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

As mentioned above, PSR J1640−4631 is located inside G338.1−0.0, the pulsar wind nebula (PWN), which is primarily responsible for the γ-ray emission of HESS J1640−465. The detection of the pulsar in the SNR provides long-awaited evidence that a PWN powers the γ-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 a 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	Referee
${F}_{{\rm{X}}}$[2–25 keV]	$1.0\times {10}^{-12}$	NuSTAR	Gotthelf et al. (2014)
${F}_{\mathrm{HESS}}$[10–500 GeV]	$1.6\times {10}^{-11}$	H.E.S.S.	Gotthelf et al. (2014)

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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 $\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. Then, the total luminosity (the currently measured photon luminosity) is calculated as

$$\begin{eqnarray}&&{L}_{{\rm{X}},\gamma }=({F}_{\mathrm{PSR}}+{F}_{\mathrm{PWN}})4\pi {d}^{2}=2.93\times {10}^{35}{d}_{12}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 7 }$$

where d₁₂ is a dimensionless distance in units of 12 kpc.

In order to estimate P₀ of PSR J1634−4731, we introduce an energy transformation coefficient, ς, which is the ratio of the total photon luminosity, ${L}_{{\rm{X}},\gamma }$, to the spin-down luminosity of the pulsar ${L}_{\mathrm{spin}}$ (${L}_{\mathrm{spin}}=I\dot{{\rm{\Omega }}}(t){\rm{\Omega }}(t)$),

$$\begin{eqnarray}&&\zeta =\displaystyle \frac{{L}_{{\rm{X}},\gamma }}{{L}_{\mathrm{spin}}}=\displaystyle \frac{({F}_{\mathrm{PSR}}+{F}_{\mathrm{PWN}})4\pi {d}^{2}}{4{\pi }^{2}I\dot{P}{P}^{-3}}.\end{eqnarray} \tag{ 8 }$$

From the values of current P and $\dot{P}$, we get the current luminosity ${L}_{\mathrm{spin}}\sim 4.36\times {10}^{36}{I}_{45}$ erg s⁻¹, where I₄₅ is the moment of inertia in units of ${10}^{45}\,{\rm{g}}\,{\mathrm{cm}}^{2}$. Inserting the values of ${L}_{{\rm{X}},\gamma }$ and ${L}_{\mathrm{spin}}$ into Equation (8) yields a general solution for the current coefficient, $\zeta =0.067{d}_{12}^{2}{I}_{45}^{-1}$. Recently, Getthelf et al. (2014) obtained the best-fit source distance to G338.3−0.0, d ∼ 12 kpc. This distance agrees well with the typical supernova explosion scenario (e.g., the order of magnitude of explosion energy is 10⁵¹ erg). In this paper, we adopt the distance of ${d}_{12}=1$.

Since ζ is variable with time, in order to estimate the initial spin parameters, including P₀, a constant ζ should be assumed in a specific model. We define a mean rotation energy transformation coefficient $\overline{\zeta }$,

$$\begin{eqnarray}&&\overline{\zeta }=\displaystyle \frac{{E}_{{\rm{X}},\gamma }}{{E}_{\mathrm{spin}}}=\displaystyle \frac{{\int }_{0}^{t}{L}_{{\rm{X}},\gamma }(t){dt}}{{E}_{\mathrm{spin}}},\end{eqnarray} \tag{ 9 }$$

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

$$\begin{eqnarray}&&{E}_{\mathrm{spin}}={\int }_{0}^{t}{L}_{\mathrm{spin}}({t}^{{\prime} }){{dt}}^{{\prime} }\simeq \displaystyle \frac{2{\pi }^{2}I}{{P}_{0}^{2}},\end{eqnarray} \tag{ 10 }$$

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

$$\begin{eqnarray}&&{L}_{\mathrm{spin}}(t)={L}_{\mathrm{spin}}(0){\left(1+\displaystyle \frac{t}{{\tau }_{0}}\right)}^{-\tfrac{\overline{n}+1}{\overline{n}-1}},\end{eqnarray} \tag{ 11 }$$

where ${L}_{\mathrm{spin}}(0)$ is the initial spin-down luminosity and ${\tau }_{0}$ is the initial spin-down timescale of the star, which is defined as

$$\begin{eqnarray}&&{\tau }_{0}=\displaystyle \frac{{P}_{0}}{(\overline{n}-1)\dot{{P}_{0}}}=\displaystyle \frac{2{\tau }_{{\rm{c}}}}{\overline{n}-1}{\left(\displaystyle \frac{{P}_{0}}{P}\right)}^{\overline{n}-1},\end{eqnarray} \tag{ 12 }$$

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

$$\begin{eqnarray}{E}_{{\rm{X}},\gamma } & = & {\displaystyle \int }_{0}^{{t}_{\mathrm{age}}}{L}_{{\rm{X}},\gamma }(0){\left(1+\displaystyle \frac{t}{{\tau }_{0}}\right)}^{-(\tfrac{\overline{n}+1}{\overline{n}-1})}{dt}\\ & = & {L}_{{\rm{X}},\gamma }({t}_{\mathrm{age}})\cdot \left[{\left(\displaystyle \frac{{P}_{0}}{P}\right)}^{-2}-1\right]\cdot {\tau }_{{\rm{c}}},\end{eqnarray} \tag{ 13 }$$

where ${L}_{{\rm{X}},\gamma }(0)$ is the initial total photon luminosity and the relations of ${\tau }_{0}+{t}_{\mathrm{age}}=\tfrac{2{\tau }_{{\rm{c}}}}{\overline{n}-1}$ and ${\tau }_{0}=\tfrac{2{\tau }_{{\rm{c}}}}{\overline{n}-1}{\left(\tfrac{{P}_{0}}{P}\right)}^{\overline{n}-1}$ are utilized.

Inserting Equations (10) and (13) and ${\tau }_{c}\simeq 3360\,\mathrm{yr}$ into Equation (9), we obtain the initial spin period

$$\begin{eqnarray}&&{P}_{0}\simeq 20.8{(\overline{\zeta }/0.067)}^{-1}{d}_{12}^{-2}{I}_{45}\,\,\mathrm{ms}.\end{eqnarray} \tag{ 14 }$$

It is interesting to note that, although $\overline{n}$ has been used in the derivation above, the ultimate value of P₀ 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 ζ for some sources (here ζ 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 ζ is about $(0.01\mbox{--}1.0){I}_{45}^{-1}$, but its average value is about $0.08{I}_{45}^{-1}$, close to our estimate of $\zeta =0.067{d}_{12}^{2}{I}_{45}^{-1}$. In Equation (14), we have adopted fiducial NS parameters: mass $M=1.4\,{M}_{\odot }$, 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 a specific NS nuclear equation of state (EOS).

From Equation (14), it is clear that P₀ 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 P₀, we need to take into account the M−R relation of the uniformly rotating NS and feasible EOSs.

Very recently, to investigate the possibility that some soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) 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 P₀ 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_⊙ 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 less than $1.0\,{M}_{\odot }$ from the viewpoint of the EOS, it is difficult to explain their formations from the supernova explosion mechanism. Pulsars having mass of $\sim 2.0\,{M}_{\odot }$ are confirmed by Demorest et al. (2010) and Antoniadis (2013). The typical nuclear matter EOSs obtained from Özel & Freire (2016) predict that pulsars have maximum mass larger than $2.0\,{M}_{\odot }$, 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 the APR EOS) instead of the RMF model. The main reasons are as follows:

• 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.  
  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.  
  In this work, we prefer the APR3 model, e.g., the Av18+$\delta v$+UIX* model, which provides a constraint on the maximum NS mass $M\leqslant 2.2\,{M}_{\odot }$. This model has also been preferred and included in recent works (e.g., Özel & Freire 2016; Xia & Zhou 2017), which limit the range of NS masses to $\lt 2.2\,{M}_{\odot }$ 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 $M\geqslant 1\,{M}_{\odot }$, provided by Steiner et al. (2016),

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

where $\beta =(\tfrac{M}{R}/\tfrac{\mathrm{km}}{{M}_{\odot }})$. The above equation shows less uncertainty, especially for compactness typical of 1.4 M_⊙ stars, and the smaller uncertainties result from an assumption that ${M}_{\max }\gt 1.97\,{M}_{\odot }$ (Steiner et al. 2016). Combining Equation (15) with the APR3 EOS, we obtain ${I}_{45}\sim 0.81(1)\mbox{--}2.07(3)$, corresponding to $M\sim 1.0\mbox{--}2.2{M}_{\odot }$ and $R\sim 11.6\mbox{--}10.3$ 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.

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However, to effectively constrain the value of P₀, it is necessary to combine the EOS and the spin-down evolution and diploe magnetic field evolution equations. For details, see the next section.

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:

$$\begin{eqnarray}\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t} & = & -{\rm{\nabla }}\times \left[\displaystyle \frac{{c}^{2}}{4\pi \sigma }{\rm{\nabla }}\times ({e}^{\nu }{\boldsymbol{B}})\right.\\ & & \left.\times +\displaystyle \frac{c}{4\pi {{en}}_{e}}[{\rm{\nabla }}\times ({e}^{\nu }{\boldsymbol{B}})]\times {\boldsymbol{B}}\right],\end{eqnarray} \tag{ 16 }$$

where σ 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 is the electron charge. This equation contains two different effects that act on two distinct timescales, which can be estimated as

$$\begin{eqnarray}&&{t}_{\mathrm{Hall}}=\displaystyle \frac{4\pi {n}_{e}{{eL}}^{2}}{{cB}},\,\,\,{t}_{\mathrm{Ohm}}=\displaystyle \frac{4\pi \sigma {L}^{2}}{{c}^{2}},\end{eqnarray} \tag{ 17 }$$

where ${t}_{\mathrm{Ohm}}$ is the ohmic dissipation timescale with a typical value of $\sim {10}^{6}$ yr or more (Viganò et al. 2013), ${t}_{\mathrm{Hall}}$ is the Hall drift timescale, and L is a characteristic length scale of variation, which can be taken to be the thickness of the NS crust (e.g., Haensel et al. 1990; Pons & Gepport 2007; Aguilera et al. 2008; Pons et al. 2009; Ho 2011). There is a typical value of several $\times ({10}^{4}\mbox{--}{10}^{5})$ yr for ${t}_{\mathrm{Hall}}$, 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,

$$\begin{eqnarray}&&\displaystyle \frac{{{dB}}_{{\rm{p}}}}{{dt}}=-\displaystyle \frac{{B}_{{\rm{p}}}}{{\tau }_{{\rm{D}}}},\end{eqnarray} \tag{ 18 }$$

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

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{D}}}}=\displaystyle \frac{1}{{t}_{\mathrm{Hall}}}+\displaystyle \frac{1}{{t}_{\mathrm{Ohm}}}.\end{eqnarray} \tag{ 19 }$$

The Hall timescale ${t}_{\mathrm{Hall}}$ 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

$$\begin{eqnarray}&&{B}_{{\rm{p}}}(t)={B}_{{\rm{p}}}(0){e}^{-\tfrac{t}{{\tau }_{{\rm{D}}}}}.\end{eqnarray} \tag{ 20 }$$

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.  
  In general, the effective timescales of pulsars are in the range of about ${10}^{4}\mbox{--}{10}^{7}$ yr (see Muslimov & Page 1995, 1996; Geppert & Rheinhardt 2002a; Lyutikov 2015; Mereghetti et al. 2015, and references therein). However, ${\tau }_{{\rm{D}}}$ of PSR J1640−4631 is determined mainly by ${t}_{\mathrm{Hall}}$. To exactly constrain ${\tau }_{{\rm{D}}}$ of the star requires both observational and theoretical investigations.
• 2.  
  Though there are several methods for roughly measuring the magnetic field strength of a pulsar, such as magnetohydrodynamic pumping, Zeeman splitting, cyclotron lines, magnetar bursts, etc., a direct estimate of the strength ${B}_{{\rm{p}}}(0)$ from the observations is unavailable.
• 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.

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 ${B}_{{\rm{p}}}(0)$, ${t}_{\mathrm{age}}$, 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

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

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

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

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

$$\begin{eqnarray}&&P(t)={\left[{P}_{0}^{2}+\left(\displaystyle \frac{2{\pi }^{2}{R}^{6}{B}_{{\rm{p}}}^{2}(0)}{3{{Ic}}^{3}}\right)\cdot {\tau }_{{\rm{D}}}\cdot (1-{e}^{\tfrac{-2t}{{\tau }_{{\rm{D}}}}})\right]}^{1/2}.\end{eqnarray} \tag{ 23 }$$

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

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

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

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

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

If we combine Equations (14) and (15) with the APR3 EOS, take $t={t}_{\mathrm{age}}$, 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 ${\tau }_{{\rm{D}}}$, ${t}_{\mathrm{age}}$, and ${B}_{{\rm{p}}}(0)$ given a specific value of P₀ (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.

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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 $M=2.0\,{M}_{\odot }$ ($I=1.92(5)$ and R = 10.94 km) predicted by the APR3 EOS. Also, we obtain the initial spin period ${P}_{0}\sim (17\mbox{--}44)$ ms, the true age ${t}_{\mathrm{age}}\sim 2800\mbox{--}3100\,\mathrm{yr}$, the initial dipole magnetic field strength ${B}_{{\rm{p}}}(0)\sim (1.84\mbox{--}4.20)\times {10}^{13}$ 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 ${\tau }_{\mathrm{Hall}}$ for young pulsars to about 100−200 kyr. This timescale is close to that given by us.

After obtaining ${B}_{{\rm{p}}}(0)$, ${t}_{\mathrm{age}}$, 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 $n=2-\tfrac{P\ddot{P}}{{\dot{P}}^{2}}$, we get

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

From the above equation, it is easily seen that n = 3 if ${\tau }_{{\rm{D}}}\to \infty$. In order to investigate the evolution of n, we plot the diagrams of n versus t for the pulsar. 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 an NS mass $M=1.4\,{M}_{\odot }$, radius R = 10 km, and moment of inertia $I={10}^{45}$ g cm², corresponding to ${P}_{0}\sim 21$ ms, ${t}_{\mathrm{age}}\sim 3220\,\mathrm{yr}$, ${\tau }_{{\rm{D}}}\sim 1.07\times {10}^{5}$ 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.

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By using Equation (20) and the constrained parameters of ${t}_{\mathrm{age}}$, ${\tau }_{{\rm{D}}}$, and ${B}_{{\rm{p}}}(0)$, we estimate the present value of the dipole magnetic field strength, ${B}_{{\rm{p}}}({t}_{\mathrm{age}})\sim (1.77\mbox{--}4.10)\times {10}^{13}$ 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 moment of inertia ${I}_{45}=1$. Then we obtain a mean magnetic field decay rate of the pulsar, ${\rm{\Delta }}{B}_{{\rm{p}}}/{\rm{\Delta }}t=({B}_{{\rm{p}}}({t}_{\mathrm{age}})\,-{B}_{{\rm{p}}}(0))/{t}_{\mathrm{age}}\approx -(2.41\mbox{--}3.51)\times {10}^{7}$ G yr⁻¹, and plot ${B}_{{\rm{p}}}$ versus t of PSR J1634−4631 in Figure 4(a).

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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,

$$\begin{eqnarray}&&\displaystyle \frac{{{dB}}_{{\rm{p}}}(t)}{{dt}}=-\displaystyle \frac{{B}_{{\rm{p}}}(0)}{{\tau }_{{\rm{D}}}}{e}^{-\tfrac{t}{{\tau }_{{\rm{D}}}}}.\end{eqnarray} \tag{ 27 }$$

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

The constraints on three parameters (B₀, t_age, and ${\tau }_{{\rm{D}}}$) of the pulsar also can be useful in accounting for the age difference between ${\tau }_{{\rm{c}}}$ and ${t}_{\mathrm{age}}$. Inserting Equations (23) and (24) into ${\tau }_{{\rm{c}}}=P/2\dot{P}$, we have

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

If ${\tau }_{{\rm{D}}}\to \infty$, then Equation (28) is approximated as

$$\begin{eqnarray}&&{\tau }_{{\rm{c}}}=\displaystyle \frac{{\pi }^{2}{R}^{6}{B}_{{\rm{p}}}^{2}(0)}{3{{Ic}}^{3}}\times \left[{P}_{0}^{2}+\left(\displaystyle \frac{2{\pi }^{2}{R}^{6}{B}_{{\rm{p}}}^{2}(0)}{3{{Ic}}^{3}}\right)\times 2t\right].\end{eqnarray} \tag{ 29 }$$

From Equations (28) and (29), we plot the diagrams of ${\tau }_{{\rm{c}}}$ versus t for the pulsar in Figure 5. In Figure 5, the red dotted line denotes ${\tau }_{{\rm{c}}}={t}_{\mathrm{age}}$. It is easy to see that ${\tau }_{{\rm{c}}}$ increases with age, but the fitted curves given by Equation (28) are always above the boundary of ${\tau }_{{\rm{c}}}={t}_{\mathrm{age}}$. This is because the characteristic age of a pulsar is always higher than its inferred age if $n\geqslant 3$, which can be seen from Equation (5) in Section 2.

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

$$\begin{eqnarray}&&N=\displaystyle \frac{{dL}}{{dt}}=I\dot{{\rm{\Omega }}(t)},\end{eqnarray} \tag{ 30 }$$

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

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

If ${\tau }_{{\rm{D}}}\to \infty$, Equation (31) is rewritten as

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

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.

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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:

$$\begin{eqnarray}&&P(t)={\left[{P}_{0}^{2}+\left(\displaystyle \frac{2{\pi }^{2}{R}^{6}{B}_{{\rm{p}}}^{2}(t)}{3{{Ic}}^{3}}\right)\times 2t\right]}^{1/2},\end{eqnarray} \tag{ 33 }$$

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

and

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

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.

In Figure 6, the measured values of N, P, $\dot{P}$, and $\ddot{P}$ are shown with the red dots, and the errors in the data point are smaller than the sizes of the symbols. In 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.

In our previous work (Gao et al. 2016), we investigated the braking indices of eight magnetars. We attributed the larger braking indices ($n\gt 3$) 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 we also calculated the dipolar field decay rates, which are compatible with the measured values of n for the following three magnetars: 1E 1841−045, SGR 0501+4516, and 1E 2259+586.

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 et al. 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).

Table 2.  Dipole Field Decay Rates of the Three Magnetars

Source	Braking Index n	$\tfrac{{{dB}}_{{\rm{p}}}}{{dt}}$ (G yr−1)
1E 1841−045	9−17	$-(6.55\times {10}^{11}\,\mbox{--}\,1.32\times {10}^{12})$
SGR 0501+4516	4.6−8.0	$-(5.19\times {10}^{9}\,\mbox{--}\,9.66\times {10}^{10}$
1E 2259+586	22−42	$-(2.44\times {10}^{9}\,\mbox{--}\,4.76\times {10}^{9})$

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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 region of the $P-\dot{P}$ diagram in $\sim {10}^{5}\mbox{--}{10}^{6}\,\mathrm{yr}$ and will eventually come down to the death valley, based on the dipole magnetic field decay model.

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Recently, to understand better the existence of 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. 2016a, 2016b; 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.  
  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 P₀ requires a smaller braking index ($n\lt 3$) and a higher true age (${t}_{\mathrm{age}}\gt {\tau }_{{\rm{c}}}$) (Gotthelf et al. 2014), which departures from the observed n and ${\tau }_{{\rm{c}}}$ of the pulsar obviously. In this paper, we first 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 parameters ${t}_{\mathrm{age}}$, ${\tau }_{{\rm{D}}}$, and ${B}_{{\rm{p}}}(0)$, according to our model. In this work, the initial spin period (${P}_{0}\sim 17\mbox{--}44$ ms) is obtained by combining the pulsar's high-energy observations with the EOS.
• 2.  
  True age estimate. Recently, using the leptonic PWN model,⁸ Slane et al. (2010) assumed the age of G338.3−0.0 to be ${t}_{\mathrm{SNR}}\,\sim$ 10 kyr, and Gotthelf et al. (2014) gave an age of ${t}_{\mathrm{SNR}}\sim 6.8$ 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 (${t}_{\mathrm{age}}\sim 2900\mbox{--}3100$ yr) is obtained by solving the pulsar spin-down evolution equation set.
• 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 ($M\sim 0.1\,{M}_{\odot }$) NS candidate, employing the vacuum gap model, Chen (2016) proposed that a low-mass NS ($M\sim 0.1{M}_{\odot }$) 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 α 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 α from the future timing and polarization observations.

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, ${{dE}}_{B}/{dt}=\tfrac{d}{{dt}}(\tfrac{{B}^{2}}{8\pi })\times \tfrac{4}{3}\pi {R}^{3}=(1.2\mbox{--}2.8)\times {10}^{32}$ erg s⁻¹, where ${B}_{{\rm{p}}}^{2}/8\pi$ 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 $n\lt 3$. 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.  
  The outflowing particle winds (the relativistic outflow mainly composed of the electron–positron pairs) and 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.⁹ 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.  
  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 pulsars, 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.  
  For PSR J1640−4631, though it has an associated PWN with a 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 ($n\lt 3$) 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.

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 P₀, ${t}_{\mathrm{age}}$, ${\tau }_{{\rm{D}}}$, and ${B}_{{\rm{p}}}(0)$ and numerically simulate 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 of ${{dB}}_{{\rm{p}}}/{dt}\sim -(1.66\mbox{--}3.85)\times {10}^{8}$ G yr⁻¹. Considering the uncertainties and assumptions in this work, 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.