Back to the Starting Point: On the Simulation of the Initial Magnetic Fields and Spin Periods of Nonaccreting Pulsars

We make the following assumptions in order to select a sample of NSs from the ATNF pulsar catalog ⁸ (Manchester et al. 2005) that is as large as possible.

• 1.  
  All of the NSs can be simply divided into two groups: the accreting or accreted ones and the nonaccreting ones.
• 2.  
  The nonaccreting NSs (NANSs) spin down, dominated only by magnetic dipole radiation.
• 3.  
  The magnetic fields of all of the NANSs decay in a similar way, which can be described in a simple mathematical form; i.e., all nonstandard mechanisms of formation and evolution of the magnetic fields are neglected, such as the reemergence.

However, in practice, it is still very hard to select NANSs using the observed spin period and period derivative, as well as the deduced magnetic field and spin-down age.

Figure 1 shows the B–P diagram of ATNF pulsars, where the dots and circles indicate the isolated NSs (INSs) and the NSs in binary systems, respectively. The histograms of the surface dipole magnetic fields and spin periods are also displayed on the right and top, both of which exhibit bimodal features. It shows that most of the isolated pulsars located in the upper right of the B–P diagram are the major contributors of the main peaks in both histograms, while most of the pulsars in binaries located in the lower left mainly contribute to the subpeaks. It gives the idea that the sub- and main peaks are dominated by accreting or accreted NSs and NANSs, respectively. So one can select NANS candidates by setting lower limits of B and P. For simplicity, we assume that all of the pulsars in binaries and the isolated pulsars with spin periods smaller than 0.03 s or a surface dipole magnetic field strength smaller than 10¹⁰ G possibly suffered or are suffering from accretion; i.e., the isolated pulsars with P > 0.03 s and B > 10¹⁰ G belong to the nonaccreting candidate groups. In addition, since the highly magnetized NSs, especially the magnetars, express specificities in observation (e.g., De Luca et al. 2006; Olausen & Kaspi 2014; Xu & Li 2019), which indicates that they may belong to a special group, we rule them out by setting an upper limit of 10¹⁴ G for the magnetic fields; then, we get the NANSs candidate sample that is studied in this paper. ⁹ The magnetic fields and spin periods of NANSs both favor lognormal distributions with $\langle \mathrm{log}({B}_{{\rm{i}}}/{\rm{G}})\rangle \sim 12.04$, ${\sigma }_{\mathrm{log}\,{B}_{{\rm{i}}}}\sim 0.6$ and $\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \sim -0.22$, ${\sigma }_{\mathrm{log}\ {P}_{{\rm{i}}}}\sim 0.4$, which are shown by a gray histogram in Figure 3.

Zoom In Zoom Out Reset image size

Since it is assumed that the NANSs have not been exerted by other torques except for the dipole magnetic radiation after birth, we are likely to trace back to the starting point of these pulsars and obtain the general distribution for their initial spin periods and magnetic fields in a simple way.

The dipole radiation could be simply written as

$$\begin{eqnarray}&&{N}_{{\rm{B}}}=-\displaystyle \frac{2{\mu }^{2}{{\rm{\Omega }}}^{3}}{3{c}^{3}},\end{eqnarray} \tag{ 1 }$$

where Ω and $\mu =B(t){R}_{\mathrm{NS}}^{3}$ represent the angular velocity and magnetic dipole moment of the NS, and c is the speed of light. If the radiation loss is the only torque, the spin evolution of the NS could be determined by the basic equation $I\dot{{\rm{\Omega }}}={N}_{{\rm{B}}}$, in which I = 10⁴⁵ g cm² is the moment of inertia of an NS with a typical mass of M = 1.4 M_⊙ and assumed to be time-independent. By integrating both sides of the equation simultaneously, we can easily derive the relation between the spin period P at time t and the birth time, that is,

$$\begin{eqnarray}&&{P}^{2}-{P}_{{\rm{i}}}^{2}=2{C}_{1}\times {\int }_{{t}_{0}}^{t}{B}^{2}(t)\ {dt},\end{eqnarray} \tag{ 2 }$$

where we have packed all the constant parameters into the coefficient C₁:

$$\begin{eqnarray}&&{C}_{1}=\displaystyle \frac{8{\pi }^{2}{R}_{\mathrm{NS}}^{6}}{3{c}^{3}I}.\end{eqnarray} \tag{ 3 }$$

Therefore, we can calculate the initial spin period P_i once the exact form of magnetic field evolution is given.

The magnetic fields of NSs are thought to decay under the effects of ohmic diffusion, Hall drift, Joule heating, and so on (e.g., Goldreich & Reisenegger 1992; Pons & Geppert 2007; Konar 2017). There are three forms of NANS magnetic field evolution from the literature: the exponential form (e.g., Usov 1988; Livio et al. 1998; Gonthier et al. 2002, 2004; Kiel et al. 2008; Osłowski et al. 2011), the power-law form (e.g., Fu & Li 2012; Xu et al. 2022), and a composite form combining the first two (e.g., Aguilera et al. 2008; Dall'Osso et al. 2012; Zhang & Xie 2012; Jawor & Tauris 2022). In fact, these categories can be found by following the analytical expression of magnetic field evolution introduced by Colpi et al. (2000),

$$\begin{eqnarray}&&\displaystyle \frac{{dB}(t)}{{dt}}=-{AB}{\left(t\right)}^{1+\alpha },\end{eqnarray} \tag{ 4 }$$

where A and α are the model parameters. Then, one can get the exponential form with α = 0 (Dall'Osso et al. 2012; Jawor & Tauris 2022),

$$\begin{eqnarray}&&B(t)={B}_{{\rm{i}}}{e}^{-t/{\tau }_{{\rm{d}},{\rm{i}}}}+{B}_{{\rm{m}}},\end{eqnarray} \tag{ 5 }$$

and the power-law form with α ≠ 0,

$$\begin{eqnarray}&&B(t)={B}_{{\rm{i}}}{\left(1+\alpha t/{\tau }_{{\rm{d}},{\rm{i}}}\right)}^{-1/\alpha }+{B}_{{\rm{m}}},\end{eqnarray} \tag{ 6 }$$

where B_i and B_m are the initial and minimal magnetic fields of an NS, respectively; ${\tau }_{{\rm{d}},{\rm{i}}}={\tau }_{{\rm{d}}}/{\left({B}_{{\rm{i}}}/{10}^{15}\,{\rm{G}}\right)}^{\alpha }$, where τ_d is the field decay timescale; and α is the decisive parameter of the form, which is relative to the decay rate of the magnetic field.

We compare these categories and find that the lines of the exponential form cannot cover the NANS dots well because they decay too fast, ¹⁰ while the power-law form is favored by the distribution trend of the scatter dots and the lines can cover all of the dots with a feasible parameter space (upper panel of Figure 2).

Zoom In Zoom Out Reset image size

Although the favored form of magnetic field evolution has been found, we cannot ascertain the age of the NS, since it is hard to detect its growth ring in observations. But in theory, one can calculate its characteristic (or spin-down) age, ${\tau }_{{\rm{c}}}=P/2\dot{P}$, which is the only age that can be easily derived for NSs. We assume that there is an unknown relation between the physical age τ and τ_c of NSs, and we can express it as τ = ξ τ_c, where the parameter ξ indicates the relation. Then, we can get the initial magnetic fields, ¹¹

$$\begin{eqnarray}&&{B}_{{\rm{i}}}\approx {\left({B}_{\mathrm{dip}}^{-\alpha }(\tau )-\displaystyle \frac{\alpha t}{{\tau }_{{\rm{d}}}{\left({10}^{15}\,{\rm{G}}\right)}^{\alpha }}\right)}^{-1/\alpha },\end{eqnarray} \tag{ 7 }$$

using the surface dipole magnetic field strength ${B}_{\mathrm{dip}}=3.2\times {10}^{19}{\left(P\dot{P}\right)}^{1/2}\,{\rm{G}}$, as well as the initial spin period,

$$\begin{eqnarray}&&{P}_{{\rm{i}}}\approx {\left\{{P}^{2}-2{C}_{1}\times \displaystyle \frac{{\tau }_{{\rm{d}},{\rm{i}}}{B}_{{\rm{i}}}^{2}}{\alpha -2}\times \left[{\left(1+\alpha \tau /{\tau }_{{\rm{d}},{\rm{i}}}\right)}^{-\tfrac{2}{\alpha }+1}-1\right]\right\}}^{1/2}.\end{eqnarray} \tag{ 8 }$$

We first consider the model without magnetic field decay and compare the results with the similar model from FK 2006 in Figure 3, where the initial spin period is

$$\begin{eqnarray}&&{P}_{{\rm{i}}}\approx {\left({P}^{2}-2{C}_{1}\times {B}_{{\rm{i}}}^{2}\tau \right)}^{1/2}.\end{eqnarray} \tag{ 9 }$$

The gray histograms in the upper and lower panels indicate the surface dipole magnetic field B_dip and the observed spin period P_obs distribution of the NANSs, respectively. The black histogram profile and the gray line indicate the initial magnetic field B_i and spin period P_i distribution from our model and FK 2006, respectively. It shows that the distribution of B_i in our results is similar to that of FK 2006, both of which follow lognormal distribution, while we obtain a smaller mean value. However, P_i also pursues lognormal distribution in our model, which is $\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \sim -1.58$, ${\sigma }_{\mathrm{log}\ {P}_{{\rm{i}}}}\sim 0.4$ for ξ = 1. It is much different from the normal distribution of FK 2006. When we take ξ < 1 (Zhang & Xie 2011), i.e., we think the actual age of the NS is smaller than its spin-down age, the mean value of the P_i distribution becomes larger ($\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \sim -0.71$ for ξ = 0.9 and $\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \sim -0.37$ for ξ = 0.5), while the standard deviation does not change. When we take ξ > 1, we cannot get any value of P_i.

Zoom In Zoom Out Reset image size

Then we introduce magnetic field decay of NSs into the model, and it (top panel of Figure 2) shows that the power-law form (Equation (6)) is favored. We limit the parameters by fitting the distribution trend of NANSs in the B_dip–τ_c diagram (lower panel of Figure 2) in a rough range: 1.2 ≤ α ≤ 1.8 and 10³ ≤ τ_d yr^–1 ≤ 10⁷. Then we try to simulate the initial values of the NANS magnetic field and spin period within the above parameter intervals.

Figure 4 displays the results of initial magnetic field distributions obtained by the control variable method. The variables are α, τ_d, and ξ from top to bottom, and other parameters are set to be α = 1.2, τ_d = 10³ yr, and ξ = 1.0 when they are not variables. This shows that all B_i in our model can well fit the lognormal distributions with $\langle \mathrm{log}({B}_{{\rm{i}}}/{\rm{G}})\rangle \,\sim 12.04\mbox{--}12.47$ and ${\sigma }_{\mathrm{log}\,{B}_{{\rm{i}}}}\sim 0.5\mbox{--}0.6$. In the case with α = 1.2, τ_d = 10³ yr, and ξ = 1.0, we get $\langle \mathrm{log}({B}_{{\rm{i}}}/{\rm{G}})\rangle \sim 12.47$, ${\sigma }_{\mathrm{log}\,{B}_{{\rm{i}}}}\sim 0.5$, which is consistent with the last results in Igoshev et al. (2022).

Zoom In Zoom Out Reset image size

When using the same parameters to simulate the spin periods, we can only get the initial values for some of the NANSs. Figure 5 shows the initial spin period distributions under the variables α, τ_d, and ξ from top to bottom, where the nonvariable parameters are set to be α = 1.2, τ_d = 10⁵ yr, and ξ = 0.5. The percentage numbers in the figure labels indicate the proportion of the valid sample, which includes the NANSs for which we can get initial spin periods, to the total sample, which includes all of the NANSs. It shows that all of the P_i histograms favor lognormal distributions with $\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \,\sim (-2.37)-(-0.37)$ and ${\sigma }_{\mathrm{log}\,{P}_{{\rm{i}}}}\sim 0.2\mbox{--}0.4$ whether full or partial samples. The top row, the middle and middle right panels, and the bottom right panel share the same results as $\langle \mathrm{log}({P}_{{\rm{i}}}/{\rm{s}})\rangle \,\sim -0.37$ and ${\sigma }_{\mathrm{log}\,{P}_{{\rm{i}}}}\sim 0.4$.

Zoom In Zoom Out Reset image size