A PROPELLER MODEL FOR THE SUB-LUMINOUS STATE OF THE TRANSITIONAL MILLISECOND PULSAR PSR J1023+0038

Energy conservation dictates that the energy available to power the radiative emission from the disk (L_disk), the inner disk boundary (L_prop), and the NS surface (L_NS), as well as the kinetic energy of the outflow launched by a propelling NS (${\dot{M}}_{\mathrm{ej}}{v}_{\mathrm{out}}^{2}/2$) and the energy converted in internal energy of the flow and advected (${\dot{E}}_{\mathrm{adv}}$), should be equal to the sum of the gravitational energy liberated by the infall of matter ${\dot{E}}_{g}$, and the energy released by the NS magnetosphere through the torque N applied at the inner disk radius

$$\begin{eqnarray}&&{L}_{\mathrm{prop}}+{L}_{d}+{\dot{E}}_{\mathrm{adv}}+{L}_{\mathrm{NS}}+\displaystyle \frac{1}{2}{\dot{M}}_{\mathrm{ej}}{v}_{\mathrm{out}}^{2}={\dot{E}}_{g}+N{{\rm{\Omega }}}_{*},\end{eqnarray} \tag{ 7 }$$

where ${{\rm{\Omega }}}_{*}=2\pi /P$ is, as before, the NS angular velocity. In a stationary state, mass conservation is expressed by:

$$\begin{eqnarray}&&{\dot{M}}_{d}={\dot{M}}_{\mathrm{NS}}+{\dot{M}}_{\mathrm{ej}}.\end{eqnarray} \tag{ 8 }$$

We define the fraction of mass ejected as

$$\begin{eqnarray}&&{k}_{\mathrm{ej}}=\displaystyle \frac{\dot{{M}_{\mathrm{ej}}}}{{\dot{M}}_{d}}=1-\displaystyle \frac{{\dot{M}}_{\mathrm{NS}}}{{\dot{M}}_{d}}.\end{eqnarray} \tag{ 9 }$$

Considering the reasoning developed in the previous section, we consider values k_ej > 0.95 for the case of PSR J1023+0038. According to these definitions, the gravitational energy liberated by the mass infall is:

$$\begin{eqnarray}&&{\dot{E}}_{g}=\displaystyle \frac{{GM}{\dot{M}}_{d}}{{R}_{\mathrm{in}}}+{GM}{\dot{M}}_{\mathrm{NS}}\left(\displaystyle \frac{1}{{R}_{\mathrm{NS}}}-\displaystyle \frac{1}{{R}_{\mathrm{in}}}\right).\end{eqnarray} \tag{ 10 }$$

We assume that the NS luminosity is given by efficient conversion of the infalling gravitational energy, so that

$$\begin{eqnarray}&&{L}_{\mathrm{NS}}={GM}{\dot{M}}_{\mathrm{NS}}\left(\displaystyle \frac{1}{{R}_{\mathrm{NS}}}-\displaystyle \frac{1}{{R}_{\mathrm{in}}}\right).\end{eqnarray} \tag{ 11 }$$

Furthermore, we express the disk luminosity as a fraction η of the energy radiated by an optically thick, geometrically thin disk

$$\begin{eqnarray}&&{L}_{d}=\eta \displaystyle \frac{{GM}{\dot{M}}_{d}}{2{R}_{\mathrm{in}}}.\end{eqnarray} \tag{ 12 }$$

The case of a radiatively efficient disk is realized for η = 1. For values of η lower than unity, we assume that the energy that is not radiated by the disk is partly advected, and partly made available to power the propeller emission. To express the latter, we introduce a parameter, ξ, that will represent the fraction of gravitational energy that is liberated in the disk and can be used to power the propeller emission; for ξ = 0 no such energy is used, while for ξ = 1 − η, all the disk energy that is not radiated is converted into propeller luminosity. This implicitly means that we express

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{adv}}=(1-\eta -\xi )\displaystyle \frac{{GM}{\dot{M}}_{d}}{2{R}_{\mathrm{in}}}.\end{eqnarray} \tag{ 13 }$$

Substituting the previous formulae in Equation (7), we obtain an expression for the energy that can be radiated from the disk–magnetospheric boundary:

$$\begin{eqnarray}&&{L}_{\mathrm{prop}}=\left(\displaystyle \frac{1+\xi }{2}\right)\displaystyle \frac{{GM}{\dot{M}}_{d}}{{R}_{\mathrm{in}}}+N{{\rm{\Omega }}}_{*}-\displaystyle \frac{1}{2}{k}_{\mathrm{ej}}{\dot{M}}_{d}{v}_{\mathrm{out}}^{2}.\end{eqnarray} \tag{ 14 }$$

Similarly, the conservation of angular momentum at the inner disk boundary yields:

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{ej}}{R}_{\mathrm{in}}{v}_{\mathrm{out}}=N+{\dot{M}}_{d}{{\rm{\Omega }}}_{K}{R}_{\mathrm{in}}^{2}.\end{eqnarray} \tag{ 15 }$$

The left-hand side represents the angular momentum imparted to eject matter, while the right-hand side represents the torque exerted by the NS magnetosphere and the angular momentum advected by disk matter in Keplerian rotation appear. In order to express the velocity of the outflow v_out we follow Ekşi et al. (2005), who treated the interaction at the inner disk boundary as a collision of particles, obtaining

$$\begin{eqnarray}&&{v}_{\mathrm{out}}={{\rm{\Omega }}}_{K}({R}_{\mathrm{in}}){R}_{\mathrm{in}}[{\omega }_{*}(1+\beta )-\beta ],\end{eqnarray} \tag{ 16 }$$

where the elasticity parameter β has been introduced. The case of completely anelastic collision is given by β = 0, while the totally elastic case is described by β = 1. Inserting the expression for the outflow velocity (Equation (16)) into Equations (14) and (15), and solving for L_prop and N gives

$$\begin{eqnarray}{L}_{\mathrm{prop}} & = & \displaystyle \frac{{GM}{\dot{M}}_{d}}{{R}_{\mathrm{in}}}\left\{\displaystyle \frac{1+\xi }{2}-{\omega }_{*}+{k}_{\mathrm{ej}}\right.\\ & & \left.\left[{\omega }_{*}[{\omega }_{*}(1+\beta )-\beta ]-\displaystyle \frac{1}{2}{[{\omega }_{*}(1+\beta )-\beta ]}^{2}\right]\right\}\\ N & = & {\dot{M}}_{d}\sqrt{{{GMR}}_{\mathrm{in}}}\{{k}_{\mathrm{ej}}[{\omega }_{*}(1+\beta )-\beta ]-1\}.\end{eqnarray} \tag{ 17 }$$

If most of the in-flowing matter is ejected from the inner disk boundary (k_ej ≃ 1), the equation for the propeller luminosity simplifies to

$$\begin{eqnarray}&&{L}_{\mathrm{prop}}=\displaystyle \frac{{GM}\dot{{M}_{d}}}{2{R}_{\mathrm{in}}}[\xi +{({\omega }_{*}-1)}^{2}(1-{\beta }^{2})].\end{eqnarray} \tag{ 18 }$$

We can immediately note that if the energy advected in the disk is not available to power the propeller (ξ = 0) and the propeller collision is completely elastic (β = 1), all the available energy goes into powering the kinetic energy of the outflow, and the propeller luminosity vanishes. On the other hand, if the collision is completely anelastic (β = 0), the outflow velocity equals the velocity of the magnetosphere at the inner disk radius, and the propeller luminosity is maximized. In what follows, we only consider the latter case. Using Equation (4) to relate the disk mass accretion rate to the inner disk radius and Equation (3) to express the latter in terms of the corotation radius and the fastness, we obtain for the parameters of PSR J1023+0038

$$\begin{eqnarray}{L}_{\mathrm{prop}} & = & 1.75\times {10}^{35}\;{\omega }_{*}^{-3}\left[\xi +{({\omega }_{*}-1)}^{2}\right.\\ & & \times \left.(1-{\beta }^{2})\right]\;{{\rm{erg}}\;{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 19 }$$

This relation is plotted in Figure 2 for β = 0 and ξ equal to 0 (i.e., no disk gravitational energy available to power the propeller emission, red solid line), and 0.15 (red dashed line). Green and blue lines in the same figure represent the cases k_ej = 0.99 and 0.95, respectively. Values of the propeller luminosity of few ×10³⁴ erg s⁻¹ are obtained for a fastness ≳1.5.

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Like in P14, we assume that at the inner disk boundary the propelling magnetosphere is able to accelerate electrons to relativistic energies. To simplify, the electron energy distribution is assumed to be described by a power law with an exponential cutoff:

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{e}}{d\gamma }={N}_{0}{\gamma }^{-\alpha }\mathrm{exp}\left(-\displaystyle \frac{\gamma }{{\gamma }_{\mathrm{max}}}\right).\end{eqnarray} \tag{ 20 }$$

These electrons produce synchrotron emission by interacting with the magnetic field at the disk–magnetosphere interface

$$\begin{eqnarray}&&\bar{B}=\displaystyle \frac{\mu }{{R}_{\mathrm{in}}^{3}}=\displaystyle \frac{\mu }{{R}_{c}^{3}{\omega }_{*}^{2}}.\end{eqnarray} \tag{ 21 }$$

At high energies the synchrotron emission is cut off at ${E}_{\mathrm{syn}}\simeq 10(\bar{B}/{10}^{6}\;{\rm{G}})\;{({\gamma }_{\mathrm{max}}/{10}^{4})}^{2}$ MeV, while at low energies the emission is self-absorbed below a few eVs (see Papitto et al. 2014 and references therein). We further assume that the electron distribution lies in a region of volume V (giving an electron density n_e = N₀/V), and up-scatters the synchrotron photons up to an energy ${E}_{\mathrm{ssc}}={\gamma }_{\mathrm{max}}{m}_{e}{c}^{2}\simeq 5({\gamma }_{\mathrm{max}}/{10}^{4})$ GeV, to give a self-synchrotron Compton (SSC) component. Therefore, in our model, the X-ray emission is given by the sum of two components: the synchrotron emission powered by the propeller luminosity L_prop, and the X-rays produced by the accretion flow, either from the disk, the NS surface, or a corona surrounding the system. On the other hand, the gamma-ray emission falling in the 0.1–10 GeV range could only arise as the SSC component of the propeller emission.

The parameters of the electron distribution (α, γ_max, n_e) and the volume V of the region of acceleration can be then adjusted such that:

(i) they model the gamma-ray emission as the self-synchrotron Compton component of the same electron population that produces a good part of the X-ray emission; and

(ii) for a fixed value of the fastness ω_*, they give a total propeller luminosity (i.e, the sum of synchrotron and self-synchrotron Compton contributions) that matches the value given above by Equation (17).

As the X-ray emission is given by the sum of the contribution of the propeller and the accretion flow luminosity in that energy range, and no cutoff is observed from 0.5 to 80 keV, we cannot disentangle the relative weight of these components. At low luminosities (≲10³⁴–10³⁵ erg s⁻¹), the X-ray spectrum of the accretion flow onto compact objects is generally dominated by a power law with index Γ between 1.5 and 2.5, extending to high energies (≳100 keV; see, e.g., Reynolds et al. 2010, Armas Padilla et al. 2013a, 2013b). A similar spectral shape is also observed for the pulsed emission of accreting millisecond pulsars (see Patruno & Watts 2012 and references therein). We then describe the contribution of the accretion flow in the X-ray band as a power law with index Γ = 1.65, cut off at an energy outside the observed energy band (we arbitrarily chose 300 keV as an example), that, summed to the synchrotron and SSC component, gives an 0.3–79 keV X-ray flux equal to the one observed in that energy band. Then, we require that once the propeller contribution to the X-ray emission is fixed by our modeling of the gamma-rays, the accretion flow X-ray emission does not exceed the observed emission. A lower limit on the contribution of the accretion flow to the X-ray flux is given by the pulsed flux amplitude, $\sqrt{2}{A}_{\mathrm{rms}}{L}_{X}\simeq 6\times {10}^{32}$ erg s⁻¹, while an upper limit to the accretion flow X-ray luminosity is given by the total luminosity observed in that band (7.3 × 10³³ erg s⁻¹).

The generic idea behind the model used here to interpret the emission observed from PSR J1023+0038 in the sub-luminous disk state has been previously successfully applied to the emission of XSS J12270−4859 in the same state (P14). However, in that paper it was assumed that all of the gravitational energy given off by the matter in-flow was available to power the propeller emission and outflow, and consequently, the interplay between the disk and the propeller luminosities was not explored as done above. Considering the formalism developed in Section 3, that assumption corresponds to the case η = 0 and ξ = 1. This case is extreme and probably unlikely, since it would require an X-ray dark accretion flow.

However, the similarities between the average SED observed from these two sources, and the results obtained in the case of PSR J1023+0038 (see Section 4), indicate that our modeling also works in the case of XSS J12270−4859. This is emphasized in Figure 3, where the high-energy SED of XSS J12270−4859 is overplotted as a yellow stripe on the SED of PSR J1023+0038. A distance of 1.4 kpc was considered, equal to the value indicated by the dispersion measure of the radio pulsed signal observed by Roy et al. (2014), and at the lower bound of the 1.4–3.6 kpc range determined by De Martino et al. (2013).

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The spin period of XSS J12270−4859 is very similar to that of PSR J1023+0038 (P = 1.69 ms), and the magnetic moment can be estimated as ${\mu }_{26}=1.13\;{I}_{45}^{1/2}\;{(1+{\mathrm{sin}}^{2}\alpha )}^{-1/2}$ G cm³ from the observed spin-down rate (Roy et al. 2014). We modeled the SED of XSS J12270−4859 using parameters similar to those used for PSR J1023+0038, and ω_* = 2.5 (see Table 1), obtaining the model that is plotted as a black solid line in Figure 3.

Coti Zelati et al. (2014), Li et al. (2014), Stappers et al. (2014), and Takata et al. (2014) discussed the phenomenology observed from PSR J1023+0038 in terms of a rotation-powered pulsar active even in the presence of an accretion disk, the radio coherent pulsation being washed out by the enshrouding of the system by intra-binary material. According to these models, the relativistic wind of particles emitted by the pulsar in this state is responsible for the emitted gamma-rays.

Coti Zelati et al. (2014) and Stappers et al. (2014) proposed that the shock between the pulsar wind of particles and the mass in-flow would be the most likely region where the gamma-ray emission is generated, as this would explain the correlation between the observed increase of the gamma-ray emission and the formation of a disk in the system. On the other hand, Li et al. (2014) and Takata et al. (2014) interpreted the gamma-ray emission observed in the sub-luminous disk state through inverse Compton scattering of UV disk photons off of the cold pulsar wind. Furthermore, they assumed that the X-rays were due to synchrotron emission taking place in the shock between the pulsar wind and the plasma in-flow. Such a shock would be expected to be stronger in the sub-luminous disk state than in the pulsar state, as a larger fraction of the pulsar wind would be intercepted, thus yielding the increased X-ray emission observed.

These ideas are similar to those used for gamma-ray binaries such as LS 5039 or LS I +61 303, where wind and inter-wind shocks are also studied as possible providers of accelerated electrons (e.g., Dubus 2006; Sierpowska-Bartosik & Torres 2008).

The main problematic aspect of applying any rotation-powered scenario to the sub-luminous disk state of PSR J1023+0038 is the observation of coherent X-ray pulsations with an rms amplitude of ≈6% during the accretion disk state. Even if X-ray pulsations were also observed during the rotation-powered state, the pulsed 0.5–10 keV X-ray luminosity was 1.2 × 10³¹ erg s⁻¹ (Archibald et al. 2010), roughly one order of magnitude lower than the pulsed luminosity observed during the disk state, 2.5 × 10³² erg s⁻¹ (Archibald et al. 2014). Synchronous switching of the radio and X-ray pulsation properties has been observed from rotation-powered pulsars, and interpreted as being due to global changes to the magnetospheric conditions (Hermsen et al. 2013). However, to interpret the X-ray pulsations observed from PSR J1023+0038 in the disk state as rotation-powered, one must admit that when a disk is present, the pulsed flux increases by more than one order of magnitude with respect to the case of an unperturbed magnetosphere. This is contrary to the expectation that high-density plasma from the accretion process shorts out the electric fields that power the electron/positron acceleration in the vacuum gaps, switching off rotation-powered pulsations (see, e.g., Illarionov & Sunyaev 1975). We consider such a scenario highly unlikely. Furthermore, the X-ray pulsations observed from PSR J1023+0038 during the disk state are sinusoidal and have a non-thermal energy distribution (Archibald et al. 2014), whereas non-thermal X-ray pulsations observed from rotation-powered pulsars are typically narrow-peaked (Zavlin 2007).

In addition to X-ray pulsation, the high conversion efficiency of the spin-down power required to explain the observed radiation also disfavors a rotation-powered scenario. If a radio pulsar is switched on, the spin-down power of 4.4 × 10³⁴ erg s⁻¹ is the dominant source of energy for the system; as a matter of fact, for the radio pulsar to be active, the inner disk radius must lie beyond the light-cylinder radius R_lc (80.6 km in the case of PSR J1023+0038), and in such a case, the implied mass in-flow rate of less than ≈10⁻¹² M_⊙ yr⁻¹ (see Equation (4)) would yield an accretion luminosity of <10³³ erg s⁻¹. The sum of the average luminosity observed from PSR J1023+0038 in the 0.3–79 keV and 0.1–100 GeV energy bands amounts to ≃1.7 × 10³⁴ erg s⁻¹, a value that implies a spin-down power conversion efficiency of >40%. A similar value is already larger than the values observed from rotation-powered pulsars, which typically convert 0.1 (Possenti et al. 2002; Vink et al. 2011) and 10% (Abdo et al. 2013) of their spin-down power into observable X-rays and gamma-rays, respectively. In addition, if one considers that the SED is most likely flat (or even peaks) in the 1–10 MeV energy range where observations are not available (see Figure 1 of this paper and Figure 18 of Tendulkar et al. 2014), the total power obtained modeling the SED with two smooth components easily attains a value of the order of the spin-down power or larger (indeed, the power of most of the models listed in Table 1 exceeds such a threshold). Note that a total luminosity significantly exceeding the spin-down power would directly exclude a rotation-powered scenario.

Furthermore, the strong flickering observed in X-rays makes the case for the spin-down power as the lone source of energy even more unlikely. The peak observed X-ray luminosity is of the order of ≈2 × 10³⁴ erg s⁻¹ (Patruno et al. 2014; Tendulkar et al. 2014), i.e., roughly 40% of the spin-down power, alone. No information is available about the correlation between the X-ray and the gamma-ray luminosity on short timescales, but it is clear that unless they are strictly anti-correlated, this likely implies that the limit set by the spin-down power is exceeded at the peak of the flares. Furthermore, the power emitted through synchrotron emission only depends on the acceleration efficiency, the solid angle under which the shock is seen by the isotropic pulsar wind, and the bow-shock geometry (Arons & Tavani 1993). It seems highly unlikely that these could produce the variability of the X-ray emission by up to two orders of magnitude observed over timescales of a few tens of seconds from PSR J1023+0038.

The most immediate interpretation of the coherent X-ray pulsations observed from PSR J1023+0038 is then in terms of an accretion of at least part of the disk in-flow close to the NS magnetic poles. However, in Section 3 we already showed that assuming that the observed average X-ray luminosity L_X is entirely due to accretion onto the NS surface, the implied mass accretion rate onto the NS surface is ${\dot{M}}_{\mathrm{NS}}\lt 6\times {10}^{-13}$ M_⊙ yr⁻¹ (see Equation (5)). According to Equation (4), and considering the value of the dipole magnetic moment of PSR J1023+0038 measured from the spin-down rate, at such a mass accretion rate the disk would be truncated beyond the light-cylinder radius. Such a large value clearly violates the criterion for accretion onto the NS surface to proceed. In order to keep the accretion disk radius closer to the corotation radius, we thus had to assume that the disk mass accretion rate was much larger than the rate at which mass is effectively accreted onto the NS, and that the excess disk mass is ejected by the system.

In addition, the simultaneous observation of a bright gamma-ray emission would be unexplained by a fully accreting scenario, considering that the transitonal pulsars PSR J1023+0038 and XSS J12270−4859 are the only LMXBs from which a Fermi gamma-ray counterpart could be securely identified among a population of >200 known accreting LMXBs, and this is unlikely to be the result of a selection effect. The 5σ sensitivity flux level above 100 MeV of the 3FGL Fermi catalog (The Fermi-LAT Collaboration 2015) for sources at high (low) galactic latitude⁵ is in fact 2 × 10⁻⁹ cm⁻² s⁻¹ (10⁻⁸ cm⁻² s⁻¹), assuming a power-law spectrum with index equal to 2. This corresponds to ≃1.5 × 10⁻¹² erg cm⁻² s⁻¹ (≃7.5 × 10⁻¹² erg cm⁻² s⁻¹). A steady 0.1–100 GeV flux like the one observed from PSR J1023+0038 in the sub-luminous disk state, $F=(4.6\pm 0.6)\times {10}^{-11}$ erg cm⁻² s⁻¹ (see Takata et al. 2014, and Section 2), would have been observed by Fermi/LAT up to ≃7 kpc (3.3 kpc if the source were at a lower latitude). On the other hand, a high-latitude low-mass X-ray binary like Cen X-4 is instead located at a distance similar to PSR J1023+0038, 1.2 kpc, and would have then been easily seen by Fermi/LAT, yet it has not been detected so far.

A scenario based on a propelling NS with an accretion rate of a few × 10⁻¹¹ M_☉ yr⁻¹ naturally reproduces the bolometric luminosity between few × 10³⁴ erg s⁻¹, observed from transitional pulsars like PSR J1023+0038 and XSS J12270−4859 in the disk sub-luminous state (see Figure 1 of this paper and Figure 2 of Campana et al. 1998). We showed that we could reproduce the gamma-ray part of the SED as being due to the self-synchrotron Compton emission that originated at the turbulent boundary between a propelling magnetosphere and the disk in-flow, assuming that there is a region where electrons can be accelerated to relativistic energies. The X-ray emission is instead due to the sum of the synchrotron emission that originated from the same region and the luminosity emitted by the accretion flow. For the accretion flow luminosity not to exceed the observed X-ray luminosity, an X-ray disk radiative efficiency of less than 20% is requested.

The observed SED was reproduced by our modeling considering values of the fastness between 1.5 and 2.5, and assuming that 99% of the disk flow is ejected by the NS, and that 15% of the gravitational energy advected in the disk is available to power the propeller radiative emission. For the considered values of the fastness, the disk–magnetospheric boundary is highly magnetized (≈10⁶ G; resulting from the dipolar contribution from the pulsar) to produce a synchrotron emission cutoff at energies of about 1–10 MeV. If the acceleration region is small enough it can then give a sizable SSC contribution, enough to explain the observed gamma-ray emission. Our modeling indicates that a region with a volume of V ≲ 10¹⁵ cm³ is needed, corresponding to a sphere with a radius equal to ≲1 km. Such a relatively small size could be attained if the acceleration process takes place, e.g., in filaments along the magnetic field lines at localized spots of the disk–magnetospheric boundary. As the volume of the emitting region decreases with decreasing ω_*, we find the higher values of ω_* considered in this work (i.e., between 2 and 2.5) more likely. For similar values of ω_*, 3D MHD simulations performed by Lii et al. (2014) showed that accretion and ejection of matter can coexist, with most of the matter being flung out by the fast rotating magnetosphere.

However, the impossibility of observationally separating contributions of the accretion flow and the propeller just at the X-ray domain is partially limiting to the model testing. This gives a larger phase space of plausible parameters for the accretion flow component, which can accommodate several different values of the fastness (see Table 1). This translates into a range of possible mass in-flow rates, between 1 and 3 × 10⁻¹¹ M_⊙ yr⁻¹. It is also true that the more direct, testable model predictions, happen in a range of energies with no sensitive coverage (at the tens of MeV regime), or at timescales for which Fermi/LAT is not sensitive enough to track them (e.g., searches for correlations of gamma-ray and X-ray fluxes at the 100 s timescales cannot proceed). We also note that this model predicts no detectable TeV counterparts, which can be proven by extrapolating the predicted (or fitted) gamma-ray spectra to this domain and comparing them with the sensitivities of current or foreseen instruments (Actis et al. 2012).

A Fermi process is a possible mechanism for accelerating electrons in the magnetized, shocked environment expected at the boundary between the disk and a propelling magnetosphere (Bednarek 2009; Papitto et al. 2012; Torres et al. 2012). In P14, we assumed that a first-order process injected energy in the electron distribution at a rate

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{acc}}=1.4\times {10}^{5}{\xi }_{0.01}\ {\bar{B}}_{6}{{\rm{erg}}\;{\rm{s}}}^{-1},\end{eqnarray} \tag{ 22 }$$

where ξ_0.01 is the acceleration parameter in units of 0.01, and ${\bar{B}}_{6}$ is the strength of the magnetic field at the interface $\bar{B}$, in units of 10⁶ G. In P14 we showed that the most efficient radiative processes at the boundary between an accretion disk and the propelling magnetosphere of a millisecond pulsar are synchrotron interaction of the accelerated electrons with the NS field lines and Compton up-scattering of the synchrotron photons by the same population of relativistic charges. Synchrotron losses proceed at a rate

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{syn}}=1.1\times {10}^{5}\;{\bar{B}}_{6}^{2}\;{\gamma }_{4}^{2}\;{\text{erg s}}^{-1},\end{eqnarray} \tag{ 23 }$$

where γ₄ is the maximum electron energy γ_max in units of 10⁴, while we assume SSC losses to be ${{\ell }}_{\mathrm{SSC}}=(f-1){{\ell }}_{\mathrm{syn}}$, where f  is implicitely defined as the ratio between the total luminosity (${{\ell }}_{\mathrm{syn}}+{{\ell }}_{\mathrm{SSC}}$) and the synchrotron luminosity ${{\ell }}_{\mathrm{syn}}$. Equating these radiative losses to the energy input of the Fermi process of electron acceleration (Equation (22)) yields the maximum electron energy 

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max}}=8.2\times {10}^{3}\ {f}_{2}^{-1/2}{\xi }_{0.01}^{1/2}\ {\bar{B}}_{6}^{-1/2},\end{eqnarray} \tag{ 24 }$$

where f₂ is the ratio between the total (synchrotron+SSC) and the synchrotron luminosity f, in units of 2. The observed high-energy cutoff of the observed SED indicates γ_max = 10⁴, and the values of f and $\bar{B}$ of our modeling (see Table 1) indicate that the previous relation is satisfied for an acceleration parameter ξ ranging from 0.03 to 0.1.

Reconnection of magnetic lines twisted in the turbulent region could also provide accelerated particles. In the case of a white dwarf propeller, such possibilities were studied by Meintjes & de Jager (2000) for the case of AE Aqr.

According to our model, a relatively strong magnetic torque is needed to power a propeller emission of $\simeq 2.5\times {10}^{34}$ erg s⁻¹, i.e., of the order of that observed in X-rays and gamma-rays from a system like PSR J1023+0038 in the sub-luminous disk state. For ω_* = 2.5, the torque expressed by Equation (17) corresponds to an expected spin-down rate, $\dot{\nu }\;={N}_{\mathrm{mag}}/2\pi I\simeq -4\times {10}^{-15}$ Hz s⁻¹, where an NS moment of inertia I = 10⁴⁵ g cm² was assumed. Lower values of the fastness give a larger expected spin-down rate. If the system is in a propeller state, a $\dot{\nu }$ larger by more than a factor of 2 with respect to that observed when the system is observed as a radio pulsar (${\dot{\nu }}_{\mathrm{dip}}=-1.9\times {10}^{-15}$ Hz s⁻¹, Archibald et al. 2013) is then expected. The spin evolution during the disk state can be estimated by following the variations of the spin frequency measured from X-ray pulsations emitted in such a state and/or by a comparison with the frequency of radio pulsations that will be observed when the rotation-powered pulsar will be back on. Such a measure will then allow us to estimate the torque acting onto the NS and test our assumption that the system lies in a propeller state when it has a disk.

The variability of the X-ray emission over timescales of few tens of seconds is a characteristic feature of millisecond pulsars in the disk sub-luminous state. In the context of the propeller model we propose, it can be attributed to variations of the mass in-flow rate (see Equation (17)) and the related response of the fastness and of the location of the inner disk radius (Equation (4)). The observed variability timescales are indeed compatible with the viscous timescales in the inner parts of an accretion disk, as noted by Patruno et al. (2014). However, the lack of information on the possible correlation between X-ray and gamma-ray emission on timescales of few hundreds of seconds prevents us from modeling the variability of the SED, as it was done for the average SED. Linares et al. (2014) and Patruno et al. (2014) argued that at the lowest X-ray luminosity observed in the case of M28I and PSR J1023+0038, respectively, ${L}_{X}\simeq 5\times {10}^{32}$ erg s⁻¹, the inner disk radius could expand beyond the corotation surface and a radio pulsar turn-on. Even in this case, the same energy budget-based considerations made in Section 5.2 suggest that we should exclude a scenario in which the observed gamma-rays are only due to intervals during which a rotation-powered pulsar is turned on. Tendulkar et al. (2014) showed that 1023 spends less than 25% of the time in the low/dipping state; if gamma-rays are only produced in this state a 0.1–100 GeV luminosity equal to four times the observed one (i.e., $\approx 4\times {10}^{34}$ erg s⁻¹) would be implied. This would require a conversion efficiency of spin-down power into gamma-rays of 90%, not taking into account the emission not falling into the LAT energy band.

Also the bright, flat-spectrum radio emission observed from transitional millisecond pulsars in the sub-luminous disk state (Hill et al. 2011; Deller et al. 2014) can be taken as an indication of synchrotron emission originating from an outflow from the system. As we already noted in P14, the properties of the emitting region where we assumed that the high-energy emission originate are such that synchrotron emission would be self-absorbed. The observed radio emission should then be produced by a wider, optically thinner region such as a compact jet.

The necessity that a pulsar in a LMXB passes through a propeller regime when the mass accretion rate decreases was already identified decades ago (Illarionov & Sunyaev 1975) and reproduced by magneto-hydrodynamical simulation (Romanova et al. 2014 and references therein). However, observational evidence has been limited and indirect, among which the rapid decrease of the X-ray emission at the end of X-ray outbursts of Aql X-1 (Campana et al. 1998, 2014), accompanied by a hardening of the X-ray spectral shape (Zhang et al. 1998), is probably the most remarkable. Transitional  millisecond pulsars have proven to be exceptional laboratories for studying not only the evolutionary link between radio and X-ray millisecond pulsars, but also the centrifugal inhibition of accretion. During its 2013 outburst IGR J18245−2452 showed marked flux and spectral variability that were interpreted by Ferrigno et al. (2014) as products of the onset of propeller reduction of the mass in-flow. Furthermore, we showed how the sub-luminous disk state of PSR J1023+0038 can be naturally interpreted with a propeller scenario, similar to the case of XSS J12270−4859 (Papitto et al. 2014). Future observations will be fundamental for detecting direct evidence of out-flowing plasma, such as spectral lines and a possible correlation between the radio and X-ray emission. Also, such observations will be used to extend knowledge of the SED and possibly detect an excess of emission with respect to the spin-down power, which would rule out a rotation-powered pulsar interpretation.