aainstitutetext: Department of Physics,
University of Florida ,
Gainesville, FL 32611,
United States

# Radiation from Axion star-Neutron star binaries with a tilted rotation axis
in the presence of plasma

A. Kyriazis<EMAIL_ADDRESS>
###### Abstract

We investigate the form of the radiation emitted by an axion star-neutron star
binary using a $f(r)=\text{sech(r/R)}$ profile for the axion star. Our
analysis takes into account the co-rotating plasma of the neutron star. We
find that that there is significant enhancement to the radiated power if the
neutron star’s spin is tilted towards the plane of the axion star-neutron star
orbit, compared to the case where it is perpendicular. We also examine whether
the neutron star’s co-rotating plasma can play a role in the emitted power and
we find that even though dilute axion stars can in principle radiate more
efficiently than dense axion stars, they will be pulled apart by the tidal
forces of the neutron star

##  1 Introduction

The axion, initially proposed as a solution to the strong CP problem Peccei
and Quinn [1977], Weinberg [1978], is now one of the most well-motivated
candidates of dark matter Preskill et al. [1983], Abbott and Sikivie [1983],
Dine and Fischler [1983]. The axions are stable bosons, with large occupation
numbers and can rethermalize through their gravitational interactions forming
a Bose-Einstein condensate (BEC) Sikivie and Yang [2009], Erken et al. [2012].
Owing to the large occupation number of the ground state, the BEC condensate
has been treated classically as a localised, coherently oscillating clump
called an axion star, if the kinetic pressure is balanced by gravity and
axiton or oscillon, if it is balanced by self-interactions. Braaten et al.
[2016], Eby et al. [2016a, b], Levkov et al. [2017], Feinblum and McKinley
[1968], Chavanis [2018], Visinelli et al. [2018].

The interaction of axions with electromagnetic field has been proposed as a
venue for their detection Sikivie [1983]. Of particular interest it the
Primakoff effect Primakoff [1951], which is the interaction of an axion with a
virtual photon to produce a real photon. One possible "laboratory" that
exploits this effect for the detection of the axion is that of neutron stars,
where some of the strongest magnetic fields in nature can be found Pshirkov
and Popov [2009].

The interaction between a dilute axion star (AS) and a neutron star (NS) has
also been proposed to explain Fast Radio Bursts Lorimer et al. [2007], Keane
et al. [2010], Thornton et al. [2013], Spitler et al. [2014], where the
magnetic field of the neutron star induces an electric field in the axion star
which then accelerates electrons in the atmosphere of the neutron star Iwazaki
[2015] or induces a dipole moment in neutrons on the upper upper core of the
neutron star Raby [2016]. However, due to tidal forces, the AS will break
apart in a time scale of the order of seconds, much longer than the
millisecond duration of FRBs Pshirkov [2017]. It has been pointed out though
that the axion to photon conversion in the collision of an dense AS with a NS
may still be able to explain FRBs Buckley et al. [2021], Bai and Hamada
[2018].

Recently, the electromagnetic power emitted by an AS-NS binary system was
derived analytically Kouvaris et al. [2022]. It was assumed that the AS is
described by an exponential profile, that the axis of rotation of the NS is
perpendicular to the AS-NS plane and that plasma effects were negligible due
to the low frequency of the plasma. For circular orbits, it was found that the
spectral flux density of the emitted radiation is modulated mainly by the high
frequency spin of the NS. There is also significant enhancement of emitted
radiation for elliptical orbits due to the periodic approach of the AS to the
NS.

In this work, we include the plasma co-rotating with the NS through the
equation $k=\sqrt{\omega^{2}-\omega^{2}_{p}}$, where k is the wave-number of
the photon, $\omega$ is its frequency and $\omega_{p}$ is the plasma
frequency. We also assume that the rotation axis of the neutron star has a
tilt with respect to the the normal to the NS-AS plane, use a profile for the
AS that is more accurate than the exponential in the case of attractive self-
interactions Eby et al. [2018] and allows us to compare with previous works
where a constant magnetic field was assumed Amin et al. [2021], and finally
explore the relevance of plasma effects when we consider an axion mass
$m_{a}\sim\omega_{p}$.

The paper is organised as follows: In section 2, we review the interactions
between the axion and electromagnetism. In section 3, we establish the profile
of the axion star that we will be using and describe the magnetic field of the
pulsar. In section 4, we calculate the power emitted by the axion condensate
and provide analytic formulae. In section 5, we summarise the results of this
paper with some remarks for future research.

##  2 Axion Electrodynamics

The Lagrangian of the axion field coupled to electromagnetism is:

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\partial^{\mu}\varphi-\frac{1}{2}m^{2}_{\varphi}\varphi^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{g_{a\gamma}}{4}\varphi
F_{\mu\nu}\tilde{F}^{\mu\nu}$ (1)

where $\phi$ is the scalar field of the axion and:

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,\qquad\tilde{F^{\mu\nu}}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$
(2)

Note that in the above Lagrangian we have neglected gravity and assumed that
there are no free currents and charges. Since the occupation number of the
axion field is huge, we may treat its field as a classical field. The
equations of motion are:

$\displaystyle(\Box+m^{2}_{a})\varphi=$ $\displaystyle
g_{a\gamma}\bf{E}\cdot\bf{B}$ (3a) $\displaystyle\nabla\cdot\bf{E}$
$\displaystyle=\rho$ (3b) $\displaystyle\nabla\times\bf{B}-\pdv{\bf{E}}$
$\displaystyle=\bf{J}$ (3c) $\displaystyle\nabla\cdot\bf{B}$ $\displaystyle=0$
(3d) $\displaystyle\nabla\times\bf{E}+\pdv{\bf{B}}{t}$ $\displaystyle=0$ (3e)

where the charge density $\rho$ and current density $\bf{J}$ are:

$\displaystyle\rho$ $\displaystyle=-g_{a\gamma}\nabla\varphi\cdot\textbf{B}$
(3da) J
$\displaystyle=g_{a\gamma}(\pdv{\varphi}{t}\textbf{B}+\nabla\varphi\times\textbf{E})$
(3db)

This is a set of coupled, partial differential equations. To make progress, we
follow Amin et al. [2021] and expand the electromagnetic fields, charge
density and current density in the small parameter $g_{a\gamma}\varphi_{0}$:

$\displaystyle\bf{E}=\bf{E}^{(0)}+\bf{E}^{(1)}+...,\qquad\bf{B}=\bf{B}^{(0)}+\bf{B}^{(1)}+...$
(3ea)

where $\bf{E}^{(0)}$ and $\bf{B}^{(0)}$ are the background fields that mix
with the axion field. The 2 dynamical Maxwell equations for the first order
electromagnetic fields are:

$\displaystyle\nabla\cdot\textbf{E}^{(1)}$ $\displaystyle=\rho^{(1)},$ (3fa)
$\displaystyle\nabla\times\textbf{B}^{(1)}-\pdv{\textbf{E}^{(1)}}{t}$
$\displaystyle=\textbf{J}^{(1)}$ (3fb)

Introducing now the vector and scalar potential through the standard equations
$\textbf{B}^{(1)}=\nabla\times\textbf{A}$, $\textbf{E}^{(1)}=-\nabla
A^{0}-\pdv{\bf{A}}{t}$, our problem is reduced to calculating $A^{0}$ and A
from the equations:

$(-\nabla^{2}+\frac{\partial^{2}}{\partial t^{2}})A^{0}=\rho$ (3ga)
$(-\nabla^{2}+\frac{\partial^{2}}{\partial t^{2}})\textbf{A}=\textbf{J}^{(1)}$
(3gb)

We now turn to the field of the AS and the background fields
$\textbf{E}^{(0)}$ and $\textbf{B}^{(0)}$ of the NS.

##  3 Axions stars and Neutron stars

###  3.1 Axion stars

For the axion star, we use the Ansatz proposed in Schiappacasse and Hertzberg
[2018] for a spherical symmetric star, oscillating coherently with frequency
$\omega$:

$\varphi=\varphi_{0}f(r)cos(\omega t)$ (3h)

For the non-relativistic case that we examine here, we may set $\omega=m_{a}$.
The spatial profile $f(r)$ must satisfy the boundary condition
$f(r)\xrightarrow[]{}0$ when $r\xrightarrow[]{}\infty$ and have an inverted
parabola at $r=0$. We will consider the profile:

$f(r)=\text{sech}(r/R)$ (3i)

Although we should include a polynomial that captures the behaviour of the
field at $r\xrightarrow[]{}0$, these profiles give us the correct qualitative
behaviour of the field. It was found in Visinelli et al. [2018], where the
potential considered was that of the QCD axion:

$V(\varphi)=\Lambda^{4}(1-cos(\varphi/f_{a}))$ (3j)

that if $|\varphi_{0}|\ll f_{a}\,(\frac{10^{-6}eV}{m_{a}})$, where $f_{a}$ is
the axion decay constant, we are in the diluted, stable branch where
$R\omega\gg 1$ and we can ignore the self-interactions. In this regime, the
axion star is supported by its kinetic pressure acting against gravity. Its
typical mass and radius is Chavanis [2018], Visinelli et al. [2018]:

$\displaystyle M_{dilute}$ $\displaystyle\leq 1.0\times
10^{-9}M_{\odot}\left(\frac{f_{a}}{10^{13}\,\text{GeV}}\right)\left(\frac{10^{-6}\,\text{eV}}{m_{a}}\right)$
(3k) $\displaystyle R_{dilute}$
$\displaystyle=27km\left(\frac{10^{13}\,\text{GeV}}{f_{a}}\right)\left(\frac{10^{-6}\,\text{eV}}{m_{a}}\right)$
(3l)

If $|\varphi_{0}|=\mathcal{O}(1)f_{a}$, we are in the dense branch where
$R\geq$few$\times\omega^{-1}$ and we have to include the self-interactions,
while the non-relativistic approximation breaks down and our Ansatz 3h does
not capture the correct profile of the oscillon. In addition, its lifetime was
determined to be short, of the order $\mathcal{O}(\frac{10^{3}}{m})$ and
therefore not relevant to cosmological investigations. The radius of a dense
axion star is given by Braaten et al. [2016]:

$R_{dense}=3m\times\left(\frac{10^{13}\,\text{GeV}}{f_{a}}\right)^{1/2}\left(\frac{10^{-6}\,\text{eV}}{m_{a}}\right)^{1/2}\left(\frac{M_{dense}}{10^{-12}M_{\odot}}\right)^{0.3}$
(3m)

In Zhang et al. [2020], Ollé et al. [2020], Cyncynates and Giurgica-Tiron
[2021], in addition to 3j, a number of other potentials were studied and the
lifetime of oscillons was found to be up to 5 orders of magnitude larger.
However, they arrived to similar results regarding the single frequency
approximation i.e. we need a more sophisticated profile than 3h to study dense
oscillons-axion stars. In this work, we will assume that the profile in 3h is
also correct for dense axion stars and leave a multiple frequency profile for
future work.

###  3.2 Neutron Stars

We are modelling the neutron star as a rotating magnetic dipole of angular
frequency $\Omega$ surrounded by co-rotating plasma. In the near zone
$r\ll\frac{1}{\Omega}$ the magnetic field is given by:

$\textbf{B}_{NS}=\frac{B_{0}}{2}\left(\frac{R_{NS}}{r}\right)^{3}(3(\hat{m}\cdot\hat{r})\hat{r}-\hat{m})$
(3n)

where $R_{NS}$ is the neutron star’s radius, $B_{0}$ is the intensity of the
magnetic field on the neutron star’s surface and $\hat{m}$ is the unit vector
in the direction of the magnetic moment. The number density of the electrons
was derived by Goldreich and Julian assuming perfect conductivity of the
plasma Goldreich and Julian [1969]:

$n_{c}=\frac{2\bf{\Omega}\cdot\textbf{B}}{e}\frac{1}{1-\Omega^{2}r^{2}\sin^{2}\theta}$
(3o)

while the plasma frequency is given by:

$\omega_{p}=\sqrt{\frac{4\pi\alpha n_{c}}{m_{e}}}\approx
10^{-5}\text{eV}\sqrt{\frac{\hat{\Omega}\cdot\textbf{B}}{10^{12}G}\frac{1s}{T_{s}}}$
(3p)

where $T_{s}$ is the period of the NS.

Figure 1: Schematic of the NS showing the different angles. The rotation axis
makes a $\tilde{\theta}$ angle with the z axis and the magnetic dipole moment
makes a constant $\gamma$ angle with the rotation axis, while precessing
around it with $\Omega$ angular frequency

We now turn to the rotation axis of the NS. For simplicity, we will assume
that it lies on the y-z plane, at an angle $\tilde{\theta}$ from the z axis:

$\hat{\Omega}=\begin{pmatrix}0\\\ \sin\tilde{\theta}\\\ \cos\tilde{\theta}\\\
\end{pmatrix}$ (3q)

We can now perform two successive rotations using SO(3) matrices to obtain the
magnetic moment unit vector $\hat{m}$ that precesses around the rotation axis.
One is around an axis perpendicular to $\hat{\Omega}$ by an angle $\gamma$ and
the next one is around $\hat{\Omega}$ by an angle $\Omega t$. The details are
given in Appendix A. Here, we simply write down the final answer:

$\hat{m}=\begin{pmatrix}\cos(\Omega t)\sin\gamma\\\
\cos\gamma\sin\tilde{\theta}+\sin\gamma\cos\tilde{\theta}\sin(\Omega t)\\\
\cos\gamma\cos\tilde{\theta}-\sin\gamma\sin\tilde{\theta}\sin(\Omega
t)\end{pmatrix}$ (3r)

Furthermore, we argue that we can ignore the background electric field in 3db.
Its value at the surface of the star is $E_{0}$=$B_{0}\Omega a$ and in order
to ignore the second term in the right hand side of 3db, it must hold:

$\nabla\varphi\times\textbf{E}\ll\pdv{\varphi}{t}\textbf{B}\Rightarrow\frac{\textbf{E}_{0}}{R}\ll
m_{a}\textbf{B}_{0}\Rightarrow{}\frac{\Omega R_{NS}}{c}\ll m_{a}R$ (3s)

For typical values of $R_{NS}$=10 km, $\Omega=10^{-3}-1s$, this inequality is
true for the range of $m_{a}R$ that we are interested in.

###  3.3 AS-NS binary

Assuming that $M_{NS}\gg M_{AS}$, we can place the NS in the origin of our
coordinate system and the AS will be orbiting around it. The AS’s motion is
described by the familiar equation for elliptical orbits:

$r_{a}=\frac{(1-e^{2})r_{0}}{1+e\cos\phi_{a}},\
r_{0}=\frac{l^{2}_{0}}{GM_{NS}(1-e^{2})}$ (3t)

where $r_{0}$ is the semi-major axis, $l_{0}$ is the angular momentum per unit
mass, $\phi_{a}$ is the azimuthal angle of the AS and $e$ is the eccentricity.
We will restrict our analysis to the cases where $e=0$ and $0<e<1$, to
circular and elliptical orbits respectively. Since the AS lies on the $x-y$
plane, its position vector is:

$\textbf{r}_{a}=r_{a}(\cos\phi_{a},\sin\phi_{a},0)$ (3u)

Later on, we will also need the time dependence of the angle $\phi_{a}$ which
is given by the parametric equations Herbert Goldstein [2001]:

$\omega t=\psi-e\sin\psi,\hskip
5.69046pt\cos\phi_{a}=\frac{\cos\psi-e}{1-e\cos\psi}$ (3v)

where $0\leq\psi\leq 2\pi$ and $\omega$ is the angular frequency of the AS’s
rotation around the NS. The period of rotation is
$T_{AS}=\frac{2\pi}{\omega}$, which we will assume to be 10s.

. We will also assume for simplicity that Earth’s position vector lies on the
z axis.

$M_{NS}(M_{\odot})$ | $M_{dense}(M_{\odot})$ | $R_{NS}(km)$ | $r_{0}(km)$
---|---|---|---
1.4 | $10^{-12}$ | 15 | $7.8\times 10^{3}$
$\gamma(rad)$ | $T_{AS}(s)$ | $T_{s}(s)$ | $m_{a}(\text{eV})$
0.3 | 10 | 1 | $10^{-6}$
$f_{a}(\text{GeV})$ | $c_{\gamma}$ | $r_{e}(kpc)$ | $B_{0}(G)$
$10^{14}$ | $10^{5}$ | 1 | $10^{12}$
Table 1: A table with the relevant constants of our system

Table 1 summarises the constants relevant to our system, which we take to be
the same as in Kouvaris et al. [2022], except for $f_{a}$ which we take to be
$f_{a}=10^{13}\text{GeV}$, so that we are consistent with the equation
$m_{a}f_{a}=m_{\pi}f_{\pi}=(10^{8}\text{eV})^{2}$ that must be true for the
QCD axion.

##  4 Analytic calculation of emitted power

The radiating fields are given by the known equations:

B $\displaystyle=\nabla\times\textbf{A},$ (3ac) E
$\displaystyle=i\omega\textbf{A}-\nabla
A^{0}=\frac{i}{\omega}\left(\omega^{2}\textbf{A}-\textbf{k}(\textbf{k}\cdot\textbf{A})\right)$
(3ad)

where we have taken the Fourier transform in time and used the Lorentz gauge
$\nabla\cdot\textbf{A}+\partial_{0}A^{0}=0$. Here,
$|\textbf{k}|=\sqrt{m_{a}^{2}-\omega_{p}^{2}}$. It is therefore sufficient to
solve 3gb. Using the retarded Green’s function, the solution is:

$\textbf{A}(\textbf{r}_{e})=\frac{g_{a\gamma}\varphi_{0}\omega\textbf{B}_{NS}}{4\pi
r_{e}}\int
d^{3}r^{\prime}e^{ik|\textbf{r}_{e}-\textbf{r}^{\prime}|}\,f(|\textbf{r}^{\prime}-\textbf{r}_{a}|)$
(3ae)

The magnetic field varies negligibly within the AS, so we can take it out of
the integral. We will only write the final result of this integration here and
leave the details for Appendix B.

$\textbf{A}(\textbf{r}_{e})\approx\frac{g_{a\gamma}\varphi_{0}\omega(\pi
R)^{2}\textbf{B}_{NS}(\textbf{r}_{a})}{4kr_{e}}\frac{\tanh(\pi
kR/2)}{\cosh(\pi kR/2)}e^{ik|\textbf{r}_{e}-\textbf{r}_{a}|}$ (3af)

The time averaged Poynting vector is given by:

$\langle\textbf{S}\rangle=\frac{1}{2}\Re(\textbf{E}\times\textbf{B}^{\star})=\frac{1}{2\omega}\Re\left(\textbf{k}(\omega^{2}|\textbf{A}|^{2}-|\textbf{k}\cdot\textbf{A}|^{2})-\omega^{2}_{p}(\textbf{k}\cdot\textbf{A})\textbf{A}^{\star}\right)$
(3ag)

Inserting in 3ag the expression in 3af, we find for the Poynting vector and
the power per solid angle :

$\langle\textbf{S}\rangle=\frac{\pi^{4}\omega}{32r^{2}_{e}}\left(\frac{g_{a\gamma}\varphi_{0}R^{2}}{k}\right)^{2}\left(\frac{\tanh(\pi
kR/2)}{\cosh(\pi
kR/2)}\right)^{2}\left(\textbf{k}(\omega^{2}|\textbf{B}_{NS}|^{2}-(\textbf{k}\cdot\textbf{B}_{NS})^{2})-\omega^{2}_{p}\textbf{B}_{NS}(\textbf{k}\cdot\textbf{B}_{NS})\right)$
(3ah)

$\frac{dP}{d\Omega}=r^{2}_{e}\hat{k}\cdot\langle\textbf{S}\rangle=\frac{\pi^{4}(g_{a\gamma}\varphi_{0}\omega^{2}R^{2})^{2}}{32k\omega}\left(\frac{\tanh(\pi
kR/2)}{\cosh(\pi kR/2)}\right)^{2}|\textbf{B}_{NS}|^{2}\sin^{2}\beta$ (3ai)

where $\beta$ is the angle between the magnetic field at the AS’s position and
Earth’s position vector from the AS. It comes as no surprise that this is the
same expression as the one derived in Amin et al. [2021], where a constant
magnetic field was assumed.

Figure 2: Plot of the function $\left(\frac{\tanh(\pi\omega
R/2)}{\cosh(\pi\omega R/2)}\right)^{2}$ versus $\omega R$. The function has a
peak at $\omega R$=0.56 and is exponentially suppressed for $\omega R\gg 1$

.

###  4.1 AS-NS binary in vacuum

For the benchmark values considered in Table 1, $\omega_{p}\sim
10^{-9}\text{eV}$, so we can neglect plasma effects for the moment, since
$m_{a}\sim 10^{-6}\text{eV}$. The power per solid angle, equation 3ai, is
exponentially suppressed with respect to $\omega R$, which means that only
dense axion stars can radiate efficiently in vacuum. We can understand this by
considering energy-momentum conservation in the axion to photon conversion
process Sikivie [2021].

Let us examine the conversion rate of axions to photons in the presence of a
magnetic field that has only spatial variation, $\textbf{B}_{0}(\textbf{x})$.
The conversion rate is proportional to $|\int
d^{3}xe^{-i(\textbf{k}-\textbf{k}_{a})\cdot\textbf{x}}\hat{e}_{\lambda}(\hat{n})\cdot\textbf{B}_{0}(\textbf{x})|^{2}$,
where $\hat{e}_{\lambda}(\hat{n})$, with $\lambda=1,2$ are the two final state
polarization vectors of a photon travelling in the $\hat{n}$ direction, k is
the momentum of the photon and $\textbf{k}_{a}$ is the momentum of the axion.
A sum over $\lambda$ is implied. Energy conservation is satisfied because
$\textbf{B}_{0}(\textbf{x})$ is time-independent. However, the axion satisfies
the dispersion relation $\omega=\sqrt{k_{a}^{2}+m_{a}^{2}}$, while the photon
satisfies $\omega=|\textbf{k}|$. Thus their momenta will in general be
different, but if we decompose the magnetic field into its Fourier modes,
$\tilde{\textbf{B}}_{0}(\textbf{q})$, momentum transfer is guaranteed because
then $\textbf{q}=\textbf{k}-\textbf{k}_{a}$. In our case, we assumed that the
magnetic field is constant within the range of integration, therefore
$|\textbf{k}_{a}|\approx\omega\Rightarrow\frac{1}{R}\approx\omega$.

We may define the spectral flux density as
$\mathcal{S}=\frac{dP}{d\Omega}\left(r^{2}_{e}\mathcal{B}\right)^{-1}$, where
$\mathcal{B}=\frac{0.1m_{a}}{2\pi}$ is an estimate for the Doppler width.
Inserting the benchmark values we are considering, the result is:

$\displaystyle S$ $\displaystyle=2.36\times
10^{-5}\text{Jy}\left(\frac{c_{\gamma}}{10^{5}}\right)^{2}\left(\frac{10^{13}\text{GeV}}{f_{a}}\right)^{2}\left(\frac{10^{-6}\text{eV}}{m_{a}}\right)\left(\frac{M_{dense}}{10^{-12}M_{\odot}}\right)^{1.2}\left(\frac{B_{0}}{10^{12}\text{GeV}}\right)^{2}\times$
$\displaystyle\times\left(\frac{1kpc}{r_{e}}\right)^{2}\sin^{2}\beta\left(\frac{1+e\cos\phi_{a}}{1-e^{2}}\right)^{6}|3(\hat{m}\cdot\hat{r})\hat{r}-\hat{m}|^{2}$
(3aj)

Figure 3: Spectral flux density versus time for e=0. The blue colour
represents the rotation axis on the z axis, while orange and green represent
the rotation axis at $\tilde{\theta}=\frac{\pi}{4}$ and
$\tilde{\theta}=\frac{\pi}{2}$ respectively

In Figure 3, we plot the spectral flux density versus time for three different
$\tilde{\theta}$ angles. For $\tilde{\theta}=0$, our results are similar to
what was found in Kouvaris et al. [2022] i.e., the signal is modulated by the
rotation of the AS around the NS and by the spinning of the NS around its
axis. The envelope at the top is approximately flat because the separation
between the AS and NS does not change with time. This picture changes if we
allow the NS’s rotation axis to be tilted. The position of the AS now plays a
more important role in modulating the signal, as we can see from the peaks and
troughs in the plot that have a period of 5 seconds. The locations of these
maxima and minima can be explained if we remember that the AS starts its orbit
on the x axis, where it receives the minimum amount of magnetic field lines
from the NS, since the latter’s rotation axis is now on the y-z plane. At 2.5
seconds, the AS has made one quarter of the orbit and it is found on the
y-axis, where it now receives the maximum number of magnetic field lines. At 5
seconds, it is on the $x$ axis again, where it receives the minimum amount and
so on. We would expect that $S$ would be larger the closer the NS rotation
axis is to the x-y plane. This also confirmed by Figure 3, where we see that
as $\tilde{\theta}$ increases, $S$ also increases.

Figure 4: Spectral flux density versus time for e=0.5. The blue color
represents the rotation axis on the z axis, while orange and green represent
the rotation axis at $\tilde{\theta}=\frac{\pi}{4}$ and
$\tilde{\theta}=\frac{\pi}{2}$ respectively

In the case where $e=0.5$, Figure 4, the situation is different, because now
the separation between the NS and the AS changes periodically with time. The
orbit stars from the perihelion, where the signal is maximum. At 5 seconds, it
is on the aphelion, where it is minimum and so on. As it was pointed out in
Kouvaris et al. [2022], this is easily explained if we remember that
$|\textbf{B}_{NS}|^{2}\sim\frac{1}{r^{6}}$, so there is significant
enhancement to the signal if the AS is close to the NS. Hence, the modulation
due to the NS spinning is not as important. This is confirmed by our results
when we allow the NS’s rotation axis to be tilted. In Figure 4, it is evident
that $S$ is slightly enhanced for the cases
$\tilde{\theta}=\frac{\pi}{4},\frac{\pi}{2}$, but not by much compared to the
$\tilde{\theta}=0$ case.

###  4.2 AS-NS binary in plasma

We would like to investigate how the plasma surrounding the NS affects the
emitted power. To that end, we consider a lower axion mass, of the order
$m_{a}=10^{-9}-10^{-8}\text{eV}$. This is outside of the QCD axion range, but
it is valid for ALPs, where $f_{a}$ and $m_{a}$ are independent. We will keep
the value of $f_{a}=10^{13}\text{GeV}$ for the axion decay constant.

It was shown in Amin et al. [2021] that when $\omega\rightarrow\omega_{p}$,
the emitted power per solid angle becomes:

$\frac{dP}{d\Omega}({\omega\rightarrow\omega_{p}})\approx\frac{(g_{a\gamma}\varphi_{0})^{2}}{128}\frac{(\pi\omega
R)^{6}}{\omega^{2}}\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1/2}|\textbf{B}_{NS}|^{2}\sin^{2}\beta$
(3ak)

There is significant enhancement in power now if $\omega R\gg 1$,due to the
factor $(\omega R)^{6}$. Therefore dilute axion stars can radiate much more
efficiently than dense axion stars. Inserting in 3k $m_{a}=10^{-9}\text{eV}$,
we find for the radius $R_{AS,dilute}=27000\text{km}$ and
$M_{AS,dilute}\approx 10^{-6}M_{\odot}$. The Roche limit for this AS is then
$R_{Roche}=R_{AS,dilute}\left(\frac{2M_{NS}}{M_{AS.dilute}}\right)^{1/3}\approx
38\times 10^{5}\text{km}$, much larger than the semi-major axis
$r_{0}=7.8\times 10^{3}\text{km}$. Hence, even though dilute axion stars could
theoretically radiate in the presence of plasma, they will be pulled apart by
tidal forces and disintegrate. We tried to avoid this problem by increasing
$r_{0}$. However, if we increase $r_{0}$, the plasma frequency, equation 3p,
becomes smaller. This means that we have to lower the axion mass too,
increasing the AS radius, via equation 3k, leading again to a Roche limit
larger than $r_{0}$. It is an open question what happens to the remnants of
this disintegrated AS which we leave for future work.

##  5 Conclusions and Outlook

In this paper, we have derived analytical expressions for the spectral flux
density emitted by an AS that is gravitationally bound to a NS. We have used a
sech profile to describe the AS that captures its essential characteristics
and took into account plasma effects using the Goldreich-Julian model for the
NS. We have also allowed the rotation axis of the NS to be tilted towards the
plane of the AS-NS, placing it on the y-z plane. If the orbit is circular, we
find that there is an order of magnitude increase to the spectral flux density
whenever the AS is on the y axis, because of the increased number of magnetic
field lines that now pass through the axion star. If, on the other hand, the
orbit is an ellipse, there is no significant enhancement to the flux, since
now the main modulation comes from the AS’s orbit.

We also considered the effect that the plasma may have on the flux. Given that
in the presence of plasma only dilute stars can radiate efficiently, we
lowered the axion mass to $m_{a}=10^{-9}\text{eV}$ to investigate this
possibility. We found that by lowering the mass, we increase the AS radius,
which in turn increases the Roche limit by several orders of magnitude more
than the AS-NS distance, meaning that the dilute AS will be torn apart by
tidal forces before it interacts with the NS’s magnetic field.

Some interesting venues for future research would be to investigate the effect
of back-reaction to the AS profile. This would probably require solving
numerically the set of equations 3. Back-reaction could also shed some light
to the time scale of disintegration of a dilute AS in the gravitational field
of a neutron star or a black hole. Another interesting direction of research
would be to include a multi-frequency profile for the dense AS. As it was
mentioned in the introduction, given that there have been attempts to explain
FRBs with dense axion stars, a more accurate profile for the AS would be a
step in that direction.

###### Acknowledgements.

I would like to thank David Sadek, Jose Alberto Ruiz Cembranos, Dennis
Maseizik and especially Pierre Sikivie for useful discussions.

##  Appendix A Appendix

We choose the vector perpendicular to $\hat{\Omega}$ to be:

$\hat{\Omega}_{\perp}=\begin{pmatrix}0\\\ -\cos\tilde{\theta}\\\
\sin\tilde{\theta}\end{pmatrix}$ (3al)

The rotation matrix about this vector and by an angle $\gamma$ is given by
$R(\hat{\Omega}_{\perp},\gamma)=e^{i\gamma\,\hat{\Omega}_{\perp}\cdot\textbf{L}}$,
where L are the generators of SO(3). They are given by:

$L_{1}=\begin{pmatrix}0&0&0\\\ 0&0&-i\\\ 0&i&0\\\ \end{pmatrix},\
L_{2}=\begin{pmatrix}0&0&i\\\ 0&0&0\\\ -i&0&0\\\ \end{pmatrix},\
L_{3}=\begin{pmatrix}0&-i&0\\\ i&0&0\\\ 0&0&0\\\ \end{pmatrix}$ (3am)

Using these expressions, we find the rotation matrix to be:

$R(\hat{\Omega}_{\perp},\gamma)=\begin{pmatrix}\cos\gamma&\sin\tilde{\theta}\sin\gamma&\cos\tilde{\theta}\sin\gamma\\\
-\sin\tilde{\theta}\sin\gamma&\cos^{2}\tilde{\theta}+\sin^{2}\tilde{\theta}\cos\gamma&-\sin
2\tilde{\theta}\sin^{2}(\frac{\gamma}{2})\\\
-\cos\tilde{\theta}\sin\gamma&-\sin(2\tilde{\theta})\sin^{2}(\frac{\gamma}{2})&\sin^{2}\tilde{\theta}+\cos^{2}\tilde{\theta}\cos\gamma\\\
\end{pmatrix}$ (3an)

As it was mentioned in the text, we now perform a second rotation around
$\hat{\Omega}$ by an angle $\Omega t$. The matrix is $R(\hat{\Omega},\Omega
t)=e^{-i\,(\Omega t)\,\hat{\Omega}\cdot\textbf{L}}$ and using the generators
3am, we get:

$R(\hat{\Omega},\Omega t)=\begin{pmatrix}\cos\Omega
t&-\cos\tilde{\theta}\sin\Omega t&\sin\tilde{\theta}\sin\Omega t\\\
\cos\tilde{\theta}\sin\Omega
t&\sin^{2}\tilde{\theta}+\cos^{2}\tilde{\theta}\cos\Omega t&\sin
2\tilde{\theta}\sin^{2}(\frac{\Omega t}{2})\\\ -\sin\tilde{\theta}\sin\Omega
t&\sin(2\tilde{\theta})\sin^{2}(\frac{\Omega
t}{2})&\cos^{2}\tilde{\theta}+\sin^{2}\tilde{\theta}\cos\Omega t\\\
\end{pmatrix}$ (3ao)

Applying these two rotations successively, we obtain the magnetic moment unit
vector in equation 3r.

##  Appendix B Appendix

The integral of equation 3ae is:

$\mathcal{I}=\int
d^{3}r^{\prime}e^{ik|\textbf{r}_{e}-\textbf{r}^{\prime}|}f(|\textbf{r}^{\prime}-\textbf{r}_{a}|)$
(3ap)

The easiest way to do it is to shift the coordinate system and place in the
centre of the AS, $\textbf{r}^{\prime}\rightarrow\textbf{r}_{a}+\textbf{s}$.
The exponential can be written as
$|\textbf{r}_{e}-\textbf{r}_{a}-\textbf{s}|\approx|\textbf{r}_{e}-\textbf{r}_{a}|-\frac{(\textbf{r}_{e}-\textbf{r}_{a})\cdot\tilde{\textbf{r}}}{|\textbf{r}_{e}-\textbf{r}_{a}|}\approx|\textbf{r}_{e}-\textbf{r}_{a}|-\hat{r}_{e}\cdot\textbf{s}$,
where we have assumed that $r_{e}\gg r_{a}$. Of course,
$\omega_{p}(r^{\prime})\sim\frac{1}{(r^{\prime})^{3/2}}=\frac{1}{|\textbf{s}+\textbf{r}_{a}|^{3/2}}$.
But, because of the exponentially decaying profile $\text{sech}(r/R)$ that we
are using for the axion star, the main contribution to the integral 3ap will
come from $0<s\lesssim 5R$, much smaller than $r_{a}$. We therefore set
$\omega_{p}(r^{\prime})=\omega_{p}(r_{a})$.The angular part of the integral
is:

$2\pi\int^{\pi}_{0}d\theta_{s}\sin\theta_{s}e^{-iks\cos\theta_{s}}=\frac{4\pi\sin(ks)}{ks}$
(3aq)

while, using the profile $f(r)=\text{sech}(r/R)$, the radial part becomes:

$\int^{\infty}_{0}ds\,s\,\sin(ks)\,\text{sech}(s/R)=\frac{\pi^{2}R^{2}}{4}\,\frac{\tanh\left(\frac{\pi
kR}{2}\right)}{\cosh\left(\frac{kR\pi}{2}\right)}$ (3ar)

Plugging equations 3aq and 3ar back into 3ap, we get:

$\mathcal{I}=\frac{\left(\pi
R\right)^{3}}{kR}\frac{\tanh\left(\frac{kR\pi}{2}\right)}{\cosh\left(\frac{kR\pi}{2}\right)}e^{ik|\textbf{r}_{e}-\textbf{r}_{a}|}$
(3as)

which gives equation 3af.

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