Exploring the accretion-induced evolution of the spin period and magnetic field strength of Be/X-ray Pulsars

Abstract

Based on the detected spin periods ($P$) and inferred magnetic field strengths ($B$) by the cyclotron resonance scattering features, we analyze the $B-P$ properties of Be/X-ray Pulsars (BeXPs). We find that the $P$ distribution of BeXPs exhibits a bimodal feature separated at $P\sim 40$ s, where the average spin period of the BeXPs with $P>40$ s ($\langle P\rangle \sim 267$ s) is larger than that of the sources with $P<40$ s ($\langle P\rangle \sim 10$ s) by about one magnitude of order. Meanwhile, the average magnetic field strength of the long period BeXPs ($\langle B\rangle \sim 4.9\times 10^{12}$ G) is higher than that of the short period sources ($\langle B\rangle \sim 2.7\times 10^{12}$ G) by a factor of $\sim 2$. We try to explain these phenomena by the accretion-induced evolution process, and find that for the neutron star (NS) with the initial magnetic field strength of $B_{0}\sim 10^{12.2}-10^{13}$ G, when it accretes about $\Delta M\sim 10^{-6.5}\,\mathrm{M_{\odot }}$ companion matter, its spin period can shorten from $P_{0}\sim 1000$ s to $P\sim 260$ s, while its magnetic field strength decays little. Furthermore, when the NS accretes about $\Delta M\sim 10^{-5.5}\,\mathrm{M_{\odot }}$ matter, its spin period can shorten to $P\sim 10$ s, while its magnetic field strength decays by half. Finally, we also notice that as the continuing of the accretion process in Be/X-ray binary, when its NS accretes about $\sim 10^{-3}\,\mathrm{M_{\odot }}$ mater, it has the possibility to evolve to the double neutron star.

Appendix

Firstly, the magnetosphere-disk radius $R_{\mathrm{M}}$ is given by

$$ R_{\mathrm{M}}=\phi R_{\mathrm{A}}, $$

(9)

where $\phi =0.5$ is the empirical ratio coefficient, and $R_{\mathrm{A}}$ is the Alfvén radius (Shapiro and Teukolsky 1983):

$$ R_{\mathrm{A}}= 1.7\times 10^{8} {\mathrm{(cm)}} \dot{M}_{18}^{-\frac{2}{7}}[B_{12}R_{6}^{3}]^{ \frac{4}{7}}m^{-\frac{1}{7}}, $$

(10)

where the reduced parameters are given as $\dot{M}_{18}=\frac{\dot{M}}{10^{18}\,\mathrm{g\cdot s^{-1}}}$, $B_{12}= \frac{B}{10^{12}\,{\mathrm{G}}}$, $R_{6}=\frac{R}{10^{6}\,{\mathrm{cm}}}$ and $m=\frac{M}{1\,\mathrm{M}_{\odot }}$. Substituting equation (9) and (10) into equation (3), and adopting the NS moment of inertia as $I=\frac{2}{5}MR^{2}$, one can obtain the time derivative of the NS spin angular velocity:

$$\begin{aligned} \dot{\Omega }\equiv \frac{d \Omega }{dt}=k\times B_{12}^{\frac{2}{7}}, \end{aligned}$$

(11)

where $k = 1.87\times 10^{-10} \dot{M}_{18}^{\frac{6}{7}}m^{-\frac{4}{7}}{R_{6}^{- \frac{8}{7}}}\phi ^{\frac{1}{2}}$.

Secondly, equation (2) can be described as:

$$\begin{aligned} B_{12}=\frac{B_{012}}{1+\frac{\Delta M}{m_{B}}}= \frac{B_{012}}{1+\frac{\dot{M}t}{m_{B}}}, \end{aligned}$$

(12)

where $t$ is the accretion time. Substituting equation (12) into equation (11), one can obtain

$$\begin{aligned} \dot{\Omega }=k \times [\frac{B_{012}}{1+\frac{\dot{M}t}{m_{B}}}]^{ \frac{2}{7}}. \end{aligned}$$

(13)

By integrating the above equation with $t$, it can infer the expression of the NS spin angular velocity:

$$\begin{aligned} \begin{aligned} \Omega -\Omega _{0} & = \xi \times B_{012}^{\frac{2}{7}}[(1+ \frac{\dot{M}t}{m_{B}})^{\frac{5}{7}}-1] \\ &= \xi \times B_{012}^{\frac{2}{7}}[(1+\frac{\Delta M}{m_{B}})^{ \frac{5}{7}}-1], \end{aligned} \end{aligned}$$

(14)

where $\xi =\frac{7}{5}\,k\times (\frac{m_{B}}{\dot{M}})$, and $\Omega _{0}=\frac{2\pi }{P_{0}}$ is the initial NS angular velocity. If setting $\Omega _{0}\approx 0$ (the initial NS spin period is set $P_{0}=1000$ s in the calculation), then the $B-P$ evolution of the accretion NSs can be described as:

$$\begin{aligned} \begin{aligned} P &= 2\pi \xi ^{-1} \times B_{012}^{-\frac{2}{7}}[(1+ \frac{\dot{M}t}{m_{B}})^{\frac{5}{7}}-1]^{-1} \\ &= 2\pi \xi ^{-1} \times B_{012}^{-\frac{2}{7}}[( \frac{B_{012}}{B_{12}})^{\frac{5}{7}}-1]^{-1}. \end{aligned} \end{aligned}$$

(15)