ON THE SPIN PERIOD DISTRIBUTION IN Be/X-Ray BINARIES

High-mass X-ray binaries (HMXBs) usually consist of a neutron star (NS) and an optical companion star of a mass higher than about 8 M_☉. According to the spectral characteristics of the optical companions, HMXBs can be further divided into supergiant X-ray binaries (SGXBs) and Be/X-ray binaries (BeXBs; see Reig 2011 for a recent review). Most BeXBs are transient systems and present moderately eccentric orbits (e ≳ 0.3). The NS captures the wind material from its companion, producing X-ray radiation. Meanwhile, the spin of the NS changes with time, and both spin-up and spin-down have been observed when accretion took place (Nagase 1989; Bildsten et al. 1997).

Corbet (1984, 1985, 1986) first noted that different subgroups of HMXBs appear to be located in different regions in the spin period (P_s) versus the orbital period (P_orb) diagram (also called the Corbet diagram). In particular, there exists a positive correlation between P_s and P_orb for BeXBs, although with a large observed scatter. The relations between P_s and P_orb in HMXBs are likely to reflect the wind structure and accretion processes in HMXBs (e.g., Stella et al. 1986; van den Heuvel & Rappaport 1987; Waters & van Kerkwijk 1989; King 1991; Li & van den Heuvel 1996).

It is generally thought that the interaction between the NS magnetic field and the captured material from its binary companion can lead to a so-called equilibrium spin period P_eq of the NS (Bhattacharya & van den Heuvel 1991). However, the derived values of P_eq for wind-accreting SGXBs are always lower than the observed ones of P_s. It was suggested that the present P_s distribution may result from the equilibrium spin period when the companion star was still on the main sequence with a much weaker wind, and the wind of an SG is unable to transfer enough angular momentum to move the NS toward a new equilibrium value (e.g., Stella et al. 1986). The situation is more complicated in BeXBs. Be star winds are known to be disklike rather than spherically expanding as in SGs, and a Be star can transform to be a B star and vice versa from time to time. The mechanism for this transition is still unknown. The varying Be star wind and the eccentric orbit imply that there does not exist a stable equilibrium spin period. Waters & van Kerkwijk (1989) argued that the observational selection effects, that is, BeXBs are more likely to be observed when the NS moves within the dense equatorial disk wind and the difference between the spin-up and spin-down timescales when the NS accretes within and outside of the disk wind, imply that P_s is potentially correlated with the accretion rate during outbursts, and thus P_orb.

Recently, Knigge et al. (2011) showed that the P_s–P_orb correlation in BeXBs become more dispersed, and a bimodal distribution for both P_s and P_orb seems to exist. While the bimodality is somewhat marginal in P_orb, the P_s distribution has a clear gap at ∼40 s with two peaks around 10 s and 200 s, respectively. Knigge et al. (2011) proposed that two types of supernovae (SNe) may be responsible for the two subpopulations of BeXBs. The electron-capture supernovae (ECS) usually produce NSs with shorter spin periods and lower eccentricities, while iron core-collapse supernovae (CCS) are preferred for NSs with longer spin periods and higher eccentricities.

The original idea for the ECS candidates in BeXBs stems from a subclass of BeXBs that can be explained by low NS kicks (Pfahl et al. 2002). These BeXBs are characterized by long spin periods, persistent low X-ray luminosities (∼10³⁴–10³⁵ erg s⁻¹), wide binary orbits (P_orb > 30 days), and low eccentricities (e ≲ 0.2). The prototype is X Persei (P_s = 837 s, White et al. 1976). Other sources include RX J0146.9+6121 (1412 s, Haberl et al. 1998), RX J1037.5+5647 (860 s), and RX J0440.9+4431 (202.5 s) (Reig & Roche 1999). Recently discovered BeXBs SXP 1062 (1062 s; Hénault-Brunet et al. 2012), 1RXS J225352.8+624354 (47 s; Esposito et al. 2013), and SWJ2000.6+3210 (890 s; Pradhan et al. 2013) may also belong to this subclass. Podsiadlowski et al. (2004) and van den Heuvel (2004) suggested that the ECS mechanism may account for the low kicks in these BeXBs. This is different from the proposal by Knigge et al. (2011) that ECS-BeXBs may have short spin periods, relatively narrow orbits, and low eccentricities.

A thorough investigation on the origin of the subpopulations of BeXBs requires a population synthesis incorporating stellar and binary evolution, SN explosions, the Be star wind structure, mass transfer processes, and the NS evolution, which is beyond the scope of this paper. Here, we focus on the origin of the P_s distribution, which shows a much clearer bimodal feature than the P_orb distribution in the current sample. We expect that the orbital periods of BeXBs may be largely dependent on the initial parameters of the progenitor binaries and the SN mechanisms because tidal interaction is unable to change them effectively in wide orbits, while the NS spin periods are likely to be determined by the accretion processes in BeXBs. In Section 2, we compare the statistical characteristics of the outbursts in the two subpopulations and present a qualitative argument that the bimodal P_s distribution can be ascribed to different accretion modes of the NSs. In Section 3, we discuss the possible implications on the SN mechanisms, and in Section 4, we summarize.

It is uncertain how different types of SNe can influence the initial parameters of the newborn NSs. However, since the current spin periods of the NSs in BeXBs are generally much longer than the initial periods, their distribution must be determined by the interaction between the NSs and the captured material during the evolution, which has erased any feature in the initial distribution. Therefore, the bimodal P_s distribution is likely to result from different spin histories of the NSs.

The NS spin evolution in HMXBs depends on the angular momentum transfer between the NS and the captured matter from its companion star. It can be briefly outlined as follows (Davies & Pringle 1981, see also Dai et al. 2006). A newborn NS usually spins rapidly (with P_s much less than 1 s) so that the transferred matter from its companion star is stopped by the strong magnetic dipole radiation outside the light cylinder radius (or the Bondi accretion radius). The NS acts as a radio pulsar with magnetic dipole radiation responsible for its spin-down. This is called the ejector phase. When the magnetic dipole radiation can no longer prevent the wind matter from penetrating the light cylinder, the accretion flow starts to interact with the NS magnetic field, ceasing the pulsar activity, and the NS enters the propeller phase. In this phase, the accretion flow is balanced by the rotating magnetic field at the magnetospheric radius R_m. Accretion is inhibited, and the rotating NS loses its angular momentum by ejecting the material at R_m. This propeller phase ends when the accretion flow overcomes the centrifugal barrier and falls onto the NS. At this time, the NS spin period evolves to the equilibrium period given by (Davies & Pringle 1981)

$$\begin{equation} P_{\rm eq,w}\simeq 62B_{12}^{6/7}R_6^{18/7} M_1^{-5/7}\dot{M}_{14}^{-3/7}\,{\rm s}, \end{equation} \tag{ 1 }$$

where B = 10¹²B₁₂ G is the NS surface magnetic field strength, M = M₁ M_☉ the NS mass, R = 10⁶ R₆ cm the NS radius, and $\dot{M}=10^{14}\ \dot{M}_{14}$ g s⁻¹ the mass capture rate. If the value of B is constant, the maximum of P_{eq, w} is attained when $\dot{M}$ takes its lowest value. This occurs when the Be star wind changes from disklike to spherical and/or the NS moves around the apastron in an eccentric orbit. In the following accretor phase, if the captured material possesses enough angular momentum, it will evolve into an accretion disk, so the NS can be spun up or down to a new equilibrium period (Pringle & Rees 1972; Ghosh & Lamb 1979) as follows.

$$\begin{equation} P_{\rm eq,d}\simeq 4.8(\omega _{\rm c}/0.5)^{-1}B_{12}^{6/7}R_6^{18/7} M_1^{-5/7}\dot{M}_{17}^{-3/7}\,{\rm s}, \end{equation} \tag{ 2 }$$

where ω_c is the "fastness" parameter ranging between 0 and 1 and $\dot{M}_{17}=\dot{M}/10^{17}$ g s⁻¹. Since most BeXBs are in eccentric orbits and transient, the NS does not evolve along the above track monotonously, but transits between the propeller (sometimes even the ejector) and the accretor phases, with its spin period lying between P_{eq, w} and P_{eq, d}. If there is efficient disk accretion and spin-up, P_s is likely to be close to P_{eq, d}. Otherwise, it stays around P_{eq, w}.

Observational and theoretical developments since the 1990s have shown that the winds from Be stars are in the form of Keplerian disks, held by viscosity and with small radial velocities (e.g., Lee et al. 1991; Wood et al. 1993; Okazaki 2001; Porter & Rivinius 2003; Carciofi et al. 2009; Jones et al. 2009; Sigut et al. 2009; McGill et al. 2013). Because of its eccentric orbit, the NS can capture gas from the Be star disk only for a short span of time when it moves close to the disk, giving rise to transient X-ray outbursts. There are two types of X-ray outbursts in BeXBs. Normal (Type I) X-ray outbursts occur at or near the periastron passage. The X-ray luminosity (L_X) increases from its quiescent value by about one order of magnitude to ∼10³⁶–10³⁷ erg s⁻¹. The duration is a small fraction of the orbital period, typically (0.2–0.3)P_orb. Giant (Type II) outbursts are significantly brighter (L_X > 10³⁷ erg s⁻¹) and less frequent than normal outbursts. The duration is about tens of days (≳ 0.5P_orb, sometimes over one orbital period), and no orbital modulation has been detected. Spin-up episodes of the NS in BeXBs have been seen during both giant and normal outbursts (e.g., Parmar et al. 1989; Bildsten et al. 1997; Wilson et al. 2008), suggesting the existence of an accretion disk.

The detailed process of how an NS captures matter from the Be star disk is not clear. Angular momentum transfer in this regime is also difficult to quantify; however, some information can be obtained from the smoothed particle hydrodynamics simulations (Haigh & Okazaki 2004, 2006). Negueruela & Okazaki (2001) and Okazaki & Negueruela (2001) argued that the Be star disk is tidally truncated by the NS so that mass transfer occurs preferentially via leakage from the disk at the inner Lagrangian point near the periastron passage (Okazaki et al. 2002), resulting in normal outbursts. Giant outbursts are likely to be caused by accretion from a warped Be star disk that is misaligned with the binary orbital plane (Negueruela et al. 2001; Okazaki et al. 2013). The reason for this is because in misaligned systems, the tidal torque is weaker than in coplanar systems, so the truncation radius could be larger than the periastron separation; therefore, the NS could capture material at a high enough rate when passaging through the warped part of the Be disk (Martin et al. 2011). There is observational evidence for warped Be star disks before or during giant outbursts. For example, Negueruela et al. (2001) and Reig et al. (2007) found that before and during the giant outbursts of 4U 0115+634, the Hα emission line from the optical counterpart changed from the usual double-peaked profile to a single-peaked or shell-line profile on a timescale of a year or so. Complicated changes in the Hα line profiles during and after the 2009 giant outburst of A0535+262 have also been observed (Moritani et al. 2011) and interpreted by a precessing, warped Be star disk (Moritani et al. 2013). The misalignment between the spin axes of the Be star and the binary orbit is thought to originate from the SN explosion, especially the associated kick (Martin et al. 2009).

To investigate whether there is any potential relation between the outburst characteristics and the bimodal P_s distribution, we plot in Figure 1 the distribution of BeXBs in the Galaxy, the Large and Small Magellanic Clouds (LMC and SMC, respectively) with known outburst behavior. The green, red, blue, and orange symbols denote sources with type I outbursts only, with type II outbursts only, persistent sources, and sources with both type I and type II outbursts, respectively (data are taken from Raguzova & Popov 2005; Rajoelimanana et al. 2011; Townsend et al. 2011 and references therein). The dashed horizontal and vertical lines correspond to P_s = 40 s and P_orb = 60 days, which separate the population in P_s and P_orb, respectively. Among the 30 short-P_s BeXBs, 23 showed giant outbursts, while only 10 out of the 39 long-P_s BeXBs showed giant outbursts. This seems to indicate that the P_s distribution may be related to the occurrence of giant outbursts, which are characterized by long episodes of spin-up. We also plot the same distribution in Figure 2, differentiating systems with different peak luminosity (in units of 10³⁷ erg s⁻¹) during outbursts. Obviously short-P_s systems tend to have higher peak luminosities. To show this feature more clearly, we plot in Figure 3 the distribution of the peak (or persistent) luminosity for BeXBs, where P_s > 40 s (blue line) and ⩽40 s (red line). A bimodal distribution with characteristic luminosities of a few 10³⁶ erg s⁻¹ and a few 10³⁸ erg s⁻¹ is seen. Note that this result may be subject to the differences in the star formation histories and the metallicities of the Galaxy, LMC and SMC, which seem to play a role in the formation of Be stars, as they are more frequent in the SMC. Although a complete census of the outbursts in BeXBs is lacking, these figures indicate that the current spin period distributions in the two subpopulations are likely related to the different outburst characteristics.

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However, the difference in the NS accretion rate itself is not large enough to explain the difference in the spin periods in the two subpopulations. According to Equations (1) or (2), changing $\dot{M}$ by a factor of 10–100 can lead to the change of P_s by a factor of ∼3–7. A more important factor may lie in different accretion modes of the NSs. Although during both normal and giant outbursts there may be (transitional) accretion disks formed around the NSs, the structure of the disks can be quite different. It was pointed out by Okazaki et al. (2013) that if the accretion disk during normal outbursts is geometrically thin and radiatively cooling dominated (Shakura & Sunyaev 1973), the accretion timescale (i.e., the viscous timescale) will be several times the orbital period, much longer than the outburst duration. In such a situation, the system will neither exhibit rapid nor large X-ray flux changes seen in the outbursts. This implies that the accretion flow should be radiatively inefficient. Indeed, it is already known that accretion disks with $\dot{M}<\sim 10^{16}$ g s⁻¹ are likely to be in the form of geometrically thick, advection-dominated accretion flows (ADAFs; Narayan & Yi 1994, 1995). The radial velocity in ADAFs is comparable with the Keplerian velocity so that the accretion timescale is much shorter than in thin disks, which is consistent with the duration of normal outbursts. Meanwhile, the angular velocity in ADAFs is significantly lower than the Keplerian velocity in thin disks. The consequence is that the spin-up torque is relatively smaller and the equilibrium period becomes longer (Dai & Li 2006), i.e.,

$$\begin{eqnarray} P_{\rm eq,ADAF} &\simeq & 64.4 (A/0.2)^{-1}(\omega _{\rm c}/0.5)^{-1} \nonumber\\ && \times B_{12}^{6/7}R_6^{18/7} M_1^{-5/7}\dot{M}_{16}^{-3/7}\,{\rm s}, \end{eqnarray} \tag{ 3 }$$

where A = Ω_ADAF(R_m)/Ω_K(R_m) and $\dot{M}_{16}=\dot{M}/10^{16}$ g s⁻¹. Taking a typical value of A ∼ 0.2–0.3 (Yi et al. 1997), we find that the equilibrium period P_{eq, ADAF} is several times larger than P_{eq, d} with the same values of B and $\dot{M}$.

According to the above arguments, we tentatively propose an explanation for the bimodality in the P_s distribution. In BeXBs that tend to experience giant outbursts, the NS accretes from a thin disk with a relatively long lifetime and the efficient mass and angular momentum transfer results in spin-up during the outbursts so that its spin period reaches ∼P_{eq, d} with a typical value of ∼10 s. In BeXBs where normal outbursts dominate or there are no outbursts at all, the accretion flow around the NS is an ADAF (or quasi-spherical) so that there is relatively infrequent effective spin-up. The spin period lies between ∼P_{eq, w} and ∼P_{eq, ADAF}, with a typical value of ∼100 s. Thus, the subpopulations of BeXBs may originate from different accretion modes of the NSs.

In the above picture, disk warping plays an important role in determining the spin evolution of the NSs. In the Be star disks, the tidal torque exerted by the NS balances the viscous torque at the tidal warp radius R_tw. Outside of R_tw, the disk is dominated by the tidal torque, which flattens the disk and causes it to align with the binary orbital plane (Martin et al. 2009).³ With standard parameters (i.e., 1.4 M_☉ NS and 17 M_☉ Be star) for a BeXB, the tidal warp radius can be expressed as follows (Martin et al. 2011),

$$\begin{equation} R_{\rm tw}\simeq 9.3\times 10^{10}(1-e^2)^{3/4}\alpha \left(\frac{P_{\rm orb}}{1\,{\rm d}} \right) \,{\rm cm}, \end{equation} \tag{ 4 }$$

where α is the viscous parameter corresponding to the vertical shear in the disk. For BeXBs, the mass ratio of the NS and the Be star is approximately 0.1–0.2, so the truncation radius is ≲ 0.5 d, where d is the binary separation. A necessary condition for the interaction between a warped disk and an NS at periastron is R_tw < 0.5a(1 − e), where a is the semi-major axis of the binary. Combining this with Equation (4) yields

$$\begin{equation} P_{\rm orb}<(72.5\,{\rm days}) \alpha ^3\frac{(1-e)^3}{(1-e^2)^{9/4}}. \end{equation} \tag{ 5 }$$

In Figure 4 we show the distribution of BeXBs in the P_orb − e diagram, with blue, red, and green symbols representing binaries with P_s > 40 s, ⩽40 s, and unknown spin periods, respectively. The two curves correspond to the limit given by Equation (5) with α = 1 (upper) and 0.5 (lower). We see that most of the short-P_s BeXBs are in the regions confined by the curves, suggesting possible existence of disk warping.

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We emphasize that the argument in Section 2 is based on the global properties of the BeXB population rather than the individual source characteristics. Actually, Figure 1 shows that there is no strict one-to-one correspondence between short/long P_s and the occurrence of giant/normal outbursts. The transient nature of BeXBs means that it is impossible to monitor all outbursts for each source; therefore, our classification of the outburst behavior is incomplete. Evolution of the Be star disk also influences the characteristics of outbursts and the NS spin evolution. Nevertheless, the features in Figures 1–3 strongly suggest that different accretion modes of the NSs may be behind the origin of the subpopulations.

The SN mechanisms can be related to the subpopulations through their influence on the initial orbital period, eccentricity, and misalignment of the Be star disk. Population synthesis calculations by Linden et al. (2009) showed that the ECS channel may be efficient at forming BeXBs, especially in the SMC, in which the population of HMXBs has been found to have relatively wide orbits and low eccentricities. However, they did not compare the characteristics of BeXBs formed through the CCS and ECS channels. To examine the influence of different kinds of SNe on the orbital period distribution, we employ a Monte-Carlo method to simulate the formation of BeXBs. We adopt the binary population synthesis (BPS) code developed by Hurley et al. (2000, 2002) to calculate the evolutions of a large number of the primordial binaries, which is similar to the code StarTrack used by Linden et al. (2009). We consider solar abundance for the stars, with most of the input parameters (i.e., the distributions of the orbital separation, mass ratio, the initial mass function of the mass of the primary star) to be the same as the standard ones described by Hurley et al. (2002). Detailed descriptions of the method and calculated results will be presented elsewhere (Shao & Li 2014). Some relevant key points in the calculations are listed below.

• 1.  
  We consider both CCS and ECS for the NS formation. For ECS, we adopt the following criterion suggested by Fryer et al. (2012). If the core mass M_{c, bagb} of the primary star (i.e., the NS's progenitor) at the base of asymptotic giant branch is between 1.83 M_☉ and 2.25 M_☉, the CO core will nonexplosively burn into an ONe core, and the core mass is accumulated gradually. If its mass can reach M_esc = 1.38 M_☉, the ONe core will collapse due to electron capture into Mg and form a NS. If the mass is less than M_esc, it will leave an ONe WD.
• 2.  
  We apply a Maxwellian distribution for the SN kick velocity imparted to the newborn NS, with one-dimensional rms velocity σ = 265 km s⁻¹ for CCS (Hobbs et al. 2005) and 50 km s⁻¹ for ECS (Pfahl et al. 2002), respectively.
• 3.  
  We define a BeXB to be a binary consisting of an NS and a main-sequence (i.e., core H burning) companion star with a mass between 8 M_☉ and 20 M_☉, which does not fill its Roche-lobe. Since Be stars are rapidly rotating, we also consider the influence of tidal synchronization on the Be stars. Only systems with a synchronization timescale greater than the main-sequence lifetime of the Be star are taken into account.

The calculated normalized orbital period distributions of BeXBs are plotted in Figure 5. The red and black curves denote systems formed through CCS and ECS, respectively. In the left panel, the secondary star is regarded as a Be star when its rotational velocity is accelerated to 80% of its break-up velocity due to previous mass transfer. In the right panel, we assume that a constant fraction of B stars are Be stars. Note that in both cases the orbital periods have similar, wide distributions, with CCS–BeXBs peaked at P_orb ∼ 40–50 days and ECS–BeXBs at ∼100 days. It seems that CCS–BeXBs dominate at P_orb < ∼20–50 days, but at longer P_orb, the numbers of the two classes of objects are comparable. We need to caution that the number and distribution of ECS–BeXBs depend on the adopted mass range of the ECS progenitors (Nomoto 1984, 1987; Podsiadlowski et al. 2004; Siess 2007; Poelarends et al. 2008), which is not well understood. Nevertheless, a bimodal orbital period distribution from the two types of SN channels does not seem to exist.

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Another factor that can influence the P_s distribution and might be related to the SN mechanisms is the NS magnetic field. In the above estimates, we assume that all of the NSs possess a magnetic field of the order of 10¹² G. It is not known how NSs born in CCS and ECS differ in their magnetic fields. For BeXBs with measured cyclotron resonance scattering features in their X-ray spectra, the characteristic line energies range from 10 to 55 keV (Pottschmidt et al. 2012 and references therein), suggesting comparable field strengths (a few 10¹² G) in the NSs. However, there is indirect evidence that the NS magnetic field strengths may occupy a wide range. For example, from the measured spin-up rate in the 9.28 s Be/X-ray pulsar 2S1553−542, Pahari & Pal (2012) derived a relatively low field B ∼ 5 × 10¹¹ G for the NS. In another case, the spin-down rate measured in the 1062 s Be/X-ray pulsar SXP1062 implies that the NS possesses a magnetic field B ≳ 10¹⁴ G (Fu & Li 2012). The large scatter of BeXBs in the Corbet diagram may be partly due to the distribution and evolution of the NS magnetic field.

In this paper, we argue that the bimodal P_s distribution in BeXBs may not be directly linked to the two SN channels for the NS formation and is more likely to be ascribed to the difference in the accretion flows onto the NSs. This is indicated by the occurrence of giant/normal outbursts in the short- and long-P_s subpopulations, which reflect different spin-up efficiencies. We point out that the difference in the accretion rate during normal and giant outbursts is not enough to account for the range of P_s in the two subpopulations, and the structure of the accretion flows may play a more vital role. Normal outbursts are thought to be triggered by the mass transfer from a tidally truncated disk at or near periastron passage, while giant outbursts are somehow associated with the warping episodes of the Be star disk (Okazaki et al. 2013 and references therein). On the one hand, if the accretion disk during the normal outbursts are transitional and in the form of ADAF, the NS spin period is relatively longer due to the lower accretion rate, the shorter spin-up duration, and especially the lower angular velocity in the ADAF (the persistent low-luminosity BeXBs usually have long spin periods, and the NSs may be fed by low-velocity winds). On the other hand, the accretion disks formed during giant outbursts are radiatively cooling dominated, and the NSs experience longer episodes of spin-up with higher accretion rates, evolving to shorter spin periods. The two types of SN mechanisms can influence the NS spin evolution through the configuration of the Be star disk, but they do not seem to result in a bimodal distribution of the orbital period.

We are grateful to an anonymous referee for helpful comments. This work was supported by the Natural Science Foundation of China under grant Nos. 11133001 and 11333004.