Evolutionary Origin of Ultra-long 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 a 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 $t_{\rm prop}$ (see also Eq. 13, Ho & Andersson, 2017).

In the most 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 $\alpha=19/16$ 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 same to Chatterjee et al. (2000) as follows

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 $T=1~{}\rm ms$. Similar to Menou et al. (2001), we take $T=2000~{}\rm s$ (Tong et al., 2016; Ronchi et al., 2022), which can be derived from a typical viscous timescale that the fallback material with excess angular momentum circularizes to form a disk. Since $T=6.6\times 10^{-5}~{}\rm yr$ is much smaller than the age ($10^{3}-10^{4}~{}\rm yr$) 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

we have $\dot{M}_{0}=(\alpha-1)M_{\rm d,i}/(\alpha T)$ (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; Ghosh & Lamb, 1979)

where $G$ is the gravitational constant, $M$ is the NS mass, $\dot{M}$ is the accretion rate, $\mu=BR^{3}/2$ ($B$ and $R$ are the surface magnetic field and the radius of the NS) is the the magnetic dipole moment of the NS, and $\xi\approx 0.5$ is a corrective factor (Wang, 1996; Long et al., 2005). In equation 3, $M_{1.4}=M/1.4~{}M_{\odot}$, $R_{6}=R/10^{6}~{}\rm cm$, $B_{14}=B/10^{14}~{}\rm G$, and $\dot{M}_{18}=\dot{M}/10^{18}~{}\rm g\,s^{-1}$.

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

where $c$ is the speed of light in vacuo, $P_{-3}=P/10^{-3}~{}\rm s$ 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, that is 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

where $\Omega=2\pi/P$ is spin angular velocity of the NS, $\theta$ 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. $\theta=\pi/2$, thus $\beta\equiv 2\mu^{2}/3c^{3}I$. According to the law of rotation $N_{\rm md}=I\dot{\Omega}$, we have $\dot{\Omega}=-\beta\Omega^{3}$. Therefore, in the ejector phase, the spin period of the NS satisfies

where $P_{0}$ is the initial spin period of the NS. Due to magnetic dipole radiation, the nascent NS spin down on the timescale

where $I_{45}=I/10^{45}~{}\rm g\,cm^{2}$.

With the spin-down of the NS, the light-cylinder radius gradually increases. Once $R_{\rm m}<R_{\rm lc}$, 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

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

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

where $T$ in units of second.

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

During the propeller phase ($R_{\rm c}\leq R_{\rm m}\leq R_{\rm lc}$), the Keplerian angular velocity ($\Omega_{\rm K}(R_{\rm m})=\sqrt{GM/R_{\rm m}^{3}}$) at the magnetosphere radius of the fallback disk is smaller than the spin angular velocity of the NS. Therefore, the accreted material cannot keep co-rotating with the NS and was thought to be ejected at $R_{\rm m}$ 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). The detailed numerical simulations indicate that the two braking torques mentioned above is approximately equal (Daumerie, 1996). As a consequence, a braking torque exerting on the NS in the propeller phase can be expressed as

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.

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

where

It is worth noting that $t_{\rm prop}$ changes with time in our model, which is different with 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 $\xi_{0.5}=M_{1.4}=R_{6}=I_{45}=1$. The spin period of the nascent NS is thought to be $P_{0}=10~{}\rm ms$ ($P_{0,-3}=10$). 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 $\dot{M}_{\rm 0}$ of the fallback disk.

Figure 1 shows the evolutionary tracks of NSs with an initial mass-accretion rate $\dot{M}_{0}=1.5\times 10^{24}~{}\rm g\,s^{-1}$ 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 $\bigtriangleup{\rm log}(P/{\rm s})/\bigtriangleup{\rm log}(t/\rm yr)=1/2$. This law arises from a simple relation $P\approx P_{0}(t/t_{\rm em})^{1/2}$ when $t/t_{\rm em}\gg 1$ according to Equation (6). The final period in the ejector phase is $P_{\rm ej,max}$, and the duration of the ejector phase $t_{\rm ej}\propto B_{14}^{-8/3}$ for a fixed $\dot{M}_{0}$. Therefore, a strong magnetic field naturally results in a short duration of the ejector phase like in Figure 1. In contrast, the NS with a weak magnetic field of $2.0\times 10^{13}$ G is always in the ejector phase in a timescale of $10^{5}$ 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 $P_{\rm ej,max}$ as

As a consequence, a strong magnetic field produces a short $P_{\rm ej,max}$. After $P_{\rm ej,max}$, the NS transitions to the propeller phase, and its spin period increases at an exponential rate. The two NSs with strong magnetic field $B=2.0\times 10^{15}$ and $2.0\times 10^{14}$ G can evolve to the present period (1091 s) of J1627 at $t=748$ and $3.7\times 10^{4}$ yr, respectively.

The model with an initial mass-accretion rate $\dot{M}_{0}=1.5\times 10^{24}~{}\rm g\,s^{-1}$ and a surface magnetic field $B=2.0\times 10^{14}~{}\rm G$ 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, $R_{\rm m}>R_{\rm lc}>R_{\rm co}$, and the NS spins down due to magnetic dipole radiation. At $t=9.7\times 10^{3}~{}\rm yr$, $R_{\rm m}=R_{\rm lc}>R_{\rm co}$, and the NS transitions to the propeller phase. Subsequently, $R_{\rm lc}>R_{\rm m}>R_{\rm co}$, 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 $R_{\rm co}$ increases to be approximately equal to $R_{\rm m}$, the NS reaches a quasi-spin equilibrium, and the quasi-equilibrium period is given by

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 $\bigtriangleup{\rm log}(P_{\rm eq}/{\rm s})/\bigtriangleup{\rm log}(t/\rm yr)=57/112\approx 1/2$ , 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).

Figure 3 plots the evolutionary tracks of NSs with $\dot{M}_{0}=1.5\times 10^{24}~{}\rm g\,s^{-1}$ 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 spin-period derivative. In the ejector phase, the evolution of the period derivative is governed by

When $t/t_{\rm em}\gg 1$, Equation (17) can derive to $\dot{P}\propto(t/t_{\rm em})^{-1/2}$, hence the spin-period derivatives evolve along lines with a slope of $\bigtriangleup{\rm log}(\dot{P}/{\rm s\,s^{-1}})/\bigtriangleup{\rm log}(t/\rm yr)=-1/2$. The period derivatives decrease to $\dot{P}\sim 10^{-12}-10^{-9}~{}\rm s\,s^{-1}$, 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.

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

where $\dot{\Omega}$ is derived from Equation (13). Therefore, a rapidly decreasing angular velocity produces a quick increasing $\dot{P}$. For the NS with $B=2.0\times 10^{14}~{}\rm G$, our simulated $\dot{P}=4.8\times 10^{-10}~{}\rm s\,s^{-1}$ at the current age ($3.7\times 10^{4}~{}\rm yr$) of J1627, which is less than the observed upper limit ($1.2\times 10^{-9}~{}\rm s\,s^{-1}$) of 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 $\tau_{\rm c}=P/(2\dot{P})>1.4\times 10^{4}~{}\rm yr$. Our simulated age is compatible with the characteristic age of J1627. The NS with a strong magnetic field of $B=2.0\times 10^{15}~{}\rm G$ can evolve to 1091 s at an age of 748 yr, while the corresponding period derivative is much higher than the observed value. From Equation (16), it can derive $\dot{P}_{\rm eq}\propto t^{-55/112}$, thus, the slope of the evolutionary lines of $\dot{P}_{\rm eq}$ is $\bigtriangleup{\rm log}(\dot{P}_{\rm eq})/\bigtriangleup{\rm log}(t/\rm yr)=-55/112\approx-1/2$. 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 $\dot{M}=\dot{M}_{0}(t/2000~{}\rm s)^{-19/16}=5.8\times 10^{13}~{}\rm g\,s^{-1}$ when its current age $t=3.7\times 10^{4}~{}\rm yr$. Therefore, the current X-ray luminosity of the NS can be calculated by

where $\delta$ is the accretion efficiency of the NS in the propeller phase. Since J1627 was observed an upper limit ($L_{\rm X}<1.0\times 10^{32}~{}{\rm erg\,s^{-1}}$) of X-ray luminosity (Hurley-Walker et al., 2022), it implies that the accretion efficiency of the NS is smaller than $1\%$ if the observed X-ray luminosity originates from an accretion from the fallback disk. Such an accretion efficiency is slightly lower than the estimated values ($0.01-0.05$) in the observation and theoretical researches (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. From Equation (10), it can derive $t_{\rm ej}\propto\dot{M}_{0}^{-16/9}$, 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 a 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 $\dot{M}_{0}=1.0\times 10^{25}~{}\rm g\,s^{-1}$ spends a timescale longer than the NS with $\dot{M}_{0}=1.5\times 10^{24}~{}\rm g\,s^{-1}$ to evolve to the current period of J1627.

Figure 5 summarizes the parameter space that can produce the ultra-long period radio transient J1627 in the magnetic field versus the initial mass-accretion rate diagram. Those magnetars with a magnetic field of $(2-5)\times 10^{14}~{}\rm G$ and a fallback disk with an initial mass-accretion rate of $(1.1-30)\times 10^{24}~{}\rm g\,s^{-1}$ can evolve toward the radio transient J1627 with a spin period of $1091~{}\rm s$, a period derivative less than $1.2\times 10^{-9}~{}\rm s\,s^{-1}$, and an X-ray luminosity less than $10^{32}~{}\rm erg\,s^{-1}$ (taking a relatively low accretion efficiency $\delta=0.001$, we calculate the X-ray luminosity of the NS in the propeller phase according to equation 19) in a timescale shorter than $10^{5}~{}\rm yr$. It is worth emphasizing that the parameter space strongly depends on the accretion efficiency during the propeller phase. The parameter space would sharply reduce to an ultra-small zone in the lower-left shaded area if the accretion efficiency $\delta=0.01$. To form the ultra-long spin period, a strong magnetic field tends to require a high initial mass-accretion rate. If $t_{\rm prop}$ is a constant, it can derive to $\Omega=[\Omega_{\rm ej,max}-\Omega_{\rm k}(R_{\rm m})]e^{-(t-t_{\rm ej})/t_{\rm prop}}+\Omega_{\rm k}(R_{\rm m})$ from Equation (13). Therefore, $t_{\rm prop}$ is a characteristic timescale that the spin period changes in the propeller phase. A suitable $t_{\rm prop}$ will determine whether the NS can evolve to an ultra-long spin period. According to Equation (14), a strong magnetic field naturally requires a high initial mass-accretion rate for a fixed $t_{\rm prop}$. According to Figure 4, a high initial mass-accretion rate can not produce ultra-long period NS in a timescale shorter than $10^{5}~{}\rm yr$, 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.

Assuming an initial spin period $P_{0}=10~{}\rm ms$ and an initial mass-accretion rate $\dot{M}_{0}\sim 10^{23}~{}\rm g\,s^{-1}$, Ronchi et al. (2022) found that a magnetar with initial magnetic field $B_{0}\sim 10^{14}$ G can spin down to a spin period of 1091 s in a timescale of $10^{3}-10^{5}$ yr. However, our models require a relatively high initial mass-accretion rate of $\gtrsim 10^{24}~{}\rm g\,s^{-1}$. 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 slightly spin up after the spin equilibrium. However, our simulated spin period slowly increases during the quasi-spin equilibrium.

Adopting a same model in Tong et al. (2016), Tong (2023a) found that a magnetar with a magnetic field $B=4.0\times 10^{14}$ G can be spun down to 1091s by a self-similar fallback disk with initial mass of $M_{\rm d,i}=10^{-3}-10^{-4}~{}M_{\odot}$. From $\dot{M}_{0}=(\alpha-1)M_{\rm d,i}/(\alpha T)$, the corresponding initial mass-accretion rates are $1.6\times(10^{25}-10^{26})~{}\rm g\,s^{-1}$. According to our simulated parameter space forming J1627, an NS with $\dot{M}_{0}=1.6\times 10^{25}~{}\rm g\,s^{-1}$ and $B=4.0\times 10^{14}$ G can evolve into J1627, while an NS with $\dot{M}_{0}=1.6\times 10^{26}~{}\rm g\,s^{-1}$ and $B=4.0\times 10^{14}$ G is unlikely to evolve toward J1627 in a timescale less than $10^{5}~{}\rm yr$. This discrepancy may be caused by different determining criterion forming J1627, in which the spin period is a unique criterion in Tong (2023a), while both $P$ and $\dot{P}$ are criterions forming J1627 in our simulations.

Gençali et al. (2022) shown that an NS with an initial period of 0.3 s, a magnetic field of $\sim 10^{12}~{}\rm G$, and a fallback disk with an initial mass of $1.6\times 10^{-5}~{}M_{\odot}$ 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 $7\times 10^{5}~{}\rm yr$. Our magnetic field is much stronger than that in their model, and the evolutionary timescale is one 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 $\dot{P}-P$ 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 $2.0\times 10^{13}~{}\rm G$ is always in the ejector phase and evolves into a normal pulsar. The two NSs with $B=2.0\times 10^{14}~{}\rm G$, and $\dot{M}_{0}=1.5\times 10^{24}$ and $4.0\times 10^{24}~{}\rm g\,s^{-1}$ evolve toward radio transients with an ultra-long period and a relatively high $\dot{P}\sim 10^{-10}-10^{-9}\rm s\,s^{-1}$ 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 $\dot{P}<3.6\times 10^{-13}~{}\rm s\,s^{-1}$. Therefore, the evolutionary state of J1839 should be different with J1627. Assuming a fallback-disk active timescale of $10^{5}~{}\rm yr$, 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 $\dot{P}=2.25\times 10^{-13}~{}\rm s\,s^{-1}$ (Caleb et al., 2022). If this NS is spinning down by a pure magnetic dipolar radiation, the dipolar magnetic field can be estimated to be $B=1.3\times 10^{14}~{}\rm G$ (Ronchi et al., 2022). Our simulations indicate that an NS with a strong magnetic field of $B=1.0\times 10^{14}~{}\rm G$ and a high initial mass-accretion rate of $\dot{M}_{0}=1.0\times 10^{27}~{}\rm g\,s^{-1}$ 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 $\sim 10^{12}~{}\rm G$ 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 $\dot{P}=2.7\times 10^{-14}~{}\rm s\,s^{-1}$ (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 $B=4.0\times 10^{13}~{}\rm G$ and $\dot{M}_{0}=2.0\times 10^{28}~{}\rm g\,s^{-1}$. The required magnetic field in our model is approximately equal to the inferred one ($B=2.6\times 10^{13}~{}\rm G$, Tan et al., 2018). To evolve into the current states of J0901 and J0250, it requires extremely high initial mass-accretion rates of $10^{27}-10^{28}~{}\rm g\,s^{-1}$. According to Equation (9), the duration of the first ejector phase $t_{\rm ej}\propto\dot{M}^{-4/7}$, thus, these two sources merely experience a very short timescale in the first ejector.

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 shown that the decay of an initial magnetic field of $10^{14}$ G cannot influence the spin-period evolution of the ejector phase in a timescale of $10^{4}$ yr (see also Figure 1 of Ronchi et al., 2022). However, the decay of magnetic fields would produce a long $t_{\rm prop}$ 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 ultra-long 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 $t_{\rm em}\propto B^{-2}$ according to Equation (7), a decaying magnetic field leads to a relatively long $t_{\rm em}$. Therefore, the increase of the period is slower than the case without magnetic field decay. Accordingly, the evolutionary timescale of J1839 is also slightly longer than our calculation.