12}$GeV. 
%It is easy to see from this interaction that the coherent axion
%field may have electric charge density 
%$-c\vec{\partial}a\cdot\vec{B}/f$ in the magnetic field $\vec{B}$\cite{Si}. 
%We assume that the field $\vec{B}=(0,0,B)$ is spatially uniform and 
%that the geometry of the axion field $a$  
%representing the boson stars is spherical. Then we understand easily that 
%this axion object has a charge distribution such that 
%it has negative charges on a hemisphere 
%( $z>0$ ) and positive charges on the other side of the sphere ( $z<0$ ). 
%Net charge is zero. Therefore the star possesess the electric field, $E$,
%parallel to the magnetic field associated with 
%the charge distribution; $E=caB/f$.  
%This field induces an electric current in conducting media.

%Denoting the conductivity of the media by $\sigma$ and assuming 
%the Ohm law, we can see easily that
%the axion star with it's radius $R$ releaeses an energy $W$ per unit time, 

%\begin{equation}
%W=\pi\sigma c^2B^2a_0^2R^3= c^2B^2a_0^2R^3/4\nu_m
%\end{equation}
%where we have assumed that the distribution of the axion field representing
%a boson star is given such that
%$a=fa_0\exp(-r/R)$; $r$ denotes a radial coordinate. 
%We have taken account of the fact that the field $a$ oscillates 
%with a frequency given approximately by
%the mass of the axion $m$; $a\propto \sin{mt}$. Hence we have 
%taken an average in time over the period, $m^{-1}$ . 
%$\nu_m=1/4\pi\sigma$ is magnetic
%diffusivity. 

%We comment that the formula may apply to the conducting media where the 
%Ohm law is hold even for oscillating electric fields with their frequencies
%$m= 10^{10}\sim 10^{12}$ Hz. The law is hold in the media where 
%electrons interact sufficiently many times with each others or charged 
%particles in enviroment, and
%diffuse their energies acquired from the electric field 
%in a period of $m^{-1}$. Actually the law is 
%hold in the convection zone of the sun, 
%neutron stars, white dwarfs, e.t.c..

%In order to see the existence of axionic boson stars, 
%we have obtained numerically solutions 
%of the axionic boson stars \cite{iwaza,real} in a 
%limit of a weak gravitational field by solving a free field equation
%of the real scalar field $a$ along with Einstein equations. 
%It means that our solutons represent
%the axion stars with small masses, e.g. $10^{-12}M_{\odot}$ 
%whose gravitational fields are sufficiently weak. 
%We have confirmed that the spherical distribution of the field assumed above 
%is hold practically.
%In the limit of the small mass of the axion star we 
%have found a relation between the mass, $M$ and the radius, $R$ 
%of the axion star, 

%\begin{equation}
%M=6.4\,\frac{m_{pl}^2}{m^2R}
%\end{equation} 
%with Planck mass $m_{pl}$.
%Numerically we can see that for example, 
%$R=1.6\times10^5m_5^{-2}\mbox{cm}$ 
%for $M=10^{-9}M_{\odot}$, 
%$R=1.6\times10^8m_5^{-2}\mbox{cm}$ for $M=10^{-12}M_{\odot}$,
%e.t.c. with $m_5\equiv m/10^{-5}\mbox{eV}$ 

%We have also found a relation between the radius and parameter, $a_0$,

%\begin{equation}
%a_0=1.73\times 10^8 \frac{(1\mbox{cm})^2}{R^2}\,
%\frac{10^{-5}\mbox{eV}}{m}
%\end{equation}

%Using these formulae we rewrite $W$ such that

%\begin{equation}
%W=2.2\times 10^{31}\mbox{erg/s} 
%\,\frac{c^2}{\nu_{m}/\mbox{cm$^2$s$^{-1}$}}\,\frac{M}{10^{-9}M_{\odot}}
%\,\frac{B^2}{(1G)^2}
%\end{equation}
%Here length scales of the media are assumed to be larger than the radius 
%of the axion star $R$; the whole of the star is involved in the media. 
%On the contrary when the scale $L$ of the medium is smaller than $R$,
%we need to put a volume factor of $(L/R)^3$ on $W$.  


%In order to evaluate the value of $W$, 
%we need to know the mass of the axion star realized in 
%the Universe. According to a creation mechanism of the axion star by 
%Kolb and Tkachev\cite{kolb}, 
%some of miniclusters are contracted gravitationally 
%to the axionic boson stars. As the mass of the 
%minicluster has been shown to be typically $10^{-12}M_{\odot}$, we 
%consider such an amount of the mass $M$ as an order of a scale.
%Forthermore we need to know how frequently the axion stars collide 
%in a galaxy with 
%stars like the sun, neutron stars and white dwarfs, which are taken as 
%the media as explicit examples. The rate of the encounter is easily
%obtained by assuming that the halo 
%( its density = $5\times 10^{-25}$ g/cm$^3$ ) in our galaxcy 
%is composed of the axion stars whose velocities of order of 
%$10^{-3}$ times light velocity are 
%produced gravitationally. For example, the rate of the encounter 
%of an axion star with stars like the sun is 
%approximately once per $10^7\,(M/10^{-13}M_{\odot})$ years: We note that 
%the cross section of the collision is given by that of the sun 
%because the radius of the sun ( $\sim 7\times 10^{10}$ cm ) 
%is much larger than that of 
%the axion star with mass $M$ larger than $10^{-15}M_{\odot}$.   
%The rate is so large that MACHO group\cite{macho} 
%may observe the encounter if the 
%energy dissipation is sufficiently large. 



%%we wish to consider smaller mass of the star because
%%the rate of the encounter of such a star with 
%%the sun, neutron star or the earth becomes higher. For example,
%%suppose that the dark matter with density 
%%$\sim 5\times 10^{-25}$g/cm$^3$ in our galaxy is composed of the axion stars.
%%Then, the rate of the encounter of a star like the sun 
%%( its radius is $7\times 10^{10}$ cm )
%%with an axion star 
%%with mass $\sim 10^{-13}M_{\odot}$ ( its radius is $10^{9}m_5^{-2}$ cm )
%%and with its relative velocity $\sim 3\times 10^7$ cm/s 
%%is approximately several per $10^7$ years. 
%%The rate is so large that MACHO group may observe the encounter if the 
%%energy dissipation is sufficiently large. 


%Now let us discuss how large the energy of the axion star is released 
%in a magnetized conducting medum. First we take the sun 
%which is a typical star with 
%a strong magnetic field $10^3\sim10^4$ G in its convection zone.
%Assuming the depth of the convection zone being $\sim 2\times 10^{10}$ cm,
%and the magnetic diffusivity $\nu_m\sim 10^7$ cm$^2$s$^{-1}$\cite{Zeld},
%it follows that 

%\begin{equation}
%W=0.6\times 10^{28}\mbox{erg/s}\,
%\frac{B^2}{(5\times 10^3\mbox{G})^2}\,\frac{M}{10^{-13}M_{\odot}}
%\end{equation}
%where we have taken the value $c^2=1$ ( DFSZ axion model ). 
%Hence the total energy released 
%from the axion star passing through the sun is given such that

%\begin{equation}
%W_t=4\times 10^{10}\mbox{cm}\times W/v=
%0.8\times10^{31}\mbox{erg}\,
%\frac{B^2}{(5\times 10^3\mbox{G})^2}\,\frac{M}{10^{-13}M_{\odot}}
%\end{equation}
%where the velocity, $v$, of the axion star is assumed to be 
%$3\times 10^7\mbox{cm/s}$ ; this is a value obtained by equating
%a kinetic energy with a gravitational one of the axion star in our galaxy.
%We have assumed nonexistence of the magnetic field 
%in the radiation zone. Therefore, the effect of the collision 
%of the axion star with a star like the sun is difficult to be observed;
%remember that the luminocity of the sun is the order of $10^{33}$ erg/s. 
%Assuming a larger mass of the axion star leads to larger dissipation
%of energy whose effects may be obsevable, but it leads to lower rate of 
%its collision with a star. Thus as far as we are concerned with 
%such stars with the same physical parameters as those of the sun, 
%it is difficult to detect the effect of such collisions.  


%We see from the formula $W$ that 
%as $W$ is proportional is $B^2$, it is sensitive to the value of the magnetic 
%field. On the other hand  the strength of the field on spots of other stars 
%is not known, 
%although most of the other stars of late type 
%with a convection zone near a surface in main sequence 
%may have magnetic fields
%with similar strength to the one of the magnetic field of the sun.
%It might vary from $10^3\sim 10^5$ G in the convection zone. 
%If among the stars of late type there exist a fairly large number of 
%stars with strong magnetic fields $\sim10^5$ G, 
%the amount of the energy release of the axion star is 
%$W_t\sim 10^{34}$ erg in such stars.   
%Furthermore, their luminocities are smaller than the one of the sun.
%Thus the effect of the energy dissipation in the stars is 
%so large relatively to be observable.
%We know that the energy deposited in the convection zone is transported 
%by the convection to the surface of the star. 
%Since the speed of the convection 
%is approximately $10^5$ cm/s in the case of the sun, it takes 
%several days or more to be transported. 
%We imagine that the release of radiations from the star 
%resulted from this deposit of the energy in convection zone would continues 
%for several days or several ten days. 
%Therefore, it follows from these facts and the rate of the encounter
%( once per $10^7$ years ) that this phenomenenon 
%may be detectable in the stars of late type with strong 
%magnetic field by a 
%search like one performed by MACHO groups. 


%Next we go on to discuss the case of neutron stars.
%The magnetic fields of the neutron stars are typically the order of 
%$10^{12}$ G so that the amount of the energy released by the axion star
%are so large,

%\begin{eqnarray}
%W&=&0.5\times 10^{42}\mbox{erg/s} 
%\,\frac{c^2}{\nu_{m}/\mbox{cm$^2$s$^{-1}$}}\,\frac{M^4}{(10^{-13}M_{\odot})^4}
%\,\frac{B^2}{(10^{12}\mbox{G})^2}\,\frac{m^6}{(10^{-5}\mbox{eV})^6},\\
%W_t&\sim&10^2\,W\sim 10^{44}\mbox{erg}
%\,\frac{c^2}{\nu_{m}/\mbox{cm$^2$s$^{-1}$}}\,\frac{M^4}{(10^{-13}M_{\odot})^4}
%\,\frac{B^2}{(10^{12}\mbox{G})^2}\,\frac{m^6}{(10^{-5}\mbox{eV})^6}
%\end{eqnarray}
%where we have taken account of the volume factor, 
%$(L/R)^3\sim 10^{-9}M^3/(10^{-13}M_{\odot})^3$, 
%because the radius of the neutron star, $L=10^6$ cm is much smaller than 
%that of the axion star, $R=1.6\times 10^{9}\,m_5^{-2}$ cm 
%with $M=10^{-13}M_{\odot}$. 
%In order to estimate 
%the value of $W$, we need to know the conductivity or the magnetic diffusivity
%$\nu_m$ of the neutron star. But the knowledge is poor, 
%so we simply make a conjecture on it. The conductivity may be larger than
%that of normal metals 
%because the neutron star possesses free electrons 
%whose density is much higher than that of the normal 
%metals. Furthermore there is a region in which supercnductivity arises
%owing to a proton pairs condensation. Thus it is reasonable to think that 
%$\nu_m$ is much smaller than normal metals. 
%That is $\nu_m\ll 1$. Actually there are some 
%theoretical estimations\cite{con} 
%of $\nu_m$ in the crusts of the neutron stars;
%$\nu_m\sim O(10^{-6})$ cm$^2$/s. 

%We note that $10^{-13}M_{\odot}\sim 10^{41}$ erg. 
%Thus such a large 
%amount of the energy dissipation $W\gg 10^{41}$ erg/s 
%in the neutron star leads to 
%a rapid decay of the axion star. 
%In such a case actual energy dissipation is limited to the value of 
%$W\sim 10^{33}(M/10^{-13}M_{\odot})^4$ erg/s.
%The reason is that 
%only a fraction 
%$L^2v/R^3\sim(10^6)^2\times3\times10^7/R^3\sim 10^{-8}
%\,(M/10^{-13}M_{\odot})^3$ 
%of the mass of the axion star is 
%transmuted into the thermal energy per unit time. 
%Thus when the whole of the energy released is transmuted into radiations, 
%its luminosity is 
%$\sim L_{\odot}\,(M/10^{-13}M_{\odot})^4$ 
%where $L_{\odot}$ is 
%the luminosity of the sun. This energy release will continue until
%the whole of the energy of the axion star is exhausted. 
%( The axion star is trapped by the neutron star because of the dissipation
%of the kinetic energy, $Mv^2/2\sim 10^{35}\,(M/10^{-13}M_{\odot})$ erg. )
%Thus it takes 
%$10^8\,(M/10^{-13}M_{\odot})^{-3}$ sec.
%In order to see precisely how it decays
%we need to know the back reaction of the axion star caused by
%the dissipation.
%In any way this energy release may be detectable. 



%As a final example, we consider the case of white dwarfs which may possess
%strong magnetic field $\sim 10^8$ G with their radius 
%$L\sim10^8$ cm. Their number is expected to be roughly
%the same order of magnitude as 
%that of stars existing in our galaxy. 
%The rate of the collison of a white dwarf
%with axion stars
%is about one per $10^8$ years in the case of $M\sim 10^{-14}M_{\odot}$.
%Thus in our galaxy the collisions will occur 
%$1000$ times per 1 year. Thus the rate of  
%their collision with the axion stars in our galaxy is so large for 
%their effects to be detectable.
%The release of the energy in such a collision is
%given such that

%\begin{eqnarray}
%\label{w}
%W&=&0.5\times 10^{39}\mbox{erg/s} 
%\,\frac{c^2}{\nu_{m}/\mbox{cm$^2$s$^{-1}$}}\,\frac{M^4}{(10^{-14}M_{\odot})^4}
%\,\frac{B^2}{(10^8\mbox{G})^2}\,\frac{m^6}{(10^{-5}\mbox{eV})^6},\\
%W_t&\sim&10^2\,W\sim 10^{41}\mbox{erg}
%\,\frac{c^2}{\nu_{m}/\mbox{cm$^2$s$^{-1}$}}\,\frac{M^4}{(10^{-14}M_{\odot})^4}
%\,\frac{B^2}{(10^8\mbox{G})^2}\,\frac{m^6}{(10^{-5}\mbox{eV})^6}
%\end{eqnarray}
%where we have taken account of the volume factor, 
%$(L/R)^3\sim10^{-6}(M/10^{-14}M_{\odot})^3$.
%As in the case of the neutron stars, the knowledge of $\nu_m$ is poor. 
%But we expect that $\nu_m$
%is much larger than ones of normal metals because 
%the number density of electrons is much larger than that of the metals.
%Theoretical evaluations show that $\nu_m\sim O(10^{-2})$ in the case of 
%a crystalized white dwarft. So anyway it is reasonable 
%to think that $\nu_m \ll 1$. 

%We note that $10^{-14}M_{\odot}\sim10^{40}$ erg. Thus such a large value 
%of $W$ in eq(\ref{w})
%implies as in the case of the neutron stars 
%that an actual release of the energy is limited. Only a fraction 
%$L^2v/R^3\sim 10^{-7}\,(M/10^{-14}M_{\odot})^3$ 
%of the mass is dissipated per unit time; 
%$10^{33}\,(M/10^{-14}M_{\odot})^4$ erg/s. 
%This amount of the energy release, 
%$\sim10^{33}\,(M/10^{-14}M_{\odot})^4$ erg/s, 
%will continue untill the whole energy 
%of the axion star is exhausted. It takes 
%$10^7\,(M/10^{-14}M_{\odot})^{-3}$ sec. When this energy is transmuted into 
%radiations, the luminosity is rouphly $L_{\odot}(M/10^{-14}M_{\odot})^4$.
%Therefore we expect that 
%this phenomenon is observable.










%%the earth, inside of which
%%a liquid iron core exists with magnetic field $\sim 10^2$ G. Assuming that 
%%the radius of the core being $3.5\times 10^8$ cm and 
%%that $\nu_m=3\times 10^4$ cm$^2$s$^{-1}$,
%%we obtain the amount of the energy dissipation,

%%\begin{eqnarray}
%%W&=&0.7\times 10^{25}\mbox{erg/s}\,\frac{M}{10^{-13}M_{\odot}}\,
%%\frac{B^2}{10^2\mbox{G}}\,\frac{m^6}{(10^{-5}\mbox{eV})^6}\\
%%W_t&=&10^3\,W\sim 10^{28}\mbox{erg}\,\frac{M}{10^{-13}M_{\odot}}\,
%%\frac{B^2}{10^2\mbox{G}}\,\frac{m^6}{(10^{-5}\mbox{eV})^6}
%%\end{eqnarray}
%%with $c^2=1$,
%%where we have also taken account of the volume factor.
%%The amount of this energy deposit is much larger than the heat energy per
%%year coming from the inside of the earth.
%%The rate of a collision of 
%%the axion star with the earth is approximately once per $10^9$ years.
%%So, it is interesting to see that $W_t\sim 10^{28}$ erg 
%%with a choice of axion mass $m=5\times 10^{-5}$ eV 
%%is comparable to a energy released from a collision of a meteorite
%%which might have resulted in extermination of dinosaurs, $6.5\times10^7$
%%years ago.


%Finally we wish to point out a possible mechanism of 
%producing extremely high energy cosmic rays. It is caused by 
%oscillating electric field of the axion star, $aB/f\sim Ba_0$ in a 
%tube of a magnetic vortex in a superconducting medium of a neutron star.  
%The superconductor is resulted from the condensation of the proton pairs in a 
%neutron star. Thus the radius of the magnetic flux tube is very small;
%it is order of $1$ GeV$^{-1}\sim 10^{-14}$ cm. Then, the strength of 
%the magnetic field is roughly 
%$\pi/\mbox{e}(10^{-14}\mbox{cm})^2\sim 10^{20}$ G; e is the charge 
%of the proton.
%Therefore, an energy acquired by a proton or an electron 
%from the oscillating electric field in a magnetic flux tube is order of 
%$a_0Bm^{-1}\sim 10^{11}$ GeV$\,(M/10^{-10}M_{\odot})^2m_5^2$.
%This may account events of the extremely high energy cosmic rays.


%As explained in above examples, the axion stars are possible sources
%for generating energies in magnetized conducting media. We may apply 
%the idea to systems such as accretion disks with strong magnetic fields 
%around black holes e.t.c.. Probably the existence of the axion
%will be confirmed indirectly by observing these phenomena. 




%In summary, we have shown that the coherent axion stars release their 
%energies in the magnetized conducting media such as stars, neutron stars,
%or white dwarfs. Among them, radiations from the white dwarfs seems to be 
%detectable most probably since their luminosities and rate of the events are 
%sufficiently large. We have also pointed out a possible mechanism 
%for generating extremely high energy cosmic rays. 

  





 
 


%Principal part of this work have been down when the author has visited 
%the Theoretical Physics Group at LBNL. He would 
%like to express his thank for the hospitality in LBNL as well as in  
%Tanashi KEK.










%%%%%%%%%%%%%%%%%%%%%%
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%\begin{figure}
%\caption{
%  The solid ( dashed ) curve represents $I-V$ characteristic with
% $\beta = 0$ ( $\beta = 4$ ) in DC current feed with normalized
%  voltage $V/R_qI_c$ in the horizontal axis.  The effect of the
%  capacitance ($\beta \ne 0$ ) leads to a hysteresis; the voltage
%  raises linealy with current $I$ ($\leq I_c$ ), but suddenly jumps (
%  indicated with the dotted line ) when the current reaches at the
%  critical current $I_c$. Then it follows the dashed curve with the
%  current increasing furthermore. Conversely, the voltage decreases
%  continuously following the dashed curve without passing the dotted
%  line even if the current becomes less than $I_c$.  }
%\end{figure}

%\begin{figure}
%\caption{
%  The dotted curve shows Shapiro-like steps ( voltage jumps in the
%  unit of $\omega /e$ ) in $I-V$ characteristic with AC current feed;
%  its frequency $\omega$ is assumed to be $0.16 e R_q I_c$.  }
%\end{figure}

%\begin{figure}
%\caption{The solid curve represents $I-V$ characterisic in 
%  DC voltage feed with $R_q/R =2$. The dotted curve shows Shapiro-like
%  steps ( current drops in the unit of $\omega/eR$ ) in $I-V$
%  characteristic with AC voltage feed; its frequency $\omeis
% assumed to be $0.16 e V_c R_q /( R + R_q )$.  }
%\end{figure}

\end{document}




























