State Switching of the X-Ray/Radio Transitional Millisecond Pulsar

Since the discovery of the first radio millisecond pulsar (MSP; Backer et al. 1982) in the 1980s, more than 200 MSPs have been discovered (Pan et al. 2015). They exhibit fast spin periods ($P\lt 20$ ms) and weak magnetic fields ($B\sim {10}^{8}$ G) and are formed in the binary accretion phases (Lyne et al. 2004; Zhang & Kojima 2006; Lorimer 2008). The classical recycling scenario (Srinivasan 2010) suggests that low-mass X-ray binaries (LMXBs) are the incubators of MSPs, i.e., LMXBs are responsible for the conversion of slow ($\geqslant 30\,\mathrm{ms}$) neutron stars (NSs) with a large magnetic field (B $\sim \,{10}^{12}\,{\rm{G}}$) into fast spinning MSPs with a relatively weak magnetic field (B $\sim {10}^{8}\,{\rm{G}}$). In LMXBs, accreting X-ray pulsars (AXMPs) with a spin period of several milliseconds accrete matter from an evolved stellar companion (with mass $M\sim 1{M}_{\odot }$) via an accretion disk or stellar wind. The accreting matter is channeled out of the accretion disk and onto the magnetic pole of the AXMP (Bisnovatyi-Kogan & Komberg 1974; Alpar et al. 1982; Bhattacharya & van den Heuvel 1991; Burderi et al. 2001, 2002, 2002, 2003; Zhang et al. 2011), giving rise to the X-ray pulsation at the spin frequency; the X-ray luminosity is thus related to the accretion rate. Based on observations and simulations, the connections between AMXPs and MSPs have been firmly established (Campana et al. 1998; Wijnands & van der Klis 1998; van der Klis 2006; Burderi & Di Salvo 2013).

The first AMXP (SAX J1808.4−3658), with a weak magnetic field ($\sim {10}^{8}\,{\rm{G}}$) and coherent millisecond X-ray pulsations at 2.5 ms (Wijnands & van der Klis 1998), was found by the Rossi X-ray Timing Explorer (RXTE) in 1998. This was regarded as the first discovery of the long-sought progenitors of MSPs. Searching for the radio pulses of SAX J1808.4−3658 was very important in studying the link between AXMPs and MSPs. However, it is regrettable that no radio pulses from SAX J1808.4−3658 have been detected. To date, over two dozen AMXPs with spin frequencies from about 100 to $600\,\mathrm{Hz}$ have been found  (Patruno & Watts 2012; Wang et al. 2014), providing strong confirmation of the recycling scenario.

Other important milestones were reached with the discoveries of three systems, PSR J1023+0038 (J1023 hereafter; Archibald et al. 2009; Bogdanov et al. 2014; Patruno et al. 2014; Archibald et al. 2015; Baglio et al. 2016), PSR J1824–2452I in the globular cluster M28 (J1824 hereafter; Papitto et al. 2013, 2014; Linares 2014; Linares et al. 2014), and XSS J1227024859 (J1227 hereafter; de Martino et al. 2010, 2013; Bassa et al. 2014; Bogdanov et al. 2014; de Martino et al. 2015; Xing & Wang 2015). These systems were detected through multiband observations in X-ray and radio bands, and share some similar characteristics, such as accreting NSs with spin periods of 1.687, 1.686, 3.932 ms lying in close binaries with orbital periods, ${P}_{\mathrm{orb}}$, of 6.91, 4.80, 11.03 hr, respectively; having low-mass companions with masses, ${M}_{{\rm{c}}}$, of 0.15–0.36, 0.24, 0.17–0.2 ${M}_{\odot }$, respectively; and transitioning between the AMXP and MSP phases. Finally, these systems confirm that AMXPs and radio MSPs are indeed related. In particular, J1023 was the first NS observed to have switched on as a radio MSP (i.e., in the MSP state) with low X-ray activity after being an accreting X-ray binary (i.e., in the AXMP state) with brighter X-ray activity (Patruno et al. 2014; Stappers et al. 2014). Sometime between 2013 June 15 and June 30, the radio pulse from the MSP of J1023 disappeared from radio frequencies, where it was otherwise easily detected at radio bands from 0.3 to 5.0 GHz. While J1023 was detected as an MSP, a low X-ray luminosity of $\sim {10}^{32}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ was observed; however, the active state of an AMXP corresponds to a luminosity of $\sim {10}^{34}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Coti Zelati et al. 2014; Patruno et al. 2014; Takata et al. 2014; Archibald et al. 2015; Bogdanov et al. 2015). The above transitional phenomenon was also observed in the binary systems J1824 and J1227.

The observed characteristics of J1023, like the switch from an AMXP to an MSP state, provide a unique platform for better understanding the formation process of MSPs. Various scenario models have been proposed, such as the propeller mechanism (Illarionov & Sunyaev 1975; Campana et al. 1998), which is exploited to account for the observed transition phenomena (Papitto et al. 2015; Papitto & Torres 2015; Tsygankov et al. 2016). The propeller mechanism proposed that inflowing matter is propelled outward by the rapidly rotating NS magnetosphere, and electrons accelerated to high energies account for the X-ray emissions. A number of researchers (Coti Zelati et al. 2014; Li et al. 2014; Patruno et al. 2014; Takata et al. 2014; Deller et al. 2015; Jia & Li 2015) discussed the observations of these three transitional systems and pointed out that each system has a radio pulsar in a high accretion state engulfed by a great amount of ejected matter. A possible radio ejection model provided by Burderi et al. (2001) is also used to discuss the formation conditions of extremely fast rotating pulsars. In the radio ejection model, the strong dipole radiation power prevents an NS from endlessly accreting matter inward from its companion, and so the spin period of the NS is not too fast.

In this paper, we propose a plausible physical model for the disappearance and reappearance of the radio pulse when the pulsar switches states between AMXP and MSP by considering the plasma oscillation frequency at the magnetosphere boundary, which depends on the plasma density and, accordingly, on the accretion rate. The basic physical picture and formula are derived in Section 2. Two interesting magnetosphere boundaries—the critical magnetosphere boundary where the accreting luminosity equals the magnetic dipole radiation luminosity and the light cylinder—are calculated and discussed in Section 3. Finally, the discussions and conclusions are summarized in Section 4.

The observations of J1023 indicate that the accretion rate plays a significant role in the transition between AMXP and MSP states (Archibald et al. 2009, 2010; de Martino et al. 2010, 2014, 2015; Bogdanov et al. 2011; Papitto et al. 2013; Bogdanov et al. 2014, 2015), and that a high (low) accretion rate corresponds to the occurrence of the AMXP state in the X-ray band (MSP state in the radio band).

Based on the above facts, we think that the appearance of the MSP state can be ascribed to the low accretion rate that induces the plasma oscillation frequency (${\nu }_{p}$) to decline to several GHz at the magnetosphere boundary. In other words, the higher accretion rate results in a higher number density of plasma particles, which makes ${\nu }_{p}$ observable in the radio bands of a telescope, where the transitional source is detected in the AMXP state instead of the radio MSP state. The details of our model are described in the following subsections.

2.1. The Magnetosphere Parameters of Transitional MSPs

In general, the magnetosphere radius R_m depends on the accretion rate ($\dot{M}$) and NS magnetic moment (μ) for AMXPs in LMXB systems (Shapiro & Teukolsky 1983; Frank et al. 2002; Wang et al. 2015),

$$\begin{eqnarray}&&{R}_{m}\simeq 0.6\times {10}^{7}(\mathrm{cm}){\mu }_{26}^{\tfrac{4}{7}}{m}^{-\tfrac{1}{7}}{\dot{M}}_{15}^{-\tfrac{2}{7}},\end{eqnarray} \tag{ 1 }$$

where $m=M/{M}_{\odot }$ is the NS mass in solar mass, with the magnetic moment ${\mu }_{26}=\mu /({10}^{26}\,{\rm{G}}\,{\mathrm{cm}}^{3})$ and accretion rate ${\dot{M}}_{15}=\dot{M}/({10}^{15}\,{\rm{g}}\,{{\rm{s}}}^{-1})$.

To make a comparison, the co-rotational radius (R_co; Bhattacharya & van den Heuvel 1991; Frank et al. 2002) of this system is defined as

$$\begin{eqnarray}&&{R}_{\mathrm{co}}={\left(\displaystyle \frac{{GM}}{{{\rm{\Omega }}}^{2}}\right)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 2 }$$

where Ω and G are the NS rotational angular velocity and the gravitational constant, respectively. We found that the co-rotational radius, ${R}_{\mathrm{co}}\sim 23.8$ km for J1023, is far less than the magnetosphere radius at both the high ($\sim 70\,\mathrm{km}$ for $\dot{M}\sim 5\times {10}^{14}\,{\rm{g}}\,{{\rm{s}}}^{-1}$) and low ($\sim 100\,\mathrm{km}$ for $\dot{M}\sim {10}^{12}\,{\rm{g}}\,{{\rm{s}}}^{-1}$) states.

2.2. The Plasma Oscillation Frequency at the Magnetosphere Radius

The propagation of an electromagnetic (EM) wave in a charged medium is influenced by the plasma oscillation frequency ${\nu }_{p}$ (Chen 1984; Melrose 1986), which is described by the following equation:

$$\begin{eqnarray}&&{\nu }_{p}=\sqrt{\displaystyle \frac{{{ne}}^{2}}{\pi {m}_{e}}}=8980(\mathrm{Hz}){n}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 3 }$$

where n is the number density of electrons in units of ${{\rm{c}}{\rm{m}}}^{-3}$, and e and m_e are the electron charge and mass, respectively. When an EM wave propagates in a plasma, its cutoff frequency can be defined by the condition ${\nu }_{c}={\nu }_{p}$; above this, the EM can propagate in the media freely. Now, we exploit the concept of plasma cutoff frequency to explain the disappearance and reappearance of the radio pulse of the NS in its transition between the AMXP and MSP states. Accreting plasma constantly accumulates around the magnetosphere boundary, and its accretion rate is so high that the plasma oscillation frequency ${\nu }_{p}$ continuously increases, so much so that it prohibits low-frequency radio waves from penetrating. Hence, only the AMXP state can be detected. If the accretion rate declines, the magnetosphere boundary will expand and the density of the accreting plasma there will be reduced. Thus, GHz radio waves can escape from the NS magnetosphere and the NS can be detected in the MSP state. The observed phenomena do not contradict the above arguments, based on the fact that high (low) X-ray luminosity corresponds to the AMXP (MSP) state (Archibald et al. 2009, 2010; Deller et al. 2012; Coti Zelati et al. 2014; Patruno et al. 2014; Linares 2014; Takata et al. 2014; Baglio et al. 2016).

To illustrate our model, we first calculate the plasma number density (n_p) at the magnetosphere boundary R_m for a certain accretion rate $\dot{M}$ (Shapiro & Teukolsky 1983),

$$\begin{eqnarray}&&{n}_{p}({R}_{m})=\displaystyle \frac{\rho ({R}_{m})}{{m}_{p}},\end{eqnarray} \tag{ 4 }$$

where m_p is the proton mass and $\rho ({R}_{m})$ is the plasma particle density at the magnetosphere boundary,

$$\begin{eqnarray}&&\rho ({R}_{m})=\displaystyle \frac{\dot{M}}{4\pi {R}_{m}^{2}{\upsilon }_{f}({R}_{m})},\end{eqnarray} \tag{ 5 }$$

with ${\upsilon }_{f}({R}_{m})$ the free-fall velocity,

$$\begin{eqnarray}&&{\upsilon }_{f}({R}_{m})=\sqrt{\displaystyle \frac{2{GM}}{{R}_{m}}}.\end{eqnarray} \tag{ 6 }$$

The plasma number density n_p can then be written as follows:

$$\begin{eqnarray}&&{n}_{p}({R}_{m})=1.5\times {10}^{13}{\mu }_{26}^{2}{m}^{-1}{\left(\displaystyle \frac{{R}_{m}}{{10}^{7}}\right)}^{-5}({\mathrm{cm}}^{-3}).\end{eqnarray} \tag{ 7 }$$

After a convenient rearrangement, the number density of plasma can be written in the following form:

$$\begin{eqnarray}&&{n}_{p}(\dot{M})=15.6\times {10}^{13}{\mu }_{26}^{-\tfrac{6}{7}}{m}^{-\tfrac{2}{7}}{\dot{M}}_{15}^{\tfrac{10}{7}}({\mathrm{cm}}^{-3}).\end{eqnarray} \tag{ 8 }$$

By a simple treatment of Equations (7) or (8), the plasma oscillation frequency ${\nu }_{p}$ can be written as follows:

$$\begin{eqnarray}&&{\nu }_{p}({R}_{m})=3.5\times {10}^{2}(\mathrm{GHz}){\mu }_{26}{m}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{R}_{m}}{{10}^{7}}\right)}^{-\tfrac{5}{2}}\end{eqnarray} \tag{ 9 }$$

or

$$\begin{eqnarray}&&{\nu }_{p}(\dot{M})=1.1\times {10}^{2}(\mathrm{GHz}){\mu }_{26}^{-\tfrac{3}{7}}{m}^{-\tfrac{1}{7}}{\dot{M}}_{15}^{\tfrac{5}{7}}.\end{eqnarray} \tag{ 10 }$$

Using Equation (10), the evolution of ${\nu }_{p}$ versus accretion rate is given in Figure 1, where we notice that the plasma oscillation frequency ${\nu }_{p}$ increases with accretion rate. Note that Equation (10) is valid while the magnetosphere boundary lies inside the light cylinder; beyond this, we have to reconsider the emission region of the radio waves.

Zoom In Zoom Out Reset image size

2.3. Plasma Oscillation Frequency at the Light Cylinder

With the accretion rate declining, the magnetosphere radius will increase until it touches the light cylinder R_L (Ruderman & Sutherland 1975). The light cylinder is the last radius where the magnetosphere co-rotates with the NS. So, when the magnetosphere radius equals the light cylinder, the accretion rate (${\dot{M}}_{L}$) can be given by Equation (1). If $\dot{M}$ is higher than ${\dot{M}}_{L}$, then the magnetosphere radius lies inside the light cylinder. If $\dot{M}$ is less than ${\dot{M}}_{L}$, then ${R}_{m}\equiv {R}_{L}$ can be achieved. In this case where $\dot{M}\lt {\dot{M}}_{L}$, the expression for the plasma density and plasma oscillation frequency have to be rewritten.

For a pulsar with a known period ${P}_{-3}$ (P in units of milliseconds), R_L and ${\dot{M}}_{L}$ can be written as

$$\begin{eqnarray}&&{R}_{L}=c/{\rm{\Omega }}={cP}/(2\pi )\simeq 47.8(\mathrm{km}){P}_{-3},\end{eqnarray} \tag{ 11 }$$

with the speed of light c, and

$$\begin{eqnarray}&&{\dot{M}}_{L}=2.6\times {10}^{15}({\rm{g}}\,{{\rm{s}}}^{-1}){\mu }_{26}^{2}{m}^{-\tfrac{1}{2}}{P}_{-3}^{-\tfrac{7}{2}}.\end{eqnarray} \tag{ 12 }$$

In other words, for a sufficiently low accretion rate, i.e., $\dot{M}\leqslant {\dot{M}}_{L}$, the magnetosphere radius R_m stops at R_L, beyond which the stable dynamical process determining the accretion flow is invalid. So, for $\dot{M}\lt {\dot{M}}_{L}$, the plasma density at the light cylinder R_L should be rewritten as follows:

$$\begin{eqnarray}&&\rho ({R}_{L})=\displaystyle \frac{\dot{M}}{4\pi {R}_{L}^{2}{\upsilon }_{f}({R}_{L})},\end{eqnarray} \tag{ 13 }$$

with the free-fall velocity ${\upsilon }_{f}({R}_{L})$ at the light cylinder

$$\begin{eqnarray}&&{\upsilon }_{f}({R}_{L})=\sqrt{\displaystyle \frac{2{GM}}{{R}_{L}}}.\end{eqnarray} \tag{ 14 }$$

Then, the plasma number density n_p at the light cylinder can be written as

$$\begin{eqnarray}&&{n}_{p}({R}_{L})=\displaystyle \frac{\rho ({R}_{L})}{{m}_{H}}=\displaystyle \frac{\dot{M}}{4\pi {(2{GM})}^{\tfrac{1}{2}}{R}_{L}^{\tfrac{3}{2}}{m}_{p}},\end{eqnarray} \tag{ 15 }$$

which can be simplified as

$$\begin{eqnarray}&&{n}_{p}({R}_{L})=2.8\times {10}^{14}{\dot{M}}_{15}{m}^{-\tfrac{1}{2}}{P}_{-3}^{-\tfrac{3}{2}}.\end{eqnarray} \tag{ 16 }$$

Finally, the plasma oscillation frequency ${\nu }_{p}$ can be rewritten as follows:

$$\begin{eqnarray}&&{\nu }_{p}=150.2(\mathrm{GHz}){m}^{-\tfrac{1}{4}}{\dot{M}}_{15}^{\tfrac{1}{2}}{P}_{-3}^{-\tfrac{3}{4}};(\dot{M}\lt {\dot{M}}_{{\rm{L}}}).\end{eqnarray} \tag{ 17 }$$

Thus, for the system J1023 with the mass parameter m = 1.4 and a high accretion rate of $\sim {10}^{14}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, the plasma oscillation frequency at the magnetosphere radius is ${\nu }_{p}\sim 30$ GHz, which is observable in the observational wavebands (∼GHz) favored for a radio pulsar (Lyne & Graham-Smith 2012). However, in a low state with accreting rate $\sim {10}^{12}$ g s⁻¹, we obtain the plasma oscillation frequency ${\nu }_{p}\simeq 2.9\,\mathrm{GHz}$, which is consistent with the wavebands of most radio telescopes utilized for pulsar observations.

2.4. Plasma Skin Depth at the Low Accretion Luminosity State

Generally, for a pulsar, most of the power produced by the rotating magnetic dipole is emitted in the form of ultra low-frequency EM radiation (ULF hereafter), namely, EM waves with frequency equal to the rotational frequency of an NS, i.e., about 600 Hz for J1023, well below the plasma oscillation frequencies of several GHz. In this situation, the collective motion of plasma particles has to be taken into account. In this regime, it is possible to define the so-called Plasma Skin Depth (Stenson et al. 2017) through which a low-frequency EM wave can effectively penetrate into the plasma. We derive and calculate the Plasma Skin Depth from the Dispersion Relation (DR hereafter) of the wave. In the DR, an important characteristic of the wave is its cutoff phenomenon, i.e., if the angular frequency of the EM wave ω is lower than the plasma oscillation angular frequency ${\omega }_{{pe}}$, the EM cannot propagate in the plasma. Then, the cutoff angular frequency ${\omega }_{c}$ is defined as ${\omega }_{c}={\omega }_{{pe}}$, which can be described as follows (Chen 1984; Melrose 1986):

$$\begin{eqnarray}&&{\omega }^{2}={\omega }_{{pe}}^{2}+{\kappa }^{2}{c}^{2},\end{eqnarray} \tag{ 18 }$$

where κ is the wave number. For plane EM waves with $\omega \lt {\omega }_{c}$, the wave number $\kappa =i\omega (1-{\omega }_{{pe}}^{2}/{\omega }^{2})/c$ is imaginary so that the wave decays exponentially in spatial coordinates:

$$\begin{eqnarray}&&E(x,t)={E}_{0}{\exp }^{i(\kappa x-\omega t)},\end{eqnarray} \tag{ 19 }$$

where $E(x,t)$ is the amplitude of the EM wave and t (x) is the propagating time (space) of the wave, ${E}_{0}=E(x,0)$. The skin depth l_s of an EM wave in a plasma is defined by the attenuation of the wave amplitude by a factor of $1/e$ (Stenson et al. 2017):

$$\begin{eqnarray}&&{l}_{s}=\displaystyle \frac{c}{{\omega }_{{pe}}}{\left(1-\displaystyle \frac{{\omega }^{2}}{{\omega }_{{pe}}^{2}}\right)}^{-\tfrac{1}{2}}.\end{eqnarray} \tag{ 20 }$$

For a low-frequency EM radiation, $\omega \ll {\omega }_{{pe}}$, we have

$$\begin{eqnarray}&&{l}_{s}=\displaystyle \frac{c}{{\omega }_{{pe}}},\end{eqnarray} \tag{ 21 }$$

where ${\omega }_{{pe}}=2\pi {\nu }_{p}\simeq 1.8\times {10}^{4}\pi {n}_{e}^{\tfrac{1}{2}}$ and ${n}_{e}=2.8\times {10}^{14}{\dot{M}}_{15}{m}^{-1/2}{P}_{3}^{-3/2}$ at the light cylinder. Then,

$$\begin{eqnarray}&&{\omega }_{{pe}}\simeq 3.01\times {10}^{11}\pi {m}^{-1/4}{P}_{-3}^{-\tfrac{3}{4}}{\dot{M}}_{15}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 22 }$$

Finally, we obtain the relation between l_s and the accretion rate $\dot{M}$

$$\begin{eqnarray}&&{l}_{s}=10\pi {m}^{-1/4}{P}_{-3}^{-\tfrac{3}{4}}{\dot{M}}_{15}^{\tfrac{1}{2}}\simeq 19.48{\dot{M}}_{15}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 23 }$$

With the expression above, we calculated the skin depth in the case of J1023 for accretion rates corresponding to luminosities of ${10}^{32}\mbox{--}{10}^{34}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and found that depths of 6.2–0.62 cm are well below the light cylinder radius. Therefore, the penetration of an EM wave with frequency less than the plasma oscillation frequency of several GHz can be ignored.

When the accretion rate is high (low) enough, the transitional MSPs behave as AMXPs in X-ray bands (MSPs in radio bands). There might exist a particular critical magnetosphere boundary at which the AMXP and MSP switch into the other. As there are two kinds of emission powers, accreting X-ray luminosity in AMXPs and dipole electromagnetic radiation in MSPs, to account for, comparing both powers is likely to provide a critical condition for evaluating the switching between AMXPs and MSPs.

It is well known that the rotating magnetic dipole model predicts the pulsar EM emission luminosity L_R (Pacini 1967; Shapiro & Teukolsky 1983):

$$\begin{eqnarray}&&{L}_{R}=\displaystyle \frac{2{\mu }^{2}{{\rm{\Omega }}}^{4}}{3{c}^{3}}.\end{eqnarray} \tag{ 24 }$$

Moreover, for the accreting NS, the kinetic energy of the accreting matter is converted into the X-ray luminosity L_x (Shapiro & Teukolsky 1983; Frank et al. 2002), and at the magnetosphere radius, L_x can be given as

$$\begin{eqnarray}&&{L}_{x}=\displaystyle \frac{{GM}\dot{M}}{{R}_{m}}.\end{eqnarray} \tag{ 25 }$$

As is known, the spin-down power of an MSP is about ${10}^{32-34}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which is generally considered to be dominated by the rotating magnetic dipole radiation with luminosity L_R. When ${L}_{x}\gt {L}_{R}$, the system is dominated by accretion power (AMXP state); otherwise, it is dominated by the rotation power (MSPs state) for ${L}_{x}\lt {L}_{R}$. Then, the critical accretion rate ${\dot{M}}_{\mathrm{crit}}$ and the critical magnetosphere boundary R_crit can be obtained from the condition that both powers are equal to each other, i.e., L_x = L_R. Therefore,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{crit}}=9.4\times {10}^{15}({\rm{g}}\,{{\rm{s}}}^{-1}){\mu }_{26}^{2}{m}^{-\tfrac{8}{9}}{P}_{-3}^{-\tfrac{28}{9}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{R}_{\mathrm{crit}}\simeq 33.0(\mathrm{km}){m}^{\tfrac{1}{9}}{P}_{-3}^{\tfrac{8}{9}}.\end{eqnarray} \tag{ 27 }$$

Since ${\dot{M}}_{\mathrm{crit}}\gt {\dot{M}}_{L}$ or ${R}_{\mathrm{crit}}\lt {R}_{L}$, Equations (1)–(10) are still applicable. Therefore, the plasma oscillation frequency ${\nu }_{p}$ at the critical magnetosphere boundary R_crit is

$$\begin{eqnarray}&&{\nu }_{p}=545.2(\mathrm{GHz}){\mu }_{26}{m}^{-\tfrac{7}{9}}{P}_{-3}^{-\tfrac{20}{9}};(\dot{M}={\dot{M}}_{\mathrm{crit}}).\end{eqnarray} \tag{ 28 }$$

The critical accretion rates ${\dot{M}}_{\mathrm{crit}}$ and magnetosphere boundaries R_crit of J1023 are calculated using Equations (26) and (27), and their values are $\sim 1.4\times {10}^{15}$ g s⁻¹ and 54.6 km, respectively.

Furthermore, we also find that the light cylinder is an important boundary. When the luminosity is lower than $\sim {10}^{14}$ g s⁻¹, the magnetosphere radius will reach the light cylinder, and the transitional source will enter the radio MSP state. We calculate the pressure due to magnetic dipole radiation and the inflowing ionized hydrogen plasma, which has been studied by Burderi et al. (2001). The outward radiation pressure P_R at a radius r can be described by the dipole radiation luminosity L_R and the area factor $S=4\pi {r}^{2}$,

$$\begin{eqnarray}&&{P}_{R}=\displaystyle \frac{{L}_{R}}{{Sc}}=\displaystyle \frac{2{\mu }^{2}}{3{{SR}}_{L}^{4}}.\end{eqnarray} \tag{ 29 }$$

The inward ram pressure P_x at a radius r can be expressed by the X-ray luminosity ${L}_{x}={GM}\dot{M}/r=\dot{M}{v}_{f}^{2}/2$ and free-fall velocity ${v}_{f}=c\sqrt{{R}_{s}/r}$ with the Schwarzschild radius ${R}_{s}=2{GM}/{c}^{2}$,

$$\begin{eqnarray}&&{P}_{x}=\displaystyle \frac{{L}_{x}}{{{Sv}}_{f}}=\displaystyle \frac{\dot{M}c\sqrt{{R}_{s}/r}}{2S},\end{eqnarray} \tag{ 30 }$$

The ratio of P_R to P_x at r can be obtained as

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{R}}{{P}_{x}}=\displaystyle \frac{4}{3}{\left(\displaystyle \frac{r}{{R}_{L}}\right)}^{4}.\end{eqnarray} \tag{ 31 }$$

Thus, for ${P}_{R}\simeq {P}_{x},r\simeq {R}_{L};$ if the accretion rate is sufficiently low with ${R}_{m}\gt {R}_{L}$, we find that the outward pressure is higher than the inward ram pressure, ${P}_{R}\gt {P}_{x}$. Burderi et al. (2001) noted that the equilibrium point is stable at the intersection between the magnetic pressure and accretion matter pressure, but unstable when P_R = P_x. If the accretion rate decreases below a critical value $\sim {10}^{14}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ the magnetosphere radius becomes larger than the light cylinder radius, and the radio pulsar will switch on (please see Burderi et al. 2001 for more details).

By calculating the radii and plasma oscillation frequencies of the light cylinder and critical magnetosphere, we find that the critical radius is less than that of the light cylinder's ${R}_{L}\sim 80.7$ km, and notice that at the critical magnetosphere boundary, the plasma oscillation frequency is about 130.9 GHz, which is higher than the observational radio frequency for pulsars, i.e., the Lband (1–2 GHz), S band (2–4 GHz), and C band (4–8 GHz) (Lyne & Graham-Smith 2012). Thus, we set ${R}_{m}\lt {R}_{\mathrm{crit}}$ as the condition for the AMXP state, inside which the radio pulse is not easily detected by radio telescopes in the L–S–Cbands. However, when R_m is close or equal to the light cylinder R_L, if the accretion rate is low enough, the system may enter the MSP state, where the radio emission of the pulsar may be detected by the radio telescope at the L–S–C bands.

Based on observations of the transitional pulsar J1023, we propose a possible physical explanation for the switching phenomenon between the AMXP and MSP states of this source by ascribing the plasma oscillation frequency at the magnetosphere boundary to the cutoff frequency of pulsar. We find that for a transitional MSP with a spin period of $\sim 2\,\mathrm{ms}$, there exist two special magnetosphere boundaries: one is the critical magnetosphere boundary R_crit of ∼63.4 km, and the other is the light cylinder radius R_L of ∼95.6 km. The plasma oscillation frequency declines from $\sim 100\,\mathrm{GHz}$ at the critical magnetosphere radius to ∼2 GHz at the light cylinder, corresponding to the accretion rates of $\sim {10}^{15}$ g s⁻¹ (high state) and $\sim {10}^{12}$ g s⁻¹ (low state), respectively. It is generally thought that the radio pulse radiation of a pulsar is generated inside the light cylinder (Dyks et al. 2004; Mitra & Rankin 2011), which shows a steep power-law spectrum with an index of about −1.6 (Sieber 1973; Malofeev et al. 1994; Lyne & Graham-Smith 2012), raising the radio emission of the pulsar to be a low radio emission flux at the higher frequency (e.g., ∼10 GHz). In addition, after considering the influence of interstellar medium scattering and low-frequency radio interference (i.e., ∼100 MHz) on Earth (Cordes & Lazio 1997), the pulsar searching bands usually adopted by ground radio telescopes are near 1.4 GHz (Lorimer & Kramer 2004; Manchester et al. 2005; Burgay et al. 2006). Therefore, for a high state, the magnetosphere is near the critical radius, implying a high plasma oscillation frequency of ∼100 GHz, which means that the radio emission of a pulsar will be hard to detect with the radio telescope. However, for a low state, the magnetosphere radius will move to the light cylinder radius, where the lower plasma oscillation frequency of about several GHz can be obtained, and it could be detected by the radio telescope.

As can be noticed, the light cylinder may be a special boundary for the transitional pulsar. The magnetosphere radius will expand to, or even exceed, the light cylinder when it is in the low X-ray luminosity state ($\lt {10}^{34}\,{\rm{g}}\,{{\rm{s}}}^{-1}$). With the accretion luminosity decreasing, the transitional source exhibits the observational property of a radio MSP. We calculate the Plasma Skin Depth for low-frequency EM (LFEM) wave radiation at the light cylinder of a pulsar, and find that it is on the order of a few centimeters. We suggest that the pressure exerted by the LFEM on the plasma must be taken into account since the LFEM cannot penetrate the plasma. Thus, we estimate the pressure exerted by the rotating magnetic dipole radiation on the inflowing ionized hydrogen plasma and infer the ratio of the outward radiation pressure P_R and inward pressure P_x using Equations (29)–(31). We find that in the direction of the accretion disk plane, when the system enters the low accretion-rate state ($\lt {10}^{14}$ g s⁻¹), the magnetosphere boundary expands to the light cylinder, where the outward radiation pressure P_R, magnetic field pressure P_mag, and inward pressure P_x are equal. As the accretion rate gradually declines, the inward pressure exerted by the inflowing ionized hydrogen plasma gradually becomes lower than the outward radiation pressure, and the LFEM radiation may push the plasma out. However, considering accretion beyond the disk-plane direction, as well as a pulsar with a non-perpendicular magnetic angle ($\lt 90^\circ$) between the magnetic axis and the spin axis, the accretion plasma may reach the light cylinder, beyond which the co-rotation velocity of the magnetic field line will be limited by the speed of light. In the direction of the disk plane, the plasma may be swept out; therefore, the light cylinder cannot be an equilibrium position at which the disk may be truncated.

Furthermore, we tried to seek other transitional pulsar candidates, and find that the source SAX J1808.4−3658(Wijnands & van der Klis 1998; Wijnands et al. 2002), with an NS spin period of 2.49 ms and which might exhibit a radio MSP state if it exhibits the low accretion-rate state of $\sim {10}^{12}$ g s⁻¹ for a period of time, has properties similar to transitional pulsars. In the future, with the multiband observations by ground/space-based telescopes, we hope that more transitional pulsars will be discovered, from which more information to distinguish and constrain the mechanisms of transitional pulsars can be gleaned.

This work is supported by National Key R&D Program of China (No. 2017YFA0402600), the International Partnership Program of the Chinese Academy of Sciences (grant No.114A11KYSB20160008), the Strategic Priority Research Program of the Chinese Academy of Sciences (grant No. XDB23000000), the National Natural Science Foundation of China (grant Nos. U1731238, 11373038, 11565010, 11173034, 11173024, 11303047, 11703003), and the Interdisciplinary Innovation Team of the Chinese Academy of Sciences. This work was supported in part by the Science and Technology Foundation of Guizhou Province (grant No. J[2015]2113), the Doctoral Starting up Foundation of Guizhou Normal University 2014, the Innovation Team Foundation of the Education Department of Guizhou Province (grant Nos. [2014]35), the Science and Technology Department of Guizhou Province (the innovation talent team (grant No. (2015)4015), the excellent young talents program (grant No. 2013-29), the high level creative talents (grant No. (2016)-4008)), and the education department of Guizhou Province (the innovation team (grant No. [2014]35)). We thank the anonymous referee who provided many comments and valuable suggestions. In particular, we would like to thank Dr. James E. Wicker for his kind help in carefully improving the language.