\section{Introduction}

A hypothetical pseudoscalar particle called axion is predicted by the theory
related to solving the CP-invariance violation problem in QCD. The most
important parameter determining the axion properties is the energy scale $f_a$
of the so-called U(1) Peccei-Quinn symmetry violation. It determines both the
axion mass and the strength of its coupling to fermions and gauge bosons
including photons. However, in spite of the numerous direct experiments, they
have not been discovered so far. Meanwhile, these experiments together with the
astrophysical and cosmological limitations leave a rather narrow band for the
permissible parameters of invisible axion (e.g.
$10^{-6} eV \leqslant m_a \leqslant 10^{-2} eV$~\citep{ref01,ref02}), which is
also a well-motivated cold dark matter candidate in this mass region
\citep{ref01,ref02}.

A whole family of axion-like particles (ALP) with their own features may exist
along with axions having the similar Lagrangian structure relative to the
Peccei-Quinn axion, as well as their own distinctive features. It consists in
the fact that if they exist, the connection between their mass and their
constant of coupling to photons must be highly weakened, as opposed to the
axions. It should be also mentioned that the phenomenon of photon-ALP mixing in
the presence of the electromagnetic field not only leads to the classic
neutrino-like photon-ALP oscillations, but also causes the change in the
polarization state of the photons (the $a \gamma \gamma$ coupling acts like a
polarimeter \citep{ref03}) propagating in the strong enough magnetic fields. It
is generally assumed that there are light ALPs coupled only to two photons,
although the realistic models of ALPs with couplings both to photons and to
matter are not excluded \citep{ref04}. Anyway, they may be considered a
well-motivated cold dark matter candidate \citep{ref01,ref02} under certain
conditions, just like axions.

It is interesting to note that the photon-ALP mixing in magnetic fields of
different astrophysical objects including active galaxies, clusters of
galaxies, intergalactic space and the Milky Way, may be the cause of the
remarkable phenomena like dimming of stars luminosity (e.g. supernovae in the
extragalactic magnetic field \citep{ref06,ref07}) and ``light  shining through
a wall'' (e.g. light from very distant objects, travelling through the Universe
\citep{ref03,ref05}). In the former case the luminosity of an astrophysical
object is dimmed because some part of photons transforms into axions in the
object's magnetic field. In the latter case photons produced by the object are
initially converted into axions in the object's magnetic field, and then after
passing some distance (the width of the ``wall'') are converted back into
photons in another magnetic field (e.g. in the Milky Way), thus emulating the
process of effective free path growth for the photons in astrophysical medium
\citep{ref08,ref09}.

For the sake of simplicity let us hereinafter refer to all such particles as
axions if not stated otherwise.

In the present paper we consider the possible existence of the axion mechanism
of Sun luminosity variations\footnote{Let us point out that the axion mechanism of Sun
luminosity used for estimating the axion mass was described for the first time
in 1978
by \cite{ref10}.} based on the ``light shining through a wall'' effect. To be
more exact, we attempt to explain the axion mechanism of Sun luminosity variations by the
``light shining through a wall'', when the photons born mainly in the solar
core are at first converted into axions via the Primakoff effect \citep{ref11}
in its magnetic field, and then are converted back into photons after passing
the solar radiative zone and getting into the magnetic field of the overshoot
tachocline. We estimate this magnetic field within the framework of the 
Ettingshausen-Nernst effect. In addition to that we obtain the consistent 
estimates for the axion mass ($m_a$) and the axion coupling constant to photons
($g_{a \gamma}$), basing on this mechanism, and verify their values against the
axion model results and the known experiments including CAST, ADMX, RBF.



\section{Photon-axion conversion and the case of maximal mixing}

Let us give some implications and extractions from the photon-axion
oscillations theory which describes the process of the photon conversion into
an axion and back under the constant magnetic field $B$ of the length $L$. It
is easy to show \citep{ref05,Raffelt-Stodolsky1988,ref07,Hochmuth2007} that in
the case of the negligible photon absorption coefficient
($\Gamma _{\gamma} \to 0$) and axions decay rate ($\Gamma _{a} \to 0$) the
conversion probability is
\begin{equation}
P_{a \rightarrow \gamma} = \left( \Delta_{a \gamma}L \right)^2 \sin ^2 \left( \frac{ \Delta_{osc}L}{2} \right) \Big/ \left( \frac{ \Delta_{osc}L}{2}
\right)^2 \label{eq01}\, ,
\end{equation}
where the oscillation wavenumber $\Delta_{osc}$ is given by
\begin{equation}
\Delta_{osc}^2 = \left( \Delta_{pl} + \Delta_{Q,\perp} - \Delta_{a} \right)^2 + 4 \Delta_{a \gamma} ^2
\label{eq02}
\end{equation}
while the mixing parameter $\Delta _{a \gamma}$, the axion-mass parameter
$\Delta_{a}$, the refraction parameter $\Delta_{pl}$ and the QED dispersion
parameter $\Delta_{Q,\perp}$ may be represented by the following expressions:
\begin{equation}
\Delta _{a \gamma} = \frac{g_{a \gamma} B}{2} = 540 \left( \frac{g_{a \gamma}}{10^{-10} GeV^{-1}} \right) \left( \frac{B}{1 G} \right) ~~ pc^{-1}\, ,
\label{eq03}
\end{equation}
\begin{equation}
\Delta _{a} = \frac{m_a^2}{2 E_a} = 7.8 \cdot 10^{-11} \left( \frac{m_a}{10^{-7} eV} \right)^2 \left( \frac{10^{19} eV}{E_a} \right) ~~ pc^{-1}\, ,
\label{eq04}
\end{equation}
\begin{equation}
\Delta _{pl} = \frac{\omega ^2 _{pl}}{2 E_a} = 1.1 \cdot 10^{-6} \left( \frac{n_e}{10^{11} cm^{-3}} \right) \left( \frac{10^{19} eV}{E_a} \right) ~~ pc^{-1},
\label{eq05}
\end{equation}
\begin{equation}
\Delta _{Q,\perp} = \frac{m_{\gamma, \perp}^2}{2 E_a} .
\label{eq06}
\end{equation}

Here $g_{a \gamma}$ is the constant of axion coupling to photons; $B$ is the
transverse magnetic field; $m_a$ and $E_a$ are the axion mass and energy;
$\omega ^2 _{pl} = 4 \pi \alpha n_e / m_e$ is an effective photon mass in terms
of the plasma frequency if the process does not take place in vacuum, $n_e$ is
the electron density, $\alpha$ is the fine-structure constant, $m_e$ is the
electron mass; $m_{\gamma, \perp}^2$ is the effective mass square of the
transverse photon which arises due to interaction with the external magnetic
field.

The conversion probability (\ref{eq01}) is energy-independent, when
$2 \Delta _{a \gamma} \approx \Delta_{osc}$, i.e.
\begin{equation}
P_{a \rightarrow \gamma} \cong \sin^2 \left( \Delta _{a \gamma} L \right)\, ,
\label{eq07}
\end{equation}
or, whenever the oscillatory term in (\ref{eq01}) is small
($\Delta_{osc} L / 2 \to 0$), implying the limiting coherent behavior
\begin{equation}
P_{a \rightarrow \gamma} \cong \left( \frac{g_{a \gamma} B L}{2} \right)^2\, .
\label{eq08}
\end{equation}

It is worth noting that the oscillation length corresponding to (\ref{eq07})
reads
\begin{equation}
L_{osc} = \frac{\pi}{\Delta_{a \gamma}} = \frac{2 \pi}{g_{a \gamma} B} \cong 5.8 \cdot 10^{-3}
\left( \frac{10^{-10} GeV^{-1}}{g_{a \gamma}} \right)
\left( \frac{1G}{B} \right)  ~pc
\label{eq13}
\end{equation}
\noindent assuming a purely transverse field. In the case of the appropriate
size $L$ of the region a complete transition between photons and axions is
possible.

From now on we are going to be interested in the energy-independent case
(\ref{eq07}) or (\ref{eq08}) which plays the key role in determination of the parameters
for the axion mechanism of Sun luminosity variations hypothesis (the axion coupling
constant to photons $g_{a \gamma}$, the transverse magnetic field $B$ of length
$L$ and the axion mass $m_a$).



\section{Axion mechanism of Sun luminosity variations}

Our hypothesis is that the solar axions which are born in the solar core
\citep{ref01,ref02} through the known Primakoff effect \citep{ref11}, may be
converted back into $\gamma$-quanta in the magnetic field of the solar
tachocline (the base of the solar convective zone). The magnetic field
variations in the tachocline cause the converted $\gamma$-quanta intensity
variations in this case, which in their turn cause the variations of the Sun
luminosity known as the active and quiet Sun states. Let us consider this
phenomenon in more detail below.

As we noted above, the expression (\ref{eq01}) for the probability of the
axion-photon oscillations in the transversal magnetic field was obtained for
the media with the quasi-zero refraction, i.e. for the media with a negligible
photon absorption coefficient ($\Gamma_{\gamma} \to 0$). It means that in order
for the axion-photon oscillations to take place without any significant losses,
a medium with a very low or quasi-zero density is required, which would
suppress the processes of photon absorption almost entirely.

Surprisingly enough, it turns out that such ``transparent'' media can take
place, and not only in plasmas in general, but straight in the convective zone
of the Sun. Here we generally mean the so-called magnetic flux tubes, the
properties of which are examined below.



\subsection{Ideal photon channeling conditions inside the magnetic flux tubes}

\label{subsec-channeling}

The idea of the energy flow channeling along a fanning magnetic field has been
suggested for the first time by
\cite{ref12} as an explanation for
darkness of umbra of sunspots. It was incorporated in a simple sunspot model by
\cite{ref13}.
\cite{ref14} extended this suggestion to smaller
flux tubes to explain the dark pores and the bright faculae as well.
Summarizing the research of the convective zone magnetic fields in the form of
the isolated flux tubes,
\cite{ref15} suggested a simple mathematical model for the behavior of thin
magnetic flux tubes, dealing with the nature of the solar cycle, sunspot
structure, the origin of spicules and the source of mechanical heating in the
solar atmosphere. In this model, the so-called thin tube approximation is used
(see \cite{ref15} and references therein), i.e. the field is conceived to exist
in the form of slender bundles of field lines (flux tubes) embedded in a
field-free fluid (Fig.~\ref{fig01}). Mechanical equilibrium between the tube
and its surrounding is ensured by a reduction of the gas pressure inside the
tube, which compensates the force exerted by the magnetic field.  In our
opinion, this is exactly the kind of mechanism
\cite{Parker1955} was thinking about when he wrote about the problem of flux
emergence: ``Once the field has been amplified by the dynamo, it needs to be
released into the convection zone by some mechanism, where it can be
transported to the surface by magnetic buoyancy''~\citep{ref17}.


\begin{figure*}
\begin{center}
\includegraphics[width=12cm]{TachoclineFluxTubes-3.pdf}
\end{center}
\caption{(a) Vertical cut through an active region illustrating the connection
between a sunspot at the surface and its origins in the toroidal field layer at
the base of the convection zone. Horizontal fields stored at the base of the
convection zone (the overshoot tachocline zone) during the cycle. Active
regions form from sections brought up by buoyancy (one shown in the process of
rising). After eruption through the solar surface a nearly potential field is
set up in the atmosphere (broken lines), connecting to the base of the
convective zone via almost vertical flux tube. Hypothetical small scale
structure of a sunspot is shown in the inset (Adopted from
\cite{ref18}
and
\cite{ref15}).
(b) Detection of emerging sunspot regions in the solar interior~\citep{ref18}.
Acoustic ray paths with lower turning points between 42 and 75 Mm
(1 Mm=1000 km) crossing a region of emerging flux. For simplicity, only four
out of total of 31 ray paths used in this study (the time-distance
helioseismology experiment) are shown here. Adopted from~\cite{ref19}.
(c) Emerging and anchoring of stable flux tubes in the overshoot tachocline
zone, and its time-evolution in the convective zone. Adopted from \cite{ref20}.
(d) Vector magnetogram of the white light image of a sunspot (taken with SOT on
a board of the Hinode satellite -- see inset) showing in red the direction of
the magnetic field and its strength (length of the bar). The movie shows the
evolution in the photospheric fields that has led to an X class flare in the
lower part of the active region. Adopted from~\cite{ref21}.}
\label{fig01}
\end{figure*}



In order to understand magnetic buoyancy, let us consider an isolated
horizontal flux tube in pressure equilibrium with its non-magnetic surroundings
so that
in cgs units
\begin{equation}
p_{ext} = p_{int} + \frac{\vert \vec{B} \vert^2}{8 \pi} ,
\label{eq21}
\end{equation}
\noindent where $p_{int}$ and $p_{ext}$ are the internal and external gas
pressures respectively, $B$ denotes the uniform field strength in the flux
tube. If the internal and external temperatures are equal so that $T_{int} =
T_{ext}$ (thermal equilibrium), then since $p_{ext} > p_{int}$, the gas in the
tube is less dense than its surrounding ($\rho _{ext} > \rho _{int}$), implying
that the tube will rise under the influence of gravity.

In spite of the obvious, though turned out to be surmountable, difficulties of
the application to real problems, it was shown (see~\cite{ref15} and Refs.
therein) that strong buoyancy forces act in magnetic flux tubes of the required
field strength ($10^4 - 10^5 ~G$~\citep{ref23}). Under their influence tubes
either float to the surface as a whole (e.g. Fig.~1 in \citep{ref24}) or they
form loops of which the tops break through the surface (e.g. Fig.~1
in~\citep{ref14}) and lower parts descend to the bottom of the convective zone,
i.e. to the overshoot tachocline zone. The convective zone, being unstable,
enhanced this process~\citep{ref25,ref26}. Small tubes take longer to erupt
through the surface because they feel stronger drag forces. It is interesting
to note here that the phenomenon of the drag force which raises the magnetic
flux tubes to the convective surface with the speeds about 0.3-0.6~km/s, was
discovered in direct experiments using the method of time-distance
helioseismology~\citep{ref19}. Detailed calculations of the
process~\citep{ref27} show that even a tube with the size of a very small spot,
if located within the convective zone, will erupt in less than two years. Yet,
according to~\cite{ref27}, horizontal fields are needed in overshoot tachocline
zone, which survive for about 11~yr, in order to produce an activity cycle.

A simplified scenario of magnetic flux tubes (MFT) birth and space-time
evolution (Fig.~\ref{fig01}a) may be presented as follows. MFT is born in the
overshoot tachocline zone (Fig.~\ref{fig01}c) and rises up to the convective
zone surface (Fig.~\ref{fig01}b) without separation from the tachocline (the
anchoring effect), where it forms the sunspot (Fig.~\ref{fig01}d) or other
kinds of active solar regions when intersecting the photosphere.  There are
more fine details of MFT physics expounded in overviews by
\cite{ref17} and
\cite{ref24}, where certain fundamental questions, which need to be addressed
to understand the basic nature of magnetic activity, are discussed in detail:
How is the magnetic field generated, maintained and dispersed? What are its
properties such as structure, strength, geometry? What are the dynamical
processes associated with magnetic fields? \textbf{What role do magnetic fields
play in energy transport?}

Dwelling on the last extremely important question associated with the energy
transport, let us note that it is known that thin magnetic flux tubes can
support longitudinal (also called sausage), transverse (also called kink),
torsional (also called torsional Alfv\'{e}n), and fluting modes
(e.g.~\cite{ref28,ref29,ref30,ref31,ref32}); for the tube modes supported by
wide magnetic flux tubes, see
\cite{ref31}. Focusing on the longitudinal tube waves known to be an important
heating agent of solar magnetic regions, it is necessary to mention the recent
papers by
\cite{ref33}, which showed that the longitudinal flux tube waves are identified
as insufficient to heat the solar transition region and corona in agreement
with previous studies~\citep{ref34}.
\textbf{In other words, the problem of generation and transport of energy by
magnetic flux tubes remains unsolved in spite of its key role in physics of
various types of solar active regions.}

It is clear that this unsolved problem of energy transport by magnetic flux
tubes at the same time represents another unsolved problem related to the
energy transport and sunspot darkness (see 2.2 in \cite{Rempel2011}). From a
number of known concepts playing a noticeable role in understanding of the
connection between the energy transport and sunspot darkness, let us consider
the most significant theory, according to our vision. It is based on the
Parker-Biermann cooling effect \citep{ref41-3,Biermann1941,ref43-3} and
originates from the works of~\cite{Biermann1941} and~\cite{Alfven1942}.

The main point of the Parker-Biermann cooling effect is that the classical
mechanism of the magnetic tubes buoyancy (e.g. Fig.~\ref{fig04-3}a,
\cite{ref41-3}), emerging as a result of the shear flows instability
development in the tachocline, should be supplemented with the following
results of the~\cite{Biermann1941} postulate and the theory developed by
\cite{ref41-3,ref43-3}: the electric conductivity in the strongly ionized
plasma may be so high that the magnetic field becomes frozen into plasma and
causes the split magnetic tube (Fig.~\ref{fig04-3}b,c) to cool inside.

\begin{figure}[tb]
\begin{center}
\includegraphics[width=12cm]{Parker-empty-tubes-6.pdf}
\end{center}
\caption{The possible ways of a toroidal magnetic flux tube development into a
sunspot.
(a) A rough representation of the form a tube can take after the rise to the
surface by magnetic buoyancy (adopted from Fig.~2a in \cite{ref41-3});
(b) demonstrates the ``crowding'' below the photosphere surface because of
cooling (adopted from Fig.~2b in \cite{ref41-3});
(c) demonstrates the tube splitting as a consequence of the inner region
cooling under the conditions when the tube is in the thermal disequilibrium
with its surroundings and the convective heat transfer is suppressed
\mbox{\citep{Biermann1941}} above $\sim 0.71 R_{Sun}$. This effect as well as
the mechanism of the neutral atoms appearance inside the magnetic tubes
are discussed further in the text (see \mbox{Fig.~\ref{fig-lampochka}a}).
Adopted from Fig.~2c in \cite{ref41-3}.
} 
\label{fig04-3}
\end{figure}

Biermann understood that the magnetic field within the sunspots might itself be
a reason of their darkness. Around the sunspots, the heat is transported up to
the surface of the Sun by means of convection (see 2.2.1 in~\cite{Rempel2011}),
while~\cite{Biermann1941} noted that such transport is strongly inhibited by
the nearly vertical magnetic field within the sunspot, thereby providing a
direct explanation for the reduced temperature at the visible surface. Thus,
the sunspot is dark because it is cooler than its surroundings, and it is
cooler because the convection is inhibited underneath.

Still, the missing cause of a very high conductivity in strongly ionized
plasma, which would produce a strong magnetic field ``frozen'' into this
plasma, has been the major flaw of the so called~\cite{Biermann1941} postulate.

Let us show a solution to the known problem of the Parker-Biermann cooling
effect, which is defined by the nature of the very large poloidal magnetic
fields in the tachocline (determined by the thermomagnetic Ettingshausen-Nernst
effect) and provides the physical basis for the photon channeling conditions 
inside the magnetic flux tubes.


\subsubsection{The thermomagnetic Ettingshausen-Nernst effect and poloidal magnetic field in the tachocline}

For the dynamo theories of planetary, stellar and spiral galactic magnetism the
Coriolis force is of crucial importance. However, the assumed large solar
dynamo leads to very large magnetic fields ($\sim 5 \cdot 10^7$ gauss
\citep{Fowler1955,Couvidat2003}, not observed on the surface of the Sun. This
requires an explanation of how these fields are screened from reaching the
surface.

As is known~\citep{Schwarzschild1958}, the temperature dependence of the
thermonuclear reaction rate in the region of 10$^7$K goes in proportion to
T$^{4.5}$. This means there is a sharp boundary between a much hotter region
where most of the thermonuclear reactions occur and a cooler region where they
are largely absent~\citep{Winterberg2015}. This boundary between radiative and
convective zones is the tachocline. It is the thermomagnetic
Ettingshausen-Nernst
effect~\citep{Ettingshausen1886,Sondheimer1948,Spitzer1956,Kim1969} which by
the large temperature gradient in the tachocline between the hotter and cooler
region leads to large currents shielding the large magnetic field of the dynamo
\citep{Winterberg2015}.

Subject to a quasi-steady state characterized by a balance of the magnetic
field of the dynamo, in the limit of weak collision (the collision frequency
much less than the cyclotron frequency of positive ions), a thermomagnetic
current can be generated in a magnetized
plasma~\citep{Spitzer1962,Spitzer2006}. For a fully ionized gases plasma the
thermomagnetic Ettingshausen-Nernst effect leads to a current density given by
(see Eqs.~(5-49) in~\citep{Spitzer1962,Spitzer2006}):

\begin{equation}
\vec{j} _{\perp} = \frac{3 k n_e c}{2 B^2} \vec{B} \times \nabla T
\label{eq06-01}
\end{equation}
\noindent where $n_e$ is the electron number density, $B$ is the magnetic
field, and $T$ is the absolute temperature (K). With $n_e = \left[ Z / (Z+1)
\right] n$, where $n = n_e + n_i$, and $n_i = n_e / Z$ is the ion number
density for a $Z$-times ionized plasma, the following is obtained:
\begin{equation}
\vec{j} _{\perp} = \frac{3 k n c}{2 B^2} \frac{Z}{Z+1} \vec{B} \times \nabla
T\, . \label{eq06-02}
\end{equation}
It exerts a force on plasma, with the force density $F$ given by
\begin{equation}
\vec{F} = \frac{1}{c} \vec{j} _{\perp} \times \vec{B} =
\frac{3 n k}{2 B^2} \frac{Z}{Z+1} \left( \vec{B} \times \nabla T \right)
\times \vec{B}
\label{eq06-03}
\end{equation}
or with $\nabla T$ perpendicular to $\vec{B}$
\begin{equation}
\vec{F} = \frac{3 n k}{2} \frac{Z}{Z+1} \nabla T
\label{eq06-04}
\end{equation}
leading to the magnetic equilibrium condition (see Eqs.~(4-1)
in~\citep{Spitzer1962})
\begin{equation}
\vec{F} = \frac{1}{c} \vec{j} _{\perp} \times \vec{B} = \nabla p
\label{eq06-05}
\end{equation}
with $p = (\rho / m) kT = nkT$. And by equating~(\ref{eq06-04})
and~(\ref{eq06-05}),
\begin{equation}
\frac{3 n k}{2} \frac{Z}{Z+1} \nabla T = nk \nabla T + kT \nabla n
\label{eq06-06}
\end{equation}
\noindent or
\begin{equation}
a \frac{\nabla T}{T} + \frac{\nabla n}{n} = 0,
~~~ where ~~ a = \frac{2 - Z}{2(Z+1)} ,
\label{eq06-06a}
\end{equation}
\noindent we obtain the condition:
\begin{equation}
T ^a n = const .
\label{eq06-07}
\end{equation}

For a singly-ionized plasma with $Z=1$, one has
\begin{equation}
T ^{1/4} n = const .
\label{eq06-08}
\end{equation}

For a doubly-ionized plasma ($Z=2$) one has $n=const$. Finally, in the limit $Z
\rightarrow A$, one has $T^{-1/2}n = const$. Therefore, $n$ does not strongly
depend on $T$, unlike in a plasma of constant pressure, in which $Tn=const$. It
shows that the thermomagnetic currents may change the pressure distribution in
magnetized plasma considerably.

Taking a Cartesian coordinate system with $z$ directed along $\nabla T$, the
magnetic field in the $x$-direction and the Ettingshausen-Nernst current in the
$y$-direction, and supposing a fully ionized hydrogen plasma with $Z=1$ in the
tachocline, one has
\begin{equation}
{j} _{\perp} = {j} _y = - \frac{3 n k c}{4 B} \frac{dT}{dz}. 
\label{eq06-09}
\end{equation}

From Maxwell's equation $4 \pi \vec{j}_{\perp}/ c = curl \vec{B}$, one has
\begin{equation}
{j} _y = \frac{c}{4 \pi} \frac{dB}{dz}, 
\label{eq06-10}
\end{equation}
and thus by equating~(\ref{eq06-09}) and~(\ref{eq06-10}) we obtain:
\begin{equation}
2B \frac{dB}{dz} = -6 \pi k n \frac{dT}{dz}.
\label{eq06-11}
\end{equation}

From~(\ref{eq06-08}) one has
\begin{equation}
n = \frac{n _{OT} T_{OT}^{1/4}}{T^{1/4}},
\label{eq06-12}
\end{equation}
\noindent where the values $n = n_{OT}$ and $T = T_{OT}$ correspond to the
overshoot tachocline. Inserting~(\ref{eq06-12}) into~(\ref{eq06-11}), one finds
\begin{equation}
dB^2 = -\frac{6 \pi k n _{OT} T_{OT}^{1/4}}{T^{1/4}} dT,
\label{eq06-13}
\end{equation}
\noindent and hence, as a result of integration in the limits $[B_{OT},0]$ in
the left-hand side and $[0,T_{OT}]$ in the right-hand side,
\begin{equation}
\frac{B_{OT}^2}{8 \pi} = n _{OT} kT_{OT}
\label{eq06-14}
\end{equation}
which shows that the magnetic field of the thermomagnetic current in the
overshoot tachocline neutralizes the magnetic field of the dynamo reaching the
overshoot tachocline (see Fig.~\ref{fig-R-MagField}).

\begin{figure}[tb]
\begin{center}
\includegraphics[width=12cm]{MagField-Radius-TurckChieze-1.pdf}
\end{center}
\caption{The reconstructed solar magnetic field (in blue) simulation
from~\cite{Couvidat2003}: 10$^3$-10$^4$~Tesla (left), 30-50~Tesla (middle) and
2-3~Tesla (right), with a temperature of $\sim$9~MK, $\sim$2~MK
and~$\sim$200~kK, respectively. The thin lines show the estimated range of
values for each magnetic field component. Internal rotation was not included in
the calculation. An additional axion production at those places can modify both
intensity and shape of the solar axion spectrum (Courtesy Sylvaine 
Turck-Chi\`{e}ze (see Fig.~2 in~\cite{Zioutas2007})). The reconstructed solar 
magnetic field (in red) simulation from~(\ref{eq06-16}): $4 \cdot 10^3$~T in 
tachocline ($\sim0.7 R_{Sun}$).} 
\label{fig-R-MagField}
\end{figure}

Hence, it is not hard to understand what forces compress the field into intense
filaments, in opposition to the enormous magnetic pressure
\begin{equation}
\frac{B_{OT}^2}{8 \pi} = p_{ext} \approx 6.5 \cdot 10^{13} \frac{erg}{cm^3} ~~
at ~~ 0.7 R_{Sun}, 
\label{eq06-15}
\end{equation}
\noindent where the gas pressure $p_{ext}$ at the tachocline of the Sun ($\rho
\approx 0.2 ~g\cdot cm^{-3}$ and $T \approx 2.3 \cdot 10^6 K$ \citep{ref45-3}
at~$0.7 R_{Sun}$) gives rise to a poloidal magnetic field

\begin{equation}
B_{OT} \simeq 4100 T. 
\label{eq06-16}
\end{equation}

According to (\ref{eq06-16}), a magnetic flux tube anchored in tachocline (see
Fig.~\ref{fig-twisted-tube}) has a significant toroidal magnetic field
($\sim$4100~T), within a layer near the base of the convection zone, where $0.7
R_{Sun}$ and $d \sim 0.05 R_{Sun}$ are constants defining the mean position and
thickness of the layer where the field is concentrated. Each of these anchored
magnetic flux tubes forms a pair of sunspots on the surface of the Sun.



Let us now show the theoretical possibility of the sunspot activity correlation
with the variations of the $\gamma$-quanta of axion origin, induced by the 
magnetic field variations in the overshoot tachocline.



\subsubsection{The Parker-Biermann cooling effect, Rosseland mean opacity and\\ axion-photon oscillations in twisted magnetic tubes}

\label{parker-biermann}

Several local models are known to have been used with great success to 
investigate the formation of buoyant magnetic transport, which transforms 
twisted magnetic tubes generated through shear amplification near the base 
tachocline (e.g.~\cite{Nelson2014}) and the structure and evolution of 
photospheric active regions (e.g.~\cite{Rempel2011a}).

Because these models assume the anchored magnetic flux tubes depending on the
poloidal field in the tachocline, it is not too hard to show that the magnetic
field $B_{OT}$ reaching $\sim 4100 T$ (see~(\ref{eq06-16})) may be at the same
time the reason for the Parker-Biermann cooling effect in the twisted magnetic
tubes (see~Fig.~\ref{fig-twisted-tube}b). The theoretical consequences of such
reasoning of the Parker-Biermann cooling effect are considered below.

\begin{figure}[tb]
\begin{center}
\includegraphics[width=15cm]{MagTube-twisted-05.pdf}
\end{center}
\caption{An isolated and anchored in tachocline (a) magnetic flux tube (adopted
from~\cite{Parker1979}) and (b) twisted magnetic flux tube
(e.g.~\citep{Stein2012}, Fig.~2 in~\citep{Gold1960}, Fig.~1 and Fig.~2
in~\citep{Sturrock2001}) bursting through the solar photosphere to form a
bipolar region. \textbf{Inset in panel (b)}: topological effect of the magnetic
reconnection in the magnetic tube (see~\cite{Priest2000}), where the
$\Omega$-loop reconnects across its base, pinching off the $\Omega$-loop to
form a free $O$-loop (see Fig.~4 in~\cite{Parker1994}). The buoyancy of the
$O$-loop is limited by the magnetic tube interior with Parker-Biermann
cooling.} 
\label{fig-twisted-tube}
\end{figure}

First of all, we suggest that the classic mechanism of magnetic tubes buoyancy
(Fig.~\ref{fig-twisted-tube}а), appearing as a result of the shear instability
development in the tachocline, should be supplemented by the rise of the
twisted magnetic tubes in a stratified medium (Fig.~\ref{fig-twisted-tube}b
(see Fig.~1 and Fig.~2 in \citep{Sturrock2001}), where the magnetic field is
produced by dynamo action throughout the convection zone, primarily by
stretching and twisting in the turbulent downflows (see~\citep{Stein2012}).

Second, the twisting of the magnetic tube may not only promote its splitting,
but also may form a cool region under a certain condition 

\begin{equation}
p_{ext} = \frac{B^2}{8\pi}
\label{eq06v2-01}
\end{equation}

\noindent
when the tube (inset in~Fig.~\ref{fig-twisted-tube}b) is in the thermal
disequilibrium with its surroundings and the convective heat transfer is
suppressed \citep{Biermann1941}.

It is interesting to explore how the cool region stretching from the 
tachocline to the photosphere, where the magnetic tube is in thermal 
non-equilibrium~(\ref{eq06v2-01}) with its surroundings, deals with the 
appearance of the neutral atoms (e.g. hydrogen) in the upper convection zone 
(see Fig.~\ref{fig-lampochka}a in contrast to Fig.~2c in \cite{Parker1955}). In
other words, how does this very cool region prevent the neutral atoms to 
penetrate from the upper convection zone to the base of the convection zone, 
i.e. tachocline?


\begin{figure*}[tbp]
  \begin{center}
    \includegraphics[width=12cm]{Sun-mag_tube-Nernst-10.pdf}
  \end{center}
\caption{(a) Topological effects of the magnetic reconnection inside the
magnetic tubes with the ``magnetic steps''. The left panel shows the
temperature and pressure change along the radius of the Sun from the tachocline
to the photosphere \citep{ref45-3}, $L_{MS}$ is the height of the magnetic
shear steps. At $R \sim 0.72~R_{Sun}$ the vertical magnetic field reaches $B_z
\sim 3600$~T, and the magnetic pressure $p_{ext} = B^2 / 8\pi 
\simeq 5.21 \cdot 10^{13}~erg/cm^3$ \citep{ref45-3}. The very cool regions 
along the entire convective zone caused by the Parker-Biermann cooling effect 
have the magnetic pressure (\mbox{\ref{eq06v2-01}}) in the twisted magnetic tubes.
\newline (b) All the axion flux, born via the Primakoff effect (i.e. the real
thermal photons interaction with the Coulomb field of the solar plasma) comes
from the region  $\leq 0.1 R_{Sun}$~\citep{ref36}. Using the angle
$\alpha = 2 \arctan \left( 0.1 R_{Sun} / 0.7 R_{Sun} \right)$ marking the
angular size of this region relative to tachocline, it is possible to estimate
the flux of the axions distributed over the surface of the Sun. The flux of the
X-ray (of axion origin) is defined by the angle
$\gamma = 2 \arctan \left( 0.5 d_{spot} / 0.3 R_{Sun} \right)$, where
$d_{spot}$ is the diameter of a sunspot on the surface of the Sun (e.g.
$d_{spot} \sim 11000~km$~\citep{Dikpati2008}).}
\label{fig-lampochka}
\end{figure*}

It is essential to find the physical solution to the problem of solar convective zone which would fit the opacity experiments. The full calculation of solar opacities, which depend on the chemical composition, pressure and temperature of the gas, as well as the wavelength of the incident light, is a complex endeavour. The problem can be simplified by using a mean opacity averaged over all wavelengths, so that only the dependence on the gas physical properties remains (e.g. \cite{Rogers1994,Ferguson2005,Bailey2009}). The most commonly used is the Rosseland mean opacity $k_R$, defined as:

\begin{equation}
\frac{1}{k_R} = \left. \int \limits_{0}^{\infty} d \nu \frac{1}{k_\nu} \frac{dB_\nu}{dT} \middle/ 
\int \limits_{0}^{\infty} d \nu \frac{dB_\nu}{dT} \right.
\label{eq06v2-02}
\end{equation}

\noindent
where $dB_\nu / dT$ is the derivative of the Planck function with respect to 
temperature, $k_{\nu}$ is the monochromatic opacity at frequency $\nu$ of the 
incident light or the total extinction coefficient,  including stimulated 
emission plus scattering. A large value of the opacity indicates strong 
absorption from beam of photons, whereas a small value indicates that the beam 
loses very little energy as it passes through the medium.

Note that the Rosseland opacity is an harmonic mean, in which the greatest 
contribution comes from the lowest values of opacity, weighted by a function 
that depends on the rate at which the blackbody spectrum varies with 
temperature (see Eq.~(\ref{eq06v2-02}) and Fig.~\ref{fig-opacity}), and the
photons are most efficiently transported through the ``windows'' where $k_\nu$ 
is the lowest (see Fig.2 in \cite{Bailey2009}).


\begin{figure}[tbp!]
\begin{center}
\includegraphics[width=15cm]{rosseland_opacity-01.pdf}
\end{center}
\caption{Rosseland mean opacity $k_R$, in units of $cm^2 g^{-1}$, shown versus 
temperature (X-axis) and density (multi-color curves, plotted once per decade),
computed with the solar metallicity of hydrogen and helium mixture X=0.7 and 
Z=0.02. The panel shows curves of $k_R$ versus temperature for several 
``steady'' values of the density, labelled by the value of $\log {\rho}$ (in 
$g/cm^3$). Curves that extend from $\log {T} = 3.5$ to 8 are from the Opacity 
Project (opacities.osc.edu). Overlapping curves from $\log {T} = 2.7$ to 4.5 
are from \cite{Ferguson2005}. The lowest-temperature region (black dotted 
curve) shows an estimate of ice-grain and metal-grain opacity from 
\cite{Stamatellos2007}. Adapted from \cite{Cranmer2015}.}
\label{fig-opacity}
\end{figure}


Taking the Rosseland mean opacities shown in Fig.~\ref{fig-opacity}, one may 
calculate, for example, four consecutive cool ranges within the convective 
zone (Fig.~\ref{fig-lampochka}a), where the internal gas pressure $p_{int}$ is 
defined by the following values:

\begin{equation}
p_{int} = n k_B T, ~where~ 
\begin{cases}
T \simeq 10^{3.48} ~K, \\
T \simeq 10^{3.29} ~K, \\
T \simeq 10^{3.20} ~K, \\
T \simeq 10^{3.11} ~K, \\
\end{cases}
\rho = 10^{-7} ~g/cm^3
\label{eq06v2-03}
\end{equation}

Since the inner gas pressure~(\ref{eq06v2-03}) grows towards the tachocline so 
that

\begin{align}
p_{int} &(T = 10^{3.48} ~K) \vert _{\leqslant 0.85 R_{Sun}}  > 
p_{int} (T = 10^{3.29} ~K) \vert _{\leqslant 0.9971 R_{Sun}} > \nonumber \\
& > p_{int} (T = 10^{3.20} ~K) \vert _{\leqslant 0.99994 R_{Sun}} > 
p_{int} (T = 10^{3.11} ~K) \vert _{\leqslant R_{Sun}} ,
\label{eq06v2-04}
\end{align}

\noindent
it becomes evident that the neutral atoms appearing in the upper convection 
zone ($\geqslant 0.85 R_{Sun}$) cannot descend deep to the base of the 
convection zone, i.e. tachocline (see Fig.~\ref{fig-lampochka}a).

Therefore it is very important to examine the connection between the Rosseland 
mean opacity and axion-photon oscillations in twisted magnetic tube.


















Let us consider the qualitative nature of the $\Omega$-loop formation and
growth process, based on the semiphenomenological model of the magnetic
$\Omega$-loops in the convective zone.

\vspace{0.3cm}

\noindent $\bullet$ A high concentration azimuthal magnetic flux 
($B_{OT} \sim 4100$~T, see Fig.~\ref{fig-lampochka}) in the overshoot 
tachocline through the shear flows instability development.


An interpretation of such link is related to the fact that helioseismology
places the principal rotation $\partial \omega / \partial r$ of the Sun in the
overshoot layer immediately below the bottom of the convective zone
\citep{Parker1994}. It is also generally believed that the azimuthal magnetic
field of the Sun is produced by the shearing $r \partial \omega / \partial r$
of the poloidal field $B_{OT}$ from which it is generally concluded that the
principal azimuthal magnetic flux resides in the shear layer
\citep{Parker1955,Parker1993}.

\vspace{0.3cm}

\noindent
$\bullet$ If some ``external'' factor of the local shear perturbation appears
against the background of the azimuthal magnetic flux concentration, such
additional local density of the magnetic flux may lead to the magnetic field
strength as high as, e.g. $B_z \sim 3600$~T (see Fig.~\ref{fig-lampochka}a
and \mbox{Fig.~\ref{fig-Bz}b}). Of course, this brings up a question about the 
physics behind such ``external'' factor and the local shear perturbation.

\begin{figure}[tb]
  \begin{center}
    \includegraphics[width=15cm]{Bz-11.pdf}
  \end{center}
\caption{
(a) Normalized external temperature, density and gas pressure as functions of 
the solar depth $R/R_{Sun}$. The standard solar model with $He$ diffusion 
\citep{ref45-3} was used  for $R < 0.95 R_{Sun}$ (solid lines). The dotted 
lines mark extrapolated values.
(b) Variation of the magnetic field strength $B_z$  along the emerging
$\Omega$-loop as a function of the solar depth $R / R_{Sun}$ throughout the
convection zone. The solid blue line marks the permitted values for
the same standard solar model with $He$ diffusion \citep{ref45-3} starting at
the theoretical estimate of the magnetic field 
$B_{OT} \approx B_z(0) = 4100~T$. The dashed line is the continuation, 
according to the existence of the very cool regions inside the magnetic tube. 
Red point marks the up-to-date observations showing the mean magnetic field 
strength at the level $\sim 0.25~T = 2500 ~G$ \citep{Pevtsov2011,Pevtsov2014}.}
\label{fig-Bz}
\end{figure}

In this regard let us consider the superintense magnetic $\Omega$-loop
formation in the overshoot tachocline through the local shear caused by the
high local concentration of the azimuthal magnetic flux. The buoyant force
acting on the $\Omega$-loop decreases slowly with concentration so the vertical
magnetic field of the $\Omega$-loop reaches $B_z \sim 3600$~T at about 
$R / R_{Sun} \sim 0.72$ (see Fig.~\ref{fig-lampochka}a and Fig.~\ref{fig-Bz}b).
Because of the magnetic pressure 
see analog \mbox{(\ref{eq06-15})} and Fig.~\ref{fig-lampochka}a)
$p_{ext} = B_{0.72 R_{Sun}}^2 / 8\pi = 5.21\cdot 10^{13}~erg/cm^3$
\citep{ref45-3} this leads
to a significant cooling of the $\Omega$-loop tube (see
Fig.~\ref{fig-lampochka}a).

In other words, we assume the effect of the $\Omega$-loop cooling to be the
basic effect responsible for the magnetic flux concentration. It arises from
the well known suppression of convective heat transport by a strong magnetic
field~\citep{Biermann1941}. It means that although the principal azimuthal
magnetic flux resides in the shear layer, it predetermines the additional local
shear giving rise to a significant cooling inside the $\Omega$-loop.

Thus, the ultralow pressure is set inside the magnetic tube as a result of the
sharp limitation of the magnetic steps buoyancy inside the cool magnetic tube
(Fig.~\ref{fig-lampochka}a). This happens because the buoyancy of the magnetic
flows requires finite \textbf{superadiabaticity} of the convection zone
\citep{ref47-3,ref35-3}, otherwise, expanding according to the magnetic
\textbf{adiabatic} law (with the convection being suppressed by the magnetic
field), the magnetic clusters may become cooler than their surroundings, which
compensates the effect of the magnetic buoyancy of superintense magnetic O-loop.

Eventually we suppose that the axion mechanism based on the X-ray
channeling along the ``cool'' region of the split magnetic tube
\sethlcolor{pink}
\sethlcolor{yellow}
(Fig.~\ref{fig-lampochka}a) effectively supplies the necessary energy flux
``channeling'' in magnetic tube to the photosphere while the convective heat transfer is heavily
suppressed.


In this context it is necessary to have a clear view of the energy transport by the X-ray of axion origin, which are a primary transfer mechanism. The recent improvements in the calculation of the radiative properties of solar matter have helped to resolve several long-standing discrepancies between observations and the predictions of theoretical models (e.g. \cite{Rogers1994,Ferguson2005,Bailey2009}), and now it is possible to calculate the photon mean free path (Rosseland length) for Fig.~\ref{fig-opacity}:

\begin{equation}
l_{photon} = \frac{1}{k_R \rho} \sim 
\begin{cases}
2 \cdot 10^{10} ~cm  ~~ & for ~~ k_R \simeq 5 \cdot 10^{-4} ~cm^2/g, \\
10^{10} ~cm          ~~ & for ~~ k_R \simeq 10^{-3} ~cm^2/g, \\
1.5 \cdot 10^{8} ~cm ~~ & for ~~ k_R \simeq 6.7 \cdot 10^{-2} ~cm^2/g, \\
10^{7} ~cm           ~~ & for ~~ k_R \simeq 1 ~cm^2/g,
\end{cases}
~~ \rho = 10^{-7} ~g/cm^3
\label{eq06v2-05}
\end{equation}

\noindent
where the Rosseland mean opacity values $k_R$ and density $\rho$ are chosen so 
that the very low internal gas pressure $p_{int}$ (see Eq.~(\ref{eq06v2-04})) 
along the entire magnetic tube almost does not affect the external gas pressure
$p_{ext}$ (see (\ref{eq06v2-05}) and Fig.~\ref{fig-opacity}).

Let us now examine the appearance  of the X-ray of axion origin, induced by the magnetic field variations near the tachocline (Fig.~\ref{fig-lampochka}a) and their impact on the Rosseland length (see~(\ref{eq06v2-05})) inside the cool region of the magnetic tubes.

Let us remind that the magnetic field strength $B_{OT}$ in the overshoot 
tachocline of $\sim 4100~T$ (see Fig.~\ref{fig-lampochka}a) and the 
Parker-Biermann cooling effect in~(\ref{eq06v2-01}) lead to the corresponding 
value of the magnetic field strength $B(z = 0.72 R_{Sun}) \sim 3600 ~T$ 
(see Fig.~\ref{fig-lampochka}a), which in its turn assumes virtually zero 
internal gas pressure of the magnetic tube.

As it is shown above (see~\cite{Priest2000}), the topological effect of the
magnetic reconnection inside the $\Omega$-loop results in the formation of the
so-called O-loops (Fig.~\ref{fig-twisted-tube} and Fig.~\ref{fig-lampochka}a)
with their buoyancy limited from above by the strong cooling inside the
$\Omega$-loop (Fig.~\ref{fig-lampochka}a). It is possible to derive the value
of the horizontal magnetic field of the magnetic steps at the top of the O-loop:
$\vert B_{MS} \vert \approx \vert B(z = 0.72 R_{Sun}) \vert \sim 3600 ~T$.

So in the case of a large enough Rosseland lengh (see Eq.~(\ref{eq06v2-05})), 
X-ray of axion origin induced by the horizontal magnetic field in O-loops, reach
the photosphere freely, while in the photosphere itself, according to the 
Rosseland length

\begin{equation}
l_{photon} \approx 100 ~km < l \approx 300 \div 400 ~km,
\label{eq06v2-06}
\end{equation}

these photons undergo a multiple Compton scattering (see 
Section~\ref{subsec-osc-parameters}) producing a typical directional pattern
(Fig.~\ref{fig-lampochka}a).

Aside from the X-rays of axion origin with mean energy of 4.2~keV, there are 
only $h \nu \sim 0.95 ~keV$ X-rays (originating from the tachocline, according 
to a theoretical estimate by \cite{Bailey2009}) inside the magnetic tube. Such 
X-rays would produce the Compton-scattered photons with mean energy of 
$\leqslant 0.95~keV$ which contradicts the known measurements of the photons 
with mean energy of 3-4~keV (see Fig.~4 in \cite{Rieutord2014}). Our suggested 
theoretical model thus removes these contradictions by involving the X-rays of 
axion origin \textit{plus} the axions of the thermal X-ray origin, both 
produced in the magnetic field of O-loops (see Fig.~\ref{fig-lampochka}a and 
Fig.~\ref{app-b-fig01} in Appendix~\ref{appendix-luminosity}).




And finally, let us emphasize that we have just shown a theoretical possibility
of the time variation  of the  sunspot activity to correlate with the flux 
of the X-rays of axion origin; the latter being controlled by the magnetic 
field variations near the overshoot tachocline. As a result, it may be 
concluded that the the axion mechanism for solar luminosity variations 
based on the lossless X-ray ``channeling'' along the
magnetic tubes allows to explain the effect of the almost complete suppression of the
convective heat transfer, and thus to understand the known puzzling darkness of
the sunspots \citep{Rempel2011}.




\subsection{Estimation of the solar axion-photon oscillation parameters on the basis of the hadron axion-photon coupling in white dwarf cooling}

\label{subsec-osc-parameters}


It is known \citep{Cadamuro2012} that astrophysics provides a very interesting
clue concerning the evolution of white dwarf stars with their small mass
predetermined by the relatively simple cooling process. It is related to the
fact that recently it has been possible to determine their luminosity function
with the unprecedented precision \citep{Isern2008}. It seems that if the DFSZ
axion \citep{ref47,Dine1981} has a direct coupling to electrons and a decay
constant $f_a \sim 10^{9} ~GeV$, it provides an additional energy-loss channel
that permits to obtain a cooling rate that better fits the white dwarf
luminosity function than the standard one~\citep{Isern2008}. On the other hand,
the KSVZ axion \citep{ref46,ref46a}, i.e. the hadronic axion (with the mass in 
the $meV$ range and $g_{a\gamma \gamma} \sim 10^{-12} ~GeV^{-1}$) would also 
help in fitting the data, but in this case a stronger value for 
$g_{a\gamma \gamma}$ is required to perturbatively produce an electron coupling
of the required strength (\cite{Cadamuro2012}, Fig.~1 in \cite{Srednicki1985}, 
Fig.~1 in \cite{Turner1990}, Eq.~82 in \cite{Kim2010}).

Our aim is to estimate the solar axion-photon oscillation parameters basing on
the hadron axion-photon coupling derived from white dwarf cooling (see 
\mbox{Appendix~\ref{appendix-wd-cooling}}). The estimate
of the horizontal magnetic field in the O-loop is not related to the
photon-axion conversion in the Sun only, but also to the axions in the model of
white dwarf evolution. Therefore along with the values of the magnetic field
strength 
$B_{MS} \sim 3600 ~T$
and the height of the magnetic shear steps 
$L_{MS} \sim 1.28 \cdot 10^4 ~km$
(Fig.~\ref{fig-lampochka}a,b) we use the following parameters of the hadronic
axion (from the White Dwarf area in Fig.~\ref{fig05}a \citep{Irastorza2013,
Carosi2013}):

\begin{figure}[tbp!]
  \begin{center}
    \begin{minipage}[h]{0.44\linewidth}
      \includegraphics[width=7.6cm]{gagamma_ma-limits-2.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}[h]{0.53\linewidth}
      \includegraphics[width=8.8cm]{Y-g_agamma-4.pdf}
    \end{minipage}
  \end{center}
\caption{\textbf{(a)} Summary of astrophysical, cosmological and laboratory
constraints on axions and axion-like particles. Comprehensive axion/ALP
parameter space, highlighting the two main front lines of direct detection
experiments: helioscopes (CAST~\citep{ref58,ref72,CAST2011,Arik2013}) and
haloscopes (ADMX~\citep{ref50} and RBF~\citep{ref51}). The astrophysical bounds
from horizontal branch and massive stars are labeled ``HB''~\citep{ref02} and
``Cepheids''~\citep{Carosi2013} respectively. The QCD motivated models
(KSVZ~\citep{ref46,ref46a} and DFSZ~\citep{ref47,Dine1981}) for axions lay in
the yellow diagonal band. The orange parts of the band correspond to
cosmologically interesting axion models: models in the ``classical axion
window'' possibly composing the totality of DM (labelled ``Axion CDM'') or a
fraction of it (``WIMP-axion CDM''~\citep{Baer2011}). For more generic ALPs,
practically all the allowed space up to the red dash line may contain valid ALP
CDM models~\citep{Arias2012}. The region of axion masses invoked in the WD
cooling anomaly is shown by the blue dash line~\citep{Irastorza2013}. The red
star marks the values of the axion mass $m_a \sim 3.2 \cdot 10^{-2} eV$ and the
axion-photon coupling constant $g_{a\gamma} \sim 4.4 \cdot 10^{-11} GeV^{-1}$
chosen in the present paper on the basis of the suggested relation between the
axion mechanisms of the Sun's and the white dwarf luminosity variations.
\newline
\textbf{(b)} $R$ parameter constraints to $Y$ and $g_{a \gamma}$ (adopted from
\cite{Ayala2014}. The dark purple area delimits the 68\%~C.L. for $Y$ and
$R_{th}$ (see Eq.~(1) in \cite{Ayala2014}). The resulting bound on the axion
($g_{10} = g_{a \gamma \gamma}/(10^{-10} ~GeV^{-1})$) is somewhere between a
rather conservative $0.5 < g_{10} \leqslant 0.8$ and most aggressive $0.35 <
g_{10} \leqslant 0.5$ \citep{Friedland2013}. The red line marks the values of
the axion-photon coupling constant $g_{a \gamma} \sim 4.4 \cdot 10^{-11}
~GeV^{-1}$ chosen in the present paper.
The blue shaded area represents the bounds from Cepheids
observation. The yellow star corresponds to $Y$=0.254 and the bounds from HB
lifetime (yellow dashed line).}
\label{fig05}
\end{figure}

\begin{equation}
g_{a \gamma} \sim 4.4 \cdot 10^{-11} ~ GeV^{-1}, ~~~ m_a \sim 3.2 \cdot 10^{-2} ~eV.
\label{eq3.30}
\end{equation}

The choice of these values is also related to the observed solar luminosity
variations in the X-ray band (see (\ref{eq3.35})). The theoretical
estimate and the
consequences of such choice are considered below.


As it is shown above, the $\sim 4100~T$ magnetic field in the overshoot
tachocline and the Parker-Biermann cooling effect in~(\ref{eq06v2-01}) may 
produce the O-loops with the horizontal magnetic field of
$\vert B_{MS} \vert \approx \vert B(z = 0.72 R_{Sun}) \vert \sim 3600 ~T$
stretching for about $L_{MS} \sim 1.28 \cdot 10^4 ~km$, and surrounded by virtually
zero internal gas pressure of the magnetic tube (see Fig.~\ref{fig-lampochka}a).


It is not hard to use the expression (\ref{eq08})
for the conversion probability\footnote{Hereinafter we use rationalized natural
units to convert the magnetic field units from $Tesla$ to $eV^2$, and the
conversion reads $1\,T = 195\,eV^2$~\citep{Guendelman2009}.}

\begin{equation}
P_{a \rightarrow \gamma} = \frac{1}{4} \left( g_{a \gamma} B_{MS} L_{MS} \right)^2 \sim 1
\label{eq3.31}
\end{equation}
for estimating the axion coupling constant to photons (\ref{eq3.30}).


Thus, it is shown that the hypothesis about the possibility for the solar
axions born in the core of the Sun to be efficiently converted back into
$\gamma$-quanta in the magnetic field of the magnetic steps of the O-loop
(above the solar overshoot tachocline) is relevant. Here the variations of the
magnetic field in the solar tachocline are the direct cause of the converted
$\gamma$-quanta intensity variations. The latter in their turn may be the cause
of the overall solar luminosity variations known as the active and quiet Sun phases.

It is easy to show that the theoretical estimate for the part of the axion
luminosity $L_a$ in the total luminosity of the Sun $L_{Sun}$ with respect to
(\ref{eq3.30}) is~\citep{ref58}

\begin{equation}
\frac{L_a}{L_{Sun}} = 1.85 \cdot 10 ^{-3} \left( 
\frac{g_{a \gamma}}{10^{-10} GeV^{-1}} \right)^2 \sim 3.6 \cdot 10^{-4} .
\label{eq3.32}
\end{equation}

As opposed to the classic mechanism of the Sun modulation, the
axion mechanism is determined by the magnetic tubes rising to the photosphere,
and not by the over-photosphere magnetic fields. In this case the solar
luminosity modulation is determined by the axion-photon oscillations in the
magnetic steps of the O-loop causing the formation and channeling of the
$\gamma$-quanta inside the almost empty magnetic $\Omega$-tubes (see
Fig.~\ref{fig-twisted-tube} and Fig.~\ref{fig-lampochka}a). When the magnetic
tubes cross the photosphere, they ``open'' (Fig.~\ref{fig-lampochka}a), and the
$\gamma$-quanta are ejected to the photosphere, where their comfortable journey
along the magnetic tubes (without absorption and scattering) ends. As the
calculations by \cite{ref36} show, the further destiny of the $\gamma$-quanta
in the photosphere may be described by the Compton scattering, which actually
agrees with the observed solar spectral shape (Fig.~\ref{fig06}b,c).

\begin{figure*}
  \begin{center}
    \includegraphics[width=14cm]{Sun_total_spectra-13.pdf}
  \end{center}

\caption{(a) Reconstructed solar photon spectrum below 10~keV from the active
Sun (red line) and quiet Sun (blue line) from accumulated observations
(spectral bin is 6.1~eV wide). Adopted from~\cite{ref59}.
\newline
(b) Reconstructed solar photon spectrum fit in the active phase of the Sun by
the quasi-invariant soft part of the solar photon spectrum (grey shaded area;
see \mbox{Eq.~(\ref{eq06-34})}) and three spectra (\ref{eq3.33}) degraded to
the Compton scattering for column densities above the initial conversion place
of 16 (adopted from~\cite{ref36}) and 2~$g / cm^2$ (present paper).
\newline
(c) The similar curves for the quiet phase of the Sun (grey shaded area 
corresponds to \mbox{Eq.~(\ref{eq06-35})})
\newline
(d) Cartoon showing the interplay between magnetic field expansion and the EUV
loop. A coalescent flow forming the sunspot drags the magnetic field in the
photosphere near the solar surface into the sunspot. In response, a hot spot of
enhanced upward directed Poynting flux, $S$, forms (red arrow). The expanding
field lines (blue) move upwards and to the side. When they transverse the hot
spot of Poynting flux, the plasma on that field line gets heated and brightens
up. As the field line expands further, it leaves the hot spot and gets darker
again. In consequence a bright coronal EUV loop forms (orange) and remains
rather stable as the successively heated field lines move through (adopted from
\cite{Chen2015}). X-ray emission is the $\gamma$-quanta of axion origin coming
from the magnetic tubes and not related to the magnetic reconnection as
conjectured by e.g. \cite{Shibata2011}.}
\label{fig06}
\end{figure*}

From the axion mechanism point of view it means that the solar spectra during
the active and quiet phases (i.e. during the maximum and minimum solar
activity) differ from each other by the smaller or larger part of the Compton
spectrum, the latter being produced by the $\gamma$-quanta of the axion origin
ejected from the magnetic tubes into the photosphere (see Fig.~4 in
\cite{Chen2015}).

A natural question arises at this point: ``What are the real parts of the
Compton spectrum of the axion origin in the active and quiet phases of the Sun,
and do they agree with the experiment?'' Let us perform the
mentioned estimations basing on the known experimental results by ROSAT/PSPC,
where the Sun's coronal X-ray spectra and the total luminosity during the
minimum and maximum of the solar coronal activity were obtained~\citep{ref59}.

Apparently, the solar photon spectrum below 10~keV of the active and quiet Sun
(Fig.~\ref{fig06}a) reconstructed from the accumulated ROSAT/PSPC observations
may be described by three Compton spectra for different column densities rather
well (Fig.~\ref{fig06}b,c). This gives grounds for the assumption that the hard
part of the solar spectrum is mainly determined by the axion-photon conversion
efficiency:
\begin{align}
\left( \frac{d \Phi}{dE} \right)^{(*)} \simeq
\left( \frac{d \Phi}{dE} \right)^{(*)}_{corona} +
\left( \frac{d \Phi _{\gamma}}{dE} \right)^{(*)}_{axions} ,
\label{eq06-33}
\end{align}
\noindent where $\frac{d \Phi}{dE}$ is the observed solar spectra during the
active (red line in Fig.~\ref{fig06}a,b) and quiet (blue line in
Fig.~\ref{fig06}a,c) phases, $\left( \frac{d \Phi}{dE} \right)_{corona}$
represents the power-like theoretical solar spectra

\begin{equation}
\left( \frac{d \Phi}{dE} \right)_{corona} \sim E^{-(1+\alpha)} e^{-E/E_0} ,
\label{eq06-33a}
\end{equation}

\noindent
where a power law decay with the ``semi-heavy tail'' takes place in practice 
\citep{Lu1993} instead of the so-called power laws with heavy tails 
\citep{Lu1991,Lu1993} (see e.g. Figs.~3 and~6 in \cite{Uchaikin2013}). 
Consequently, the observed corona spectra 
($0.25 ~keV < E \leqslant 2.5 ~keV$) (shaded area in Fig.~\ref{fig06}b)

\begin{align}
\left( \frac{d \Phi}{dE} \right)^{(active)}_{corona} \sim
5 \cdot 10^{-3} \cdot (E~[keV])^{-3} \cdot \exp{\left(-\frac{E}{1 keV} \right)} 
~~for~the~active~Sun
\label{eq06-34}
\end{align}

\noindent and (shaded area in Fig.~\ref{fig06}c)

\begin{align}
\left( \frac{d \Phi}{dE} \right)^{(quiet)}_{corona} \sim
1 \cdot 10^{-4} \cdot (E~[keV])^{-3} \cdot \exp{\left(-\frac{E}{0.5 keV} \right)} 
~~for~the~quiet~Sun ;
\label{eq06-35}
\end{align}

\noindent $\left( \frac{d \Phi _{\gamma}}{dE} \right)_{axions}$ is the
reconstructed solar photon spectrum fit ($0 ~keV < E < 10 ~keV$) constructed
from three spectra (\ref{eq3.33}) degraded to the Compton scattering for
different column densities (see Fig.~\ref{fig06}b,c for the active and quiet
phases of the Sun respectively).

As is known, this class of flare models (Eqs.~(\ref{eq06-34})
and~(\ref{eq06-35})) is based on the recent paradigm in statistical physics
known as self-organized criticality
\citep{Bak1987,Bak1988,Bak1989,Bak1996,Aschwanden2011}. The basic idea is that
the flares are a result of an ``avalanche'' of small-scale magnetic
reconnection events cascading \citep{Lu1993,Charbonneau2001,Aschwanden2014} 
through the highly intense coronal magnetic structure \citep{Shibata2011} driven
at the critical state by the accidental photospheric movements of its magnetic
footprints. Such models thus provide a natural and computationally convenient
basis for the study of Parker hypothesis of the coronal heating by nanoflares
\citep{Parker1988}.

Another significant fact discriminating the theory from practice, or rather 
giving a true understanding of the measurements against some theory, should be 
recalled here (e.g. (\ref{eq06-33a}) (see Eq.~(5) in \cite{Lu1993})). The 
nature of power laws is related to the strong connection between the consequent
events (this applies also to the ``catastrophes'', which  in turn gives rise to
a spatial nonlocality related to the appropriate structure of the medium (see 
page 45 in \cite{Uchaikin2013})). As a result, the ``chain reaction'', i.e. the
avalanche-like growth of perturbation with more and more resource involved,
leads to the heavy-tailed distributions. On the other hand, obviously, none of 
the natural events may be characterized by the infinite values of mean and 
variance. Therefore, the power laws like (\ref{eq06-33a}) are approximate and
must not hold for the very large arguments. It means that the power law decay
of the probability density rather corresponds to the average asymptotics, and
the ``semi-heavy tails'' must be observed in practice instead.

In this regard we suppose that the application of the power-law distributions 
with semi-heavy tails leads to a soft attenuation of the observed corona 
spectra (which are not visible above $E > 2 \div 3 ~keV$), and thus to a close
coincidence between the observed solar spectra and $\gamma$-spectra of axion 
origin (Fig.~\ref{fig06}). I.e.

\begin{equation}
\left( \frac{d \Phi}{dE} \right)^{(*)} \simeq
\left( \frac{d \Phi _{\gamma}}{dE} \right)^{(*)}_{axions} 
~~~ \text{for energies} ~~ E > 2 \div 3 ~keV.
\label{eq06-35a}
\end{equation}

It means that the physics of the formation and ejection of the $\gamma$-quanta 
above $2 \div 3 ~keV$ through the sunspots into corona is not related to the 
magnetic reconnection theory by e.g. \cite{Shibata2011} (Fig.~\ref{fig06}d),
and may be of the axion origin.

With this in mind, let us suppose that the part of the differential solar axion
flux at the Earth~\citep{ref58}
\begin{align}
\frac{d \Phi _a}{dE} = 6.02 \cdot 10^{10} \left( \frac{g_{a\gamma}}{10^{10} GeV^{-1}} \right)^2 E^{2.481} \exp \left( - \frac{E}{1.205} \right) ~~cm^{-2}
s^{-1} keV^{-1} ,
\label{eq3.33}
\end{align}
\noindent which characterizes the differential $\gamma$-spectrum of the axion
origin $d \Phi _{\gamma} / dE$
(see $[ d \Phi _{\gamma} / dE ]_{axions}$ in (\ref{eq06-33}) and 
(\ref{eq06-35a}))

\begin{align}
\frac{d \Phi _{\gamma}}{dE}  \cong P_{\gamma} \frac{d \Phi _{a}}{dE}
~~ cm^{-2} s^{-1} keV^{-1} \approx
6.1 \cdot 10^{-3} P_{\gamma} \frac{d \Phi _{a}}{dE}
~ ph\cdot cm^{-2} s^{-1} bin^{-1}
\label{eq3.34}
\end{align}
\noindent
where the spectral bin width is 6.1~eV (see Fig.~\ref{fig06}a);
the probability $P_{\gamma}$ describing the relative portion of $\gamma$-quanta
(of axion origin) channeling along the magnetic tubes may be defined, according
to~\cite{ref59}, from the observed solar luminosity variations in the X-ray
band, recorded in ROSAT/PSPC experiments (Fig.~\ref{fig06}):
$\left(L_{corona}^X \right) _{min} \approx 2.7
\cdot 10^{26} ~erg/s$ at minimum and
$\left( L_{corona}^X \right) _{max} \approx 4.7 \cdot 10^{27} ~erg/s$
at maximum,

\begin{equation}
P_{\gamma} = P_{a \rightarrow \gamma} \cdot \dfrac{\Omega \cdot (0.5 d_{spot})^2}
{(\tan \left( \alpha / 2 \right) \cdot 0.3 R_{Sun})^2} \cdot \Lambda_a
\approx 3.4 \cdot 10^{-3},
\label{eq3.35}
\end{equation}

\noindent directly following from the geometry of the system
(Fig.~\ref{fig-lampochka}b), where the conversion probability 
$P_{a \rightarrow \gamma} \sim 1$ (\ref{eq3.31}); 

\begin{equation}
\Omega = (I_{\gamma ~CZ} / I_0) \cdot (I_{\gamma ~photo} / I_{\gamma ~CZ})
\cdot (I_{\gamma ~corona} / I_{\gamma ~photo}) \approx 0.23
\end{equation}

\noindent
is the total relative intensity of
$\gamma$-quanta, where $(I_{\gamma ~CZ} / I_0) \sim 1$ is the relative
intensity of $\gamma$-quanta ``channeling'' through the 
magnetic tubes in the convective zone,
$I_{\gamma ~photo} / I_{\gamma ~CZ} = \exp {[-(\mu l)_{photo}]} \sim 0.23$ (see
Eq.~\ref{eq06-43}) is the relative intensity of the Compton-scattered
$\gamma$-quanta in the solar photosphere, and $I_{\gamma ~corona} / I_{\gamma
~photo} = \exp {[-(\mu l)_{corona}]} \approx 1$ (see Eq.~\ref{eq06-44})
is the relative intensity of the Compton-scattered $\gamma$-quanta in the solar
corona;
$d_{spot}$ is the measured diameter of the sunspot
(umbra) \citep{Dikpati2008,Gough2010}. Its size determines the relative portion
of the axions hitting the sunspot area. Further,

\begin{equation}
\dfrac{(0.5 d_{spot})^2}{(\tan \left( \alpha / 2 \right) \cdot 0.3 R_{Sun})^2} \cong 0.034,
\end{equation}


\noindent where 

\begin{equation}
0.5 d_{spot} = \left[ \frac{1}{\pi} \left( 
\frac{\langle sunspot ~area \rangle _{max}}{\left\langle N_{spot} \right\rangle _{max}} 
\right) \right] ^{(1/2)} \cong 5500~km,
\end{equation}

\noindent and the value $\Lambda_a$ characterizes the portion of the 
axion flux going through the total $(2\left\langle N_{spot}
\right\rangle_{max})$ sunspots on the photosphere:

\begin{equation}
\Lambda_a = \dfrac{\left( sunspot\ axion\ flux \right)}{(1/3)\left( total\ axion\ flux \right)} \approx 
\dfrac{2 \left\langle N_{spot} \right\rangle _{max} (\tan \left( \alpha / 2 \right) \cdot 0.3 R_{Sun})^2}{(4/3) R_{Sun} ^2} \sim 0.42 , 
\label{eq3.36}
\end{equation}

\noindent and $\left\langle N_{spot} \right\rangle _{max} \approx 150$ is the
average number of the maximal sunspot number, and 
$\langle sunspot ~area \rangle _{max} \approx 7.5 \cdot 10^9 ~km^2 \approx 2470 ~ppm ~of ~visible ~hemisphere$
is the sunspot area (over the visible
hemisphere~\citep{Dikpati2008,Gough2010}) for the cycle 22 experimentally
observed by the Japanese X-ray telescope Yohkoh (1991)~\citep{ref36}.

On the other hand, from the known observations (see~\cite{ref59} and
Appendix~\ref{appendix-luminosity})
\begin{equation}
\frac{(L_{corona}^X)_{max}}{L_{Sun}} \cong 1.22 \cdot 10^{-6},
\label{eq3.37}
\end{equation}
\noindent where $L_{Sun} = 3.8418 \cdot 10^{33} erg / s$ is the solar
luminosity~\citep{ref63}. Using the theoretical axion impact estimate
(\ref{eq3.32}), one can clearly see that the obtained value (\ref{eq3.35}) is
in good agreement with the observations (\ref{eq3.37}):

\begin{equation}
P_{\gamma} =  \left. \frac{(L_{corona}^X)_{max}}{L_{Sun}} \middle/ 
\frac{L_a}{L_{Sun}} \sim 3.4 \cdot 10^{-3} \right. ,
\label{eq3.38}
\end{equation}


\noindent derived independently.

In other words, if the hadronic axions found in the Sun are the same particles
found in the white dwarfs with the known strength of the axion coupling to
photons (see (\ref{eq3.30}) and Fig.~\ref{fig05}a,b),
it is quite natural that the independent observations give the same estimate of
the probability $P_{\gamma}$ (see (\ref{eq3.35}) and (\ref{eq3.38})). So the
consequences of the choice (\ref{eq3.30}) are determined by the independent
measurements of the average sunspot radius, the sunspot
number~\citep{Dikpati2008,Gough2010}, the model estimates of the horizontal
magnetic field and the height $L_{MS}$ of the magnetic steps (see
Fig.~\ref{fig-lampochka}), and the hard part of the solar photon spectrum
mainly determined by the axion-photon conversion efficiency, and the
theoretical estimate for the part of the axion luminosity $L_a$ in the total
luminosity of the Sun $L_{Sun}$ (\ref{eq3.38}).






\section{Axion mechanism of the solar Equator -- Poles effect}

The axion mechanism of Sun luminosity variations is largely validated by the experimental
X-ray images of the Sun in the quiet (Fig.~\ref{fig-Yohkoh}a) and active
(Fig.~\ref{fig-Yohkoh}b) phases~\citep{ref36} which clearly reveal the
so-called Solar Equator -- Poles effect (Fig.~\ref{fig-Yohkoh}b).

\begin{figure*}
\centerline{\includegraphics[width=12cm]{Sun_X-ray_image_spectrum-3.pdf}}

\caption{\textbf{Top:} Solar images at photon energies from 250~eV up to a few
keV from the Japanese X-ray telescope Yohkoh (1991-2001) (adopted
from~\cite{ref36}). The following is shown:
\newline
(a) a composite of 49 of the quietest solar periods during the solar minimum in 1996;
\newline
(b) solar X-ray activity during the last maximum of the 11-year solar cycle.
Most of the X-ray solar activity (right) occurs at a wide bandwidth of
$\pm 45^{\circ}$ in latitude, being homogeneous in longitude. Note that
$\sim$95\% of the solar magnetic activity covers this bandwidth.
\newline
\textbf{Bottom:} (c) Axion mechanism of solar irradiance variations 
above $2 \div 3 ~keV$, which is independent of the cascade reconnection
processes in corona (see shaded areas and Fig.~\ref{fig06}b,c,d),
and the red and blue curves characterizing the irradiance increment in the
active and quiet phases of the Sun, respectively;
\newline
(d) schematic picture of the radial travelling  of the axions inside the Sun.
Blue lines on the Sun designate the magnetic field. Near the tachocline 
(Fig.~\ref{fig-lampochka}a) the axions are converted into $\gamma$-quanta, 
which form the experimentally observed Solar photon spectrum after passing the 
photosphere (Fig.~\ref{fig06}). Solar axions that move towards the poles (blue 
cones) and in the equatorial plane (blue bandwidth) are not converted by 
Primakoff effect (inset: diagram of the inverse coherent process). The 
variations of the solar axions may be observed at the Earth by special 
detectors like the new generation CAST-helioscopes~\citep{ref68}. }
\label{fig-Yohkoh}
\end{figure*}

The essence of this effect lies in the following. It is known that the axions
may be transformed into $\gamma$-quanta by inverse Primakoff effect in the
transverse magnetic field only. Therefore the axions that pass towards the
poles (blue cones in Fig.~\ref{fig-Yohkoh}b) and equator (the blue band in
Fig.~\ref{fig-Yohkoh}b) are not transformed into $\gamma$-quanta by inverse
Primakoff effect, since the magnetic field vector is almost collinear to the
axions' momentum vector. The observed nontrivial X-ray distribution in the
active phase of the Sun may be easily and naturally described within the
framework of the axion mechanism of the solar luminosity variations.

As described in Section~\ref{subsec-channeling}, the photons of axion origin 
travel through the convective zone along the magnetic flux tubes, up to the 
photosphere. In the photosphere they are Compton-scattered, which results in a 
substantial deviation from the initial axions directions of propagation 
(Fig.~\ref{fig07a}).


\begin{figure*}
\centerline{\includegraphics[width=15cm]{axion-channaling-scattering-Yohkoh-3.pdf}}

\caption{The formation of the high X-ray intensity bands on the Yohkoh
matrix. \label{fig07a}}
\end{figure*}


Let us make a simple estimate of the Compton scattering efficiency in terms of
the X-ray photon mean free path (MFP) in the photosphere:
\begin{equation}
l_{\mu} = (\mu)^{-1} = \left( \sigma_c \cdot n_e \right)^{-1} ,
\label{eq-compt-01}
\end{equation}
\noindent where
$\mu$ is the total linear attenuation coefficient
(cm$^{-1}$),
the total Compton cross-section $\sigma_c = \sigma_0 = 8 \pi r_0^2 / 3$ for the
low-energy photons \citep{ref81,ref82}, $n_e$ is the electrons density in the
photosphere, and $r_0 = 2.8\cdot10^{-13}~cm$ is the so-called classical
electron radius.

Taking into account the widely used value of the matter density in the solar
photosphere $\rho \sim 10^{-7} ~g/cm^3$ and supposing that it consists of the
hydrogen (for the sake of the estimation only), we obtain that
\begin{equation}
n_e \approx \frac{\rho}{m_H}  \approx 6 \cdot 10^{16} ~ electron / cm^3\, ,
\label{eq-compt-02}
\end{equation}
which yields the MFP of the photon \citep{ref81,ref82}
\begin{align}
l_{\mu} = \left( 7 \cdot 10^{-25} ~cm^2 \cdot 6 \cdot 10^{16} ~electron/cm^3 \right)^{-1}
\approx  2.4 \cdot 10^7 ~cm = 240 ~km .
\label{eq-compt-03}
\end{align}

Since this value is smaller than the thickness of the solar photosphere
($l_{photo} \sim 300 \div 400 ~km$),
the Compton scattering is efficient
enough to be detected at the Earth
(see \mbox{Fig.~\ref{fig07a}} and \mbox{Fig.~\ref{fig06}} adopted from
\mbox{\cite{ref59}});
\begin{equation}
\frac{I_{\gamma ~photo}}{I_{\gamma ~CZ}}
= \exp {\left[ - (\mu l)_{photo} \right]} \sim 0.23,
\label{eq06-43}
\end{equation}
\noindent which follows the particular case of the Compton scattering: Thomson
differential and total cross section for unpolarized photons
\citep{Griffiths1995}.

And finally taking into account that $l_{chromo} \sim 2 \cdot 10^3 ~km$ and
$n_e \sim 10^{13} ~electron/cm^3$ (i.e. $l_{\mu} \sim 1.4 \cdot 10^6 ~km$) and
$l_{corona} \sim 10^5 ~km$ and $n_e < 10^{11} ~electron/cm^3$ (i.e. $l_{\mu} >
1.4 \cdot 10^8 ~km$) (Fig.~12.9 in \cite{Aschwanden2004}), one may calculate
the relative intensity of the $\gamma$-quanta by Compton scattering in the
solar corona
\begin{equation}
\frac{I_{\gamma ~corona}}{I_{\gamma ~photo}}
= \frac{I_{\gamma ~chromo}}{I_{\gamma ~photo}} \cdot
\frac{I_{\gamma ~corona}}{I_{\gamma ~chromo}} =
\exp {\left[ - (\mu l)_{chromo} \right]} \cdot
\exp {\left[ - (\mu l)_{corona} \right]} \approx 1 ,
\label{eq06-44}
\end{equation}
\noindent which depends on the
total relative intensity of $\gamma$-quanta (see Eq.~(\ref{eq3.35})).

A brief summary is appropriate here. Coronal activity is a
collection of plasma  processes manifesting from the passage of magnetic fields
through it from below, generated by the solar dynamo in cycles of approximately
11 years (Fig.~\ref{fig-Yohkoh}). This global process culminating in the
reversal of the solar magnetic dipole at the end of each cycle involves the
turbulent dissipation of the magnetic energy, the flares  and heating of the
corona. The turbulent, highly dissipative, as well as largely ideal MHD
processes play their distinct roles, each liberating a comparable amount of
energy stored in the magnetic fields.

This mechanism is illustrated
in Fig.~\ref{fig06}d. When the magnetic flux erupts through the photosphere, it
forms a pair of sunspots pushing the magnetic field up and aside. The magnetic
field inside the sunspots is very high and the convection is suppressed.
Therefore the coalescence of the magnetic field is also suppressed. When some
magnetic field line crosses the region of the high Poynting flux, the energy is
distributed along this line in the form of plasma heating. This makes such line
visible in EUV band for some short time. While the magnetic field is being
pushed to the sides, next field line crosses the region of high Poynting flux
and flares up at the same position as the previous one and so on. This creates
an illusion of a static flaring loop, while the magnetic field is in fact
moving. It is interesting to note that \cite{Chen2015} expect the future
investigation to show to what extent this scenario also holds for X-ray
emission (see Supplementary Section~3 in \cite{Chen2015}).

In this context it is very important to consider the experimental observations
of solar X-ray jets (e.g. solar space missions of Yohkoh and Hinode
satellites), which show, for example, a gigantic coronal jet, ejecting from a
compact active region in a coronal hole \citep{Shibata1994} and tiny
chromospheric anemone jets \citep{Shibata2007}.

These jets are believed to be an indirect proof of small-scale ubiquitous
reconnection in the solar atmosphere and may play an important role in heating
it, as conjectured by Parker (\cite{Parker1988,Zhang2015,Sterling2015}).

Our main supposition here is that in contrast to EUV images (see orange line in
Fig.~\ref{fig06}d) and the coronal X-ray below $\sim 2 \div 3~keV$, the hard X-ray
emission above $\sim 3~keV$ is in fact the $\gamma$-quanta of axion origin,
born inside the magnetic tubes (see sunspot in Fig.~\ref{fig06}d), and is not
related to the mentioned indirect evidence (see e.g. Fig.~42 and Fig.~47 in
\cite{Shibata2011}) of the coronal jet, generated by the solar dynamo in cycles
of approximately 11 years (Fig.~\ref{fig-Yohkoh}). It will be interesting to
see if the proposed picture will ultimately be confirmed, modified, or rejected
by future observations and theoretical work to pin down the underlying physical
ideas.

Taking into account the directional patterns of the resulting radiation as well
as the fact that the maximum of the axion-originated X-ray radiation is
situated near 30 -- 40 degrees of latitude (because of the solar magnetic field
configuration), the mechanism of the high X-ray intensity bands formation on
the Yohkoh matrix becomes obvious. The effect of these bands widening near the
edges of the image is discussed in Appendix~\ref{appendix-widening} in detail.



\section{Summary and Conclusions}

In the given paper we present a self-consistent model of the axion mechanism of
the Sun's luminosity variations, in the framework of which we estimate the values of the axion
mass ($m_a \sim 3.2 \cdot 10^{-2} ~eV$) and the axion coupling constant to
photons ($g_{a \gamma} \sim 4.4 \cdot 10^{-11} ~GeV^{-1}$). A good
correspondence between the solar axion-photon oscillation parameters and the
hadron axion-photon coupling derived from white dwarf cooling (see
Fig.~\ref{fig05}) is demonstrated.

One of the key ideas behind the axion mechanism of Sun luminosity variations is the effect
of $\gamma$-quanta channeling along the magnetic flux tubes (waveguides inside
the cool region) above the base of the Sun convective zone
(Figs.~\ref{fig04-3}, \ref{fig-twisted-tube} and~\ref{fig-lampochka}). The low
refraction (i.e. the high transparency) of the thin magnetic flux tubes is
achieved due to the ultrahigh magnetic pressure (Fig.~\ref{fig-lampochka}a)
induced by the magnetic field of about 4100~T (see Eq.~(\ref{eq06-16}) and
Fig.~\ref{fig-lampochka}a). So it may be concluded that the axion mechanism of
Sun luminosity variations based on the lossless $\gamma$-quanta channeling along the
magnetic tubes allows to explain the effect of the partial suppression of the
convective heat transfer, and thus to understand the known puzzling darkness of
the sunspots (see 2.2.1 in \cite{Rempel2011}).





It is shown that the axion mechanism of luminosity variations (which means that
they are produced by adding the intensity variations of the $\gamma$-quanta of
the axion origin to the coronal part of the solar spectrum 
(Fig.~\ref{fig-Yohkoh}c)) easily explains the physics of the so-called Solar 
Equator -- Poles effect observed in the form of the anomalous X-ray 
distribution over the surface of the active Sun, recorded by the Japanese X-ray
telescope Yohkoh (Fig.~\ref{fig-Yohkoh}, top).

The essence of this effect consists in the following: axions that move towards
the poles (blue cones in Fig.~\ref{fig-Yohkoh}, bottom) and equator (blue
bandwidth in Fig.~\ref{fig-Yohkoh}, bottom) are not transformed into
$\gamma$-quanta by the inverse Primakoff effect, because the magnetic field
vector is almost collinear to the axions' momentum in these regions (see the
inset in Fig.~\ref{fig-Yohkoh}, bottom). Therefore the anomalous X-ray
distribution over the surface of the active Sun is a kind of a ``photo'' of the
regions where the axions' momentum is orthogonal to the magnetic field vector
in the solar over-shoot tachocline. The solar Equator -- Poles effect is not
observed during the quiet phase of the Sun because of the magnetic field
weakness in the overshoot tachocline, since the luminosity increment of the
axion origin is extremely small in the quiet phase as compared to the active
phase of the Sun.

In this sense, the experimental observation of the solar Equator -- Poles
effect is the most striking evidence of the axion mechanism of Sun luminosity
variations. It is hard to imagine another model or considerations which would
explain such anomalous X-ray radiation distribution over the active Sun surface
just as well (compare Fig.~\ref{fig-Yohkoh}a,b with Fig.~\ref{app-b-fig01}a
).

And, finally, let us emphasize one essential and the most painful point of the
present paper. It is related to the key problem of the axion mechanism of the solar
luminosity variations and is stated rather simply: ``Is the process of axion conversion
into $\gamma$-quanta by the Primakoff effect really possible in the magnetic
steps of an O-loop near the solar overshoot tachocline?'' This question is
directly connected to the problem of the hollow magnetic flux tubes existence
in the convective zone of the Sun, which are supposed to connect the tachocline
with the photosphere. So, either the more general theory of the Sun or the
experiment have to answer the question of whether there are the waveguides in
the form of the hollow magnetic flux tubes in the cool region of the convective
zone of the Sun, which are perfectly transparent for $\gamma$-quanta, or our
model of the axion mechanism of Sun luminosity variations is built around simply guessed
rules of calculation which do not reflect any real nature of things.



\section*{Acknowledgements}

\noindent The work of M. Eingorn was supported by NSF CREST award HRD-1345219
and NASA grant NNX09AV07A.

