# Neutron Stars as the Dark Matter detectors

Ariel Zhitnitsky<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of British Columbia, Vancouver, V6T 1Z1, BC, Canada

###### Abstract

It has been known for quite sometime that the Neutron Stars (NS) can play a
role of the Dark Matter (DM) detectors due to many uniques features of NS. We
apply these (previously developed) ideas to a specific form of the DM when it
is represented by a composite object, rather than by a local fundamental field
(such as WIMPs). To be more precise we consider the so-called axion quark
nuggets (AQN) dark matter model, when the “non-baryonic” dark matter in fact
is made of quarks and gluons which are in dense quark phase (similar to the
old idea of the Witten’s strangelets). We argue that the interaction of the
AQNs with NS material may lead to many profound observable effects, which
dramatically different from conventional picture when DM particles are
represented by weakly interacting WIMPs. In particular, we argue that the AQNs
may serve as the triggers for the magnetic reconnection to heat the NS
surface. This effect may strongly alleviate (or even completely remove) the
observed inconsistencies between the predicted and observed surface
temperatures for many old NS. This heating mechanism is always accompanied by
the hard X ray emission, which may serve as an indicator of the proposed
mechanism.

##  I Introduction

It has been known for a long time that the dynamics of the Neutron Stars (NS)
can be modified by the influence of the Dark Matter (DM) particles [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12]. Furthermore, it has been also known that the
basic properties of the DM particles can be strongly constrained by
considering DM-NS interactions. In particular, the conventional cooling
pattern of NS can be modified due to the capturing and consequent annihilation
of the DM particles [1, 2, 3]. There are many other processes which could
potentially become observable as a result of interaction of the DM particles
with very dense NS environment. We refer to several original papers [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12] devoted to analysis of many possible physical
phenomena which could result from such DM-NS interactions. This is obviously
very broad area of research, and we refer to the recent paper [13] for review
on this topic. In the present work we consider a specific model for the DM
which is dramatically different from conventional paradigm when DM particles
are assumed to be new (yet to be discovered) fundamental weakly interacting
massive particles (WIMPs). Before we elaborate on this specific form of the DM
represented by macroscopically large nuclear density composite objects made of
quarks and gluons we detour with overview of the most important features the
DM particles must satisfy.

Observational precision data gathered during the last quarter of century have
guided the development of the so called concordance cosmological model
$\Lambda$CDM of a flat universe, $\Omega\simeq 1$, wherein the visible
hadronic matter represents only $\Omega_{B}\simeq 0.05$ a tiny fraction of the
total energy density, see recent review [14], and interesting historical
comments [15]. Most of the matter component of the universe is thought to be
stored in some unknown kind of cold dark matter, $\Omega_{DM}\simeq 0.25$. The
largest contribution $\Omega_{\Lambda}\simeq 0.70$ to the total density is
cosmological dark energy with negative pressure, another mystery which will
not be discussed in the present work.

There is a fundamental difference between dark matter and ordinary matter
(aside from the trivial difference dark vs. visible). Indeed, DM played a
crucial role in the formation of the present structure in the universe.
Without dark matter, the universe would have remained too uniform to form the
galaxies. Ordinary matter could not produce fluctuations to create any
significant structures because it remains tightly coupled to radiation,
preventing it from clustering, until recent epochs. On the other hand, dark
matter, which is not coupled to photons, would permit tiny fluctuations to
grow for a long, long time before the ordinary matter decoupled from
radiation. The required material is called the Cold Dark Matter, and the
obvious candidates are the WIMPs of any sort which are long-lived, cold and
weakly interacting with visible hadronic material. The key parameter which
enters all the cosmological observations is the corresponding cross section
$\sigma$ to mass $M_{\rm DM}$ ratio which must be sufficiently small to play
the role of the DM as briefly mentioned above, i.e.

$\frac{\sigma}{M_{\rm DM}}\ll 1\frac{\rm cm^{2}}{\rm g},$ (1)

and WIMPs obviously satisfy to the criteria (1) to serve as DM particles.
However, the WIMP framework which has been the dominant paradigm for the last
40 years has failed as dozen of dedicated instruments could not find any
traces of WIMPs though the sensitivity of the instruments had dramatically
improved by many orders of magnitude during the last decades.

In the present work we consider a fundamentally different type of the DM which
is in form of the dense macroscopically large composite objects, similar to
the Witten’s quark nuggets [16, 17, 18]. The corresponding objects are called
the axion quark nuggets (AQN) and behave as chameleons: they do not interact
with the surrounding material in dilute environment, such that the AQNs may
serve as proper DM candidates as the corresponding condition (1) is perfectly
satisfied for the AQNs during the structure formation when the ratio
$\sigma/M_{\rm AQN}\lesssim 10^{-10}{\rm cm^{2}}{\rm g^{-1}}$. However, the
same AQNs become strongly interacting objects in sufficiently dense
environment, such as planets and stars. The interaction of AQNs with NS
environment also dramatically deviates from conventional WIMP -NS
interactions. As a result many observable consequences discussed previously
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] are dramatically modified as a
result of strong interactions of the AQNs with NS environment.

In the present work we address a single but very important question related to
the NS cooling scenario: why is the observed surface temperature of many old
NS are well above the conventional theoretical predictions? The proposed
answer is that this excess of heating is the result of the AQN-NS interaction,
which is the topic of the present work.

Before we present our arguments supporting this claim we briefly overview
several previously proposed heating mechanisms in next Sect. II which could in
principle generate some extra heat and could potentially answer the question
formulated above. We argue, however, that standard astrophysical sources are
very unlikely to be responsible for the observed excess of heating and there
must be some unconventional sources of heat to explain the anomalies in
observations.

The rest of our presentation is organized as follows. In Sect. III we overview
the basic features of the AQN model relevant for the present studies. In Sect.
IV we formulate the main lessons to be learned from our previous studies of
the AQN interactions with solar corona. We apply these ideas to AQN-NS
interaction in Sect. V where we argue that DM in form of the AQNs may serve as
the triggers igniting the large explosive events (due to magnetic
reconnections, similar to solar flares). The dynamics of magnetic reconnection
as the heating source of NS is elaborated in details in sections VI and VII.
In Sect. VIII we argued that the study of the hard X ray emission from NS
(using Magnificent Seven stars as an example) can serve as the indicator of
the proposed heating mechanism. We list our basic results in Sect. IX where we
also mention some other manifestations and observational consequences of the
AQN framework. Many technical details on physics of the magnetic helicity
$\cal{H}$ powering the magnetic reconnection are discussed separately in
Appendix A.

##  II possible heating mechanisms of old NS

We start with brief overview of the minimal cooling theory, see original paper
[19] and recent review [13]. The observations in general are in very good
agreement with minimal cooling paradigm when the neutrino emission from the
core dominates at early times ($t\lesssim 10^{5}\rm yr$). The photon emission
from the surface dominates at $t\gtrsim 10^{5}\rm yr$, when the neutrino
emission rate gets highly suppressed and the NS cools down. As a result, it is
expected that the NS surface temperature rapidly decreases to
$T_{s}^{\infty}\lesssim 10^{4}$ K at $t\gtrsim 10^{6}\rm yr$, which represents
a generic consequence of the minimal cooling theory111The $T_{s}^{\infty}$ in
this work is defined as the observed surface temperature at infinity. To be
more precise, the $T_{s}^{\infty}$ is defined as
$T_{s}^{\infty}=T_{s}\sqrt{1-\frac{2GM}{R}}$, where $T_{s}$ is the surface
temperature in its local reference frame, while $R$ and $M$ are the radius and
the mass of the NS..

However, some recent studies of old pulsars apparently are inconsistent with
this canonical cooling theory, see e.g. [20, 21, 22] for the references on the
numerous original results. These results suggest that some new sources of
heating must be operational to explain the observed surface temperature being
higher than expected. Many mechanisms which could potentially heat the old NS
have been suggested. In particular, it includes magnetic field decay, DM
accretion, crust cracking, vortex creep, roto-chemical heating, to name just a
few, see [20] for a brief review and references. It has been argued previously
that some NS which are slightly older than $\sim 10^{6}\rm yr$ can be
explained by some (or the combination) of these mechanisms. However, there are
still many cases when suggested mechanisms are not capable to explain the
data, see details below.

In the rest of this section we critically overview some of the most promising
mechanisms suggested previously. We also identify some cases when these
mechanisms still fail to explain the observed data. Precisely this dramatic
failure in explanation of the observed data was the main motivation for the
present work to suggest a novel heating mechanism which has the potential to
explain the observed anomalies. First, we start with a brief overview of
previously suggested heating mechanisms, see e.g. [20].

####  II.0.1 Rotochemical heating

The rotochemical heating is considered to be the most promising heating
mechanism [20, 21, 22]. The basic idea is that the chemical equilibrium is
altered when the rotation of the star is slowing down. The relaxing to the new
equilibrium state enforces the emission of photons which eventually heat the
surface. There is a number of uncertainties in the estimates which could be
sensitive to many parameters of the models, such as the gap, the EoS, or
initial conditions for young pulsars expressed in terms of the initial period
$P_{0}$. The fitting all these parameters in principle allows to explain the
observed surface temperatures of ordinary pulsars. However, in many cases this
mechanism fails to explain the observed surface temperatures.

For example, the $T_{s}^{\infty}$ of the so-called Magnificent Seven stars
cannot be explained by this mechanism with reasonable changes of the
parameters, see Fig. 3 in [21]. Another set of examples includes the old
pulsars with $t\gtrsim 10^{9}\rm yr$ with $T_{s}^{\infty}\sim 10^{5}$ K, see
e.g. two first lines in Table 1 in [21]. Such old and warm pulsars obviously
cannot be explained by the rotochemical heating mechanism with any reasonable
modifications of the parameters, see Fig. 3 and 4 in [21]. There are many
similar cases when the observed temperatures $T_{s}^{\infty}$ dramatically
exceeds the theoretical estimates, and we shall not discuss all these cases in
details.

For the purposes of the present work the most important outcome of these
estimates is that the rotochemical heating is obviously could be efficient and
operational in many cases. However, there are also many cases when it
dramatically fails, which obviously implies that: a). there is no unified and
simple mechanism which is capable to explain the observed data; b). there must
be some other mechanisms which could be also important and which become
especially pronounced at later times of the NS evolution.

####  II.0.2 Magnetic field decay

Another mechanism of heating which was widely discussed in the literature in
the past is the heating due to the magnetic field decay, see e.g. review [20].
It is common and generally accepted view that the magnetic field cannot play a
role of heating for relatively old stars with $t\gtrsim 10^{6}\rm yr$.
Nevertheless, we opted to present the conventional arguments (on irrelevance
of the magnetic field) below as the AQN-induced mechanism of heating of NS to
be introduced later in Sect. V will be based precisely on the transferring the
magnetic energy to the heat. We will show in Sect. V where and why the
conventional arguments (on irrelevance of the magnetic field) fail for this
specific AQN model.

The basic idea of estimate is to assume that the decaying magnetic field
strength $\cal{B}$ transfers its energy to the surface on the time scale $t$.
In this case one can equalize the luminosity $L$ of the NS with decreasing
magnetic energy in the entire NS, i.e.

$L=4\pi R^{2}\sigma T_{s}^{4}\approx\frac{4\pi
R^{3}}{3}\cdot\frac{\langle{\cal{B}}^{2}\rangle}{8\pi}\cdot\frac{1}{t}.$ (2)

The corresponding numerical estimate [20] suggests that the required magnetic
field to explain the observations with $T_{s}\sim 10^{5}$K is too high.
Indeed,

$\sqrt{{\langle{\cal{B}}^{2}\rangle}}\sim 10^{13}\cdot\sqrt{\frac{t}{10^{7}\rm
yr}}\cdot\sqrt{\frac{10\rm~{}km}{R}}\cdot\left(\frac{T_{s}}{10^{5}\rm
K}\right)^{2}~{}~{}\rm G,$ (3)

which is much higher than the observed magnetic field in classical pulsars
$\sim 10^{11}G$ and millisecond pulsars $\sim 10^{8}$ G. As a result of this
simple estimate the magnetic field as the source of heating was largely
ignored in the literature, in spite of the fact that the magnetic field
potentially represents enormous energy reservoir.

As we argue in this work precisely this enormous magnetic energy reservoir
will play a key role in heating of the NS within the AQN framework. However,
the relevant time scale entering the right hand side of (2) will be very
different from $(t)$ entering (2) which dramatically modifies the conventional
estimate (3).

####  II.0.3 Dark Matter accretion

We want to mention one more heating mechanism which occurs due to the DM
accretion. This mechanism was also widely discussed in the literature. Similar
to the previous case reviewed above in Sect. II.0.2 it was also thought that
the DM cannot play any essential role of heating for NS with $T_{s}\sim
10^{5}$K as maximum temperature which could be achieved by DM accretion cannot
exceed $T_{s}\approx 3\cdot 10^{3}$K for any reasonable parameters, see e.g.
review [20]. Therefore, this mechanism could potentially play a role but for
very old stars of age $t\gtrsim 10^{8}\rm yr$.

The basic idea of the estimate is to observe that the maximum possible
accretion rate onto a NS is given by

$\dot{M}\approx\rho_{\rm DM}\pi b^{2}_{\infty}v_{\infty},$ (4)

where $\rho_{\rm DM}$ is the local DM density in the vicinity of the NS, and
the velocity $v_{\infty}$ and impact parameter $b_{\infty}$ are defined at
very large distance from NS. Assuming that entire energy of the DM particles
is released in form of the heat one can infer that

$\dot{M}c^{2}\lesssim 5\cdot 10^{22}\rm\frac{erg}{s},$ (5)

which is many orders of magnitude smaller than the radiation from NS with
temperature $T_{s}\sim 10^{5}$K. Indeed,

$L=4\pi R^{2}\sigma T_{s}^{4}\approx 7\cdot
10^{28}\left(\frac{T_{s}}{10^{5}\rm K}\right)^{4}\rm\frac{erg}{s},$ (6)

which is 6 orders of magnitude higher than DM accretion heating mechanism can
provide according to estimate (5). As a result of this simple estimate it has
been concluded that the Dark Matter accretion mechanism may play a role only
for very old stars of age $t\gtrsim 10^{8}\rm yr$, and it can be safely
ignored for younger NS with temperature $T_{s}\gtrsim 10^{5}$K.

The irrelevance of the DM physics for NS with temperature $T_{s}\gtrsim
10^{5}$K was based on the canonical assumption that the DM particles are some
kind of fundamental (yet to be discovered) new particles in form of WIMPs.
Precisely this type of DM particles was previously considered in the
literature in context of DM-NS interaction [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13].

In contrast with this WIMP framework we are advocating the AQN framework where
DM is in form of the dense macroscopically large composite objects, similar to
the Witten’s quark nuggets [16, 17, 18], as mentioned in Sect. I. In this case
the macroscopically large AQNs can play the dual role: they obviously can
inject the energy directly, similar to WIMPs, in which case the constraint (5)
holds (with minor numerical modifications).

However, the same AQNs could also play the role of the triggers which can
initiate and ignite magnetic reconnections such that the enormous magnetic
energy reservoir stored in NS atmosphere and crust can heat the NS surface as
will be discussed in Sect.V. Precisely this dual role of the AQNs dramatically
modifies the conclusion of Sect. II.0.2 on irrelevance of the magnetic field
as a possible heating mechanism of sufficiently old NS.

Before we present the main ingredients on the AQN-induced heating mechanism of
the NS in Sects. V, VI and VII we have to make a detour to introduce the
basics of the AQN framework in Sect. III and the lessons to be learned from
similar processes (though in dramatically different environment) of
interactions of AQNs with solar corona in Sect IV.

##  III The AQN dark matter model

We overview the fundamental ideas of the AQN model in subsection III.1, while
in subsection III.2 we list some specific features of the AQNs relevant for
the present work.

###  III.1 The basics

As we already mentioned the AQN construction in many respects is similar to
the Witten’s quark nuggets, see [16, 17, 18]. This type of DM is
“cosmologically dark” as a result of smallness of the parameter (1) relevant
for cosmology. This numerically small ratio scales down many observable
consequences of an otherwise strongly-interacting DM candidate in form of the
AQN nuggets.

There are several additional elements in the AQN model in comparison with the
older well-known and well-studied theoretical constructions [16, 17, 18].
First, there is an additional stabilization factor for the nuggets provided by
the axion domain walls which are copiously produced during the QCD transition.
This additional element helps to alleviate a number of problems with the
original Witten’s model. In particular, a first-order phase transition is not
a required feature for nugget formation as the axion domain wall (with
internal QCD substructure) plays the role of the squeezer.

Another problem of the old construction [16, 17, 18] is that nuggets likely
evaporate on the Hubble time-scale. For the AQN model, this is not the case
because the vacuum-ground-state energies inside (the colour- superconducting
phase) and outside the nugget (the hadronic phase) are drastically different.
Therefore, these two systems can coexist only in the presence of an external
pressure, provided by the axion domain wall, which is inevitable feature of
the AQN construction. This should be contrasted with the original model [16,
17, 18], which is assumed to be stable at zero external pressure. This
difference has dramatic observational consequence relevant for the present
work- the Witten’s nugget will turn a NS into the quark star if it hits the
NS. In contrast, a matter type AQN will not turn an entire star into a new
quark phase because the quark matter in the AQNs is supported by external
axion domain wall pressure, and therefore, can be extended only to relatively
small distance $\sim m_{a}^{-1}$, which is much shorter than the NS size.

Finally, the nuggets can be made of matter as well as antimatter during the
QCD transition. The presence of the antimatter nuggets in the AQN framework is
an inevitable and the direct consequence of the $\cal{CP}$ violating axion
field which is present in the system during the QCD time. As a result of this
feature the DM density, $\Omega_{\rm DM}$, and the visible density,
$\Omega_{\rm visible}$, will automatically assume the same order of magnitude
densities $\Omega_{\rm DM}\sim\Omega_{\rm visible}$ irrespective to the
parameters of the model, such as the axion mass $m_{a}$. This feature
represents a generic property of the construction [23] as both component, the
visible, and the dark are proportional to one and the same fundamental
dimensional constant of the theory, the $\Lambda_{\rm QCD}$.

We refer to the original papers [24, 25, 26, 27] devoted to the specific
questions related to the nugget’s formation, generation of the baryon
asymmetry, and survival pattern of the nuggets during the evolution in early
Universe with its unfriendly environment. We also refer to a recent brief
review article [28] which explains a number of subtle points on the formation
mechanism, survival pattern of the AQNs during the early stages of the
evolution, including the Cosmic Microwave Background (CMB) Big Bang
Nucleosynthesis (BBN), and recombination epochs.

The only comment we would like to make here is that in this work we take the
agnostic viewpoint, and assume that such nuggets made of antimatter are
present in our Universe today irrespective to their formation mechanism. This
assumption is consistent with all presently available cosmological,
astrophysical and terrestrial constraints as long as the average baryon charge
of the nuggets is sufficiently large as we review below.

Property |  |  | Typical value or feature
---
AQN’s mass $[M_{N}]$ |  |  $M_{N}\approx 16\,g\,(B/10^{25})$ [28]
baryon charge constraints $[B]$ |  |  $B\geq 3\cdot 10^{24}$ [28]
annihilation cross section $[\sigma]$ |  | $\sigma\approx\kappa\pi R^{2}\simeq 1.5\cdot 10^{-9}{\rm cm^{2}}\cdot\kappa(R/2.2\cdot 10^{-5}\rm cm)^{2}$
density of AQNs $[n_{\rm AQN}]$ |  |  $n_{\rm AQN}\sim 0.3\cdot 10^{-25}{\rm cm^{-3}}(10^{25}/B)$ [28]
survival pattern during BBN |  |  $\Delta B/B\ll 1$ [29, 30, 31, 32]
survival pattern during CMB |  |  $\Delta B/B\ll 1$ [29, 33, 31]
survival pattern during post-recombination |  |  $\Delta B/B\ll 1$ [27]
Table 1: Basic properties of the AQNs adopted from [34]. The parameter
$\kappa$ in Table is introduced to account for possible deviation from
geometric value $\pi R^{2}$ as a result of ionization of the AQNs due to
interaction with environment. The ratio $\Delta B/B\ll 1$ in the Table implies
that only a small portion $\Delta B$ of the total (anti)baryon charge $B$
hidden in form of the AQNs get annihilated during big-bang nucleosynthesis
(BBN), Cosmic Microwave Background (CMB), or post-recombination epochs
(including the galaxy and star formation), while the dominant portion of the
baryon charge survives until the present time.  Figure 1: AQN-structure (not
in scale), adopted from [35]. The dominant portion of the energy $\sim 2$ GeV
produced as a result of a single annihilation process inside the anti-nugget
is released in form of the bremsstrahlung radiation with frequencies
$\omega\leq T$, see description and notations in the main text.

We conclude this brief review subsection with Table 1 which summarizes the
basic features and parameters of the AQNs. Important point here is that only a
small portion $\Delta B\ll B$ of the total (anti)baryon charge $B$ hidden in
form of the AQNs get annihilated during long evolution of the Universe. The
dominant portion of the baryon charge survives until the present time.
Independent analysis [32] and [31] also support our original claims as cited
in the Table 1 that the anti-quark nuggets survive the BBN and CMB epochs.

We draw the AQN-structure on Fig 1, where we use typical parameters from the
Table 1. There are several distinct length scales of the problem: $R\sim
10^{-5}$ cm represents the size of the nugget filled by dense quark matter
with total baryon charge $B\sim 10^{25}$ in CS phase. Much larger scale
$R_{\rm DW}\sim m_{a}^{-1}$ describes the axion DW surrounding the quark
matter. The axion DW has the QCD substructure surrounding the quark matter and
which has typical width of order $R_{\rm QCD}\sim 10^{-13}\rm cm$. Finally,
there is always electro-sphere which represents a very generic feature of
quark nuggets, including the Witten’s original construction. In case of
antimatter-nuggets the electro-sphere comprises the positrons. The typical
size of the electrosphere is order of $10^{-8}\rm cm$, see below.

###  III.2 When the AQNs start to interact with dense environment

For our present work, however, the most relevant studies are related to the
effects which may occur when the AQNs made of antimatter propagate in the
environment with sufficiently large visible matter density $n(r)$ such as
density of the solar corona, or NS atmosphere. In this case the annihilation
processes start and a large amount of energy will be injected to surrounding
material, which may be manifested in many different ways. What is more
important for the present studies is that the same annihilation processes
become much more important if the AQN enters the region of highly ionized
plasma because the ions are much more likely to interact with the AQNs in
comparison with neutral atoms due to the long-ranged Coulomb attraction.

The related computations on the AQN-visible matter interaction originally have
been carried out in [36] in application to the galactic neutral environment at
present time with a typical density of surrounding baryons of order $n_{\rm
galaxy}\sim{\rm cm^{-3}}$ in the galaxy. We review these computations with few
additional elements which must be implemented in case of propagation of the
AQN in denser and ionized environment such as NS atmosphere.

When the AQN enters the region of the baryon density $n$ the annihilation
processes start and the internal temperature increases. A typical internal
temperature $T$ of the AQNs can be estimated from the condition that the
radiative output must balance the flux of energy onto the nugget

$F_{\rm{tot}}(T)(4\pi R^{2})\approx\kappa\cdot(\pi R_{\rm
eff}^{2})\cdot(2~{}{\rm GeV})\cdot n\cdot v_{\rm AQN},$ (7)

where $n$ represents the baryon number density of the surrounding material,
and $F_{\rm{tot}}(T)$ is total surface emissivity, see below. The left hand
side accounts for the total energy radiation from the AQN’s surface per unit
time while the right hand side accounts for the rate of annihilation events
when each successful annihilation event of a single baryon charge produces
$\sim 2m_{p}c^{2}\approx 2~{}{\rm GeV}$ energy. If the environment is
represented by neutral atoms and molecules the interaction of the AQNs with
environment can be approximated by the geometrical cross section $\pi R^{2}$
for macroscopically large object of size $R$. However, if the surrounding
material is highly ionized the effective cross section $\pi R_{\rm eff}^{2}$
could be dramatically larger than the geometric value $\pi R^{2}$ due to the
long range Coulomb interaction as the AQN assumes a large negative charge at
sufficiently high temperature $T$, see below for estimates. The factor
$\kappa$ in (7) accounts for large theoretical uncertainties related to the
annihilation processes of the (antimatter) AQN colliding with surrounding
material.

The total surface emissivity due to the bremsstrahlung radiation from
electrosphere at temperature $T$ has been computed in [36] and it is given by

$F_{\rm{tot}}\approx\frac{16}{3}\frac{T^{4}\alpha^{5/2}}{\pi}\sqrt[4]{\frac{T}{m}}\,,$
(8)

where $\alpha\approx 1/137$ is the fine structure constant, $m=511{\rm\,keV}$
is the mass of electron, and $T$ is the internal temperature of the AQN. One
should emphasize that the emission from the electrosphere is not thermal, and
the spectrum is dramatically different from blackbody radiation.

From (7) and (8) one can estimate a typical internal nugget’s temperature for
the neutral environment when $R_{\rm eff}\approx R$ and the density $n$
assumes a typical galactic values $n\sim\rm cm^{-3}$:

$T\sim 0.4~{}{\rm eV}\cdot\left(\frac{n}{\rm
cm^{-3}}\right)^{\frac{4}{17}}\cdot\left(\frac{v_{\rm
AQN}}{10^{-3}c}\right)^{\frac{4}{17}}\cdot\kappa^{\frac{4}{17}}.$ (9)

Another feature which is relevant for our present studies is the ionization
properties of the AQN itself (along with ionization of the surrounding
plasma). Ionization, as usual, occurs in a system as a result of the high
internal temperature $T$, in which case a large number of weakly bound
positrons from the electrosphere get excited and can easily leave the system.
As a result, the anti-nugget assumes a negative electric charge. Its absolute
value $Q(T)$ strongly depends on the environment (density and the
temperature).

Precisely this feature of ionization of the AQN dramatically enhances the
visible-DM interaction in highly ionized sufficiently dense environment when
cosmologically relevant ratio ($\sigma/M)$ from (1) could become very large.
This feature may dramatically modify many previously obtained results [19, 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] which were made under assumption that
DM particles are fundamental weakly interacting objects such as WIMPs.

The emergence of very strong interaction of DM with surrounding material at
the star’s surface is a direct manifestation of the AQN construction when the
dark matter in form of the AQNs behave as chameleon like composite objects.
Indeed, the AQNs are the perfect DM particles in dilute environment as
reviewed in Sect. III.1 but become very strongly interacting objects in
relatively dense environment.

Finally, one should mention here that the AQN model with the same set of
parameters to be used in the present work may explain a number of puzzling and
mysterious observations which cannot be explained by conventional
astrophysical phenomena. These mysterious puzzles occur at many different
scales in dramatically different environments, including BBN epoch, dark ages,
as well as galactic, Solar and Earth environments at present time, see
concluding section IX.2.

##  IV Lessons from the solar corona heating puzzle

Before we consider the dynamics of the AQN-NS interaction in next section V we
would like to make a short detour in this section to overview the application
of the AQN framework to the solar corona heating problem [37, 38, 39, 40]. The
basic lesson from the studies [37, 38, 39, 40] is that the energy which heats
the corona may come from the annihilation processes of the DM particles in
form of the AQNs in solar corona. Furthermore, the same AQNs may play the role
of triggers which may ignite the large solar flares. In other words, the AQNs
entering the solar corona may activate the magnetic reconnection in active
regions of the sun and initiate the large solar flares.

We shall use the corresponding lessons from [37, 38, 39, 40] for the case when
AQNs hit the NS, which is the topic of the present studies. Precisely this
analogy between the AQN dynamics in NS’s atmosphere versus solar corona will
be our guiding principle in our studies of the AQN-NS interactions in Sects.
V, VI and VII.

###  IV.1 Solar corona heating puzzle: the observations

We start with a few historical remarks. The solar corona is a very peculiar
environment. Starting at an altitude of 1000 km above of the photosphere, the
highly ionized iron lines show that the plasma temperature exceeds a few
$10^{6}$ K. The total energy radiated away by the corona is of the order of
$L_{\rm corona}\sim 10^{27}{\rm erg~{}s^{-1}}$, which is about
$10^{-6}-10^{-7}$ of the total energy radiated by the photosphere. Most of
this energy is radiated at the extreme ultraviolet (EUV) and soft X-ray
wavelengths. There is a very sharp transition region, located in the upper
chromosphere, where the temperature suddenly jumps from a few thousand degrees
to $10^{6}$ K. This transition layer is relatively thin, 200 km at most. This
transition happens uniformly over the Sun, even in the quiet Sun, where the
magnetic field is small ($\sim 1~{}{\rm G}$), away from active spots and
coronal holes. The reason for this uniform heating of the corona remains to be
a mystery.

A possible solution to the heating problem in the quiet Sun corona was
proposed in 1983 by Parker [41], who postulated that a continuous and uniform
sequence of miniature flares, which he called “nanoflares”, could happen in
the corona.

The term “nanoflare” has been used in a series of papers by Benz and coauthors
[42, 43, 44, 45, 46], and many others, to advocate the idea that these small
“micro-events” might be responsible for the heating of the quiet solar corona.
In most recent studies for the purpose of the modelling the term “nanoflare”
describes a generic event for any impulsive energy release on a small scale,
without specifying its cause and its physics nature, see review papers [47,
48] with references on recent activities in the field. The list below shows
the most important constraints on nanoflares from the observations of the EUV
iron lines with SoHO/EIT:
1. The EUV emission is highly isotropic [43, 45], therefore the nanoflares have to be distributed very “uniformly in quiet regions”, in contrast with flares which have a highly non-isotropic spatial distribution because they are associated with small active regions;
2. According to [44], in order to reproduce the measured EUV excess, the observed range of nanoflares needs to be extrapolated from the observed events interpolating between $(3.1\cdot 10^{24}-1.3\cdot 10^{26})~{}{\rm erg}$ to sub-resolution events with much smaller energies, see item 3 below.
3. In order to reproduce the measured radiation loss, the observed range of nano flares needs to be extrapolated to energies as low as $10^{22}$erg and in some models, even to $10^{20}$erg (see table 1 in [44]);
4. The flares, in contrast with nano-flares, originate at sunspot areas, with locally large magnetic fields ${\cal{B}}\sim(10^{2}-10^{3})$ G, while the EUV emission (which is observed even in very quiet regions where ${\cal{B}}\sim 1$G) is isotropic and covers the entire solar surface;
5. The temporal evolution of flares and nanoflares also appears different. The typical ratio between the maximum and minimum EUV irradiance during the solar cycle does not exceed a factor of 3 between its maximum in 2000 and its minimum in 2009 (see Fig. 1 from [49]), while the same ratio for flares and sunspots is much larger, of the order of $10^{2}$. If the magnetic reconnection (as Parker originally conjectured) was fully responsible for both the flares and nanoflares, then the variation during the solar cycles should be similar for these two phenomena. It is not what is observed: the modest variation of the EUV with the solar cycles in comparison to the flare fluctuations suggests that the EUV radiation does not directly follow the magnetic field activity, and that the EUV fluctuation is a secondary, not a primary effect of the magnetic activity.

###  IV.2 The nanoflares as the AQN annihilation events

All the puzzles (such as isotropic features of the EUV emission over the
entire solar surface with very modest variations during the solar cycles) as
mentioned above can be naturally understood if the EUV emission from the solar
corona is related to the DM particles. The corresponding conjecture that the
nanoflares heating the corona can be identified with AQN annihilation
events222In this sense our proposal fulfills a key missing ingredient on the
nature of the nanoflares as conventional scaling arguments suggest that the
typical time scales for the magnetic reconnection in the background of a
typical magnetic field ($\sim 1$ G), must be very long, dramatically longer
than the observations suggest, see Appendix in [50] for the details. has been
explicitly formulated in [37]. The main argument supporting this conjecture is
amazing numerical coincidence between the observed total luminosity ($\sim
10^{27}{\rm erg\cdot s^{-1}}$) radiated from corona in form of the soft X rays
and EUV and the injected energy resulting from the annihilation events when
the AQNs hit the Sun.

Indeed, the impact parameter for capture of the nuggets by the Sun can be
estimated as

$b_{\rm cap}^{\odot}\simeq
R_{\odot}\sqrt{1+\gamma_{\odot}},~{}~{}~{}~{}\gamma_{\odot}\equiv\frac{2GM_{\odot}}{R_{\odot}v^{2}},$
(10)

where $v\simeq 10^{-3}c$ is a typical velocity of the nuggets. Assuming that
$\rho_{\rm DM}\sim 0.3~{}{\rm GeVcm^{-3}}$ and using the capture impact
parameter (10), one can estimate the total energy being injected due to the
complete annihilation of the nuggets in solar corona as follows:

$L_{\odot}^{\rm AQN}\sim(\pi b^{2}_{\rm cap})\cdot v\cdot\rho_{\rm DM}\simeq
10^{30}\frac{\rm GeV}{\rm s}\simeq 10^{27}\frac{\rm erg}{\rm s},$ (11)

where we substitute constant $v\simeq 10^{-3}c$ for numerical estimate333A
more proper estimation should include $n_{\rm AQN}(2~{}m_{p}\cdot
B\cdot{3}/{5})$ where coefficient $3/5$ reflects the portion of the anti-
nuggets, $n_{\rm AQN}\approx\rho_{\rm DM}/M_{N}$ is the density of the AQNs
while $M_{N}\approx m_{p}B$ is their typical mass. We ignore all these
numerical coefficients in estimate (11) in front of $\rho_{\rm DM}$..
Precisely this “accidental numerical coincidence” between the observed
luminosity $L_{\rm corona}\sim 10^{27}{\rm erg~{}s^{-1}}$ and the AQN-induced
luminosity (11) was the main motivation to put forward the idea that the AQNs
represent a new source of energy feeding the EUV radiation from solar corona,
which is very hard to explain in terms of conventional astrophysical sources
as highlighted above in section IV.1, see also footnote 2 for a comment.

Based on this amazing numerical coincidence the nanoflares have been
identified with the AQN annihilation events (within the AQN framework). An
immediate self-consistency check of this conjecture is that the lower limit
for the AQN baryonic charge (see Table 1) approximately coincides with
nanoflare’s low energy. Indeed, according to this identification the AQN
annihilation of baryon charge $B$ produces the energy $W\simeq 2m_{p}c^{2}B$.
One can check that the smallest AQN baryonic charge $B\sim 10^{24}$ as given
in Table 1 is indeed close to the lowest nanoflare’s energy $W\sim 10^{21}\rm
erg$ as given by (LABEL:dN). We emphasize that this numerical similarity
represents a highly nontrivial self-consistency check of proposal [37], as the
acceptable range for the AQNs and nanoflares have been constrained from
dramatically different physical systems.

Encouraged by this self-consistency check and the highly nontrivial energetic
consideration, a full scale of the Monte Carlo (MC) simulations had been
performed in [39]. It has been shown that the annihilation mostly occurs at
the altitudes around 2000 km where the most of the injected energy is
released. This represents a highly nontrivial explanation of the emergence of
a very narrow transition region of order 200 km width within the AQN
framework.

To summarize the proposal on identification of the AQN annihilation events
with nanoflares: the uniformity of the EUV emission is naturally understood in
the AQN framework as DM is distributed very uniformly over the Sun, making no
distinction between quiet and active regions. Furthermore, our proposal
explains an insignificant role of the magnetic field for the EUV radiation as
the AQN events do not depend on the strength of the magnetic field, which is
also consistent with observations. It should be contrasted with original
conjecture [41] where nanoflares are thought to be scaled down configurations
of their larger cousins, which are known to be localized exclusively in the
area with large magnetic field, and fed by the magnetic field energy.

Finally, the temporal modulation of the EUV irradiance over a solar cycle is
very modest, as opposed to the very dramatic changes in flare activity on the
level of $10^{2}$ over the same time scale. This is perfectly consistent with
our interpretation of nanoflares being associated with AQN annihilation events
which are not related to the solar activity, nor to dynamics of the magnetic
field itself during the cycles.

###  IV.3 AQNs as the triggers of large solar flares

In this section we overview the basic results from [38] where it was argued
that the same AQNs (which are identified with nanoflares as overviewed in
previous section IV.2) could serve as the triggers for large Solar flares. The
basic reason for necessity for a trigger (which can initiate the magnetic
reconnection) to be present in the system is related to very large numerical
values of the so-called the Lundquist number $S\sim(10^{12}-10^{14})$ in solar
corona. Precisely this parameter $S$ determines the theoretical value for
reconnection time which is much longer than observations show, see Appendix in
[50] for details and references. Though in the last 10-15 years many new ideas
have been pushed forward to speed up the reconnection, a large number of
fundamental questions remains444One of the question is the observation of the
dramatic variation of time scales when pre-flare X ray radiation lasts for
seconds, flare itself lasts for about an hour while the preparation phase of
the magnetic configurations to be reconnected could last for months. The
presence of a trigger may automatically resolve this and many other hard
questions, see [38]..

The basic idea of [38] is as follows:

1\. The AQNs entering the solar corona with typical velocity of the nuggets
$v\sim(600-800)~{}{\rm km/s}$ in the vicinity of the surface will inevitably
generate shock waves as the typical velocities $v$ of the dark matter
particles much larger than the speed of sound $c_{s}$ such that the Mach
number $M\equiv v/c_{s}>1$, see estimates below;

2\. When the AQNs (distributed uniformly) enter regions with a strong magnetic
field in active regions, they trigger magnetic reconnection of preexisted
magnetic configurations;

3\. Technically, the AQNs are capable sparking magnetic reconnections due to
the large discontinuities of the pressure $\Delta p/p\sim M^{2}$ and
temperature $\Delta T/T\sim M^{2}$ when the AQN-induced shock front passes
through the magnetic reconnection regions, see estimates for these parameters
below.

Now we estimate the relevant parameters suggesting that the AQNs indeed could
serve as the triggers igniting and initiating the large solar flares.

We start our estimate with the speed of sound $c_{s}$ in the corona at
$T\simeq 10^{6}K$,

$\displaystyle\left(\frac{c_{s}}{c}\right)^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{3T\cdot\Gamma}{m_{p}c^{2}},~{}~{}~{}c_{s}\simeq 7\cdot
10^{-4}c\cdot\sqrt{\frac{T}{10^{6}~{}{\rm K}}}$ (12)

where $\Gamma=5/3$ is a specific heat ratio, $c$ is the speed of light, and we
approximate the mass density $\rho_{p}$ of plasma by the proton’s number
density density $n$ as follows $\rho_{p}\simeq nm_{p}$. The crucial
observation here is that the Mach number $M$ is always much larger than one
for a typical dark matter velocities at the surface:

$M\equiv\frac{v}{c_{s}}\simeq 4\sqrt{\frac{10^{6}~{}{\rm K}}{T}}>1.$ (13)

As a result, a strong shock waves will be generated when the AQNs enter the
solar corona. In the limit when the thickness of the shock wave can be
ignored, the corresponding formulae for the discontinuities of the pressure
$p$, temperature $T$, and the density $\rho_{p}$ are well known and given by,
see e.g. [51]:

$\displaystyle\frac{{\rho_{p}}_{2}}{{\rho_{p}}_{1}}$ $\displaystyle\simeq$
$\displaystyle\frac{(\Gamma+1)}{(\Gamma-1)},~{}~{}~{}~{}~{}\frac{p_{2}}{p_{1}}\simeq
M^{2}\cdot\frac{2\Gamma}{(\Gamma+1)}$ $\displaystyle\frac{T_{2}}{T_{1}}$
$\displaystyle\simeq$ $\displaystyle
M^{2}\cdot\frac{2\Gamma(\Gamma-1)}{(\Gamma+1)^{2}},~{}~{}~{}~{}M\gg 1,$ (14)

where we assume $M\gg 1$ and keep the leading terms only in the corresponding
formulae.

Another important parameter which determines importance of the magnetic
pressure in comparison with kinetic pressure is dimensionless parameter
$\beta$:

$\displaystyle\beta\equiv\frac{8\pi p}{{\cal{B}}^{2}}\sim
0.05\left(\frac{n}{10^{10}~{}{\rm
cm^{-3}}}\right)\left(\frac{T}{10^{6}K}\right)\left(\frac{100~{}G}{{\cal{B}}}\right)^{2}~{}~{}~{}~{}~{}$
(15)

where for numerical estimates we use typical parameters for the active regions
in corona when $\beta\ll 1$. The same relation (15) also explicitly shows that
the magnetic field cannot play any essential role (including a very unlikely
possibility of magnetic reconnection) outside the active regions when typical
value of ${\cal{B}}\sim 1~{}G$ and $\beta\gg 1$, which was the topic of the
EUV radiation in previous sections IV.1 and IV.2.

The main assumption for the estimates presented above is that the AQNs can be
treated as the macroscopically large objects such that conventional classical
hydrodynamics applies. In different words, the size of the objects must be
much larger than the average distance $a$ between the particles in surrounding
plasma, which itself is determined by the density $a\approx n^{-1/3}$. This
condition is perfectly satisfied for the AQNs because their effective size
$R_{\rm eff}\gg 1\rm~{}cm$ is indeed much larger than the geometric size of
the quark nugget $R\sim 10^{-5}\rm cm$ due to very strong interaction with
surrounding ionized plasma when positively charged ions are captured by the
negatively charged nugget, see [39] for the estimates of parameter $R_{\rm
eff}$ which indeed satisfies $R_{\rm eff}\gg a$ for typical density in corona.

The idea that the shock waves may dramatically increase the rate of magnetic
reconnection is not new, and has been discussed previously in the literature,
though a quite different context, see references in Appendix in [50]. The new
element which was advocated in [38] is that the small shock waves resulting
from entering AQNs are widespread and generic events in the solar corona.
These small events identified with nanoflares and the corresponding
annihilation energy is sufficient to heat corona as estimated by (11).
However, they do not generate large flares as $\beta\gg 1$ in quiet regions.

The situation becomes very different however if the nuggets hit the active
regions with strong magnetic field ${\cal{B}}\sim 10^{2}~{}G$ and $\beta\ll
1$. In this case the AQN- induced shock waves may ignite large flare. This
proposal may answer many questions of the complicated dynamics of the flares,
including a dramatic variation of the time scales as mentioned in footnote 6.

###  IV.4 Energetics of large solar flares

Now we want to estimate the size and the energy scales associated with such
events. We consider separately two different stages. First, we estimate the
scales related to the initial phase of the evolution when the AQNs produce the
shock waves, but the magnetic reconnection has not started yet. The estimation
for the second phase assumes that the magnetic reconnection, leading to a
large solar flare, is already fully developed.

In the first, initial stage of the evolution, the magnetic reconnection has
not started yet, and entire energy is related to the shock wave, which itself
forms as a result of AQN entering the solar atmosphere from outer space. In
this case a typical time scale when AQN completely annihilates its baryon
charge is of order of $\tau\sim 10~{}{\rm sec}$, see [37, 39]. A typical
length scale is determined by the initial velocity of the AQN which is of
order $v_{\rm AQN}\sim(600-700){\rm km/s}$ such that $L\sim v_{\rm
AQN}\cdot\tau\sim 5\cdot 10^{3}~{}{\rm km}$. At the same time, a typical
radius $R$ of the cone formed by the shock wave is determined by the speed of
sound $c_{s}$, such that $R\sim M^{-1}L$, where Mach number $M$ is estimated
in (13). For numerical estimates below we take $M\simeq 10$. The affected area
$A$ due to the shock wave (where the magnetic reconnection starts) is
estimated as $A\sim M^{-1}L^{2}$. We summarize the parameters of the initial
stage as follows

$\displaystyle\tau$ $\displaystyle\approx$ $\displaystyle 10~{}{\rm
sec},~{}~{}L\approx v_{\rm AQN}\cdot\tau\approx 5\cdot 10^{3}~{}{\rm
km}~{}~{}$ $\displaystyle R$ $\displaystyle\approx$ $\displaystyle
M^{-1}L\approx 10^{-1}L,~{}~{}A\approx R\cdot L\sim 10^{-1}L^{2}.$ (16)

We are now in position to estimate the typical energetic characteristics of
the system during this initial stage. The key element is the observation that
the temperature $T$ experiences a large discontinuity resulting form the shock
according to (IV.3). Therefore, we estimate a typical internal temperature of
the nugget $T_{2}$ as follows

$\frac{T_{2}}{T_{1}}\approx M^{2}\approx 10^{2},~{}~{}~{}T_{2}\approx
M^{2}T_{1}\approx 10^{8}~{}{\rm K}$ (17)

where $T_{1}\sim 10^{6}K$ corresponds to unperturbed temperature of the solar
corona before the shock passage through the area.

Important comment here is that formula (17) shows that the AQN’s internal
temperature could reach very high values on the level $T\sim 10^{8}~{}{\rm
K}$. As a result, the AQNs could be the source of the $(1-10)~{}{\rm keV}$
x-rays. Interestingly enough, the x rays emission had indeed been recorded for
many large flares few moments (pre-cursor) before the flare starts [52].
Furthermore, one can explicitly see that pre-flare enhancement propagates from
higher levels of corona into lower corona and chromosphere [52]. It is very
hard to explain such X rays emission pattern within conventional MHD. In our
framework the X ray emission before the large flare starts is a natural
consequence of the proposal when the AQNs (moving from outer space to the
surface) generate the shock and play the role of the triggers initiating and
igniting large flares. We further comment on similarity of the x ray emission
from solar corona during the flare and from NS in Sect. VIII.

The second stage of the flare (after the initial stage described above ends)
in this framework is represented by the magnetic reconnection ignited by the
shock wave. We have nothing new to say about this conventional phase of the
evolution. We present the corresponding estimate below for the total flare’s
energy for completeness and following discussions,

$\displaystyle W_{\rm flare}\sim\frac{{\cal{B}}^{2}V_{\rm flare}}{8\pi}\sim
3\cdot 10^{30}(\frac{{\cal{B}}}{300\rm G})^{2}(\frac{V_{\rm
flare}}{10^{13}{\rm km^{3}}})~{}{\rm erg},~{}~{}$ (18)

where $V_{\rm flare}\approx L_{\perp}^{2}L$ with $L\sim 5\cdot 10^{3}~{}{\rm
km}$ being a typical length scale (IV.4) where shock waves develops in solar
corona555to simplify the estimates we assume the nugget trajectory is
perpendicular to the solar surface such that $L$ is oriented along $z$
direction., while $L_{\perp}^{2}$ is the area within active region (sunspots)
which eventually becomes a part of magnetic reconnection producing the large
flares. Numerically $L_{\perp}\sim(10^{3}-10^{4})~{}{\rm km}$ for microflares,
and it could be as large as $L_{\perp}\sim 10^{5}~{}{\rm km}$ for large
flares. It is assumed that precisely this region of volume $V_{\rm
flare}=L_{\perp}^{2}L$ with large average magnetic field
${\cal{B}}\sim{300~{}\rm G}$ feeds the solar flare as a result of magnetic
reconnection. It is also known that the magnetic reconnection is always
accompanied by presence of non-trivial topological structures which manifest
of a variety of complex processes during the flare, see [38] for references.

It is quite obvious that the energy (18) of a fully developed flare is many
orders of magnitude larger than the initial energy of the AQN which serves as
a trigger of a large flare. Nevertheless, this initial stage in the flare
evolution plays a key role in future development of the system because it
provides a very strong impulse with $\Delta T/T\gg 1$ and $\Delta p/p\gg 1$ in
very small and very localized area for very short period of time (IV.4) in the
region where the magnetic reconnection eventually develops. Precisely the
presence of a trigger explains a large number of puzzles related to
dramatically different time scales which are known to exist in the system, see
[38] for references.

We conclude this section on solar corona heating puzzle with the following
remark. Our main topic of this work is analysis of possible effects which may
occur when the AQNs hit NS surface. We shall use many lessons from the present
section as the magnetic reconnection in NS triggered and initiated by AQNs may
generate many profound effects as we will discuss in next section V.

##  V When AQN hits the NS

###  V.1 Energy injection due to the AQN’s annihilation

First of all, we would like to estimate a total power being injected as a
result of AQN hitting the NS surface and get annihilated along its path close
to the surface. For simplicity we ignore the DM velocity distribution and
assume that $v_{\infty}\simeq 10^{-3}c$ is a typical velocity of the nuggets
at a large distance from NS. In this case the impact parameter for capture and
consequent annihilation of the AQNs by NS can be estimated as follows,

$\displaystyle\frac{b_{\rm cap}^{\rm NS}}{R_{\rm
NS}}\simeq\frac{c}{v_{\infty}}\cdot\left(\frac{\rm 10^{6}~{}cm}{R_{\rm
NS}}\right)^{1/2}\cdot\left(\frac{M_{\rm NS}}{2~{}M_{\odot}}\right)^{1/2}$
(19)

which replaces formula (10) estimated for the Sun. The total energy being
injected due to the complete annihilation of the nuggets in NS as follows:

$L_{\rm NS}^{\rm AQN}\sim(\pi b^{2}_{\rm cap})\cdot v_{\infty}\cdot\rho_{\rm
DM}\simeq 10^{23}\frac{\rm erg}{\rm s},$ (20)

which replaces formula (11) with estimated for the Sun, see also footnote 3
with a comment. The estimate (20) of course gives the same order of magnitude
for the AQN model as for any other DM model, see e.g. [10] where only kinetic
energy of a DM particle contributes to the heating. It should be contrast with
our case when entire energy due to the annihilation will be released. The
numerical difference, however, is a very minor effect as all DM particles on
NS surface become relativistic objects irrespective to the models. As a
result, the difference in equilibration temperatures on the surface does not
lead to any qualitative observable effects in comparison with previous
analysis. In particular, instead of $T\approx 1750K$ from WIMP type models
[10] we would get $T\sim 3000K$ from AQN model, if all released energy is
thermalized.

Now we would like to make few comments on comparison of (20) with analogous
estimates for the solar corona (11) where the same annihilation events of the
AQNs in solar atmosphere generate fundamentally new phenomenon representing
the resolution of the solar corona heating puzzle within AQN framework as
explained in previous sections IV.1 and IV.2.

The dramatic differences in luminosities (four orders in magnitude) between
(20) and (11) is related to the fact that the impact parameters are very
different for this two cases. Indeed, factor $10^{4}$ between (11) and (20)
can be understood as the ratio:

$\left(\frac{b^{\odot}_{\rm cap}}{b^{\rm NS}_{\rm
cap}}\right)^{2}\sim\left(\frac{10^{6}~{}\rm km}{10^{4}~{}\rm
km}\right)^{2}\sim 10^{4}.$ (21)

As a result, the luminosity ($\sim 10^{27}{\rm erg\cdot s^{-1}}$) radiated
from corona (11), though represents only $\sim 10^{-6}$ fraction of the total
luminosity of the Sun, nevertheless produces the profound observable effects
in form of the EUV and X rays emission from corona. In contrast, the
observation of the emission (20) from NS is unlikely to be directly observed
anytime soon.

Indeed, the value for the surface temperature $T\approx 3\cdot 10^{3}\rm K$ as
estimated above is way below of the present observational capabilities, and we
shall not elaborate on this effect of heating due to the direct AQN
annihilations in NS atmosphere and the crust in the present work. Precisely
this effect (heating of the very old stars due to the direct energy injection
by DM particles in form of WIMPs) was the main subject of the analysis in most
of previous studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. This is
precisely the main conclusion of Sect.II.0.3 that DM accretion cannot play any
role in heating of the NS to temperatures in the range $T\sim 10^{5}K$ as
observed. This conclusion was entirely based on canonical assumption within
the 40 year old paradigm that the DM is represented by a fundamental field in
form of a microscopical particle such as WIMP.

In contrast, the AQN is a complex macroscopical object, outside of this
canonical paradigm. Therefore, it may play another role (as a trigger, see
below) along with the effect mentioned above.

This work is focused precisely on another consequence of the AQN framework
when the nuggets play the role of the triggers which may ignite and initiate
much larger events similar to the flares in the Sun as discussed in sections
IV.3 and IV.4. This effect is not shared by any other DM models when the DM
particles are represented by fundamental local quantum fields, such as WIMPs.

When AQN serves as a triggers of a large event such as flare in the Sun, the
dominant portion of the energy feeding such event is coming from a strong
magnetic field of NS (not from the AQN itself) converting its energy into the
radiation in broad frequency bands. If this happens there could be many
dramatic observable effects, which precisely represents the topic of the
present studies.

###  V.2 Mach number and shock waves

The goal of the present section is to argue that the AQNs can serve as the
triggers which may initiate the magnetic reconnection similar to our
discussions in sections in sections IV.3 and IV.4 in context of the solar
flare physics. Our arguments are based on estimation of different parameters
such as $\beta$ and the Mach number $M$ for NS environment.

In what follows it is convenient to parametrize the velocity of an AQN when it
enters the NS atmosphere in terms of the proper $\eta_{\mu}$ velocity and
4-momentum $p_{\mu}$ defined in the usual way:

$\eta_{\mu}=\gamma(c,\vec{v}),~{}~{}~{}\gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}},~{}~{}~{}p_{\mu}=M_{N}\cdot\eta_{\mu},$
(22)

where $M_{N}\simeq m_{p}B$ is the AQN’s rest mass expressed in terms of the
proton mass as reviewed in section III.1.

The key observation here is that the Mach number $M\gg 1$ is always very large
for a typical AQNs entering the NS atmosphere,

$\displaystyle M$ $\displaystyle\equiv$
$\displaystyle\frac{\sqrt{v^{2}_{\perp}+v^{2}_{\parallel}}}{c_{s}}$
$\displaystyle\simeq$ $\displaystyle 6\cdot
10^{3}\cdot\left({\frac{{10^{5}K}}{{\rm}T}}\right)^{1/2}\cdot\left(\frac{\rm
10^{6}~{}cm}{R_{\rm NS}}\right)^{1/2}\cdot\left(\frac{M_{\rm
NS}}{2~{}M_{\odot}}\right)^{1/2}$

is much larger than one. The shock wave may initiate a large event similar to
solar flares considered in sections IV.3 and IV.4.

Another important parameter which determines importance of the magnetic
pressure in comparison with kinetic pressure is dimensionless parameter
$\beta$:

$\displaystyle\beta\equiv\frac{8\pi p}{{\cal{B}}^{2}}\sim
10^{-18}\left(\frac{n}{10^{10}~{}{\rm
cm^{-3}}}\right)\left(\frac{T}{10^{6}K}\right)\left(\frac{10^{10}~{}G}{{\cal{B}}}\right)^{2}~{}~{}~{}~{}~{}$
(24)

where for numerical estimates we use typical parameters for NS. The crucial
difference with the solar corona here is that $\beta\ll 1$ is very small
everywhere on the NS surface. It is not the case for the solar corona, where
$\beta\ll 1$ only in active regions, while $\beta\gg 1$ in quiet regions of
the solar surface. As a result, the solar flares occur only in active regions
with $\beta\ll 1$ when magnetic reconnection could in principle take place as
discussed in sections IV.3 and IV.4, while in NS the magnetic reconnection
could occur everywhere at any given moment, and may occupy entire NS surface
as condition $\beta\ll 1$ holds everywhere.

One more parameter which characterizes the NS atmosphere is the electron
number density $n_{e}$ where the AQN-induced shock wave may propagate. The
uncompensated charge density of electrons and ions in the NS atmosphere is not
vanishing due to the so-called Goldreich-Julian (GJ) effect when the magnetic
field spinning through a very good conductor produces the electric field which
separates the charges. To be more precise the GJ number density is
proportional to $n_{GJ}\propto{\mathbf{\Omega}}\cdot{\cal{B}}$. Numerically,
it can be estimated as follows, see e.g. [53]:

$n_{e}\approx\frac{2{\cal{B}}}{ecP}\frac{R_{\rm
NS}^{3}}{r^{3}}\sim\frac{10^{10}}{\rm
cm^{3}}\left(\frac{1s}{P}\right)\left(\frac{{\cal{B}}}{10^{10}G}\right)\left(\frac{R_{\rm
NS}^{3}}{r^{3}}\right),$ (25)

where $P$ is the pulsar period, and $r$ distance from the centre of the star.

If we assume that magnetic reconnection indeed occurs as a result of the AQN
triggering event, what could be the energy injection rate in this case? Is it
sufficient to heat the old NS to the high temperatures such as
$T\sim(10^{5}-10^{6})K$ as observed? The corresponding energy is determined by
the energy of magnetic field as a result of successful reconnection. This
injected energy is dramatically larger in comparison with the energy released
due to the direct annihilation events of the AQNs in the NS atmosphere
discussed in previous subsection V.1. The corresponding estimates will be
presented in the next section.

##  VI Magnetic Reconnection as the heating source of NS

We are now prepared to present our order of magnitude estimates to argue that
the heat being released as a result of the AQN-induced magnetic reconnection
events is sufficient to heat the old NS to high temperature $T\sim 10^{5}K$ as
observed. First, we estimate the total hitting rate $\dot{N}$ of NS by AQNs.
It can be estimated by dividing formulae (4) to $M_{N}$, i.e.

$\frac{dN}{dt}\approx\pi b^{2}_{\infty}v_{\infty}\left(\frac{\rho_{\rm
DM}}{M_{N}}\right)\approx\frac{10}{\rm s}\left(\frac{10^{25}}{\langle
B\rangle}\right)\cdot\left(\frac{M_{\rm NS}}{2~{}M_{\odot}}\right),$ (26)

where $\rho_{\rm DM}$ is the local DM density in the vicinity of the NS and
$M_{N}\approx m_{p}B$ is the mass of the AQN, see table 1.

Our next task is to estimate the minimal required energy to heat the NS
surface to the temperature $T\approx 10^{5}K$. The required energy rate to be
injected to heat the NS’s surface has been estimated previously in (6) and is
given by

$L=4\pi R^{2}\sigma T_{s}^{4}\approx 10^{29}\left(\frac{T_{s}}{10^{5}\rm
K}\right)^{4}\rm\frac{erg}{s}.$ (27)

The next task is to estimate the total available magnetic energy above the
NS’s surface. After that one can estimate the portion of the energy which
should be converted to the heat to equalize the radiation loss (27). The total
magnetic energy $E^{\rm tot}_{\rm mag}(A)$ above the NS’s surface can be
estimated as follows

$\displaystyle E^{\rm tot}_{\rm mag}\simeq\frac{1}{8\pi}\int_{r\geq R_{\rm
NS}}d^{3}x{\cal{B}}^{2}\simeq 10^{36}{\rm erg}\left(\frac{{\cal{B}}_{\rm
surf}}{10^{10}~{}{\rm G}}\right)^{2},$ (28)

where we used a simple dipole formula ${\cal{B}}\simeq{\cal{B}}_{\rm
surf}{R_{\rm NS}^{3}}{r^{-3}}$ for the estimate.

To make further progress with our computations we assume that every hit by the
AQN of the NS triggers and initiates a shock wave which consequently generates
the magnetic reconnection. This assumption is very reasonable as a similar
assumption for solar flare gives very reasonable estimates for the rate and
strength of solar flares as reviewed in sections IV.3 and IV.4. Indeed, in
both cases the relevant parameters $M\gg 1$ and $\beta\ll 1$ and the shock
waves are very likely to be formed. The difference between the Sun and NS is,
of course, that flares in the Sun can be generated in active regions only
(which accounts for very tiny portion of the solar surface), while in NS the
shock wave may develop anywhere on the entirely NS’s surface.

With this assumption we introduce parameter $\epsilon\ll 1$ which describes a
small portion of the total magnetic energy (28) which will be converted to the
heat after AQN struck the NS and triggers the shock wave leading to the
magnetic reconnection with consequent heating the surface, i.e $E_{\rm
heat}\equiv\epsilon E^{\rm tot}_{\rm mag}$. The corresponding parameter
$\epsilon$ is estimated from the following condition:

$\epsilon E^{\rm tot}_{\rm
mag}\cdot\frac{dN}{dt}=L\;\;\;\;\Rightarrow\;\;\;\;\epsilon\approx\frac{L}{\dot{N}E^{\rm
tot}_{\rm mag}},$ (29)

where parameters $\dot{N}$, and $L$ are given by (26) and (27)
correspondingly. Numerically, parameter $\epsilon$ can be estimated as
follows:

$\epsilon\equiv\frac{E_{\rm heat}}{E^{\rm tot}_{\rm
mag}},\;\;\;\epsilon\approx 10^{-8}\left(\frac{T_{s}}{10^{5}\rm
K}\right)^{4}\left(\frac{10^{10}~{}{\rm G}}{{\cal{B}}_{\rm
surf}}\right)^{2}\left(\frac{\langle B\rangle}{10^{25}}\right),$ (30)

which implies that the total magnetic energy is more than sufficient to heat
old NS to explain the puzzling observations reviewed in Sect.VI. In fact, only
very tiny portion of the magnetic energy ($\sim 10^{-8}$) needs to be
converted to the heat at each event of the magnetic reconnection triggered by
the AQN. In what follows we shall argue that this condition can be indeed
naturally satisfied.

At this point one could wonder what went wrong with the old (and naively, very
generic) argument from Sect. II, suggesting that magnetic field cannot play
any role in heating of old NS according to (2)? The answer is as follows.
There is a new element which was completely missing in old estimates (2). It
is the relevant time scale which enters (26) and which is dramatically
different from time scale entering the old estimate (2). Furthermore, as we
argue below in Sect. VII the energy which is powering the magnetic
reconnection is related to the magnetic helicity $\cal{H}$, see Appendix A for
definition and basic features of the magnetic helicity $\cal{H}$. This portion
of the magnetic energy could be quickly restored after every AQN-induced event
of reconnection with rate (26). A possible mechanism for such “refill” of the
magnetic helicity is discussed in items 10, 11 in next Sect. VII and in
Appendix A.

##  VII Magnetic Reconnection in NS. Basic ingredients

In this section we want to formulate the basic ingredients of the proposal
supporting our main result formulated in previous section. It suggests that
the magnetic reconnection triggered by the AQN may indeed heat the old NS. In
this sense we suggest an alternative mechanism, which we claim, is capable to
generate enough energy to heat the old NSs to explain the puzzling
observations listed in Sect. II.

First of all, we would like to notice that the computation of the parameter
$\epsilon$ defined by (30) from the first principles is not feasible at this
point due to very complicated dynamics of the strongly coupled system AQN-NS.
In particular, it includes the evolution of the shock waves, developing of the
turbulence and many other accompanied phenomena when the energy transfer
occurs from a body moving with enormous Mach number as estimated by (V.2).

Nevertheless, there are many systems where the magnetic reconnection is known
to occur and is believed to power very energetic events such as solar flares
discussed in previous sections IV.3 and IV.4. One can use the corresponding
observations to support or refute some of the assumptions on dynamical
features of the magnetic reconnection in NS based on experience with the solar
flare events.

Another system where magnetic reconnection is believed to play a crucial role
(see e.g. [53]) is the magnetars where the so-called Fast Radio Bursts (FRB)
are erupted as a result of the magnetic reconnection, see recent review on the
topic [54]. While the idea that FRBs are powered by the magnetic field
transferring an enormous energy to the radio emission is commonly accepted in
the community, the suggested triggering mechanisms which could initiate the
magnetic reconnection dramatically vary: from a crust cracking at the NS’s
surface to sudden triggers from an external event, see [54] for review.

Our proposal [50] that the DM particles in form of the AQNs play the role of
the triggers for FRBs is exactly from the last category when an external
object initiates the FRB. In our case the external object is the AQN. In many
respects our present proposal that the AQNs could be the triggers of the
magnetic reconnections in old NS which may heat the NS’s surface to explain
the puzzling observations as reviewed in Sect. VI, is very similar to proposal
[50] in context of FRBs.

There are many quantitative differences between these two cases, of course:
the magnetars from proposal [50] are much younger, have much stronger magnetic
field, have much higher surface temperature than the old NS which is the topic
of the present work. However, the basic fundamental concept in these two cases
is the same: the AQNs can serve as the triggers to ignite and initiate the
magnetic reconnection which may feed the very energetic explosions in both
cases.

The two systems mentioned above (flares in solar corona and FRB in magnetars)
will be considered as a tool box which allows us to test the main proposal of
the present work. Indeed, by comparing one or another assumption from the
present proposal with similar (analogous) studies of the flares in solar
corona or FRB in magnetars one can support or refute a corresponding
assumption. This is precisely the approach we are advocating in this section.

We present below the basic elements of our proposal, item by item. In many
items we explicitly point out some similarities between our present system and
previously considered systems –flares in solar corona and FRB in magnetars.
Therefore, our assumptions can be confronted with available observations.

1\. The basic conditions such as $M\gg 1$ and $\beta\ll 1$ are satisfied
according to (V.2) and (24), similar to analysis in active regions in the Sun
and FRB;

2\. As a result of these conditions the AQN may serve as the trigger to
initiate the magnetic reconnection as environment in all cases is also very
similar. Indeed, the density of highly ionized plasma estimated as (25) which
is close to the density of the solar corona;

3\. Therefore, we expect a strong shock wave generated by propagating AQN,
such that the pressure $p$ and temperature $T$ experience strong
discontinuities according to (IV.3). This very strong impulse in very small
and very localized area (determined by the AQN’s path) for very short period
of time $\tau$ may lead to a successful magnetic reconnection;

4\. A typical time scale $\tau$ where the shock wave develops is determined by
the velocity $v_{\rm AQN}$ of the AQN in vicinity of the NS surface, which is
close to $c$. Therefore

$\tau\sim\frac{d}{c}\sim\frac{10\rm~{}km}{c}\sim 0.3\cdot 10^{-4}\rm s,$ (31)

where distance $d\sim 10{\rm~{}km}$ is determined by the region where the
density of the ionized gas is sufficiently large (25);

5\. The time scale $\tau$ determines the size of the cone where the shock wave
develops and where the pressure $p$ and temperature $T$ experience strong
discontinuities, similar to the solar flare analysis (17);

6\. This AQN-induced shock will trigger the magnetic reconnection in the area
$A$ estimated as

$\displaystyle A\approx dc_{s}\tau\approx\frac{d^{2}}{M}\sim 10^{-4}d^{2},$
(32)

which represents a small portion $\sim(10^{-4}-10^{-5})$ of the NS’s surface.
This area has the same physical meaning as estimate (IV.4) for the solar flare
which defines the area for initial stage of the magnetic reconnection. The
difference with the solar flare is that the magnetic reconnection could only
occur in a small active region of the Sun where magnetic field is large and
condition $\beta\ll 1$ is satisfied, while the reconnection in NS may occur
anywhere on the surface as condition $\beta\ll 1$ is satisfied everywhere;

7\. Therefore, if magnetic reconnection starts in one location it may quickly
sweep out (potentially) entire NS’s surface. In this case the total energy of
the event in NS (second stage in classification (18) in context of solar
flares) represents a finite portion of the integral (28). It must be
contrasted with the solar flare estimate (18) which represents a very tiny
portion of the surface in comparison with total surface of the Sun (the so-
called active regions, the sunspots);

8\. The speed of magnetic reconnection is governed by dimensionless parameter
$\beta_{\rm in}$ in notations [53] and [50], where “in” in $\beta_{\rm in}$
stands for inflow speed. It must be sufficiently high for fast successful
reconnection, but it must be much slower than the speed of light $\beta_{\rm
in}\ll 1$. If we formulate this condition in terms of the reconnection typical
time scale $\tau_{\rm in}\equiv d\cdot\beta^{-1}_{\rm in}$ the following
hierarchy scales must be satisfied666Parameter $\beta_{\rm in}$ was estimated
in [53] in context of FRB physics. It determines the duration of the magnetic
reconnection as $\tau_{\rm FRB}\propto\beta_{\rm in}^{-1}\approx 10^{-3}$ s.
Parameter $\tau_{\rm FRB}$ plays the same role as $\tau_{\rm in}$ in (33).
Similar parameter (duration of the large flare) in context of the solar
physics could be as long as few hours, while the duration of the initial stage
of flare lasts about 10s, see (IV.4). In all cases the ratio between duration
of the trigger event (due to the AQN) and the magnetic reconnection itself is
about $\sim 1\%$.:

$\tau\ll\tau_{\rm in}\ll\dot{N}^{-1}\;\;\;\rightarrow\;\;\;\rm 0.3\cdot
10^{-4}s\ll\tau_{\rm in}\ll 0.1s,$ (33)

where parameter $\tau$ is determined by the AQN serving as a trigger, see
(31), while time scale $\dot{N}^{-1}$ is a typical time scale between two
independent consecutive events according to (26);

9\. The dynamics of magnetic reconnection studied in context of FRB suggests
that the energy powering FRB is the magnetic helicity ${\cal{H}}$, see
Appendix A for the definition and short introduction into the subject. We
assume that this feature on the dominant role of the ${\cal{H}}$ holds for the
present proposal as well. The basic argument for this assumption is that the
environments of NS and magnetars are very similar.

The magnetic reconnection implies that the electric field $E_{\parallel}$
during the reconnection time $\tau_{\rm in}$ will be induced, see [53, 50].
Its direction should be parallel to the original static magnetic field
$\cal{B}$ with the coefficient to be proportional to $\beta_{\rm in}$, i.e.

$E_{\parallel}(\rm induced)\propto\beta_{\rm in}\cal{B}.$ (34)

The condition (34) unambiguously implies that very specific $E\&M$
configuration with $\sim\vec{E}\cdot\vec{{\cal{B}}}\propto\beta_{\rm
in}{\cal{B}}^{2}$ must be generated during the reconnection. The main feature
of this configuration is that it enters the formula (42) which describes the
dissipation of the magnetic helicity ${\cal{H}}$. Precisely the dissipation of
the magnetic helicity ${\cal{H}}$ powers the magnetic reconnection. Assuming
$\beta_{\rm in}\sim 1\%$ (which is within the window (33)) one can infer that
every event of the magnetic reconnection will convert $\sim 1\%$ of its
magnetic helicity into the heat along with other radiation losses such as
X-rays. Such relatively high efficiency rate is obviously more than sufficient
to generate energy heating the NS surface according to (30) even if one
assumes that only a small finite portion of the surface (rather than entire
NS) will be experiencing the magnetic reconnection.

10\. It is very likely that there are some mechanisms which restore the energy
associated with magnetic helicity dissipation (due to the magnetic
reconnection) and restore its equilibrium value. Indeed, there are some
observations in FRB context suggesting that frequency of some FRB repeaters is
enormously high. For example, rFRB 20121102A emitted a total amount of energy
$\sim 3.4\cdot 10^{41}\rm erg$ in the radio band from 1652 bursts detected in
59.5 hours in a 47 day time span [55, 54]. Assuming that this activity
represents a typical behaviour of the FRB repeaters one could infer that there
must be a mechanism which restores the magnetic source of the energy777The
total energy emitted would exceed $6.4\cdot 10^{45}\rm erg$ assuming a radio
efficiency $10^{-4}$, which we consider is already too high. This is a
substantial fraction of the available magnetic energy from a magnetar [54].
This estimate again strongly suggests that there must be a mechanism restoring
the magnetic energy after the eruptions as example of the rFRB 20121102A
already poses significant energy budget issues [54]..

Based on this observation (in FRB context) we assume that there should be an
operational mechanism which restores the energy reservoir (28) by
equilibrating the system after the events of reconnection which heat the NS’s
surface;

11\. A possible mechanism of equilibration which could potentially restore the
magnetic energy had been discussed previously in a very different context in
[56]. Important conclusion of these studies was that the magnetic helicity
${\cal{H}}$ which is powering the magnetic reconnection (see item 9) is a very
generic feature of the NS, and in fact there are many observational evidences
suggesting that the magnetic helicity ${\cal{H}}$ must be present in NS. We
overview the basic results of these studies below and refer for the details to
the Appendix A;

12\. The basic argument presented in [56] is that the so-called topological
non-dissipating currents will be induced in the NS as a result of quantum
anomalies in high density QCD. These currents might be responsible for many
observed phenomena such as NS kicks, toroidal magnetic field, the magnetic
helicity, to name just a few. This is very generic and well known phenomenon
in QCD. It is related to the asymmetry between left handed and right handed
chemical potentials which could be generated by $\cal{P}$-odd weak
interactions. Formally, it could be expressed as generation of the so-called
axial chemical potential $\mu_{5}\equiv 1/2(\mu_{R}-\mu_{L})$.

The $\mu_{5}\neq 0$ in context of the heavy ion physics in QCD leads to a
number of $\cal{P}$ odd effects, such as chiral magnetic effect, chiral
vortical effect, charge separation effect, to name just a few. This field of
research initiated in [57] became a hot topic in recent years as a result of
many interesting theoretical and experimental advances, see recent review
papers [58, 59] on the subject.

In context of the present work of NS physics, one should mention that there is
a strong observational evidence, see e.g.[60] and references therein,
supporting the presence of the toroidal magnetic field which unambiguously
suggests that the magnetic helicity $\mathcal{H}$ must be non-zero in neutron
stars. We consider this as an indirect observational evidence supporting claim
that $\cal{P}$-odd topological currents [56] had been induced at some early
moment in the star’s evolution.

13\. If one assumes that the magnetic helicity $\mathcal{H}$ is present in the
system at the moment of magnetic reconnection, it is expected from (34) that
the energy feeding the magnetic reconnection comes from the magnetic helicity
${\cal{H}}$, which is directly related to the toroidal magnetic field in the
system, see Appendix A for additional comments. One should emphasize that the
toroidal magnetic field is generated by non-dissipating topological current
(47), in contrast with typical dipole type field (28) which is generated by
usual dissipating currents satisfying the conventional Ohm’s law (46) in the
core of the NS.

14\. In order to understand what happens when the magnetic reconnection event
occurs, one should recall that the NS system (which includes the dipole type
magnetic field, the toroidal magnetic field, the magnetic helicity
${\cal{H}}$, and the non-dissipating topological current with many other
conventional components) is a very complicated dynamical self-interacting and
self-equilibrating system. It implies that if some elements of the system
suddenly get changed the other elements of the system will adjust their values
to restore the equilibrium of the system.

In context of the reconnection events it implies that the removing (due to
reconnection) some value of the magnetic helicity ${\cal{H}}$ will result in
modification of the currents and chemical potential $\mu_{5}$ to restore its
equilibrium values. This equilibration is formally expressed by (45).

The reservoir of the chemical potential $\mu_{5}$ is truly enormous as
estimate (51) suggests. Therefore, we propose that the mechanism of
equilibrating the magnetic helicity $\cal{H}$ as suggested above, in principle
is capable to restore the energy associated with magnetic helicity after the
events of reconnection, see Appendix A with additional comments and
clarifications.

##  VIII X-rays as the indicators of the magnetic reconnection in NS

First, we start by mentioning that it has been known for quite some time that
the solar flares are normally accompanied by strong X ray radiation, see e.g.
[52] for review of the last complete solar cycle No -24. Furthermore, the
X-rays are considered to be a good indicator for large flares because the X
ray intensity dramatically increases few moments before a flare starts.

In addition to that it has been point out in [52] that the solar pre-flare
enhancement in form of the X rays propagates from higher levels of corona into
lower corona and chromosphere, see Fig. 8 in [52]. This pre-flare enhancement
is very puzzling and unexpected phenomenon as pre-flare propagates from top to
bottom. Nevertheless, this unusual temporal and spatial patterns of
propagation have very natural explanation within AQN framework (reviewed in
Sect. IV.4 in context of the solar flares) because the AQNs propagate (and
ignite the magnetic reconnection) from outer space to the surface.

Now we return to our main topic of the NS. The leitmotiv of the proposal
advocated in this work is that two naively very different phenomena (in
dramatically different systems):
1\. the heating mechanism of relatively old NS
2\. the solar flares
are in fact very similar as they both powered by the same mechanism of the
magnetic reconnection (according to the proposal) triggered and initiated by
the dark matter AQN particles. Therefore, we expect that a number of
accompanying effects associated with magnetic reconnections must also manifest
themselves in very similar ways in both cases. One of such profound
accompanying effect of large solar flares is the X-ray emission, see first
paragraph of this section. Therefore, we expect that a similar X-ray emission
must be also present in NS if the NS heating is indeed powered by magnetic
reconnection as advocated in this work.

The main goal of this section is to estimate the X ray intensity from NS by
using (for normalization) the observed and well recorded intensity of the
X-ray radiation during the solar flares. In this estimate we use the same
logical steps as in previous sections V\- VII by comparing one and the same
phenomenon in two different systems.

###  VIII.1 X rays from solar flare as normalization point

The starting point of our estimates is as follows. The peak of the X rays in
the band $0.1-0.8~{}\rm nm$ of the flare of August 9. 2011 is recorded as
follows, see Fig 8 in ref.[52]:

$\displaystyle F^{\odot}_{X}$ $\displaystyle\approx$ $\displaystyle
10^{-3}\rm\frac{W}{m^{2}}\approx\frac{erg}{s\cdot cm^{2}},$ (35)
$\displaystyle E$ $\displaystyle\in$ $\displaystyle(1.5-12.4)~{}\rm keV,$

while the average $\langle F^{\odot}_{\rm X}\rangle$ of the X ray emission
during the flare can be estimated as $\langle F^{\odot}_{\rm X}\rangle\approx
10^{-2}~{}\rm erg\cdot s^{-1}\cdot cm^{-2}$ which is almost two orders of
magnitude smaller than the peak value (35). At the same time the total flux
from the flare emitted from all frequency bands can be estimated as

$\displaystyle F^{\odot}_{\rm tot}$ $\displaystyle\approx$ $\displaystyle
10\left(\frac{E_{\rm tot}}{10^{32}\rm erg}\right){\rm\frac{erg}{s\cdot
cm^{2}}},$ $\displaystyle\xi$ $\displaystyle\equiv$
$\displaystyle\frac{\langle F^{\odot}_{\rm X}\rangle}{F^{\odot}_{\rm
tot}}\approx(10^{-2}-10^{-3})$ (36)

assuming that the flare lasts about $\rm 1h$ and the total released energy
$E_{\rm tot}$ during a large flare was between $10^{31}\rm erg$ and
$10^{32}\rm erg$. The order of magnitude estimate (VIII.1) suggests that the X
ray flux represents less than 1% portion of a large flare. We use this ratio
$\xi$ to estimate the X ray flux from NS due to the magnetic reconnection in
next subsection.

###  VIII.2 X rays from NS originated from magnetic reconnection

In our estimates which follows we assume that the observed temperatures of
sufficiently old NS are entirely saturated by the magnetic reconnection
mechanism as advocated in this work. This is obviously a strong assumption.
However, as reviewed in Sect. II any common mechanisms such as roto-chemical
heating cannot explain many observations, see Sect.II.0.1. In particular, the
so-called Magnificent Seven stars cannot be explained by this mechanism with
reasonable changes of the parameters, see Fig. 3 in [21]. There are many
similar cases when the observed temperature of NS dramatically exceeds the
theoretical estimates and inconsistent with canonical picture of cooling,
including additional mechanisms reviewed in Sect. II.

With this assumption in mind and assuming the black-body radiation spectrum we
estimate the total flux of E&M radiation (powered by magnetic reconnections,
as presented in previous Sect. VII) as follows:

$\displaystyle F^{\rm NS}_{\rm tot}$ $\displaystyle=$
$\displaystyle\frac{L}{4\pi r^{2}}\sim\frac{7\cdot 10^{32}}{4\pi
r^{2}}\left(\frac{T_{s}}{10^{6}\rm K}\right)^{4}\rm\frac{erg}{s}$
$\displaystyle\sim$ $\displaystyle 6\cdot 10^{-10}\left(\frac{T_{s}}{10^{6}\rm
K}\right)^{4}\left(\frac{0.1\rm kpc}{r}\right)^{2}{\rm\frac{erg}{s\cdot
cm^{2}}}$

where $r$ is the distance to NS, while the luminosity $L$ is estimated in (6).

We need two more elements to complete our estimate for $\langle F^{\rm
NS}_{\rm X}\rangle$ analogous to $\langle F^{\odot}_{\rm X}\rangle$ entering
(VIII.1). First, we assume that the ratio $\xi\sim 10^{-3}$ defined by
(VIII.1) is the same for the solar flares and NS because (according to our
proposal) two different phenomena are originated from the same physics of the
magnetic reconnection as argued above. Another element which also important
for our estimate of $\langle F^{\rm NS}_{\rm X}\rangle$ is the ratio of two
different time scales: first, the $\tau_{\rm in}$ is reconnection time scale,
which represents the duration of the magnetic reconnection, while
$\dot{N}^{-1}$ is a typical time scale between two independent consecutive
events as defined in (33). As a result, we arrive to the following estimate
for the X ray flux from NS due to the magnetic reconnection888The time scale
$\tau_{\rm in}$ enters the formula (VIII.2) because X rays can be emitted
exclusively during the reconnection period according to our proposal, similar
to discussions of the X ray radiation during the solar flare from Sect.VIII.1.
At the same time the $\tau_{\rm in}$ is normalized by a typical time scale
$\dot{N}^{-1}$, which represents a time scale before a next AQN hits the NS
when new reconnection starts and new portion of the energy is injected into
the NS’s atmosphere. We assume that the energy will be eventually thermalized,
which justifies our formula for luminosity (VIII.2) with black body radiation
spectrum.:

$\displaystyle\langle F^{\rm NS}_{\rm X}\rangle$ $\displaystyle\sim$
$\displaystyle\xi\cdot\left(\frac{\tau_{\rm in}}{\dot{N}^{-1}}\right)\cdot
F^{\rm NS}_{\rm tot}\sim(10^{-4}-10^{-5})F^{\rm NS}_{\rm tot}~{}~{}~{}$
$\displaystyle\sim$
$\displaystyle(10^{-13}-10^{-14})\left(\frac{T_{s}}{10^{6}\rm
K}\right)^{4}\left(\frac{0.1\rm kpc}{r}\right)^{2}{\rm\frac{erg}{s\cdot
cm^{2}}},$

where for the numerical estimate we use $\tau_{\rm in}\sim 10^{-3}\rm s$
(motivated by the FRB analysis) and $\dot{N}^{-1}\simeq 0.1\rm s$, see
footnote 6 with some comments.

One should emphasize that this is really an order of magnitude estimate- it is
very hard to improve it as the estimate (VIII.2) includes large number of
elements with unknown physics. Furthermore, many NS are known to be emitters
of the X rays due to many other reasons (for example, due to the conversion of
the spin-down power into the X rays). Therefore, our estimation (VIII.2)
should be considered as an additional contribution to the X ray emission. It
obviously implies that it is very hard to discriminate the X ray emission flux
as given by (VIII.2) from many other canonical astrophysical mechanisms.
However, there are known special cases when conventional astrophysical
mechanisms produce very tiny contribution to the X ray radiation, in which
case (VIII.2) could play the dominant role, and in principle could be
discriminated from other mechanisms, see example below.

It is instructive to present a numerical value of the flux $\langle F^{\rm
J1856}_{\rm X}\rangle$ for $J1856$ from Magnificent Seven (M7) stars which is
the closest to Earth. This NS belongs to the category when the star’s
temperature greatly exceeds an anticipated value as reviewed in Sect. II.0.1.
The corresponding flux $\langle F^{\rm J1856}_{\rm X}\rangle$ is estimated
from (VIII.2) as:

$\langle F^{\rm J1856}_{\rm X}\rangle\approx(4\cdot 10^{-15}-4\cdot
10^{-16})\rm\frac{erg}{s\cdot cm^{2}},[\rm estimation]$ (39)

where we use $r\approx 0.123~{}\rm kpc$ and $T_{s}\approx 0.5\cdot 10^{6}K$
for the numerical estimates. It is quite remarkable that the X ray emission
has indeed been observed [61] from this star with $5\sigma$ excess in
$(2-8)\rm~{}keV$ energy range with result:

$\displaystyle\langle F^{\rm J1856}_{\rm X}\rangle=(1.5^{+0.7}_{-0.6})\cdot
10^{-15}\rm\frac{erg}{s\cdot cm^{2}},\;\;\;[\rm observations].\;\;\;$ (40)

According to [61] it is very hard to explain this excess of X ray radiation by
any conventional astrophysical sources or any systematic effects. This
observation nicely falls into the interval (39). We consider this result as a
highly nontrivial consistency check for the application of the AQN framework
to NS heating problem as the parameters entering the estimate (VIII.2) are
based on dramatically different physics describing enormous range of scales in
drastically different contexts, from DM distribution to solar physics. All the
corresponding parameters had been fixed long ago in application to different
systems, without any attempt to modify or fit them to match the present
observations. Therefore, (VIII.2) could be potentially many orders of
magnitude off from the observed value (40).

In many aspects the similarity of the numerical values between (39) and (40)
is analogous to similarity between the observed solar corona luminosity
$L_{\rm corona}\sim 10^{27}{\rm erg~{}s^{-1}}$ in EUV and the AQN-induced
luminosity (11) which is entirely determined by the DM parameters, not related
to solar corona. Therefore, we think it is very hard to interpret the
numerical agreement between (39) and (40) as simply an “accidental numerical
coincidence”. We think it should be interpreted, similar to mysterious solar
corona EUV radiation, as a result of some deep roots and connections between
DM physics and NS physics, which is naturally incorporated by the AQN
framework.

##  IX Concluding comments and Future Developments

The presence of the antimatter nuggets in the system implies, as reviewed in
Sect.III, that there will be annihilation events (and continues injection of
energy at different frequency bands) leading to large number of observable
effects on different scales: from Early Universe to the galactic scales to the
Sun and the terrestrial rare events.

In the present work we focus on manifestations of these annihilation events on
physics of the NS. We proposed that DM in form of the AQNs may serve as the
triggers igniting the large explosive events powered by the magnetic
reconnection. The released energy as a result of these events may serve as the
heater of NS as suggested in Sect. VI. This is precisely the additional source
of energy which may resolve the mysterious and puzzling observations as
reviewed in Sect. II when the NS temperature is inconsistent with canonical
cooling mechanisms.

We do not need to repeat the arguments presented in Sect. VII item by item in
this Conclusion. Instead, we want to mention below several phenomena which
should accompany the proposed mechanism when the magnetic reconnection events
triggered by AQNs heat the old NS. As such these additional emissions should
be considered as possible tests and predictions for proposed mechanism of
heating.

In Sect. IX.1 we list some possible new tests which can substantiate or refute
our proposal. Finally, in Sect. IX.2 we describe several other mysterious and
puzzling observations (outside the NS system), which can be understood within
the same AQN framework. We consider this as indirect support for our proposal
as the computations are based on the same set of parameters of the AQN model
reviewed in Sect. III.

###  IX.1 Possible tests of the proposal

As mentioned in Sect. VIII there will be radiation in very broad frequency
bands (from radio to hard X rays, and likely to gamma rays) which always
accompany the heating mechanism suggested in sections VI and VII. There is
enormous energy reservoir (51) which could be converted to the magnetic
helicity and eventually to the heat and $E\&M$ radiation.

We specifically focus on hard X ray radiation in Sect. VIII because we
consider this frequency band is the most promising channel where this heating
mechanism can be directly tested. Another possible radiation in the radio
frequency bands which also accompanies the magnetic reconnection is expected
to be less promising as the estimates in Appendix B imply.

Essentially we suggest to study the hard X ray emission from other M7 stars as
estimation (VIII.2) applies to all of them. As we mentioned in Sect. II all M7
stars have the temperatures which greatly exceed the expectations. We
interpret this inconsistency as the presence of the additional heating
mechanism in form of the magnetic reconnection for all M7 stars. Therefore we
predict that all M7 stars should emit the hard X rays which always accompany
the reconnection with flux being estimated in (VIII.2). As mentioned in [61]
the observation of the hard X ray from $J1856$ with $5\sigma$ excess (and not
observations of a similar signal in other M7 stars) could be related to the
fact that $J1856$ has the most exposure time across all of the X-ray cameras
that were considered in [61].

Another possible class of NS where proposed heating mechanism could manifest
itself is represented by vey old pulsars. As mentioned in Sect. II the
observed temperatures (well exceeding $T_{s}\gtrsim 10^{5}$ K) of many old
pulsars with $t\gtrsim 10^{8}\rm yr$ cannot be explained by conventional
mechanisms, see review [21] with details. At the same time, there are many
cases where such high (and even much higher) surface temperatures have been
observed. As an illustrative sample from this class one could consider PSR
J0108-1431which is a nearby ($r\simeq 0.13~{}\rm kpc$), 170 Myr old pulsar.
Its surface temperature is known to be very high: $(1-5)\cdot 10^{5}K$ and it
is hard to explain even with additional heating mechanisms [21].

It is also interesting to note that this pulsar is observed in X ray with flux
$\langle F^{\rm J0108}_{\rm X}\rangle=(9\pm 2)\cdot
10^{-15}\rm\frac{erg}{s\cdot cm^{2}}$ in the $(0.3-8)~{}\rm keV$ band. It is
very similar in value to (40) observed in $(2-8)~{}\rm keV$ band, if one
excludes the soft X ray segment from $\langle F^{\rm J0108}_{\rm X}\rangle$
representing its dominant portion as the corresponding spectrum has a power
law with index $\gamma\simeq 2.2$, see [62]. This similarity between two
different cases is consistent with our formula (VIII.2) as the temperatures
and distances for both NS are almost the same.

One should note that in the original paper [62] the relatively large flux
$\langle F^{\rm J0108}_{\rm X}\rangle$ was entirely attributed to spin-down
mechanism with enormously high X ray efficiency $\eta_{X}\simeq 0.4\cdot
10^{-2}$, while for typical younger pulsars similar X ray efficiency is two
orders of magnitude smaller: $\eta_{X}\simeq(10^{-5}-10^{-4})$. In our view it
is very hard to justify such dramatic increase of the X ray efficiency for
older pulsars from the theoretical side. We more incline to interpret a
sufficiently high X ray flux $\langle F^{\rm J0108}_{\rm X}\rangle$ as a
manifestation of the magnetic reconnection which powers the heating of this
old pulsar, and accompanied to this heating the X ray emission as estimated in
Sect.VIII.

Another possible test of the proposed heating mechanism is a study of thermal
pattern on the NS’s surface. The main observation here is that the magnetic
reconnection is powered by magnetic helicity $\cal{H}$ (on large and small
scales). Consequently, the toroidal magnetic field is expected to play an
essential role in the dynamics and heating of the NS as discussed in Sect.
VII. It implies that the thermal pattern on the NS’s surface must be very
different from canonical poloidal dipolar magnetic field (in which case the
cold region is always localized along the equator while the hot regions are
always localized around the poles). The same conclusion also follows from an
observation that the magnetic field during the solar flares in active regions
always demonstrates very complex topological structure as mentioned in Sect.
IV.4. Therefore, an observation of any correlations between the complex
thermal pattern, higher than expected temperature of a NS, and the excess of
the hard X ray could (implicitly) support a proposed heating mechanism. In
fact, some recent studies apparently indicate, see review [63] for references,
that the thermal patterns of the NS very often display a complicated
structure, dramatically different from canonical poloidal dipolar magnetic
field pattern.

###  IX.2 Other (indirect) evidences for DM in form of the AQN

There are many hints (outside the NS physics which represents the topic of the
present work) suggesting that the annihilation events, which is inevitable
feature of this framework, may indeed took place in early Universe as well as
in present epoch at very different scales. In particular, in early Universe
the AQNs do not affect BBN production for H and He, but might be responsible
for a resolution of the “Primordial Lithium Puzzle” due to its large electric
charge $Z=3$, see [30] for the details.

The very same interaction of the visible-DM components may lead to many
observable effects during the galaxy formation epoch. Indeed, while Cold Dark
Matter model works very well on large scales, a number of discrepancies have
arisen between numerical simulations and observations on sub-galactic scales,
see e.g. recent review [64] and references on original papers therein. Such
discrepancies have stimulated numerous alternative proposals including, e.g.
Self-Interacting dark matter, Self-Annihilating dark matter, Decaying dark
matter, and many others, see [64] and references therein. Our comment here is
that the AQN model represents a specific example of a strongly interacting
chameleon-like model: the AQNs do not interact with the surrounding material
in dilute environment, but strongly interact with baryonic material in
sufficiently dense environment at the galactic scale, which helps to resolve
many observed discrepancies during the structure formation epoch [65].

The very same interaction of the visible-DM components may lead to large
number of observable effects also at present epoch. In particular, the recent
studies [66, 67, 68] suggest that there is a strong component of the diffuse
far-ultraviolet (FUV) background which is very hard to explain by conventional
physics in terms of the dust-scattered starlight. As argued in [69] the
mysterious and puzzling observations of the diffuse FUV could be directly
related to the annihilation events of the AQNs propagating in the galactic
media. There are numerous similar examples in many frequency bands (from radio
to optical bands to UV to X rays) when the observations require an additional
energy injection into the system. The AQN annihilation events may provide this
required (by observations) an additional source of radiation.

We conclude this work with the following final comment. We advocate the idea
that study of specific features of NS as mentioned in Sect.IX.1 could shed
some light on the nature of DM. It is very unexpected turn of our studies as
it allows (implicitly) study the nature of the DM by analyzing some subtle
features of the NS.

The new paradigm on the nature of cold DM (when it represented in form of
macroscopical large objects as reviewed in Sect. III instead of commonly
accepted WIMPs) has many consequences which are mentioned above, and which are
consistent with all presently available cosmological, astrophysical, satellite
and ground-based observations. In fact, it may even shed some light on the
long standing puzzles and mysteries (outside of the NS physics) as mentioned
above and in Sect. IV.

## Acknowledgements

The motivation for this work emerged as a result of discussions with Ben Safdi
during the conference “Axions across boundaries between Particle Physics,
Astrophysics, Cosmology and forefront Detection Technologies” which took place
at the Galileo Galilei Institute in Florence, June 2023. I am thankful to him
for the detail explanations of analysis carried out in studies for the hard X
ray excess, which was the topic of Sect. VIII.2.

This research was supported in part by the Natural Sciences and Engineering
Research Council of Canada.

##  Appendix A Magnetic Helicity ${\cal{H}}$ and its role in magnetic
reconnection

The main goal of this Appendix is to overview some important results on the
magnetic helicity which is a topological invariant, and represents the
observable which characterizes the dynamics of the magnetic reconnection.
Needless to say that the magnetic reconnection, according to our proposal,
plays the crucial role in transforming the static magnetic energy into the
flare type events (similar to the solar flares) which could heat the old NS.

There are several elements which we would like to overview in this Appendix.
First of all, we would like to explain why magnetic helicity ${\cal{H}}$ plays
a key role in reconnection. Secondly, we want to argue that the magnetic
energy related to the helicity ${\cal{H}}$ could be potentially restored as a
result of coupling of the helicity ${\cal{H}}$ with a reservoir of the
chirality which could be generated when the NS was sufficiently hot. Finally,
we also want to argue that the chirality reservoir is truly enormous in NS.

We start with definition999In particle physics literature the magnetic
helicity is defined with additional coefficient $e^{2}/(4\pi^{2})$ in front of
the integral (41). This normalization factor becomes obvious from eq.(45). of
the magnetic helicity in volume $V$ which can be represented as follows [70]

${\cal{H}}\equiv\int_{V}\vec{\cal{A}}\cdot\vec{{\cal{B}}}~{}dV,$ (41)

where $\vec{\cal{A}}$ is the vector potential corresponding to the magnetic
field $\vec{{\cal{B}}}=\vec{\nabla}\times\vec{\cal{A}}$. It is known that the
magnetic helicity ${\cal{H}}$ in general is not a gauge invariant observable
because the gauge potential $\vec{\cal{A}}$ is not a gauge invariant object.
However, if one requires that the magnetic field is tangent on the surface
boundary $\partial V$ of $V$, i.e. $\vec{{\cal{B}}}\cdot\vec{n}|_{\partial
V}=0$, the magnetic helicity becomes well defined gauge invariant object, see
e.g. [70].

In simplest case when the magnetic configuration can be represented in form of
two interlinked (but not overlapping) tubes with fluxes $\Phi_{1}$ and
$\Phi_{2}$, the magnetic helicity ${\cal{H}}$ counts its linking number, i.e.
${\cal{H}}=2\Phi_{1}\Phi_{2}$ is proportional to an integer linking number if
fluxes $\Phi_{1}$ and $\Phi_{2}$ are quantized. This is precisely the reason
why the magnetic helicity is the topological invariant and cannot be easily
changed during its evolution. In fact, the crucial property of the magnetic
helicity ${\cal{H}}$ is that it is exactly conserved during the time evolution
in ideal MHD [70]. It is also known that the magnetic helicity ${\cal{H}}$ is
odd under the $\cal{P}$ symmetry corresponding to :
$\vec{x}\rightarrow-\vec{x}$ transformations. This implies that the magnetic
helicity can be only induced if there are $\cal{P}$ violating processes
producing a large coherent effect on macroscopic scales. We refer to one of
the proposals [56] with specific estimates on how it could happen.

In what follows we also need the expression for the temporal variation of
magnetic helicity as it is directly related to the dissipation rate.
Differentiating of eq. (41) one arrives to

$\frac{d{\cal{H}}}{dt}=-2\int_{V}\vec{E}\cdot\vec{{\cal{B}}}~{}dV$ (42)

where we ignored the surface boundary term, see e.g. [71],[72] with explicit
derivations. A key observation here is that the dissipation term in (42) is
proportional to $\sim\vec{E}\cdot\vec{{\cal{B}}}$ which is precisely the
$E\&M$ configuration which emerges as a result of the magnetic reconnection as
formula (34) states. As explained in the text the induced electric field
parallel to the original static magnetic field is absolutely required feature
for the successful magnetic reconnection. The relation (42) answers the
question of why the magnetic helicity ${\cal{H}}$ is the key player of the
reconnection101010Important observation here is that the integrand entering
(42) which describes the dissipation of the magnetic helicity ${\cal{H}}$
identically vanishes in ideal MHD where
$\vec{E}=-\vec{v}\times\vec{{\cal{B}}}$. .

There is one more important element here on relation between the magnetic
helicity and the chirality which needs to be explained. In the chiral limit
the axial current is not conserved as a result of quantum anomaly, see e.g.
review papers [58, 59] on the subject, i.e.

$\partial_{\mu}J^{\mu}_{5}=\frac{e^{2}}{2\pi^{2}}\vec{E}\cdot\vec{{\cal{B}}},$
(43)

where $J^{\mu}_{5}$ is the density of the axial current. In the integral form
the same equation can be written as follows

$\frac{dQ_{5}}{dt}=2\int_{V}\frac{e^{2}}{4\pi^{2}}\vec{E}\cdot\vec{{\cal{B}}}dV\;\;{\rm
where}\;\;Q_{5}\equiv\int_{V}J^{0}_{5}dV,$ (44)

where the surface term has been ignored. Comparing (42) and (44) one arrives
to the following result, see e.g. review papers [58, 59] on the subject

$\frac{d}{dt}\left(Q_{5}+\frac{e^{2}}{4\pi^{2}}{\cal{H}}\right)=0,$ (45)

which implies that the magnetic helicity (41) in combination with the axial
charge (44) becomes a conserved quantity, while they are not conserved
separately. Important element here is that the axial current is the
combination of right handed and left handed currents, i.e.
$J^{\mu}_{5}=J^{\mu}_{R}-J^{\mu}_{L}$, while $\mu_{5}$ is the chemical
potential for $J^{\mu}_{5}$. The significance of $\mu_{5}$ is explained in
item 12 in section VII. One should comment here is that the $\mu_{5}\neq 0$ is
not a true chemical potential as it is not associated with any exactly
conserved charges (in contrast with $\mu$ which corresponds to the conserved
baryon charge).

The significance of equation (45) is that the $Q_{5}$ and ${\cal{H}}$ are
strongly coupled with each other such that decrease of $Q_{5}$ will lead to
increase of ${\cal{H}}$ and vice versa. This implies that the magnetic
helicity ${\cal{H}}$ can be refilled and restored (in principle) after the
reconnection, and the source of the refill of the magnetic helicity
${\cal{H}}$ is the chiral charge $Q_{5}$ determined by parameter $\mu_{5}$.

Now we want to make several comments on the currents and their properties
which could generate the chiral asymmetry and consequently the magnetic
helicity ${\cal{H}}$, which are obviously belong to the class of the
topological effects. Normally, the topological phenomena are also associated
with the topological features of the sources, such as non-dissipating
currents. Well known example of such relation is the quantization of the
magnetic flux and associated with the quantized flux the non-dissipating
super-current.

It is very instructive to explain the differences between the currents by
analyzing their symmetric properties. We start with analysis of the
conventional Ohm’s law

$\vec{J}^{\rm ohm}=\sigma\vec{E},$ (46)

where $\sigma$ is the ohmic conductivity. Both electric current $\vec{J}$ and
electric field $\vec{E}$ are normal vectors ($\cal{P}$ -odd) under the
$\cal{P}$ symmetry. Therefore, the $\sigma$ has to be $\cal{P}$ even. If we
consider the time reversal symmetry ${\cal{T}}:t\rightarrow-t$ we observe that
the current $\vec{J}^{\rm ohm}$ is ${\cal{T}}$ odd, while the electric field
$\vec{E}=-\vec{\nabla}V$ is ${\cal{T}}$ even. Therefore, the Ohmic
conductivity $\sigma$ has to be odd under the ${\cal{T}}$ reversal for the
Ohm’s law (46) to make sense. This is an anticipated result since the ohmic
conductivity describes processes of dissipation that produce entropy, and
entropy production by the second law of thermodynamics is an irreversible
process which generates an arrow of time. In fact, all conventional transport
coefficients are odd under ${\cal{T}}$ reversal being consistent with
presented argument.

Now we consider the so-called Chiral Magnetic Effect (CME) when the electric
current is induced due to the chiral asymmetry expressed in terms of the
chemical potential $\mu_{5}$, see reviews [58, 59]:

$\vec{J}^{\rm
top}=\sigma_{5}\vec{\cal{B}},\;\;\;\;\;\sigma_{5}=\frac{e^{2}}{2\pi^{2}}\mu_{5}.$
(47)

The difference with previous case of the Ohm’s law (46) is that the magnetic
field $\vec{{\cal{B}}}=\vec{\nabla}\times\vec{\cal{A}}$ is ${\cal{T}}$ odd
because the vector potential $\vec{\cal{A}}$ is ${\cal{T}}$ odd. From (47) we
infer that $\sigma_{5}$ has to be ${\cal{T}}$ even, and the topological
current $\vec{J}^{\rm top}$ entering (47) is expected to be non-dissipating.

It is very instructive to compare this analysis with another type of non-
dissipating current which is induced in superconducting materials. This is
also important from phenomenological viewpoint as the NS is believed to be a
large superconductor111111It remains to be a matter of debates whether
superconductivity realized in NS is the type -I or type II superconductor
[73].. The corresponding physics is captured by the London relation between
the electric current and gauge potential

$\displaystyle\vec{J}^{\rm
London}=\lambda^{-2}\vec{\cal{A}},\;\;\;\;\;\vec{\nabla}\cdot\vec{\cal{A}}=0,$
(48)

where $\lambda$ is the penetration length. The vector potential
$\vec{\cal{A}}$ as well as $\vec{J}^{\rm London}$ are ${\cal{T}}$ odd
functions, which suggests that the London current $\vec{J}^{\rm London}$ is
non-dissipating. Indeed, the magnetic flux through an Abrikosov vortex for
type II superconductor is quantized. This means that the circulating super-
current is topologically protected– it is not allowed to dissipate as the flux
is quantized.

The non-dissipating nature of the topological current (47) is also supported
by analysis [56] where it has been argued that the conventional ${\cal{P}}-$
and ${\cal{T}}$-even QED processes (which are normally incorporated in MHD
analysis) cannot eliminate the current (47). Indeed, the correlation
$\langle\vec{J}^{\rm top}\cdot\vec{\cal{B}}\rangle$ is ${\cal{P}}$ -odd
correlation which cannot be changed by conventional ${\cal{P}}$ -even QED
processes121212We emphasize that the claim is not that the transitions
$L\rightarrow R$ and $R\rightarrow L$ do not occur all the time. These
transitions of course occur in QED as for example, the mass term flips the
chirality. The claim is that the expectation value $\langle\vec{J}^{\rm
top}\cdot\vec{\cal{B}}\rangle$ in equilibrium (including magnetic portion of
the helicity $\cal{H}$) cannot be washed out without ${\cal{P}}$ -odd weak
interactions.. The ${\cal{P}}$ odd weak interactions are capable to diminish
the induced current (47) which was generated by weak interactions when the
NS’s temperature was sufficiently high, see [56] for the details. The surface
effects can also play a role as helicity may leak through the surface.
However, these surface effects are expected to be subdominant, and not
considered here.

We finish this Appendix with order of magnitude estimate of the total energy
reservoir which potentially can refill the magnetic helicity. As we mentioned
after (45) the strength of helicity ${\cal{H}}$ can be restored after every
event of reconnection by conversion of the energy from $Q_{5}$. By definition,
$J^{0}_{5}=\int_{V}(J^{0}_{R}-J^{0}_{L})dV$ is the difference between right
handed and left handed densities. The source of this asymmetry is $\mu_{5}$,
which was estimated in [56]. The most important parameters for the present
estimate from [56] is the asymmetry parameter $P_{\rm asym}$ and the Fermi
momenta $k_{e}\sim\mu$ for electrons numerically assume the following values:

$P_{\rm asym}\sim\frac{\mu_{5}}{\mu}\approx 2\cdot 10^{-5},\;\;\;\;\;\
\mu\approx 100\rm~{}MeV.$ (49)

From (49) one can estimate the total difference between right handed and left
handed electrons in the entire NS as follows:

$\displaystyle\left(\frac{\mu_{R}^{3}}{3\pi^{2}}-\frac{\mu_{L}^{3}}{3\pi^{2}}\right)\left(\frac{4\pi
R^{3}}{3}\right)\approx\frac{\mu^{2}\mu_{5}}{\pi^{2}}\left(\frac{4\pi
R^{3}}{3}\right)\sim 10^{51}.~{}~{}~{}~{}$ (50)

The total energy reservoir which is available to refill the magnetic helicity
${\cal{H}}$ can be estimated by multiplication eq. (50) to a typical energy of
electrons, which is determined by $\mu$. Therefore, we arrive to the following
estimate for the total available energy

$\displaystyle E^{\rm
tot}_{(R-L)}\sim\mu\cdot\left(\frac{\mu^{2}\mu_{5}}{\pi^{2}}\right)\cdot\left(\frac{4\pi
R^{3}}{3}\right)\sim 10^{47}\rm erg.$ (51)

It is instructive to compare this total energy (produced at early times due to
the generation of the right-left asymmetry) with total magnetic energy as
given by (28). It is clear that the total energy reservoir (51) is enormous.
This amount of energy is more than sufficient to refill the magnetic helicity
${\cal{H}}$ after each magnetic reconnection to heat the NS surface, which is
precisely the proposal mentioned in items 11-14 in Sect. VII. As we mentioned
above this asymmetry cannot be washed out by conventional QED $\cal{P}$ even
processes incorporated into MHD. Only the $\cal{P}$ odd weak interactions are
capable to diminish this asymmetry. But these processes are very slow at low
temperatures [56], and unlikely to play a role.

##  Appendix B  Radio bands as indicator of the magnetic reconnection

It is known that the magnetic reconnection is also accompanied by emission in
radio frequency bands as the solar flare observations suggest. Therefore, one
could expect that the search in radio frequency bands could be an alternative
way to study the heating mechanism due to the magnetic reconnection in NS
along with X ray as suggested in Sect. VIII. The main goal of this Appendix is
to make an order of magnitude estimates for expected the radio signal from NS
which accompanies the magnetic reconnection. We follow the same logic in these
estimates as we used previously for the X ray analysis in Sect. VIII.

It is known that along with X rays there is another indicator for solar
flares, the so-called $F_{10.7}$ flux which reflects the solar activity in
radio wave band with wave length $10.7$cm. Typical enhancement during the
solar flare is about 200 sfu (solar flux units), see Fig 2 in ref.[52].
Converting sfu to conventional units we arrive to the following estimate for
the radio emission during the solar flare

$\displaystyle\langle F^{\odot}_{\rm radio}\rangle$ $\displaystyle\approx$
$\displaystyle 6\cdot 10^{-8}{\rm\frac{erg}{s\cdot cm^{2}}},\;\;\;\;\rm
sfu\equiv 10^{-22}\rm\frac{W}{m^{2}Hz}$ $\displaystyle\xi^{2.7\rm GHz}_{\rm
radio}$ $\displaystyle\equiv$ $\displaystyle\frac{\langle F^{\odot}_{\rm
radio}\rangle}{\langle F^{\odot}_{X}\rangle}\approx 6\cdot 10^{-6}.$ (52)

We assume that this ratio $\xi_{\rm radio}$ holds for NS as well. Therefore,
assuming the flat spectrum we arrive to the following estimate for $\langle
F^{\rm NS}_{\rm radio}\rangle$:

$\displaystyle\langle F^{\rm NS}_{\rm radio}\rangle\sim\xi^{1.4\rm GHz}_{\rm
radio}\cdot\langle F^{\rm NS}_{\rm X}\rangle$ $\displaystyle\sim$
$\displaystyle 3\cdot(10^{-19}-10^{-20})\left(\frac{T_{s}}{10^{6}\rm
K}\right)^{4}\left(\frac{0.1\rm kpc}{r}\right)^{2}{\rm\frac{erg}{s\cdot
cm^{2}}},$

where we use our estimate for $\langle F^{\rm NS}_{\rm X}\rangle$ from
(VIII.2).

It is instructive to present a numerical value of the flux $\langle F^{\rm
J1856}_{\rm radio}\rangle$ for $J1856$ from M7 stars where X ray flux is
recorded according to eq. (40).

$\displaystyle\langle F^{\rm J1856}_{\rm radio,~{}1.4\rm GHz}\rangle\sim
4\cdot 10^{-21}\rm\frac{erg}{s\cdot cm^{2}},\;\;\;[\rm prediction],\;\;\;$
(54)

while the observed upper limit for this star in $\rm 1.4~{}GHz$ band is
$10^{-4}\rm(mJy)(kpc^{2})$, which is almost two orders of magnitude higher
than (54), see Fig. 14 in [61]. Therefore, we are not optimistic with possible
observations of the radio signals from NS which always accompany the magnetic
reconnection, but we quite optimistic with possible observations in hard X ray
energy band as discussed in the main body of the text in Sect. VIII.

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