Evolution of LMXBs under Different Magnetic Braking Prescriptions
aa r X i v : . [ a s t r o - ph . H E ] J a n Evolution of LMXBs under Different Magnetic BrakingPrescriptions
Zhu-Ling Deng , , , , , Xiang-Dong Li , ∗ , Zhi-Fu Gao , ∗ , Yong Shao , Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150, Science 1-Street,Urumqi, Xinjiang 830011, China Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008,China University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China;[email protected] School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China;[email protected] Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry ofEducation, Nanjing 210023, China
ABSTRACT
Magnetic braking (MB) likely plays a vital role in the evolution of low-massX-ray binaries (LMXBs). However, it is still uncertain about the physics of MB,and there are various proposed scenarios for MB in the literature. To examineand discriminate the efficiency of MB, we investigate the LMXB evolution withfive proposed MB laws. Combining detailed binary evolution calculation withbinary population synthesis, we obtain the expected properties of LMXBs andtheir descendants binary millisecond pulsars. We then discuss the strength andweakness of each MB law by comparing the calculated results with observations.We conclude that the τ -boosted MB law seems to best match the observationalcharacteristics. Subject headings: stars: evolution – stars: neutron – pulsars: individual: mag-netic braking – X-rays: binaries
1. INTRODUCTION
Low-mass X-ray binaries (LMXBs) contain an accreting compact star (a black hole ora neutron star) and a low-mass donor. Mass transfer (MT) in LMXBs proceeds via Roche-lobe overflow (RLOF). There are about 200 LMXBs discovered in the Galaxy (Liu et al. 2 –2007), and their formation remains to be a controversial topic (Tauris & van den Heuvel2006; Li 2015, for reviews). Here, we focus on the evolution of LMXBs with a neutronstar (NS). For these LMXBs there exists a critical value for the initial orbital period, i.e.,the bifurcation period P bif , separating the formation of converging LMXBs from divergingLMXBs (Pylyser & Savonije 1988, 1989). The origin of the bifurcation period comes fromthe competition between orbital expansion caused by MT and orbital shrinking due to an-gular momentum loss (AML) by magnetic braking (MB) and gravitational radiation (GR),both of which strongly depend on the orbital period. Binaries with P orb > P bif will becomerelatively wide binaries containing a recycled NS and a He or CO white dwarf (WD), whilethose with P orb < P bif mainly keep going to contract along the cataclysmic variable (CV)-like or ultra-compact X-ray binary (UCXB) evolutionary tracks (Rappaport et al. 1995;Kalogera & Webbink 1996; King & Ritter 1999; Tauris & Savonije 1999; Kolb et al. 2000;Podsiadlowski et al. 2000; Tauris et al. 2000; Li 2002; Podisadlowski et al. 2002; Lin et al.2011; Istrate et al. 2014a; Jia & Li 2014). Theoretically, the value of the bifurcation period P bif is ∼ . − P orb ) distribution of binary millisecond pulsars (BMSPs), which are thoughtto be the descendants of LMXBs, does not match the observations, especial for BMSPswith P orb ∼ . −
10 d (Pfahl et al. 2003; Istrate et al. 2014a; Shao & Li 2015). It was also 3 –found that UCXBs can hardly form within a Hubble time under the Skumanich MB law(Podisadlowski et al. 2002; van der Sluys et al. 2005a).Some alternative AML mechanisms have been proposed to account for the discrepancybetween theory and observation. For example, Chen & Li (2006) proposed that a circumbi-nary disk can extract orbital AM from the inner LMXB. Justham et al. (2006) suggestedthat the secondaries of black hole intermediate-mass X-ray binaries (BHIMXBs) may possessanomalously high magnetic fields to enhance the efficiency of MB. The effect of consequen-tial AML was also investigated by Nelemans et al. (2016) and Schreiber et al. (2016) for theCV evolution. Pavlovskii & Ivanova (2016) reproduced the formation of Sco X-1 by usinga modified Skumanich MB law. Recently, Van et al. (2019) added two additional “boosted”factors (i.e., convective turnover time and stellar wind) to the Skumanich MB, in order toexplain the measured MT rates of LMXBs. Van & Ivanova (2019, hereafter VI19) suggesteda new MB formula (named as Convection And Rotation Boosted MB) for persistent NSLMXBs.Meanwhile, various MB models have been developed to investigate the rotational evolu-tion of single low-mass stars (e.g., Matt et al. 2012; Reiners & Mohanty 2012; Gallet & Bouvier2013; Van Saders et al. 2013; Sadeghi Ardestani et al. 2017). In principle, these MB mech-anisms may also work in LMXBs. So it is important to examine and discriminate differentMB formulae by comparing the observations of LMXBs and BMSPs with the theoreticalresults, and this is the objective of our study.The structure of the paper is organized as follows. We describe the stellar evolution codeand summarize several typical MB formalisms in Section 2. The calculated binary evolutionresults are demonstrated in Section 3. We present our discussion and summary in Section 4.
2. EVOLUTION CODE AND BINARY MODEL2.1. The stellar evolution code
All calculations were carried out by using the stellar evolution code Modules for Ex-periments in Stellar Astrophysics (MESA; version number 11554; Paxton et al. 2011, 2013,2015, 2018, 2019). The binaries initially consist of an NS (of mass M ) and a main-sequence(MS) secondary/donor star (of mass M ) with Solar chemical compositions ( X = 0 . Z = 0 . α = 2, anddo not consider semiconvection and overshooting. The effective radius of the RL for thesecondary is calculated with the Eggleton (1983) formula, 4 – R L , a = 0 . q / . q / + ln (1 + q / ) , (1)where a is the orbital separation of the binary and q = M /M is the mass ratio. We adoptthe Ritter (1988) scheme to calculate the MT rate via RLOF, − ˙ M = ˙ M , exp (cid:18) − R − R L , H (cid:19) , (2)where H is the scale-height of the atmosphere evaluated at the surface of the donor, R isthe radius of the donor, and ˙ M , = 1 e / ρc th Q, (3)where ρ and c th are the mass density and the sound speed on the surface of the star respec-tively, and Q is the cross section of the mass flow via the L point. In LMXBs the binary orbital revolution and the spin of the secondary star are assumedto be synchronized, because the tidal synchronization timescale is usually much shorter thanthe evolutionary timescale of LMXBs (King 1988) . The orbital AM can be expressed as J orb = M M M + M Ω a , (4)where the orbital angular velocity Ω = p GM/a , and M = M + M is the total mass.Taking the logarithmic derivative of Eq. (4) with respect to time gives the rate of change inthe orbital separation ˙ aa = 2 ˙ J orb J orb − M M − M M + ˙ MM . (5)Here the total rate of change in the orbital AM is determined by˙ J orb = ˙ J gr + ˙ J ml + ˙ J mb , (6)where the three terms on the right-hand-side of Eq. (6) represent AML caused by GR, massloss, and mb, respectively. The GR-induced AML rate ˙ J gr is given by (Landau & Lifshitz1959; Faulkner 1971) ˙ J gr J orb = − G c M M Ma , (7) We actually treated the spin and orbital evolution separately in our calculation, since in a few cases thetidal torques are not strong enough to synchronize the spin and orbital revolution. G and c are the gravitational constant and the speed of light, respectively. Theaccretion rate of the NS is assumed to be limited by the Eddington accretion rate ˙ M Edd ,˙ M = min (cid:16)(cid:12)(cid:12)(cid:12) ˙ M (cid:12)(cid:12)(cid:12) , ˙ M Edd (cid:17) , (8)and the mass loss rate from the binary system is˙ M = ˙ M − (cid:12)(cid:12)(cid:12) ˙ M (cid:12)(cid:12)(cid:12) . (9)In the case of super-Eddington MT, we adopt the isotropic reemission model, assuming thatthe extra material leaves the binary in the form of isotropic wind from the NS. Therefore,the AML rate due to mass loss is˙ J ml = − (cid:16)(cid:12)(cid:12)(cid:12) ˙ M (cid:12)(cid:12)(cid:12) − ˙ M (cid:17) a Ω , (10)where a is the distance between the NS and the center of mass of the binary.Magnetized stellar winds coupled with the secondary star can reduce the stellar spinspeed effectively, then carry away the orbital AM through tidal torques. In the following,we list several MB prescriptions we will use in calculating the evolution of LMXBs.1. The Skumanich ModelBased on the widely used Skumanich (1972) MB model, Verbunt & Zwaan (1981) andRappaport et al. (1983) proposed an MB formula for LMXBs:˙ J mb , Sk = − . × − M R ⊙ (cid:18) R R ⊙ (cid:19) γ mb Ω dyne cm . (11)Here R ⊙ is the Solar radius and γ mb is a dimensionless parameter ranging from 0 to 4. Inthis paper we adopt the default value γ mb = 4 in MESA. Note that for CV evolution smallervalues of γ mb are often used (Knigge et al. 2011), so we also calculate the evolution of LMXBwith γ mb = 2 and 3, and find that the main results do not change significantly comparedwith those with γ mb = 4.2. The Matt12 ModelMatt & Pudritz (2008, hereafter MP08) carried out 2D axisymmetric magnetohydro-dynamical (MHD) simulations of wind outflows from a rigidly rotating star with a aligneddipolar field, to determine the dependence of the wind torque on the magnetic field and massoutflow rate. Matt et al. (2012, hereafter Matt12) extended the parameter study of MP08by including variations in both the magnetic field strength and the stellar rotation rate, andderived the following stellar wind torque formula,d J d t = − K (2 G ) m ¯ B ˙ M − , W (cid:18) R M m2 (cid:19) Ω( K + 0 . u ) m , (12) 6 –where u is the equatorial rotation speed divided by the break-up speed, K = 6 . K = 0 . m = 0 .
17 are adjustable parameters to fit the observations, which were taken fromGallet & Bouvier (2013). The wind loss rate ˙ M , W is evaluated using the Reimers (1975)wind mass-loss prescription˙ M , W = 4 × − M ⊙ yr − (cid:18) R R ⊙ (cid:19) (cid:18) L L ⊙ (cid:19) (cid:18) M ⊙ M (cid:19) . (13)where L is the donor’s luminosity and L ⊙ is the Solar luminosity. The mean magnetic field¯ B s = f B s , where B s is the surface magnetic field strength of the donor star, and f is thefilling factor expressing the magnetized fraction of the stellar surface (Saar 1996; Amard et al.2016). Here we set B s as the Solar surface magnetic field strength B s = B ⊙ = 1 G. Theexpression of f is re-calibrated to reach the Solar mass-loss value at the age of the Sun forsolid-body rotating models, f = 0 . x/ . . ] . , (14)with x being the normalised Rossby number x = ( Ω ⊙ Ω )( τ ⊙ , conv τ conv ) (Amard et al. 2016). Herethe rotation rate of the Sun Ω ⊙ ≃ × − s − , τ conv is the turnover time of the convectiveeddies of the donor star (Noyes et al. 1984), and τ ⊙ , conv ≃ . × s.3. The RM12 ModelReiners & Mohanty (2012, hereafter RM12) proposed a formalism in which MB is re-lated to mean surface magnetic field strength ( f B s ∝ Ω a ) instead of magnetic flux ( B s R ∝ Ω a ) as suggested by King (1988). In this model there is a critical surface angular velocityΩ sat = 3Ω ⊙ , above which the magnetic field reaches saturation (Stauffer & Hartmann 1987).This saturation modifies the braking law and the relation between the AML rate and therotation rate, and the magnetic field strength likely stops increasing even if the star stillspins up (Vilhu 1984; O’dell et al. 1995). The AML rate is given byd J d t = − C (cid:18) R M (cid:19) / Ω s for Ω ≥ Ω sat , (15)d J d t = − C (cid:18) R M (cid:19) / (cid:18) Ω s Ω sat (cid:19) Ω for Ω < Ω sat , (16)with C = 23 B G K V ˙ M , W ! / . (17)Here Reiners & Mohanty (2012) assumed that the saturation field strength B crit , the windmass-loss rate ˙ M , W and the velocity scaling factor K V are all constant, independent of 7 –stellar mass. By comparing with observations they got a best-fit choice with C = 2 . × (g cm − s ) / .4. The τ -boosted ModelVan et al. (2019) suggested that the surface magnetic field B s of the secondary is con-nected with the Rossby number R as (Noyes et al. 1984; Ivanova 2006), B s B s , ⊙ = R R , ⊙ = (cid:18) ΩΩ ⊙ (cid:19) (cid:18) τ conv τ ⊙ , conv (cid:19) . (18)In addition, they added winds in the Skumanich MB and proposed a modified MB law:˙ J MB , boost = ˙ J MB , Sk (cid:18) ΩΩ ⊙ (cid:19) β (cid:18) τ conv τ ⊙ , conv (cid:19) ξ ˙ M , W ˙ M ⊙ ! α . (19)This MB law can be divided into three forms depending on the power indices ( ξ , α , β ). Theyare classified into Convection-boosted ( τ -boosted, with ξ = 2, α = 0, β = 0), Intermediate(with ξ = 2, α = 1, β = 0) and Wind-boosted (with ξ = 4, α = 1, β = 2), respectively. The τ -boosted and Intermediate MB schemes were found to be more effective to reproduce theobserved NS LMXBs (Van et al. 2019). Here, we adopt the τ -boosted scheme.5. The VI19 ModelThe Intermediate prescription in Van et al. (2019) was found to have difficulty in ex-plaining the effective temperature of Sco X-1. Van & Ivanova (2019) then considered therotational effects on the Alfv´en radius and the magnetic field dependence on the convectiveturnover time, and presented a modified MB prescription called Convection And RotationBoosted (CARB),˙ J MB , CARB = −
23 ˙ M − / , W R / (cid:0) v + 2Ω R /K (cid:1) − / Ω ⊙ B / ⊙ (cid:18) ΩΩ ⊙ (cid:19) / (cid:18) τ conv τ ⊙ , conv (cid:19) / , (20)where v esc is the surface escape velocity, and K = 0 .
07 is a constant obtained from a gridof simulations by R´eville et al. (2015).Finally, by default, we assume that MB operates only when the star has a convectiveenvelope and a radiative core (Paxton et al. 2015).
3. RESULTS OF EVOLUTION CALCULATIONS
In our calculations, we choose the initial NS mass M = 1 . M ⊙ and the donor’s massranging from M = 1 . M ⊙ to M = 4 . M ⊙ . We adopt the initial binary orbital period in 8 –the range − . ≤ log( P orb , i / d) ≤ P orb , i / d) = 0 . We first demonstrate the evolutionary sequences for an LMXB that consists of a 1 . M ⊙ donor star with different MB laws in Fig. 1. The left, middle, and right panels correspondto the initial orbital period P orb , i = 1.0, 10.0, and 100.0 d, respectively. In the upper panelswe depict the evolution of the orbital period as a function of the donor mass, and in thelower panels we compare the the AML rates for different MB laws. MB stops working whenthe donor star becomes full convective, with mass ∼ . − . M ⊙ . Note that there is a jumpin the AML rate for the Matt12 MB at the final evolutionary stage with P orb ≃ P orb , i = 1 . P orb , i is set to be 10 d,binaries in the Skumanich MB model join the divergent sequences. In contrast, the orbitalperiod still decreases with the RM12 MB law. In the other two cases with the τ -boosted andVI19 MB, the final period does not significantly deviate from its initial value. When P orb , i is increased to 100 d, the binaries evolve to divergent systems in all cases. The lower panelsof Fig. 1 display the AML rate due to MB relative to the total AML rate. Generally, forthe same initial parameters, stronger MB leads to forming converging systems, while weakerMB law result in forming divergent binary pulsar systems.Table 1 presents the final parameters of the evolutionary products, including the NSmass, companion mass, orbital period and hydrogen abundance. For the LMXB with P orb , i =1 d, only the Matt12 MB leads to the formation of a binary pulsar with a 0 . M ⊙ WD(with the final orbital period P orb , f = 13 .
367 d), while in other cases the binaries evolve toconverging binaries ( P orb , f < . P orb , i = 10 . P orb , i = 100 d form divergent widebinary pulsars with a WD companion. 9 – We compare the properties of LMXBs from our evolution calculations with observations.In Fig. 2-6 we present the evolution of the orbital period as a function of the donor mass withdifferent MB laws. In each figure we consider initial donor mass to be 1.0, 1.1, 1.2, 1.4, 2.0,and 3.0 M ⊙ . We use different colors to represent the ranges of the MT rates. In IMXBs, MTproceeds on a (sub)thermal timescale because of the relatively high mass ratio ( q = M /M > . > − M ⊙ yr − . We adopt theobservational parameters from Tables 4 and 5 of Van et al. (2019) for the binned propertiesof selected persistent and transient NS LMXBs. Note that some systems are in globularclusters (GCs) and probably formed via dynamical encounters rather than from primordialbinaries (Van et al. 2019). We use circles, squares and triangles to represent persistent, GC,and transient LMXBs , respectively.Fig. 2 compares the calculated results with the Skumanich MB law with observations.It shows that most LMXBs can be covered in the P orb − M diagram, but the accretion ratesin persistent NS LMXBs are at least an order of magnitude higher than the theoreticallyexpected ones (see also Podisadlowski et al. 2002; Shao & Li 2015; Van et al. 2019). Mean-while, the mean accretion rates in part of the transient NS LMXB systems are lower thanthe calculations, and there are very few systems that can form UCXBs.Fig. 3 shows the results with the τ -boosted MB law. With this MB law most LMXBscan be reproduced. In addition, the calculated rates can match the observed accretion ratesfor almost all LMXBs. UCXBs can be produced more effectively compared with the caseusing the Skumanish MB.Fig. 4 shows the results with the Matt12 MB law. Its strength of AML is so weak that ithardly reproduces converging systems with P orb < P orb − M diagram, the calculated MT rates for most LMXBs do not match theobservations, but UCXBs can form effectively. For transient LMXBs we adopt long-term, averaged MT rates inferred from observations for comparison.
10 –Fig. 6 shows the results with the VI19 MB law. The evolutionary tracks can cover allLMXBs in the P orb − M diagram. The calculated MT rates can match the observations ofpersistent LMXBs, but are too high for some transient systems.To summarize the results in Figs. 2-6 in a more clear way, we compare the effectivenessof the five MB models in explaining the LMXBs in Table 2. The symbols △ , N , NN and NNN represent that the LMXB can be reproduced by 0, 1 −
2, 3 −
5, and more than 5 evolutionarytracks under a specific MB model, respectively. It seems that both the τ -boosted and theVI19 MB laws are more preferred. A large fraction of LMXBs will evolve into BMSPs by the “recycling” process (Alpar et al.1982). So a successful MB model should be able to reproduce the properties of not onlyLMXBs but also their descendants BMSPs. Shao & Li (2015) used the binary populationsynthesis (BPS) code developed by Hurley et al. (2002) to investigate the formation andevolution of Galactic intermediate- and low-mass X-ray binaries (I/LMXBs). Some key as-sumptions of the initial parameters in that study are briefly described as follows. A constantstar formation rate of 5 M ⊙ yr − in the Galaxy was assumed for the primordial binaries withSolar metallicity (Smith et al. 1978). The primary star’s mass is distributed according tothe initial mass function of Kroupa et al. (1993) in the range of 3 − M ⊙ , and the initialmass ratio of the secondary to the primary masses has a uniform distribution between 0 and1. If the mass transfer is dynamically unstable, a common envelope (CE) phase follows. Thestandard energy conservation equation (Webbink 1984) was used to treat the CE evolutionwith an energy efficiency α CE = 1 . λ of the primary’senvelope taken from the numerical results in Xu & Li (2010). NSs formed from core-collapseand electron-capture SNe were assumed to be imparted a kick with a Maxwellian distributionfor the kick velocity with σ = 265 km s − and 50 km s − , respectively (Hobbs et al. 2005;Dessart et al. 2006).Combining the birthrate and the distribution of the orbital period and the donor mass forthe incipient I/LMXBs in Shao & Li (2015) with our detailed binary evolution calculations,we then obtain the orbital period distributions for BMSPs, which are displayed in Fig. 7with red line. From top to bottom are the results by using the Skumanich, the τ -boosted,the Matt12, the RM12 and the VI19 MB laws, respectively.The NSs in LMXBs may be fully or partially recycled depending on the amount of 11 –accreted mass ∆ M NS . Accretion of 0 . M ⊙ material is sufficient to spin up an NS’s spinperiod to milliseconds (e.g., Tauris et al. 2012). The actual value of ∆ M NS may be smallerby a factor of 2 if considering the effect of the NS magnetic field-accretion disk interaction(Deng et al. 2020). So we use this criterion to distinguish fully and partially recycled pulsars.In the left and right panels of Fig. 7, we show the calculated distributions of binary pulsarswith ∆ M NS > M NS > . M ⊙ , respectively. The observed distributions of of binarypulsars with spin periods P s ≤ P s ≤
30 ms are plotted in blue line correspondingly(data are from the ATNF pulsar catalog , Manchester et al. 2005).Fig. 7 shows that adopting the Skumanich MB law cannot reproduce binary pulsarsystems with P orb ∼ . − τ -boosted and the VI19 MB laws bestmatch the observations. The AML efficiency with the Matt12 MB law is too weak to formbinary pulsars with P orb .
10 days. The RM12 MB scheme produces a double-peaked orbitalperiod distribution of binary pulsars which is likely related to the saturation of MB whenthe angular velocity of the donor becomes large enough. While the calculated orbital perioddistribution obviously deviates from observations, this model can effectively produce tightbinary pulsars with P orb < F n , orb and F m , orb , respectively. We measure the maximum distance D nm = max | F n , orb − F m , orb | by comparing the CDFs. The null hypothesis is rejected at theconfidence level α = 0 .
05 if D nm > D α, nm ≃ . p nmn + m , where n and m are the numbers ofcalculated and observed binary pulsar systems, respectively. Table 3 shows the calculatedvalues of D nm and D α, nm in the case of the five MB laws. When adopting the τ -boostedMB and the VI19 MB laws, D nm < D α, nm , indicating that the null hypothesis cannot berejected, and the two distributions are not significantly different. Using the other threeMB laws, D nm > D α, nm , allowing rejection of the null hypothesis and suggesting that thedifferences are significant.
12 –
In this subsection, we investigate the influence of the MB laws on the bifurcation period.Here we adopt the definition of the bifurcation period used by Podisadlowski et al. (2002),which is the orbital period ( P rlof ) when RLOF just begins, and we express this bifurcationperiod as P bifrlof . Fig. 8 shows the results of the bifurcation periods in the five models describedin Section 2.2 with solid lines. We also plot the minimum initial period P ZAMS for a lobe-filling ZAMS donor star with the dashed line.Several features of the bifurcation periods can be seen in Fig. 8. First, when excludingthe RM12 MB law, the magnitude of the bifurcation periods P bifrlof is about 0 . − . . M ⊙ to 3 . M ⊙ . Second, when adopting the Matt12 MBlaw, there does not exist any bifurcation period with the donor mass M ∼ . − . M ⊙ ,which means that the binary orbit always increases in these cases. Third, when adoptingthe RM12 MB law, the final orbital period does not significantly change with P rlof ∼ −
4. DISCUSSION AND CONCLUSIONS4.1. Forming Binary Pulsars with Orbital Periods . d < P orb < d Previous studies on the LMXB evolution suggested that it is hard to form BMSPs withorbital periods of 0 . − P orb ∼ − . M ⊙ donor star and the range of the initial orbitalperiod being described as in Section 3. There are totally 51 models in each case. Weregard the binary systems to be low-mass binary pulsars when the mass transfer rate ˙ M tr < − M ⊙ yr − , the donor mass M > . M ⊙ , and the hydrogen abundance < . < P orb <
24 h and 2 h < P orb < < P orb <
24 h. 13 –Three and eight models using the RM12 MB law form binary pulsars with 2 h < P orb < < P orb <
24 h, respectively, and the numbers of successive models are correspondinglyone and four when using the τ -boosted MB law, and two and eight when using the VI19 MBlaw. Note that there is one model adopting the VI19 MB law can form binary pulsar with2 h < P orb < τ -boosted MB laws are effective in producing BMSPswith orbital periods 0.1 d < P orb < MB plays a vital role in the LMXB evolution, however, its mechanism has not been wellunderstood. The traditional MB law faces difficulties in explaining the observational charac-teristics of LMXBs and MSPs, such as the discrepancy between the calculated MT rates andthe observed mass accretion rates of LMXBs, and the conflict between the calculated andobserved orbital period distribution of BMSPs (Podisadlowski et al. 2002; Pfahl et al. 2003;van der Sluys et al. 2005a; Istrate et al. 2014a; Shao & Li 2015; Van et al. 2019). There arevarious alternative proposals for the MB laws, and we have made a systematic study to ex-amine their influence on the LMXB evolution. We mainly compare the MT rates of LMXBsand the orbital period distribution of binary pulsars with observations. In addition, we ex-plore the formation efficiency of binary pulsars with P orb ∼ . − τ -boosted MB law may be more suitable for the LMXBs evolution compared withother models. The observed properties of most NS LMXBs could be reproduced, includingthe distributions of the orbital periods, donor masses and MT rates, as well as the orbitalperiod distribution of binary pulsars, especially for the binary pulsars with orbital periods P orb ∼ . − P orb < REFERENCES
Alpar, M. A., Cheng, A. F., Ruderman, M. A., Shaham, J., 1982, Nat, 300,728Amard, L., Palacios, A., Charbonnel, C., Gallet, F., & Bouvier, J. 2016, A&A, 587, A105Bhattacharya, D., & van den Heuvel, E. P. J. 1991, Phys. Rev., 203, 1Chen, W.-C., Li, X.-D., 2006, MNRAS, 373, 305Carroll B.W., Ostlie D. A., 1995, An Introduction to Modern Astrophysics. Pearson Educa-tion Limited, Boston, MADubus, G., Lasota, J.-P., Hameury, J.-M., & Charles, P. 1999, MNRAS, 303, 139Deng, Z.-L., Gao, Z.-F., Li, X.-D., & Shao, Y. 2020, ApJ, 892, 4Dessart, L., Burrows, A., Ott, C. D., et al. 2006, ApJ, 644, 1063Eggleton, P. P., 1983, ApJ, 268, 368Faulkner, J., 1971, ApJ, 170, L99Feigelson, E. D., & Jogesh Babu, G. 2012, Modern Statistical Methods for Astronomy (Cam-bridge: Cambridge Univ. Press)Gallet, F., & Bouvier, J. 2013, A&A, 556, A36Hobbs, G., Lorimer, D. R., Lyne, A. G., & Kramer, M. 2005, MNRAS, 360, 974Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS, 329, 897Hut P., 1981, A&A, 99, 126Iben, I. Jr., & Livio, M. 1993, PASP, 105, 1373Istrate, A. G., Tauris, T. M., Langer, N. 2014a, A&A, 571, A45Istrate, A. G., Tauris, T. M., Langer, N., & Antoniadis, J. 2014b, A&A, 571, L3Ivanova N., 2006, ApJ, 653, L137Jia, K., & Li, X.-D. 2014, ApJ, 791, 127Justham S., Rappaport S., Podsiadlowski P., 2006, MNRAS, 366, 1415Kalogera, V., & Webbink, R. F. 1996, ApJ, 458, 301 16 –King, A. R., & Ritter, H. 1999, MNRAS, 309, 253Kiel, P. D., & Hurley, J. R. 2006, MNRAS, 369, 1152King, A. R., 1988, QJRAS, 29, 1Kolb, U., Davies, M. B., King, A., & Ritter, H. 2000, MNRAS, 317, 438Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545Knigge, C., Baraffe, I., & Patterson, J. 2011, ApJS, 194, 28Landau, L. D., & Lifshitz, E. M., 1959, The Classical Theory of Fields (Oxford: Pergamon)Lasota, J.-P. 2001, New Astron. Rev., 45, 449LIGO Scientific Collaboration; VIRGO Collaboration, 2016, Phys. Rev. Lett., 116, 241103LIGO Scientific Collaboration; Virgo Collaboration, 2017, Phys. Rev. Lett., 119, 161101Li, X.-D. 2002, ApJ, 564, 930Li, X.-D. 2015, NewAR, 64, 1Lin, J., Rappaport, S., Podsiadlowski, P., Nelson, L., Paxton, B., Todorov, P., 2011, ApJ,732, 70Liu, Q. Z., van Paradijs, J., & van den Heuvel, E. P. J. 2007, A&A, 469, 807Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993Matt, S. P., & Pudritz, R. E. 2008, ApJ, 678, 1109Matt, S. P., MacGregor, K. B., Pinsonneault, M. H., & Greene, T. P. 2012, ApJ, 754, L26Ma, B., & Li, X.-D. 2009, ApJ, 691, 1611Nelemans, G., Siess, L., Repetto, S., Toonen, S., Phinney, E.S., 2016, ApJ, 817, 69Noyes R. W., Hartmann L. W., Baliunas S. L., Duncan D. K., Vaughan A. H., 1984, ApJ,279, 763O’dell, M. A., Panagi, P., Hendry, M. A., & Collier Cameron, A. 1995, A&A, 294, 715Paczy´nski, B. 1976, in IAU Symp. 73, Structure and Evolution in Close Binary Systems, ed.P. p. Eggleton, S. Mitton, & J. Whealan (Dordrecht: Reidel), 75 17 –Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJ, 192, 3Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10Pavlovskii K., Ivanova N., 2016, MNRAS, 456, 263Pfahl, E., Rappaport, S., & Podsiadlowski, P. 2003, ApJ, 597, 1036Podsiadlowski, Ph., & Rappaport, S. 2000, ApJ, 529, 946Podisadlowski, Ph., Rappaport, S., & Pfahl, E. D., 2002, ApJ, 565, 1107Pylyser, E., & Savonije, G. J. 1988, A&A, 191, 57Pylyser, E., & Savonije, G. J. 1989, A&A, 208, 52Rappaport, S., Verbunt, F., & Joss, P. C. 1983, ApJ, 275, 713Rappaport, S., Podsiadlowski, P., Joss, P. C., Di Stefano, R., & Han, Z., 1995, MNRAS,273, 731R´eville, V., Brun, A. S., Matt, S. P., Strugarek, A., & Pinto, R. F. 2015, ApJ, 798, 116, doi:10.1088/0004-637X/798/2/116Reimers D., 1975, Mem. Soc. R. Sci. Liege, 8, 369Reiners, A., & Mohanty, S. 2012, ApJ, 746, 43Ritter, H., 1988, A&A, 202, 93Saar, S. H. 1996, in Stellar Surface Structure, eds. K. G. Strassmeier, & J. L. Linsky, IAUSymp., 176, 237Sadeghi Aedestani, Leila., Guillot, Tristan., Morel, Pierre., 2017, MNRAS, 472, 2590Schreiber, M. R., Zorotovic, M., & Wijnen, T. P. G. 2016, MNRAS, 455, 16Shao, Y., & Li, X.-D. 2012, ApJ, 756, 85Shao, Y., & Li, X. -D., 2015, ApJ, 809, 99 18 –Skumanich A., 1972, ApJ, 171, 565Smith, L. F., Biermann, P., & Mezger, P. G. 1978, A&A, 66, 65Stauffer, J. R., & Hartmann, L. W. 1987, ApJ, 318, 337Tauris, T. M., & Savonije, G. J., 1999, A&A, 350, 928Tauris, T. M., van den Heuvel, E. P. J., & Savonije, G. J. 2000, ApJL, 530, L93Tauris, T. M., & van den Heuvel, E. P. J. 2006, in Compact Stellar X-Ray Sources, ed. W.H. G. Lewin & M. van der Klis (Cambridge: Cambridge Univ. Press), 623Tauris, T. M., Langer, N., & Kramer, M. 2011, MNRAS, 416, 2130Tauris, T. M., Langer, N., & Kramer, M. 2012, MNRAS, 425, 1601van der Sluys, M. V., Verbunt, F., & Pols, O. R. 2005a, A&A, 440, 973van der Heuvel, E. P. J., & van Paradijs, J. 1988, Natur, 334, 227van der Heuvel, E. P. J. 1994, in Saas-Fee Advanced Course 22: Interacting Binaries, eds. S.N. Shore, M. Livio, E. P. J. van den Heuvel, H. Nussbaumer. & A. Orr, 263Van Saders, J. L., & Pinsonneault, M. H. 2013, ApJ, 776, 67Van, K. X., Ivanova, N., & Heinke, C. O. 2019, MNRAS, 483, 5595, doi: 10.1093/mn-ras/sty3489Van, K. X., & Ivanova, N. 2019, ApJ, 886, 31Verbuant, F., & Zwaan, C., 1981, A&A, 100, L7Vilhu, O. 1984, A&A, 133, 117Webbink, R. F. 1984, ApJ, 277, 355Weisberg J. M., Huang Y., 2016, ApJ, 829, 55Xu, X.-J., & Li, X.-D. 2010, ApJ, 716, 114
This preprint was prepared with the AAS L A TEX macros v5.2.
19 – (M ⊙ ⊙10 −1 P o r b ( d a y ⊙ Initial P orb =1.0 d
Skumanichτ-boostedVI19RM12Matt12 0.00.20.40.60.81.01.2 M (M ⊙ ⊙0.00.20.40.60.81.0 ̇ J m b / ̇ J t o t (M ⊙ ⊙10 P o r b ( d a y ⊙ Initial P orb =10.0 d (M ⊙ ⊙0.00.20.40.60.81.0 ̇ J m b / ̇ J t o t (M ⊙ ⊙10 P o r b ( d a y ⊙ Initial P orb =100.0 d (M ⊙ ⊙0.00.20.40.60.81.0 ̇ J m b / ̇ J t o t Fig. 1.— Evolution of a LMXB by using different MB formulae. The red, blue, green, cyanand yellow curves are for the Skumanich, τ -boosted, VI19, RM12, and Matt12 MB laws,respectively. In the left, middle, and right panels, the initial orbital period is taken to be1, 10, and 100 d, respectively. The top and bottom panels show the evolution of P orb andthe rate of AML due to MB divided by the total AML rate as a function of the donor mass,respectively. l o g P o r b ( h o u r s ) M =1.0M ⊙ =1.1M ⊙ =1.2M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 l o g P o r b ( h o u r s ) M =1.4M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =2.0M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =3.0M ⊙ −11 −10 −9 −8 −7 −6 −5 −4log ̇Ṁ(M ⊙ yr −1 ⊙ Fig. 2.— I/LMXB evolution under the Skumanich MB law. Different colors denote the magnitude of the MT rates.The symbols with errorbars represent observed systems, with circles, squares and triangles corresponding to persistent,GC, and transient LMXBs, respectively. l o g P o r b ( h o u r s ) M =1.0M ⊙ =1.1M ⊙ =1.2M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 l o g P o r b ( h o u r s ) M =1.4M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =2.0M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =3.0M ⊙ −11 −10 −9 −8 −7 −6 −5 −4log ̇Ṁ(M ⊙ yr −1 ⊙ Fig. 3.— Same as Fig. 2 but under the τ -boosted MB law. l o g P o r b ( h o u r s ) M =1.0M ⊙ =1.1M ⊙ =1.2M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 l o g P o r b ( h o u r s ) M =1.4M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =2.0M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =3.0M ⊙ −11 −10 −9 −8 −7 −6 −5 −4log ̇Ṁ(M ⊙ yr −1 ⊙ Fig. 4.— Same as Fig. 2 but under the Matt12 MB law. l o g P o r b ( h o u r s ) M =1.0M ⊙ =1.1M ⊙ =1.2M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 l o g P o r b ( h o u r s ) M =1.4M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =2.0M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =3.0M ⊙ −11 −10 −9 −8 −7 −6 −5 −4log ̇Ṁ(M ⊙ yr −1 ⊙ Fig. 5.— Same as Fig. 2 but under the RM12 MB law. l o g P o r b ( h o u r s ) M =1.0M ⊙ =1.1M ⊙ =1.2M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 l o g P o r b ( h o u r s ) M =1.4M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =2.0M ⊙ (M ⊙ )−1.0−0.50.00.51.01.52.02.53.0 M =3.0M ⊙ −11 −10 −9 −8 −7 −6 −5 −4log ̇Ṁ(M ⊙ yr −1 ⊙ Fig. 6.— Same as Fig. 2 but under the VI19 MB law. 25 – −1 0 1 2 3logP orb (days)0.000.010.020.030.040.050.060.07 R a e Skumanich
Calcula ionObserva ion −1 0 1 2 3logP orb (days)0.000.020.040.060.08 R a t e Sk manich
Calc lationObservation−1 0 1 2 3logP orb (da s)0.000.010.020.030.040.050.060.070.08 R a t e Convection boosted
CalculationObservation −1 0 1 2 3logP orb (days)0.000.020.040.060.08 R a t e Convection boosted
CalculationObservation−1 0 1 2 3logP orb (days)0.000.020.040.060.080.10 R a t e Matt12
CalculationObservation −1 0 1 2 3logP orb (days)0.000.020.040.060.080.10 R a t e Matt12
CalculationObservation−1 0 1 2 3logP orb (days)0.000.020.040.060.080.100.12 R a t e RM12
CalculationObservation −1 0 1 2 3logP orb (days)0.000.020.040.060.080.10 R a t e RM12
CalculationObservation−1 0 1 2 3logP orb (days)0.000.010.020.030.040.050.060.07 R a t e VI19
Calc lationObservation −1 0 1 2 3logP orb (days)0.000.020.040.060.08 R a t e VI19
CalculationObser ation
Fig. 7.— Comparison of the calculated orbital period distributions of binary pulsars withobservations, which are shown with the red and blue curves, respectively. The left panelsshow the calculated distributions of binary pulsars with accreted mass ∆ M NS > P s ≤ M NS > . M ⊙ and theobserved distribution of binary pulsars with the spin periods P s ≤
30 ms. 26 – (M ⊙ ⊙10 −1 B i f u r c a t i o n e r i o d ( d a y s ⊙ Bifurcation eriods (RLOF⊙ in LMXBs
Skumanichτ−boostedRM12VI19Matt12
Fig. 8.— Bifurcation periods as a function of the secondary mass in an LMXB under the fivekinds of MB laws. The dotted line shows the minimum initial period P ZAMS that correspondsto a Roche-lobe filling ZAMS secondary star. M d o n o r ( M ⊙ ⊙ l o g ( P e r i o d / h o r s ⊙ Sk manich 0.050.100.150.200.400.600.80 M d o n o r ( M ⊙ ⊙ l o g ( P e r i o d / h o u r s ⊙ Con ection boosted 0.050.100.150.200.400.600.80 M d o n o r ( M ⊙ ⊙ l o g ( P e r i o d / h o u r s ⊙ Matt120.050.100.150.200.400.600.80 M d o n o r ( M ⊙ ⊙ l o g ( P e r i o d / h o u r s ⊙ M d o n o r ( M ⊙ ⊙ l g ( P e r i d / h u r s ⊙ ORLF again
VI19
Fig. 9.— Orbital period evolution for LMXBs with a donor of mass of 1 . M ⊙ in the five MB models. The dashed anddotted curves denote forming BMSPs with orbital periods P orb = 9 −
24 h and P orb = 2 − M , i = 1 . M ⊙ and M , i = 1 . M ⊙ by using different MB laws. P orb , i =1.0 d P orb , i =10.0 d P orb , i =100.0 dMB M , f M , f P orb , f H f M , f M , f P orb , f H f M , f M , f P orb , f H f model ( M ⊙ ) ( M ⊙ ) (days) (%) ( M ⊙ ) ( M ⊙ ) (days) (%) ( M ⊙ ) ( M ⊙ ) (days) (%)Skumanich 2.261 0.013 0.096 68.2 1.879 0.302 76.264 0.3 1.403 0.380 508.447 0.06 τ -boosted 2.185 0.088 0.051 62.0 1.578 0.201 2.758 0.5 1.467 0.349 267.892 0.2Matt12 2.034 0.250 13.367 0.2 1.892 0.305 83.024 0.4 1.402 0.380 509.214 0.1RM12 2.230 0.051 0.046 65.2 1.607 0.195 0.8 0.004 1.424 0.373 445.8 0.001VI19 1.979 0.104 0.294 46.7 1.492 0.187 1.327 0.5 1.516 0.338 203.295 0.3Note: M , f , M , f , P orb , f and H f are the masses of the NS and the companion star, the orbital period andthe H abundance of companion star at the end of the mass transfer, respectively.
29 –Table 2: Comparison of the effectiveness of different MB models in reproducing the propertiesof persistent (upper) and transient (lower) LMXBs
Source Skumanich τ -boosted Matt12 RM12 VI194U 0513-40 N NN △ NNN NN
2S 0918-549
N NN △ NNN NN
4U 1543-624
N NN △ NNN NN
4U 1850-087
N NN △ NNN NN
M15 X-2
N NN △ NNN NN
4U 1626-67
NN NNN △ N NN
4U 1916-053
NN N △ △ NN
4U 1636-536
NNN NNN △ △
NNN
GX 9+9 △ NNN △ △
NNN
4U 1735-444 △ NNN △ △
NNN
2A 1822-371 △ △ △ △
NNN
Sco X-1 △ NNN △ △
NNN
GX 349+2
N NNN N N NN
Cyg X-2
NNN NNN NNN △ NNN
HETE J1900.1-2455
NNN NNN △ NNN △
1A 1744-361
NNN NN NN NNN N
SAX J1808-3658 △ △ △ △ △
IGR 00291+5394
NNN NN △ NNN N
EXO 0748-678 △ NN △ △ △
4U 1254-69
NNN NNN △ △
NNN
XTE J1814-338
NNN NN N NNN N
XTE J2123-058 △ △
NNN NNN △ X 1658-298
NNN NNN △ △
NNN
SAX J1748.9-2021
NNN NN NNN NNN N
IGR J18245-2452 △ △ △
NNN NN
Cen X-4 △ N N NNN N
Her X-1
NN NN NNN NN N
GRO J1744-28
NNN NNN NNN △ NNN
30 –Table 3: Kolmogorov-Smirnov test for the measured and calculated orbital period distribu-tions of binary pulsars. P s maximum distance Skumanich τ -boosted Matt12 RM12 VI19 ≤ D nm D α, nm ≤ D nm D α, nmnm