[1]

a]organization=Indian Institute of Technology, addressline=Jodhpur,
city=Rajasthan, postcode=342030, country=India

[2]

a]organization=Indian Institute of Technology, addressline=Jodhpur,
city=Rajasthan, postcode=342030, country=India

[3]

a]organization=Indian Institute of Technology, addressline=Jodhpur,
city=Rajasthan, postcode=342030, country=India

# Thermal properties of the core of magnetar

Trisha Sarkar<EMAIL_ADDRESS>[  Shalu Yadav<EMAIL_ADDRESS>[  Monika
Sinha<EMAIL_ADDRESS>[

###### Abstract

During very early age of neutron stars, the core cools down faster compared to
the crust creating a large thermal gradient in the interior of the star.
During $10-100$ years, a cooling wave propagates from the core to the crust
causing the interior of the star to thermalize. During this duration thermal
properties of the core material is of great importance to understand the
dynamics of the interior of the star. The heat capacity and thermal
conductivity of the core depends on the behaviour of matter inside the core.
We investigate these two properties in case of magnetars. Due to presence of
large magnetic field, the proton superconductivity is quenched partially
inside the magnetars depending upon the comparative values of upper critical
field and the strength of the magnetic field present. This produces non-
uniformity in the behaviour of matter throughout the star. Moreover, such non-
uniformity arises from the variation of nature of the pairing and values of
the pairing gap energy. We find that the heat capacity is substantially
reduced due to the presence of superfluidity. On the other hand, the thermal
conductivity of neutron is enhanced due to proton superconductivity and gets
reduced due to neutron superfluidity. Hence, the variation of the thermal
properties due to superfluidity in presence of magnetic field is different at
different radius inside the star. However, in all the cases the maximum
variation is of the order one. This affects the thermal relaxation time of the
star and eventually its the thermal evolution.

###### keywords:

Neutron Star Dense matter Equation of state Specific heat Thermal conductivity

##  1 Introduction

Neutron stars (NS) are unique astrophysical objects born in supernova
explosions with initial temperature $\sim~{}10^{11}$ K. Just after its birth a
large temperature gradient is set up inside the star. However, within a few
days its surface temperature goes down to $\sim~{}10^{9}$ K with the inner
region of the star remaining hotter compared to its surface. The crust and the
core cool down independently with different cooling mechanisms. Due to the
lack of sufficient thermal conduction in the interior of the star, the
thermalization inside the star is not able to take place at this stage. The
cooling is faster around the neutron drip density of the crust while the inner
crust cools comparatively at a slower rate , as a consequence of which the
inner crust remains hotter in comparison with the outer crust. The core of the
star cools down quickly by neutrino emission, while the crust continues to be
hotter as compared the core. This results in heat flow from the outer to inner
region of the star. The entire phenomena can be pictured as the propagation of
a cooling wave from the core towards the surface (Gnedin et al., 2001) which
occurs within a short period of time, $10-100$ years, known as the era of
thermal relaxation.

At a latter stage of thermal relaxation, the surface temperature drops down
very fast due to the large value of thermal conductivity and the
thermalization between the inner and the outer core. This results in the
formation of a nearly isothermal region within the entire core, which may even
extend to some parts of the inner crust. In this context the heat capacity and
thermal conductivity are two of the most crucial ingredients to understand and
construct the cooling model of NS, which are able to estimate the total loss
of thermal energy of the star. It is to be noted that both of these quantities
are completely dependent on the state of the interior matter compositions. For
the detailed review on the thermal properties in the interior of a NS, see the
work by Yakovlev et al. (2001) and the references mentioned in it.

The direct observation of NS cooling is detected for $\sim 330$ years old NS
Cassiopeia A. Analyzing nine years of data ($2000-2009$) obtained by NASA’s
_Chandra_ satellite its temperature is found to decline from $2.12\pm
0.01\times 10^{6}$ K to $2.04\pm 0.01\times 10^{6}$ K ($\sim 5.4\sigma$)
Heinke and Ho (2010). Before this the observation of thermal radiation emitted
from NS was only limited to the estimation of the temperature at any
individual time _e.g._ Crab pulsar Weisskopf et al. (2004), Vela pulsar Pavlov
et al. (2001), Geminga pulsar Halpern and Wang (1997) and so on. The observed
cooling rate of Cassiopeia A is much faster than the standard cooling
calculated from modified URCA process. One possible explanation of such fast
cooling could possibly be the coexistence of neutron superfluidity and proton
superconductivity in the stellar interior Page et al. (2011). In this work we
consider the transition of normal neutrons and protons to superfluid and
superconducting states respectively. The neutrons may reside in superfluid
state pairing via triplet (${}^{3}P_{2}$) channel at higher density of the
inner core, while the singlet pairing (${}^{1}S_{0}$) of neutrons are active
in the lower density regions, in the vicinity of the crust (Hoffberg et al.,
1970). The proton may form ${}^{1}S_{0}$ pairing even in the inner part of the
core, due to its lower density fraction as compared to the neutrons. The
presence of superfluidity/superconductivity substantially affects the thermal
properties, heat capacity and thermal conductivity (De Blasio, 2000; Chugunov
and Haensel, 2007; Sajad et al., 2014; Manuel et al., 2014; Page and Reddy,
2012; Cumming et al., 2017).

The magnetars belong to a special class of NSs which possess extremely high
surface magnetic field ($\gtrsim 10^{13}$ G). Several astrophysical
observation such as soft gamma repeaters and anomalous X-ray pulsars are
related to such a distinctive stellar body (Woods, 2008; Blumer and Safi-Harb,
2020; Coti Zelati et al., 2020; Rea et al., 2020), in which the structural
properties and behaviour of the highly dense matter are affected
extraordinarily (Sinha and Sedrakian, 2015). Consequently the thermal
properties of the matter inside such stars are also expected to exhibit
diverse characteristics. Due to the presence of large magnetic field the
proton superconductivity in magnetars is subdued significantly (Sinha and
Sedrakian, 2015) which results in astounding repercussions in its thermal
properties. In this work we analyse and estimate the heat capacity and thermal
conductivity of dense matter inside the core of the magnetar, considering the
possibilities of its matter configuration to be both normal and
superfluid/superconducting. We present a comparative analysis of these two
quantities at different times during its thermal evolution.

The plan of the work is as follows. In section 2 we describe the complete
model of the stars including its superfluid/superconducting pairing gap
profiles both in presence and in absence of the magnetic field. We also
illustrate the heat capacity and thermal conductivity of different particle
species in this section and describe the concerned results. In section 4.2 the
effect of magnetic field on these two quantities in addition to
superfluidity/superconductivity is explained. Finally in section 5 we conclude
and briefly discuss our results.

##  2 Model

We consider magnetars with sufficiently strong surface magnetic field ($B\sim
10^{15}$ G). The field inside the core is assumed to be at least one order of
magnitude greater than its surface value. In the core, the matter is
considered to be composed of only nucleons and electrons ($npe$), governed by
DDME2 equation of state (EoS) (Lalazissis et al., 2005) which satisfies
several astrophysical observables. The interior matter is charge neutral and
in $\beta$-equilibrium. We consider two model magnetars having two extreme
mass limits, $2.3~{}M_{\odot}$ and $1.4~{}M_{\odot}$. The radii of the massive
($M=2.3~{}M_{\odot}$) and canonical mass star ($M=1.4~{}M_{\odot}$) are
$12.72$ and $12.30$ km respectively.

We consider two magnetic field profile inside the model magnetars which are
discussed as follows.

(a) Exponential profile:
$B{{}_{exp}}(n_{b}/n_{0})=B_{s}+B_{c}\\{1-\beta~{}e^{(n_{b}/n_{0})^{\gamma}}\\}$
(Bandyopadhyay et al., 1997). Here $B_{c}$ and $B_{s}$ are the magnetic field
at the core and the surface of the star respectively. $\beta$ and $\gamma$ are
two arbitrary parameters which are selected in such a way that the desired
magnetic field on the surface can be obtained. In this work we have considered
$\beta=0.04$ and $\gamma=3.22$, to obtain $B_{s}\sim 10^{15}$ G and $B_{c}\sim
10^{16}$ G.

(b) Universal profile:
$B{{}_{uni}}(x)=B_{c}(1-1.6x^{2}-x^{4}+4.2x^{6}-2.4x^{8})$ (Chatterjee et al.,
2019). Here $x=r/R$, r and R are the radial distance from the centre and the
radius of the star respectively.

Table 1: Different ranges of radii ($R$) expressed in km in which ${}^{1}S_{0}$ and ${}^{3}P_{2}$ pairing of nucleons are active, following the lower curves of $\Delta_{n}$ and $\Delta_{p}$ as shown in fig. 1 for the two stars, both in presence and absence of magnetic field. Mass ($M_{\odot}$) |  $\Delta_{n}(^{1}S_{0})$ |  $\Delta_{n}(^{3}P_{2})$ |  $\Delta_{p}(^{1}S_{0})$
($B=0$) |  $\Delta_{p}(^{1}S_{0})$
($B_{uni}$) |  $\Delta_{p}(^{1}S_{0})$
($B_{exp}$)
---|---|---|---|---|---
1.4 |  12.3 - 11.8 |  11.8 - 8.48 |  11.8-3.13 |  11.8 - 8.71 |  11.8- 9.27
2.3 |  12.72 - 12.5 |  12.5 - 11 |  12.5-9.4 |  12.5-10.85 |  12.5-10.98
Table 2: Same as Table 1 following the upper curves of $\Delta_{n}$ and $\Delta_{p}$ Mass ($M_{\odot}$) |  $\Delta_{n}(^{1}S_{0})$ |  $\Delta_{n}(^{3}P_{2})$ |  $\Delta_{p}(^{1}S_{0})$
($B=0$) |  $\Delta_{p}(^{1}S_{0})$
($B_{uni}$) |  $\Delta_{p}(^{1}S_{0})$
($B_{exp}$)
---|---|---|---|---|---
1.4 |  12.30 - 11.8 |  11.8 - 0.8 |  11.8 - 0.8 |  11.8 - 6.45 |  11.8 - 7.22
2.3 |  12.72-12.5 |  12.5-0.8 |  12.5-5.69 |  12.5-9.44 |  12.5-9.86

In generally, immediately after the birth, the initial temperature of the star
is $\sim 10^{10}-10^{11}$ K and gradually it cools down. The matter inside the
core is highly dense, and neutrons (protons) may form Cooper pairs with strong
interaction leading to superfluidity (superconductivity). The concept of
superfluidity in NS is well established Migdal (1959) and have been studied
extensively in many previous works (Sedrakian and Clark, 2019; Sauls, 2019).
Inside NS the superfluidity of the baryons is considered to be of BCS type.
The neutrons form singlet pairing (${}^{1}S_{0}$) near the crust at lower
density regions where $\rho\lesssim\rho_{0}$ ($\rho_{0}\sim 0.17fm^{-3}$ is
the nuclear saturation density), while at the higher density regions of the
core the pairing is formed via triplet channel (${}^{3}P_{2}$), since the
singlet state pairing becomes repulsive at larger densities (Wolf, 1966).
Although at higher densities both ${}^{1}D_{0}$ and ${}^{3}P_{2}$ pairing
channels are dominant, ${}^{3}P_{2}$ is the most attractive among them (Sauls,
2019). The triplet pairing is originally understood by
${}^{3}P_{2}~{}-~{}^{3}F_{2}$ channel. However, because of the smallness of
${}^{3}F_{2}$ phase, only ${}^{3}P_{2}$ wave is considered for simplicity
(Baldo et al., 1992). The protons form ${}^{1}S_{0}$ state pairing even inside
the deeper layers of the core, due to its smaller number density fraction
compared to the neutrons (Yakovlev and Pethick, 2004). There are several
phenomenological models for the superfluid energy gap ($\Delta$) in both
${}^{1}S_{0}$ and ${}^{3}P_{2}$ channels for the protons and neutrons,
depending on the nature of the strong nuclear interactions. From the overall
picture of different models, two extreme bounds of energy gap for both of the
neutrons ($\Delta_{n}$) and protons ($\Delta_{p}$) are considered Sedrakian
and Clark (2019). For these two extreme cases the variation of $\Delta_{n}$
and $\Delta_{p}$ inside the two model magnetars are shown in the left and
right panels respectively, of the top row of fig. 1. It can be pointed out
comparing the left and right panels that the proton superconductivity is very
prominent near the surface of both the stars, although for the massive star
the region containing the proton pairs extends up to a smaller depth as
compared to the canonical mass star.

The neutron superfluidity in ${}^{1}S_{0}$ state is also restricted in a
narrow region near the surface of the magnetars. It is also evident from fig.
1 that for the massive star the neutron superfluidity is confined within some
specific portion of the star. In case of the canonical mass star, in absence
of magnetic field, the neutron superfluidity remains active throughout almost
the entire star for the upper boundaries of the pairing energy gap, while for
the lower boundaries it is only effective in a small radial zone inside the
star. The critical temperatures ($T_{c}$) corresponding to the various
scenarios of both $\Delta_{n}$ and $\Delta_{p}$ variation inside the stars are
shown in the middle row of the same figure.

In the magnetars, the protons are considered to be type-II superconductors,
which causes the magnetic field to penetrate by constructing the array of
quantized fluxtubes in the limit, $B\leq H_{c2}$, $H_{c2}$ being the upper
critical field. With increase in magnetic field, the fluxtubes become more and
more densely packed. Finally when the magnetic field exceeds $H_{c2}$ $i.e.$
$B\geq H_{c2}$ the fluxtubes are contacted with each other (Lander, 2014) and
this leads to the destruction of the proton superconductivity. Depending on
the variation of $H_{c2}$ and the magnetic field of the star, the proton
superconductivity is quenched at those specific regions of the star, where the
magnetic field is larger than $H_{c2}$ (Sinha and Sedrakian, 2015). In terms
of the coherence length $\zeta_{p}$ the upper critical field $H_{c2}$ is given
by,

$H_{c2}=\frac{\phi_{0}}{2\pi\zeta_{p}^{2}},$ (1)

where $\phi_{0}$ is the quantum of flux at $B\leq H_{c2}$. The lower critical
field $H_{c1}$ is insignificant in case of NSs due to their large electrical
conductivity, unlike ordinary type-II superconductors constructed at
laboratories (Lander, 2014; Chamel, 2017; Haskell and Sedrakian, 2018).

The variation of $H_{c2}$ for two extreme profiles of $\Delta_{p}$ and the
profiles of the magnetic field inside the magnetars are illustrated in the
bottom row of fig. 1. It is clearly evident from the plots that in presence of
the magnetic field, the region where proton pairing is functional, is narrowed
down and shifted more towards the surface. In case of the the massive star,
the suppression is nearly similar for the two kinds of magnetic field
profiles, while for the canonical mass star the suppression appears to be
occurring a little further with the consideration of the exponential profile
of the magnetic field in comparison with the universal one. .

The notion of different pairing interaction of the nucleons at the different
radial regions of the model stars, both in presence and absence of magnetic
field, is illustrated in table 1 and 2.

During the thermal relaxation stage the effective surface temperature of the
star does not reflect the thermal state of its core, since the core and crust
remain thermally decoupled. At the end of this period the temperature is
dropped drastically when the cooling wave coming from the core hits the
surface of the star, initiating the process of neutrino cooling. At this stage
the core becomes nearly isothermal and in case of the model stars, the
temperature is even reduced to as low as $\sim 10^{8}$ K. As the overall
critical temperature of neutrons ($T_{cn}$) and protons ($T_{cp}$) in the
pairing channels is much greater than $\sim 10^{8}$ K, as shown in the middle
row of fig. 1, for both of the massive and canonical mass stars, it can be
concluded that the neutrons (protons) inside the stars are converted to
superfluid (superconducting) phase, soon after the thermal relaxation period
is ended. However, it is to be noted that in case of the lower boundary of the
triplet pairing channel of the neutrons, $T_{cn}$ is quite low, even lower
than $10^{8}$ K at some parts of the stars, as compared to the other pairing
channels. It is needless to say that in the NS interior superfluid neutrons
and superconducting protons can coexist. The thermal relaxation period lasts
around $\sim 100$ years for the massive star, while it extends a little more
in case of the canonical mass star. The duration of the thermal relaxation
period impacts the thermal evolution of the magnetars remarkably. Due to the
shorter thermal relaxation time, the massive star starts to cool down faster
than the canonical mass star, because of which the massive star becomes
superfluid at an earlier stage, as compared to the canonical mass star. Once
the stellar matter is converted from normal to superfluid/superconducting,
both the heat capacity and the thermal conductivity are changed exceptionally.
In our model stars, the baryons start forming superfluid pairs at the later
stage of the thermal relaxation process which might be responsible for slowing
down their overall cooling rates.

Figure 1: Top panel: Variation of the energy gap of neutron ($\Delta_{n}$) and
proton ($\Delta_{p}$) with radius ($R$) for different pairing types; middle
panel: Variation of the critical temperature of neutrons ($T_{cn}$) and
protons ($T_{cp}$) with radius ($R$) for different pairing interaction
channel; bottom panel: Variation of the critical magnetic field ($H_{c2}$) and
magnetic field profiles with radius ($R$). Left panels are for star with mass
$M\sim 2.3M_{\odot}$ and right panels for the star of mass $M\sim
1.4M_{\odot}$, the interior matter governed by DDME2 EoS. $H_{c2}$ is plotted
for the upper ($\Delta_{p}^{upper}$) and lower ($\Delta_{p}^{lower}$) curve of
proton pairing.

##  3 Thermal properties in absence of magnetic field

###  3.1 Heat capacity

The total heat capacity of the core of a NS is taken as the sum of the partial
heat capacity of each component particle $i.e.$
$C_{v}=\sum\limits_{i=n,p,e}C_{i}$, since the core is consisted of strongly
degenerate Fermions. Here the partial heat capacity of the $i$-th particle
species is given by (Lifshitz et al., 1981),

$C_{i}=\frac{2\hbar}{(2\pi\hbar)^{3}}\int
d^{3}k_{i}(\epsilon_{i}-\mu_{i})\frac{\mathrm{d}f_{i}}{\mathrm{d}T},$ (2)

where $f_{i}=1/(1+e^{z})$ is Fermi Dirac distribution function with
$z=({\epsilon_{i}-\mu_{i}})/k_{B}T$. $\epsilon_{i}$ and $\mu_{i}$ are energy
and chemical potential of $i$-th particles respectively, $k_{B}$ is Boltzmann
constant and $T$ is the temperature. Inside the NS, the electrons form an
ultra relativistic highly degenerate Fermi gas, while the nucleons form a non
relativistic non-ideal Fermi liquid. Solving eq. (2) for these two cases, one
obtains the general expression of partial heat capacity for $i$-th particle as
follows,

$C_{{i}{0}}=\frac{m_{i}^{*}k_{F_{i}}k_{B}^{2}T}{3\hbar^{3}},$ (3)

where $i=n,p,e$ and $m_{i}^{*}$ and $k_{F_{i}}$ are effective mass and Fermi
momentum of $i$-th species respectively. The subscript ’$0$’ implies the
presence of normal nucleons.

####  3.1.1 Effect of superfluidity/superconductivity

If any particle species ($i$) is considered to be superfluid then below its
critical temperature ($T_{c}$) all the particles of that particular species
transit to the ground state which reduces their free energy and consequently,
the entropy. When the temperature reaches below $T_{c}$ the energy gap
increases. In NS, only those nucleons having energy
$|\epsilon_{i}-\mu_{i}|\lesssim k_{B}T$ contribute to the heat capacity. In
presence of strong superfluidity $i.e.$ at $T<<T_{c}$, the energy gap
$\Delta(T)$ exceeds $k_{B}T$, as a consequence of which the corresponding
nucleons cannot contribute to the heat capacity for which it gets lowered
(Yakovlev et al., 1999). The concept is realised mathematically by introducing
a multiplicative reduction factor, denoted by $\mathcal{R}(T)$, so that the
suppression of heat capacity in presence of superfluidity can be expressed as,
$C=\sum_{i=n,p,e}C_{i0}\mathcal{R}(T)$, $C_{i0}$ is mentioned in eq. (3).

Figure 2: Variation of heat capacity inside the star for DDME2 EoS. Solid line
without superfluidity, long-dashed curves for total $C_{v}$ with lower
$\Delta$ curve and dash-dotted curves for with upper $\Delta$ curve for total
$C_{v}$. For neutron contribution to $C_{v}$ dotted curves are without
superfluidity, short-dashed curve with lower $\Delta$ curve and dash-dot-
dotted curves with upper $\Delta$ curves. Top panel: for the massive star
$M\sim 2.3M_{\odot}$; bottom panel: for $M\sim 1.4M_{\odot}$. Left panel: for
$t=100$ years; right panel: for $t=1000$ years.

The reduction factor depends on the nature of the pairing energy gap of the
nucleons, $\Delta_{n}$ and $\Delta_{p}$ (Yakovlev et al., 1999) . For
${}^{1}S_{0}$ pairing, $\Delta$ is isotropic, denoted by $\Delta=\delta_{0}$
and is independent of the orientation of the particle momenta with respect to
the quantization axis. The dependence of $\delta_{0}$ with $T$ is obtained
from the BCS theory which can be expressed as (Levenfish and Yakovlev, 1994;
Yakovlev and Levenfish, 1995),

$\displaystyle y_{n}(^{1}S_{0})\equiv\frac{\delta_{0}(T)}{k_{B}T}\equiv
y_{p}(^{1}S_{0})$
$\displaystyle=\sqrt{1-T/T_{c}}\left(1.456-\frac{0.157}{\sqrt{T/T_{c}}}+\frac{1.764}{T/T_{c}}\right)$
(4)

For ${}^{3}P_{2}$ pairing the energy gap becomes anisotropic (Hoffberg et al.,
1970) because of its dependence on the direction of particle momenta. However
the energy gap can still be assumed to be isotropic if the orientation of the
quantization axis is arbitrary instead of having a fixed direction. Therefore
for triplet pairing the energy gap, denoted by $\delta_{1}$, is expressed as
(Levenfish and Yakovlev, 1994; Yakovlev and Levenfish, 1995; Haensel et al.,
2000; Baiko et al., 2001),

$y_{n}(^{3}P_{2})\equiv\frac{\delta_{1}(T)}{k_{B}T}=\sqrt{1-T/T_{c}}\left(0.7893+\frac{1.188}{T_{c}}\right).$
(5)

The quantities $y_{n}$ and $y_{p}$ mentioned in eqn. (3.1.1) and (5) are
dimensionless. It is evident that for ${}^{1}S_{0}$ and ${}^{3}P_{2}$ pairing
the reduction factors are different, that are expressed in (Yakovlev et al.,
1999) as algebraic expressions. All the reduction factors tend to be unity in
the absence of superfluidity/superconductivity.

The variation of heat capacity with radius is shown in fig. 2 for the massive
and the canonical mass stars at two different ages, $t=100$ and $1000$ years.
The reason for selecting these two specific times is that at $t\gtrsim 100$
years in both of the stars the thermal equilibrium is about to be established
and the neutrons (protons) just start to convert into superfluid
(superconducting), while at a much latter time, $t=1000$ years, the effect of
the nucleon pairing is expected to be manifested much more prominently on the
thermal properties. The total heat capacity of the matter with contribution
from all the particle species is denoted by $C_{v(0)}$, while $C_{n(0)}$
represents the contribution from the neutrons only. The heat capacity of the
protons and electrons are relatively much smaller, hence they are not shown
separately in the figure.

Near the crust, all the nucleons form ${}^{1}S_{0}$ pairing and the strength
of neutron pairing dominates over proton pairing $i.e.$
$\Delta_{n}^{lower(upper)}(^{1}S_{0})>\Delta_{p}^{lower(upper)}(^{1}S_{0})$,
in case of both the massive and canonical mass stars, as observed from fig. 1.
In case of the massive star, the neutrons form ${}^{1}S_{0}$ pairing till
$R\sim 12.5$ km  and transit to ${}^{3}P_{2}$ pairing state after that, till
$R\sim 11$ km, for $\Delta_{n}^{lower}$, while in case of $\Delta_{n}^{upper}$
the neutron pairing exists till the inner core . It is to be noticed that the
pairing strength in ${}^{3}P_{2}$ state is much weaker as compared to
${}^{1}S_{0}$ state. The lower bound of ${}^{3}P_{2}$ provides very small
value of pairing gap energy inside the star. Any particular particle species
($i$) may not exist in the pairing state, if the temperature of the star
remains greater than the critical temperature of the particles $i.e.$
$T>T_{ci}$, even though its pairing energy gap is non-zero.

Therefore, the resultant heat capacity $C_{v}$ is highly suppressed near the
surface region of the star $i.e.$ at $R\gtrsim 12.5$ km, for both
$\Delta_{n}^{lower}$ and $\Delta_{p}^{upper}$. The protons are superconducting
within the radial distance $12.5\gtrsim R\gtrsim 9.4$ km ($12.5\gtrsim
R\gtrsim 5.69$ km), corresponding to $\Delta_{p}^{lower(upper)}$. The region
of active proton superconductivity is altered in presence of magnetic field
which is discussed in detail in section 4.2.

Within the radial zone $12.5~{}\text{km}\gtrsim R\gtrsim 10.98$ km, the
pairing energy gaps of ${}^{1}S_{0}$ protons and ${}^{3}P_{2}$ neutrons are
non-zero in case of their respective lower boundaries. However, since the
internal temperature of the star varies within $T\sim(6-9)\times 10^{8}$ K at
$t=100$ years and $\sim(4-6)\times 10^{8}$ K at $t=1000$ years, the critical
temperature of ${}^{3}P_{2}$ neutrons for $\Delta_{n}^{lower}$ becomes smaller
than the temperature of the star $(T_{cn}^{lower}<T)$, in these regions. As a
consequence, the neutrons remain non-superfluid, despite of the non-zero value
of $\Delta_{n}^{lower}$ for triplet state which is extremely small. Proton
superconductivity is further extended till $R\sim 9.4$ km, above which both
the baryons are normal. Unlike the triplet neutrons, the corresponding
critical temperature $T_{cp}^{lower}$ is larger than $T$ which activates the
proton superconductivity. This results in the order of $C_{v}$ to stay nearly
constant at $\sim 6\times 10^{19}$ erg cm-3 K-1 within the range $12.5\gtrsim
R\gtrsim 9.4$ km at $t=100$ years which is lower than the range of $C_{v0}$ in
this region. At $R\sim 9.36$ km, $C_{v}$ is suddenly lifted to $2\times
10^{20}$ to merge with $C_{v0}$ and is slightly increased to $3\times 10^{20}$
erg cm-3 K-1 till the inner core. In this scenario, the neutron contribution
to specific heat, $C_{n}$ remains the same as $C_{n0}$ at $R\lesssim 12.5$ km,
as observed from the plot, due to the loss of the triplet pairing of the
neutrons.

In case of $\Delta_{n}^{upper}$ and $\Delta_{p}^{upper}$, the scenario is
quite different. Here, at $R\gtrsim 12.5$ km neutrons are in ${}^{1}S_{0}$
pairing state and at $R\lesssim 12.5$ km neutron superfluidity is converted
from singlet to triplet pairing state. The proton pairing is also turned on at
$R\sim 12.5$ km. At $R\lesssim 5.69$ km the protons are normal, while the
neutrons remain in the ${}^{3}P_{2}$ state till the inner core. Since
$T_{cn}^{upper}$ for the ${}^{3}P_{2}$ neutrons remains higher than the
stellar temperature due to the presence of sizable $\Delta_{n}^{upper}$, the
${}^{3}P_{2}$ neutrons remain intact nearly all throughout the star. The
proton superconductivity also remains activated in the radial zone
$12.5\gtrsim R\gtrsim 5.69$ km, as the condition $T_{cp}^{upper}>T$ remains
satisfied. This causes $C_{v}$ to vary within $(2-5)\times 10^{19}$ erg cm-3
K-1 for $12.5\gtrsim R\gtrsim 5.69$ km and jump suddenly to $\sim 2.8\times
10^{20}$ erg cm-3 K-1 at $R\sim 5.69$ km, due to the conversion of singlet
protons to normal ones and remain constant till the inner core. At $t=1000$
years both $C_{v}$ and $C_{v0}$ are reduced to some extent in all the regions
of the star.

In case of the canonical mass star the neutrons form ${}^{1}S_{0}$ pairing in
the narrow region around the crust $R\gtrsim 11.8$ km for $\Delta_{n}^{lower}$
which causes $C_{v}$ to have very small values near the surface of the star.
For the ${}^{3}P_{2}$ neutrons, although $\Delta_{n}^{lower}$ is non-zero
within the radial range $11.8\gtrsim R\gtrsim 8.48$ km, due to its very small
value, $T_{cn}^{lower}<T$ with $T\sim(6-9)\times 10^{8}$ K at $t=100$ years
and $T\sim(4-6)\times 10^{8}$ K at $t=1000$ years, which causes the neutrons
to exist in normal state in this area of the star, similar to the scenario of
the massive star. For the protons, although $\Delta_{p}^{lower}$ is non-zero
inside the radial zone $R\sim 11.8$ to $R\sim 3.13$ km, the proton
superconductivity is actually active inside the star from $R\sim 11.86$ km to
$R\sim 3.63$ km as the conditions $T_{cp}^{lower}>T$ is no longer valid at
$R\leq 3.63$ km. Therefore, $C_{v}$ varies from $6\times 10^{19}$ to $\sim
1.5\times 10^{20}$ erg cm-3 K-1 within the radial range $12\gtrsim R\gtrsim
3.63$ km which is suddenly raised to $\sim 2.5\times 10^{20}$ erg cm-3 K-1 at
$R\sim 3.63$ km and remains nearly the same beyond that.

For the upper boundaries of the nucleon pairing energy, within the radial zone
at $R\gtrsim 11.8$ km neutrons form ${}^{1}S_{0}$ pairing and reside in
${}^{3}P_{2}$ state in the rest of the star. The proton pairing is also active
in the entire star starting from $R\sim 11.8$ km. Both
$T_{cn}^{upper},~{}T_{cp}^{upper}$ in different pairing channels are greater
than the stellar temperature in all the region of the star. The range of
$C_{v}$ prevails within $(2-3)\times 10^{19}$ erg cm-3 K-1 in almost through
the entire star $i.e.$ $R\lesssim 11.8$ km. Further, it is needless to mention
that both $C_{v}$ and $C_{v0}$ are lower at $t=1000$ years as compared to
$t=100$ years.

The distinct features of $C_{v}$ in the different radial regions of the
massive star at different time are illustrated in table 3. The entire star is
divided into different radial zones depending on the nature of nucleon pairing
at different zones which are different for the lower and upper boundaries of
the pairing energy gap. Although the variation of $C_{v0}$ is continuous
throughout the star, its value is mentioned separately in the distinct zones
to point out how much $C_{v}$ is lesser than $C_{v0}$ in
superfluid/superconducting nuclear matter.

$M=2.3~{}M_{\odot}$
---
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
Radial zone$\rightarrow$ |  $12.5-9.4$ km |  $\leq 9.4$ km |  $12.5-9.4$ km |  $\leq 9.4$ km
$C_{v0}$ | $5\times 10^{19}-2\times 10^{20}$ | $(2-3)\times 10^{20}$ | $3\times 10^{19}-1.5\times 10^{20}$ | $(1.5-2)\times 10^{20}$
$C_{v}$ ($\Delta_{n}^{lower},\Delta_{p}^{lower}$) | $6\times 10^{19}$ | $(2-3)\times 10^{20}$ | $4\times 10^{19}$ | $(1.5-2)\times 10^{20}$

Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
---|---|---
Radial zone$\rightarrow$ |  $12.5-5.69$ km |  $\leq 5.69$ km |  $12.5-5.69$ km |  $\leq 5.69$ km
$C_{v0}$ |  $5\times 10^{19}-2.8\times 10^{20}$ |  $(2.8-3)\times 10^{20}$ |  $3\times 10^{19}-1.8\times 10^{20}$ |  $2\times 10^{20}$
$C_{v}$($\Delta_{n}^{upper},\Delta_{p}^{upper}$) |  $(2-5)\times 10^{19}$ |  $2.8\times 10^{20}$ |  $8\times 10^{18}-2\times 10^{19}$ |  $1.8\times 10^{20}$

Table 3: Estimation of the resultant heat capacity of all the particle
species, $C_{v}=\sum_{i=n,p,e}C_{i)}$, in the unit of erg cm-3 K-1 in the
different radial zones of the massive star ($M=2.3M_{\odot}$), in presence of
normal ($C_{v0}$) and superfluid/superconducting nucleons ($C_{v}$),
considering the lower and upper boundaries of their pairing energies
($\Delta_{n}$, $\Delta_{p}$) at the two different ages, $t=100$ and $t=1000$
years. Different radial zones are selected accordingly as the existence of the
nucleon pairing, as mentioned in Tables 1 and 2.

$M=1.4~{}M_{\odot}$
---
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
Radial zone$\rightarrow$ |  $11.8-3.63$ km |  $\leq 3.63$ km |  $11.8-3.63$ km |  $\leq 3.63$ km
$C_{v0}$ | $7\times 10^{19}-2.8\times 10^{20}$ | $2.8\times 10^{20}$ | $6\times 10^{19}-1.8\times 10^{20}$ | $1.8\times 10^{20}$
$C_{v}(\Delta_{n}^{lower},\Delta_{p}^{lower})$ | $6\times 10^{19}-1.5\times 10^{20}$ | $(2.5-2.8)\times 10^{20}$ | $(4-9)\times 10^{19}$ | $1.8\times 10^{20}$

Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
---|---|---
Radial zone$\rightarrow$ |  $\leq 11.8$ km |  $\leq 11.8$ km
$C_{v0}$ |  $8\times 10^{19}-2.8\times 10^{20}$ |  $6\times 10^{19}-1.8\times 10^{20}$
$C_{v}(\Delta_{n}^{upper},\Delta_{p}^{upper})$ |  $(2-3)\times 10^{19}$ |  $(1-1.8)\times 10^{19}$

Table 4: Same as table 3, for the canonical mass star ($M=1.4M_{\odot}$)

###  3.2 Thermal conductivity

Neutrons are the most significant heat carriers in the neutron star core being
the most abundant particles, while the protons have much smaller number
density. Therefore, the contribution of neutrons is much larger to the thermal
conductivity as compared to the protons. The leptons also play a crucial role
in heat conduction along with neutrons due to their high mobility and large
mean free path. In the envelope and outer crust, the heat conduction by
electrons is much significant where electron-ion scattering takes place. In
the inner crust and core free neutrons start to appear. Heat conduction by
neutrons occur by neutron-neutron ($nn$) and neutron-proton ($np$) collisions
via strong interaction, while the electrons undergo Coulomb collision with
protons ($ep$) and other electrons ($ee$). Neutron-electron ($ne$) scattering
is neglected in this work. It is to be mentioned that the thermal conductivity
for neutrons and electrons can be studied independently (Potekhin et al.,
2015). Theoretical development of thermal conductivity in dense matter with
$npe$ configuration, along with other transport coefficients are analysed in
previous works Flowers and Itoh (1976, 1979). The key ingredient of thermal
conductivity calculation is the evaluation of nucleon-nucleon scattering which
occurs via the strong interaction. The problem is discussed with different
approaches and nuclear potentials in several literatures (Flowers and Itoh,
1979; Wambach et al., 1993; Sedrakian et al., 1994; Baiko et al., 2001;
Shternin and Baldo, 2020).

As mentioned earlier, due to plenty of abundance, neutrons contribute more
compared to protons to the thermal conductivity inside the stellar core, in
which the nucleons form a strongly degenerate non-relativistic Fermi liquid
while the electrons form a strongly degenerate ultra-relativistic gas. In the
limit of strong degeneracy as the inter-particle interaction is taken into
account, the Fermi liquid is considered to be composed of $quasiparticles$
Landau (1957). Since neutrons and electrons are the only significant heat
carriers, they are treated as quasiparticles while the protons are considered
to be scatterers. The calculation of thermal conductivity of Fermi liquid was
done previously in the low temperature limit Abrikosov and Khalatnikov (1959);
Sykes and Brooker (1970). The quasiparticles are treated as a weakly
interacting dilute gas having non-equilibrium probability distribution
function $F_{c}(\mathbf{r},\mathbf{v},t)$ where $c=n,e$. The dynamics of the
quasiparticles is determined by Boltzmann transport equation, given by

$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{c}.\mathbf{\nabla}_{r}+\mathbf{g}_{c}.\mathbf{\nabla}_{v}\right)F_{c}(\mathbf{r},\mathbf{v},t)=J_{ci},$
(6)

where $\mathbf{v}_{c}$ is the velocity of the heat carrier ($c$),
$\mathbf{g}_{c}$ is the force per unit mass exerted by it. The collisions
occurring among different particle species restore the non-equilibrium to
equilibrium. Here $\mathbf{\nabla}_{r}$ and $\mathbf{\nabla}_{v}$ denote
partial differentiation in configuration and momentum space respectively.
$J_{ci}$ is the collision integral for collision occurring between the
quasiparticle heat carrier $c$ and $i$-th particle species. To estimate the
thermal conductivity of the neutrons, the total contribution to the collision
integral, $J_{ci}$ is represented by, $J_{n}=J_{nn}+J_{np}$, while in case of
the electrons it is expressed as, $J_{e}=J_{ep}+J_{ee}$. Since the protons are
not treated as quasiparticles, they are described by equilibrium Fermi-Dirac
(FD) distribution function, given by (Baiko et al., 2001)

$F_{p}=f_{p}=[1+exp(\epsilon_{p}-\mu_{p})]^{-1},$ (7)

where $\epsilon_{p}$ and $\mu_{p}$ are the energy and chemical potential of
the proton respectively. The distribution function for neutron and electron
$F_{c}$ is expressed as (Flowers and Itoh, 1979; Gnedin and Yakovlev, 1995;
Baiko et al., 2001)

$F_{c}=f_{c}-\phi_{c}\frac{\partial f_{c}}{\partial\epsilon_{c}}.$ (8)

Here $f_{c}$ ($c=e,n$) is the equilibrium FD distribution function, similar to
eqn. (7). The small deviation from the equilibrium is introduced in terms of
the factor $\phi_{c}$, which is represented by (Ziman, 1960)

$\phi_{c}=-\tau_{c}(\epsilon_{c}-\mu_{c})\frac{\mathbf{v}_{c}.\mathbf{\nabla}{T}}{T},$
(9)

where $\tau_{c}$ is the effective relaxation time corresponding to the
collision of the quasiparticle heat carrier ($c$) and $T$ is the temperature.
Now for the scattering process $c+i\rightarrow
c^{{}^{\prime}}+i^{{}^{\prime}}$, the collision integral has the general form
given by the following expression

$\displaystyle J_{ci}=\frac{1}{(2\pi\hbar)^{9}}\int
d^{3}\mathbf{k}_{c}^{{}^{\prime}}d^{3}\mathbf{k}_{i}d^{3}\mathbf{k}_{i}^{{}^{\prime}}\sum_{\sigma_{c}^{{}^{\prime}},\sigma_{i},\sigma_{i}^{{}^{\prime}}}\mathcal{P}_{ci}$
$\displaystyle\times[F_{c}^{{}^{\prime}}F_{i}^{{}^{\prime}}(1-F_{c})(1-F_{i})-F_{c}F_{i}(1-F_{c}^{{}^{\prime}})(1-F_{i}^{{}^{\prime}})],$
(10)

where $\mathcal{P}_{ci}$ is the differential transition probability. The
unprimed (primed) quantities correspond to the particles before (after)
collision and $\sigma_{i}^{(^{\prime})}$ is the spin of the
$i^{(^{\prime})}$-th particle. Since the particles remain in strongly
degenerate state at the core, only those particles near the vicinity of the
Fermi surface, $\hbar k_{F}|k-k_{F}|\lesssim k_{B}T$ contribute to the
collision integral significantly, given by eq. (3.2). Using eqn. (7), (8),
(9), (3.2) in eq. (6), we get

$\displaystyle\mathbf{v}_{c}\cdot\frac{\partial f_{c}}{\partial
T}\mathbf{\nabla}T=\frac{\tau_{c}\nabla{T}}{(2\pi)^{9}T^{2}}\sum_{i}\sum_{\sigma_{c}^{{}^{\prime}},\sigma_{i},\sigma_{i}^{{}^{\prime}}}\int\int\int
d\mathbf{k}_{c}^{{}^{\prime}}d\mathbf{k}_{i}^{{}^{\prime}}d\mathbf{k}_{i}$
$\displaystyle\times
f_{c}f_{i}(1-f_{c}^{{}^{\prime}})(1-f_{i}^{{}^{\prime}})\mathcal{P}_{ci}(\phi_{c}^{{}^{\prime}}+\phi_{i}^{{}^{\prime}}-\phi_{c}-\phi_{i}).$
(11)

Solving eq. (3.2), one arrives at the following expression

$\tau_{c}=\frac{1}{\sum\limits_{i}\nu_{ci}},$ (12)

where $\nu_{ci}$ denotes the collision frequency corresponding to the
collision occurring between the heat carrier ($c$) and $i$-th particle species
($i=n,p$ if $c=n$ ; $i=e,p$ for $c=e$).Thus, in case of neutrons,
$\tau_{n}=(\nu_{nn}+\nu_{np})^{-1}$ and for electrons,
$\tau_{e}=(\nu_{ee}+\nu_{ep})^{-1}$.

Thermal conductivity $\kappa_{c}$ of a particular heat carrier $c$ can be
expressed by Fourier’s heat conduction equation as

$\mathbf{q_{c}}=-\kappa_{c}\mathbf{\nabla}T=\frac{2}{(2\pi\hbar)^{3}}\int
d^{3}\mathbf{k}_{c}(\epsilon_{c}-\mu_{c})\mathbf{v}_{c}F_{c},$ (13)

where $\mathbf{q_{c}}$ is the heat flux. Using eqs. (8) and (9) into (3.2) and
comparing with eq. (13), one can obtain the expression of thermal conductivity
as

$\displaystyle\kappa_{c0}=\frac{\pi^{2}Tn_{c}\tau_{c}}{3m_{c}^{*}}$
$\displaystyle=\frac{\pi^{2}Tn_{c}}{3m_{c}^{*}\sum\limits_{i}\nu_{ci}}$ (14)

The subscript ’$0$’ implies that both the nucleons are normal. $n_{c}$ is the
number density and $m_{c}^{*}$ is its effective mass of the heat carrier
($c$). The collision frequency of the baryon and lepton are calculated
independently.

##### Baryon:

The expression of collision frequency for the neutrons ($c=n$) in the dense
matter is given by (Baiko et al., 2001)

$\nu_{ni}=\frac{64m_{n}^{*}m_{i}^{*_{2}}k_{B}^{2}T^{2}S_{ni}}{5m_{n}^{2}\hbar^{3}},$
(15)

where $m^{*}_{i}$ is the effective mass of $i$-th species. The quantity
$S_{ni}$ has the dimension of the cross section and are expressed as
analytical fit expressions for $nn$ and $np$ scattering, given by Baiko et al.
(2001)

$\displaystyle
S_{nn}=\frac{7.880(1-0.2241k_{Fn}+0.2006k_{Fn}^{2}}{k_{Fn}^{2}(1-0.1742k_{Fn})}\times\left(\frac{m_{n}}{m_{n}^{*}}\right)^{2}\times$
$\displaystyle(0.4891+1.111u^{2}-0.2283u^{3}+0.01589k_{Fp}$
$\displaystyle-0.02099k_{Fp}^{2}+0.2773uk_{Fp}),$ $\displaystyle
S_{np}=\frac{0.8007k_{Fp}}{k_{Fn}^{2}}(1+31.28k_{Fp}-0.0004285k_{Fp}^{2}+$
$\displaystyle 26.85k_{Fn}+0.08012k_{Fn}^{2})(1-0.5898k_{Fn}+$ $\displaystyle
0.2368k_{Fn}^{2}+0.5838k_{Fp}^{2}+0.884k_{Fn}k_{Fp})^{-1}$
$\displaystyle\times\left(\frac{m_{n}}{m_{p}^{*}}\right)^{2}(0.04377+1.1v^{2}+0.118v^{3}+0.1626k_{Fp}$
$\displaystyle+0.3871vk_{Fp}-0.299v^{4})+\frac{0.383k_{Fp}^{4}}{k_{Fn}^{5.5}}(1+102k_{Fp}+$
$\displaystyle 53.91k_{Fn})(1-0.7087k_{Fn}+0.2537k_{Fn}^{2}+$ $\displaystyle
9.404k_{Fp}^{2}-1.589k_{Fn}k_{Fp})\times\left(\frac{m_{n}}{m_{p}^{*}}\right)^{2}$
$\displaystyle(0.0001313+1.248w^{2}+0.2403w^{3}+0.3257k_{Fp}$
$\displaystyle+0.5536wk_{Fp}-0.3237w^{4}+0.09786w^{2}k_{Fp})$
$\displaystyle=S_{np,1}+S_{np,2}$ (16)

Here, $u=k_{Fn}-1.556$, $v=k_{Fn}-2.126$ and $w=k_{Fn}-2.116$, provided
$k_{Fn}$ and $k_{Fp}$ are the respective Fermi momenta of neutron and proton.
In eqn. (3.2) both $S_{nn}$ and $S_{np}$ are expressed in the unit of
$millibarn$. The fit expressions are consisted within the range, $1.1\leq
k_{Fn}2.6$ fm-1 and $0.3\leq k_{Fp}\leq 1.2$ fm-1, which are consisted with
the two model stars considered in this analysis.

##### Lepton:

For the leptonic component, the collision frequencies corresponding to $ep$
and $ee$ collision are given by Gnedin and Yakovlev (1995)

$\displaystyle\nu_{ep}=\frac{4\pi^{2}}{5\hbar}\left(\frac{e^{2}}{\hbar
c}\right)^{2}\left(\frac{\hbar
k_{F_{e}}}{q_{0}}\right)^{3}\frac{\mu_{e}m_{p}^{*2}k_{B}^{2}T^{2}\hbar^{2}}{k_{F_{e}}^{2}}$
$\displaystyle\nu_{ee}=\frac{2\pi^{2}}{\hbar}\left(\frac{e^{2}}{\hbar
c}\right)^{2}\left(\frac{\hbar
k_{F_{e}}}{q_{0}}\right)^{3}\frac{k_{B}^{2}T^{2}}{\mu_{e}},$ (17)

where $q_{0}$ is the screening momentum transfer given by,
$q_{0}^{2}=(4e^{2}/\pi)(m_{e}^{*}k_{F_{e}}+m_{p}^{*}k_{F_{p}})$ (Gnedin and
Yakovlev, 1995). $k_{F_{e}}$ and $\mu_{e}$ are the Fermi momentum and chemical
potential of the electron respectively. Using eqn. (15) and (17) in eqn. (14),
the thermal conductivity of neutron and electron are estimated respectively,
in normal nuclear matter inside the core of a magnetar.

####  3.2.1 Effect of Superfluidity

The presence of nucleon superfluidity/superconductivity the thermal excitation
of the particles is minimized and consequently the inter-particle interaction
is subdued. As the dispersion relation of the particles is modified due to the
formation of nucleon pairing, the collision integral is also revised
accordingly due to the change in the available phase space of the particles.
The electrons remain non-superconducting throughout the star during the entire
thermal evolution period as their critical temperature is much lower than the
usual neutron star temperature.

##### Baryon:

Due to the modification in phase space the collision integral corresponding to
$nn$ and $np$ collisions, $J_{ci}$ ($c=n$), mentioned in eq. (3.2) is altered.
The phenomena of suppression of the internucleon interactions due to the
change in phase space is reflected in collision integral by the introduction
of a reduction factor $\mathcal{R}_{ni}$;
$J_{ni}^{sf}=J_{ni}\mathcal{R}_{ni}$. This in turn affects the quantity
$S_{ni}$ in eqn. (15). Thus in presence of neutron (proton) superfluidity
(superconductivity), eqn. (15) is modified as

$\nu_{ni}^{sf}=\frac{64m_{n}^{*}m_{i}^{*_{2}}k_{B}^{2}T^{2}S_{ni}^{sf}}{5m_{n}^{2}\hbar^{3}}.$
(18)

Here $S_{ni}^{sf}$ is the modified form of $S_{ni}$ due to the presence of
neutron and proton pairing, which are represented as

$\displaystyle S_{nn}^{sf}=S_{nn}\mathcal{R}_{n2}(y_{n})+\frac{3\times
14.57(1-0.0788k_{Fn}+0.0883k_{Fn}^{2})}{k_{Fn}^{1.5}(1-0.1114k_{Fn})}$
$\displaystyle\times\left(\frac{m_{n}}{m_{n}^{*}}\right)^{2}(0.4583+0.892s^{2}-0.5497s^{3}-0.06205k_{Fp}$
$\displaystyle+0.04022k_{Fp}^{2}+0.2122sk_{Fp})(\mathcal{R}_{n1}(y_{n})-\mathcal{R}_{n2}(y_{n})),$
$\displaystyle
S_{np}^{sf}=S_{np,2}\mathcal{R}_{p2}(y_{n},y_{p})+0.5S_{np,1}\times\\{3\mathcal{R}_{p1}(y_{n},y_{p})$
$\displaystyle-\mathcal{R}_{p2}(y_{n},y_{p})\\}$ (19)

Here $s=k_{Fn}-1.665$. The factors $\mathcal{R}_{n1}$, $\mathcal{R}_{n2}$,
$\mathcal{R}_{p1}$ and $\mathcal{R}_{p2}$ are mentioned in the ref. Baiko et
al. (2001). $y_{n}$ and $y_{p}$ are the energy gap functions for neutrons and
protons respectively, described in the section 3.1.1. The relaxation time of
collision is modified in presence of neutron and proton pairing accordingly
which is expressed as

$\tau_{n}^{sf}=\frac{\mathcal{R}(y_{n})}{\nu_{nn}^{sf}+\nu_{np}^{sf}}.$ (20)

$\mathcal{R}(y_{n})$ is another reduction factor different from
$\mathcal{R}_{ni}$ mentioned earlier. $\mathcal{R}(y_{n})$ arises from the
kinematic correction which is expresse as

$\mathcal{R}(y_{n})=3\sqrt{2}\left(\frac{y_{n}}{\pi}\right)^{1.5}e^{-y_{n}}$
(21)

In case of strong superfluidity ($T<<T_{c}$), $\mathcal{R}(y_{n})<<1$ while at
$T\geq T_{c}$, $\mathcal{R}(y_{n})=1$ . Consequently the expression of thermal
conductivity of neutron is changed as follows

$\kappa_{n}^{sf}=\frac{\pi^{2}Tn_{n}\tau_{n}^{sf}\mathcal{R}^{2}(y_{n})}{3m_{n}^{*}}$
(22)

One crucial point to be pointed out here is that while the superfluid neutrons
suppresses $\kappa_{n}$, it is enhanced by proton superconductivity, as proton
pairing leads to the reduction in collision frequency between neutron and
proton ($\nu_{np}^{sf}<\nu_{np}$).

Figure 3: Variation of neutron thermal conductivity with radius ($R$) in
normal matter ($\kappa_{n0}$) and in presence of nucleon
superfluidity/superconductivity ($\kappa_{n}$) for the lower and upper bounds
of $\Delta_{n}$ and $\Delta_{p}$ . Top row: for the massive star
($M=2.3M\odot$); bottom row: for the canonical mass star ($M=1.4M_{\odot}$).
Left panels: at $t=100$ years; right panels: at $t=1000$ years.

The scenario for neutron thermal conductivity $\kappa_{n}$ in absence of the
magnetic field is depicted in the fig. 3. The left and right panels of the
upper row correspond to the massive star at two different times $t=100$ and
$t=1000$ years respectively, while those in the lower row represent the
results for the canonical mass star. In all the panels, the variation of
$\kappa_{n}$ for the upper limits of $\Delta_{n}$ and $\Delta_{p}$ are
presented by the solid curve, while the dashed curve represents the
contribution in case of their lower bounds. The scenario of normal matter is
expressed by the dot-dashed curves.

In case of the massive star, $\kappa_{n}$ is very small at $R\gtrsim 12.5$ km
where the neutrons exist in ${}^{1}S_{0}$ pairing state. The protons start
forming ${}^{1}S_{0}$ pairing at $R\sim 12.5$ till $R\sim 9.4$ km ($R\sim
5.69$ km), corresponding to $\Delta_{p}^{lower(upper)}$. The pair formation of
neutrons via ${}^{3}P_{2}$ channel is only possible for $\Delta_{n}^{upper}$
within the radial zone $R\lesssim 12.5$ km up to the inner core, while such
pairing is not possible for $\Delta_{n}^{lower}$ within the region
$12.5\lesssim R\lesssim 11$ km because of extremely low value of energy gap,
as discussed in section 3.1.1. Overall, $\kappa_{n}$ exceeds its normal
counterpart $\kappa_{n0}$, because of the dominance of strong proton
superconductivity in this region of the star.

Thus, at $t=100$ years in the radial zone $12.5\gtrsim R\gtrsim 9.4$ km,
$\kappa_{n}$ varies within the range $10^{21}-6\times 10^{23}$ erg cm-1 s-1
K-1. At $R\sim 9.4$ km $\kappa_{n}$ is suddenly dropped to $3.5\times 10^{23}$
erg cm-1 s-1 K-1 and overlaps with $\kappa_{n0}$, increasing up to $8\times
10^{23}$ erg cm-1 s-1 K-1 towards the centre of the star.  In case of the
upper boundaries of $\Delta_{n}$ and $\Delta_{p}$ the scenario is somewhat
different. Here, at $R\sim 5.69$ km the superconducting protons become normal,
while the neutrons continue to exist in ${}^{3}P_{2}$ pairing state till the
inner core. This results in a sudden reduction in $\kappa_{n}$ and falls below
$\kappa_{n0}$ at $R\sim 5.69$ km from $2\times 10^{24}$ to $1.5\times 10^{23}$
erg cm-1 s-1 K-1 which increases slightly to $\sim 2.5\times 10^{23}$ erg cm-1
s-1 K-1 towards the deeper layers inside the star.

Comparing the left and right panels in the upper row of fig. 3 it is observed
that $\kappa_{n0}$ is reduced slightly at $t=1000$ years in comparison with
$t=100$ years. The distinction between $\kappa_{n}$ and $\kappa_{n0}$ at
different radial regions of the star becomes greater at $t=1000$ years, as
compared to $t=100$ years $i.e.$ in the region where $\kappa_{n}>\kappa_{n0}$,
$\kappa_{n}$ exhibits slight enhancement with time, while in the region where
$\kappa_{n}<\kappa_{n0}$, $\kappa_{n}$ is reduced with time.

Now for the canonical mass star, at $t=100$ years $\kappa_{n}$ is very low
($10^{16(15)}$ erg cm-1 s-1 K-1), around the surface, $R\gtrsim 11.8$ km in
which the neutrons exist in the singlet pairing state .

Within the range $11.8\gtrsim R\gtrsim 8.48$ km, the neutrons are unable to
form pairing in ${}^{3}P_{2}$ channel due to very small value of
$\Delta_{n}^{lower}$, while protons are superconducting in this region which
extends further till $R\sim 3.63$ km. This causes $\kappa_{n}$ to vary within
$\sim 10^{22}-5\times 10^{23}$ erg cm-1 s-1 K-1. At $R\sim 3.63$ km, as the
superconducting protons are converted to the normal ones, $\kappa_{n}$
experiences sudden drop to $3\times 10^{23}$ erg cm-1 s-1 K-1 and is
superposed with $\kappa_{n0}$. For the upper bounds of $\Delta_{n}$ and
$\Delta_{p}$, at $R\lesssim 11.8$ km neutron superfluidity is converted from
${}^{1}S_{0}$ to ${}^{3}P_{2}$ pairing state and both the nucleons remain in
their respective pairing state till the inner core. This leads to the
variation of $\kappa_{n}$ residing within the range $\sim 10^{22}-6.5\times
10^{23}$ erg cm-1 s-1 K-1, which is higher than $\kappa_{n0}$ as proton
superconductivity in ${}^{1}S_{0}$ channel is stronger than ${}^{3}P_{2}$
neutron pairing. The thermal evolution in the canonical mass star occurs in a
similar fashion as the massive star.

$M=2.3~{}M_{\odot}$
---
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
Radial zone$\rightarrow$ |  ${\color[rgb]{0,0,0}12.5-9.4}$ km |  ${\color[rgb]{0,0,0}\leq 9.4}$ km |  ${\color[rgb]{0,0,0}12.5-9.4}$ km |  ${\color[rgb]{0,0,0}\leq 9.4}$ km
$\kappa_{n0}$ | $10^{19}-3.5\times 10^{23}$ | $(3.5-8)\times 10^{23}$ | $10^{18}-3\times 10^{23}$ | $(3-5)\times 10^{23}$
$\kappa_{n}(\Delta_{n}^{lower},\Delta_{p}^{lower}$) | $10^{19}-6\times 10^{23}$ | $(3.5-8)\times 10^{23}$ | $10^{18}-9\times 10^{23}$ | $(3-5)\times 10^{23}$
$\kappa_{e0}$ | $3.5\times 10^{23}-5\times 10^{23}$ | $5\times 10^{23}$ | $1.5\times 10^{22}-3.5\times 10^{23}$ | $3.5\times 10^{23}$
$\kappa_{e}$($\Delta_{p}^{lower})$ | $1.5\times 10^{22}-10^{23}$ | $5\times 10^{23}$ | $(1-7)\times 10^{22}$ | $3.5\times 10^{23}$
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
---|---|---
Radial zone$\rightarrow$ |  $12.5-5.69$ km |  $\leq 5.69$ km |  $12.5-5.69$ km |  $\leq 5.69$ km
$\kappa_{n0}$ |  $10^{19}-6\times 10^{23}$ |  $(6-8)\times 10^{23}$ |  $10^{18}-4\times 10^{23}$ |  $(4-5)\times 10^{23}$
$\kappa_{n}(\Delta_{n}^{upper},\Delta_{p}^{upper})$ |  $10^{19}-2\times 10^{24}$ |  $(1.5-2.5)\times 10^{23}$ |  $10^{18}-3\times 10^{24}$ |  $(1-1.8)\times 10^{23}$
$\kappa_{e0}$ |  $(3.5-5)\times 10^{23}$ |  $5\times 10^{23}$ |  $2\times 10^{22}-3.5\times 10^{23}$ |  $3.5\times 10^{23}$
$\kappa_{e}(\Delta_{p}^{upper})$ |  $10^{22}-1.5\times 10^{23}$ |  $5\times 10^{23}$ |  $10^{22}-10^{23}$ |  $3.5\times 10^{23}$
Table 5: Estimation of the thermal conductivity, in the unit of erg cm-1 s-1
K-1, for the neutrons ($\kappa_{n}$) and electrons ($\kappa_{e}$) in the
different radial zones of the massive star ($M=2.3M_{\odot}$) governed by
DDME2 EoS, in presence of normal and superfluid/superconducting nucleons,
considering the lower and upper boundaries of their pairing energies
($\Delta_{n}$, $\Delta_{p}$) at the two different ages, $t=100$ and $t=1000$
years. Different radial zones are selected accordingly as the existence of
pairing of the different particle species, as mentioned in table 1 and 2.
$M=1.4~{}M_{\odot}$
---
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
Radial zone$\rightarrow$ |  ${\color[rgb]{0,0,0}11.8-3.63}$ km |  ${\color[rgb]{0,0,0}\leq 3.63}$ km |  ${\color[rgb]{0,0,0}11.86-3.63}$ km | ${\color[rgb]{0,0,0}\leq 3.63}$
$\kappa_{n0}$ | $10^{22}-3\times 10^{23}$ | $(3-3.2)\times 10^{23}$ | $10^{22}-2\times 10^{23}$ | $(2-2.2)\times 10^{23}$
$\kappa_{n}(\Delta_{n}^{lower},\Delta_{p}^{lower}$) | $10^{22}-5\times 10^{23}$ | $(3-3.2)\times 10^{23}$ | $10^{22}-7\times 10^{23}$ | $(2-2.2)\times 10^{23}$
$\kappa_{e0}$ | $(3-4)\times 10^{23}$ | $4\times 10^{23}$ | $(2-2.5)\times 10^{23}$ | $2.5\times 10^{23}$
$\kappa_{e}(\Delta_{p}^{lower})$ | $6\times 10^{21}-10^{23}$ | $4\times 10^{23}$ | $4\times 10^{21}-8\times 10^{22}$ | $2.5\times 10^{23}$
Time$\rightarrow$ |  $t=100$ yrs |  $t=1000$ yrs
---|---|---
Radial zone$\rightarrow$ |  $\leq 11.8$ km |  $\leq 11.8$ km
$\kappa_{n0}$ |  $10^{22}-3.2\times 10^{23}$ |  $10^{22}-2.2\times 10^{23}$
$\kappa_{n}(\Delta_{n}^{upper},\Delta_{p}^{upper})$ |  $10^{22}-6.5\times 10^{23}$ |  $10^{22}-1.2\times 10^{24}$
$\kappa_{e0}$ |  $(3-4)\times 10^{23}$ |  $(2-2.5)\times 10^{23}$
$\kappa_{e}(\Delta_{p}^{upper})$ |  $6\times 10^{21}-5\times 10^{22}$ |  $4\times 10^{21}-4.5\times 10^{22}$
Table 6: Same as table 5, for the canonical mass star $(M=1.4M_{\odot})$

Figure 4: Variation of electron thermal conductivity with radius ($R$) in
normal matter ($\kappa_{e0}$) and in presence of superconducting protons
($\kappa_{n}$). Top left panel: for the massive star ($M\sim 2.3M_{\odot}$) at
$t=100$ years; top right panel: same as the left panel at $t=1000$ years;
bottom panels: same as upper panels for the canonical mass star ($M\sim
1.4M_{\odot}$).

##### Leptons:

When the protons become superconducting, the screening momentum $q_{0}$is
modified as, $q_{0}^{sf}=(4e^{2}/\pi)(m_{e}^{*}k_{Fe}+m_{p}^{*}k_{Fp}Z_{p})$.
The reduction factor $Z_{p}$ is multiplied with the second term due to proton
superconductivity which results in overall enhancement of the screening
momenta $i.e.$ $q_{0}^{sf}>q_{0}$. This is understood by the fact that when
the proton pairing is formed, they are condensed to the ground state and the
electric field screening mainly occurs due to the electrons. Both $ee$ and
$ep$ collisions are affected by proton superconductivity. The corresponding
collision frequencies are expressed as Gnedin and Yakovlev (1995)

$\displaystyle\nu_{ep}^{sf}=R(y_{p})\frac{4\pi^{2}}{5\hbar}\left(\frac{e^{2}}{\hbar
c}\right)^{2}\left(\frac{\hbar
k_{F_{e}}}{q_{0}^{sf}}\right)^{3}\frac{\mu_{e}m_{p}^{*2}k_{B}^{2}T^{2}}{p_{F_{e}}^{2}}$
$\displaystyle\nu_{ee}^{sf}=\frac{2\pi^{2}}{\hbar}\left(\frac{e^{2}}{\hbar
c}\right)^{2}\left(\frac{\hbar
k_{F_{e}}}{q_{0}^{sf}}\right)^{3}\frac{k_{B}^{2}T^{2}}{\mu_{e}}$ (23)

The additional reduction factor $R(y_{p})$ comes from the modification in
phase space due to proton pairing. Although both the screening momenta
$q_{0}^{sf}$ and $R(y_{p})$ affect the $ep$ collision, it is observed that the
effect of $R(y_{p})$ is more dominating which ultimately causes reduction in
$\nu_{ep}$. On the other hand, as $q_{0}^{sf}>q_{0}$, $\nu_{ee}^{sf}$ becomes
lower than $\nu_{ee}$. In presence of proton pairing the relaxation time is
modified and is expressed as, $\tau^{sf}_{e}=1/(\nu_{ee}^{sf}+\nu_{ep}^{sf})$.
Since the contribution of $\nu_{ep}^{sf}$ becomes almost negligible and
$\nu_{ee}^{sf}$ exceeds $\nu_{ee}$, $\tau^{sf}_{e}<\tau_{e}$. Therefore,
following (3.2), the thermal conductivity of the electrons undergoes depletion
eventually, in presence of the proton superconductivity. Electrons remain
normal all throughout the star. As the $en$ interaction is not taken into
consideration, neutron superfluidity does not affect the electron thermal
conductivity $\kappa_{e}$.

Figure 5: Left panel: Variation of parallel ($||$) and perpendicular ($\perp$)
components of the electron thermal conductivity with radius ($R$) in presence
of magnetic field with universal profile for the star of mass
$M=2.3M_{\odot}$, in the normal matter; right panel: Same for the star of mass
$M=1.4M_{\odot}$.

The variation of $\kappa_{e}$ with radius is described in fig. 4, similar to
fig. 3, the upper and lower row corresponding to the massive and canonical
mass stars, with the left and right panel exhibiting the results at two
different times, $t=100$ and $t=1000$ years respectively.

As mentioned earlier, for the massive star the protons are normal within a
very narrow region in the vicinity of the surface ($R>12.5$ km) and
$\kappa_{e}$ remains in the range $\sim(2.5-7)\times 10^{22}$ erg cm-1 s-1
K-1. The normal protons shift to ${}^{1}S_{0}$ superconducting state in the
radial zone $12.5\gtrsim R\gtrsim 9.4$ km ($12.5\gtrsim R\gtrsim 5.69$ km) for
$\Delta_{p}^{\text{lower(upper)}}$ which causes $\kappa_{e}$ to be decreased
suddenly at $R=12.5$ km from $\sim 7\times 10^{22}$ to $\sim 1.5\times
10^{22}$ erg cm-1 s-1 K-1 at $t=100$ years and then increases gradually
further towards the inner core of the star and reach at $\sim 10^{23}$ ($\sim
1.5\times 10^{23}$) erg cm-1 s-1 K-1 at $R\sim 9.36$ km ($R\sim 5.69$ km). The
proton superconductivity is absent beyond $R\sim 9.4$ km ($R\sim 5.69$ km)
corresponding to $\Delta_{p}^{lower(upper)}$, for which $\kappa_{e}$ is
drastically increased from $10^{23}(1.5\times 10^{23})$ erg cm-1 s-1 K-1, to
overlap with $\kappa_{e0}$ and remains nearly constant at a value $\sim
5\times 10^{23}$ erg cm-1 s-1 K-1 till the inner core.

In case of the canonical mass star, the protons reside in ${}^{1}S_{0}$
pairing state within the range $11.8\gtrsim R\gtrsim 3.63$ km, corresponding
to $\Delta_{p}^{lower}$, while they remain superconducting approximately all
through the entire star $i.e.$ in the region $R\lesssim 11.8$ km for
$\Delta_{p}^{upper}$. In the narrow radial zone around the surface $R\gtrsim
11.8$ km, for $t=100$ years the value of $\kappa_{e}$ dwells within $\sim
3\times 10^{22}-10^{23}$ erg cm-1 s-1 K-1. At $R\sim 11.8$ km $\kappa_{e}$
declines instantly from $\sim 10^{23}$ to $\sim 6\times 10^{21}$ erg cm-1 s-1
K-1. Beyond that in case of $\Delta_{p}^{\text{lower}}$ , in the region
$11.8\gtrsim R\gtrsim 3.63$ km, $\kappa_{e}$ is confined within the range
$\sim 6\times 10^{21}-10^{23}$, rising instantly to $\sim 4\times 10^{23}$ erg
cm-1 s-1 K-1 at $R\sim 3.63$ km, as at $R\lesssim 3.63$ km the
$\Delta_{p}^{lower}$ becomes zero. $\kappa_{e}$ remains the same as
$\kappa_{e0}$, nearly constant at $4\times 10^{23}$ erg cm-1 s-1 K-1 till the
inner core. However, in case of $\Delta_{p}^{\text{upper}}$ the protons exist
in the singlet pairing state till the inner core, starting from $R\sim 11.8$
km. In this scenario $\kappa_{e}$ varies within the range $\sim 6\times
10^{21}-5\times 10^{22}$ erg cm-1 s-1 K-1 which is lower than $\kappa_{e0}$ by
an order approximately. At the latter time $t=1000$ years both $\kappa_{e}$
and $\kappa_{e0}$ are observed to be reduced a little as expected similar to
the previous cases.

##  4 Thermal properties in presence of magnetic field

###  4.1 In normal matter

Till now the effect of magnetic field was not taken into account in our
analysis. Since the magnetic field is associated with the motion of charged
particles, its effect is discernible only in case of the electron thermal
conductivity. It is to be noted that the magnetic field affects the electrons
much more than the protons as the electron mass is much smaller than the mass
of the proton. In presence of magnetic field the motion of the electrons along
the field direction remains the same, while along the direction perpendicular
to the magnetic field it is highly suppressed. As a consequence, the thermal
conductivity of electrons becomes anisotropic. In this context it is to be
mentioned that although the neutrons are affected by the magnetic field due to
its non-zero spin, the trajectory of its motion remains unchanged in presence
of the magnetic field. This implies that unlike the electrons, the thermal
conductivity of neutrons are not decomposed into distinct components along the
different directions. The heat capacity, given by eqn. (2) also remains
unaffected in normal matter, since it is not associated with the dynamical
motion of the particles. However, in presence of neutron superfluidity and
superconductivity, both the specific heat and neutron thermal conductivity are
modified, along with the thermal conductivity of the electrons, which are
discussed in the next section.

The expression of the thermal conductivity tensor of the electrons in presence
of magnetic field is given by (Potekhin, 1999)

$\kappa_{e0,ij}=\int\frac{(\mu-\epsilon)^{2}}{T}\frac{\mathcal{N}_{B}(\epsilon)}{\epsilon}\tau_{ij}(\epsilon)(-\frac{\partial
f_{0}}{\partial\epsilon})d\epsilon.$ (24)

Here $\mu$ and $\epsilon$ represent the chemical potential and energy of the
integral respectively, while $T$ is the temperature. $f_{0}$ is the
equilibrium Fermi-Dirac distribution function, given by eqn (7) as $f_{p}$ for
the protons. $\mathcal{N}_{B}$ is the baryon number density in presence of the
magnetic field.

In this context it is to be mentioned that in presence of the magnetic field
the electrons are distributed into several Landau levels along the direction
transverse to the magnetic field. However, in our analysis, the number of
Landau levels are quite large, even in the presence of the strong magnetic
field $B_{c}=10^{16}$ G. Therefore, the effect of magnetic field quantization
becomes negligible and the system can be treated classically, although the
transverse motion of the electron is affected Potekhin (1999).

In eqn. (24) $\tau_{ij}$ is the relaxation time corresponding to the electron
scattering in the magnetic field. The subscript ’$0$’ implies the case of
normal matter. Due to the symmetry property of $\kappa_{ij}$,
$\tau_{xx}=\tau_{yy}$ which are the transverse components of the electron
conductivity and $\tau_{zz}$ is its longitudinal counterpart, provided that
the magnetic field is pointed along the $z$-direction. From now onwards, we
denote $\tau_{zz}$ and $\tau_{xx}$ or $\tau_{yy}$ as $\tau_{||}$ and
$\tau_{\perp}$ respectively. To determine the components of the relaxation
time, we follow the formalism described in the ref. (Potekhin, 1999) in which
the longitudinal and transverse components of the relaxation time are
expressed as

$\displaystyle\tau_{||}=\tau_{0},$ (25)
$\displaystyle\tau_{\perp}=\frac{\tau_{0}}{1+(\omega_{g}\tau_{0})^{2}}.$

Here, $\omega_{g}=eB/\epsilon$ which is known as the gyrofrequency of
electron. $\tau_{0}$ is the relaxation time in absence of magnetic field. Here
the nucleon superfluidity/superconductivity effects are not taken into
account.

In fig. 5, the two components of $\kappa_{e0}$ have been illustrated at two
different times, $t=100$ years and $1000$ years considering the universal
profile of magnetic field with the central magnetic field $B_{c}=10^{16}$ G.
The left and right panel correspond to the massive and canonical mass star
respectively. In both the panels, the solid and the dashed curve correspond to
the parallel component $\kappa_{e0,||}$ at $t=100$ and $t=1000$ years
respectively. The dot-dashed and dot double-dashed line represent the same for
the perpendicular component $\kappa_{e0,\perp}$.

It can be observed from the left panel, in case of the massive star both
$\kappa_{e0,||}$ and $\kappa_{e0,\perp}$ varies slowly throughout the star,
falling rapidly near the surface of the star. The order of $\kappa_{e0,\perp}$
is much smaller compared to $\kappa_{e0,||}$ in case of both the stars. At
$t=100$ yrs, in case of the massive star $\kappa_{e0,||}$ varies within $\sim
4-7\times 10^{5}$ erg cm-1 s-1 K-1 starting from its surface to the centre,
while at the latter time $t=1000$ yrs, its order is reduced even further
residing within the range, $1-10^{4}$ erg cm-1 s-1 K-1. The order of
$\kappa_{e,\perp}$ is nearly $10$ at the centre of the star at $t=100$ years
which decreases to a much smaller value towards the surface. Its order is
reduced even further at the latter time, $t=1000$ years. It is to be mentioned
that in absence of the magnetic field in normal matter, $\kappa_{e0}$ remains
nearly constant throughout the core and its order is much higher ($\sim
10^{23}$), which is observed to be diminished remarkably in presence of the
magnetic field.

In case of the canonical mass star, the order of $\kappa_{e0,||}$ and
$\kappa_{e0,\perp}$ are lower by at least two orders, as compared to the
massive star. At $t=100$ years, $\kappa_{e,||}$ acquires a value of $\sim
7\times 10^{3}$ erg cm-1 s-1 K-1 at the inner core which is decreases towards
the surface at $t=1000$ years. $\kappa_{e,\perp}$ is much lower as compared to
$\kappa_{e,||}$, similar to the case of the massive star. In both the stars,
the order of both $\kappa_{e0,||}$ and $\kappa_{e0,\perp}$ are reduced by one
or two orders at the latter time.

Figure 6: Variation of $C_{v}$ inside the star in the presence of magnetic
field following the universal profile ($B_{uni}$) at $t=100years$ for upper
and lower delta. Left panel: For star of mass $M\sim 1.4M_{\odot}$. Right
panel: For $M\sim 2.3M_{\odot}$. Solid lines and dotted lines corresponds to
total heat capacity and partial heat capacity of neutrons with superfluidity
respectively. Long dash-and small dash curves are for lower delta while long
dash - dotted and dash - double dotted are for upper delta curves.

Figure 7: Left panel: Variation of $\kappa_{n}$ in presence of magnetic field
following the universal profile ($B_{uni}$) for the upper and lower bounds of
$\Delta_{n}$ and $\Delta_{p}$ at $t=100$ years for the massive star ($M\sim
2.3M_{\odot}$); right panel: Same for the low mass star ($M\sim
1.4M_{\odot}$). The solid and the dashed line correspond to the nucleon
pairing with pairing energy gap functions, $\Delta_{n}$ and $\Delta_{p}$
following upper and lower bounds respectively. The dot-dashed line is related
to the case of non-superfluid/superconducting matter.

###  4.2 In superfluid/superconducting matter

Figure 8: Left panel : Variation of parallel ($\kappa_{e,||}$) of electron
thermal conductivity with radial distance inside the massive star
($M=2.3M_{\odot}$); right panel: Variation of parallel ($\kappa_{e,\perp}$) of
electron thermal conductivity with radial distance inside the massive star

Figure 9: Same as fig. 8, for the canonical mass star ($M=1.4M_{\odot}$)

In this section the effect of magnetic field on the thermal properties is
analysed considering the stellar matter to be consisted of superfluid neutrons
and superconducting protons. The large magnetic field of the magnetar
suppresses the proton superconductivity which causes the proton pairing to
remain active in a comparatively narrower radial section towards the surface
(Sinha and Sedrakian, 2015). In this region the frequency of $ep$ collision is
also expected to be suppressed. On the other hand, in case of the neutrons,
the effect of magnetic field on superfluidity is still unknown (Haskell and
Sedrakian, 2018). Therefore, in this work, we focus only on the effect of
proton superconductivity in presence of magnetic field and analyze the thermal
properties accordingly. We present our results only for the universal profile
of magnetic field, as the results do not differ significantly in the two kinds
of field profiles. For example, as observed from the table 1 and 2, in the
massive star the active region of proton superconductivity is suppressed to
the radial range $\sim 12.5-10.85$ km ($\sim 12.5-9.44$ km) and $\sim
12.5-10.98$ km ($\sim 12.5-9.86$ km) for the universal and exponential profile
respectively, in case of $\Delta_{p}^{lower(upper)}$. Similarly, for the
canonical mass star, the universal profile suspends the proton
superconductivity in the radial zone $12\gtrsim R\gtrsim 8.71(6.45)$ km for
the universal profile and $12\gtrsim R\gtrsim 9.27(7.22)$ km for the
exponential profile of the magnetic field, in case of
$\Delta_{p}^{lower(upper)}$.

The combined effect of the magnetic field and proton superconductivity is
illustrated in case of the heat capacity in fig. 6. The results are exhibited
only at time $t=100$ years. For the massive star, at $R\sim 10.85$ km ($R\sim
9.44$ km) $C_{v}$ experiences a sudden jump from $\sim 6\times 10^{19}$ ($\sim
2\times 10^{19}$) to $\sim 2\times 10^{20}$ ($\sim 1.8\times 10^{20}$) erg
cm-3 K-1, corresponding to $\Delta_{p}^{lower(upper)}$. The rest of the
features are observed to be identical to the non-magnetic case shown in fig.
2. Within the radial range $10.85~{}\text{km}\gtrsim R\gtrsim 9.36$ km
($9.44~{}\text{km}\gtrsim R\gtrsim 5.69$ km) in case of
$\Delta_{p}^{lower(upper)}$, the order of $C_{v}$ is $\sim 2\times 10^{20}$
($\sim 1.8\times 10^{20}$) and $\sim 6\times 10^{19}$ ($\sim(2-5)\times
10^{19}$) erg cm-3 K-1 respectively, in presence and absence of the magnetic
field.

For the canonical mass star, at $R\sim 8.71(6.45)$ km $C_{v}$ is changed
promptly from $\sim 7\times 10^{19}$ ($2\times 10^{19}$) to $\sim 2\times
10^{20}$ ($1.5\times 10^{20}$) erg cm-3 K-1 in case of
$\Delta_{p}^{lower(upper)}$. Also, within the radial zone
$8.71~{}\text{km}\lesssim R\lesssim 3.63$ km ($R\lesssim 6.45$ km),
corresponding to $\Delta_{p}^{lower(upper)}$, the value of $C_{v}$ is $\sim
2.8\times 10^{20}$($\sim 2\times 10^{20}$) erg cm-3 K-1 in presence of the
magnetic field, while it is somewhat lesser, $\sim 6\times 10^{19}-1.5\times
10^{20}$ ($\sim 3\times 10^{19}$) erg cm-3 K-1, in absence of the magnetic
field. This implies the overall enhancement in the heat capacity in presence
of the magnetic field, if the protons present in the stellar matter are
superconducting.

Fig. 7 depicts the variation of $\kappa_{n}$ in presence of both proton
superconductivity and the magnetic field at time $t=100$ years. For the
massive star, at $R\sim 10.85$ km ($R\sim 9.44$ km) $\kappa_{n}$ is dropped
from $\sim 3.5\times 10^{23}$ ($\sim 8\times 10^{23}$) to $\sim 2\times
10^{23}$ ($\sim 7\times 10^{22}$) erg cm-1 s-1 K-1 corresponding to
$\Delta_{p}^{lower(upper)}$ respectively. Now it can be pointed out that for
$\Delta_{p}^{upper}$, within the radial range $R\sim 9.4-5.69$ km $\kappa_{n}$
is lower by at least an order in the scenario of magnetic case as compared to
the non-magnetic case, as observed by comparing fig. 3 (left panel in the
upper row ) and 7. Such distinction in the order of $\kappa_{n}$ is not
significant corresponding to $\Delta_{p}^{lower}$ within the range $R\sim
10.85-9.4$ km.

Now in case of the canonical mass star, at $R\sim 8.71$ km there is a rapid
reduction in $\kappa_{n}$ from $\sim 3\times 10^{23}$ erg cm-1 s-1 K-1 to
$\sim 1.5\times 10^{23}$ erg cm-1 s-1 K-1 for $\Delta_{p}^{lower}$, while for
$\Delta_{p}^{upper}$ the jump occurs at $R\sim 6.45$ km from $\sim 8\times
10^{23}$ erg cm-1 s-1 K-1 to $\sim 8\times 10^{22}$ erg cm-1 s-1 K-1. Within
the range $R\sim 8.71-3.63$ km there is no significant difference in the order
of $\kappa_{n}$ in presence of the magnetic field compared to the case of zero
magnetic field for $\Delta_{p}^{lower}$, while in the scenario of
$\Delta_{p}^{upper}$ the difference is about an order in the region $R\lesssim
6.45$ km.

In fig. 8, the left and right panel exhibit the variation of $\kappa_{e,||}$
and $\kappa_{e,\perp}$ with radius for the massive star at the time $t=100$
years, while for the canonical mass star similar results are presented in fig.
9. In case of the massive star the value of $\kappa_{e,||}$ is raised
instantaneously from $\sim 2$ ($\sim 1$) erg cm-1 s-1 K-1 to $\sim 5\times
10^{3}$ ($\sim 4\times 10^{4}$) erg cm-1 s-1 K-1 at $R\sim 10.85$ km ($R\sim
9.4$ km) and coincide with $\kappa_{e0,||}$, as proton superconductivity is
limited in the region $12.5~{}\text{km}\gtrsim R\gtrsim 10.85$ km
($12.5~{}\text{km}\gtrsim R\gtrsim 9.4$ km), corresponding to
$\Delta_{p}^{lower(upper)}$ while, $\kappa_{e,\perp}$ also experiences abrupt
elevation from $\sim 0.08$ ($\sim 0.1$) erg cm-1 s-1 K-1 to $\sim 0.4$ ($\sim
1.5$) erg cm-1 s-1 K-1.

Similarly for the canonical mass star, $\kappa_{e,||}$ is suddenly increased
from $\sim 6(\sim 2)$ to $2\times 10^{3}$ ($4\times 10^{3}$) erg cm-1 s-1 K-1,
while there is a sudden rise in $\kappa_{e,\perp}$ to $\sim 0.4$ ($\sim 0.8$)
erg cm-1 s-1 K-1 from $\sim 0.002$ ($\sim 0.004$) erg cm-1 s-1 K-1 at $R\sim
8.71(6.45)$ km.

$M=2.3~{}M_{\odot},t=100$ yrs
---
Radial zone$\rightarrow$ |  $12.5-10.85$ km |  $\leq 10.85$ km
$\kappa_{e0,||}$ | $4-7\times 10^{3}$ | $7\times 10^{3}-6\times 10^{5}$
$\kappa_{e,||}(\Delta_{p}^{lower})$ | $0.2-2$ | $7\times 10^{3}-6\times 10^{5}$
$\kappa_{e0,\perp}$ | $0.3-0.5$ | $0.5-5.5$
$\kappa_{e,\perp}(\Delta_{p}^{lower})$ | $0.03-0.08$ | $0.4-5.5$
Radial zone$\rightarrow$ |  $12.5-9.44$ km |  $\leq 9.44$ km
$\kappa_{e0,||}$ | $4-4\times 10^{4}$ | $4\times 10^{4}-6\times 10^{5}$
$\kappa_{e,||}(\Delta_{p}^{upper})$ | $0.2-1$ | $4\times 10^{4}-6\times 10^{5}$
$\kappa_{e0,\perp}$ | $0.3-1.5$ | $1.5-5.5$
$\kappa_{e,\perp}(\Delta_{p}^{upper})$ | $0.01-0.15$ | $1.5-5.5$
Table 7: Estimation of the two components of $\kappa_{e0}$ and $\kappa_{e}$,
in presence of magnetic field, following the universal profile having the
value at its core $B_{c}=10^{16}$ G.

$M=1.4~{}M_{\odot},t=100$ yrs
---
Radial zone$\rightarrow$ |  $11.8-8.71$ km |  $\leq 8.71$ km
$\kappa_{e0,||}$ | $10-1.5\times 10^{3}$ | $(1.5-7)\times 10^{3}$
$\kappa_{e,||}(\Delta_{p}^{lower})$ | $\sim 6$ | $(2-7)\times 10^{3}$
$\kappa_{e0,\perp}$ | $0.001-0.4$ | $0.4-1$
$\kappa_{e,\perp}(\Delta_{p}^{lower})$ | $0.0007-0.002$ | $0.4-1$
Radial zone$\rightarrow$ |  $11.8-6.45$ km |  $\leq 6.45$ km
$\kappa_{e0,||}$ | $10-4\times 10^{3}$ | $(4-7)\times 10^{3}$
$\kappa_{e,||}$($\Delta_{p}^{upper}$) | $\sim 2$ | $(4-7)\times 10^{3}$
$\kappa_{e0,\perp}$ | $0.001-0.8$ | $0.8-1$
$\kappa_{e,\perp}$($\Delta_{p}^{upper}$) | $0.0002-0.004$ | $0.8-1$
Table 8: Same as table 7, in presence of magnetic field, with the universal
profile having the value at the core $B_{c}=10^{16}$ G.

##  5 Discussion

We study the behaviour of heat capacity and thermal conductivity inside the
magnetars having different mass limits $M\sim 2.3M_{\odot}$ and $M\sim
1.4M_{\odot}$ with their interior matter having $npe$ configuration governed
by DDME2 EoS. We analyse the effect of superfluidity/superconductivity,
considering two extreme limits of the pairing energy gaps of the nucleons both
in absence and presence of magnetic field. We conduct our analysis taking the
universal profile of the magnetic field into consideration which generates the
field value at the core of the magnetars about $10^{16}$ G. Different nucleons
form pairs in different layers inside the magnetars having different
strengths, once their respective critical temperatures are declined beyond
certain critical limits, $e.g.$ the neutrons are in singlet state around the
surface, but form triplet pairing in the deeper layers of the magnetars. The
protons are only capable of existing in the singlet state in certain regions
of the star, which is further dwindled if the effect of the magnetic field is
taken into account. The nucleons remain in the paired state up to a much
deeper layer inside the star from the surface for the upper boundaries of
$\Delta_{n}$ and $\Delta_{p}$, as compared to their lower limits. In the
massive star, all the nucleons become normal around the inner core. In the
canonical mass star the neutrons remain superfluid till the inner core, while
the proton pairing remains limited to specific region of the star. It should
be noted that the electrons remain normal all throughout the star at any
instant.

In absence of magnetic field, the resultant heat capacity of all the particle
species is diminished overall to some extent in the domains where the nucleons
are superfluid/superconducting. When the effect of the magnetic field is
implemented along with the nucleon pairing, in a certain region of the star
the heat capacity is increased relatively. This is observed more prominently
for the upper boundaries of the $\Delta_{n}$ and $\Delta_{p}$ in comparison
with their lower boundaries.

On the other hand, the thermal conductivity exhibits much diverse behaviour.
Unlike the scenario of heat capacity, in absence of the magnetic field, the
thermal conductivity of neutrons increases in the regions of the star where
the proton superconductivity dominates over neutron superfluidity, but
decreases in the area in which the strength of neutron pairing oversteps the
proton pairing. The thermal conductivity of the electrons are reduced in the
regions in which the protons are superconducting. Such enhancement or
declination is much greater in case of the upper bounds of $\Delta_{n}$ and
$\Delta_{p}$, as compared to their lower bounds. The electron thermal
conductivity is reduced in presence of proton superconductivity. Needless to
say, the reduction is higher in case of the upper bounds of the pairing
energies of the nucleons.

Additionally, if the effect of magnetic field is incorporated, the neutron
thermal conductivity is curtailed in some specific regions of the stars. The
electron conductivity is splitted into two components, in the directions along
and perpendicular to the magnetic field. Both the components are reduced even
in normal matter configuration, by at least eighteen and twenty four orders of
magnitude in case of the massive and canonical mass star at their inner core
regions.In superfluid/superconducting matter, both of the components are
minimized in certain regions near the surface due to the existence of proton
superconductivity. This implies a slower rate of heat transfer inside the star
due to the inclusion of magnetic field. Therefore, the magnetar may cool down
at a slower rate.

We also inspect the thermal evolution of the two properties to elucidate the
cooling phenomena of the magnetars studying the two quantities at an earlier
time $t=100$ years and a latter time $t=1000$ years. The order of the heat
capacity is lowered everywhere inside the stars with time, irrespective of
whether the matter configuration is normal or superfluid/superconducting. In
normal matter the thermal conductivity is decreased with time as expected.
However, in presence of nucleon pairing, the deviation of the quantity from
its normal counterpart becomes larger with time. For example, $\kappa_{n}$
attains larger value than its normal counterpart at certain regions of the
star which gets even larger at $t=1000$ years than at $t=100$ years. The same
scenario remains intact both in presence and absence of magnetic field, which
possibly exhibits the cooling phenomenon occurring in the magnetars. The
massive star undergoes faster cooling than the canonical mass star, due to its
larger value of thermal conductivity.

In case of electron thermal conductivity, $\kappa_{e}$, its value remains the
same for the upper and lower bounds of $\Delta_{p}$. However the presence of
proton superconductivity causes one order of magnitude change in $\kappa_{e}$
compared to the case when protons are normal. Overall both $\kappa_{n}$ and
$\kappa_{e}$ for the massive star are a little higher compared to the low mass
star, although there is no significant change in their order of magnitude.
This implies a shorter thermal relaxation period of the massive star
($t\lesssim 100$ years) in comparison with the low mass star ($t\sim 100-200$
years). Larger conductivity of the massive star also indicates faster cooling.
In presence of magnetic field $\kappa_{e}$ is anisotropic. The component
parallel to the magnetic field ($\kappa_{e}^{||}$) is about $\sim 4-5$ orders
of magnitude higher than the normal component ($\kappa_{e}^{\perp}$) in normal
$npe$ matter. $\kappa_{n}$ is studied in presence of magnetic field when the
nucleons inside the star form Cooper pairs. Magnetic field does not affect the
magnitude of $\kappa_{n}$. In presence of magnetic field, the parallel and
perpendicular components of $\kappa_{e}$ are not expected to show much
deviation when proton superconductivity is taken into account. In case of
different EoSs the order of magnitude of the thermal properties are varied
along with the superfluid/superconducting energy gap profile inside the star.

## Acknowledgements

The authors would like to thank the anonymous referee for their constructive
comments which enhance the quality of the manuscript. The authors acknowledge
the funding support from Science and Engineering Research Board, Department of
Science and Technology, Government of India through Project No.
CRG/2022/000069.

## Data Availability

Data sharing is not applicable to this article as no data sets were generated
during this study.

## References

  * Abrikosov and Khalatnikov (1959) Abrikosov, A.A., Khalatnikov, I.M., 1959\.  The theory of a fermi liquid (the properties of liquid 3He at low temperatures).  Reports on Progress in Physics 22, 329–367.  doi:10.1088/0034-4885/22/1/310.
  * Baiko et al. (2001) Baiko, D.A., Haensel, P., Yakovlev, D.G., 2001.  Thermal conductivity of neutrons in neutron star cores.  aap 374, 151–163.  doi:10.1051/0004-6361:20010621, arXiv:astro-ph/0105105.
  * Baldo et al. (1992) Baldo, M., Cugnon, J., Lejeune, A., Lombardo, U., 1992\.  Proton and neutron superfluidity in neutron star matter.  nphysa 536, 349--365.  doi:10.1016/0375-9474(92)90387-Y.
  * Bandyopadhyay et al. (1997) Bandyopadhyay, D., Chakrabarty, S., Pal, S., 1997.  Quantizing magnetic field and quark-hadron phase transition in a neutron star.  Phys. Rev. Lett. 79, 2176--2179.  URL: https://link.aps.org/doi/10.1103/PhysRevLett.79.2176, doi:10.1103/PhysRevLett.79.2176.
  * Blumer and Safi-Harb (2020) Blumer, H., Safi-Harb, S., 2020\.  Chandra Observations of the Newly Discovered Magnetar Swift J1818.0-1607.  apjl 904, L19.  doi:10.3847/2041-8213/abc6a2, arXiv:2011.00324.
  * Chamel (2017) Chamel, N., 2017.  Superfluidity and Superconductivity in Neutron Stars.  Journal of Astrophysics and Astronomy 38, 43.  doi:10.1007/s12036-017-9470-9, arXiv:1709.07288.
  * Chatterjee et al. (2019) Chatterjee, D., Novak, J., Oertel, M., 2019.  Magnetic field distribution in magnetars.  prc 99, 055811.  doi:10.1103/PhysRevC.99.055811, arXiv:1808.01778.
  * Chugunov and Haensel (2007) Chugunov, A.I., Haensel, P., 2007\.  Thermal conductivity of ions in a neutron star envelope.  mnras 381, 1143--1153.  doi:10.1111/j.1365-2966.2007.12301.x, arXiv:0707.4614.
  * Coti Zelati et al. (2020) Coti Zelati, F., Borghese, A., Rea, N., Israel, G.L., Esposito, P., et al., 2020\.  NuSTAR observation of the newly discovered magnetar SGR 1830-0645.  The Astronomer’s Telegram 14092, 1\.
  * Cumming et al. (2017) Cumming, A., Brown, E.F., Fattoyev, F.J., Horowitz, C.J., Page, D., Reddy, S., 2017\.  Lower limit on the heat capacity of the neutron star core.  prc 95, 025806.  doi:10.1103/PhysRevC.95.025806, arXiv:1608.07532.
  * De Blasio (2000) De Blasio, F.V., 2000.  A dense two-component plasma in a strong gravity field and thermal conductivity of neutron stars.  aap 353, 1129--1133.
  * Flowers and Itoh (1976) Flowers, E., Itoh, N., 1976\.  Transport properties of dense matter.  apj 206, 218--242.  doi:10.1086/154375.
  * Flowers and Itoh (1979) Flowers, E., Itoh, N., 1979\.  Transport properties of dense matter. II.  apj 230, 847--858.  doi:10.1086/157145.
  * Gnedin and Yakovlev (1995) Gnedin, O.Y., Yakovlev, D.G., 1995\.  Thermal conductivity of electrons and muons in neutron star cores.  nphysa 582, 697--716.  doi:10.1016/0375-9474(94)00503-F.
  * Gnedin et al. (2001) Gnedin, O.Y., Yakovlev, D.G., Potekhin, A.Y., 2001.  Thermal relaxation in young neutron stars.  Mon. Not. Roy. Astron. Soc. 324, 725\.  doi:10.1046/j.1365-8711.2001.04359.x, arXiv:astro-ph/0012306.
  * Haensel et al. (2000) Haensel, P., Levenfish, K.P., Yakovlev, D.G., 2000.  Bulk viscosity in superfluid neutron star cores. I. Direct Urca processes in npemu matter.  aap 357, 1157--1169.  arXiv:astro-ph/0004183.
  * Halpern and Wang (1997) Halpern, J.P., Wang, F.Y.H., 1997\.  A Broadband X-Ray Study of the Geminga Pulsar.  apj 477, 905--915.  doi:10.1086/303743.
  * Haskell and Sedrakian (2018) Haskell, B., Sedrakian, A., 2018\.  Superfluidity and Superconductivity in Neutron Stars. volume 457.  p. 401.  doi:10.1007/978-3-319-97616-7_8.
  * Heinke and Ho (2010) Heinke, C.O., Ho, W.C.G., 2010\.  Direct Observation of the Cooling of the Cassiopeia A Neutron Star.  apjl 719, L167--L171.  doi:10.1088/2041-8205/719/2/L167, arXiv:1007.4719.
  * Hoffberg et al. (1970) Hoffberg, M., Glassgold, A.E., Richardson, R.W., Ruderman, M., 1970\.  Anisotropic Superfluidity in Neutron Star Matter.  prl 24, 775--777.  doi:10.1103/PhysRevLett.24.775.
  * Lalazissis et al. (2005) Lalazissis, G.A., Nikšić, T., Vretenar, D., Ring, P., 2005\.  New relativistic mean-field interaction with density-dependent meson-nucleon couplings.  prc 71, 024312.  doi:10.1103/PhysRevC.71.024312.
  * Landau (1957) Landau, L., 1957.  The Theory of a Fermi Liquid.  Sov. Phys. JETP 3, 920\.
  * Lander (2014) Lander, S.K., 2014.  The contrasting magnetic fields of superconducting pulsars and magnetars.  mnras 437, 424--436.  doi:10.1093/mnras/stt1894, arXiv:1307.7020.
  * Levenfish and Yakovlev (1994) Levenfish, K.P., Yakovlev, D.G., 1994\.  Specific heat of neutron star cores with superfluid nucleons.  Astronomy Reports 38, 247--251.
  * Lifshitz et al. (1981) Lifshitz, E.M., Pitaevskii, L.P., Nyboer, B.R.A., 1981.  Book-Review - Statistical Physics - Part Two.  ssr 28, 450.
  * Manuel et al. (2014) Manuel, C., Sarkar, S., Tolos, L., 2014.  Thermal conductivity due to phonons in the core of superfluid neutron stars.  prc 90, 055803.  doi:10.1103/PhysRevC.90.055803, arXiv:1407.7431.
  * Migdal (1959) Migdal, A.B., 1959.  Superfluidity and the moments of inertia of nuclei.  Nuclear Physics 13, 655--674.  doi:10.1016/0029-5582(59)90264-0.
  * Page et al. (2011) Page, D., Prakash, M., Lattimer, J.M., Steiner, A.W., 2011\.  Rapid Cooling of the Neutron Star in Cassiopeia A Triggered by Neutron Superfluidity in Dense Matter.  prl 106, 081101\.  doi:10.1103/PhysRevLett.106.081101, arXiv:1011.6142.
  * Page and Reddy (2012) Page, D., Reddy, S., 2012\.  Thermal and transport properties of the neutron star inner crust.  arXiv e-prints , arXiv:1201.5602arXiv:1201.5602.
  * Pavlov et al. (2001) Pavlov, G.G., Zavlin, V.E., Sanwal, D., Burwitz, V., Garmire, G.P., 2001.  The X-Ray Spectrum of the Vela Pulsar Resolved with the Chandra X-Ray Observatory.  apjl 552, L129--L133.  doi:10.1086/320342, arXiv:astro-ph/0103171.
  * Potekhin (1999) Potekhin, A.Y., 1999.  Electron conduction in magnetized neutron star envelopes.  aap 351, 787--797.  arXiv:astro-ph/9909100.
  * Potekhin et al. (2015) Potekhin, A.Y., Pons, J.A., Page, D., 2015.  Neutron Stars—Cooling and Transport.  ssr 191, 239--291.  doi:10.1007/s11214-015-0180-9, arXiv:1507.06186.
  * Rea et al. (2020) Rea, N., Coti Zelati, F., Viganò, D., Papitto, A., Baganoff, F., Borghese, A., Campana, S., Esposito, P., Haggard, D., Israel, G.L., Mereghetti, S., Mignani, R.P., Perna, R., Pons, J.A., Ponti, G., Stella, L., Torres, D.F., Turolla, R., Zane, S., 2020.  The X-Ray Outburst of the Galactic Center Magnetar over Six Years of Chandra Observations.  apj 894, 159.  doi:10.3847/1538-4357/ab8387, arXiv:2003.07235.
  * Sajad et al. (2014) Sajad, A., Carlson, J., Duan, H., Reddy, S., 2014\.  Thermal Conductivity of the Neutron Star Crust, in: APS April Meeting Abstracts, p. C8.006.  doi:10.1103/BAPS.2014.APRIL.C8.6.
  * Sauls (2019) Sauls, J.A., 2019.  Superfluidity in the Interiors of Neutron Stars.  arXiv e-prints , arXiv:1906.09641arXiv:1906.09641.
  * Sedrakian and Clark (2019) Sedrakian, A., Clark, J.W., 2019\.  Superfluidity in nuclear systems and neutron stars.  European Physical Journal A 55, 167\.  doi:10.1140/epja/i2019-12863-6, arXiv:1802.00017.
  * Sedrakian et al. (1994) Sedrakian, A.D., Blaschke, D., Röpke, G., Schulz, H., 1994\.  Nuclear in-medium effects on the thermal conductivity and viscosity of neutron star matter.  Physics Letters B 338, 111--117.  doi:10.1016/0370-2693(94)91352-8.
  * Shternin and Baldo (2020) Shternin, P.S., Baldo, M., 2020\.  Transport coefficients of nucleon neutron star cores for various nuclear interactions within the Brueckner-Hartree-Fock approach.  prd 102, 063010\.  doi:10.1103/PhysRevD.102.063010, arXiv:2004.14909.
  * Sinha and Sedrakian (2015) Sinha, M., Sedrakian, A., 2015\.  Magnetar superconductivity versus magnetism: Neutrino cooling processes.  prc 91, 035805.  doi:10.1103/PhysRevC.91.035805, arXiv:1502.02979.
  * Sykes and Brooker (1970) Sykes, J., Brooker, G.A., 1970\.  The transport coefficients of a fermi liquid.  Annals of Physics 56, 1--39.  doi:10.1016/0003-4916(70)90002-3.
  * Wambach et al. (1993) Wambach, J., Ainsworth, T.L., Pines, D., 1993.  Quasiparticle interactions in neutron matter for applications in neutron stars.  nphysa 555, 128--150.  doi:10.1016/0375-9474(93)90317-Q.
  * Weisskopf et al. (2004) Weisskopf, M.C., O’Dell, S.L., Paerels, F., Elsner, R.F., Becker, W., Tennant, A.F., Swartz, D.A., 2004.  Chandra Phase-Resolved X-Ray Spectroscopy of the Crab Pulsar.  apj 601, 1050--1057.  doi:10.1086/380600, arXiv:astro-ph/0310332.
  * Wolf (1966) Wolf, R.A., 1966.  Some Effects of the Strong Interactions on the Properties of Neutron-Star Matter.  apj 145, 834.  doi:10.1086/148829.
  * Woods (2008) Woods, P.M., 2008.  Observations of Magnetars, in: Bassa, C., Wang, Z., Cumming, A., Kaspi, V.M. (Eds.), 40 Years of Pulsars: Millisecond Pulsars, Magnetars and More, pp. 227--233.  doi:10.1063/1.2900149.
  * Yakovlev et al. (2001) Yakovlev, D.G., Kaminker, A.D., Gnedin, O.Y., Haensel, P., 2001\.  Neutrino emission from neutron stars.  physrep 354, 1--155.  doi:10.1016/S0370-1573(00)00131-9, arXiv:astro-ph/0012122.
  * Yakovlev and Levenfish (1995) Yakovlev, D.G., Levenfish, K.P., 1995\.  Modified URCA process in neutron star cores.  aap 297, 717.
  * Yakovlev et al. (1999) Yakovlev, D.G., Levenfish, K.P., Shibanov, Y.A., 1999.  REVIEWS OF TOPICAL PROBLEMS: Cooling of neutron stars and superfluidity in their cores.  Physics Uspekhi 42, 737--778.  doi:10.1070/PU1999v042n08ABEH000556, arXiv:astro-ph/9906456.
  * Yakovlev and Pethick (2004) Yakovlev, D.G., Pethick, C.J., 2004\.  Neutron Star Cooling.  araa 42, 169--210.  doi:10.1146/annurev.astro.42.053102.134013, arXiv:astro-ph/0402143.
  * Ziman (1960) Ziman, J., 1960.  Electrons and Phonons: The Theory of Transport Phenomena in Solids.  International series of monographs on physics, Clarendon Press.  URL: https://books.google.co.in/books?id=_HEsAAAAYAAJ.