High-energy Emissions from the Pulsar/Be Binary System PSR J2032+4127/MT91 213

The relativistic pulsar wind, which may have a bulk Lorentz factor of ${{\rm{\Gamma }}}_{\mathrm{PW},0}\sim {10}^{3-6}$, interacts with the stellar wind and forms a cone-shaped shock separating the pulsar and the companion star. The radial distance to and its opening angle of the cone-shaped shock are determined by the ratio of the two winds (e.g., Canto et al. 1996 and reference therein),

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{L}_{\mathrm{sd}}}{{\dot{M}}_{w}{v}_{w}c},\end{eqnarray} \tag{ 3 }$$

where we assume that the pulsar wind carries the spin-down power L_sd of the pulsar, ${\dot{M}}_{w}$ and v_w is the mass loss rate and wind velocity of the companion star, respectively. The wind velocity of the stellar wind at a distance R from the stellar surface may be expressed by (Waters et al. 1988)

$$\begin{eqnarray}&&{v}_{w}(R)={v}_{0}+({v}_{\infty }-{v}_{0})(1-{R}_{* }/R).\end{eqnarray} \tag{ 4 }$$

where ${v}_{0}\sim \sqrt{3{{kT}}_{s}/{m}_{p}}\sim 28$ km with ${T}_{s}\,\sim$ 30,000 K being the temperature of the B star. The terminal velocity is estimated from

$$\begin{eqnarray*}&&{v}_{\infty }\sim \sqrt{\displaystyle \frac{2{{GM}}_{* }}{{R}_{* }}}\sim 8\times {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}\ {\left(\displaystyle \frac{{M}_{* }}{15{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{R}_{* }}{10{R}_{\odot }}\right)}^{-1/2}\end{eqnarray*}$$

with M_* and R_* being the mass and radius of the B star, respectively. Since the shock is far away from the surface of the B star, we may assume ${v}_{w}({R}_{s})={v}_{\infty }$.

With ${L}_{\mathrm{sd}}=1.7\times {10}^{35}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ of this pulsar (Lyne et al. 2015) and a typical mass loss rate ${\dot{M}}_{w}\,\sim {10}^{-9}\mbox{--}{10}^{-7}{M}_{\odot }\,{\mathrm{yr}}^{-1}$ of an O-type or B-type star (e.g., Snow 1981; Smith 2006), the momentum ratio of this system will be

$$\begin{eqnarray}\eta & \sim & 0.085\left(\displaystyle \frac{{L}_{\mathrm{sd}}}{1.7\cdot {10}^{35}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{\dot{M}}_{w}}{{10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{{v}_{w}}{{10}^{-8}\mathrm{cm}{{\rm{s}}}^{-1}}\right)}^{-1},\end{eqnarray} \tag{ 5 }$$

indicating that the stellar wind is stronger than the pulsar wind with a reasonable mass loss rate of the former. The distance to the apex of the shock cone, r_s, and the opening angle of the shock, ${\theta }_{s}$, measured by the pulsar are calculated from (Canto et al. 1996)

$$\begin{eqnarray}&&{r}_{s}(\phi )=\displaystyle \frac{{\eta }^{1/2}}{1+{\eta }^{1/2}}a(\phi )\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{\theta }_{s}(\eta )-\tan {\theta }_{s}(\eta )=\displaystyle \frac{\pi }{1-\eta },\end{eqnarray} \tag{ 7 }$$

respectively, where ϕ represents the orbital phase and $a(\phi )$ is the separation between the two stars. Figure 5 summarizes the evolution of the distance (left panel) and the magnetic field strength (right panel) at the shock apex along the orbit between −4000 days and +4000 days from periastron (we did not plot the results for the other orbital phase, since we are interested in the emission processes around periastron). In the figure, we used $\sigma ({r}_{s})=0.1$ to calculate the magnetic field strength. As the figure shows, the shock distance and the magnetic field at the shock change by more than factor of ten along the orbit. Since the pulsar orbit velocity (${v}_{p}\sim {10}^{6-7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$) is at least about a factor of ten slower than the stellar wind velocity (${v}_{w}\sim {10}^{8}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$), we assume that the shape of the shock cone is axially symmetric about the axis connecting the two stars.

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Since the Be-type star, MT91 213, forms a dense equatorial disk outflow, the pulsar may interact with the Be disk at the periastron passage, as in the case with the gamma-ray binary PSR B1259–63/LS2883 system. The decretion disk model has been explored to describe the structure of the Be disk (Lee et al. 1991; Carciofi & Bjorkman 2006; Okazaki et al. 2011), and it implies a disk mass density (${\rho }_{d}$) and a scale height (H) described by

$$\begin{eqnarray}&&{\rho }_{d}(R,z)={\rho }_{0}{\left(\displaystyle \frac{{R}_{* }}{R}\right)}^{n}\exp \left(-\displaystyle \frac{{z}^{2}}{2{H}^{2}}\right),\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&H(R)={H}_{0}{\left(\displaystyle \frac{R}{{R}_{* }}\right)}^{\beta },\end{eqnarray} \tag{ 9 }$$

respectively, where $n\sim 3\mbox{--}3.5$ and $\beta \sim 1\mbox{--}1.5$ (n = 3.5 and $\beta =1.5$ for an isothermal disk). In addition, we apply ${H}_{0}\sim 0.02{R}_{* }$ and the base density ${\rho }_{0}\sim {10}^{-11}\mbox{--}{10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

Takata et al. (2012) studied the orbital modulations of the X-ray/TeV emission from the PSR B1259–63/LS2883 system. They argued that with the larger base density (${\rho }_{0}\sim {10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$) for the Be disk, the pulsar wind creates a cavity in the disk gas and this causes a significant increase in the conversion efficiency from pulsar spin-down power to the shock-accelerated particle energy. This explains the double-peak structure of the X-ray light curves of PSR B1258-63/LS 2883. The pulsar/Be disk interaction with a smaller base density (${\rho }_{0}\lt {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$) causes no cavity in the pulsar wind and hence no enhancement in the emission.

With the elongated orbit and the long orbital period of PSR J2032+4127, a strong pulsar/Be disk interaction will be possible only at periastron passage. The radius of the shock from the pulsar may be obtained from the pressure balance condition,

$$\begin{eqnarray}&&{r}_{s}={\left(\displaystyle \frac{{L}_{\mathrm{sd}}}{2\pi {\rho }_{d}{v}_{r}^{2}c}\right)}^{1/2},\end{eqnarray} \tag{ 10 }$$

where v_r is the relative velocity between the pulsar and the disk rotation. If the pulsar interacts with the Be disk at periastron, where the separation of the two stars is $a\sim 1$ au, the mass density and scale height of the disk at the pulsar are $\rho (a)\sim 2\,\times \,{10}^{-5}{\rho }_{0}$ and $H(a)\sim {10}^{12}\,\mathrm{cm}$, where we used n = 3.5, $\beta =1.5$, ${H}_{0}=0.02{R}_{* }$ and ${R}_{* }=10{R}_{\odot }$. The orbital velocity of the pulsar and circular velocity of the disk are ${v}_{p}\sim {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ and ${v}_{d,K}(a)\,={{v}_{d,K}({R}_{* })({R}_{* }/1\mathrm{au})}^{1/2}\sim {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$, respectively, where we used ${v}_{d,K}({R}_{* })=5\,\times \,{10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. The radial velocity of the disk matter, ${v}_{d,r}\sim 0.1{c}_{s}\sim {10}^{5}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ with ${c}_{s}\sim 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ being the sound speed (Okazaki et al. 2011), is much slower than the pulsar motion and the disk's circular motion. Hence, by assuming the relative velocity ${v}_{r}\sim \sqrt{{v}_{p}^{2}+{v}_{d,K}^{2}}$, the shock distance calculated from Equation (10) is ${r}_{s}\,\sim {10}^{12}\,\mathrm{cm}{({\rho }_{0}/{10}^{-10}{\rm{g}}{\mathrm{cm}}^{-3})}^{-1/2}$ at periastron. A pulsar/Be disk interaction may cause a cavity in the wind around the pulsar, provided ${r}_{s}/H(a)\leqslant 1$, which yields ${\rho }_{0}\geqslant {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

Figure 6 summarizes the ratio of the shock distance given by Equation (10) and the scale height of the disk, provided the pulsar interacts with the disk at the given orbital phase in the horizontal axis. The figure summarizes the shock distance for the epoch between −100 days and +100 days from periastron. The different lines show the case for a base density of ${\rho }_{0}={10}^{-9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (solid line), ${10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (dashed line) and ${10}^{-11}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (dotted line), respectively. We find from the figure that a cavity will form (${r}_{s}/H\lt 1$) at the periastron passage if ${\rho }_{0}\gt {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$. If the base density is less than ${\rho }_{0}\lt {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$, on the other hand, the pulsar wind will strip off an outer part of the Be disk, truncating it at a radius smaller than the pulsar orbit, and the pulsar wind/Be disk interactions will not affect the observed emission.

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We calculate the emissions from the shock due to the interaction between the pulsar wind and stellar wind, as follows. The magnetic field at the shock radius but before the shock is calculated from

$$\begin{eqnarray}&&{B}_{1}=\sqrt{\displaystyle \frac{{L}_{\mathrm{sd}}{\sigma }_{0}}{{r}_{s}^{2}c(1+{\sigma }_{0})}},\end{eqnarray} \tag{ 11 }$$

where we defined ${\sigma }_{0}\equiv \sigma ({r}_{s})$. At the shock, the kinetic energy of the pulsar wind is converted into the internal energy of the shocked pulsar wind. Applying a jump condition of a perpendicular magnetohydrodynamic shock, we calculate the velocity ${v}_{{pw},2}$, the magnetic field B₂, and the gas pressure P₂ of the shocked pulsar wind at the shock (Kennel & Coroniti 1984). For the particle kinetic energy dominating the un-shocked flow, that is, for the low ${\sigma }_{0}$ regime, we obtain

$$\begin{eqnarray}&&{v}_{{pw},2}\sim \displaystyle \frac{c}{3}\sqrt{\displaystyle \frac{1+9{\sigma }_{0}}{1+{\sigma }_{0}}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{B}_{2}\sim 3{B}_{1}(1-4{\sigma }_{0})\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&{P}_{2}\sim \displaystyle \frac{2(1-2{\sigma }_{0})}{3{(1+9{\sigma }_{0})}^{1/2}(1+{\sigma }_{0})}\displaystyle \frac{{L}_{\mathrm{sd}}}{4\pi {r}_{s}^{2}c}.\end{eqnarray} \tag{ 14 }$$

We assume that the post-shock pulsar wind flows along the shock surface. Along the downstream flow, we assume a conservation of the magnetic flux $B(r)={r}_{s}{B}_{2}/r$. For the spherical symmetric flow, the bulk velocity decreases with distance from the shock (Kennel & Coroniti 1984). However, numerical simulations imply that the post-shock bulk flow for binary systems does not simply decrease but increases with distance from the shock, because of a rapid expansion of the flow in the downstream region (Bogovalov et al. 2008). Furthermore, the high-energy emissions take place in the vicinity of the shock radius. In this study, therefore, we assume the velocity field of the post-shock pulsar wind with ${v}_{\mathrm{pw}}(r)\,=$ constant.

We assume that the electrons and positrons in the pulsar wind are accelerated by the shock and form a power-law distribution over several decays in energy;

$$\begin{eqnarray}&&{f}_{0}(\gamma )={K}_{0}{\gamma }^{-p},\,\,{{\rm{\Gamma }}}_{\mathrm{PW},0}\leqslant \gamma \leqslant {\gamma }_{\max },\end{eqnarray} \tag{ 15 }$$

where we assume that the minimum Lorentz factor of the accelerated particles is equal to the average Lorentz factor of the particles forming the cold-relativistic pulsar wind (Section 3.4). We assume the maximum Lorentz factor by balancing between the acceleration timescale $\sim \,{\gamma }_{\max }{m}_{e}c/(\xi {{eB}}_{2})$ and the synchrotron loss timescale $\sim \,9{m}_{e}^{3}{c}^{5}/(4{e}^{4}{B}_{2}^{2}{\gamma }_{\max })$, yielding ${\gamma }_{\max }\,={[9\xi {m}_{e}^{2}{c}^{4}/(4{e}^{3}{B}_{2})]}^{1/2}$, where ξ represents the efficiency of the acceleration, which will be $\xi \leqslant 1$. The efficiency of the acceleration in gamma-ray binaries has been explained from the observed spectra of very high-energy emissions; for example, a high efficiency is required to explain the TeV emission of LS 5039 (Zabalza et al. 2013). For PSR J2032+4127/MT91 213, it is difficult to discuss efficiency, since TeV emission from this source has not yet been detected. In the calculations, we found that the predicted X-ray flux and 0.1–10 TeV flux are insensitive to the efficiency if $\xi \gt {10}^{-4}$. In this study, therefore, we present the results with $\xi =1$. Future TeV observations at the periastron passage may provide addition information constraining the efficiency.

For the injected particles at the shock, we assume p = 2 for the power-law index of the distribution. Since the particle energy density $\epsilon$ is related with the pressure P₂ as ${P}_{2}=\epsilon /3$, the normalization factor K₀ of Equation (15) is calculated from the relation ${P}_{2}={m}_{e}{c}^{2}{\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}\gamma {f}_{0}(\gamma )d\gamma /3$.

We solve the evolution of the Lorentz factor after the shock with a simple one-dimensional treatment, that is,

$$\begin{eqnarray}&&\displaystyle \frac{d{\gamma }_{e}}{{dr}}=\displaystyle \frac{1}{{v}_{\mathrm{pw}}}\left[{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{ad}}+{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{syn}}+{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{ICS}}\right].\end{eqnarray} \tag{ 16 }$$

We assume the adiabatic loss given by

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{ad}}=\displaystyle \frac{\gamma }{3n}\displaystyle \frac{{dn}}{{dt}}=-\displaystyle \frac{2\gamma {v}_{\mathrm{pm}}}{3r},\end{eqnarray*}$$

where n ∝ r⁻² is the particle number density. The synchrotron loss is given as

$$\begin{eqnarray}&&{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{syn}}=-\displaystyle \frac{4{e}^{2}{B}^{2}{\gamma }^{2}}{9{m}_{e}^{3}{c}^{5}}.\end{eqnarray} \tag{ 17 }$$

We calculate the ICS energy loss rate using

$$\begin{eqnarray}&&{\left(\displaystyle \frac{d\gamma }{{dt}}\right)}_{\mathrm{ICS}}=-\int \int (E-{E}_{s})\displaystyle \frac{{\sigma }_{\mathrm{ICS}}c}{{m}_{e}{c}^{2}{E}_{s}}\displaystyle \frac{{{dN}}_{s}}{{{dE}}_{s}}{{dE}}_{s}{dE},\end{eqnarray} \tag{ 18 }$$

where ${{dN}}_{s}/{{dE}}_{s}$ is the stellar photon field distribution and ${\sigma }_{\mathrm{ICS}}$ is the cross-section with the isotropic photon field. For the soft-photon field, we consider the blackbody radiation from the B star and apply the Planck function with temperature T_s = 30,000 K, which is the case for the companion star of PSR B1259–63 (Negueruela et al. 2011). Figure 7 summarizes the timescale of the cooling at periastron and at the apex of the shock cone.

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The TeV gamma-ray produced by the ICS process may be converted into an electron–positron pair through the photon–photon pair-creation process with the stellar photon. The cross section of this process is calculated from

$$\begin{eqnarray}&&{\sigma }_{\gamma \gamma }({E}_{s},{E}_{\gamma })\\ &&\quad =\displaystyle \frac{3}{16}{\sigma }_{T}(1-{v}^{2})\left[(3-{v}^{4}\mathrm{ln}\displaystyle \frac{1+v}{1-v}-2v(2-{v}^{2})\right],\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray*}&&v({E}_{s},{E}_{\gamma })=\sqrt{1-\displaystyle \frac{2}{1-\cos {\theta }_{\gamma \gamma }}\displaystyle \frac{{({m}_{e}{c}^{2})}^{2}}{{E}_{s}{E}_{\gamma }}},\end{eqnarray*}$$

where ${\theta }_{\gamma \gamma }$ is the collision angle between the soft photon and gamma-ray, and is a function of position. The optical depth is calculated from $\tau ({E}_{\gamma })=\int \int {\sigma }_{\gamma \gamma }({E}_{s},{E}_{\gamma }){n}_{s}({E}_{s}){{dE}}_{s}d{\ell }$, where ${n}_{s}({E}_{s})$ represents the energy distribution of the stellar radiation number density (see Table 1 for the parameters of the companion star). Figure 8 summarizes the optical depth of the high-energy photons emitted from the shock apex ($\eta =0.085$); different lines in the figure represent difference system inclination angles ${\theta }_{E}$ and different positions of the pulsars along the orbit. We find in Figure 8 that the optical depth of the 0.1–1TeV photons emitted around periastron and the SUPC exceeds unity, indicating the TeV photons emitted during the periastron passage are significantly absorbed by the stellar photons.

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Table 1.  Parameters of the Binary Systems

PSR/Companion	P (s)	L35	Po (years)	e	a (lt-s)	T*	R*
J2032+4127/MT91 213	0.143	1.7	25–50	0.96	9022	30,000 K	10R⊙
B1259–63/LS2883	0.048	8	3.4	0.83	1296	∼30,000 K	$\sim 9{R}_{\odot }$

Note.  P: spin period of the pulsars. L₃₅: spin-down luminosity in units of ${10}^{35}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. P_o: orbital period. e: eccentricity. a: projected semimajor axis. T_*: surface temperature of companion star. R_*: radius of B star. T_* and R_* of MT91 213 represent the values used in the calculation. References: Ho et al. (2017) for J2032+3127 and Negueruela et al. (2011) for B1259–63.

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The pulsar binaries will be a laboratory to search for evidence of the high-energy gamma-rays produced by the ICS process of the cold-relativistic pulsar wind (Cerutti et al. 2008). In this study, we assume the pulsar wind is isotropic and is formed near the light cylinder of the pulsar, which is ${r}_{\mathrm{lc}}\sim 6.8\times {10}^{8}$ cm for PSR J2032+4127. Since there is a theoretical uncertainty in the particle distribution of the cold-relativistic pulsar wind, we explore the process with a relativistic Maxwell distribution of the form,

$$\begin{eqnarray}&&f({{\rm{\Gamma }}}_{\mathrm{PW}})\propto {{\rm{\Gamma }}}_{\mathrm{PW}}^{2}\exp \left(-\displaystyle \frac{3{{\rm{\Gamma }}}_{\mathrm{PW}}}{{{\rm{\Gamma }}}_{\mathrm{PW},0}}\right),\end{eqnarray} \tag{ 20 }$$

which provides an average Lorentz factor of $\sim {{\rm{\Gamma }}}_{\mathrm{PW},0}$. The normalization is calculated from the condition that

$$\begin{eqnarray}&&{m}_{e}{c}^{2}{\int }_{0}^{\infty }{{\rm{\Gamma }}}_{\mathrm{PW}}f({{\rm{\Gamma }}}_{\mathrm{PW}})d{{\rm{\Gamma }}}_{\mathrm{PW}}=\displaystyle \frac{{L}_{\mathrm{sd}}}{4\pi {r}^{2}c(1+\sigma )}.\end{eqnarray} \tag{ 21 }$$

Due to the unknown energy conversion process from magnetic energy to particle energy, the magnetization parameter will evolve with radial distance from the pulsar. From the energy conservation, the average Lorentz factor of the cold-relativistic pulsar wind just before the shock may be described as ${{\rm{\Gamma }}}_{\mathrm{PW},0}({r}_{s})={{\rm{\Gamma }}}_{L}(1+{\sigma }_{L})/[1+{\sigma }_{0}]$, which provides ${{\rm{\Gamma }}}_{\mathrm{PW},0}({r}_{s})\sim {{\rm{\Gamma }}}_{L}{\sigma }_{L}$ in the limit of ${\sigma }_{L}\gg 1$ and ${\sigma }_{0}\ll 1$, where ${{\rm{\Gamma }}}_{L}$ and ${\sigma }_{L}$ are the Lorentz factor and the magnetization parameter at the light cylinder, respectively. The pairs created inside the light cylinder lose their momentum perpendicular to the magnetic field line via synchrotron radiation, and they will eventually escape from the light cylinder with a Lorentz factor of ${{\rm{\Gamma }}}_{L}\sim 10$. Hence, the magnetization parameter at the light cylinder may be estimated from

$$\begin{eqnarray}&&{\sigma }_{L}=\displaystyle \frac{{B}_{L}^{2}}{4\pi {{\rm{\Gamma }}}_{L}\kappa {n}_{{GJ},L}{m}_{e}{c}^{2}}\sim 9\times {10}^{2}\,{\left(\displaystyle \frac{{{\rm{\Gamma }}}_{L}}{10}\right)}^{-1}{\left(\displaystyle \frac{\kappa }{{10}^{5}}\right)}^{-1}\end{eqnarray} \tag{ 22 }$$

where κ is the multiplicity and we applied $P=0.143{\rm{s}}$ and ${B}_{L}=4.3\times {10}^{3}{\rm{G}}$ of PSR J2032+4127. From observations of pulsar wind nebulae, the multiplicity is expected to be $\kappa \sim {10}^{5-6}$ for Crab-like young pulsars (Tanaka & Takahara 2010) and $\kappa \sim {10}^{2-3}$ for Vela-like pulsars (Sefako & de Jager 2003). Theoretical studies have implied a  multiplicity $\kappa \sim {10}^{2-5}$ (Hibschman & Arons 2001; Timokhin & Harding 2015). Hence we assume the typical value of the Lorentz factor of the pulsar wind is in the range of ${{\rm{\Gamma }}}_{\mathrm{PW},0}\sim {{\rm{\Gamma }}}_{L}{\sigma }_{L}\sim {10}^{3-6}$.

The power per unit energy per unit solid angle of the ICS process with an anisotropic soft-photon field may be calculated from

$$\begin{eqnarray}&&\displaystyle \frac{{{dP}}_{\mathrm{ICS}}}{d{\rm{\Omega }}}={{ \mathcal D }}_{\mathrm{ICS}}\int (1-\beta \cos {\theta }_{0}){I}_{b}/h\displaystyle \frac{d{\sigma }_{\mathrm{ICS}}^{\prime }}{d{\rm{\Omega }}^{\prime} }d{{\rm{\Omega }}}_{* },\end{eqnarray} \tag{ 23 }$$

where $d{{\rm{\Omega }}}_{* }$ is the solid angle of the sky covered by the B star measured from the emission region, ${{ \mathcal D }}_{\mathrm{ICS}}={{\rm{\Gamma }}}^{-1}{(1-\beta \cos {\theta }_{1})}^{-1}$ with ${\theta }_{1}$ (or ${\theta }_{0}$) describing the angle between the direction of the particle motion and the propagating direction of the scattered photons (or background photons). In addition, I_b is the Planck function of the stellar photon field, and $d{\sigma }_{\mathrm{ICS}}^{\prime }/d{\rm{\Omega }}^{\prime}$ is the differential Klein–Nishina cross section.

Figure 9 shows the expected light curves (left panel) and spectra at periastron (right panel) for the different system inclination angles relative to the Earth viewing angle (${\theta }_{E}$) and the average Lorentz factor of the pulsar wind ${{\rm{\Gamma }}}_{\mathrm{PW},0}$, respectively. As the left panel indicates, the calculated emissions for a larger viewing angle are suppressed at around −150 days from periastron. This is because the INFC occurs at ∼−150 days from periastron (Figure 2), and because around the INFC the ICS process between a soft photon from the B star and a pulsar wind particle moving toward the Earth is a tail-on collision process, which reduces the ICS efficiency. Since the SUPC is $\sim +10$ days after periastron, the calculated flux measured from the Earth becomes maximum around, but after, periastron. As the right panel shows, the energy flux in the GeV energy bands is $\sim {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ for ${{\rm{\Gamma }}}_{\mathrm{PW},0}\sim {10}^{4-5}$, which is close to the observed flux level ($\sim 2\times {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$) of the pulsed emission from Fermi-LAT. Hence, we expect that if ${{\rm{\Gamma }}}_{\mathrm{PW},0}\sim {10}^{4-5}$, Fermi-LAT will observe an increase in the GeV flux during the next periastron passage.

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As described in the previous section, Swift observed a rapid increase in the emissions in the 0.3–10 keV energy bands after ∼MJD56250 (see Figure 3). These X-ray emissions are probably a result of the pulsar wind/stellar wind interaction, since the pulsar/Be disk interaction will be important only around periastron (say, between −100 days and +100 days), as discussed in Section 3.2. In this section, therefore, we will discuss the X-ray and TeV emissions from the shock caused by the pulsar wind and stellar wind interaction, and will examine the pulsar wind properties by comparing the model results with the observed X-ray light curve.

4.1.1. Constant σ

First, we consider the case where the magnetization parameter is constant with the radial distance from the pulsar, namely, the magnetization parameter at the shock does not depend on the shock distance. We also ignore the Doppler boosting effect caused by the finite velocity of the shocked pulsar wind, which will be discussed in Section 4.1.2. Figure 10 summarizes the calculated flux and orbital modulation in the 0.3–10 keV energy bands (left panel) and in the 0.1–10 TeV energy bands (right panel); the solid, dashed, and dotted lines are for ${\sigma }_{0}=0.5$, 0.1, and 0.01, respectively. In the calculation, we assumed $\eta =0.085$ (that is, ${\dot{M}}_{W}\sim {10}^{-8}{M}_{\odot }\,{\mathrm{yr}}^{-1}$), and ${{\rm{\Gamma }}}_{\mathrm{PW},0}={10}^{4}$. In the left panel of the figure, the results of the Swift observations are also displayed.

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We find in Figure 10 that the calculated light curve with constant magnetization parameter predicts an increase in the X-ray flux slower than that of the Swift observations after $\sim -2000$ days from periastron. The synchrotron luminosity is roughly proportional to ${L}_{\mathrm{syn}}\propto {r}_{s}{B}^{2}({r}_{s})\propto 1/{r}_{s}$ for a constant magnetization parameter. As Figure 5 shows, the shock distance decreases by about a factor of ∼2 during $\sim -2000$ days and $\sim -1000$ days, and therefore, the calculated flux slightly increases during that epoch, which cannot explain the increase in the flux (a factor of $\sim 5\mbox{--}10$) measured by Swift.

Figure 11 shows how the calculated flux depends on the model parameters, namely, the magnetization parameter (left panel) and the average Lorentz factor of the un-shocked pulsar wind ${{\rm{\Gamma }}}_{\mathrm{PW},0}$ (right panel). As seen in the left panel, the calculated X-ray flux (solid line) reaches its maximum value at the magnetization parameter ${\sigma }_{0}\sim 0.1$. This is because the internal energy (∝P₂) of the post-shock flow at the shock given by the jump condition decreases as the assumed magnetization parameter increases (e.g., Figure 4 of Kennel & Coroniti 1984), while the magnetic field at the shock (B₂) increases as the magnetization parameter increases. The former tends to decrease the calculated X-ray flux, while the latter increases it. These two effects compensate each other and the calculated X-ray flux becomes maximum at the magnetization parameter ${\sigma }_{0}\sim 0.1$. We find therefore that it is difficult to explain the observed X-ray flux of Swift after $\sim -1000$ days using the magnetization parameter ${\sigma }_{0}\sim 1$. By comparing the calculated flux and Swift observations, therefore, we conclude that the magnetization parameters at the shock is of order of ${\sigma }_{0}\sim 0.1$ at $\sim -1000$ days. It is possible that the Lorentz factor ${{\rm{\Gamma }}}_{\mathrm{PW},0}$ of the cold-relativistic pulsar wind evolves with the shock distance from the pulsar. In the right panel, however, we can see that the calculated fluxes in X-ray/TeV energy bands are less dependent on the minimum Lorentz factor of the shocked particles at the shock (${{\rm{\Gamma }}}_{\mathrm{PW},0}$).

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In the calculated TeV light curve (right panel in Figure 10), we can see an asymmetry relative to the periastron. Since the TeV photons from the shock are produced by the ICS process, the asymmetry is introduced by the dependence of the collision angle between the stellar photons and the shocked pulsar wind particles that emit the photons toward the Earth, as in the case with the orbital modulation of the ICS of the cold-relativistic pulsar wind (Figure 9).

For the TeV photons, the optical depth of the pair-creation process depends on the orbital phase, and it becomes maximum at the SUPC for photons traveling toward the Earth. This effect can be seen as a rapid drop in the calculated flux around $\sim +10$ days. The calculated TeV emission from the shock depends on the Earth viewing angle, as summarized in Figure 12.

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4.1.2. Doppler Boosting Effect

In the previous sections, we ignored the effect of the motion of the post-shocked pulsar wind. In such a case, as Figure 10 indicates, the orbital modulation of the X-ray emission is symmetric relative to the periastron. Asymmetry in the X-ray light curve as well as in the TeV light curve will be introduced by the finite velocity of the shocked pulsar wind.

The finite velocity of the shocked pulsar wind will cause the Doppler boosting effect, which enhances or suppresses the observed emissions. It has been suggested that the observed modulation of the X-ray emissions from LS 5039 is a result of this effect (Dubus et al. 2010; Takata et al. 2014a). The Doppler factor is calculated from

$$\begin{eqnarray}&&{ \mathcal D }=\displaystyle \frac{1}{{{\rm{\Gamma }}}_{\mathrm{pw}}\sqrt{1-({v}_{\mathrm{pw}}/c)\cos {\theta }_{\mathrm{pw}}}},\end{eqnarray} \tag{ 24 }$$

where ${{\rm{\Gamma }}}_{\mathrm{pw}}$ is the Lorentz factor of the bulk motion of the post-shocked flow, and ${\theta }_{\mathrm{pw}}$ is the angle of the flow direction and the line of sight. We assume that the velocity of the post-shocked pulsar wind flow is constant.

Figure 13 summarizes the Doppler boosting effect with different velocities of the post-shocked flow, ${v}_{\mathrm{pw}}={v}_{{pw},2}$ (solid lines) given by the jump condition, $c/3$ (dashed line) in the limit of ${\sigma }_{0}=0$, and $0.6c$ (dotted line). We can see that the Doppler boosting effect enhances the emissions before periastron, while suppressing after it. With $\eta \lt 1$, namely for the stellar wind stronger than the pulsar wind, the shock-cone wraps around the pulsar. Around the INFC ($\sim -150$ days), the post-shocked pulsar wind moves toward the Earth and hence enhances the emissions. These are suppressed around the SUPC ($\sim +10$ days) where the post-shocked pulsar wind moves away from the Earth. However, the Doppler boosting effect will not be the reason for the rapid increase in the X-ray emissions observed by Swift.

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4.1.3. Dependence on Wind Momentum Ratio

It has been considered that mass loss from the companion star will be driven by the radiation pressure, and the rapidly rotating main-sequence star, such that the angular velocity of the star rotation is close to its Keplerian angular velocity, and produces a stellar wind enhanced in the polar region (Georgy et al. 2011 and references therein). It is also considered that anisotropy of the pulsar wind explains the torus-like and jet-like structures of the pulsar wind nebulae, and theoretical models suggest that angular distribution of the pulsar wind energy is proportional to ${\sin }^{2}\theta$, where θ is the angle measured from the spin axis (Bogovalov & Khangoulian 2002). The anisotropy of the two winds could change the momentum ratio η at their interaction region along the orbit.

Besides the large structure of the anisotropy, an irregularity of the stellar wind from the high-mass star could be formed. It has been discussed that a radiatively non-stationary acceleration process produces clumping (small-scale density inhomogeneity) of the stellar wind from the high-mass star (Runacres & Owocki 2002; Owocki & Cohen 2006). It was analyzed that the typical size of a clump at the stellar surface is $\sim 0.01{R}_{* }$, and may linearly expand with radial distance. A larger structure of clumps could be formed in the wind due to difference production mechanisms, magnetic field inhomogeneities, and the star's rotation/pulsation (see Bosch-Ramon 2013). Although the clumps fill only a small fraction of the volume of the wind region, it is thought that they carry most of mass ejected from the star, and have a mass density larger than the average density of the wind. Studies with hydrodynamic simulations have shown that the clumping could develop in size and mass density due to merging between dense shells traveling from the star, although it was also pointed out that break-up of the dense shell as a consequence of the Rayleigh–Taylor or thin-shell instability limits the clump growth (Bozzo et al. 2016, and references therein). If a clumpy wind is formed in a high-mass binary system hosting a compact object, it is expected that the irregularity of the mass density in the stellar wind will cause a temporal evolution of the momentum ratio (η), yielding temporal variations in the shock emissions (Owocki et al. 2009; Bosch-Ramon 2013; de la Cita et al. 2017).

The calculated X-ray fluxes depend on the wind momentum ratio, which determines the location of the shock. Figure 14 shows the calculated light curves for $\eta =0.85$ (solid lines), 0.085 (dashed lines), and 0.0085 (dotted lines), respectively, with parameters ${\sigma }_{0}=0.1$ and ${{\rm{\Gamma }}}_{\mathrm{PW},0}={10}^{4}$. As we can see, the calculated X-ray flux increases with decreasing momentum ratio, because the shock distance from the pulsar decreases and hence the magnetic field at the shock increases with decreasing momentum ratio. As the figure shows, a decrease in the momentum ratio (that is, the stellar wind becomes stronger relative to the pulsar wind) by a factor of ten can give an increase in the observed X-ray fluxes during $\sim -2000$ days to $\sim -1000$ days from periastron. Such momentum evolution with time could be caused by the anisotropy of the stellar wind/pulsar wind. A shorter timescale variability will be also possible as a consequence of the pulsar wind/clumpy wind interaction.

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The observed flux ratio between the X-ray and TeV energy bands may provide additional information for the magnetization parameter and momentum ratio of the pulsar wind to the stellar wind. As we expect, the flux ratio of X-ray and TeV does not greatly depend on ${{\rm{\Gamma }}}_{\mathrm{PW},0}$, but it is sensitive to the energy densities of the magnetic field and soft-photon field at the emission region, which mainly depends on the magnetization parameter and the momentum ratio of the two winds. Figure 15 presents the dependence of ${F}_{0.3-10\mathrm{keV}}/{F}_{0.1-10\mathrm{TeV}}$ on the magnetization parameter and the wind momentum ratio. If future observations could measure the flux ratio in the X-ray and TeV energy bands, we could discuss the momentum ratio of the two winds; for example, if future observations provide ${F}_{0.3-10\mathrm{keV}}/{F}_{0.1-10\mathrm{TeV}}\sim 1$ at periastron, the current model predicts that the magnetization ${\sigma }_{0}\sim 1$ if the momentum ratio $\eta \sim 0.0085$.

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4.1.4. Radial-dependent σ of the Pulsar Wind

As seen in Figures 10, 13, and 14, the calculated X-ray light curve cannot explain the Swift observations after $\sim -2000$ days, if the momentum ratio does not significantly change along the orbit. The radial-dependent magnetization parameter has been discussed in previous studies to explain the orbital modulations of gamma-ray binaries (Takata & Taam 2009; Takata et al. 2014a; Kong et al. 2011 ). In this paper, we explore the radial dependence with a function form of

$$\begin{eqnarray}&&\sigma (r)\propto {r}^{-\alpha }.\end{eqnarray} \tag{ 25 }$$

The left panel in Figure 16 shows the results of fitting the Swift X-ray data with different power law indices. Other parameters are $\eta =0.02\,(\dot{M}\sim 4\times {10}^{-8}{M}_{\odot }$ yr⁻¹) and ${{\rm{\Gamma }}}_{\mathrm{PW},0}={10}^{4}$. In addition, we applied ${v}_{\mathrm{pw}}={v}_{{pw},2}$ taking into account the Doppler boosting effect discussed in Section 4.1.2. To explain the observed flux level from Swift, we normalized the magnetization parameter to be $\sigma \sim 0.01$ at $r\sim 2.5$ au for each power law index. Figure 17 shows the evolution of the fitting magnetization parameters at the apex of the shock cone as a function of the orbital phase.

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We can see in Figure 16 that a faster evolution (larger α) of the magnetization parameter with radial distance reproduces a X-ray light curve being more consistent with the Swift observations after −4000 days. As the figure shows, the current model predicts that the peaks of the flux in the light curve occur a day before periastron and the flux then rapidly decreases during $\sim -20$ days and $\sim +10$ days by about one order of magnitude. This is because (1) the assumed magnetization parameter determined by the relation, $\sigma (r)\propto {r}^{-\alpha }$, exceeds unity for $\alpha \gt 2$, as Figure 17 shows, and (2) the Doppler boosting suppresses the emissions around the SUPC. This feature can be tested by future observations.

Gamma-ray binaries have also been detected by Fermi-LAT and the emission will originate from the magnetospheric processes and/or the intra-binary processes. For PSR J2032+4127, the pulsed GeV emission due to the pulsar spin has been measured by Fermi-LAT with an energy flux $\sim 1.6\times {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ (Section 2). The high-energy tail of the synchrotron spectrum and the low energy tail of the ICS process of the intra-binary shock discussed in the previous sections could contribute to the energy bands (0.1–300GeV) of Fermi-LAT. In addition to the shock emissions, the ICS process of the cold-relativistic pulsar wind with ${{\rm{\Gamma }}}_{\mathrm{PW},0}\sim {10}^{4-5}$ produces GeV gamma-rays (Figure 9).

Figure 18 compares the observed flux level of the pulsed emissions with the predicted flux levels of the shocked emissions (solid line) and of the ICS process of the cold-relativistic pulsar wind (dashed line). In the calculation, we assume ${{\rm{\Gamma }}}_{0}={10}^{4}$ for the average Lorentz factor of the particles of the cold-relativistic pulsar wind, and ${\theta }_{E}=60^\circ$ for the inclination of the system. As we see, the emissions from the intra-binary space are comparable to the magnetospheric emissions around periastron. With the current parameters, therefore, we expect that Fermi-LAT measures the orbital modulation of the GeV flux during the periastron passage that will occur in late 2017 or early 2018.

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In Figure 19, we present the model broadband spectra averaged over −100 days to +100 days; the model parameters are $\alpha =2$ in Equation (25), $\eta =0.02$ and ${v}_{\mathrm{pw}}={v}_{{pw},2}$, which reproduce the observed X-ray light curves. To see the dependence of the Lorentz factor ${{\rm{\Gamma }}}_{\mathrm{PW},0}$, we calculated the spectra for ${{\rm{\Gamma }}}_{\mathrm{PW}}={10}^{4}$ (right panel) and 10⁶ (left panel), respectively. The solid line shows the calculated spectrum combining the synchrotron and ICS process at the shock, and the dashed line represents the spectrum of the ICS process of the cold-relativistic pulsar wind. The current model predicts that the magnetospheric emissions dominate in the spectral energy distribution averaged over the periastron passage. If the minimum Lorentz factor of the shocked pulsar wind particles is ${{\rm{\Gamma }}}_{\mathrm{PW}}\sim {10}^{6}$, the TeV spectrum will show a turnover at around $\sim {10}^{11}\mathrm{eV}$ and the X-ray spectrum has a break at around 10 keV.

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During the periastron passage, the TeV photons from the shocked pulsar wind may be converted into pairs by the photon–photon pair creation process with the stellar photon fields; in particular, the TeV photons emitted toward the B star will be totally absorbed. These secondary pairs created in the stellar wind will also emit non-thermal photons via the synchrotron and ICS processes, and may initiate a further pair-creation cascade. If these pairs are isotropized, they will emit photons propagating toward the observer, even though the primary TeV photons do not do this (Bednarek 1997, 2007; Sierpowska & Bednarek 2005; Sierpowska-Bartosik & Bednarek, 2008; Sierpowska-Bartosik & Torres 2007, 2008; Cerutti et al. 2010; Yamaguchi & Takahara 2010).

The emissions from the secondary pairs produced in the stellar wind side will be dominated by the ICS process. The ratio of the radiation power between the synchrotron radiation and ICS processes will be described by the energy density ratio of the magnetic field to the stellar photon. The ICS process dominates the synchrotron radiation process at a radial distance from the star of

$$\begin{eqnarray}&&R\geqslant {R}_{* }\times {6.5}^{\tfrac{1}{2(m-1)}}{\left(\displaystyle \frac{{B}_{* }}{{10}^{3}{\rm{G}}}\right)}^{\tfrac{1}{m-1}}{\left(\displaystyle \frac{{T}_{* }}{\mathrm{30,000}{\rm{T}}}\right)}^{-\tfrac{2}{m-1}},\end{eqnarray} \tag{ 26 }$$

where B_* represents the stellar magnetic field at the stellar surface. In addition, we assumed $B(R)\propto {R}^{-m}$ as the radial evolution of the stellar magnetic field. It is known that some high-mass main-sequence stars have a surface dipole magnetic field of order 10³ G (Walder et al. 2012). With a typical value of $m=2\sim 3$, we can see that the synchrotron radiation of the created pairs in the stellar wind side can be important only very near the stellar surface and its contribution to the observed X-ray emissions will be negligible.

By tracing the primary TeV gamma-rays from all emission regions considered in the calculation box, we calculated the ICS process of the secondary pairs. We assumed that the pairs produced in the stellar wind side are quickly isotropized, and we calculated the emission process with a constant photon field during the crossing time, $\sim {R}_{i}/c$, where R_i represents the distance of the pair-creation position from the stellar surface. Figure 20 compares the spectra, which are measured on the Earth, of the ICS photons from the shock region (solid line) and from the secondary pairs (dashed line) at periastron, where we assumed that all photons from the secondary pairs can escape the pair-creation process. A further pair-creation cascade will make the spectra softer. As the figure shows, the integrated flux from the secondary pairs is much less than the primary emissions, and therefore we expect that the contribution of the emissions from higher-order pairs is much less than that from the shock emissions. The emissions from the secondary pairs could contribute to  the 0.1–1 TeV bands, where the shock emissions around the periastron are significantly absorbed.

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The interaction of PSR J203+4127 and the Be disk of MT91 213 may enhance the high-energy emissions, as in the case with the PSR B1259–69/LS2883 system showing an increase in the X-ray/TeV fluxes at the pulsar/Be disk interaction phase (Chernyakova et al. 2006). As we discussed in Section 3.2, we expect that the pulsar/Be disk interaction in this system affects the observed emissions, provided that the base density is ${\rho }_{0}\gt {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and the interaction occurs at the periastron passage, say between −100 days and +100 days from periastron (Figure 6). Since we do not know the geometry (e.g., inclination relative to the orbital plane) of the disk, we cannot predict when the pulsar interacts with it. We expect however that the pulsar/Be disk interaction will occur at least once during the periastron passage during −100 days and +100 days, since the true anomaly $\pm 90^\circ$ corresponds to $\sim \pm 50$ days from periastron.

If the base density of the Be disk is close to ${\rho }_{0}\sim {10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, an interaction between pulsar and Be disk at the periastron passage will form a cavity in the pulsar wind, and most of the pulsar wind ($4\pi$ sr in solid angle) will be stopped at the radial distance from the pulsar given by Equation (10) and Figure 6. In Figure 21, we show an example of the calculated light curve for pulsar/Be disk interactions and compare the estimated X-ray/TeV fluxes with those for the latter interactions. For the calculation, we assumed a base density ${\rho }_{0}={10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, magnetization ${\sigma }_{0}=0.1$, and momentum ratio $\eta =0.085$ for the pulsar wind/stellar wind interaction.

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Figure 21 indicates that the pulsar/Be disk interaction predicts X-ray fluxes several factors larger than those due to the pulsar/Be wind interaction, if the magnetization parameter does not evolve with radial distance. Within the framework of the current calculation, the shock distance from the pulsar is closer, and the magnetic field at the shock is stronger for the pulsar/Be disk interaction, in which ${r}_{s}/a\sim 0.025$ at periastron, than the pulsar/Be wind interaction, for which ${r}_{s}/a\geqslant {\eta }^{1/2}/(1+{\eta }^{1/2})\sim 0.22$ with $\eta =0.085$. As a result, the synchrotron power is stronger for the pulsar/Be disk interaction than the pulsar/Be wind interaction. This result suggests that if the base density of the Be disk is ${\rho }_{0}\sim {10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the pulsar/Be disk interaction may give rise to a local maximum in the orbital modulation of the X-ray emissions.

As the right panel in Figure 21 shows, the calculated TeV flux for the pulsar/Be disk interaction is lower than that for the wind interaction around periastron. This is because the synchrotron cooling timescale for TeV leptons becomes shorter than the ICS cooling and adiabatic cooling timescales at the shock for the pulsar/Be disk interaction. At periastron, for example, the shock distance at the apex for the pulsar/Be wind interaction is ${r}_{s}\sim 0.22a$, for which the inverse-Compton cooling and adiabatic cooling timescales of TeV electrons are shorter than the synchrotron cooling timescale (Figure 7). For the pulsar/Be disk interaction, the shock distance becomes ${r}_{s}\sim 0.0025a$, which increases the magnetic field at the shock by a factor of ∼10, and hence decreases the synchrotron cooling timescale by about $\sim 1/100$. As a result, the ICS cooling scale, which is less sensitive to the shock distance, becomes longer than the synchrotron timescale, and the ICS emissivity is suppressed.

As we discussed in Section 3.2, if the base density of the disk is high enough, the pulsar wind will be confined within a small region by the Be disk matter. When the radius of the cavity estimated in Figure 6 is less than the radius below which the kinetic energy of the disk gas is less than the gravitational potential energy of the pulsar, the disk matter may be gravitationally captured by the pulsar, and results in the formation of a disk around the pulsar. The capture radius measured from the pulsar may be estimated as

$$\begin{eqnarray}&&{r}_{\mathrm{cap}}\sim \displaystyle \frac{2\,{{GM}}_{N}}{{v}_{r}^{2}}=0.25\,\mathrm{au}\left(\displaystyle \frac{{M}_{N}}{1.4{M}_{\odot }}\right){\left(\displaystyle \frac{{v}_{r}}{{10}^{7}\mathrm{cm}{{\rm{s}}}^{-1}}\right)}^{-2},\end{eqnarray} \tag{ 27 }$$

where M_N is the mass of the pulsar and v_r is the relative velocity of the pulsar with respect to the disk matter. Near periastron, the orbital velocity of the pulsar is ${v}_{p}\sim {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. The velocities of the Kepler motion and the radial velocity of the Be disk at the pulsar position are of the order of ${v}_{d,K}{(1\mathrm{au})\sim {v}_{d,K}({R}_{* })({R}_{* }/1\mathrm{au})}^{1/2}\sim {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ and ${v}_{d,r}\sim 0.1{c}_{s}\sim {10}^{5}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$, respectively (Okazaki et al. 2011). Hence, we expect that the relative velocity is of order of ${v}_{r}\sim {10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$, although this depends on the disk geometry and the rotation direction.

As Figure 6 shows, near periastron, where the separation is $a\sim 1$ au, the estimated shock distance due to the pulsar/Be disk interaction is of order of ${r}_{s}\sim 0.1\,\,\mathrm{au}$ for ${\rho }_{0}\sim {10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, suggesting the disk matter could be captured by the pulsar because ${r}_{s}\lt {r}_{\mathrm{cap}}$. The accretion rate may be estimated as

$$\begin{eqnarray}{\dot{M}}_{\mathrm{acc}} & \sim & f\rho (a){r}_{\mathrm{cap}}^{2}{v}_{r}\sim 4\times {10}^{17}\,{\rm{g}}\,{{\rm{s}}}^{-1}\\ & & \times f\left(\displaystyle \frac{\rho (a)}{3\cdot {10}^{-15}\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{{r}_{\mathrm{cap}}}{0.25\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{{v}_{r}}{{10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1}}\right)\end{eqnarray} \tag{ 28 }$$

where $f\sim 1$ is the geometrical factor and we used the parameter at $a\sim 1.5$ au (typical separation during the periastron passage) with ${\rho }_{0}=5\times {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. We note that for the stellar wind, ${v}_{r}\sim {10}^{8}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$, the capture radius becomes ${r}_{\mathrm{cap}}\sim 0.0025$ au, which is usually smaller than the shock radius due to the pulsar wind and stellar wind interaction (Figure 5). Hence the stellar wind will not be captured by the pulsar.

There will be a relative angular momentum of the captured matter with respect to the pulsar, indicating the captured matter forms a circular orbit around the pulsar at a distance ($\equiv {r}_{\mathrm{circ}}$) where the relative angular momentum per unit mass is equal to the angular momentum of the Kepler orbit around the pulsar, ${({{GM}}_{N}{r}_{\mathrm{icrc}})}^{1/2}$. The angular frequency of the orbital motion of the pulsar is ${\omega }_{N}\sim {v}_{p}/a\sim 5\times {10}^{-7}\,\mathrm{rad}\,{{\rm{s}}}^{-1}$. The angular frequency of the circular orbit of the Be disk around the B star is also ${\omega }_{d}\sim {\omega }_{N}$, because ${v}_{d,K}\sim {v}_{p}$. Roughly speaking, therefore, the angular frequency associated with the relative angular momentum will be of the order of $\omega \sim 5\times {10}^{-7}\,\mathrm{rad}\,{{\rm{s}}}^{-1}$, although again it depends on the disk geometry and rotation direction. The circularization radius of the captured matter may be estimated as (Frank et al. 2002)

$$\begin{eqnarray}{r}_{\mathrm{circ}} & \sim & \displaystyle \frac{{r}_{\mathrm{cap}}^{4}{\omega }^{2}}{16{{GM}}_{N}}\sim 1.7\times {10}^{10}\,\mathrm{cm}\\ & & \times {\left(\displaystyle \frac{{M}_{N}}{1.4{M}_{\odot }}\right)}^{3}{\left(\displaystyle \frac{\omega }{5\cdot {10}^{-7}\mathrm{rad}{{\rm{s}}}^{-1}}\right)}^{2}{\left(\displaystyle \frac{{v}_{r}}{{10}^{7}\mathrm{cm}{{\rm{s}}}^{-1}}\right)}^{-8}.\end{eqnarray} \tag{ 29 }$$

If we apply the standard Shakura–Sunyaev disk model, the disk matter will move inward with a dynamical timescale ${\tau }_{d}(r)\sim r/{v}_{d,r}\sim 15\,\mathrm{days}\,{\alpha }_{0.1}^{-4/5}{\dot{M}}_{17}^{-3/10}{r}_{10}^{3/4}$, where ${\alpha }_{0.1}$ is the viscosity parameter in units of 0.1, suggesting the accreting matter will take several weeks to reach the pulsar after the capture event. If the dynamical timescale (${\tau }_{d}$) is longer than the pulsar's disk crossing timescale, which may be of the order of ${\tau }_{c}\sim 2H/{v}_{p}\sim 4\,\mathrm{days}(H/0.1\,\mathrm{au})({v}_{p}/{10}^{7}\,\mathrm{cm}\,{{\rm{s}}}^{-1})$, the accretion disk may not develop around the pulsar. We note however that the circularization radius given by Equation (29) is sensitive to the relative velocity; with ${v}_{r}\sim 2\times {10}^{7}$ cm, for example, ${r}_{\mathrm{circ}}\sim 6\times {10}^{7}$ cm, which is inside the light cylinder. Moreover, we do not know the geometry of the disk. If the disk plane is parallel to the orbital plane, the pulsar/Be disk continues to interact during the periastron passage. Hence it is possible that the accretion disk develops with a radial length $\sim {10}^{10}\,\mathrm{cm}$ around the pulsar at the periastron passage.

If the accretion disk forms around the pulsar, the disk supplies UV photons in the pulsar wind region. Figure 22 summarizes the radiation power ($\nu {L}_{\nu }$) for the accretion disk (solid line), stellar surface (dashed line), and Be disk (dotted line). For the accretion disk, we assume an accretion rate $\dot{M}=4\times {10}^{17}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and the disk extends from the light cylinder $r={r}_{\mathrm{lc}}$ of the pulsar to $r={10}^{10}\,\mathrm{cm}$. For the stellar emission, we assume the Planck function with ${T}_{* }=$ 30,000 K and ${R}_{* }=10{R}_{\odot }$. For the Be disk emissions, we assumed a disk temperature ${T}_{d}=0.6{T}_{* }$ after the calculation of Carciofi & Bjorkman (2006). We ignore the effects of the emission and absorption lines on the spectra.

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As discussed in above, the captured matter will take about several weeks to reach the pulsar magnetosphere. Before the accretion disk reaches the light cylinder radius, the rotation of the pulsar is still active and produces the cold-relativistic pulsar wind. Then we can expect that the ICS process of the cold-relativistic pulsar will boost up the UV photons from the disk to higher energy. Figure 23 summarizes the calculated spectra (left panel) and the integrated flux as a function of the accretion disk inclination angle (right panel). The figure shows the results when the accretion disk extends from the light cylinder radius $r={r}_{\mathrm{lc}}$ to $r={10}^{10}\,\mathrm{cm}$. In addition, we assumed a mono-energetic distribution of the particles in the cold-relativistic pulsar wind, and took into account the energy loss of the pulsar wind due to the ICS process. As Figure 23 shows, if the initial Lorentz factor is ${{\rm{\Gamma }}}_{0}={10}^{4-5}$ which produces the scattered photons with 1–10 GeV energy, the calculated flux level can reach an order of $2\times {10}^{-10}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, which is comparable to that of the pulsed emissions.

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Pulsar binary systems may provide a valuable laboratory to study the accretion process on rapidly spinning and strongly magnetized neutron stars. One may consider that once the accretion disk crosses the light cylinder, the rotation-powered activity would be quenched. This is because copious plasma fills the acceleration region at the rotation-powered stage, where the charge density deviates from the so-called Goldreich–Julian charge density (Goldreich & Julian 1969). However, recent observation suggests the rotation-powered activity and the accretion disk can co-exist in the magnetosphere. A transient-millisecond pulsar is a binary millisecond pulsar with a low-mass companion star, and it transits between the rotation-powered stage and the accretion stage (Archibald et al. 2009). PSR J1023+0038 transited from the former stage to the latter in 2013 June, when the pulsed radio emission from this pulsar disappeared (Stappers et al. 2011). Coherent pulsed X-ray emission discovered after 2013 June suggests an accretion of matter on the neutron star surface (Archibald et al. 2015). Jaodand et al. (2016), however, found that the spin-down rate in the accretion stage is higher than that in the rotation-powered stage, and they concluded that the rotation activity is still operating in the accretion stage. Enhancement of observed GeV emission in the accretion stage also suggests the survival of rotation activity in the accretion stage (Takata et al. 2014b). This model was also applied to another transient-millisecond pulsar PSR J1227–4853 (Bednarek 2015). Gamma-ray binaries with a Be companion may also offer opportunities to study this accretion process on young pulsars.

The ICS model of the pulsar wind could provide an explanation for the origin of the flare-like GeV emissions from PSR B1259–63/LS 2883 system after the second Be disk passage of the pulsar. It has been observed that the GeV peak in the orbital modulation occurs ∼20 days after the X-ray peak (Tam et al. 2011, 2015). It has been considered that the phase of the X-ray peak corresponds to that of the pulsar/Be disk interaction. So we may expect that the Be disk matter starts to be captured by the pulsar at around the orbital phase of the X-ray peak. Since it will take several weeks to fully develop the accretion disc around the pulsar, the GeV peak position will delay the X-ray peak position.

Finally, the magnetic radius r_M from the pulsar is defined as the distance where the dipole magnetic field of the pulsar begins to dominate the dynamics of the accreting matter, and it is estimated from ${B}^{2}({r}_{M})/8\pi ={\rho }_{\mathrm{acc}}{v}^{2}$ with ${\rho }_{\mathrm{acc}}v=\dot{M}/4\pi {r}_{M}^{2}$ and $v\sim {(2{{GM}}_{N}/{r}_{M})}^{1/2}$,

$$\begin{eqnarray}{r}_{M} & \sim & 3\times {10}^{8}\,\mathrm{cm}\ {\left(\displaystyle \frac{{B}_{s}}{{10}^{12}{\rm{G}}}\right)}^{4/7}\\ & & \times {\left(\displaystyle \frac{\dot{M}}{{10}^{17}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-2/7}{\left(\displaystyle \frac{{M}_{N}}{1.4{M}_{\odot }}\right)}^{-1/7}.\end{eqnarray} \tag{ 30 }$$

This radius is smaller than the light cylinder radius of PSR J2032+4127 (${r}_{\mathrm{lc}}=6.8\times {10}^{8}$ cm). The co-rotation radius (r_co) is defined as the distance where the angular velocity of Keplerian motion is equal to the spin angular velocity of the pulsar, namely

$$\begin{eqnarray}{r}_{\mathrm{co}} & = & {\left(\displaystyle \frac{{{GMP}}^{2}}{4{\pi }^{2}}\right)}^{1/3}\\ & \sim & 4.6\times {10}^{7}\,\mathrm{cm}\ {\left(\displaystyle \frac{P}{0.143\mathrm{ms}}\right)}^{2/3}{\left(\displaystyle \frac{{M}_{N}}{1.4{M}_{\odot }}\right)}^{1/3},\end{eqnarray} \tag{ 31 }$$

which will be smaller than the magnetic radius, suggesting the accretion process is in a propeller regime, and not all infalling matter will accrete on the pulsar surface. The outflow due to the propeller effect also could cause the high-energy emission (e.g., Papitto & Torres 2015).

In summary, we analyzed Swift and Fermi-LAT data of thePSR J2032+4127/MT91 213 binary system, which is a candidate to be another TeV gamma-ray binary. The X-ray flux has rapidly increased as the pulsar approaches periastron, which will occur in late 2017 or early 2018, while the GeV flux measured by Fermi-LAT shows no significant change. We investigated the X-ray emission by the synchrotron process of the pulsar wind particles accelerated at the shock due to the pulsar wind and stellar wind interaction. We argued that the increase in the observed X-rays is caused by (1) variation of the momentum ratio of the two winds along the orbital phase or (2) the evolution of the magnetization parameter of the pulsar wind with radial distance as $\sigma (r)\propto {r}^{-\alpha }$ and $\alpha \sim 2\mbox{--}3$. The current model predicts that the peak fluxes in the 0.3–10 keV and 0.1–10 TeV energy bands are of the order of $\sim {10}^{-11}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. The gamma-ray binary system could provide a unique laboratory to study the emissions from the cold-relativistic pulsar wind. The current model suggests that the ICS process of the cold-relativistic pulsar wind could contribute to the Fermi-LAT observations at the periastron passage (Figure 18). For this binary system, the pulsar/Be disk interaction will affect the observed emissions, provided that the base density of the Be disk is ${\rho }_{0}\gt {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and the interaction occurs during $\sim -100$ days and $\sim +100$ days. For a Be disk with higher base density ${\rho }_{0}\sim {10}^{-9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the pulsar/Be disk interaction could form an accretion disk around the pulsar. The ICS process of the cold-relativistic pulsar wind of the UV from the disk also provides high-energy emission from the pulsar/Be star binary system. Future multi-wavelength observations will provide a unique opportunity to probe the pulsar wind properties and the various emission processes caused by the pulsar/Be wind and pulsar/Be disk interactions.