Anomalous orbital expansion of low-mass X-ray binary 2A 1822-371: the existence of a circumbinary disk?

The source 2A 1822-371 is an eclipsing low-mass X-ray binary (LMXB) consisting of a neutron star (NS) and a $\sim0.5~M_{\odot}$ donor star in an orbit of 5.57 hr. Based on timing of the eclipse arrival times, this source was found to be experiencing a rapid orbital expansion with an orbital-period derivative as $\dot{P}_{\rm orb}=(1.51\pm0.05)\times10^{-10}~\rm s\, s^{-1}$, implying that the mass-transfer rate should be higher than at least three times the Eddington accretion rate. The standard magnetic braking (MB) model cannot produce such a high mass-transfer rate. The modified MB model derived by Van \&Ivanova (2019) can produce a high mass-transfer rate, resulting in a high $\dot{P}_{\rm orb}$. This work proposes an alternative model to account for the anomalously high mass-transfer rate and $\dot{P}_{\rm orb}$ of 2A 1822-371. During the mass transfer, a tiny fraction of the transferred material is thought to form a circumbinary (CB) disk around the LMXB, which can efficiently extract orbital angular momentum from the system by the interaction between the CB disk and the binary. We use the MESA code to model the formation and evolution of 2A 1822-371 for different CB-disk masses. When the CB-disk mass is $2.3\times10^{-8}~ M_{\odot}$, the simulation can reproduce the observed donor-star mass, orbital period, and orbital-period derivative. Such a CB disk can accelerate the evolution of the binary and produce a high mass transfer rate of $1.9\times10^{-7}~ M_\odot\,\rm yr^{-1}$, driving the binary to evolve toward a wide-orbit system. Therefore, we propose that CB disks may be responsible for the rapid orbital changes observed in some LMXBs.


Introduction
According to the masses of the companion stars, strong Galactic X-ray sources are mainly classified into two groups: high-mass X-ray binaries and low-mass X-ray binaries (LMXBs) (Tauris & van den Heuvel 2006).In LMXBs, a neutron star (NS) or black hole accretes materials from its donor star with a mass less than 1 − 2 M ⊙ by Roche lobe overflow (RLOF) and emits strong Xray.Due to the long accretion phase in LMXBs, the accreting NSs are spun up to the millisecond period, and evolve into millisecond pulsars (Alpar et al. 1982;Radhakrishnan & Srinivasan 1982).Therefore, studying LMXBs is significant in understanding the formation and evolution of millisecond pulsars.Furthermore, LMXBs provide much information about mass exchange and angular momentum loss, thus they are ideal probes testing stellar and binary evolution theory.
The mass function of 2A 1822-371 was measured to be f (M) = (2.03 ± 0.03) × 10 −2 M ⊙ (Jonker & van der Klis 2001), resulting in a minimum companion-star mass of M d = 0.33 ± 0.05 M ⊙ (Jonker et al. 2003).By studying the K-correction for the emission lines formed in the X-ray-illuminated atmo-sphere of the donor star, the masses of the NS and the donor star were constrained to be 1.61 M ⊙ ≤ M NS ≤ 2.32 M ⊙ and 0.44 M ⊙ ≤ M d ≤ 0.56 M ⊙ , respectively (Muñoz-Darias et al. 2005).
According to a distance of 2.5 kpc, the unabsorbed X-ray luminosity of 2A 1822-371 was calculated to be L X ≃ 1.0 × 10 36 erg s −1 , yielding a mean ratio of the X-ray over optical luminosity as L X /L opt ∼ 20 (Mason & Cordova 1982).Such a ratio is much smaller than the typical luminosity ratio (L X /L opt ∼ 1000) of LMXBs, implying that the intrinsic luminosity of 2A 1822-371 may exceed the Eddington limit (Bayless et al. 2010;Burderi et al. 2010).Magnetic braking (MB) is the main mechanism driving the mass transfer of LMXBs with long orbital periods.However, the standard MB model given by Rappaport et al. (1983) is challenging to produce a super-Eddington mass transfer in this LMXB with a low-mass donor star.
The orbital evolution of the source 2A 1822-371 also remains mysterious.Based on the analysis of eclipses detected by the HEAO-1, Einstein, Exosat, and Ginga, its orbital period was thought to be increasing at a rate of Ṗorb = (2.19 ± 0.58) × 10 −10 s s −1 (Hellier et al. 1990).According to an improved ephemeris for the optical eclipses, Bayless et al. (2010) confirmed that its orbital period is rapidly changing at a rate of Ṗorb = 2.12 × 10 −10 s s −1 .Subsequently, the orbital period derivative of 2A 1822-371 was independently measured to be Ṗorb = (1.50 ± 0.07) × 10 −10 s s −1 (Burderi et al. 2010), Ṗorb = (1.59 ± 0.09) × 10 −10 s s −1 (Iaria et al. 2011), Ṗorb = (1.464± 0.041) × 10 −10 s s −1 (Chou et al. 2016), and Ṗorb = (1.475± 0.054) × 10 −10 s s −1 (Mazzola et al. 2019).Recently, the orbital period derivative was refined to be Ṗorb = (1.51±0.05)×10−10 s s −1 based on the updated orbital ephemeris joining two new eclipse times related to NuSTAR and Swift observations (Anitra et al. 2021).Such an orbital-period derivative is three orders of magnitude higher than that derived from conservative mass transfer driven by MB and gravitational radiation, implying that the mass-transfer rate should be higher than at least three times the Eddington accretion rate of a NS (Burderi et al. 2010;Bayless et al. 2010).As mentioned above, the standard MB model cannot produce such a high mass-transfer rate for a low-mass donor star.Therefore, the formation and evolution of the source 2A 1822-371 challenge the conventional binary evolution theory, which may require a new MB description or new physical mechanisms.

Analysis for the Orbital Evolution
The total orbital angular momentum of a LMXB consisting of a NS and a low-mass donor star can be written as where M NS and M d are the NS mass and the donor-star mass, respectively; Ω = 2π/P orb is the orbital angular velocity of the binary, and a is the orbital separation.Inserting Kepler's third law (G(M NS + M d )/a3 = Ω2 ) into equation ( 1) and differentiating it, the orbital period derivative satisfies Ṗorb where ẋ = dx/dt, q = M d /M NS is the mass ratio of the binary, and β = − ṀNS / Ṁd is the accretion efficiency of the NS.

conservative mass transfer
In the case of conservative mass transfer, β = 1, equation ( 2) changes into Ṗorb The first term on the right-hand side of equation (3) would produce a negative Ṗorb because J < 0, while the mass transfer from the less massive donor star to the more massive NS would cause a positive Ṗorb since Ṁd < 0 and q < 1 (see also the second term on the right-hand side).The angular-momentum-loss rate predicted by the standard MB model is Jmb where R d is the donor-star radius, γ is a dimensionless magneticbraking index from 0 to 4 (Rappaport et al. 1983).In this work, we adopt the simplest approximation as γ = 4 (Verbunt & Zwaan 1981).Since the donor star filled its Roche lobe, its radius can be estimated to be (Eggleton 1983) For a binary with an orbital period exceeding 3 hr, the angular-momentum-loss rate by gravitational radiation is weaker than that by MB.Hence the contribution of gravitational radiation can be ignored.Taking P orb = 5.57 hr, Ṗorb = 1.51 × 10 −10 s s −1 , M NS = 1.6 M ⊙ , and M d = 0.5 M ⊙ , we have R d = 0.57 R ⊙ , a = 2.0 R ⊙ , Jmb /J = −6.3× 10 −17 s −1 , and Ṗorb /P = 7.5 × 10 −15 s −1 .Because | Jmb /J| is two orders of magnitude smaller than Ṗorb /P, the contribution of angular momentum loss on Ṗorb of 2A 1822-371 is trivial.Therefore, according to equation (3), the mass transfer rate can be derived to be (1.6, 0.5), (1.8, 0.5), and (1.6, 0.6) M ⊙ , respectively.

non-conservative mass transfer
The calculated mass-transfer rate of 2A 1822-371 is higher than the Eddington accretion rate ( ṀEdd = 1.5 × 10 −8 M ⊙ yr −1 ), hence the mass transfer of 2A 1822-371 is probably nonconservative, i.e. the accretion efficiency β 1.To study the influence of the accretion efficiency, we plot the relation between the mass transfer rate and the accretion efficiency β in Figure 1 according to equation (2) (the contribution of angular momentum loss for Ṗorb is still ignored 1 ).It is clear that a low accretion efficiency tends to require a low mass-transfer rate to produce the observed orbital-period derivative.For M NS = 1.6 M ⊙ and M d = 0.5 M ⊙ , β = 1 predicts a mass-transfer rate of ≈ 5.8 × 10 −8 M ⊙ yr −1 , which is consistent with the mass-transfer rate predicted by the conservative mass transfer case.It seems that the mass-transfer rate is weakly dependent on the accretion efficiency.Even if the NS does not accrete any material (β = 0), it still requires a masstransfer rate of ∼ (4 − 5) × 10 −8 M ⊙ yr −1 to produce the observed Ṗorb of 2A 1822-371.Furthermore, the NS mass and the donorstar mass can also alter the derived mass-transfer rate.However, the effect of the latter is more significant than that of the former.Therefore, such a rapid orbital expansion implies a rapid mass transfer between the donor star and the accreting NS of 2A 1822-371.

some possible models
When q = 0.5/1.6 ≈ 0.31, the standard MB model predicts an approximate mass-transfer rate as (Pavlovskii & Ivanova 2016 For some typical parameters of 2A 1822-371, the estimated mass-transfer rate is 2.0×10 −9 M ⊙ yr −1 , which is about one order of magnitude smaller than that derived from the observed orbital period derivative.Similarly, the inferred mass-transfer rates of some LMXBs with short orbital periods are found to be at least an order of magnitude higher than theoretically expected values (Podsiadlowski et al. 2002).To solve this discrepancy problem between the observation and the theory, Van & Ivanova (2019) modified MB prescription and proposed a convection and rotation boosted (CARB) MB model (see also section 2.4).Chou et al. (2016) argued that the unusual Ṗorb may be caused by the X-ray radiation-driven mass loss proposed by Tavani & London (1993).Assuming that a fraction ( f , i.e., the efficiency of irradiation-driving wind) of the X-ray luminosity that the donor star receives conquers the gravitational bind energy of the material on the surface of the donor star and drives a strong wind (with a velocity equaling to the escape speed of the donor's surface), the loss rate of the irradiation-driving winds is (Chen & Podsiadlowski 2016) where f 0.001 = f /0.001, L X,38 = L X /10 38 erg s −1 .Tavani & London (1993) obtained the efficiencies of irradiation-driving wind in the range from 0.001 to 0.1.Therefore, it is possible to produce a relatively high wind-loss rate from the donor star because the intrinsic X-ray luminosity of 2A 1822-371 may exceed 10 38 erg s −1 .The irradiation-driving winds may also play a key role in resulting in a positive orbital period of the first-discovered accreting millisecond pulsar SAX J1808.4-3658(Chen 2017).
Considering angular momentum loss due to MB by coupling between the strong magnetic field and an irradiation-driving wind (Justham et al. 2006), Xing & Li (2019) used the MESA code to model the evolution of a LMXB consisting of a 1.4 M ⊙ NS and a 1.1 M ⊙ donor star in an initial orbit of 0.4 days.Taking a wind-driving efficiency of f = 5 × 10 −3 and a magnetic field of 900 G, their simulated total mass loss rate of the donor star is ∼ 10 −7 M ⊙ yr −1 when the donor-star mass is ∼ 0.5 M ⊙ .Such an irradiation-driving wind model can account for the donor mass, orbital period, orbital-period derivative, and high X-ray luminosity of 2A 1822-371.

the CARB MB model
The CARB MB model considered the influence of the donor-star rotation on the stellar-wind velocity (Matt et al. 2012;Réville et al. 2015) and the influences of the donor-star convectiveturnover timescale and the donor-star rotation on its surface magnetic field (Parker 1971;Noyes et al. 1984;Ivanova 2006; Van et al. 2019).Included these mechanisms, the angularmomentum-loss rate can be written as (Van & Ivanova 2019) where Ṁw , and v esc are the wind mass-loss rate, and the surface escape velocity of the donor star, respectively; K 2 = 0.07 is a constant originated from a grid of simulations (Réville et al. 2015); τ conv is the turnover time of convective eddies; B ⊙ = 1 G is the surface magnetic-field strength of the Sun, and the surfacerotation rate and the convective-turnover time of the Sun are Ω ⊙ ≈ 3.0 × 10 −6 s −1 , and τ ⊙,conv = 2.8 × 10 6 s (see also Van et al. 2019), respectively.Some information regarding the inlists, and subroutines used to simulate the mass-transfer process of LMXBs in the CARB MB model see also Mangat et al. (2022).
The CARB MB model can reproduce the observed masstransfer rates and orbital periods at the detected mass ratio for all observed persistent NS LMXBs in the Galaxy (Van & Ivanova 2019).Using the CARB MB model, Van & Ivanova (2021) investigated the potential progenitors of the observed persistent NS LMXBs, and found that these progenitors occupy a small part of the plausible parameter space in the initial donor-star mass versus initial orbital period diagram.Deng et al. (2021) also found that the CARB MB model can successfully reproduce the observed characteristics of all persistent NS LMXBs and binary pulsars.Since the CARB MB model can result in a high masstransfer rate (∼ 10 −7 M ⊙ yr −1 ) for persistent NS LMXBs (Van & Ivanova 2019, 2021), which is higher than the required masstransfer rate (see also section 2.2), it is possible to reproduce the observed parameters of 2A 1822-371.
In this paper, we attempt to explore an alternative mechanism that can result in a rapid mass transfer and produce the rapid orbital expansion observed in the source 2A 1822-371.Based on a circumbinary (CB) disk model, we present a detailed stellar evolution model for the formation of 2A 1822-371 in Section 3. In section 4, we discuss the possible influence of input parameters and the observed confirmation.Finally, we give a summary in section 5.

CB disk model
During the mass transfer of a LMXB, a tiny fraction of the transferred material may be ejected by the strong radiation pressure of the accreting NS.Since the ejecta possesses a large orbital angular momentum, they may form a CB disk surrounding the system rather than completely leave it (van den Heuvel & de Loore 1973; van den Heuvel 1994).The resonant torque produced by the interaction between the CB disk and the binary can extract orbital angular momentum from orbital motion.Based on an assumption of a standard thin disk, the angular-momentum-loss rate via a CB disk can be expressed as (Chen & Podsiadlowski 2019) where M cb , α, H, and R are the mass, viscosity parameter, thickness, and half angular-momentum radius of the CB disk, respectively.For simplicity, a CB disk with a constant mass is assumed to surround the LMXB if a mass transfer occurs.Assuming that the ratio between half angular-momentum radius and the orbital separation is a constant (R/a = 2.3) in three black-hole LMXBs, Chen & Podsiadlowski (2019) found that the derived CB-disk masses around XTE J1118 and A0620-00 are consistent with the inferred values (M cb ∼ 10 −9 M ⊙ , Muno & Mauerhan 2006) when α = 0.1, H/R = 0.1.Same to black-hole LMXBs, we also take α = 0.1, H/R = 0.1, and R/a = 2.3 for 2A 1822-371.CBdisk parameters, including α, H/R, and R/a, might be different for NS LMXBs and black hole LMXBs.However, both M cb and α(H/R)2 R/a are degenerate in equation ( 9).Therefore, some uncertainties resulting from α, H/R, and R/a can be compensated by slightly altering the CB-disk mass.
mation and evolution of the source 2A 1822-371.The progenitor of the source is assumed to be a binary system consisting of a NS and a low-mass main-sequence (MS) star in a circular orbit.The code only models the nuclear synthesis and evolution of the MS companion star, and the NS is considered a point mass.The initial chemical composition of the MS companion star is taken to be a solar composition, i.e.X = 0.70, Y = 0.28, and Z = 0.02.
During the mass transfer, the mass-growth rate of the NS is limited by Eddington accretion rate ( Ṁedd = 1.5×10 −8 M ⊙ yr −1 ), i.e. the mass-growth rate of the NS Ṁns = min( Ṁedd , Ṁtr ), where Ṁtr is the mass-transfer rate.During the accretion, the excess material in unit time ( Ṁtr − Ṁns ) is thought to be re-emitted as an isotropic fast wind, carrying away the specific orbital angular momentum of the NS (Tauris & van den Heuvel 2023).
Using the MESA code, we model the evolution of a LMXB consisting of a NS with a mass of 1.6 M ⊙ , and a MS donor star with a mass of 1.0 M ⊙ for two independent models: the CARB MB and the CB disk models.In the CARB MB model, angular momentum loss via gravitational radiation and MB (see also section 2.4) is included.For the CB-disk model, a tiny fraction of the transferred material is assumed to form a CB disk with a constant mass once the mass transfer occurs.Thus the torque originating from the resonant interaction between the CB disk and the binary can extract orbital angular momentum from the orbital motion at a rate of Jcb (see also equation 9) 2 .Furthermore, the gravitation radiation and the standard MB mechanism proposed by Rappaport et al. (1983) with γ = 4 are also considered.The MB mechanism (the CARB MB or the standard MB) would work if the donor star possesses both a convective envelope and a radiative core.
In the calculation, we alter the initial orbital period and/or the CB-disk mass to obtain a model that can match the observed properties of 2A 1822-371, as listed in Table 1.Adopting the CARB MB description, a NS binary with a donor star of 1.75 M ⊙ and an initial orbital period of 0.5 days can evolve into 2A 1822-371 (Van & Ivanova 2021).Therefore, the initial orbital period ranges from 0.31 (the donor star has already filled its Roche lobe at the evolutionary beginning if the initial orbital period is less than 0.31 days) to 2.0 days in our models.Similar to the inferred CB-disk mass of three black-hole LMXBs (Chen & Podsiadlowski 2019), the CB-disk mass is assumed to be in the range of ∼ 10 −9 − 10 −7 M ⊙ .

Simulated results
Our simulations find that the evolution of a binary consisting of a 1.6 M ⊙ NS and a 1.0 M ⊙ donor star in an initial orbit of 0.35 days can match the observed properties of 2A 1822-371.The evolu- tion of the orbital period and donor-star mass with the stellar age is shown in the upper and bottom panels of Figure 2, respectively.Before the donor star fills its Roche lobe, MB drives the orbit to shrink, and the orbital period decreases to be ∼ 0.3 days.
The evolutionary tracks with four different CB-disk masses are the same in this stage.When the stellar age is t = 1.28 × 10 7 yr, the donor star fills its Roche lobe and initiates a mass transfer.Because of a tiny outflow from the mass transfer, a CB disk surrounding the binary forms.Different CB-disk masses result in different angular-momentum-loss rates, driving the binary to evolve along different evolutionary tracks.A heavy CB disk results in a rapid angular-momentum loss and propels the system to evolve into a minimum orbital period in a short timescale.Our simulations indicate that a CB disk with M cb = 2.3×10 −8 M ⊙ can successfully account for the observed orbital period and donorstar mass of 2A 1822-371 when t = 1.59 × 10 7 yr.It seems that the models with M cb = 1.0 × 10 −7 , and 1.0 × 10 −9 M ⊙ also reproduce the observed orbital period in the orbital-expansion stage, however, the corresponding donor-star mass for M cb = 1.0×10 −9 are much smaller than the observation (see also Figure 3).It is worth emphasizing that the orbital period with M cb = 0 (i.e., the mass transfer is only driven by the standard MB mechanism) continuously decreases without experiencing an orbitalexpansion stage in the observed donor-star mass of 2A 1822-371.At the stellar age of t = 1.1 × 10 7 yr, the CARB MB model can also successfully produce the observed orbital period and donor-star mass of 2A 1822-371.At the current orbital period, the calculated NS masses by the CB-disk and CARB MB models are 1.65 M ⊙ and 1.71 M ⊙ , respectively.Figure 3 plots the evolution of LMXBs in the orbital period versus the donor-star mass diagram.Three evolutionary tracks when M cb = 2.3 × 10 −8 M ⊙ , M cb = 1.0 × 10 −7 M ⊙ , and the CARB MB are in agreement with the observed data of 2A 1822-371, while the orbit of the binary continuously lessens for M cb = 1.0 × 10 −9 M ⊙ , and M cb = 0, without experiencing an orbitalexpansion stage in observed donor-star-mass range.
Figure 4 illustrates the evolution of the orbital-period derivative of LMXBs with three different CB-disk masses and the CARB MB model in the Ṗorb − P orb diagram.To compare with the observation, we only plot the evolution of Ṗorb in the orbitalexpansion stage (the evolutionary track with M cb = 0 is not included because the orbital-period derivative is negative at the current orbital period).Four orbital periods steadily climb after the minimum orbital periods, resulting in positive orbitalperiod derivatives.Two orbital-period derivatives predicted by the CB-disk model with M cb = 2.3 × 10 −8 M ⊙ and the CARB MB model are approximately consistent with the observed value at the current orbital period, while the simulated Ṗorb are greater and smaller than the observation for M cb = 1.0 × 10 −7 M ⊙ and 1.0 × 10 −9 M ⊙ , respectively.Meanwhile, these orbital-period derivatives also emerge an increasing tendency.The factors leading to an increasing Ṗorb are very complicated, while the main two factors are the increasing orbital period and the decreasing donor-star mass.Ignoring the contribution of angularmomentum loss, equation (2) yielding Ṗorb ∝ P orb /M d .Therefore, the increasing orbital period and the decreasing donor-star mass should be responsible for the increasing Ṗorb .
The evolution of the mass-transfer rates of LMXBs is presented in Figure 5. Once the mass transfer starts, the five evolutionary tracks are different because of different angularmomentum-loss rates.It is clear that three models with M cb = at the current donor-star mass, respectively.These two masstransfer rates are approximately one order of magnitude higher than the analytical estimation ((4.3 − 5.8) × 10 −8 M ⊙ yr −1 ) in Section 2. This difference is because the torque exerted by the CB disk or the CARB MB causes a significant negative orbital-period derivative.In other words, an efficient angularmomentum loss is worthy of a rapier in influencing the orbital evolution of LMXBs as follows: first, it produces a negative Ṗorb,aml due to a rapid angular-momentum loss; second, it also results in a positive Ṗorb,mt because of a rapid mass transfer from the less massive donor star to the more massive NS (see also equation 2).To compensate for the influence of the negative Ṗorb,aml , it requires a more high mass-transfer rate to produce the observed Ṗorb .The simulated mass-transfer rates when M cb = 0, and 1.0 × 10 −9 M ⊙ are always smaller than the Eddington accretion rate, resulting in a negative orbital-period derivative in the observed donor-star-mass range.The main reason is that Ṗorb,mt is smaller than | Ṗorb,aml |, thus the total period derivative ( Ṗorb = Ṗorb,aml + Ṗorb,mt ) is still negative.As M d ∼ 0.3 M ⊙ , the mass-transfer rates produced by the CARB and the standard MB models sharply decrease, which arises from the cut-off of MB when the donor stars become fully convective.

Discussion
Our standard model proposes that the progenitor of 2A 1822-371 consists of a 1.6 M ⊙ NS and a 1.0 M ⊙ MS companion star in an orbit of 0.35 days, which is surrounded by a CB disk with M cb = 2.3 × 10 −8 M ⊙ during the mass transfer.In this section, we investigate the influence of initial parameters, including initial donor-star mass (M d,i ) and initial orbital period (P orb,i ). Figure 6 depicts the evolution of NS-MS star binaries with different initial companion-star masses and initial orbital periods in the P orb -M d diagram.All five models can evolve to the current orbital period in the orbital-expansion stage.It is insensitive to the initial orbital periods whether the binaries can evolve toward the source 2A 1822-371 in the orbital-expansion stage.The models with (M d,i , P orb,i ) = (1.0M ⊙ , 0.6 days), and (1.0 M ⊙ , 1.0 day) can also reproduce the observed orbital period and donor-star mass.A system with a relatively high donorstar mass of 1.1 M ⊙ and the same orbital period of 0.35 days is also the potential progenitor of 2A 1822-371.Three models with (M d,i , P orb,i ) = (1.0M ⊙ , 0.35 days), (1.0 M ⊙ , 0.6 days), and (1.0 M ⊙ , 1.0 day) have similar evolutionary tendency and slope, resulting in a similar Ṗorb (see also Figure 8).
To account for the current NS mass of 2A 1822-371, both the CB-disk model and the CARB MB model found that the NS was born massive (∼ 1.6 M ⊙ ).This is very similar to PSR J1614-2230 and PSR J1640+2224, which were proposed to be born with masses of ∼ 1.95 M ⊙ (Tauris et al. 2011) and > 2.0 M ⊙ (Deng et al. 2020), respectively.Because of a relatively low mass-transfer rate in the early stage (see also Figure 5), the accretion efficiency of the CARB MB model is slightly higher than that of the CB-disk model.
The similar evolutionary tendencies of the five curves in Figure 6 should originate from similar mass-transfer rates.Figure 7 summarizes the evolution of mass-transfer rates as a function of the donor-star masses.It seems that the model with (M d,i , P orb,i ) = (0.8 M ⊙ , 0.35 days) possesses a maximum masstransfer rate at the current stage of 2A 1822-371.In the observed donor-star-mass range, five models have similar mass-transfer rates (1.7 − 2.1 × 10 −7 M ⊙ yr −1 ), which are higher than theoretical estimation in section 2. A high mass-transfer rate naturally results in rapid orbital expansion, which can vanquish the orbital shrinkage originating from angular-momentum loss.Therefore, the orbital evolution of five models emerges an expansion tendency.
Figure 8 shows the evolution of LMXBs with different initial donor-star masses and initial orbital periods in the Ṗorb − P orb diagram.It seems that a small donor-star mass seems produces a high orbital-period derivative at the current orbital period of 2A 1822-371.Three models with a 1.0 M ⊙ donor star and initial orbital periods of 0.35, 0.6, and 1.0 day can approximately reproduce the observed orbital-period derivative of 2A 1822-371.In a word, the peculiar observed properties of this source are probably associated with three initial parameters, including the initial donor-star mass, the initial orbital period, and the CB-disk mass.
To produce the observed orbital-period derivative of 2A 1822-371, our CB-disk model predicts a relatively high masstransfer rate of 1.9 × 10 −7 M ⊙ yr −1 .Such a mass-transfer rate is higher than the mass-accretion rate (1.6 − 5.0 × 10 −8 M ⊙ yr −1 ) derived from the intrinsic luminosity of 2A 1822-371 (Van et al. 2019).This implies a much outflow during the mass transfer, which would absorb a large fraction of X-ray radiation, resulting in a low observed luminosity.Meanwhile, because the orbital inclination angle of this source is in the range of 81 • − 85 • (Heinz & Nowak 2001;Ji et al. 2011), an edge-on accretion disk may be responsible for the low X-ray luminosity in observations.
Compared with the work of Xing & Li (2019), NS-MS star binary with a CB disk can evolve into 2A 1822-371 in a wide initial orbital-period range (0.35 − 1.0 days).Since the angularmomentum-loss rate due to MB by coupling between the strong magnetic field and an irradiation-driving wind is smaller than that by a CB disk, it requires a relatively short initial orbital period to reproduce the observed orbital period and orbital-period derivative.As a consequence, the excited wind-driving masstransfer model found that the progenitor of 2A 1822-371 is a narrow orbit system with a short orbital period of 0.4 days when the surface magnetic field of the donor star is 900 G (Xing & Li 2019).Another distinction between these two models is the evolutionary tendency of the mass-transfer rates.In the observed donor-star-mass range, the CB disk model predicts a decreasing mass-transfer rate (see also Figures 5 and 7); however, the one obtained by the excited wind-driving mass-transfer model is increasing (Xing & Li 2019).Since the mass transfers of both models are super-Eddington, it is hard to test the validity of these two models according to the observed X-ray luminosities.
If a CB disk exists around source 2A 1822-371, the infrared radiation might confirm its existence, like GG Tau (Roddier et al. 1996).In particular, recent works performed by the Wide-Field Infrared Survey Explorer confirmed that three black-hole LMXBs XTE J1118+480, A0620-00, GRS 1915+105, and NS LMXB 3A 1728-247 are surrounded by CB disks (Wang & Wang 2014).However, the orbital period (1160.8days) of NS LMXB 3A 1728-247 is much longer than that of 2A 1822-371.Therefore, it still has no observable evidence for the existence of such a CB disk around NS LMXBs similar to 2A 1822-371.Furthermore, the orbital inclination angle of 2A 1822-371 is in the range of 81 • − 85 • (Heinz & Nowak 2001;Ji et al. 2011), it is challenging to detect an edge-on CB disk.We expect that the powerful new infrared capabilities of JWST will confirm whether or not a CB disk encloses the source 2A 1822-371.
In theory, a CB disk around LMXBs can not only cause a rapid orbital expansion but also lead to a fast orbital shrinkage.According to equation ( 10), the angular-momentum loss by a CB disk results in a negative orbital-period derivative as The orbital evolution fates of LMXBs would depend on the competition between Ṗorb,cb and Ṗorb,mt .For a LMXB like 2A 1822-371, M d = 0.5 M ⊙ , µ = 0.38 M ⊙ , P orb = 5.57 hours, we have Ṗorb,cb = −2.2× 10 −11 s s −1 if the system is surrounded by a CB disk with a mass similar to XTE J1118+480.Similarly, Ṗorb,mt = (0.26 − 2.6) × 10 −11 s s −1 if the system possesses a mass ratio (q = 0.31) same to 2A 1822-371 and a mass-transfer rate in the range from 10 −9 M ⊙ yr −1 to 10 −8 M ⊙ yr −1 .As a result, the total orbital-period derivative ( Ṗorb = Ṗorb,cb + Ṗorb,mt ) is in the range from −1.94 ×10 −11 s s −1 to 0.4 ×10 −11 s s −1 .Therefore, the LMXB would evolve toward a wide orbit binary like 2A 1822-371 for a high mass-transfer rate.Otherwise, its descendant is a narrow orbit system like black-hole LMXBs XTE J1118+480 and A0620-00 (Xu & Li 2018;Chen & Podsiadlowski 2019).

Summary
According to the observed orbital-period derivative of 2A 1822-371, our analysis finds that the mass-transfer rate must be in the range of (4.3 − 5.8) × 10 −8 M ⊙ yr −1 , which is non-sensitive to the accretion efficiency of the NS.Besides gravitational radiation and MB, it requires an additional mechanism to efficiently extract angular momentum from the system, resulting in a rapid mass transfer.
In this work, we attempt to investigate whether a surrounding CB disk can explain the anomalous orbital-period derivative of 2A 1822-371.Adopting typical CB-disk parameters as α = 0.1, H/R = 0.1, and R/a = 2.3, detailed stellar evolution models show that a CB disk with a mass of M cb = 2.3 × 10 −8 M ⊙ can reproduce the observed orbital period, donor-star mass, and orbital-period derivative of 2A 1822-371.However, the expected error of the orbital-period derivative exceeds our simulated value by at least a factor of 2. According to our simulations, the progenitor of the source 2A 1822-371 is probably a binary system including a 1.6 M ⊙ NS and a 1.0 M ⊙ MS companion star in an initial orbit of 0.35 days.Although the angular-momentum loss by the CB disk gives rise to a rapid orbital shrinkage, a high mass-transfer rate of 1.9 × 10 −7 M ⊙ yr −1 from the less massive donor star to the more massive NS causes a more significant orbital expansion.It is noteworthy that there is no direct observational evidence for 2A 1822-371 to possess such a high masstransfer rate.
Our calculations also find that CB disks can accelerate the mass exchange of LMXBs, and alter their evolutionary fates, which strongly depend on the CB-disk masses and mass-transfer rate.For a system similar to 2A 1822-371, a CB disk with a mass of M cb = 10 −9 M ⊙ would cause its orbit to widen for a high mass-transfer of 10 −8 M ⊙ yr −1 .However, the orbit of the system would continuously shrink for the same CB disk and a low masstransfer rate of 10 −9 M ⊙ yr −1 .We have to stress that observable evidence of such a CB disk is still absent in NS LMXBs similar to 2A 1822-371.
It is worth emphasizing that the CARB MB model not only can interpret the observed characteristics of all persistent NS LMXBs (Van & Ivanova 2019, 2021) and binary pulsars (Deng et al. 2021), but also reproduce the anomalous orbital-period derivative of 2A 1822-371.Therefore, it might be the best MB model to replace the standard MB model in the binary evolution model in the future.

Fig. 2 .
Fig. 2. Evolution of a LMXB consisting of 1.6 M ⊙ NS and 1.0 M ⊙ MS donor star with an initial orbit of 0.35 days in the orbital period vs. stellar age diagram (top panel), and the donor-star mass vs. stellar age diagram (bottom panel).The horizontal short dashed line in the top panel indicates the orbital period of 2A 1822-371, while the horizontal short dashed lines in the bottom panel represent the inferred upper limit and lower limit of the donor-star mass.

Fig. 3 .Fig. 4 .
Fig. 3. Same as in Figure 2, but for the orbital period vs. donor-star mass diagram.The solid circle represents the observed data of the source 2A 1822-371.

Fig. 5 .
Fig.5.Same as in Figure2, but for the evolution of mass-transfer rates as a function of the donor-star masses.The horizontal short dashed line denotes the Eddington accretion rate.

Fig. 6 .
Fig. 6.Evolutionary tracks of LMXBs with different initial donor-star masses and initial orbital periods in the orbital period vs. donor-star mass diagram.The solid curves represents our standard model with M d,i = 1.0 M ⊙ and P orb,i = 0.35 days.A CB disk with a mass of M cb = 2.3 × 10 −8 M ⊙ is included in all models.The solid circle represents the observed data of the source 2A 1822-371.

Fig. 7 .Fig. 8 .
Fig.7.Same as in Figure6, but for the evolution of mass-transfer rates as a function of the donor-star masses.The horizontal short dashed line denotes the Eddington accretion rate.
where µ = M d M NS /(M d + M NS ) is the reduced mass of LMXBs.Taking β = 1, the mass transfer produces a positive orbital-

Table 1 .
Some observed data of the source 2A 1822-371.