Can the Subsonic Accretion Model Explain the Spin Period Distribution of Wind-fed X-ray Pulsars?
aa r X i v : . [ a s t r o - ph . H E ] M a y Can the Subsonic Accretion Model Explain the Spin PeriodDistribution of Wind-fed X-ray Pulsars?
Tao Li, Yong Shao, and Xiang-Dong Li
Department of Astronomy, Nanjing University, Nanjing 210023, ChinaKey Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry ofEducation, Nanjing 210023, China [email protected]
ABSTRACT
Neutron stars in high-mass X-ray binaries (HMXBs) generally accrete fromthe wind matter of their massive companion stars. Recently Shakura et al. (2012)suggested a subsonic accretion model for low-luminosity ( < × ergs − ), wind-fed X-ray pulsars. To test the feasibility of this model, we investigate the spinperiod distribution of wind-fed X-ray pulsars with a supergiant companion star,using a population synthesis method. We find that the modeled distributionof supergiant HMXBs in the spin period - orbital period diagram is consistentwith observations provided that the winds from the donor stars have relativelylow terminal velocities ( . − ). The measured wind velocities in severalsupergiant HMXBs seem to favor this viewpoint. The predicted number ratio ofwind-fed X-ray pulsars with persistent X-ray luminosities higher and lower than4 × ergs − is about 1 : 10. Subject headings: stars: early-type — stars: evolution — stars: neutron — X-rays: binaries
1. Introduction
High-mass X-ray binaries (HMXBs) contain a neutron star (NS) or a black hole (BH)that is accreting from an O or B type donor star. There are two subtypes of HMXBs, oneconsisting of a supergiant and the other a B-emission (Be) star (see van den Heuvel 2009;Reig 2011, for reviews). The NSs are generally young ( < yr) and strongly magnetized,and can be observed as X-ray pulsars due to accretion channeled by their magnetic field linesonto the polar caps, so the X-ray pulsations reflect the NS’s spin. Most supergiant HMXBs 2 –are close binaries with orbital periods less than a few days, and the NSs capture the donor’sradially-expanding wind matter as outlined by Bondi & Hoyle (1944), but in a few of themRoche-lobe overflow can also occur with an accretion disk formed around the NSs. The Besystems are usually in wide, eccentric orbits, and the NSs accrete from the Be star’s disklikewinds, usually near periastron.Corbet (1984, 1985, 1986) noticed that in the spin period ( P s ) vs. orbital period ( P orb )diagram the distribution of various subtypes of HMXBs shows distinct features: wind-fedNSs in supergiant HMXBs have much longer spin periods than disk-fed NSs, while the Besystems seem to roughly follow a positive correlation between P s and P orb . Moreover, both P s and P orb in the Be systems seem to show a bimodal distribution (Knigge et al. 2011). Thesefeatures must be related to the properties of the wind and the accretion processes in HMXBs,and have been extensively investigated (e.g., Stella et al. 1986; van den Heuvel & Rappaport1987; Waters & van Kerkwijk 1989; King 1991; Li & van den Heuvel 1996; Zhang et al. 2004;Dai et al. 2006; Cheng et al. 2014).The classical picture of the NS spin evolution in supergiant HMXBs was constructed byDavies & Pringle (1981), who argued that the captured wind matter forms a quasi-static,spherical atmosphere around the NS. Generally speaking, the spin-down evolution of a NSbefore steady accretion occurs contains at least two phases: the radio pulsar (or ejector)phase and the propeller phase. A newborn NS is usually rapidly rotating and emits ener-getic particles. If the accreting material is stopped outside the light cylinder, the NS actsas a radio pulsar and its spin-down is caused by energy loss via magnetic dipole radiation.When the accreting matter can permeate the NS’s light cylinder, the propeller phase beginsand the NS experiences rapid spin-down due to the interaction between the NS magneto-sphere and the surrounding plasma (Pringle & Rees 1972). Although the efficiency of thepropeller spin-down torque is still in debate (e.g., Illarionov & Sunyaev 1975; Davies et al.1979; Wang & Robertson 1985; Mori & Ruderman 2003), it is generally thought that anequilibrium spin period will be reached at the end of the propeller phase, when the nettorque exerted on the NS vanishes (Bhattacharya & van den Heuvel 1991).In the subsequent quasi-spherical accretion phase, the freely-falling matter is deceler-ated by a shock formed above the NS magnetosphere. The condition of steady accretion isdetermined by whether the magnetosphere boundary is unstable with respect to Rayleigh-Taylor instability (Arons & Lea 1976) or Kelvin-Helmholtz instability (Burnard et al. 1983).Shakura et al. (2012) argued that, if the X-ray luminosity L X > × erg s − , the shockedmatter can efficiently cool via Compton processes and enter the magnetosphere; if L X < × erg s − , a subsonic settling accretion regime sets in, with a hot convective shell formedabove the NS magnetosphere. In the latter case, both the accretion torque and the equilib- 3 –rium spin period deviate from those in the traditional spherical wind accretion model. Thissubsonic accretion model has been applied to account for the observational characteristicsof some long-spin period X-ray pulsars including GX 1+4 (Gonz´alez-Gal´an et al. 2012), 4U1954+31 (Martinz et al. 2011), SXP 1062 (Popov & Turolla 2012), X Per (Lutovinov et al.2012), and GX 301 − ∼ − ergs − (Ducci et al. 2014). However, astudy on the global feature of HMXBs in this model is still lacking.This paper aims to test the model of Shakura et al. (2012) by comparing the expectedspin period distribution based on population synthesis calculation with the observations ofwind-fed HMXBs. Different from Dai et al. (2006, 2016), we consider the NS spin evolutionboth prior to and during the HMXB phase, in which the subsonic accretion may dominate.In Section 2 we describe our assumptions and theoretical considerations. Section 3 presentsthe calculated results, which are then compared with observations. Our discussion andconclusions are in Section 4.
2. Model2.1. Formation of Incipient HMXBs
The formation history of HMXBs starts from primordial massive binaries and has beenreviewed by Bhattacharya & van den Heuvel (1991) and van den Heuvel (2009). The moremassive, primary star first evolves, fills its Roche-lobe, transfers matter to the secondary,and finally leaves a NS (or a BH) with a supernova (SN) explosion. If the binary is notdisrupted after the SN, the NS will gravitationally capture the wind matter from its com-panion star and become an X-ray source. We use the rapid binary evolution code developedby Hurley et al. (2000, 2002) and updated by Kiel & Hurley (2006) to generate the incipientHMXB population. This code combines the standard assumptions for primordial binarieswith analytic prescriptions that describe stellar evolution, binary interactions and SN explo-sions. Shao & Li (2014) further modified the code for the formation and evolution of NSsand BHs. We briefly describe them as follows.If the primary star is significantly more massive than the secondary, the mass transfermay be dynamically unstable, leading to common envelope (CE) evolution, which is still notwell understand. To judge the stability of mass transfer, we adopt the calculated results byShao & Li (2014) for a grid of binaries rather the empirical relations in Hurley et al. (2002)for the critical mass ratios, and assume a 50% mass transfer efficiency (see Shao & Li 2014, 4 –for details). We use the energy conservation formalism for CE evolution (Webbink 1984),by equating the difference in the orbital energy with the binding energy of the primary’senvelope, α CE (cid:18) GM , f M a f − GM , i M a i (cid:19) = GM , i M , e λR L1 , (1)where α CE ≤ G is the gravity constant, M , M , e , M , f arethe total mass, the envelope mass, and the core mass of the primary, respectively, M is thesecondary mass, a is the binary separation, R L1 is the Roche-lobe radius of the primary at theonset of Roche-lobe overflow, the indices i and f denote the values at the beginning and endof the CE stage, respectively, and the parameter λ reflects the effect of the mass distributionwithin the envelope and the contribution from other energies besides gravitational energy(e.g., de Kool 1990; Dewi & Tauris 2000; Podsiadlowski et al. 2003). We used the calculatedvalues of λ by Xu & Li (2010) for high- and intermediate-mass stars at various evolutionarystages, and take α CE = 1 . σ k = 265 kms − for core-collapseSNe (Hobbs et al. 2005) and σ k = 50 kms − for electron-capture SNe (Dessart et al. 2006).The initial parameters in our binary population synthesis calculation are taken as fol-lows. All the binaries are tidally circularized (Hurley et al. 2002). The primary mass M isin the range of 7 − M ⊙ , following the Kroupa et al. (1993) mass function. The secondarymass M lies between 3 M ⊙ and 30 M ⊙ , with the initial mass ratio M /M uniformly dis-tributed between 0 and 1. The logarithm of the orbital separation ln a follows a uniformdistribution, with the values of a ranging from 3 R ⊙ to 10 R ⊙ . We adopt Solar metallicityfor the stars, and assume that the star formation proceeds at a constant rate (5 M ⊙ yr − )during the past 13 Gyr.Figure 1 demonstrates the distribution of P orb and M for the produced incipient NS bi-naries. To compare with the observed HMXBs we select the 4 × binaries with P orb < M ⊙ ≤ M ≤ M ⊙ for further investigation. We also calculate the luminosities,radii and effective temperatures of the companion stars, and use them to estimate the windmass loss rates and the mass transfer rates. 5 – We then investigate the spin history of a 1 . M ⊙ NS with an OB companion star basedon the model of Davies & Pringle (1981) but with considerable modifications. As brieflydescribed below, the NS evolves through three different evolutionary stages: the radio pulsar,propeller, and accretor phases.Normally a NS is born with a short spin period and a strong magnetic field. Its magneticor radiation pressure is able to to expel the wind matter outside the Bondi accretion radius r G = 2 GM/v (Bondi 1952), or the light cylinder radius, r lc = cP s / π . Here M is the NSmass and v = ( v + v ) / is the velocity of the wind relative to the NS, where v orb and v w are the orbital velocity of the NS and the wind velocity at the NS’s orbit, respectively. Thespin-down torque induced by magnetic dipole radiation in this radio pulsar phase is I ˙Ω s = − µ Ω s c , (2)where I , µ , and Ω s are the moment of inertia, the magnetic dipole moment, and the angularvelocity of the NS, respectively. The radio pulsar phase ends when the wind matter entersthe Bondi radius r G or the light cylinder radius r lc (see Davies & Pringle 1981, for details),and the spin periods are correspondingly, P a ≃ . µ / ˙ M − / ( M/M ⊙ ) / v − / s , (3) P b ≃ . M − / µ / v − / s , (4)where µ = µ/ Gcm , v = v/ cm s − , and ˙ M = 10 ˙ M g s − is the mass transferrate given by (Bondi & Hoyle 1944) ˙ M = πr ρ w v, (5)where ρ w is the density of the companion’s wind matter at the NS’s orbit. For an isotropically-expanding wind we have ρ w = − ˙ M / (4 πa v w ) . (6)Here ˙ M is the mass loss rate of the companion star, which can be estimated with theempirical formula proposed by Nieuwenhuijzen & de Jager (1990) for OB type stars, − ˙ M = 9 . × − ( R /R ⊙ ) . ( L /L ⊙ ) . ( M /M ⊙ ) . M ⊙ yr − , (7)where L and R are the luminosity and the radius of the companion star, respectively.In the propeller phase the infalling matter starts to interact with the magnetosphere,but the fast rotating magnetosphere inhibits the matter from steady accretion onto the NS. 6 –At this moment the magnetospheric radius r m = [ µ / (2 GM ˙ M )] / (Pringle & Rees 1972)is still larger than the corotation radius r co = ( GM/ Ω ) / . The infalling matter is stoppedand then expelled at the magnetosphere because of the centrifugal barrier, extracting theangular momentum of the NS. The efficiency of angular momentum loss due to the propellermechanism is still in debate (Illarionov & Sunyaev 1975; Wang & Robertson 1985; Ikhsanov2007; Romanova et al. 2005; Ustyugova et al. 2006; D’Angelo & Spruit 2010). When theejected mater is corotating with the magnetosphere, the spin-down torque is I ˙Ω s = − ˙ M r Ω s . (8)This phase ends when r co = r m , and the corresponding spin period P eq ≃ µ / ˙ M − / ( M/M ⊙ ) − / s (9)is called the equilibrium period.In the following accretor phase the NS spin period will be also changed due to mass andangular momentum transfer. However, both observations (Bildsten et al. 1997) and numeri-cal simulations (e.g., Matsuda et al. 1987; Fryxell & Taam 1988; Sawada et al. 1989; Ruffert1994; Blondin & Pope 2009; D¨onmez et al. 2011; Cruz-Osorio et al. 2012) indicate erraticspin-up/down with relatively low averaged angular momentum transfer rate during windaccretion. Thus, one may expect that the spin periods of the NSs during the accretor phasedo not change much from the period reached in the propeller phase. Therefore, Dai et al.(2006) stopped their calculation when the companion star evolves off the main-sequence, nomatter whether P eq is reached. This simple picture should be modified, because, accordingto Shakura et al. (2012), there exist two cases of quasi-spherical accretion depending on theaccretion rate. The matter heated up in the shock above the magnetosphere can fall to-ward the NS surface only when it cools down rapidly, which requires the X-ray luminosity & × erg s − . In this regime, the NS spin evolution is determined by the magnitudeof the angular momentum carried by the captured matter as discussed above. If the X-rayluminosity < × erg s − , the shocked matter is unable to cool down, a quasi-static shellis formed around the NS magnetosphere, in which the hot matter settles down subsonically.In this subsonic accretion regime the accretion rate is determined by the ability of the mat-ter to enter the magnetosphere via instabilities, which can be considerably lower than thecapture rate by the NS. The torques exerted on the NS come from both accretion and themagnetosphere-plasma interaction. In the case of moderate magnetosphere-plasma coupling,the equation that governs the NS spin evolution reads I ˙Ω s = A ˙ M / − B ˙ M / , (10) 7 –where the coefficients A and B are (in CGS units) A ≃ . × K µ / v − (cid:18) P orb
10 d (cid:19) − , (11)and B ≃ . × K µ / (cid:18) P s
100 s (cid:19) − . (12)Here K is a dimensionless numerical factor with typical value of 40 (Shakura et al. 2012).The equilibrium spin period in the subsonic accretion phase is P s , eq ≃ µ / v ˙ M − / (cid:18) P orb
10 d (cid:19) s . (13)Obviously, NSs experiencing subsonic accretion can have much longer spin period than thoseundergoing rapid accretion at the same rate.
3. Results
We then calculate how the NS spins change in the incipient HMXBs, based on thetheoretical model presented in Section 2. We assume that the NSs initially spin at a perioduniformly distributed between 10 ms and 100 ms. A log-normal distribution is set for theinitial NS magnetic fields (in units of Gauss), with a mean of 12.5 and a standard deviationof 0.3. We assume that the fields do not decay during the lifetime of a HMXB. We adoptthe standard Castor et al. (1975) formula for the wind velocity v w , v w = v ∞ (1 − R /a ) β , (14)where v ∞ is the terminal velocity of the wind, and β ∼ . −
1, taken to be 0.8 in this work.The mass capture rate depends on many parameters, among which the relative velocity v and hence the wind velocity v w are the most sensitive ones. So we set v ∞ = v esc and 3 v esc to examine its influence (e.g., Waters & van Kerkwijk 1989; Owocki 2014), where v esc is theescape velocity at the surface of the companion star.Since our calculations are limited to the stage of quasi-spherical wind accretion, we keepthe binary evolution going on until the companion star starts to fill its Roche lobe. Our calcu-lation shows that the produced NS populations are distributed in the radio pulsar, propeller,and accretor phases, depending on the activity of the NS magnetic field - wind interaction.To compare with observations of wind-fed NSs in HMXBs, we chose the calculated resultsfor NSs that have entered the accretor phase with X-ray luminosities ≥ erg s − , which 8 –are regarded to be potential X-ray pulsars. The accretion processes in Be/X-ray binariesare quite different from supergiant systems, because of the complicated structure of the Bestar’s wind and eccentric orbits, which imply that the quasi-spherical accretion model maynot apply for these sources (see Dai et al. 2006; Cheng et al. 2014, for detailed discussion).So we only pay attention to the wind-fed X-ray pulsars in the supergiant systems.Figure 2 displays the calculated spin period distribution, in which the left and rightpanels correspond to the cases of v ∞ = v esc and 3 v esc , respectively. The darkness of each ele-ment represents the relative number of the X-ray binaries. The diamonds mark the observedwind-fed supergiant HMXBs with known P s and P orb , and the triangles represent disk-fedX-ray pulsars (data from Townsend et al. 2011; Dai et al. 2016, and references therein). Ac-cording to our calculation, the wind-fed X-ray pulsars are clustered into the upper and lowersubgroups, related to different regimes of quasi-spherical accretion. Since the NS mass cap-ture rates are higher for shorter orbital periods, only HMXBs in relatively narrow orbits canhave X-ray luminosities & × erg s − . The NSs in these systems are experiencing rapidaccretion, but their spin periods still stay around the periods reached during the propellerphase, because they have got small net angular momentum during the accretor phase. Sothey are located at the lower-left shaded region of the diagram. The other sources are un-dergoing the subsonic accretion and have longer spin periods, located in the upper part ofthe diagram.We find that when v ∞ = v esc , around 8% of the NSs are in the rapid accretion stage, and89% and 3% in the subsonic accretion stage with X-ray luminosities > and < erg s − ,respectively (the latter are not displayed in the diagram). If v ∞ is increased to 3 v esc , themass transfer rates are lowered by a factor of ∼
80 according to Eqs. (5) and (6). Theluminosity condition for the subsonic accretion becomes easier to satisfy, further extendingthe subsonic accretion region in the diagram, and the corresponding numbers of the NSsbecome ∼ v ∞ = v esc better match the observations.To demonstrate the difference in the expected HMXB distribution in different models,we recalculate the NS spin evolution but without the subsonic accretion process taken intoaccount, and show the results in Fig. 3. This was previously done by Dai et al. (2006), butthey did not consider the NSs that enter the accretor phase when the companion stars haveevolved to be supergiants. The left and right panel also correspond to v ∞ = v esc and 3 v esc ,respectively. Since the equilibrium period in the rapid accretion regime is much shorter than 9 –in the subsonic accretion regime at the same v ∞ , most of the shaded regions in Fig. 3 are lowerthan in Fig. 2, except the ones with X-ray luminosities > × erg s − , which are locatedat the same positions. It seems that the results in the right panel with v ∞ = 3 v esc betterfit the observations. In this case, the percentages of the sources with X-ray luminosities L X > × ergs − and 10 ergs − < L X < × ergs − are ∼
1% and 52% respectively,while the rest 47% have X-ray luminosities L X < ergs − .
4. Discussion and conclusions
We have applied the model proposed by Shakura et al. (2012) for quasi-spherical accre-tion to explore the spin period distribution for X-ray pulsars in supergiant HMXBs. Oneof the differences between this model and the traditional one is that, some of the NSs mayexperience long-term spin-down during the subsonic accretion phase, and have much longerequilibrium period. So to explain the current spin periods, it is not required that the spin-down processes were conducted when the NS was interacting with relatively weak winds froma main-sequence companion star (Waters & van Kerkwijk 1989). The two different equilib-rium periods (i.e., Eqs. [13] and [9]) imply that, given the same P orb , the NSs in differentaccretion regimes can have distinct spin period distributions.Whether the subsonic accretion is taken into account or not, our calculated results dis-played in both Figs. 2 and 3 can be in accord with the observed supergiant HMXBs in the P s − P orb diagram. The difference lies in that, to match the observations, the model withsubsonic accretion requires a relative low wind velocity v ∞ ∼ v esc (with the average value ∼ −
800 km s − ), otherwise a higher wind velocity v ∞ ∼ v esc (with the average value ∼ − − ) is required. Since the mass capture rates strongly depend on thewind velocity, the values of v ∞ can serve as a plausible indicator to discriminate the models.They have been estimated by modelling of the ultraviolet resonance lines, and have beenextensively investigated (e.g., Howarth & Prinja 1989; Prinja et al. 1990; Lamers et al. 1995;Howarth et al. 1997; Prinja & Crowther 1998; Puls et al. 1996; Kudritzki et al. 1999). Gen-erally v ∞ /v esc ≃ . .
1) in the OB su-pergiants in HMXBs (van Loon et al. 2001). The most reliable values of v ∞ for HMXBswhich we have found in the literature are as follows: ∼ −
700 kms − for Vela X-1, ∼ − for 4U 1700 − ∼
500 kms − for LMC X-4, .
600 kms − for SMC X − ∼ − for 4U 1907+09 (Cox et al. 2005), 1500 ±
200 kms − forIGR J17544 − ∼
350 kms − for 4U2206+54 (Rib´o et al.2006), ∼
305 kms − for GX 301 − ±
100 kms − for 4U 1908+075 10 –(Mart´ınez-N´u˜nez et al. 2015). Thus quite a few systems have v ∞ ∼ v esc , lending support tothe subsonic accretion model.Our results also indicate that NSs expected to experience subsonic accretion are over-whelming in the whole population, which can also be tested by observations. According toour calculated results, the number ratio of supergiant HMXBs with L X above and below4 × ergs − is 1 : 11 in the subsonic accretion model with v ∞ = v esc , and 1 : 52 in thetraditional model with v ∞ = 3 v esc . However, almost all HMXBs are variable in X-rays with adynamical range from a few to 10 (Stella et al. 1986; Reig 2011). Long-term monitoring thesampled HMXBs and reasonable evaluation of the X-ray luminosities will also help resolvethe issue.In our calculations we employ the classical β -velocity law to model the supergiant’s wind.Modern self-consistent calculations of the wind hydrodynamics in Wolf-Rayed stars show thatthe wind acceleration is more complicated: the wind structure shows two acceleration regions,one close to the hydrostatic wind base in the optically thick part of the atmosphere, and theother farther out in the wind (Gr¨afener & Hamann 2005, 2008). Thus a simple β -velocitylaw may not be an accurate description of the velocity field, and the strong acceleration inthe inner part is best described by β ∼ β . Thistype of velocity profile in the Wolf-Rayet winds has also been found in recent calculations ofsupergiant winds (Sander et al. 2015). In this case the average wind velocity is a bit lowerthan that in our calculation with the same terminal velocity v ∞ . This will make the shadedregions in Figs. 2 and 3 downward slightly, but it does not significant affect our final results.Another issue is that, because of the kick velocity received by the NSs at their formation,one does not expect the NS’s and/or the supergiant’s spins to be aligned with the orbit(e.g., Lai et al. 1996; van den Heuvel & van Paradijs 1997). In principle this may not causeconsiderable changes in the mass and angular momentum transfer rates in the binaries, sincethe wind material is assumed to be spherically expanding from the donor star and there isnot a preferred direction for the wind distribution. Actually most supergiant HMXBs arein close, circular orbits, suggesting strong tidal interactions between the components. TheSN kick may play a more important role in Be/X-ray binaries. A misaligned Be star disk isless likely to be tidally truncated by the NS than in coplanar systems, which is more likelyto trigger giant outbursts rather than quasi-periodic, normal outbursts (Martinz et al. 2011;Okazaki et al. 2013). This might be related to the bimodal distribution of the spin periodsin Be/X-ray binaries (Knigge et al. 2011; Cheng et al. 2014).We finally caution that our approximation of a smooth stellar wind should be consid-ered unrealistic. Stellar winds are likely to have both large-scale (quasi)-cyclical structures(Kaper & Fullerton 1998) and small-scale, stochastic structures (Runacres & Owocki 2002; 11 –Puls et al. 2008), which may be caused by various instabilities in the stars. So our resultsshould be considered as averaged over long-time for the NS - wind interaction.We are grateful to the referee for helpful comments. This work was funded by theNatural Science Foundation of China under grant numbers 11133001 and 11333004, and theStrategic Priority Research Program of CAS (under grant number XDB09000000). REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
16 –Fig. 1.— Distributions of incipient HMXBs in the initial companion mass ( M ) vs. theorbital period ( P orb ) plane. The red curve represents their birthrate as a function of M . 17 – log P orb (days) l og P s ( s ) log P orb (days) l og P s ( s ) Fig. 2.— The distribution of wind-fed HMXBs in the P s − P orb diagram, with the subsonicsettling accretion taken into account. The gray scale denotes the relative number of HMXBs.The diamonds and triangles represent the observed wind-fed and Roche-lobe overflowingsupergiant HMXBs, respectively. The left and right panels correspond to v ∞ = v esc and3 v esc , respectively. log P orb (days) l og P s ( s ) log P orb (days) l og P s ( s )1E+001E+011E+021E+031E+041E+051E+06