The Maximum Accreted Mass of Recycled Pulsars

Since there are many uncertainties during supernovae and NS birth, we start our study from an NS with a zero-age main sequence star as a companion. The companion may overfill its Roche lobe and transfer material to the NS. If the mass transfer is dynamically unstable, the NS will be involved in the envelope of the companion and the binary will enter into common envelope evolution process. The common envelope evolution is complicated and whether the NS can accrete material during the common envelope phase is under debate (Ivanova et al. 2013; MacLeod & Ramirez-Ruiz 2015; Holgado et al. 2018). We therefore do not consider NS accretion in this case in our study. If the mass transfer is dynamically stable, an NS will accrete mass from the companion via stable mass transfer, and we focus on this case here.

Detailed binary evolution calculations are done with the state-of-the-art stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA version 9575; Paxton et al. 2011, 2013, 2015). For convenience, the NS is taken as a point mass. For the donor star, the initial element abundances of Population I stars, i.e., metallicity Z = 0.02, are adopted. The hydrogen mass fraction is given by X = 0.76 − 3Z (Pols et al. 1998). The mixing-length parameter is set to be α_MLT = 1.9. The mass transfer rate is calculated with the Ritter scheme (Ritter 1988), that is,

$$\begin{eqnarray}&&\dot{M}\propto \displaystyle \frac{{R}_{\mathrm{RL},{\rm{d}}}^{3}}{{M}_{{\rm{d}}}}\exp \left(\displaystyle \frac{{R}_{{\rm{d}}}-{R}_{\mathrm{RL},{\rm{d}}}}{{H}_{{\rm{p}}}}\right),\end{eqnarray} \tag{ 1 }$$

where R_d and R_RL,d are the donor radius and its Roche lobe radius, M_d is the donor mass, and H_p is the pressure scale height.

The initial NS, M_NS,i, ranges from 1.10–2.2 M_⊙, where the NS mass has a step of 0.2 M_⊙ from 1.4–2.2 M_⊙; the choice of 1.10 M_⊙ is intended to cover the minimum NS in the observations (i.e., 1.174 ± 0.004 M_⊙ for the companion of PSR J0453+1559; Martinez et al. 2015), and 1.25 M_⊙ is regarded as the mean mass of an NS from electron capture supernovae (Schwab et al. 2010). We assume that all transferred material from the donor flows to the NS, then some (β_mt) is lost from the binary, and some (δ_mt) forms a circumbinary (CB) disk (see more details in Section 2.2). The remainder (1 − β_mt − δ_mt) is defined as mass transfer efficiency f_mt. The value of f_mt is quite uncertain, and is set from 0.1–0.9 with a step of 0.1. Note that the accreted mass of an NS is also limited by the Eddington rate and an inefficient accretion stage (see more details in Section 2.3), and the real accretion efficiency ⁵ is lower than f_mt. With a given M_NS,i and f_mt, the initial donor masses range from 1.0–3.6 M_⊙ with a variable step size ⁶ , i.e., the donor mass has a step of △M_d = 0.2 M_⊙ for M_d ≤ 2.4 M_⊙, and △M_d = 0.4 M_⊙ for 2.4 < M_d ≤ 3.6 M_⊙. The initial orbital period ranges from 0.7–2.0 days with a step of 0.05 day, and from 2–20 days in a step of ${\rm{\Delta }}{\mathrm{log}}_{10}({P}_{\mathrm{orb},{\rm{i}}}/\mathrm{days})=0.025$. For binary systems in wider orbits, the donors generally enter into the red giant branch at the onset of mass transfer. The mass transfer rate is significantly larger than the Eddington rate, and most of the envelope masses of the donors are lost from the system (see more details in Section 3.1). For a given f_mt and M_NS,i, we will obtain the maximum increased mass of an NS. The evolution stops as the evolutionary age reaches 14 Gyr, but we mainly focus on the mass transfer stage, and the termination of mass transfer is defined as $\dot{M}\leqslant {10}^{-12}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Chen et al. 2017).

In this work, we mainly consider the binaries that evolve into detached NS + WD systems. The case of accreting pulsars with very low-mass nondegenerate companions, e.g., redbacks and black widows (Chen et al. 2013; Roberts 2013), are not included in our simulations. With the loss of the orbital angular momentum due to the gravitational-wave radiation, the WD will fill its Roche lobe and transfer mass to the NS. The stability of mass transfer processes is still under debate (van Haaften et al. 2012; Bobrick et al. 2017; Yu et al. 2021), therefore, we ignore the cases of NS accreting mass from the WDs.

We consider three types of angular momentum loss mechanisms, which are gravitational-wave radiation, magnetic braking, and mass loss, respectively.

The orbital angular momentum carried away by the gravitational-wave radiation can be calculated as (Landau & Lifshitz 1975)

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{GW}}=-\displaystyle \frac{32{G}^{7/2}{M}_{\mathrm{NS}}^{2}{M}_{{\rm{d}}}^{2}{({M}_{\mathrm{NS}}+{M}_{{\rm{d}}})}^{1/2}}{5{c}^{5}{a}^{7/2}},\end{eqnarray} \tag{ 2 }$$

where M_NS and M_d denote the NS and the donor star mass, a is the semimajor axis of the orbit, c is the speed of light in vacuum, and G is the gravitational constant.

The angular momentum loss because of magnetic braking is calculated from the formula (Rappaport et al. 1983)

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{MB}}=-3.8\times {10}^{-30}{M}_{{\rm{d}}}{R}_{{\rm{d}}}^{{\gamma }_{\mathrm{MB}}}{{\rm{\Omega }}}^{3}\,\mathrm{dyn}\,\mathrm{cm},\end{eqnarray} \tag{ 3 }$$

where γ_MB is the magnetic braking index, and is set to be 4 according to the standard magnetic braking prescription ⁷ (Chen et al. 2013; Van et al. 2019), R_d is the radius of the donor, Ω is the spin angular velocity, which equals the orbital angular velocity ω_orb as tidal synchronization is assumed. The magnetic braking effect can be neglected if the convective envelope becomes too thin. In this work, we switch on magnetic braking when the convective envelope fraction is larger than 0.01, as is done in Chen et al. (2017).

During mass transfer, we also consider the angular momentum extracted by the CB disk. The angular momentum loss rate under this torque can be expressed as (Spruit & Taam 2001; see also Shao & Li 2012; Chen et al. 2013)

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{CB}}=\gamma \left(\displaystyle \frac{2\pi {a}^{2}}{{P}_{\mathrm{orb}}}\right){\delta }_{\mathrm{mt}}{\dot{M}}_{{\rm{d}}}{\displaystyle \frac{t}{{t}_{\mathrm{vi}}}}^{1/3},\end{eqnarray} \tag{ 4 }$$

where ${\dot{M}}_{{\rm{d}}}$ is the mass transfer rate, γ² is the scale factor, and given by r_i/a, where r_i is the inner radius of the disk, a is the binary separation, t is the time since mass transfer began, t_vi is the viscous timescale at the inner edge of the disk, which is defined by t_vi = 2γ³ P_orb/3π α β², α is the viscosity parameter (Shakura & Sunyaev 1973), and β is the ratio of the scale height to the radius of the disk. Based on the observed results, we set γ² = 1.7 (Muno & Mauerhan 2006), α = 0.01, β = 0.03 (Belle et al. 2004), and δ_mt = 3 × 10⁻⁴ (Taam et al. 2003).

The extra material that leaves the systems is assumed to take away the specific angular momentum of the NS. Then, the angular momentum loss due to mass loss is

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{ML}}=-(1-{f}_{\mathrm{mt}}-{\delta }_{\mathrm{mt}})| {\dot{M}}_{{\rm{d}}}| {\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{{M}_{\mathrm{NS}}+{M}_{{\rm{d}}}}\right)}^{2}\displaystyle \frac{2\pi {a}^{2}}{{P}_{\mathrm{orb}}}.\end{eqnarray} \tag{ 5 }$$

First, the accretion of the NS is limited by the Eddington accretion limit, ⁸ ${\dot{M}}_{\mathrm{Edd}}$,

$$\begin{eqnarray}\begin{array}{rcl}{\dot{M}}_{\mathrm{Edd}} & = & 3.6\times {10}^{-8}\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right)\left(\displaystyle \frac{0.1}{{{GM}}_{\mathrm{NS}}/{R}_{\mathrm{NS}}{c}^{2}}\right)\\ & & \times \left(\displaystyle \frac{1.7}{1+X}\right)\,{M}_{\odot }{\mathrm{yr}}^{-1},\end{array}\end{eqnarray} \tag{ 6 }$$

where R_NS is the NS radius, and can be approximately expressed as a simple nonrelativistic degenerate Fermi-gas polytrope: ${R}_{\mathrm{NS}}=15{({M}_{\mathrm{NS}}/\,{M}_{\odot })}^{-1/3}$ (Tauris et al. 2012). Combining the limit of Eddington accretion rate, the accretion rate of the NS is

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{acc}}=\min (-{f}_{\mathrm{mt}}{\dot{M}}_{{\rm{d}}},{\dot{M}}_{\mathrm{Edd}}).\end{eqnarray} \tag{ 7 }$$

Second, the spin evolution of the NS is considered in addition to the limit of Eddington rate, which leads to an inefficient accretion during the mass transfer stage, as described below.

We define the magnetosphere radius, r_mag of the NS at which the ram pressure of the accreted material equals the magnetic pressure in the magnetosphere (Lamb et al. 1973; Ghosh & Lamb 1979a, 1979b; Liu & Chen 2011), that is,

$$\begin{eqnarray}&&{r}_{\mathrm{mag}}=1.8\times {10}^{8}{\left(\displaystyle \frac{{B}_{{\rm{s}}}}{{10}^{12}{\rm{G}}}\right)}^{4/7}{\left(\displaystyle \frac{{\dot{M}}_{\mathrm{acc}}}{{10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)}^{-2/7}\,\mathrm{cm},\end{eqnarray} \tag{ 8 }$$

where B_s is the surface magnetic field of the NS, and ${\dot{M}}_{\mathrm{in}}$ is the mass inflow rate. The evolution of magnetic field during the accretion process is described as (Shibazaki et al. 1989; Wijers 1997)

$$\begin{eqnarray}&&{B}_{{\rm{s}}}=\displaystyle \frac{{B}_{{\rm{i}}}}{1+{\rm{\Delta }}{M}_{\mathrm{acc}}/{m}_{{\rm{B}}}},\end{eqnarray} \tag{ 9 }$$

where B_i is the initial magnetic field of the NS, and is set to be 10¹² G, ΔM_acc is the accreted mass of the NS, m_B is the mass constant for the field decay, and is set to be 10⁻⁴ M_⊙, according to observations (Shibazaki et al. 1989; Zhang & Kojima 2006; Wang et al. 2011). If the magnetosphere radius is less than the corotation radius, the infalling material can be accreted onto the NS surface. The corotation radius is defined as (Liu & Chen 2011; Romanova et al. 2018)

$$\begin{eqnarray}&&{r}_{\mathrm{co}}=1.5\times {10}^{8}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{\,{M}_{\odot }}\right)}^{1/3}{P}_{\mathrm{spin}}^{2/3}\,\mathrm{cm},\end{eqnarray} \tag{ 10 }$$

where P_spin is the spin period of the NS in units of seconds. The spin-up torque during the accretion process is given by

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{acc}}={\dot{M}}_{\mathrm{acc}}\sqrt{{{GM}}_{\mathrm{NS}}{R}_{\mathrm{NS}}}.\end{eqnarray} \tag{ 11 }$$

With the spin-up of the NS, r_mag will be greater than r_co at some point. In this situation, the centrifugal barrier at r_mag prevents the infalling material from being accreted by the NS. This process is known as the propeller effect. The spin evolution during the propeller phase is approximately calculated by Alpar (2001) as

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{prop}}={\dot{M}}_{\mathrm{acc}}{r}_{\mathrm{mag}}^{2}[{\rm{\Omega }}-{{\rm{\Omega }}}_{{\rm{K}}}({r}_{\mathrm{mag}})].\end{eqnarray} \tag{ 12 }$$

Moreover, if r_mag is larger than the light cylinder radius r_lc, where r_lc = c/Ω = 48 km(P_spin/1 ms), the NS spins too fast to allow the infalling material to penetrate the light cylinder and the NS appears as a radio pulsar.

There is a maximum spin frequency of the NS, i.e., Keplerian frequency, f_K(M_NS). The accreted material is supposed to be ejected if the spin frequency of the NS equals f_K(M_NS). Here ${f}_{{\rm{K}}}{({M}_{\mathrm{NS}})\simeq C\,\mathrm{kHz}\,({M}_{\mathrm{NS}}/\,{M}_{\odot })}^{1/2}{({R}_{\mathrm{NS}}/10\,\mathrm{km})}^{-3/2}$, where C is a fitted parameter and is set to be 1.15 (Haensel et al. 2009). Since the NS is an extremely compact object, the moment of inertia of the NS should be calculated with general relativity effects and the specific equation of state of the NS considered (Arnett & Bowers 1977). For convenience, we take the NS as a point mass, and adopt I = 10⁴⁵ g cm² for all kinds of NSs. The influence of I on the NS accreted mass will be discussed in Section 4.2.

From what has been introduced above, the accretion efficiency, _acc, is given by

$$\begin{eqnarray}{\epsilon }_{\mathrm{acc}}=\left\{\begin{array}{cl}\min ({f}_{\mathrm{mt}},| {\dot{M}}_{\mathrm{Edd}}/{\dot{M}}_{{\rm{d}}}| ), & \mathrm{accretion}\ \mathrm{phase};\\ 0, & \mathrm{inefficient}\ \mathrm{accretion},\end{array}\right.\end{eqnarray} \tag{ 13 }$$

where the inefficient accretion cases include that the propeller effects occur, the NS is in the radio phase, and the NS spins at the Keplerian frequency.

The increase in NS mass is connected with the mass transfer rate during binary evolution, as shown in the left panel of Figure 1, where the binary initially contains a 1.4 M_⊙ NS and a 1.4 M_⊙ donor. The mass accretion rate of an NS can be easily obtained by using Equation (7), and is not shown for clarity. At the early stage of mass transfer, the mass transfer is on a thermal timescale. Therefore, the NS mass can increase rapidly. The propeller effect starts to work after the NS accretes a small part of masses. At this moment, the magnetosphere radius is larger than the corotation radius, as shown in the inset of the right panel of Figure 1 (where the cyan line is above the gray line). The matter is unable to be accreted onto the NS due to the centrifugal force exerted by the magnetosphere. ⁹ Meanwhile, the centrifugal barrier exerts a propeller spin-down torque on the NS in that phase. As the spin period increases, the magnetosphere radius can be less than the corotation radius at some point (where the gray line is above the cyan line in the inset of the right panel of Figure 1), resulting in repeated accretion processes at the early mass transfer phase. And the NS can accrete about 0.15 M_⊙ during that phase. After the initial thermal timescale mass transfer, the mass transfer rate decreases due to the radius expansion of a star driven by nuclear burning (Podsiadlowski et al. 2002), and the NS enters into a long-term propeller phase. The sudden decrease in mass transfer around 3.52 Gyr is due to the discontinuity of the composition gradient during the first dredge-up stage (Tauris & Savonije 1999; Istrate et al. 2016). There is still enough envelope material for burning after the dredge-up; the donor star expands again and a subsequent mass transfer occurs (Jia & Li 2014; Li et al. 2019). It is noted that an NS can only accrete very little material in that epoch and most transferred material is accreted during the thermal timescale mass transfer stage.

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As shown in Figure 1, how much mass can be accreted by an NS during the recycling process strongly depends on the mass transfer process. For low-mass donors, e.g., M_d,i = 1.4 M_⊙ in the left panel of Figure 1, the mass transfer rate is always sub-Eddington. While for more massive donors, the mass transfer rate may exceed the Eddington limit, such as in the cases shown in Figure 2. We see that the the NS can accrete relatively more mass from intermediate-mass donors (1.8 ≲M_d,i ≲ 2.4 M_⊙, e.g., the case shown in the left panel of Figure 2). The reason is that the main accretion process for an NS occurs during the thermal timescale mass transfer phase. While for massive donors (M_d,i ≳ 2.4 M_⊙, e.g., the case shown in the right panel of Figure 2), the thermal timescale mass transfer rate could be larger than the Eddington rate by several orders of magnitude. Therefore, a significant part of transferred mass will be ejected due to the Eddington limit (Podsiadlowski et al. 2002). As a result, an NS would not accrete too much material from a low-mass donor (M_d,i ≲ 1.8 M_⊙) due to the low thermal timescale mass transfer rate, and also cannot accrete too much material from a massive donor since most of the transferred material is ejected on account of the Eddington limit.

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In Figure 3, we present the spin evolution of an NS with different initial parameters, as shown in the panels. Most material has been accreted by the NS during thermal timescale mass transfer, resulting in a rapid decrease in the spin period. The NS accreted masses from left to right are 0.13, 0.22, and 0.02 M_⊙, respectively, and the NS generally rotates fast with a large accreted mass. In the middle panel, the minimum spin period of the NS is less than 1 ms. However, no sub-millisecond pulsars have been discovered yet (Hessels et al. 2006; Papitto et al. 2014; Bassa et al. 2017; Patruno et al. 2017; Haskell et al. 2018). The cause may be that the timescale of an NS in the sub-millisecond stage is very short, about 10⁷ yr, as shown in the middle panel. As a comparison, the timescale of an NS with P_spin ≲ 10 ms is about 2 × 10⁸ yr, which is 20 times larger than that of the sub-millisecond stage. At the end of mass transfer, the spin periods (as shown by blue circles) are several times larger than the minimum spin period, and the NS subsequently spins down due to the magnetic dipole radiation (Tauris et al. 2012).

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The spin period of an NS versus the accreted mass is shown in Figure 4, where the open circles denote the spin period at the end of mass transfer, and the solid circles denote the minimum spin period during the accretion process. In general, the NS rotates fast for a large ΔM_NS, consistent with the theoretical result in Tauris et al. (2012). The reason for the dispersion around the theoretical curve is that we consider the specific mass transfer process during the accretion. We also found that some NSs may have a sub-millisecond spin period, but they will spin down due to the propeller effects and the spin periods for the simulated samples at the termination of mass transfer are larger than 1 ms. Besides, at the end of mass transfer, the accreted mass is larger than that calculated in Tauris et al. (2012) for a given recycled spin period (i.e., the open circles above the green line in Figure 4). The reason is that the NS may accrete more mass and obtain a relatively shorter spin period, and then spins down to the given recycled spin period (as shown in Figure 3).

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To find the maximum accreted mass of an NS for a given f_mt and M_NS,i, we present the relation between NS accreted mass and the remnant WD mass, as shown in Figure 5, where M_NS,i = 1.4 M_⊙, f_mt = 0.3. The symbols represent the donors in the different mass ranges, as indicated in the panels. It is clear that the NSs accrete relatively more masses as M_d,i ranges from 1.8–2.4 M_⊙. The reasons are that the mass loss caused by the Eddington limit is not too much, and the NS can accrete relatively more material during the thermal timescale mass transfer phase, as discussed in Section 3.1.

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NSs can accrete relatively more masses for M_WD around 0.25–0.30 M_⊙. As discussed above, the main accretion process occurs at the early stage of the mass transfer phase. A binary with a short orbital period generally leads to a low thermal timescale mass transfer rate and a small remnant WD mass. Therefore, we can see that the NS accretes 0.1 M_⊙ at most with M_WD ≃ 0.15 M_⊙ in Figure 5, which is lower than that with M_WD ≃ 0.25–0.30 M_⊙. For binaries with large orbital periods, most of the transferred material is lost due to the Eddington limit, and the NS accreted mass is less than 0.05 M_⊙ for M_WD around 0.40 M_⊙. The fluctuation of NS accreted mass is ascribed to the propeller effect, which depends not only on the mass transfer rate, but also on the surface magnetic field of the NS and the corotation radius (see Equations (8)–(10)).

Figure 5 shows the correlation between an NS accreted mass and the remnant WD mass. If we assume all NSs have similar birth masses in binary pulsars, there should be a correlation between the final NS masses and the WD masses. For example, the NSs could be more massive with M_WD in the range of 0.25–0.30 M_⊙. However, the relation becomes uncertain when the birth masses of NSs distribute in a large range, as found in observations (Faulkner et al. 2005; Janssen et al. 2008; Rawls et al. 2011; Arzoumanian et al. 2018; Kandel & Romani 2020). In a further work, we will explore the mass distribution of an NS and its companion by combining the simulation results and binary population synthesis method, and try to find the correlation between NS mass and companion mass.

The NS accreted mass is strongly dependent on the initial binary parameters, and is difficult to determine. However, for a given f_mt and M_NS,i, there is a maximum accreted mass in the simulations. For example, when f_mt = 0.3 and M_NS,i = 1.4, the maximum accreted mass of an NS is about 0.27 M_⊙, as shown in Figure 5. By changing the values of f_mt and M_NS,i, we may get the corresponding maximum accreted mass in a similar way. In the upper panel of Figure 6, we present the maximum NS mass after the accretion with given mass transfer efficiencies and initial NS masses, where the final NS masses are shown in colors. For clarity, we plot several dotted lines to express a given final NS mass as noted in the figure. The curves are given by the linear interpolation between adjoining grids. The relations between NS maximum accreted mass and mass transfer efficiency for different initial NS masses are shown in the lower panel. We can see that the maximum accreted masses vary little for f_mt larger than ∼0.5, due to the existence of the propeller effect. In general, massive NSs can accrete more material in comparison with low-mass NSs if other parameters are fixed. For example, with a binary with a 1.25 M_⊙ NS, the NS can accrete about 0.39 M_⊙ mass for f_mt = 0.9. However, for an initial NS mass of 2.2 M_⊙, the maximum accreted mass could be ∼ 0.66 M_⊙. The reason is that the massive NS has a relatively large corotation radius, resulting in more material captured by the NS.

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The two massive pulsars with He WD companions, i.e., PSR J0348 and J0740, are shown by solid lines in the upper panel of Figure 6. It is noted that the maximum accreted mass of an NS is different for each M_NS,i. If a moderate mass transfer efficiency of 0.3 is assumed, the minimum birth mass of an NS can be obtained by interpolation between the adjoining grids, as shown in the upper panel of Figure 6. We find that the NS birth masses should be larger than 1.70 and 1.75 M_⊙ for PSR J0348 and J0740, respectively. The results of minimum NS birth masses with different mass transfer efficiencies speculated for the two observed samples are presented in Table 1.

Table 1. The Minimum NS Birth Masses (${M}_{\mathrm{NS},\min }$) of J0348 (2.01 ± 0.04 M_⊙) and J0740 (${2.072}_{-0.066}^{+0.067}\,{M}_{\odot }$) Inferred from This Work

fmt	${M}_{\mathrm{NS},\min }(\,{M}_{\odot })$	${M}_{\mathrm{NS},\min }(\,{M}_{\odot })$
	J0348	J0740
0.1	1.84	1.89
0.2	1.75	1.81
0.3	1.70	1.75
0.4	1.64	1.70
0.5	1.61	1.66
0.6	1.57	1.62
0.7	1.55	1.59
0.8	1.52	1.57
0.9	1.51	1.56

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Black widows and redbacks are one particular class of recycled pulsars that have been suggested to have significant accretion during the recycled processes (Roberts 2013). In these pulsar binaries, the very low-mass nondegenerate companions are irradiated and ablated by the NSs (Chen et al. 2013). However, the NSs in such binaries are still in the accretion phase, which are not considered in our simulations. In recent observations, two of such binaries are found with massive NSs, i.e., an NS mass of 2.18 ± 0.09 M_⊙ for PSR J1959+2048 and an NS mass of ${2.28}_{-0.09}^{+0.10}\,{M}_{\odot }$ for J2215+5135. Chen et al. (2013) studied the formation of this kind of pulsar binary, and found that low-mass companions can be produced from donors with mass around 1.0–1.2 M_⊙ by considering the evaporation effects. In the formation scenario of an NS with a low-mass companion, a degenerate core is not be developed at the onset of mass transfer. Therefore, progenitor binaries are supposed to have a short orbital period, which leads to a relatively lower mass transfer rate. As discussed in Section 3.1, an NS mainly increases its mass during the thermal timescale mass transfer, and the accreted mass of an NS in such binaries should not be larger than the maximum accreted mass calculated in this work.

Most NSs at birth may have a magnetic field higher than ∼ 10¹³ G (Haberl 2007). Before the onset of mass transfer, the magnetic field decays due to the ohmic decay of electric currents located in the NS crust or core. Then the NS may have a magnetic field weaker than 10¹² G at the onset of the mass transfer process (Aguilera et al. 2008; Gullón et al. 2014; Bransgrove et al. 2018). A weak pulsar magnetic field will lead to a small accretion radius (magnetosphere radius), and angular momentum can be effectively transferred to the NS, which results in a large spin-up rate (Longair 2011). In general, the propeller phase occurs (r_mag > r_co) slightly later for an NS with a weak magnetic field, and then more material could be accreted by the NS.

We calculated the evolution of binaries with M_NS,i = 1.4 M_⊙, B_mag,i = 0.5 × 10¹² G. For f_mt = 0.9, the NS accreted mass for models with small B_mag,i is higher than that for models with B_mag,i = 10¹² G by about 0.22 M_⊙. With the decrease in mass transfer efficiency f_mt, the influence of the initial magnetic field correspondingly decreases. For example, the difference in the maximum NS accreted mass between B_mag,i = 0.5 × 10¹² G and 10¹² G for f_mt = 0.1 is about 0.025 M_⊙. Therefore, the choice of the initial magnetic field has a limited effect for a small f_mt, but has a significant effect for a large f_mt.

The NS moment of inertia is difficult to solve due to the uncertainties of the NS equation of state. In this work, we adopt a constant value of I (10⁴⁵ g cm²) in the calculations, which is almost the lower limit for an NS. The true value of the moment of inertia for a massive NS may be larger than 3 × 10⁴⁵ g cm² (Greif et al. 2020, and references therein). Here we discuss the effect of the moment of inertia on our results.

A larger I means the acceleration of the spin-up process during the accretion phase is small, which results in a long spin period during the accretion phase. According to Equation (10), the corotation radius is larger than that for an NS with a low-value I. As a result, the binary spends a relatively shorter time in the propeller phase for an NS with a larger I, and the NS can accrete more material during the accretion phase. To illustrate this issue, we additionally calculate the cases of 1.4 M_⊙ with I = 2 × 10⁴⁵ g cm². For a large mass transfer efficiency of f_mt = 0.9, the NS accreted mass is about 0.1 M_⊙ greater than the case of I = 1 × 10⁴⁵ g cm². Similarly, the influence of I decreases when f_mt becomes small.