# The Dipole Magnetic Field and Spin-down Evolutions of The High Braking Index
Pulsar PSR J1640$-$4631

Zhi-Fu Gao11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150, Science 1-Street, Urumqi, Xinjiang, 830011, China;
<EMAIL_ADDRESS>22affiliation: Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, West Beijing Road, Nanjing, Jiangsu 210008, China  , Na
Wang11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150, Science 1-Street, Urumqi, Xinjiang, 830011, China;
<EMAIL_ADDRESS>22affiliation: Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, West Beijing Road, Nanjing, Jiangsu 210008, China  , Hao
Shan11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150, Science 1-Street, Urumqi, Xinjiang, 830011, China;
<EMAIL_ADDRESS>, Xiang-Dong Li 33affiliation: Department of Astronomy and
Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Jiangsu 210046, China ,Wei Wang 44affiliation: School of Physics and
Technology, Wuhan University, Wuhan, Hubei, 430072, China;
<EMAIL_ADDRESS>55affiliation: National Astronomical Observatories,
Chiese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012,
<EMAIL_ADDRESS>††affiliationmark:

###### Abstract

In this work, we interpreted the high braking index of PSR J1640$-$4631 with a
combination of the magneto-dipole radiation and dipole magnetic field decay
models. By introducing a mean rotation energy conversion coefficient
$\overline{\zeta}$, the ratio of the total high-energy photon energy to the
total rotation energy loss in the whole life of the pulsar, and combining the
pulsar’s high-energy and timing observations with reliable nuclear equation of
state, we estimate the pulsar’s initial spin period, $P_{0}\sim(17-44)$ ms,
corresponding to the moment of inertia $I\sim(0.8-2.1)\times 10^{45}$ g cm2.
Assuming that PSR J1640$-$4631 has experienced a long-term exponential decay
of the dipole magnetic field, we calculate the true age $t_{\rm age}$, the
effective magnetic field decay timescale $\tau_{\rm D}$, and the initial
surface dipole magnetic field at the pole $B_{p}(0)$ of the pulsar to be
$2900-3100$ yr, $1.07(2)\times 10^{5}$ yr, and $(1.84-4.20)\times 10^{13}$ G,
respectively. The measured braking index of $n=3.15(3)$ for PSR J1640$-$4631
is attributed to its long-term dipole magnetic field decay and a low magnetic
field decay rate, $dB_{\rm p}/dt\sim-(1.66-3.85)\times 10^{8}$ G yr-1. Our
model can be applied to both the high braking index ($n>3$) and low braking
index ($n<3$) pulsars, tested by the future polarization, timing, and high-
energy observations of PSR J1640$-$4631.

###### Subject headings:

stars: evolution – magnetic field: neutron – pulsars: individual
(J1640$-$4631) – supernova remnant: general

##  1\. Introduction

Pulsars are commonly recognized as highly magnetized and rapidly rotating
neutron stars (NSs). A pulsar’s secular spin-down is mainly caused by its
rotational energy losses (Lyne et al. 2015). An important and measurable
quantity closely related to a pulsar’s rotational evolution is the braking
index $n$, defined by assuming that the star spins down in the light of a
power law (Lyne et al. 1993)

$\dot{\Omega}=~{}-K\Omega^{n}~{},$ (1)

where $\Omega$ and $\dot{\Omega}$ are the angular velocity and its derivative
of the star, respectively, and $K$ is a proportionality constant.

A standard way to define the braking index is

$n\equiv~{}\frac{\nu\ddot{\nu}}{\dot{\nu}^{2}}=2-\frac{P\ddot{P}}{\dot{P}^{2}},$
(2)

where $\ddot{\Omega}$ is the second derivative of $\Omega$, $\nu=\Omega/2\pi$
the spin frequency, and $P=1/\nu$ the spin period (Manchester & Taylor 1977,
and references therein). When the magneto-dipole radiation (MDR) solely causes
the pulsar to spin down, the braking index is predicted to be $n=3$. Since the
braking index of a pulsar can provide some information about the pulsar’s
energy loss mechanisms, it has been investigated by many authors (for recent
works, see, e.g., Espinoza et al. 2011, Ferdman et al. 2015; Franzon et al.
2016; Rogers & Safi-Harb 2017).

Until now, only 9 of the $\sim$ 2500 known pulsars111see the ATNF catalogue
(Manchester et al. 2005) have reliably measured braking indices. Most of them
are remarkably lower than 3, demonstrating that the spin-down mechanism is not
pure MDR. Recently, Dupays et al. (2008, 2012) put forward an energy loss
mechanism, called quantum vacuum friction (QVF), which results from the
interaction between the magnetic dipole moment of a pulsar and its induced
quantum vacuum. Taking into account the QVF effect, the measured braking
indices of $n<3$ can be well explained (Coelho et al. 2016). To account for
the observed braking indices, several other interpretations have been put
forward (see Gao et al. 2016 for a brief summary).

Recently, Gotthelf et al. (2014) reported the discovery of PSR J1640$-$4631,
associated with the TeV $\gamma-$ray source HESS J1640$-$465 using the
$NuSTAR$ X-ray observatory. Based on its timing observational data,
$\nu=4.84341$ s-1, $\dot{\nu}=-2.2808\times 10^{-11}$ s-2 and
$\ddot{\nu}=3.38(3)\times 10^{-22}$ s-3, (corresponding to $P=206.4$ ms,
$\dot{P}=9.7228\times 10^{-13}$ s s-1 and $\ddot{P}=-5.27(13)\times 10^{-24}$
s s-2, respectively), Archibald et al. (2016) derived its braking index to be
$n=3.15(3)$ (here and following all digits in parentheses denote the standard
uncertainty). PSR J1640$-$4631 is the first pulsar with a measured braking
index larger than 3.

The possible origin of the braking index of PSR J1640$-$4631 has been
discussed in the literature. The decrease in the magnetic inclination angle
$\alpha$ (the angle between the rotational and the magnetic axes) can result
in a braking index higher than 3 (Ekşi et al. 2016). A combination of
gravitational wave (GW) and magnetic energy dipole loss mechanism radiation
could give rise to a braking index $3<n<5$ (de Araujo et al. 2016 a, b).
Alternatively, such a braking index can be accounted for if the magnetic
dipole braking still dominates but the star is experiencing a long-term dipole
magnetic field decay.

The surface dipole magnetic field has a significant effect on the spin
evolution of a pulsar, as well as on its braking index (e.g., Goldreich &
Reisenegger 1992; Tauris & Konar 2001; Jones 2009; Marchant et al. 2014). In
the case of a varying dipole magnetic field at the pole $B_{\rm p}$, the
braking index $n$ can be simply expressed as follows:

$n=3-4\frac{\dot{B}_{\rm p}}{B_{\rm p}}\tau_{\rm c}\equiv 3-4\frac{\tau_{\rm
c}}{\tau_{\rm B}},$ (3)

where $\tau_{\rm c}=P/2\dot{P}$ is the characteristic age and $\tau_{\rm
B}=B_{\rm p}/\dot{B_{\rm p}}$ is the timescale on which the dipole component
evolves. The above equation shows that any variation in $B$ can lead to a
deviation from 3. For a decreasing $B_{\rm p}$, we will always obtain $n>3$
owing to a decaying dipole braking torque (e.g., Rea & Esposito 2011; Kaminker
et al. 2014; Potekhin et al. 2015), whereas $n<3$ for an increasing $B_{\rm
p}$ (e.g., Mereghetti 2008; Gao et al. 2016).

The motivation of this work is to account for the high $n$ of PSR J1640$-$4631
within a theoretical model of a long-term dipole magnetic field decay. The
rotational evolution of PSR J1640$-$4631 is assumed to be dominated by the
magnetic field anchored to the solid crust of the star. Since the core field
evolves on much longer timescales, the crustal field evolves mainly through
ohmic dissipation and the Hall drift. The Hall drift mainly occurs in the
neutron-drip solid or out of the core, conserving the magnetic energy (e.g.,
Haensel et al. 1990; Hertman et al. 1997; Page et al. 2000), while ohmic
diffusion occurs in the range of density below the neutron-drip threshold and
dissipates magnetic energy through Joule heating (e.g., Pons et al. 2009,
2012; Gourgouliatos & Cumming 2015).

The remainder of this work is organized as follows: The key problems in
estimating the true age of PSR J1640$-$4631 are presented in Section 2, and
the initial spin period is constrained in Section 3, The initial magnetic
field strength, the true age, and the effective dipole field evolution
timescale of the pulsar are determined in Section 4, the dipole magnetic field
and spin-down evolutions are numerically simulated in Section 5, and
alternative dipole magnetic field decay models are presented in Section 6.
Comparisons with other works and discussions are given in Section 7.

##  2\. Key problems in estimating $t_{\rm age}$

To investigate the long-term spin evolution of a pulsar, it is best to know
the true age of the pulsar. It is well known that $\tau_{\rm c}$ is a poor age
approximation, and it coincides with the true age only if the current spin
period is much longer than the initial value and the braking torque and the
moment of inertia $I$ are constant during the entire pulsar life. A pulsar’s
true age $t_{\rm age}$ can be estimated by the age of its associated supernova
remanant (SNR), because it is universally considered that pulsars originate
from supernova explosions (e.g., Kouveliotou et al. 1994; Gaensler et al.
2001; Vink & Kuiper 2006).

The SNR ages are currently estimated based on the measurements of the shock
velocities, the X-ray temperatures, and/or other quantities. Assuming that an
SNR is in the Sedov phase222Sedov (1959) showed that there is a self-similar
solution for the adiabatic expansion phase, and firstly applied this solution
to estimate the age of SNRs in this phase, i.e., the Sedov age. The Sedov age
estimates rely on many assumptions, but ultimately on the SNR size and the SNR
temperature. If a SNR is too young (when the expansion is free), or too old
(when radiation energy losses brake the expansion), the Sedov expansion is no
longer applicable. (Sedov 1959), the SNR age can be estimated by a convenient
expression:

$t_{\rm SNR}\approx 435\times R_{\rm SNR}T^{-1/2}~{}\rm~{}yrs,$ (4)

where the SNR radius $R_{\rm SNR}$ is in units of pc, and the post-shock
plasma temperature $T$ is in units of keV (for details see Gao et al. 2016).

G338.3$-$0.0 is a shell-type, 8${}^{{}^{\prime}}$ diameter SNR (Shaver & Goss
1970; Whiteoak & Green 1996), and spatially relates with HESS J1640$-$465,
which is considered to be the the most luminous $\gamma$-ray source in the
Galaxy. The X-ray pulsar PSR J1640$-$4631 was recently discovered within the
shell of SNR G338.3$-$0.0 (Gotthelf et al. 2014). Unfortunately, no X-ray
emission was detected from the shell of the SNR (probably due to its large
intervening column density, $N_{\rm H}\sim 1.4\times 10^{23}$ cm-2 and low
temperature; Lemiere et al. 2009; Castelletti et al. 2011); thus, the true age
of PSR J1640$-$4631 cannot be estimated from Equation (4).

The pulsar’s true age can be inferred from Equation (1),

$\displaystyle
t=\frac{P}{(n-1)\dot{P}}~{}[1-(\frac{P_{0}}{P})^{n-1}]~{}~{}~{}~{}\left(n\neq
1\right),$ $\displaystyle t=2\tau_{\rm
c}~{}\rm{\ln}(\frac{P}{P_{0}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(n=1\right),$
(5)

where $P_{0}$ is the initial spin period of the star and the braking index is
assumed to be a constant since its birth (e.g., Manchester et al. 1985;
Espinoza et al. 2011; Rogers & Safi-Harb 2017). As discussed in Gao et al.
(2016), one indeed cannot derive an analytical expression such as Equation (5)
to estimate $t_{\rm age}$ if we consider that either $K$ or $n$ in Equation
(1) is a time-dependent quantity. Since the braking index is influenced by the
variation of braking torque during the observational intervals, it is
necessary to assume a constant mean braking index to calculate the age of a
young pulsar by using the above equation (Gao et al. 2016).

The mean braking index $\overline{n}$ of a pulsar is defined as

$\overline{n}=\int n(t)dt/\int dt.$ (6)

As seen from Equation (5), $t_{\rm age}$ is less than $\tau_{\rm c}$ if
$\overline{n}=3$. In addition, Equation (5) implies an upper limit of the true
age, $t_{\rm age}=2\tau_{\rm c}/(\overline{n}-1)$, corresponding to
$(P_{0}/P)^{\overline{n}-1}=0$. However, it is difficult to obtain the values
of $P_{0}$ and $\overline{n}$ of the pulsar; $t_{\rm age}$ cannot be directly
derived from Equation (5).

##  3\. Initial spin period of PSR J1640$-$4631

The initial spin period $P_{0}$ of a pulsar is also very important in our
ability to investigate the pulsar’s long-term spin evolution. Since one cannot
know the value of $P_{0}$ of PSR J1640$-$4631, theoretically estimating its
initial spin period becomes urgent. This section is composed of the following
two subsections.

###  3.1. X-ray and $\gamma$-ray luminosities

As mentioned above, PSR J1640$-$4631 is located inside G338.1$-$0.0, the
pulsar wind nebula (PWN), which is primarily responsible for the $\gamma$-ray
emission of HESS J1640$-$465\. The detection of the pulsar in the SNR provides
long-awaited evidence that a PWN powers the $\gamma$-ray source HESS
J1640$-$465, and its properties can be used to test the radiation mechanisms
(e.g., Reynolds 2008; Slane et al. 2014).

Recently, employing Markov Chain Monte Carlo algorithm, Gotthelf et al. (2014)
fitted the parameters of the PWN in G338.3$-$0.0 and obtained the best-fitted
values of free parameters, including the high-energy photon (“photon ” in
short) fluxes. For the purpose of illustration, in Table 1 we list the photon
fluxes of PSR J1634$-$4731 and its PWN, respectively.

Table 1 Spectrum Fluxes of PSR J1640$-$4631 and Its Wind Nebula.  Flux | Observed Flux | Telescope | Reference
---|---|---|---
$F_{\rm X}$[2-25 keV] | $1.0\times 10^{-12}$ | $NuSTAR$ | Gotthelf et at. (2014)
$F_{\rm HESS}$[10-500 GeV] | $1.6\times 10^{-11}$ | H.E.S.S | Gotthelf et at. (2014)

In Table 1, the spectrum flux of $F_{\rm X}$[$2-25$ keV] includes the X-ray
fluxes of PSR J1634$-$4731 and its PWN. All the given fluxes are in units of
$\rm erg\,cm^{-2}s^{-1}$. Then, the total luminosity (the currently measured
photon luminosity) is calculated as

$L_{X,\gamma}=(F_{\rm PSR}+F_{\rm PWN})4\pi d^{2}=2.93\times
10^{35}d_{12}^{2}\,{\rm erg\,s^{-1}},$ (7)

where $d_{12}$ is a dimensionless distance in units of 12 kpc.

###  3.2. Mean rotational energy transformation coefficient

In order to estimate $P_{0}$ of PSR J1634$-$4731, we introduce an energy
transformation coefficient, $\varsigma$, which is the ratio of the total
photon luminosity, $L_{X,\gamma}$, to the spin-down luminosity of the pulsar
$L_{\rm spin}$ ($L_{\rm spin}=I\dot{\Omega}(t)\Omega(t)$)

$\zeta=\frac{L_{X,\gamma}}{L_{\rm spin}}=\frac{(F_{\rm PSR}+F_{\rm PWN})4\pi
d^{2}}{4\pi^{2}I\dot{P}P^{-3}}.$ (8)

From the values of current $P$ and $\dot{P}$, we get the current luminosity
$L_{\rm spin}\sim 4.36\times 10^{36}I_{45}$ erg s-1, where $I_{45}$ is the
moment of inertia in units of $10^{45}\,\rm g\,cm^{2}$. Inserting the values
of $L_{X,\gamma}$ and $L_{\rm spin}$ into Equation (8) yields a general
solution for the current coefficient, $\zeta=0.067d_{12}^{2}I^{-1}_{45}$.
Recently, Getthelf et al. (2014) obtained the best-fit source distance to
G338.3$-$0.0, $d\sim$12 kpc. This distance agrees well with the typical
supernova explosion scenario (e.g., the order of magnitude of explosion energy
is $10^{51}$ erg). In this paper, we adopt the distance of $d_{12}=1$.

Since $\zeta$ is variable with time, in order to estimate the initial spin
parameters including $P_{0}$, a constant $\zeta$ should be assumed in a
specific model. We define a mean rotation energy transformation coefficient
$\overline{\zeta}$,

$\overline{\zeta}=\frac{E_{X,\gamma}}{E_{\rm
spin}}=\frac{\int^{t}_{0}L_{X,\gamma}(t)dt}{E_{\rm spin}},$ (9)

where $E_{X,\gamma}$ is the total photon energy and $E_{\rm spin}$ is the
total rotation energy loss, which can be estimated by

$E_{\rm spin}=\int^{t}_{0}L_{\rm
spin}(t^{{}^{\prime}})dt^{{}^{\prime}}\simeq\frac{2{\pi}^{2}I}{P_{0}^{2}},$
(10)

for $P_{0}\ll P$ (Tanaka Shuta 2016). For a constant braking index
$\overline{n}$, the spin-down luminosity $L_{\rm spin}(t)$ evolves in the
light of a simple power-law form,

$L_{\rm spin}(t)=L_{\rm
spin}(0)\left(1+\frac{t}{\tau_{0}}\right)^{-\frac{\overline{n}+1}{\overline{n}-1}},$
(11)

where $L_{\rm spin}(0)$ is the initial spin-down luminosity and $\tau_{0}$ is
the initial spin-down timescale of the star, which is defined as

$\tau_{0}=\frac{P_{0}}{(\overline{n}-1)\dot{P_{0}}}=\frac{2\tau_{\rm
c}}{\overline{n}-1}\left(\frac{P_{0}}{P}\right)^{\overline{n}-1}.$ (12)

with the initial period derivative $\dot{P_{0}}$. Since PSR J1640$-$4631 is a
pulsar powered by rotational energy loss, the total photon luminosity,
$L_{X,\gamma}(t)$, as well as $L_{\rm spin}(t)$, should decrease with time.
For the sake of simplicity, we assume that $L_{X,\gamma}$ and $L_{\rm
spin}(t)$ have the same evolution form. Then the total photon energy from
$t=0$ to $t=t_{\rm age}$ is calculated as

$\displaystyle E_{X,\gamma}=\int_{0}^{t_{\rm
age}}L_{X,\gamma}(0)(1+\frac{t}{\tau_{0}})^{-(\frac{\overline{n}+1}{\overline{n}-1})}dt$
$\displaystyle=L_{X,\gamma}(t_{\rm
age})\cdot\left[(\frac{P_{0}}{P})^{-2}-1\right]\cdot\tau_{\rm c}.$ (13)

where $L_{X,\gamma}(0)$ is the initial total photon luminosity and the
relations of $\tau_{0}+t_{\rm age}=\frac{2\tau_{\rm c}}{\overline{n}-1}$ and
$\tau_{0}=\frac{2\tau_{\rm
c}}{\overline{n}-1}(\frac{P_{0}}{P})^{\overline{n}-1}$ are utilized.

Inserting Equations (10) and (13) and $\tau_{c}\simeq 3360$ yrs into Eq.(9),
we obtain the initial spin period

$P_{0}\simeq 20.8(\overline{\zeta}/0.067)^{-1}d_{12}^{-2}I_{45}~{}~{}{\rm
ms}.$ (14)

It is interesting to note that, although $\overline{n}$ has been used in the
derivation above, the ultimate value of $P_{0}$ is free of $\overline{n}$.
Recently, Abdo et al. (2010, 2013) presented catalogs of high-energy gamma-ray
pulsars detected by the Large Area Telescope on the $Fermi$ satellite and
estimated the values of $\zeta$ for some sources, (here $\zeta$ is the ratio
of the gamma-ray luminosity to rotational energy loss rate of a pulsar).
According to Abdo et al. (2010, 2013), the range of $\zeta$ is about
$(0.01-1.0)I^{-1}_{45}$, but its average value is about $0.08I^{-1}_{45}$,
close to our estimate of $\zeta=0.067d^{2}_{12}I^{-1}_{45}$. In Equation (14),
we have adopted fiducial NS parameters: mass $M=1.4\,M_{\bigodot}$, radius
$R$=10 km, and the moment of inertia $I_{45}=1$. Strictly speaking, the true
value of $\overline{\zeta}$ is unknown and may be constrained by specific NS
nuclear equation of state (EOS).

From Equation (14), it is clear that $P_{0}$ is a function of moment of
inertia $I$. In order to investigate how different values of $I$ (or the mass
[$M$]-equatorial radius [$R$] relation) modify $P_{0}$, we need to take into
account the $M-R$ relation of the uniformly rotating NS and feasible EOSs.

###  3.3. Nuclear EOS and moment of inertia

Very recently, to investigate the possibility that some of soft gamma-ray
repeaters (SGRs) and anomalous X-ray pulsars (AXPs) family could be canonical
rotation-powered pulsars, Coelho et al. (2017) adopted realistic NS structure
parameters (e.g., mass, radius, and moment of inertia) instead of fiducial
values and considered the issue related to the high-energy luminosity and the
spin-down luminosity of SGRs and AXPs in detail. Their results provide useful
constraints on the initial spin period $P_{0}$ of PSR J1640$-$4631.

As we know, the maximum NS mass predicted by EoSs is model dependent (e.g.,
Dutra et al. 2014; Li et al. 2016; Xia et al. 2016; Xia & Zhou 2017; Zhou et
al. 2017). The largest sample of measured NS masses available for analysis is
publicly accessible online at http://www.stellarcollapse.org/, and has been
updated by J. Lattimer. A similar analysis can be found in Valentim et al.
(2011), from which one can get a range of about $1-2$ $M_{\bigodot}$ for the
observational NS masses. The minimum NS mass, as well as the maximum NS mass,
is still a matter of debate. Though the NS masses could be could be less than
$1.0\,M_{\bigodot}$ from the viewpoint of the EOS, it is difficult to explain
their formations from the supernova explosion mechanism. Pulsars having mass
of $\sim 2.0\,M_{\bigodot}$ are confirmed by Demorest et al. (2010) and
Antoniadis et al. (2013). The typical nuclear matter EOSs obtained from Özel &
Freire (2016) predict that pulsars have maximum mass larger than
$2.0\,M_{\bigodot}$, which prefers the APR (Akmal et al. 1998) model and the
RMF (relativistic mean-field) model (e.g., Lalazissis et al. 1997; Toki et al.
1995). In the APR model, properties of dense nucleon matter and the structure
of NSs are studied using variational chain summation methods and the new
Argonne v18 two-nucleon interaction (Av18). However, in this work, we adopt
the APR3 model (one version of APR EOS) instead of the RMF model. The main
reasons are as follows:

  1. 1.

The RMF theory is based on effective coupling constants that take the
correction effect into consideration. However, these coupling constants are
density dependent, and a microscopic theory is needed to calculate them.

  2. 2.

Akmal et al. (1998) investigated the properties of dense nucleon matter and
the structure of NSs and provided an excellent fit to all of the nucleon-
nucleon scattering data in the Nijmegen database. The authors not only
considered the nonrelativistic calculations with Av18 and Av18+UIX (Urbana IX
three-nucleon interaction) models for nuclear forces but also described the
relativistic boost interaction model (denoted as $\delta v$) with and without
three-nucleon interaction (UIX∗).

  3. 3.

In this work, we prefer the APR3 model, e.g., Av18+$\delta v$+UIX∗ model,
which provides a constraint on the maximum NS mass $M\leq 2.2\,M_{\bigodot}$.
This model has also been preferred and included in recent works (e.g., Özel &
Freire 2016; Zhou et al. 2017), which limit the range of NS masses to
$<2.2\,M_{\bigodot}$ because of the absence of any data to constrain the
relation at higher mass.

In order to calculate the moment of inertia $I$, it is of importance for us to
adopt a better $M-R-I$ relation for NSs. In this work, we prefer an improved
approximation expression of $M-R-I$ with $M\geq 1\,M_{\bigodot}$, provided by
Steiner et al. (2016),

$\displaystyle\frac{I}{MR^{2}}$ $\displaystyle\simeq
0.01+(1.200^{+0.006}_{-0.006})\beta^{1/2}-0.1839\beta$ (15)
$\displaystyle-(3.735^{+0.095}_{-0.095})\beta^{3/2}+5.278\beta^{2}.$

where $\beta=(\frac{M}{R}\frac{\rm{km}}{M_{\bigodot}})$. The above equation
shows less uncertainty, especially for compactness typical of 1.4
$M_{\bigodot}$ stars, and the smaller uncertainties result from an assumption
that $M_{max}>1.97M_{\bigodot}$ (Steiner et al. 2016). Combining Equation (15)
with the APR3 EoS, we obtain $I_{45}\sim 0.81(1)-2.07(3)$, corresponding to
$M\sim(1.0-2.2)\,M_{\bigodot}$, and $R\sim(11.6-10.3)$ km, respectively ($M$
decrease with increasing $R$). Figure 1 shows the relation of $M,R$ and $I$
for PSR J1640$-$4631 in the APR3 model.

Figure 1.— Relation of $M,R$ and $I$ for PSR J1640$-$4631 in the APR3 model.

However, to effectively constrain the value of $P_{0}$, it is necessary to
combine the EOS and the spin-down evolution and diploe magnetic field
evolution equations. For details, see the next section.

##  4\. Constraining $B_{\rm p}(0)$, $t_{\rm age}$ and $\tau_{\rm D}$ of the
pulsar

###  4.1. Hall drift and Ohmic decay

Here we restrict ourselves to the dipole magnetic field evolution in the
crusts of NSs, ignoring possible effects involving the fluid core. Assuming
that ions are locked into a column lattice, the only freely moving charged
species in the crust are electrons. In order to reconcile the measured braking
index of PSR J1640$-$4631 with the dipole magnetic field decay, it is
necessary to introduce the Hall induction equation describing the evolution of
the crust magnetic field:

$\frac{\partial\vec{B}}{\partial
t}=-\nabla\times\left[\frac{c^{2}}{4\pi\sigma}\nabla\times(e^{\nu}\vec{B})+\frac{c}{4\pi
en_{e}}[\nabla\times(e^{\nu}\vec{B})]\times\vec{B}\right],$ (16)

where $\sigma$ is the electric conductivity parallel to the magnetic field,
$e^{\nu}$ is the relativistic redshift correction, $n_{e}$ is the electron
number density, and $e$ the electron charge. This equation contains two
different effects that act on two distinct timescales, which can be estimated
as

$t_{\rm Hall}=\frac{4\pi n_{e}eL^{2}}{cB},~{}~{}~{}t_{\rm
Ohm}=\frac{4\pi\sigma L^{2}}{c^{2}},$ (17)

where $t_{\rm Ohm}$ is the ohmic dissipation timescale with a typical value of
$\sim 10^{6}$ yr or more (Viganò et al. 2013), $t_{\rm Hall}$ is the Hall
drift timescale, and $L$ is a characteristic length scale of variation, which
can be taken to be the thickness of the neutron star crust (e.g., Haensel et
al. 1990; Pons & Gepport 2007; Pons et al. 2009; Aguilera et al. 2008; Ho
2011). There is a typical value of several $\times(10^{4}-10^{5})$ yr for
$t_{\rm Hall}$, which is constrained by the magnetic field strength of high
magnetic field pulsars and magnetars (Viganò et al. 2013).

There are two field configurations: one in which the field is confined in the
crust, and another in which the field extends into the core (there could be an
electron current flowing through a superconducting core). For the purposes of
illustration, we choose a dipole magnetic field $B_{\rm p}$ constrained in the
crust and assume a simple exponential decay of $B_{\rm p}$ over time,

$\frac{dB_{\rm p}}{dt}=-\frac{B_{\rm p}}{\tau_{\rm D}},$ (18)

as done in other works (e.g., Pons et al. 2009; Viganò et al. 2013). This
assumption requires that the pulsar has an initial magnetic field at the pole
$B_{\rm p}(0)$ (corresponding to $t=0$) and that the magnetic field decays at
a rate proportional to its strength. $\tau_{\rm D}$ is an effective dipole
magnetic field timescale, which is defined as

$\frac{1}{\tau_{\rm D}}=\frac{1}{t_{\rm Hall}}+\frac{1}{t_{\rm Ohm}}.$ (19)

The Hall timescale $t_{\rm Hall}$ is universally one order of magnitude lower
than the ohmic timescale (e.g., Pons & Gepport 2007; Pons et al. 2009). This
would imply that the effective timescale we define in Equation (19) is
basically dominated by the Hall timescale. By integrating Equation (18), we
get a time-dependent field

$B_{\rm p}(t)=B_{\rm p}(0)e^{-\frac{t}{\tau_{\rm D}}}.$ (20)

With respect to two parameters of $\tau_{\rm D}$ and $B_{\rm p}(0)$, there are
three aspects that need to be addressed explicitly:

  1. 1.

In general, the effective timescales of pulsars are in the range of about
$10^{4}-10^{7}$ yrs (see Muslimov & Page 1995, 1996; Geppert & Rheinhardt
2002; Lyutikov et al. 2015; Mereghetti et al. 2015 and references therein).
However, $\tau_{\rm D}$ of PSR J1640$-$4631 is determined mainly by $t_{\rm
Hall}$. To exactly constrain $\tau_{\rm D}$ of the star requires both
observational and theoretical investigations.

  2. 2.

Though there are several methods for roughly measuring the magnetic field
strength of a pulsar, such as magneto-hydrodynamic pumping, Zeeman splitting,
cyclotron lines, magnetar bursts and etc., a direct estimate of the strength
$B_{\rm p}(0)$ from the observations is unavailable.

  3. 3.

For PSR J1640$-$4631, the initial dipole magnetic field was inherited from its
progenitor star and cannot be estimated from its initial spin parameters via a
simple MDR model. In this work, we have assumed a variable dipole magnetic
field for the pulsar.

###  4.2. Equations of $P(t)$, $\dot{P}(t)$ and $\ddot{P}(t)$

Assuming that the MDR of a pulsar with a decaying field can be responsible for
the high braking index of PSR J1640$-$4631, we here constrain three parameters
$B_{\rm p}(0)$, $t_{\rm age}$ and $\tau_{\rm D}$ of the pulsar.

If the magnetic field evolution of PSR J1640$-$4631 cannot be ignored and the
dipole braking still dominates, according to Blandford & Romani (1988), the
braking law of the pulsar is reformulated as

$\dot{\nu}(t)=-\frac{2\pi^{2}R^{6}}{3Ic^{3}}B^{2}_{\rm p}(t)\nu^{3},$ (21)

where a constant inclination angle $\alpha=90^{\circ}$ is assumed for the sake
of simplicity. Integrating Equation (21) gives the spin frequency,

$\nu^{-2}=\nu^{-2}_{0}+2\int^{t}_{0}\frac{2\pi^{2}R^{6}}{3Ic^{3}}B^{2}_{\rm
p}(t)dt^{{}^{\prime}},$ (22)

where $\nu_{0}$ is the initial spin frequency of the pulsar. Then we get the
spin period,

$P(t)=\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\cdot\tau_{\rm D}\cdot(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right]^{1/2}.$ (23)

Utilizing the differential method, we get the first-order derivative of the
spin period $\dot{P}(t)$,

$\displaystyle\dot{P}(t)=(\frac{2\pi^{2}R^{6}B^{2}_{\rm p}(0)}{3Ic^{3}})\times
e^{\frac{-2t}{\tau_{\rm D}}}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right]^{-1/2},$ (24)

and the second-order derivative of the spin period $\ddot{P}(t)$,

$\displaystyle\ddot{P}(t)=-(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})^{2}\times e^{\frac{-4t}{\tau_{\rm D}}}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right]^{-3/2}$ $\displaystyle-\frac{2}{\tau_{\rm
D}}\times(\frac{2\pi^{2}R^{6}B^{2}_{\rm p}(0)}{3Ic^{3}})\times
e^{\frac{-2t}{\tau_{\rm D}}}$
$\displaystyle\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right]^{-1/2}.$ (25)

In the above three equations there are unknown variables of $B_{\rm p}(0)$,
$t_{\rm age}$ and $\tau_{\rm D}$, which will be constrained by numerically
simulating in the next subsection.

###  4.3. The results of numerical simulations

If we combine Equations (14) and (15) with the APR3 EOS, take $t=t_{\rm age}$,
and insert the current values of $P=206.4$ ms $\dot{P}=9.7228\times 10^{-13}$
s s-1 and $\ddot{P}=-5.27(13)\times 10^{-24}$ s s-2 into Equations (23)$-$
(25), then we obtain the values of $\tau_{\rm D}$, $t_{\rm age}$ and $B_{\rm
p}(0)$ given a specific value of $P_{0}$ (corresponding to a specific $I$ of
the pulsar).

By numerically simulating, we make a plot of $B_{\rm p}(0)$ versus $I$ for PSR
J1640$-$4631, as shown in Figure 2.

Figure 2.— Initial dipole magnetic field strength $B_{\rm p}(0)$ as a function
of the moment of inertia $I$ for PSR J1640$-$4631\. The range of $I$ is taken
as $(0.80-2.09)I_{45}$, corresponding to $M\sim(1.0-2.2)\,M_{\bigodot}$.

In Figure 2, the blue solid line denotes the fitted values of $B_{\rm p}(0)$
and $I$. The red solid line stands for a typical mass $M=2.0\,M_{\bigodot}$
($I=1.92(5)$ and $R=10.94$ km) predicted by the APR3 EoS. Also, we obtain the
initial spin period $P_{0}\sim(17-44)$ ms, the true age $t_{\rm age}\sim
2800-3100$ yr, the initial dipole magnetic field strength $B_{\rm
p}(0)\sim(1.84-4.20)\times 10^{13}$ G, and the effective timescale $\tau_{\rm
D}\sim 1.07(2)\times 10^{5}$ yr for the pulsar.

As we know, the estimates of the crustal ohmic and Hall timescales represent
the choice of microscopic parameters. Recently, Gourgouliatos & Cumming (2014)
estimate the average initial value of $\tau_{\rm Hall}$ for young pulsars
about $100-200$ kyr. This timescale is close to that given by us.

##  5\.  Numerical Simulations of the Dipole Magnetic Field and Spin-down
evolutions

###  5.1. The evolution of the braking index

After obtaining $B_{\rm p}(0)$, $t_{\rm age}$ and $\tau_{\rm D}$ of the
pulsar, we then use them in numerically simulating the spin-down evolution,
especially accounting for its high braking index.

Combining Equations (23)-(25) with $n=2-\frac{P\ddot{P}}{\dot{P}^{2}}$, we get

$\displaystyle n=3+\frac{3Ic^{3}}{\pi^{2}R^{6}B^{2}_{\rm
p}(0)e^{\frac{-2t}{\tau_{\rm D}}}\times\tau_{\rm D}}$
$\displaystyle\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right].$ (26)

From the above equation, it is easily seen that $n=3$ if $\tau_{\rm
D}\rightarrow\infty$. In order to investigate the evolution of $n$, we plot
the diagrams of $n$ versus $t$ for the pulsar.

Figure 3.— Braking index as a function of $t$ for PSR J1640$-$4631.

In Figure 3, the red solid line, yellow solid line, black dashed line, and
blue solid line stand for the predictions of the dipole magnetic field decay
model with $P_{0}=17,26,34$ and 44 ms, respectively. The blue dot-dashed line
stands for the prediction of the MDR model, in which we assume the NS mass
$M=1.4\,M_{\bigodot}$, radius $R=10$ km and momentum of inertial, $I=10^{45}$
g cm2, corresponding to $P_{0}\sim 21$ ms, $t_{\rm age}\sim 3220$ yr,
$\tau_{\rm D}\sim 1.07\times 10^{5}$ yr and $B_{\rm p}(0)\sim 2.97\times
10^{13}$ G. Notice that the signs in Figure 3 are universally adopted in other
figures in the subsequent sections of this work. The horizontal blue dotted
line and the surrounding shaded region denote, respectively, the measured
braking index of $n=3.15$ and its possible range given by the uncertainty 0.03
of PSR J1640$-$4631\. From Figure 3, it is obvious that $n$ increases with $t$
owing to the decay of the dipole magnetic field.

###  5.2. The evolution of the dipole magnetic field

By using Eq.(20) and the constrained parameters of $t_{\rm age}$, $\tau_{\rm
D}$ and $B_{\rm p}(0)$, we estimate the present value of the dipole magnetic
field strength, $B_{\rm p}(t_{\rm age})\sim(1.77-4.10)\times 10^{13}$ G. This
range is different from that estimated by the characteristic magnetic field at
the polar of a pulsar $B_{\rm c}\simeq 6.4\times
10^{19}\times(P\dot{P}I_{45})^{1/2}=2.87\times 10^{13}$ G, because the latter
assumes a constant dipole braking torque and a constant momentum of inertial
$I_{45}=1$. Then we obtain a mean magnetic field decay rate of the pulsar,
$\Delta B_{\rm p}/\Delta t=(B_{\rm p}(t_{\rm age})-B_{\rm p}(0))/t_{\rm
age}\approx-(2.41-3.51)\times 10^{7}$ G yr-1, and plot $B_{\rm p}$ versus $t$
of PSR J1634$-$4631 in Figure 4(a).

The decay rate $\dot{B}_{\rm p}$ of PSR J1634$-$4631 is also an important
issue. From Equation (20), we obtain the expression of $\dot{B}_{\rm p}$ and
$t$ for the pulsar,

$\frac{dB_{\rm p}(t)}{dt}=-\frac{B_{\rm p}(0)}{\tau_{\rm
D}}e^{-\frac{t}{\tau_{\rm D}}}.$ (27)

The magnetic field decay rate $\dot{B}_{\rm p}$ of the pulsar declines with
$t$, as shown in Figure 4(b). When $t=t_{\rm age}$, we get the present value
of $dB_{\rm p}/dt\sim-(1.66-3.85)\times 10^{8}$ G yr-1.

Figure 4.— (a)Relation between $B_{\rm p}$ and $t$ of PSR J1640$-$4631\.
(b)Relation between $\dot{B}_{\rm p}$ and $t$ of PSR J1640$-$4631.

###  5.3. The relation of $\tau_{\rm c}$ and $t_{\rm age}$.

The constraints on three parameters $B_{\rm p}(0)$, $t_{\rm age}$ and
$\tau_{\rm D}$) of the pulsar also can be useful in accounting for the age
difference between $\tau_{\rm c}$ and $t_{\rm age}$. Inserting Equation (23)
and Equation (24) into $\tau_{\rm c}=P/2\dot{P}$, we have

$\displaystyle\tau_{\rm c}=$
$\displaystyle\frac{1}{2}(\frac{2\pi^{2}R^{6}B^{2}_{\rm p}(0)}{3Ic^{3}})\times
e^{\frac{2t}{\tau_{\rm D}}}\times$ (28)
$\displaystyle\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right].$

If $\tau_{\rm D}\rightarrow\infty$, then Equation (28) is approximated as

$\tau_{\rm c}=\frac{\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}}\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times 2t\right].$ (29)

From Equations (28)-(29), we plot the diagrams of $\tau_{\rm c}$ vs. $t$ for
the pulsar in Figure 5.

Figure 5.— Characteristic age as a function of $t$ for PSR J1640$-$4631\.
Here, the measured value of $\tau_{\rm c}$ is shown with the red dot. The
error in data point is smaller than the size of the symbol.

In Figure 5, the red dotted line denotes $\tau_{\rm c}=t_{\rm age}$. It is
easy to see that $\tau_{\rm c}$ increases with the age, but the fitted curves
given by Equation (28) are always above the boundary of $\tau_{\rm c}=t_{\rm
age}$. This is because the characteristic age of a pulsar is always higher
than its inferred age if $n\geq 3$, which can be seen from Equation (5) in
Section 2.

###  5.4. Diagrams of $N-t$, $P-t$, $\dot{P}-t$ and $\ddot{P}-t$.

Considering a pulsar with rotational angular momentum $L$ ($L=I\Omega(t)$),
the braking torque acting on the pulsar is given by

$N=\frac{dL}{dt}=I\dot{\Omega(t)},$ (30)

where $I$ is assumed to be constant in time. The decay of $B_{\rm p}$
inevitably causes a decrease in the dipole braking torque $N$, which is
described by

$\displaystyle N=I\dot{\Omega}=-\frac{4\pi^{3}R^{6}B^{2}_{\rm
p}(0)}{3c^{3}}\times e^{\frac{-2t}{\tau_{\rm D}}}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\tau_{\rm D}\times(1-e^{\frac{-2t}{\tau_{\rm
D}}})\right]^{-3/2}.$ (31)

If $\tau_{\rm D}\rightarrow\infty$, Equation (31) is rewritten as

$N(t)=-\frac{4\pi^{3}R^{6}B^{2}_{\rm
p}(0)}{3c^{3}}\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times 2t\right]^{-3/2}.$ (32)

Then we plot the diagrams of $N-t$ for PSR J1634$-$4631, as shown in Figure
6(a). It is obvious that $N$ decreases as $t$ increases.

As a comparison, below we give the relations of $P-t$, $\dot{P}-t$, and
$\ddot{P}-t$ in the case of the MDR model:

$P(t)=\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm p}(t)}{3Ic^{3}})\times
2t\right]^{1/2},$ (33) $\dot{P}(t)=(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(t)}{3Ic^{3}})\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times 2t\right]^{-1/2}$ (34)

and

$\ddot{P}(t)=-(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})^{2}\times\left[P^{2}_{0}+(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times 2t\right]^{-3/2}.$ (35)

Based on Equations (22)-(24) and (31)-(33), we numerically simulate the
relations of $P-t$, $\dot{P}-t$ and $\ddot{P}-t$ for PSR J1634$-$4631.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 6.— Numerical simulations of $N-t$, $P-t$, $\dot{P}-t$ and $\ddot{P}-t$
for PSR J1634$-$4631\. Panels(a), (b), (c), (d) describe the relations of
$N-t$ $P-t$, $\dot{P}-t$ and $\ddot{P}-t$ for the pulsar, respectively.

In Figure 6, the measured values of $N$, $P$, $\dot{P}$ and $\ddot{P}$ are
shown with the red dots, and the errors in data point are smaller than the
sizes of the symbols. In the terms of the dipole magnetic field decay model,
as $t$ increases, $P$ increases rapidly at the earlier evolution stage and
then increases slowly at the later evolution stage, whereas $\dot{P}$ and
$\ddot{P}$ decrease slowly at the earlier evolution stage and then decrease
rapidly at the later evolution stage. Unlike PSR J1734$-$3333, which could
evolve into a magnetar (Espinoza et al. 2011), PSR J1634$-$4631 will
eventually come down to the death valley, due to the pulsar death effect.

##  6\. Alternative models for the dipole magnetic field decay

It is worth emphasizing that our results and conclusions strongly depend on
Equation (20) in which an exponential form of dipole magnetic field decay is
assumed. In order to investigate how different magnetic field decay laws would
affect our results, we assume that the dipole magnetic is deeply buried in the
inner crust and extends into the core (see, e.g., Muslimov & Page 1995;
Gourgouliatos & Cumming 2015). If the magnetic field decays via the Hall drift
and ohmic decay with a nonlinear form:

$B_{\rm p}(t)=\frac{B_{0}}{1+\,t/\tau_{\rm D}}.$ (36)

As a comparison, we give the relations of $P-t$, $\dot{P}-t$ and $\ddot{P}-t$
in this nonlinear decay form. Inserting Equation (36) into Equation (22), we
obtain the expression of the spin period,

$P(t)=\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm D}\cdot t}{\tau_{\rm
D}+t}\right]^{1/2}.$ (37)

Utilizing the differential method, we get the first-order derivative of the
spin period $\dot{P}(t)$,

$\displaystyle\dot{P}(t)=(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times(\frac{\tau_{\rm D}}{t+\tau_{\rm D}})^{2}$
$\displaystyle\times\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm D}\cdot t}{\tau_{\rm
D}+t}\right]^{-1/2},$ (38)

and the second-order derivative of the spin period $\ddot{P}(t)$,

$\displaystyle\ddot{P}(t)=-(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})^{2}\cdot(\frac{\tau_{\rm D}}{t+\tau_{\rm
D}})^{4}\cdot[P^{2}_{0}+$ $\displaystyle(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\cdot\frac{\tau_{\rm D}\cdot t}{\tau_{\rm
D}+t}]^{-3/2}-(\frac{4\pi^{2}R^{6}B^{2}_{\rm p}(0)}{3Ic^{3}})\cdot\tau_{\rm
D}^{2}$ $\displaystyle\times(\frac{1}{t+\tau_{\rm
D}})^{3}\cdot[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\cdot\frac{\tau_{\rm D}\cdot t}{\tau_{\rm D}+t}]^{-1/2}.$ (39)

If we assume $M\sim$ 1.0-2.2$\,M_{\bigodot}$ and use the same APR3 EOS, then
we obtain the effective magnetic field decay timescale $\tau_{\rm D}\sim
0.91(8)\times 10^{5}$ yr and the true age $t_{\rm age}\sim 3100-3200$ yr, the
initial dipole magnetic field $B_{\rm p}(0)\sim(1.72-3.83)\times 10^{13}$ G,
and the dipole magnetic field decay rate $dB_{\rm p}/dt\sim-(1.76-3.94)\times
10^{8})$ G yr-1.

Alternatively, the magnetic field is assumed to decay with a power-law decay
form:

$B_{\rm p}(t)=B_{\rm p}(0)\times(t/\tau_{\rm D})^{\varepsilon},$ (40)

as done in Muslimov & Page (1995, 1996), where $\varepsilon<0$ is the magnetic
field index. In this case, Equations (37)-(39) are replaced by

$P(t)=\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm
D}}{2\varepsilon+1}\times(\frac{t}{\tau_{\rm
D}})^{2\varepsilon+1}\right]^{1/2},$ (41)
$\displaystyle\dot{P}(t)=(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times(\frac{t}{\tau_{\rm D}})^{2\varepsilon}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm
D}}{2\varepsilon+1}\times(\frac{t}{\tau_{\rm
D}})^{2\varepsilon+1}\right]^{-1/2},$ (42)

and

$\displaystyle\ddot{P}(t)=-(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})^{2}\times\frac{2\varepsilon}{\tau_{\rm
D}}\times(\frac{t}{\tau_{\rm D}})^{2\varepsilon-1}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm
D}}{2\varepsilon+1}\times(\frac{t}{\tau_{\rm
D}})^{2\varepsilon+1}\right]^{-1/2}$
$\displaystyle-(\frac{2\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})^{2}\times(\frac{t}{\tau_{\rm D}})^{4\varepsilon}\times$
$\displaystyle\left[P^{2}_{0}+(\frac{4\pi^{2}R^{6}B^{2}_{\rm
p}(0)}{3Ic^{3}})\times\frac{\tau_{\rm
D}}{2\varepsilon+1}\cdot(\frac{t}{\tau_{\rm
D}})^{4\varepsilon+1}\right]^{-3/2},$ (43)

respectively. For the sake of simplicity, we assume that the range of $t_{\rm
age}$ obtained from the power-law decay form (Equation (40)) is very close to
those obtained from the exponential and nonlinear decay forms (Equation (20)
and Equation (36)), e.g., $t_{\rm age}\sim 2800-3200$ yr. Utilizing the same
NS mass range and fitting method, we get the magnetic field index
$\varepsilon=-0.034(1)$, the effective field decay timescale $\tau_{\rm
D}\sim(0.4-2.5)\times 10^{6}$ yr, the initial dipole magnetic field strength
$B_{\rm p}(0)\sim(1.44-3.03)\times 10^{13}$ G, and the magnetic field decay
rate $dB_{\rm p}/dt\sim-(1.78-4.42)\times 10^{8})$ G yr-1.

It is worth addressing that for ordinary radio pulsars the effect of field
decay may not be very pronounced, and so it is difficult to uncover it. For
example, Mukherjee & Kembhavi (1997) synthesized a population of pulsars using
assumed theoretical distributions of age, initial magnetic field, period of
rotation, position, luminosity, and dispersion measure and concluded that the
timescale for the field decay must be greater than 160 Myr. Other studies
(Bhattacharya et al. 1992; Hartman et al. 1997; Regimbau & de Freitas Pacheco
2001) also confirm these long timescales. We recall that the pulsars
considered by these authors are older radio pulsars having the kinetic ages
and/or characteristic ages $\tau\sim 10-100$ Myr, the dipole magnetic field
strengthes $B\sim(2-5)\times 10^{12}$ G, which are different from young
X$-$ray/$\gamma-$ray pulsar PSR J1640$-$4631 with $\tau_{\rm c}\sim 3360$ yr
(the true age $t_{\rm age}\sim 2800-3200$ yr). Furthermore, using the
selection effect of pulsars’ distance distribution and velocity distribution,
Mukherjee & Kembhavi (1997) obtained the data for radio pulsars from the
catalog of pulsars in the Princeton database (Taylor et al. 1995) and dropped
binaries, X-ray pulsars and gamma-ray pulsars from the simulated pulsar
samples. Thus, the magnetic field decay timescale obtained from Mukherjee &
Kembhavi (1997) is larger than the one we give. Nevertheless, the magnetic
field decay timescale of pulsars is an interesting subject worth long-term
study.

##  7\. Comparisons and Conclusions

In this work, the high braking index of PSR J1640$-$4631 is well interpreted
within a combination of MDR and dipole magnetic field decay. The related
comparisons and conclusions are given as follows.

###  7.1. Comparisons with our previous work

In our previous work (Gao et al. 2016), we investigated the braking indices of
eight magnetars. We attributed the larger braking indice ($n>3$) of three
magnetars to the decay of external braking torque, which might be caused by
magnetic field decay, magnetospheric current decay, or the decay of magnetic
inclination angle, and also calculated the dipolar field decay rates, which
are compatible with the measured values of $n$ for the following three
magnetars:

Table 2 Dipole Field Decay Rates of the Magnetars.  Source | Braking index $n$ |  $\frac{dB_{\rm p}}{dt}$ (G yr${}^{-1})$
---|---|---
1E 1841$-$045 | $9-17$ | $-(6.55\times 10^{11}-1.32\times 10^{12})$
SGR 0501$+$4516 | $4.6-8.0$ | $-(5.19\times 10^{9}-9.66\times 10^{10}$
1E 2259$+$586 | $22-42$ | $-(2.44\times 10^{9}-4.76\times 10^{9})$

Compared with PSR J1640$-$4631, the three magnetars in Table 2 have higher
magnetic field decay rates and larger braking indices. The main reasons are
twofold:(i) magnetars are a kind of special pulsar powered by their magnetic
energy rather than their rotational energy (Duncan & Thompson 1992; Gao et al.
2013; Liu 2017); and (ii) to account for the high-value braking indices of
magnetars, we adopted the updated magnetothermal evolution model of Viganò et
al. (2013) (see Gao et al. 2016 for details).

In addition, we show the long-term rotational evolution of PSR J1634$-$4631 in
Figure 7. Five fitted lines in Figure 7 represent the evolution paths of the
pulsar in the context of dipole magnetic field decay, and the arrows represent
the spin-down evolution directions with different initial spin period in the
next 100 kyr. PSR J1634$-$4631 would be placed in the top left region of the
$P-\dot{P}$ diagram at its birth. As it spins down, the pulsar moves toward
the bottom right of the $P-\dot{P}$ diagram in $\sim 10^{5}-10^{6}$ yr and
will eventually come down to the death valley, based on the dipole magnetic
field decay model.

Figure 7.— Long-term rotational evolution of PSR J1640$-$4631 dominated by the
dipole magnetic field decay. Radio, binary and magnetars are defined by black
dot, red triangle, and blue square, respectively. The red solid line, green
solid line, black dotted line, and blue solid line stand for the predictions
of the dipole magnetic field decay model with $P_{0}=17,26,34$ and 44 ms,
respectively. The red filled circle denotes the observations of PSR
J1640$-$4631.

###  7.2. Comparsions with other models

Recently, to understand better the existence of both pulsar braking indices
both larger than 3 and smaller than 3, different braking mechanisms have been
proposed (e.g., Ekşi et al. 2016; Chen 2016; Clark et al. 2016; Coelho et al.
2016; de Araujo et al. 2016 a, b; Magalhaes et al. 2016; Tong & Kou 2017). For
the purpose of illustrating the feasibility of this theoretical model, it is
worth comparing our results with those of other models.

  1. 1.

Initial spin period estimate. Recently, by numerically simulating, Gotthelf et
al. (2014) predicted that PSR J1640$-$4631 has a short initial spin period
$P_{0}\sim 15$ ms. Such a short $P_{0}$ requires a smaller braking index
($n<3$) and a higher true age ($t_{\rm age}>\tau_{\rm c}$) (Gotthelf et al.
2014), which departures from the observed $n$ and $\tau_{\rm c}$ of the pulsar
obviously. In this paper, we firstly introduce a mean braking index
$\overline{n}$ and a mean rotation energy conversion coefficient
$\overline{\zeta}$ and then estimate the initial spin period $P_{0}=17-44$ ms.
The largest uncertainty of our model comes from $\overline{\zeta}$, because
the true value of $\overline{\zeta}$ is unable to be obtained. However, if
$P_{0}\ll P$, the uncertainty of $\overline{\zeta}$ will have little effect on
the results of the three parameter of $t_{\rm age}$, $\tau_{\rm D}$ and
$B_{\rm p}(0)$, according to our model. In this work, the initial spin period
($P_{0}\sim 17-44$ ms) is obtained by combining pulsar’s high- energy
observations with the EOS.

  2. 2.

True age estimate. Recently, using the leptonic PWN model333Slane et al.
(2010) modeled the $\gamma-$ray emission from HESS J1640$-$465 and the
associated broad-band spectrum with an evolving, one-zone PWN model, where the
$\gamma-$rays are ambient photons inverse Compton scattered by relativistic
electrons that also produce the synchrotron emission., Slane et al. (2010)
assumed the age of G338.3$-$0.0 $t_{\rm SNR}\sim$10 kyr, and Gotthelf et al.
(2014) gave an age $t_{\rm SNR}\sim 6.8$ kyr. The former requires a constant
braking index $n=3$, while the latter requires a smaller braking index
$n\approx 2.0$ (Gotthelf et al.2014). However, their methods are universally
based on supernova explosion mechanisms, due to the unknown explosion energy
$E$ and the ambient mass density $\rho_{0}$. In this work, the true age
estimate ($t_{\rm age}\sim 2900-3100$ yr) is obtained by solving the pulsar
spin-down evolution equation set.

  3. 3.

Inclination angle estimate. Based on the MDR model with corotating plasma,
Ekşi et al. (2016) found that the high braking index is consistent with two
different inclination angles, $\alpha\sim 18.5(3)^{\circ}$ and
$56(4)^{\circ}$. Very recently, based on the vacuum gap model and low mass
($M\sim 0.1\,M_{\bigodot}$) neutron star candidate, employing the vacuum gap
model, Chen (2016) proposed that a low-mass neutron star ($M\sim
0.1M_{\bigodot}$) with a large inclination angle (close to the perpendicular
case) could interpret the high braking index, as well as the radio-quiet
nature of the pulsar. Note that in these two models a constant magnetic field
has been uniformly assumed for PSR J1640$-$4631, and the discrepancy of
$\alpha$ they obtained is obvious. As we know, it is the most meaningful to
determine a pulsar’s inclination angle from its timing and polarization
observations. We expect that PSR J1640$-$4631 has a reliable measurement of
$\alpha$ from the future timing and polarization observations.

###  7.3. No detected bursts in PSR J1640$-$4631

Since the higher-order magnetic moments decay rapidly with the distance to the
stellar center, the influence of multipole magnetic moments of the star on the
braking index can be ignored (if multipole magnetic fields exist in the
pulsar). Interestingly, Chen (2016) supposed that there could be superhigh
multipole magnetic fields inside PSR J1640$-$4631 (e.g., quadrupole magnetic
field) resulting in a large stellar deformation and strong GW emission. If
this is true, the burst behaviors should have been detected. However, up to
date, no burst has been observed. If our model is correct, the undetected
bursts in PSR J1640$-$4631 will be well explained as follows: by using the
currently estimated values of $B_{\rm p}$ and $\dot{B}_{\rm p}$, we estimate
the magnetic field energy decay rate,
$dE_{B}/dt=\frac{d}{dt}(\frac{B^{2}}{8\pi})\times\frac{4}{3}\pi
R^{3}=(1.2-2.8)\times 10^{32}$ erg s-1, where $B^{2}_{\rm p}/8\pi$ is the
magnetic field energy density. This magnetic field energy decay rate is about
two orders of magnitude lower than the X-ray luminosity of PSR J1640$-$4631\.
Nevertheless, our prediction could be tested by the future high-energy
observations of the pulsar.

In this work, we interpret the high braking index of PSR J1640$-$4631 with a
combination of the dipole magnetic field decay and MDR models, ignoring the
influences of all other possible braking torques. It is suggested that the
decay of dipole magnetic field could occur universally in pulsars with $n<3$.
If so, why are the measured braking indices of the eight pulsars lower than 3
(see, e.g., Gao et al. 2016; Tong & Kou 2017)? Below we suggest two possible
reasons.

  1. 1.

The outflowing particle winds (the relativistic outflow mainly composed of the
electron-positron pairs) luminosities in the eight rotationally powered
pulsars with low braking indices may be so high that additional braking
torques provided by winds can be comparable to the magneto-dipole braking
torques in magnitude. Thus the stars’ braking indices are less than 3. A PWN
is dominated by a bright collimated feature, which is interpreted as a
relativistic jet directed along the pulsar spin axis (e.g., Gaensler et al.
2003; Grondin et al. 2013). The PWNs of five low braking index pulsars have
been detected.444PSR B0531$+$21 (Crab) with $n=2.34(2)$, Lyne et al. 1993,
2015; PSR B0833$-$45 (Vela) with $n=1.4(2)$, Lyne et al. 1996; PSR B1509$-$58
with $n=2.839(1)$, Livingstone et al. 2007; PSR J1833$-$1034 with
$n=1.857(6)$, Roy et al. 2012, and PSR B0540$-$69 with $n=2.140(9)$, Ferdman
et al. 2015 Especially, the spin-down luminosity of the Crab pulsar $\sim
5\times 10^{38}$ erg s-1 is converted into radiation with a remarkable
efficiency, approaching $\sim 30\%$ (Abdo et al. 2011). No detection of PWNs
for three pulsars (PSR J1846$-$0258 with $n=2.65(1)$, Livingstone et al. 2007;
PSR J1119-6127 with $n=2.684(2)$, Weltevrede et al. 2011; and PSR J1734$-$3333
with $n=0.9(2)$, Espinoza et al. 2011) could be explained by the tenuous
interstellar medium around the pulsars.

  2. 2.

Another possibility is that the eight pulsars are experiencing toroidal
magnetic field (e.g., octupole field) decay via Hall drift and ohmic
dissipation. For example, magnetar-like outbursts from three pulsar of PSR
J1846$-$0258, PSR J1734$-$3333 and PSR J1119-6127 were reported (e.g., Gavriil
et al. 2008; Göǧüş et al.2016). These outbursts could be caused by the decay
of initial toroidal multipole magnetic fields. The multipole magnetic fields
are merging through crustal tectonics to form a dipole magnetic field causing
a low braking index.

  3. 3.

For PSR J1640$-$4631, though it has an associated PWN with very low rotational
energy transformation coefficient (see Section 3 of this work), the pulsar may
be at an inactive wind epoch. Maybe this pulsar has experienced the stage of
multipole fields merging, i.e., its dipole magnetic field stops increasing. At
the current epoch, the emission and braking properties of the pulsar are
dominated by the dipole magnetic field decaying, which causes a high mean
braking index. In summary, attempts to explain the lower braking indices
($n<3$) of pulsars within the dipole magnetic field decay model coupled with
wind braking and/or toroidal field decaying will be considered in our future
studies.

###  7.4. Conclusions

In this work, we interpreted the high braking index of PSR J1640$-$4631 with a
combination of the MDR and dipole magnetic field decay. By introducing a mean
rotation energy conversion coefficient $\overline{\zeta}$ and combining the
EOS with the high-energy and timing observations of the pulsar, we give the
constrained values of $P_{0}$, $t_{\rm age}$, $\tau_{\rm D}$ and $B_{\rm
p}(0)$, and numerically simulat the dipole magnetic field and spin-down
evolutions of the pulsar. The high braking index of $3.15(3)$ of PSR
J1640$-$4631 is attributed to its long-term dipole magnetic field decay at a
low rate $dB_{\rm p}/dt\sim-(1.66-3.85)\times 10^{8}$ G yr-1. Considering the
uncertainties and assumptions in this work, thus our theoretical model needs
to be tested and modified by the future polarization, timing, and high-energy
observations of PSR J1640$-$4631\. Our results may apply to other pulsars with
higher braking indices.

We thank an anonymous referee for carefully reading the manuscript and
providing valuable comments that improved this paper substantially. We also
thank Prof. Andrew Lyne for useful discussions. This work was supported by
National Basic Research Program of China grants 973 Programs 2015CB857100; the
West Light Foundation of CAS through grants XBBS-2014-23, XBBS-2014-22 and
2172201302; Chinese National Science Foundation through grants
No.11673056,11622326, 11273051, 11373006, 11133004 and 11173042; the Strategic
Priority Research Program of CAS through no.XDB23000000 and National Program
on Key Research and Development Project through no. 2016YFA0400803.

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