# Photon-axion mixing in thermal emission of isolated neutron stars

Aleksei Zhuravlev<EMAIL_ADDRESS>Sergei Popov  Maxim Pshirkov

###### Abstract

Thermally emitting neutron stars represent a promising environment for probing
the properties of axion-like particles. Due to the strong magnetic fields of
these sources, surface photons may partially convert into such particles in
the large magnetospheric region surrounding the stars, which will result in
distinctive signatures in their spectra. However, the interaction depends on
the polarization state of the radiation and is rather weak due to the low
experimentally allowed values of the coupling constant $g_{\gamma a}$. In this
work, we compute the degree of photon-axion transition in the case of 100%
O-mode polarization and spectral energy distribution of an isotropic blackbody
with uniform surface temperature. The stellar magnetic field is assumed to be
dipolar. We show that with the maximum effect reached for the magnetic fields
$\sim 10^{13}$ – $10^{14}$ G (typical for X-ray dim isolated neutron stars)
and $g_{\gamma a}=2\times 10^{-11}$ GeV-1, the optical flux is reduced by 30 –
40%, while the high-energy part of the spectrum is not affected. The low-
energy decrease exceeds 5% at $g_{\gamma a}\geq 2\times 10^{-12}$ GeV-1 and
$m_{a}\leq 2\times 10^{-6}$ eV, which is below the present experimental and
astrophysical limits on axion parameters. To obtain the actual observational
constraints, rigorous treatment of the radiative surface layers is required.

###### keywords:

axion-like particles , neutron stars , polarization

###### PACS:

14.80.Va , 97.60.Jd , 97.10.Ld

††journal: Physics Letters B

[MSU]organization=Faculty of Physics, Lomonosov Moscow State University,
addressline=Vorobjevy Gory 1, city=Moscow, postcode=119234, country=Russia

[SAI]organization=Sternberg Astronomical Institute, Lomonosov Moscow State
University, addressline=Universitetskij pr. 13, city=Moscow, postcode=119234,
country=Russia

[INR]organization=Institute for Nuclear Research of Russian Academy of
Sciences, addressline=60th October Anniversary Prospect, 7a, city=Moscow,
postcode=117312, country=Russia

[PRAO]organization=P. N. Lebedev Physical Institute of the Russian Academy of
Sciences Pushchino Radio Astronomy Observatory, addressline=PRAO,
city=Pushchino, postcode=142290, country=Russia

##  1 Introduction

Pseudoscalar axion-like particles appear in various extensions of the Standard
Model of particle physics. Proposed in the late 1970s to explain the absence
of CP violation in strong interactions [1, 2], they have transcended the
original context and were employed to solve a variety of problems in physics
and astrophysics [3, 4, 5, 6]. In this work, we refer to all such particles
simply as axions.

The defining property of axions is the conversion into photons and vice versa
in the presence of an external magnetic field [7]. Many laboratory-based
searches exploited this interaction in attempts to prove their existence, but
so far no satisfactory observational evidence has been found, and the
stringent bounds on the coupling constant over a wide axion mass range were
obtained by the CERN Axion Solar Telescope (CAST), $g_{\gamma a}\leq 6.6\times
10^{-11}$ GeV-1 (95% C.L.) [8]. For this reason, studying manifestations of
axions in various astrophysical environments has become increasingly popular.
A common strategy used in such works is to search for spectral features in
sources of radiation observed through large-scale regions with relatively
strong magnetic fields. These include active galactic nuclei such as NGC 1275
[9, 10, 11] (see also [12]), blazars and quasars [13, 14], supernovae [15],
etc. By comparing the detected spectra with the ones arising after the mixing
was taken into account, several authors were able to impose astrophysical
constraints on a range of axion parameters, exceeding the CAST limit.

More recently, considerable research has been devoted to studying the axion-
photon interaction in strongly magnetized objects, namely magnetic white
dwarfs (mWDs) and neutron stars. In the case of white dwarfs, photons emitted
by the photosphere may undergo conversion as they propagate through the
encompassing gaseous atmosphere [16]. If the plasma is sufficiently tenuous,
this could lead to an enhancement of the observed degree of linear
polarization. Since in most white dwarfs its value is very low, such
interaction can be used to constrain a range of axion parameters. In
particular, for mWD PG 1015+014 the induced polarization degree should exceed
the observed value of 5% at $g_{\gamma a}\geq 2\times 10^{-11}$ GeV-1, which
is below the CAST limit.

In the case of neutron stars, the presence of axions can manifest itself in a
number of ways. First, their magnetospheres can be a site for the conversion
of dark matter axions [17, 18]. Also, axions could be produced thermally
inside the core with energies from several keV to several hundred keV through
the nucleon bremsstrahlung processes, and escape the interior due to the
feeble interaction with matter. Their subsequent transition into photons in
the magnetic field surrounding the star could lead to observable signatures in
its hard X-ray spectrum [6, 19, 20]. Magnetars are natural targets for such
investigation due to the extremely strong magnetic fields and high internal
temperatures, and their study led to further astrophysical limits on axion
parameters [21, 22, 23].

At the same time, manifestations of the reverse process have not yet been
fully explored. Spectra of neutron stars commonly include a thermal component,
and as the surface radiation propagates through the large magnetospheric
region with a relatively strong magnetic field, a fraction of the initial flux
may oscillate into axions. Although in most of the magnetosphere the
probability of conversion is suppressed by the QED effects, previous studies
have shown that the mixing can lead to distinctive spectral signatures [24,
25]. However, the coupling values they considered are now excluded
experimentally. The purpose of our work is to find out whether current
constraints on axion parameters allow for observable effects, and to estimate
their possible magnitude.

X–ray dim isolated neutron stars (XDINSs), also known as the “Magnificent
Seven”, are particularly suited for such investigation. These isolated, radio-
silent sources exhibit purely thermal spectra with no high-energy power-law
component, which is often found in other classes of neutron stars. Their X–ray
luminosity of $10^{31}-10^{32}$ erg s-1 is consistent with the surface
blackbody emission with temperatures $kT^{\infty}\sim 40-100$ eV, and the
radiation radii of a few kilometers [26, 27]. The spin periods $P\sim 3-12$ s
and period derivatives $\dot{P}\sim 10^{-14}-10^{-13}$ s s-1 translate into
spin-down magnetic fields $B\sim 10^{13}-10^{14}$ G.

In the strongly magnetized vacuum around a neutron star, the photons propagate
in two normal modes, the extraordinary (X) and ordinary (O) mode, and only
those polarized in the latter can convert into axions. Although the physical
conditions of the outermost layers of XDINSs are still a debated issue,
several models have been proposed to explain the observed properties of these
sources [28], and the polarization patterns they predict are different. The
conventional picture is that cooling neutron stars are covered with a gaseous
atmosphere that reprocesses the emission coming from the outermost surface
layers. As a result, the radiation acquires a net polarization predominantly
in the X-mode, since the strong magnetic field substantially reduces its
opacity while leaving that for the O-mode almost unchanged [29, 30, 31]. On
the other hand, it has recently been appreciated that the stars with a low
surface temperature ($\lesssim 100$ eV) and a strong magnetic field ($\gtrsim
10^{13}$ G) may have a liquid or solid condensed surface due to a phase
transition in the outermost layers [27, 32, 33, 34]. Thus, in this case, the
atmosphere may be absent. The emissivity of a naked condensed surface in both
X- and O-mode is of the same order in a broad energy range [29, 35].

Since the magnitude of photon-axion transition will depend on the composition
of stellar surface layers, it is essential to estimate its possible magnitude
at the currently allowed values of axion parameters. To this end, here we
consider a simple picture in which the thermal emission is an isotropic
blackbody with a uniform surface temperature, and is 100% polarized in the
O-mode. These conditions provide the highest degree of conversion, and our
results will show whether a more detailed study can lead to astrophysical
constraints that exceed the present experimental limits.

Our paper is organized as follows. In Section 2, we summarize the main
properties of the photon-axion interaction in a strong magnetic field. Then in
Section 3, we calculate the resulting degree of conversion and determine the
range of axion parameters $\left(m_{a},g_{\gamma a}\right)$ at which the
mixing has a significant effect. The observability of our results and the
prospects for future work are discussed in Section 4.

##  2 Theoretical framework

In this section we outline the physical basis for our calculations. Throughout
the paper we use Lorentz–Heaviside units with $\hbar=c=1$ and
$\alpha=e^{2}/4\pi\approx 1/137$.

###  2.1 Polarization of radiation

In the presence of an external magnetic field, the vacuum around the star
behaves as a birefringent medium, in which the photons are polarized in two
normal modes: the ordinary mode (O-mode), with the electric field oscillating
in the plane of the propagation vector $\boldsymbol{k}$ and the local magnetic
field $\boldsymbol{B}$, and the extraordinary mode (X-mode), with the electric
field oscillating perpendicularly to both $\boldsymbol{k}$ and
$\boldsymbol{B}$. According to the QED, the dielectric and magnetic
permeability tensors of the vacuum are modified by virtual electron-positron
pairs, which affects the polarization properties of radiation primarily in two
ways.

First, the photons emitted at the surface maintain their initial polarization
state up to a characteristic distance [36, 37, 38], the polarization radius
$r_{p}$, which depends on the photon energy and magnetic field strength. In
this region, the wave electric field can instantly adapt its direction to that
of $\boldsymbol{B}$. Around $r_{p}$, the coupling weakens, and the photon
electric field is no longer able to properly follow the variation of the local
magnetic field, until for $r\gg r_{p}$, its direction freezes. If the
radiation propagates radially in a dipolar field, its initial polarization
state is preserved because the direction of transverse part of
$\boldsymbol{B}$ is always constant. This does not hold for the photons,
trajectory of which has been modified by the gravitational effects [39]. In
general, to account for the QED-induced depolarization at $r>r_{p}$, it is
necessary to solve the system determining the evolution of the polarization
state [38, 40]. However, the magnitude of this effect is relatively small,
because the dipolar magnetic field does not significantly change its direction
outside the polarization radius. According to our calculations, for the
energies considered in this paper, the angle between the wave electric field
and the transverse part of $\boldsymbol{B}$ for both modes changes only
slightly from its initial value in the region that will be of interest to us.
For this reason, we can assume that the radiation always retains its initial
mode without losing significant physical accuracy.

Second, the refractive indices for each mode are different and can deviate
from their vacuum value of unity. For the photon energies well below the
electron rest mass, the Euler-Heisenberg effective Lagrangian can be applied
[41, 42], which for arbitrary values of $B=|\boldsymbol{B}|$ leads to

$\begin{Bmatrix}n_{\parallel}\\\
n_{\perp}\end{Bmatrix}=1+\sin^{2}\Theta\begin{Bmatrix}\xi\left(b\right)\\\
\chi\left(b\right)\end{Bmatrix},$ (1)

$\xi\left(b\right)=\frac{7\alpha
b^{2}\left(1+1.2b\right)}{45\pi\left[1+1.33b+0.56b^{2}\right]},$ (2)

$\chi\left(b\right)=\frac{4\alpha
b^{2}}{45\pi\left[1+0.72b^{5/4}+\left(4/15\right)b^{2}\right]},$ (3)

where $\Theta$ is the angle between $\boldsymbol{k}$ and $\boldsymbol{B}$, $b$
is the magnetic field strength in units of the critical field
$B_{\text{crit}}=m^{2}_{e}/e=4.4\times 10^{13}$ G.

The magnetized plasma surrounding the star modifies the dielectric tensor
similarly to the vacuum polarization. In the following, we assume that it has
the Goldreich-Julian charge density
$\rho_{\text{GJ}}=-\boldsymbol{\Omega}\cdot\boldsymbol{B}$, where
$\boldsymbol{\Omega}$ is the stellar angular velocity [43]. Note that
$\rho_{\text{GJ}}$ estimates only the net charge density, and the total charge
density may be larger due to the presence of electron-positron pairs [44].
However, since there is no reliable probe of the plasma density in the
magnetospheres of neutron stars, we will only consider the Goldreich-Julian
value as a conservative lower limit.

###  2.2 Photon-axion mixing

The oscillations of a photon with energy $\omega$ and an axion in the external
magnetic field are described by the following interaction:

$\mathcal{L}=-\frac{1}{4}g_{\gamma a}aF_{\mu\nu}\tilde{F}^{\mu\nu},$ (4)

where $F_{\mu\nu}$ is the electromagnetic stress tensor, $\tilde{F}^{\mu\nu}$
is its dual, $a$ is the axion field, $m_{a}$ is the axion mass, and $g_{\gamma
a}$ is the coupling constant [7]. By adding the above term to the Euler-
Heisenberg effective Lagrangian and linearizing the wave equations, we obtain
the following system:

$\displaystyle\left(\omega+\begin{pmatrix}\Delta_{\parallel}+\Delta_{p}&\Delta_{M}\\\
\Delta_{M}&\Delta_{a}\end{pmatrix}-i\partial_{z}\right)\begin{pmatrix}A_{\parallel}\\\
a\end{pmatrix}=0,$ (5)

where $A_{\parallel}$ and $a$ denote the amplitudes of the ordinary photon
state and the axion, respectively, and the $z-$ axis is along
$\boldsymbol{k}$. Note that only the radiation polarized in the O-mode can
convert into axions, and the X-mode flux always remains intact. The components
of the mixing matrix are:

$\displaystyle\Delta_{a}$ $\displaystyle=-\frac{m_{a}^{2}}{2\omega},$
$\displaystyle\Delta_{M}$ $\displaystyle=\frac{1}{2}g_{\gamma a}B\sin\Theta,$
(6) $\displaystyle\Delta_{p}$ $\displaystyle=-\frac{\omega_{p}^{2}}{2\omega},$
$\displaystyle\Delta_{\parallel}$
$\displaystyle=\frac{1}{2}\omega\left(n_{\parallel}-1\right),$

where $\omega_{p}^{2}=4\pi\alpha n_{e}/m_{e}$ is the plasma frequency,
$n_{e}=\rho_{\text{GJ}}/e$ is the electron density [45].

While the converted fraction of the initial photon flux can only be obtained
by numerically integrating Equations 5, the strength of the mixing can be
estimated as follows. The external magnetic field varies along the photon
trajectory over a lengthscale $l_{B}=B/|\hat{\boldsymbol{k}}\cdot\nabla
B|=r/3$, where a hat denotes a unit vector. If we take the value of $B$ to be
constant over $l_{B}$, the probability of a photon-axion transition after
traveling this distance is

$P_{\gamma_{\parallel}\rightarrow
a}=\frac{4\Delta_{M}^{2}}{\left(\Delta_{\parallel}+\Delta_{p}-\Delta_{a}\right)^{2}+4\Delta_{M}^{2}}\sin^{2}\left(\frac{1}{2}\Delta_{\text{osc}}l_{B}\right)$
(7)

where
$\Delta_{\text{osc}}^{2}=\left(\Delta_{\parallel}+\Delta_{p}-\Delta_{a}\right)^{2}+4\Delta_{M}^{2}$,
so that the oscillation length is $l_{\text{osc}}=2\pi/\Delta_{\text{osc}}$.
Strong mixing occurs when the conversion probability is close to unity, which
is satisfied if

$\displaystyle l_{\text{osc}}$ $\displaystyle\lesssim l_{B},$ (8)
$\displaystyle|\Delta_{\parallel}+\Delta_{p}-\Delta_{a}|$
$\displaystyle\lesssim 2|\Delta_{M}|.$

##  3 Numerical simulations

In this section, we calculate the observational signatures of photon-axion
mixing in the O-mode flux of a thermally emitting neutron star, and determine
the range of axion parameters $\left(m_{a},g_{\gamma a}\right)$ at which the
effect is significant.

###  3.1 Computational model

Our treatment of the surface radiation relies on a number of simplifications
which are discussed below. We stress that our primary goal is to assess
whether the mixing can have a noticeable effect on spectra of XDINSs at the
currently unconstrained values of $\left(m_{a},g_{\gamma a}\right)$ and
whether there is room for future investigations, rather than to derive
theoretical predictions to be compared with the observational data. For this
reason, we avoid dealing with the complex structure of the outermost stellar
layers and consider the most favorable conditions for the detection of photon-
axion conversion.

The stellar magnetic field is assumed to be an aligned dipole, components of
which in polar coordinates $\left(r,\theta\right)$ take the form

$\boldsymbol{B}^{\text{dip}}=\frac{B_{p}}{2}\left(\frac{R_{s}}{r}\right)^{3}\begin{pmatrix}2f_{\text{dip}}\cos\theta\\\
g_{\text{dip}}\sin\theta\\\ 0\end{pmatrix},$ (9)

where $B_{p}$ is the surface polar field strength. The functions
$f_{\text{dip}}$, $g_{\text{dip}}$ account for the relativistic corrections
[46, 47]. The stellar radius, mass, and period are $R_{s}=12$ km,
$M_{s}=1.4M_{\odot}$, and $P_{s}=7$ s, respectively.

Initially, the radiation is emitted from the cooling stellar surface with an
isotropic blackbody distribution and uniform temperature $kT^{\infty}=50$ eV.
The seed photons are $100\%$ polarized in the O-mode, which provides the
maximum degree of conversion into axions. As the radiation moves away from the
emission point, we numerically integrate Equations 5 along its trajectory,
accounting for the gravitational redshift. The observer is located at
infinity, with the line of sight inclined at an angle $\chi$ with respect to
the spin axis, and collects the contribution of each part of the surface which
is into view due to the effects of ray bending [39].

###  3.2 Results

To get a clear picture of the conversion process, let us first study the
mixing in an individual photon ray. The simplest case is the radiation emitted
in radial direction (colatitude of the emission point is the same as that of
the observer), so that its trajectory is not modified by gravity. We take the
coupling constant $g_{\gamma a}=2\times 10^{-11}$ GeV-1 (the highest value
allowed by the study of mWD PG 1015+014 [16]), axion mass $m_{a}=10^{-8}$ eV,
and photon energy $\omega_{\infty}=1$ eV. The polar magnetic field strength is
$B_{p}=10^{13}$ G (representative of XDINSs [27]) and the observer inclination
angle is $\chi=60^{\circ}$. The resulting plot of the photon and axion square
amplitudes is shown in Figure 1.

Figure 1: Evolution of the photon and axion square amplitudes in the case of
radiation emitted radially, as a function of the distance $r$ from the stellar
surface. The parameters of the system are $\omega_{\infty}=1$ eV, $g_{\gamma
a}=2\times 10^{-11}$ GeV-1, $m_{a}=10^{-8}$ eV. The magnetic field geometry is
an aligned dipole with $B_{p}=10^{13}$ G and $\chi=60^{\circ}$. The
probability of conversion beyond $r=500$ km is vanishingly small, this region
is not displayed.

As we can see, significant conversion takes place at a distance of several
stellar radii away from the surface. The system does not enter the strong-
oscillation regime (in this case, $|A_{\parallel}|^{2}$ and $|a|^{2}$ would
have the form of sinusoidal curves) because Conditions 8 are never met
simultaneously. In particular, near the emission point, where $l_{\text{osc}}$
is small compared to $l_{B}$, $\Delta_{\parallel}$ is much larger than
$2\Delta_{M}$, and therefore the mixing is suppressed. As the radiation moves
away from the surface of the star, $\Delta_{\parallel}$ eventually becomes
comparable with $2\Delta_{M}$ ($\Delta_{a}$ is negligible and
$|\Delta_{p}|\lesssim\Delta_{\parallel}$), which is due to their different
dependence on the radial coordinate: $\Delta_{\parallel}\sim 1/r^{6}$, while
$\Delta_{M}\sim 1/r^{3}$. However, in this region, $l_{\text{osc}}$ is already
5-10 times greater than $l_{B}$, which results in the conversion probability
being less than unity. Still, these conditions are sufficient for the mixing
to have a noticeable effect. At relatively large distances, the conversion
probability becomes negligible.

We then perform the calculations for a broad energy range (from 1 eV to 10
keV), integrating the intensity over the visible part of the stellar surface.
The result is conveniently expressed in terms of the differential modification
factor $\Lambda$, which is defined as the fraction of the initial photon flux
with a given energy, reaching the observer without conversion. In particular,
if $\Lambda=1$, the flux will not be modified, and if $\Lambda=0$, all photons
will turn into axions and thus no signal will be detected. The data for
$g_{\gamma a}=2\times 10^{-11}$ GeV-1, $m_{a}=10^{-8}$ eV, and $B_{p}=10^{13}$
G, $\chi=60^{\circ}$ are presented in Figure 2, as a function of the
redshifted photon energy.

Figure 2: Differential modification factor $\Lambda$ of the O-mode radiation
for a broad energy range. The remaining parameters are the same as in Figure
1. At $\omega_{\infty}>100$ eV, $\Lambda$ is almost unity, this region is not
displayed.

Our results show that the mixing is most significant at the optical energies,
where the O-mode flux is almost halved. At the same time, it has little effect
on the X-ray emission, which is due to the fact that the QED contribution
$\Delta_{\parallel}$ increases linearly with increasing $\omega$, while
$\Delta_{M}$ does not depend on it. At higher photon energies,
$\Delta_{\parallel}$ and $\Delta_{M}$ become of the same order in the region
where the oscillation length is much larger than $l_{B}$, which results in a
considerable decrease in the conversion probability. The shapes of the photon
and axion square intensities are similar to those shown in Figure 1, except
for the oscillations having smaller amplitudes. It is noteworthy that the
converted fraction of the flux varies by no more than a few percent between
different ray trajectories, which due to the fact that the system remains in
the weak-oscillation regime. For the same reason, its value does not change
noticeably for other observer inclination angles.

We then repeat our calculations for different values of $\left(m_{a},g_{\gamma
a}\right)$ to obtain the range at which the mixing has a significant effect on
the optical radiation. Figure 3 illustrates the modification factor at
$\omega_{\infty}=1$ eV for $B_{p}=10^{13}$ G and a given choice of axion
parameters. The optical photon flux is reduced by more than 5% at $g_{\gamma
a}\geq 6\times 10^{-12}$ GeV-1 and $m_{a}<2\times 10^{-6}$ eV. This result can
be explained using Conditions 8: for lower values of the coupling constant,
$\Delta_{M}$ does not get close enough to $\Delta_{\parallel}$ in the region
with a relatively small oscillation length; for higher axion masses,
$\Delta_{a}$ becomes of the same order as both $\Delta_{M}$ and
$\Delta_{\parallel}$, suppressing the conversion probability. The plasma
contribution $\Delta_{p}$ becomes important at lower values of $g_{\gamma a}$,
where $\Delta_{\parallel}\gg\Delta_{M}$ almost everywhere in the
magnetosphere. At some point, the terms $\Delta_{p}$ and $\Delta_{\parallel}$
become of the same order and cancel each other because of their opposite
signs. Even though the oscillation length in this region is much larger than
$l_{B}$, this results in a slight increase in the conversion probability.

Figure 3: Contour plot of the O-mode modification factor $\Lambda$ for
$\omega_{\infty}=1$ eV, as a function of the axion parameters
$\left(m_{a},g_{\gamma a}\right)$. The magnetic field geometry is
$B_{p}=10^{13}$ G, $\chi=60^{\circ}$. We also include the values of $g_{\gamma
a}$ allowed by CAST [8].

We also perform the same calculations for a greater polar field strength
$B_{p}=10^{14}$ G, which can also be present in XDINSs [27]. As shown in
Figure 4, in this case the converted fraction reaches 5% at even lower values
of the coupling constant, $g_{\gamma a}\geq 2\times 10^{-12}$ GeV-1 and
$m_{a}\leq 2\times 10^{-6}$ eV.

Figure 4: Same as Figure 3, but for $B_{p}=10^{14}$ G.

##  4 Discussion and conclusions

In this paper we studied the photon-axion conversion in the magnetosphere of a
thermally emitting neutron star. Assuming the dipolar magnetic field and the
spectral energy distribution of an isotropic blackbody with a uniform surface
temperature, we have shown that the mixing can result in a significant
decrease in the optical O-mode flux at the allowed values of
$\left(m_{a},g_{\gamma a}\right)$. Further investigation may lead to
astrophysical constraints on axion parameters that exceed the present
experimental and astrophysical limits.

We found that the photon-axion system exhibits only weak oscillations at the
currently allowed values of the coupling constant. The reason for the decrease
in the conversion probability is either the dominance of the QED contribution
$\Delta_{\parallel}$, or the fact that the oscillation length is much larger
than the scale length of the magnetic field. This is in agreement with the
study of [25], who focused on higher photon energies and currently excluded
values of $g_{\gamma a}$. According to their results, weak oscillations are
present in the X-ray range at $g_{\gamma a}=10^{-10}-10^{-9}$ GeV-1 and the
converted fraction depends inversely on $\omega_{\infty}$, similarly to the
case shown in Figure 2. Only at a relatively high value of $g_{\gamma
a}=10^{-8}$ GeV-1, the conversion probability becomes equal to unity in a
large area of the magnetosphere, and the system enters the strong-oscillation
mode. As a result, $\Lambda$ depends on the photon energy in a more complex
way and, in particular, can increase with increasing $\omega_{\infty}$. Such
relation should not be confusing, since it is simply a consequence of the
strong-oscillation mode, which is substantially different from the weak-mixing
regime discussed in our work [7, 24]. Overall, greater values of the coupling
constant could be more easily constrained by looking into the X-ray energies
where the emission of XDINSs typically peaks, while at lower values of
$g_{\gamma a}$ one has to consider the mixing in the optical band.

Our results exceed the limits obtained by [16] in their study of mWD PG
1015+014. At the same time, observations of the magnetic white dwarfs with
higher surface field strengths could potentially exclude a wider region of the
axion parameter space, which will be surpassed by the limits shown in Figures
3 and 4 only at high axion masses. However, these results should be
interpreted with caution, as they are highly dependent on the assumptions
about the atmospheric plasma composition. In particular, the authors of [16]
consider the case of a relatively sparse hydrogen plasma with a surface
density of $\rho_{0}=10^{-10}$ g cm-3 and a barometric profile. They also note
that there is no clear agreement on the surface plasma density of magnetic
white dwarfs, and its actual value is expected to lie between $10^{-11}$ and
$10^{-6}$ g cm-3. In this regard, the study of XDINSs may provide a more
robust method of imposing constraints on axion parameters, which, however,
will depend on another kind of assumptions.

A major challenge in assessing the effect of mixing on spectra of XDINSs is
the uncertainty in the structure of their radiative surface layers. A well-
known feature of the ”Magnificent seven” is that their optical and ultraviolet
counterparts exceed the extrapolation of X–ray blackbody at low energies by a
factor 5 – 50 [48], and numerous authors have made extensive use of
atmospheric and condensed surface models in attempts to explain its origin. A
strongly magnetized atmosphere has been suggested for RX J1605.3+3249 [49].
The observed properties of RX J0720.4-3125 were attributed to the emission of
a bare condensed surface [50], and a combination of the latter and a thin,
magnetic, partially ionized hydrogen atmosphere was used to fit the spectral
shape of J1856.5-3754 [51] and RX J1308.6+2127 [52].

The polarization pattern induced by both models can be conveniently expressed
in terms of the intrinsic polarization degree (that at the source):
$\Pi_{\text{L}}^{\text{EM}}=\left(F_{\text{X}}-F_{\text{O}}\right)/\left(F_{\text{X}}+F_{\text{O}}\right)$,
where $F_{\text{X,O}}$ is the monochromatic, phase-averaged flux in each mode,
obtained by integrating the intensity over the visible part of the stellar
surface [29]. By definition, $\Pi_{\text{L}}^{\text{EM}}=1$ for the radiation
100% initially polarized in the X-mode, $\Pi_{\text{L}}^{\text{EM}}=-1$ for
that in the O-mode, and $\Pi_{\text{L}}^{\text{EM}}=0$ for the unpolarized
emission. According to the results of [29], in the case of a fully ionized
hydrogen atmosphere, $\Pi_{\text{L}}^{\text{EM}}\approx 0.99$ in the X–ray
band and $\Pi_{\text{L}}^{\text{EM}}\approx 0.87$ in the optical band for all
possible viewing geometries. The radiation of a condensed surface in the X-ray
band is almost unpolarized, $|\Pi_{\text{L}}^{\text{EM}}|\lesssim 0.07$, while
in the optical band $\Pi_{\text{L}}^{\text{EM}}$ can reach $-0.5$ for the most
favorable viewing geometries, with the emission being mostly polarized in the
O-mode. Since the polarization state of thermal radiation directly affects the
magnitude of photon-axion conversion, the effect will be observable only if
the flux contains a considerable fraction of O-mode photons. From this point
of view, the emission of a condensed surface is a promising place to search
for its manifestations, but the result is likely to be lower than that
obtained in Section 3, since we considered the liming case of 100% O-mode
polarization. The presence of an atmosphere may make the interaction less
noticeable because its radiation is predominantly polarized in the X-mode.

Rigorous treatment of the outermost stellar layers provides several ways of
imposing constraints on a range of axion parameters. One approach is to simply
measure the change in the low-energy spectra, as shown in Figure 3. Although
the optical counterparts of XDINSs are rather faint, observational data have
been presented for all sources [48]. However, this method will not be
sensitive enough if the converted fraction is relatively small, which is
expected at low values of the coupling constant $g_{\gamma a}$. In such cases,
it may be more efficient to compute the polarization observables, linear
polarization fraction $\Pi_{\text{L}}$ and polarization angle
$\chi_{\text{p}}$ [38], which accurately describe the polarization state of
the detected radiation and can reflect a slight decrease in the O-mode flux.
Since various types of stellar surface composition are expected to have
different polarization patterns [29], constraints can be based on a comparison
of $\Pi_{\text{L}}$ and $\chi_{\text{p}}$, predicted by the models which do
and do not take the mixing into account. Additionally, one can make use of the
observational polarimetric data, as described in [53] for the case of RX
J1856.5-3754.

Another potentially promising approach is to consider the photon-axion
interaction in magnetars. Their magnetic fields are stronger than those of
XDINSs $\left(B_{p}\approx 10^{15}\text{ G}\right)$, and are believed to
include higher multipoles and global or local shears (the “twisted
magnetosphere” model, see [54]), which in general can increase the mixing
strength. In addition, their low-energy radiation may include a considerable
fraction of the O-mode photons [55, 56]. Such investigation, however, is
hampered by our present poor knowledge of both the composition of surface
layers and the magnetospheric structure of magnetars, as well as the faintness
of the optical counterparts of their thermal emission [57]. At the same time,
considering other classes of neutron stars, such as radio pulsars, is unlikely
to produce greater constraints on axion parameters. Due to their lower
magnetic field strengths, the converted fraction will likely be smaller than
in the case of XDINSs.

## Acknowledgements

We are grateful to Sergey Troitsky for numerous helpful discussions and the
anonymous referee for insightful comments and suggestions. This work was
supported by Russian Science Foundation, grant 18-12-00258.

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