Evolutionary Origin of Ultralong-period Radio Transients

Isolated NSs are the evolutionary products of massive stars through core-collapse supernovae when their progenitors exhaust all fuels. During supernovae explosions, a tiny fraction of ejecta could form a fallback disk surrounding the nascent NS rather than completely leave it because those ejecta possess sufficient angular momentum (Michel 1988; Lin et al. 1991; Perna et al. 2014). Two normal magnetars, 4U 0142+61 and 1E 2259+586, are most likely to be surrounded by fallback disks (Wang et al. 2006; Kaplan et al. 2009). Without the interaction between the fallback disk and the NS, the spin evolution of the NS is driven by magnetic dipole radiation and is in an ejector phase in the early stage. With the increase of the spin period, the fallback disk can interact with the NS magnetosphere, and the NS enters a propeller phase. In this phase, a propeller torque originated from the interaction between the fallback disk and the NS can significantly influence the spin evolution of the NS (Illarionov & Sunyaev 1975), and the spin angular velocity of the NS decreases at an exponential rate for a short time t_prop (see also Equation (13); Ho & Andersson 2017).

During most of its lifetime, the accretion rate in the fallback disk decreases self-similarly according to a power law as $\dot{M}\propto {t}^{-\alpha }$ (t is the age of the fallback disk), in which α = 19/16 for a fallback disk whose opacity is dominated by electron scattering (Cannizzo et al. 1990). In the calculation, we adopt an accretion rate whose evolutionary law is the same as that in Chatterjee et al. (2000) as follows:

$$\begin{eqnarray}\dot{M}=\left\{\begin{array}{ll}{\dot{M}}_{0}, & t\lt T\\ {\dot{M}}_{0}{(t/T)}^{-\alpha }, & t\geqslant T\end{array}\right.,\end{eqnarray} \tag{ 1 }$$

where ${\dot{M}}_{0}$ is the initial mass-accretion rate when t < T, in which T is of the order of the dynamical timescale in the inner region of the nascent fallback disk. Chatterjee et al. (2000) adopted a typical value of T = 1 ms. Similar to Menou et al. (2001), we take T = 2000 s (Tong et al. 2016; Ronchi et al. 2022), which can be derived from a typical viscous timescale during which the fallback material with excess angular momentum circularizes to form a disk. Since T = 6.6 × 10⁻⁵ yr is much smaller than the age (10³–10⁴ yr) of long-period pulsars, the accretion process in the early stage (t < T) is negligible in our simulations. Since the initial mass of the fallback disk satisfies

$$\begin{eqnarray}&&{M}_{{\rm{d}},{\rm{i}}}={\dot{M}}_{0}T+{\int }_{T}^{\infty }{\dot{M}}_{0}{\left(t/T\right)}^{-\alpha }{dt},\end{eqnarray} \tag{ 2 }$$

we have ${\dot{M}}_{0}=(\alpha -1){M}_{{\rm{d}},{\rm{i}}}/(\alpha T)$ (Chen 2022).

In general, the inner radius of the fallback disk is thought to be the magnetospheric radius, at which the magnetic energy density is equal to the kinetic energy density of the inflow material. The magnetospheric radius is given by Davidson & Ostriker (1973), Elsner & Lamb (1977), and Ghosh & Lamb (1979) as

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{m}}} & = & \xi {\left(\displaystyle \frac{{\mu }^{4}}{2{GM}{\dot{M}}^{2}}\right)}^{1/7}=7.5\times {10}^{8}\mathrm{cm}\,{\xi }_{0.5}\\ & & \times {M}_{1.4}^{-1/7}{B}_{14}^{4/7}{R}_{6}^{12/7}{\dot{M}}_{18}^{-2/7},\end{array}\end{eqnarray} \tag{ 3 }$$

where G is the gravitational constant, M is the NS mass, $\dot{M}$ is the accretion rate, μ = BR³/2 (B and R are the surface magnetic field and the radius of the NS) is the the magnetic dipole moment of the NS, and ξ ≈ 0.5 is a corrective factor (Wang 1996; Long et al. 2005). In Equation (3), M_1.4 = M/1.4 M_⊙, R₆ = R/10⁶ cm, B₁₄ = B/10¹⁴ G, and ${\dot{M}}_{18}\,=\dot{M}/{10}^{18}\,{\rm{g}}\,{{\rm{s}}}^{-1}$.

In the early stage after the NS was born, the inner radius of the fallback disk is greater than the light cylinder radius, which is defined as

$$\begin{eqnarray}&&{R}_{\mathrm{lc}}={Pc}/2\pi =4.8\times {10}^{6}\,\mathrm{cm}\,{P}_{-3},\end{eqnarray} \tag{ 4 }$$

where c is the speed of light in vacuo, P₋₃ = P/10⁻³ s is the spin period of the NS. In this stage, the rapidly rotating NS cannot interact with the accreted material, appearing as a radio pulsar, which is in the ejector phase. The NS could radiate strong radio emission by the magnetic dipole radiation, which exerts a braking torque on the NS as follows:

$$\begin{eqnarray}&&{N}_{\mathrm{md}}=-\displaystyle \frac{2{\mu }^{2}{{\rm{\Omega }}}^{3}{\sin }^{2}\theta }{3{c}^{3}}=-\beta I{{\rm{\Omega }}}^{3},\end{eqnarray} \tag{ 5 }$$

where Ω = 2π/P is the spin angular velocity of the NS, θ is the inclination angle between the magnetic axis, and the rotation axis I is the momentum of inertia. For simplicity, in this work, we assume an orthogonal rotator, i.e., θ = π/2; thus, β ≡ 2μ²/3c³ I. According to the law of rotation ${N}_{\mathrm{md}}=I\dot{{\rm{\Omega }}}$, we have $\dot{{\rm{\Omega }}}=-\beta {{\rm{\Omega }}}^{3}$. Therefore, in the ejector phase, the spin period of the NS satisfies

$$\begin{eqnarray}&&P={P}_{0}{\left(1+\displaystyle \frac{8\beta {\pi }^{2}t}{{P}_{0}^{2}}\right)}^{1/2}={P}_{0}{\left(1+\displaystyle \frac{t}{{t}_{\mathrm{em}}}\right)}^{1/2},\end{eqnarray} \tag{ 6 }$$

where P₀ is the initial spin period of the NS. Due to magnetic dipole radiation, the nascent NS spins down on the timescale of

$$\begin{eqnarray}&&{t}_{\mathrm{em}}=\displaystyle \frac{{P}_{0}}{\dot{2{P}_{0}}}=\displaystyle \frac{{P}_{0}^{2}}{8{\pi }^{2}\beta }=2.1\times {10}^{5}\,{\rm{s}}\,\displaystyle \frac{{P}_{0,-3}^{2}{I}_{45}}{{B}_{14}^{2}{R}_{6}^{6}},\end{eqnarray} \tag{ 7 }$$

where I₄₅ = I/10⁴⁵ g cm².

With the spindown of the NS, the light cylinder radius gradually increases. Once R_m < R_lc, the NS magnetosphere can interact with the fallback disk, and the propeller phase begins. The transition period (i.e., the maximum period of the ejector phase) between the ejector phase and the propeller phase is

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{ej},\max } & = & \displaystyle \frac{2\pi }{{{\rm{\Omega }}}_{\mathrm{ej},\max }}=\displaystyle \frac{2\pi {R}_{{\rm{m}}}}{c}=0.16\,{\rm{s}}\,{\xi }_{0.5}\\ & & \times {M}_{1.4}^{-1/7}{B}_{14}^{4/7}{R}_{6}^{12/7}{\dot{M}}_{18}^{-2/7}.\end{array}\end{eqnarray} \tag{ 8 }$$

Inserting Equation (8) into Equation (6), we can estimate the duration of the ejector phase to be

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{ej}} & = & {t}_{\mathrm{em}}\left[{\left(\displaystyle \frac{{P}_{\mathrm{ej},\max }}{{P}_{0}}\right)}^{2}-1\right]\approx 164\,\mathrm{yr}\,{\xi }_{0.5}^{2}\\ & & \times {M}_{1.4}^{-2/7}{R}_{6}^{-18/7}{I}_{45}{B}_{14}^{-6/7}{\dot{M}}_{18}^{-4/7}.\end{array}\end{eqnarray} \tag{ 9 }$$

In Equation (9), $\dot{M}$ is time varying, while t_ej should be a constant. According to ${\dot{M}}_{18}={\dot{M}}_{\mathrm{0,18}}{(t/T)}^{-19/16}$, t = t_ej, and Equation (8), it yields

$$\begin{eqnarray}&&{t}_{\mathrm{ej}}={\left(164\,\mathrm{yr}\,{\xi }_{0.5}^{2}{M}_{1.4}^{-\tfrac{2}{7}}{R}_{6}^{-\tfrac{18}{7}}{I}_{45}{B}_{14}^{-\tfrac{6}{7}}{\dot{M}}_{\mathrm{0,18}}^{-\tfrac{4}{7}}{T}^{-\tfrac{19}{28}}\right)}^{\tfrac{28}{9}},\end{eqnarray} \tag{ 10 }$$

where T is in units of seconds.

The NS enters the propeller phase once the inner radius of the fallback disk is located between the light cylinder radius and the corotation radius, in which the corotation radius is defined as

$$\begin{eqnarray}&&{R}_{\mathrm{co}}=\sqrt[3]{\displaystyle \frac{{{GMP}}^{2}}{4{\pi }^{2}}}=1.7\times {10}^{6}\,\mathrm{cm}\,{M}_{1.4}^{1/3}{P}_{-3}^{2/3}.\end{eqnarray} \tag{ 11 }$$

During the propeller phase (R_c ≤ R_m ≤ R_lc), the Keplerian angular velocity (${{\rm{\Omega }}}_{{\rm{K}}}({R}_{{\rm{m}}})=\sqrt{{GM}/{R}_{{\rm{m}}}^{3}}$) at the magnetosphere radius of the fallback disk is smaller than the spin angular velocity of the NS. Therefore, the accreted material cannot keep corotating with the NS and is thought to have been ejected at R_m, due to the centrifugal barrier, resulting in a negative torque exerted on the NS. Meanwhile, the interaction between the magnetic lines and the fallback disk also produces a similar negative torque (Menou et al. 1999). Detailed numerical simulations indicate that the two braking torques mentioned above are approximately equal (Daumerie 1996). As a consequence, a braking torque exerting on the NS in the propeller phase can be expressed as

$$\begin{eqnarray}&&{N}_{\mathrm{prop}}=2\dot{M}{R}_{{\rm{m}}}^{2}({{\rm{\Omega }}}_{{\rm{K}}}({R}_{{\rm{m}}})-{\rm{\Omega }}),\end{eqnarray} \tag{ 12 }$$

where the factor of 2 originates from the two nearly equal negative torques mentioned above. Our adopted braking torque is two times as large as that taken by Ho & Andersson (2017), in which the interaction between the magnetic field lines and the fallback disk is ignored. Such an interaction may quench the processes of pair production in the pulsar magnetosphere, and only allow transient radio emission (Li 2006).

According to the law of rotation, the time derivative of the NS spin can be calculated from

$$\begin{eqnarray}&&\dot{{\rm{\Omega }}}=-\displaystyle \frac{2\dot{M}{R}_{{\rm{m}}}^{2}}{I}\left({\rm{\Omega }}-{{\rm{\Omega }}}_{{\rm{K}}}({R}_{{\rm{m}}})\right)=-\displaystyle \frac{{\rm{\Omega }}-{{\rm{\Omega }}}_{{\rm{K}}}({R}_{{\rm{m}}})}{{t}_{\mathrm{prop}}},\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&{t}_{\mathrm{prop}}=28.2\,\mathrm{yr}\,{I}_{45}{\xi }_{0.5}^{-2}{M}_{1.4}^{2/7}{B}_{14}^{-8/7}{R}_{6}^{-24/7}{\dot{M}}_{18}^{-3/7}.\end{eqnarray} \tag{ 14 }$$

It is worth noting that t_prop changes with time in our model, which is different from the model of Ho & Andersson (2017). Therefore, we have to use a numerical method to calculate the spin evolution of the NS in the propeller phase.

In the ejector phase, we use Equation (6) to calculate the spin evolution of the NS. Once the NS transitions to the propeller phase, we adopt a numerical method based on Equations (13) and (14) to simulate its spin evolution. For simplicity, we take ξ_0.5 = M_1.4 = R₆ = I₄₅ = 1. The spin period of the nascent NS is thought to be P₀ = 10 ms (P_0,−3 = 10). Adopting the above input parameters, the spin evolution of the NS is governed by two input parameters, including its magnetic field B and the initial mass-accretion rate ${\dot{M}}_{0}$ of the fallback disk.

Figure 1 shows the evolutionary tracks of NSs with an initial mass-accretion rate of ${\dot{M}}_{0}=1.5\times {10}^{24}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and different surface magnetic fields in the spin period versus NS age diagram. In the ejector phase, the spin periods evolve along lines with the same slope of $\bigtriangleup \mathrm{log}(P\,{{\rm{s}}}^{-1})/\,\bigtriangleup \mathrm{log}(t\,{\mathrm{yr}}^{-1})=1/2$. This law arises from a simple relation, $P\approx {P}_{0}{(t/{t}_{\mathrm{em}})}^{1/2}$, when t/t_em ≫ 1, according to Equation (6). The final period in the ejector phase is ${P}_{\mathrm{ej},\max }$, and the duration of the ejector phase is ${t}_{\mathrm{ej}}\propto {B}_{14}^{-8/3}$ for a fixed ${\dot{M}}_{0}$. Therefore, a strong magnetic field naturally results in a short duration of the ejector phase, as shown in Figure 1. In contrast, the NS with a weak magnetic field of 2.0 × 10¹³ G is always in the ejector phase in a timescale of 10⁵ yr, and merely evolves to a period of 0.8 s. Furthermore, from Equations (2), (8), and (10), we can derive the evolutionary law of ${P}_{\mathrm{ej},\max }$ as

$$\begin{eqnarray}&&{P}_{\mathrm{ej},\max }\propto {B}_{14}^{4/7}{t}_{\mathrm{ej}}^{19/56}\propto {B}_{14}^{-1/3}.\end{eqnarray} \tag{ 15 }$$

As a consequence, a strong magnetic field produces a short ${P}_{\mathrm{ej},\max }$. After ${P}_{\mathrm{ej},\max }$, the NS transitions to the propeller phase, and its spin period increases at an exponential rate. The two NSs with strong magnetic fields of B = 2.0 × 10¹⁵ and 2.0 × 10¹⁴ G can evolve to the present period (1091 s) of J1627 at t = 748 and 3.7 × 10⁴ yr, respectively.

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The model with an initial mass-accretion rate of and a surface magnetic field of B = 2.0 × 10¹⁴ G can successfully reproduce the observed P and $\dot{P}$ of J1627. We depict the evolution of the three critical radii of the NS in Figure 2. During the ejector phase, R_m > R_lc > R_co, and the NS spins down due to magnetic dipole radiation. At t = 9.7 × 10³ yr, R_m = R_lc > R_co, and the NS transitions to the propeller phase. Subsequently, R_lc > R_m > R_co, and the spin period of the NS rapidly increases at an exponential rate due to the propeller torque. With the increase of the spin period, the corotation radius also increases at an exponential rate. When R_co increases to be approximately equal to R_m, the NS reaches quasi-spin equilibrium, and the quasi-equilibrium period is given by

$$\begin{eqnarray}&&{P}_{\mathrm{eq}}=9.3\,{\rm{s}}\,{B}_{14}^{6/7}{\dot{M}}_{18}^{-3/7}\propto {t}^{57/112}.\end{eqnarray} \tag{ 16 }$$

As a consequence, the spin period of the NS slowly increases in the quasi-spin-equilibrium stage. Because the slope of the evolutionary track is $\bigtriangleup \mathrm{log}({P}_{\mathrm{eq}}\,{{\rm{s}}}^{-1})/\,\bigtriangleup \mathrm{log}(t\,{\mathrm{yr}}^{-1})=57/112\approx 1/2$, it seems that the two evolutionary lines of spin periods in the ejector phase and the quasi-spin-equilibrium stage are parallel (see also Figure 1).

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Figure 3 plots the evolutionary tracks of NSs with ${\dot{M}}_{0}=1.5\times {10}^{24}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and different surface magnetic fields in the spin period derivative versus NS age diagram. It is clear that the evolutionary tracks in the ejector phase are lines with the same slope. This phenomenon is caused by the evolutionary law of the spin period derivative. In the ejector phase, the evolution of the period derivative is governed by

$$\begin{eqnarray}&&\dot{P}=\displaystyle \frac{4{\pi }^{2}\beta }{P}=\displaystyle \frac{4{\pi }^{2}\beta }{{P}_{0}}{\left(1+\displaystyle \frac{t}{{t}_{\mathrm{em}}}\right)}^{-1/2}.\end{eqnarray} \tag{ 17 }$$

When t/t_em ≫ 1, Equation (17) can be derived to $\dot{P}\propto {(t/{t}_{\mathrm{em}})}^{-1/2}$; hence, the spin period derivatives evolve along lines with a slope of $\bigtriangleup \mathrm{log}(\dot{P}/{\rm{s}}\,{{\rm{s}}}^{-1})/\,\bigtriangleup \mathrm{log}(t\,{\mathrm{yr}}^{-1})=-1/2$. The period derivatives decrease to $\dot{P}\sim {10}^{-12}-{10}^{-9}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, depending on the surface magnetic fields of NSs. A strong surface magnetic field tends to result in a high final period derivative in the ejector phase.

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Once the NS transitions to the propeller phase, the spin period derivatives rapidly climb to a peak, and then decline. The evolutionary law of the period derivative is

$$\begin{eqnarray}&&\dot{P}=-\displaystyle \frac{2\pi \dot{{\rm{\Omega }}}}{{{\rm{\Omega }}}^{2}},\end{eqnarray} \tag{ 18 }$$

where $\dot{{\rm{\Omega }}}$ is derived from Equation (13). Therefore, a rapidly decreasing angular velocity produces a quickly increasing $\dot{P}$. For the NS with B = 2.0 × 10¹⁴ G, our simulated $\dot{P}=4.8\times {10}^{-10}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ at the current age (3.7 × 10⁴ yr) of J1627, which is less than the observed upper limit (1.2 × 10⁻⁹ s s⁻¹) of the period derivative of J1627 (Hurley-Walker et al. 2022). Based on the observed period and period derivative, the characteristic age of J1627 can be constrained to be ${\tau }_{{\rm{c}}}=P/(2\dot{P})\gt 1.4\,\times {10}^{4}\,\mathrm{yr}$. Our simulated age is compatible with the characteristic age of J1627. The NS with a strong magnetic field of B = 2.0 × 10¹⁵ G can evolve to 1091 s at an age of 748 yr, while the corresponding period derivative is much higher than the observed value. ${\dot{P}}_{\mathrm{eq}}\propto {t}^{-55/112}$ can be derived from Equation (16); thus the slope of the evolutionary lines of ${\dot{P}}_{\mathrm{eq}}$ is $\bigtriangleup \mathrm{log}({\dot{P}}_{\mathrm{eq}})/\,\bigtriangleup \mathrm{log}(t/\mathrm{yr})=-55/112\approx -1/2$. As a consequence, it seems that the two evolutionary lines of period derivatives in the ejector phase and the quasi-spin-equilibrium stage are parallel in Figure 3.

In our best model, the current mass-accretion rate of J1627 is $\dot{M}={\dot{M}}_{0}{(t/2000\,{\rm{s}})}^{-19/16}=5.8\times {10}^{13}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ when its current age is t = 3.7 × 10⁴ yr. Therefore, the current X-ray luminosity of the NS can be calculated by

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}}} & = & \displaystyle \frac{{GM}\delta {\dot{M}}_{{\rm{c}}}}{R}=1.0\times {10}^{32}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{M}_{1.4}{R}_{6}^{-1}\\ & & \times \left(\displaystyle \frac{\delta }{0.01}\right)\left(\displaystyle \frac{{\dot{M}}_{{\rm{c}}}}{5.8\times {10}^{13}\,{\rm{g}}\,{{\rm{s}}}^{-1}}\right),\end{array}\end{eqnarray} \tag{ 19 }$$

where δ is the accretion efficiency of the NS in the propeller phase. Since J1627 was observed in an upper limit (L_X < 1.0 × 10³² erg s⁻¹) of X-ray luminosity (Hurley-Walker et al. 2022), this implies that the accretion efficiency of the NS is smaller than 1% if the observed X-ray luminosity originates from accretion from the fallback disk. Such an accretion efficiency is slightly lower than the estimated values (0.01–0.05) in observational and theoretical research (Cui 1997; Zhang et al. 1998; Papitto & Torres 2015; Tsygankov et al. 2016).

Figure 4 presents the influence of the initial mass-accretion rates of fallback disks on the spin evolution of NSs. Because of the same magnetic field, the spin periods of NSs with different initial mass-accretion rates of fallback disks evolve along the same line in the ejector phase, while their durations are different. ${t}_{\mathrm{ej}}\propto {\dot{M}}_{0}^{-16/9}$ can be derived from Equation (10); thus, a high initial mass-accretion rate results in a short duration in the ejector phase. After the NSs transition to the propeller phase, they quickly reach quasi-spin equilibrium. A high initial mass-accretion rate naturally produces a short initial quasi-equilibrium period, according to Equation (16). Therefore, the NS with ${\dot{M}}_{0}=1.0\times {10}^{25}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ spends a longer timescale than the NS with ${\dot{M}}_{0}=1.5\times {10}^{24}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ to evolve to the current period of J1627.

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Figure 5 summarizes the parameter space that can produce the ultralong-period radio transient J1627 in the magnetic field versus the initial mass-accretion rate diagram. Those magnetars with a magnetic field of (2–5) × 10¹⁴ G and a fallback disk with an initial mass-accretion rate of (1.1 − 30) × 10²⁴g s⁻¹ can evolve toward the radio transient J1627 with a spin period of 1091 s, a period derivative less than 1.2 × 10⁻⁹ s s⁻¹, and an X-ray luminosity of less than 10³² erg s⁻¹ (taking a relatively low accretion efficiency δ = 0.001, we calculate the X-ray luminosity of the NS in the propeller phase according to Equation (19)) in a timescale shorter than 10⁵ yr. It is worth emphasizing that the parameter space strongly depends on the accretion efficiency during the propeller phase. If the accretion efficiency is δ = 0.01, the parameter space would sharply reduce to an ultrasmall zone in the lower-left shaded area. To form the ultralong spin period, a strong magnetic field tends to require a high initial mass-accretion rate. If t_prop is a constant, it can be derived from Equation (13) to ${\rm{\Omega }}=[{{\rm{\Omega }}}_{\mathrm{ej},\max }-{{\rm{\Omega }}}_{{\rm{k}}}({R}_{{\rm{m}}})]{e}^{-(t-{t}_{\mathrm{ej}})/{t}_{\mathrm{prop}}}+{{\rm{\Omega }}}_{{\rm{k}}}({R}_{{\rm{m}}})$​​​​​​. Therefore, t_prop is a characteristic timescale during which the spin period changes in the propeller phase. A suitable t_prop will determine whether the NS can evolve to an ultralong spin period. According to Equation (14), a strong magnetic field naturally requires a high initial mass-accretion rate for a fixed t_prop. According to Figure 4, a high initial mass-accretion rate cannot produce an ultralong-period NS in a timescale shorter than 10⁵ yr, resulting in the right boundary of the shaded region. An NS with a strong magnetic field would evolve toward the state exceeding the upper limits of the observed period derivative and X-ray luminosity, resulting in the upper boundary. Meanwhile, the bottom and left boundaries originate from the lower limits of the magnetic field and initial mass-accretion rate, under which the NS is always in the ejector phase.

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3.3.1. J1839 Is in the Propeller Phase

We first consider that J1839 is still in the propeller phase at present. Figure 6 shows the evolutionary trajectories of P and $\dot{P}$ of NSs with the initial mass-accretion rate of ${\dot{M}}_{0}\,=1.0\times {10}^{25}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and different magnetic fields. The NS with a weak magnetic field of B = 1.0 × 10¹² G is permanently in the ejector phase in a timescale of 10⁸ yr, and cannot evolve to an ultralong period. The two NSs with magnetic fields of B = 1.0 × 10¹⁴ and 7.9 × 10¹² G can evolve to the current period of J1839 within ages of 8.3 × 10⁵ and 6.0 × 10⁷ yr, respectively. In the case of B = 7.9 × 10¹² G, the current period derivative is $\dot{P}=3.55\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$. However, the NS with B = 1.0 × 10¹⁴ G spends a longer timescale than its current age to evolve to the observed upper limit ($\dot{P}=3.6\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$) of the period derivative.

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Similar to J1627, we also investigate whether it can produce the observed properties of J1839 in a wide range of magnetic fields and initial mass-accretion rates. Those NSs with initial mass-accretion rates of ${\dot{M}}_{0}={10}^{23}\mbox{--}{10}^{30}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and magnetic fields of B = 10⁹–10¹⁶ G cannot evolve to the current period and period derivative ($\dot{P}\lt 3.6\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$) in a timescale shorter than 10⁷ yr. It is generally thought that the active lifetime of a fallback disk is ∼10⁵ yr (Gençali et al. 2022); thus, the probability that J1839 is in the propeller phase can be ruled out.

3.3.2. J1839 Is in the Second Ejector Phase

J1839 may experience the first ejector phase and the propeller phase in a timescale of 10⁵ yr when the fallback disk is active. Subsequently, the source transitions to the second ejector phase again because the fallback disk becomes inactive. Figure 7 shows the evolutionary tracks of the P and $\dot{P}$ of NSs with an initial mass-accretion rate of ${\dot{M}}_{0}=4.0\times {10}^{24}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and different magnetic fields. The NS with a weak magnetic field of B = 1.0 × 10¹³ G is consistently in the ejector phase and evolves to a period of ∼10 s in a timescale of 10⁸ yr. The NS with a strong magnetic field of B = 1.0 × 10¹⁵ G can evolve to the present period of J1839 in the quasi-spin-equilibrium stage, while the corresponding period derivative is much higher than the observed value.

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Only the NS with B = 2.0 × 10¹⁴ G successively experiences the first ejector phase, the propeller phase, the quasi-spin-equilibrium stage, and the second ejector phase. When the NS is of the age of t = 10⁵ yr, the NS was spun down to a period of 1200 s in the quasi-spin-equilibrium stage. Subsequently, it transitions to the second ejector phase because the fallback disk becomes inactive. Because the initial period P₀ = 1200 s in the second ejector phase, the t_em is approximately 10 orders of magnitude longer than that in the first phase. Since t ≪ t_em, the period of the NS increases at an extremely low rate, which is consistent with the low $\dot{P}$ observed in J1839. At t = 2.5 × 10⁷ yr, the period of the NS increases to the current period (1318 s) of J1839, and the period derivative decreases to be $\dot{P}=8.0\times {10}^{-14}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, which is less than the observed upper limit of $\dot{P}=3.6\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$. At t = 10⁵ yr, the propeller torque exerted on the NS changes into the magnetic dipole radiation torque; thus, the $\dot{P}$ sharply decreases.

To understand the influence of ${\dot{M}}_{0}$ on the spin evolution of NSs, we also chart the evolution of the spin periods of NSs with B = 2.0 × 10¹⁴ G and ${\dot{M}}_{0}=4.0\times {10}^{23},4.0\times {10}^{24}$, and 4.0 × 10²⁵ g s⁻¹ in Figure 8. Since the duration of the first ejector phase is ${t}_{\mathrm{ej}}\propto {\dot{M}}_{0}^{-16/9}$, the NS with a low initial mass-accretion rate of 4.0 × 10²³ g s⁻¹ cannot transition to the propeller phase in a timescale of 10⁸ yr. However, the NS with a high initial mass-accretion rate of 4.0 × 10²⁵ g s⁻¹ enters the propeller phase in a short timescale of ∼30 yr.

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To investigate the progenitor properties of J1839, in Figure 9, we summarize the parameter space that can produce the ultralong-period radio transient J1839 in the magnetic field versus the initial mass-accretion rate diagram. The characteristic age of J1839 is ${\tau }_{{\rm{c}}}=P/(2\dot{P})\gt 1.2\times {10}^{8}\,\mathrm{yr}$. In the simulation, we examine whether an NS can evolve to the current state of J1839 in a timescale of 1.2 × 10⁸ yr. It is clear that the potential progenitor of J1839 is also a magnetar. Those magnetars with a magnetic field of (2–6) × 10¹⁴ G and a fallback disk with an initial mass-accretion rate of 4.0 × 10²⁴–10²⁶ g s⁻¹ are possible progenitors of the radio transient J1627. Similar to J1627, a strong magnetic field tends to require a high initial mass-accretion rate in order to evolve into J1839. Furthermore, an NS with a magnetic field stronger than 6.0 × 10¹⁴ G can evolve to the observed period of J1839, while the corresponding period derivative is much higher than the observed value.

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Assuming an initial spin period of P₀ = 10 ms and an initial mass-accretion rate of ${\dot{M}}_{0}\sim {10}^{23}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, Ronchi et al. (2022) found that a magnetar with an initial magnetic field of B₀ ∼ 10¹⁴ G can spin down to a spin period of 1091 s in a timescale of 10³–10⁵ yr. However, our models require a relatively high initial mass-accretion rate of ≳10²⁴ g s⁻¹. This discrepancy should arise from their different accretion model of the fallback disk, in which $\dot{M}={\dot{M}}_{0}{(1+t/T)}^{-\alpha }$ (Menou et al. 2001; Ertan et al. 2009). Furthermore, they considered the decay of magnetic fields under the combined contribution of ohmic dissipation and the Hall effect in the NS crust, which causes the NS to spin up slightly after the spin equilibrium stage. However, our simulated spin period slowly increases during the quasi-spin equilibrium stage.

Adopting the same model as that in Tong et al. (2016), Tong (2023a) found that a magnetar with a magnetic field of B = 4.0 × 10¹⁴ G can be spun down to 1091 s by a self-similar fallback disk with an initial mass of M_d,i = 10⁻³ − 10⁻⁴ M_⊙. From ${\dot{M}}_{0}=(\alpha -1){M}_{{\rm{d}},{\rm{i}}}/(\alpha T)$, the corresponding initial mass-accretion rates are 1.6 × (10²⁵–10²⁶) g s⁻¹. According to our simulated parameter space forming J1627, an NS with ${\dot{M}}_{0}=1.6\times {10}^{25}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and B = 4.0 × 10¹⁴ G can evolve into J1627, while an NS with ${\dot{M}}_{0}=1.6\times {10}^{26}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and B = 4.0 × 10¹⁴ G is unlikely to evolve toward J1627 in a timescale less than 10⁵ yr. This discrepancy may be caused by different determining criteria forming J1627, in which the spin period is a unique criterion in Tong (2023a), while both the P and $\dot{P}$ are criteria forming J1627 in our simulations.

Gençali et al. (2022) showed that an NS with an initial period of 0.3 s, a magnetic field of ∼10¹² G, and a fallback disk with an initial mass of 1.6 × 10⁻⁵ M_⊙ can evolve into J1627. Their simulations can interpret the observed period, period derivative, and X-ray luminosity of J1627 when the disk becomes completely inactive in a timescale of 7 × 10⁵ yr. Our magnetic field is much stronger than that in their model, and the evolutionary timescale is 1 order of magnitude smaller than their result. Different torque models should be responsible for these discrepancies.

According to the NS + fallback disk model, the evolutionary fates of pulsars depend on the two input parameters: magnetic field B and initial mass-accretion rate ${\dot{M}}_{0}$ of the fallback disk. In the $\dot{P}\mbox{--}P$ diagram of pulsars, we depict the evolutionary tracks of NSs with several special B and ${\dot{M}}_{0}$. An NS with a weak magnetic field of 2.0 × 10¹³ G is always in the ejector phase and evolves into a normal pulsar. The two NSs with B = 2.0 × 10¹⁴ G, and ${\dot{M}}_{0}=1.5\times {10}^{24}$ and 4.0 × 10²⁴ g s⁻¹ evolve toward radio transients with an ultralong period and a relatively high $\dot{P}\sim {10}^{-10}\mbox{--}{10}^{-9}{\rm{s}}\,{{\rm{s}}}^{-1}$ in the propeller phase. Such a $\dot{P}$ is compatible with the observed upper limit of J1627; hence, J1627 is probably in the propeller phase. However, J1839 has a very small period derivative as $\dot{P}\lt 3.6\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$. Therefore, the evolutionary state of J1839 should be different from that of J1627. Assuming a fallback disk active timescale of 10⁵ yr, our models show that the magnetic dipole torque in the second ejector phase can produce a $\dot{P}$ that is compatible with the observed upper limit of J1839.

J0901 is a long-period pulsar with a period of P = 75.9 s and a period derivative of $\dot{P}=2.25\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ (Caleb et al. 2022). If this NS is spinning down by pure magnetic dipolar radiation, the dipolar magnetic field can be estimated to be B = 1.3 × 10¹⁴ G (Ronchi et al. 2022). Our simulations indicate that an NS with a strong magnetic field of B = 1.0 × 10¹⁴ G and a high initial mass-accretion rate of ${\dot{M}}_{0}=1.0\times {10}^{27}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ can evolve to the current state of J0901 in the second ejector phase. Recently, Gençali et al. (2023) found that an NS with a weak magnetic field of ∼10¹² G can evolve to the current state of J0901 in the strong propeller phase.

J0250 is another slow-spinning radio pulsar with a spin period of 23.5 s and a spin period derivative of $\dot{P}=2.7\times {10}^{-14}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ (Tan et al. 2018). In Figure 10, the observed P and $\dot{P}$ of J0250 can be matched by the blue evolutionary track, in which the nascent NS has a magnetic field of B = 4.0 × 10¹³ G and ${\dot{M}}_{0}=2.0\times {10}^{28}\,{\rm{g}}\,{{\rm{s}}}^{-1}$. The required magnetic field in our model is approximately equal to the inferred one (B = 2.6 × 10¹³ G, Tan et al. 2018). To evolve into the current states of J0901 and J0250, extremely high initial mass-accretion rates of 10²⁷ − 10²⁸ g s⁻¹ are required. According to Equation (9), the duration of the first ejector phase is ${t}_{\mathrm{ej}}\propto {\dot{M}}^{-4/7}$; thus, these two sources merely experience a very short timescale in the first ejector.

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In the calculations, we ignore a decay of the magnetic field. Actually, the magnetic fields of NSs ought to decay due to the ohmic dissipation and the Hall effect in their crusts (Pons & Geppert 2007; Pons et al. 2009; Viganò et al. 2013; Pons & Viganò 2019; De Grandis et al. 2020). An evolution simulation showed that the decay of an initial magnetic field of 10¹⁴ G cannot influence the spin period evolution of the ejector phase in a timescale of 10⁴ yr (see also Figure 1 of Ronchi et al. 2022). However, the decay of magnetic fields would produce a long t_prop, according to Equation (14). As a consequence, J1627 would take a relatively long timescale to evolve to the current period.

If J1839 can evolve to an ultralong period that is very close to the current period in the active stage of the fallback disk, it would naturally transition to the second ejector phase. Since t_em ∝ B⁻², according to Equation (7), a decaying magnetic field leads to a relatively long t_em. Therefore, the increase in the period is slower than in the case without magnetic field decay. Accordingly, the evolutionary timescale of J1839 is also slightly longer than our calculation.