Radio absorption in high-mass gamma-ray binaries

High-mass gamma-ray binaries (HMGBs), consisting of a compact object and a massive main-sequence star, emit broad-band radiations from radio to $\gamma$-rays (see Dubus 2013; Paredes & Bordas 2019 for reviews on HMGBs). The energy spectra of HMGBs peak above 1 MeV, distinguishing them from well-known high-mass X-ray binaries (HMXBs). So far, the proposed emission models for HMGBs can be divided into the following two classes: (1) in the micro-quasar model, the broad-band emission of HMGBs arises from the relativistic jet produced by the accretion process of the compact object (e.g. Bosch-Ramon & Khangulyan 2009); and (2) in the pulsar model, the emission is attributed to the termination shock that formed by wind interactions between the pulsar and the star (e.g. Dubus 2006). In comparison, the latter scenario seems to be favoured by growing evidence, such as the spectral and timing characteristics as well as the lack of accretion signals in HMGBs. Detecting pulsation signals from these systems would undoubtedly bring a conclusion to the arguments on the nature of compact objects in these HMGBs. Furthermore, timing observations can help to constrain the orbital parameters and the properties of the presumptive pulsar.

Until now, only nine HMGBs have been detected, and only two of them with compact objects are identified as radio pulsars, namely, PSR B1259-63/LS 2883 and PSR J2032+4127/MT91 213 (Johnston et al. 1992; Lyne et al. 2015). Recently, Yoneda et al. (2020) have reported the detection of X-ray pulsations in LS 5039 with a period of $9$ s. However, the timing analysis of Volkov et al. (2021) shows that the statistical significance of the pulsation might be too low to attribute it to the pulsar spin period. Several tentative observations have been performed to search for radio pulsations in LS 5039 and other HMGBs, but each concluded with a null detection (e.g. McSwain et al. 2011; Cañellas et al. 2012). As a natural explanation, the result of a non-detection could be just because the pulsar emission beam is not pointing towards us. Additionally, the intense outflows from the massive companion could further obscure the radio pulsations through free-free absorption (FFA), even if the line of sight (LOS) is in the pulsar radio beam. To minimise the influence of this wind obscuration, previous radio observations of HMGBs were usually set around the apastron of their orbits, where the putative pulsar encounters the most negligible wind density. However, due to the tight orbits of these HMGBs, the stellar wind can always be optically thick, even at the apastron. Furthermore, for some Be/$\gamma$-ray binaries, the presence of a stellar disc results in much more complicated absorptions of radio pulsations.

Radio absorption processes in HMGBs have been studied in previous works, where the effects of pulsar wind cavities are usually ignored (e.g. McSwain et al. 2011; Cañellas et al. 2012; Koralewska et al. 2018). Here, we take a more proper binary geometry into account and show that minimum absorption could occur around the inferior conjunction (INFC). As the pulsar wind collides with companion outflows, a bow shock cavity is formed, shielding the pulsar from dense stellar wind (e.g. Dubus 2006). In addition, a Coriolis shock is created in the opposite direction from the star due to the fast orbital motion of the pulsar (Bosch-Ramon & Barkov 2011). When the pulsar moves around the INFC, the LOS could point to the pulsar through the Coriolis shock and pulsar wind cavity if the binary orbit is appropriately inclined. Depending on the stellar outflow properties, radio emission from the putative pulsar could become detectable around the INFC under certain conditions. For a specific HMGB such as LS 5039, a detailed investigation of its radio absorption process would be beneficial for searching for possible radio pulsations in the future and identifying the nature of the compact star. Alternatively, even a null detection of radio pulsations could still be helpful for providing some constraints on the binary properties, which would be beneficial for searching the pulses at X-rays or $\gamma$-rays which are less affected by wind absorption.

The paper is structured as follows. First, in Sect. 2, we describe the basic geometry of a pulsar orbiting around a massive star and the ingredients to calculate radio absorption. Then, in Sect. 3, we apply our model to two typical and well-studied binaries, namely, PSR B1259-63/LS 2883 and LS 5039. Finally, the conclusion and discussion of the implications on other binaries are presented in Sect. 4.

For HMGBs, the obscuration of radio pulsations depends on the density and temperature of the medium distributed along the LOS, which are mainly determined by their optical companions. A massive star can lose its mass via intense outflows, including a polar wind and an equatorial disc for some Oe/Be stars. In the isotropic case, the number density of the wind is given by (e.g. Waters et al. 1988)

where $r$ is the radial distance from the centre of the star. The base density $n_{\rm w,0}$ at the stellar surface $r_{\star}$ is related to the mass-loss rate and wind velocity as $n_{\rm w,0}={\dot{M}/4\pi r_{\star}^{2}v_{\rm w}\mu_{\rm i}m_{\rm p}}$, with $\mu_{\rm i}$ being the mean ion molecular weight and $m_{\rm p}$ being the mass of protons. Because of adiabatic cooling, the wind temperature decreases slowly with radial distance from the star. By ignoring the heating due to photoionisation, the temperature distribution of the wind can be expressed by the following power-law relation (Kochanek 1993; Bogomazov 2005):

where $T_{\star}$ is the effective temperature of the star at $r_{\star}$. The value of the index $\beta$ depends on the adiabatic index of the wind gas and can be in the range of $\sim\left({2/3}-{4/3}\right)$. In the following calculations, we simply adopt $\beta=2/3$.

In addition to the stellar wind, some Oe/Be stars can further possess an equatorial decretion disc. The disc is believed to be responsible for the optical line emission and the infrared excess of these stars (see Porter & Rivinius 2003 for a review on Be stars). The density of the disc is usually assumed to follow a vertical Gaussian distribution as (e.g. Carciofi & Bjorkman 2006)

where $r_{\rm d}$ is the radial distance along the mid-plane and $z$ is the vertical distance. The scale height of the disc is given by

where $v_{\rm s}=\sqrt{kT_{\star}/\mu m_{\rm p}}$ and $v_{\rm c}=\sqrt{GM_{\star}/r_{\star}}$ are the isothermal sound speed and the critical speed of the star, respectively. We note that $M_{\star}$ is the stellar mass and $\mu\simeq 0.62$ is adopted. Assuming that the disc is vertically isothermal, the temperature profile with respect to the radial distance can be written as follows (e.g. Carciofi & Bjorkman 2006):

where the effects of viscous heating and irradiation from higher stellar latitudes on the disc are neglected.

Since we are interested in the matter obstructing along the LOS that contributes to the dispersion measure (DM) and radio absorption, it is more convenient to express the above parameters in a proper coordinate system. As shown in Fig. 1, with the origin of Cartesian coordinates located at the centre of the massive companion, we set the $x$-axis along the direction of the periastron and the $z$-axis perpendicular to the orbital plane. Then, the unit vectors in the direction of the orbital angular momentum, the pulsar, and the observer can be expressed as follows:

where $\phi$ is the true anomaly of the pulsar, $i_{\rm o}$ is the orbital inclination angle, and $\phi_{\rm o}$ is the true anomaly of the LOS projected on the orbital plane.

As shown in Fig. 2, the unit vector of the stellar wind towards an arbitrary point $O$ along the LOS with a distance of $l$ from the pulsar is given by

where $d=a(1-e^{2})/(1+e\cos\phi)$ is the binary separation and $a$ and $e$ are the semi-major axis and the eccentricity of the orbit, respectively. The above equation can be rewritten as follows:

The normal vector of the disc of the Oe/Be star is given by

where $i_{\rm d}$ is the polar angle measured from the normal direction of the orbit and $\phi_{\rm d}$ is the true anomaly of the disc normal projected on the orbital plane. With the basic geometric relation, it is easy to obtain

with

By substituting the above relationships into Eqs. (1), (2), (3), and (5), we can obtain the density and temperature distributions of the stellar wind and disc along the LOS.

When the pulsar is moving around the massive star, the relativistic wind blown from the pulsar can create a cavity inside the stellar outflows. The cavity is shaped by a bow shock that arises from the dynamical balance between two winds, as shown in Fig. 2. Denoting the distance of an arbitrary point $S$ on the shock contact discontinuity (CD) surface to the pulsar as $l_{\rm s}$, the shape of the surface can be described by the following geometric equations (Canto et al. 1996):

and

where $\theta_{\rm s}$ and $\theta_{\rm p}$ are the angles between the lines joining point $S$ on the CD surface to the pulsar and its companion related to the pulsar vector. The momentum rate ratio of the two winds is given by

with $L_{\rm sd}$ being the pulsar spin-down power and $c$ being the speed of light. For most HMGBs, the momentum rate of stellar outflows is usually much higher than that of pulsar wind (i.e. $\eta\ll 1$), and then Eq. (15) can be approximated as follows:

When the LOS cuts through the shock cavity, the polar angle of the intersection is the following:

In substituting Eqs. (18) and (17) into Eq. (14), it is easy to obtain the size of the cavity $l_{\rm s,obs}$ along the LOS. The asymptotic angle of the shock cavity can be approximated as (Eichler & Usov 1993; Bogovalov et al. 2008)

by considering that $\theta_{\rm p}\sim(\pi-\theta_{\rm s})$ at infinity.

Finally, it is easy to obtain the orbital modulations of the DM and optical depth due to FFA by integrating the electron density and the absorption coefficient over the LOS:

The contribution from the pulsar wind cavity is ignored here since the pair density of the pulsar wind is far less than that of the stellar outflows (Melatos et al. 1995). The FFA coefficient is given by (Rybicki & Lightman 1979; Ghisellini 2013)

where

is the Gaunt factor, $Z^{2}\simeq 1.4$, and $T$ is the absorbing medium temperature. The second expression of Eq. (22) is obtained under the Raleigh-Jeans approximation ($h\nu\ll kT$). The number densities of electrons and ions are related by $n_{\rm e}=n_{\rm i}\mu_{\rm i}/\mu_{\rm e}$, with $\mu_{\rm i}=4/(1+3X)\simeq 1.29,\mu_{\rm e}=2/(1+X)\simeq 1.18$ for the typical values of stellar outflows, and $X\sim 0.7$ is the hydrogen abundance (Zdziarski et al. 2010; Dubus 2013).

It is necessary to point out that in the above descriptions, the shock is assumed to have an asymptotic radial structure that is directing away from the massive star, and the effect of orbital motion on the shock structure is not considered. However, for HMGBs, in particular for LS 5039 which has a more compact orbit, the putative pulsar’s fast orbital motion would significantly deflect the shock structure (Bosch-Ramon & Barkov 2011; Zabalza et al. 2013; Bosch-Ramon 2021). Furthermore, due to the Coriolis force induced by orbital motion, the pulsar wind would also be terminated by an additional shock located in the opposite direction of the star. The presence of this Coriolis shock was predicted in Bosch-Ramon & Barkov (2011) and later confirmed by numerical simulations (Bosch-Ramon et al. 2012, 2015; Huber et al. 2021a,b). We explore and discuss the influence of this Coriolis shock in the case of LS 5039 in a later section. As for PSR B1259-63/LS 2883, due to its relatively larger binary separation, the Coriolis shock is located far from the companion star, where the density of the stellar outflows is very low. Furthermore, because of the small orbital inclination angle (i.e. $180^{\circ}-i_{\rm o}\sim 26^{\circ}$), the LOS will not directly point to this Coriolis shock. Therefore, the effect of the Coriolis shock on the DM and radio absorption of PSR B1259-63 can be ignored.

Among the detected HMGBs, PSR B1259-63/LS 2883 and LS 5039 are the most well-studied binaries in both observational and theoretical aspects. In this section, we use the model described in Sect. 2 to investigate the orbital modulations of the DM and radio absorption of these two systems. The observational parameters are listed in Table. 1.

HMGBs are unique laboratories for us to study the evolution of binaries and the properties of massive stars. The lack of accretion signal detection in HMGBs suggests that the compact objects may be non-accreting neutron stars, even though most of them are not confirmed. In this paper, we have described the basic geometry and formulae to calculate the dispersion measure and radio absorption of HMGBs containing a pulsar in binary with a massive star, and we have applied our model to two typical HMGBs. For the PSR B1259-63 system with a decretion disc around its massive companions, the radio pulsation is less affected by the polar wind due to its large binary separation, and the absorption is usually dominated by inclined discs. The position and orientation of the disc can be obtained by the obscuration of radio pulsation or the multi-wavelength fitting of light curves (e.g. Chen et al. 2019). We studied the effects of the inclined disc of LS 2883 on the radio pulsation from PSR B1259-63. The observed DM data are consistent with the equatorial disc inserted on the orbital plane with an inclination angle of $i_{\rm d}\sim 60^{\circ}$ and the true anomaly of the disc normal projected on the orbital plane is $\phi_{\rm d}\sim 188^{\circ}$. The corresponding opening angle of the disc projected on the orbital plane is about $10^{\circ}$. Our model for PSR B1259-63/LS 2883 can be easily applied to other similar Be/gamma-ray binaries, such as PSR J2032+4127/MT91 213, HESS J0632+057, and LS I+61^∘303. In particular, the last two have compact objects with an unknown nature as of yet. For LS 5039-like systems, due to their compact orbits, intense outflows from their massive companions make detecting radio pulsations unlikely. Several tentative searches for radio pulsations turn out as null detections even at the apastron. We have shown that the shock cavity formed by wind collision and orbital motion could shield the putative pulsar from the intense outflows. We studied the effects of the orbital inclination angle and wind parameters on the radio absorption process. We have shown that there could be a transparent window at higher frequencies to detect radio pulsations if the momentum rate ratio of two winds is large enough and the binary orbit is appropriately inclined.

As mentioned above, in addition to wind absorption, the non-detection of radio pulsation in some HMGBs could also be due to the pulsar emission beam not pointing towards Earth. Following Emmering & Chevalier (1989), the radio beaming fraction of pulsars can be written as follows:

where the radio beaming angle is (Kijak & Gil 2003; Takata et al. 2011)

and $\dot{P}_{-15}=\dot{P}/10^{-15}$ is the period derivative. Obviously, for young pulsars with faster rotational speeds and/or larger period derivatives, the radio beaming fraction would be larger. In substituting the parameters of PSR B1259-63 into the above equation with $P=0.0477625\ {\rm s}$ and $\dot{P}=2.27875\times 10^{-15}\ {\rm s\cdot s^{-1}}$, we can estimate the radio beaming fraction with $f\simeq 38\%$ at the frequency of $\nu=15\ {\rm GHz}$. For LS 5039 and other HMGBs, if the compact objects are indeed energetic non-accreting pulsars similar to PSR B1259-63, we would expect more than one-third of them with the radio beaming pointing towards us. However, if the unknown object in LS 5039 is a magnetar, as suggested by Yoneda et al. (2020), the radio beaming fraction would be much smaller, that is, $f\simeq 10\%$ with $P\sim 9\ {\rm s}$ and $\dot{P}\sim 3\times 10^{-10}\ {\rm s\cdot s^{-1}}$. To confirm the nature of the compact objects in LS 5039 and other HMGBs, further observations and deep analyses from radio to $\gamma$-ray data are needed.