2. Propagation of VHE -ray in the radiation field of a massive star
Let us assume that VHE -ray photon with energy is injected a distance from the center of a massive star and propagates at an angle , measured from the direction to the star (see Fig. 1). The star has a radius and emits black body radiation with temperature . The picture is axially symmetric with the main axis covering with the direction . If the star is very luminous and the -ray is injected relatively close to its surface, it may create an pair in collision with soft star photons. The secondary pairs lose energy on different processes but, in the case when the inverse Compton scattering (ICS) of star photons dominates, the cascade in an anisotropic radiation field can develop (Fig 1). In the following subsections we discuss the conditions for such a cascade to occur.
2.1. Optical depth for the -ray photon in the radiation field of a massive star
The optical depth for the -ray photon propagating in the soft photon field can be computed following the formula given by Gould & Schreder (1967). The way how to compute it in our specific case of the anisotropic soft photon field coming from the star surface is shown in Appendix A. Here we present some results of the numerical computations of the optical depth for the parameters of a massive star which seems to be characteristic for close massive binaries (see e.g. Moffat & Marchenko 1996). Fig. 2a shows the dependence of the optical depth for the -ray photon with energy eV on the angle of photon injection for selected distances from the star. As expected, the optical depth is higher for larger values of the angle up to the moment of collision of the photon path with the star surface. If the injection angle passes the angle at which the star limb is seen (), the optical depth drops drastically. For , the optical depth decreases slightly because the propagation distance of the -ray photon to the moment of collision with the star surface becomes shorter but the geometrical effects do not play important role in such a case. The optical depth decreases with the distance of injection of -ray photon from the star surface for small angles but the maximal value of the optical depth, corresponding to the marginal collision with the star surface, increases with because of the propagation distance in the star radiation is higher. In Fig. 2b the dependence of the optical depth on the energy of -ray photon is shown for selected angles of photon injection . As expected from the kinematics of the ICS process, the maximum in the optical depth shifts to lower -ray photon energies and its absolute value increases with increasing angle , if , where for the parameters in Fig. 2b. For the angles , the location of the maximum in the optical depth do not change with energy of the -ray photon since the -ray propagates towards the surface of the star. However the value of the optical depth drops for higher since the propagation distance of -ray photon from the place of its injection up to the collision with the star surface is shorter.
Based on the computations of the optical depth we are able to determine the observable parameters of massive stars (its luminosity and surface temperature) for which the absorption of the -rays can become important. The optimal absorption conditions occur for the -ray photons, with energy , which propagate along the path tangent to the star limb. This photon energy is approximately given by the threshold condition for the absorption of -ray, , where is the electron mass and k is the Boltzman constant. The optical depth for the -ray photon is proportional to and the star luminosity to . Therefore, the star parameters has to fulfil the following condition,
provided that the optical depth is equal to one for the limb crossing -ray. In this relation is expressed in erg s-1, in Kelvins, and A is a constant weakly dependent on the distance above . The value of A has been computed numerically and is plotted in Fig. 3 as a function of . The -ray photons with energies above the threshold for pair production in collision with soft photons can be absorbed in the radiation field of the massive star if its luminosity .
2.2. General conditions for the development of an ICS cascade
The ICS cascade can be initiated by a VHE -ray in the vicinity of a specific massive star if the ICS losses of secondary pairs dominates the electron losses on synchrotron and bremsstrahlung processes. The ratio of ICS pair losses in the Thomson limit to synchrotron losses is determined by the energy densities of the star radiation and the magnetic field . At the star surface is given by
where is the magnetic field at the star surface (in Gauss), and is the dilution factor of the star radiation which for isotropic electrons can be defined as a part of the sphere which is obscured by the star. For the star surface . The ICS losses dominates if the star surface magnetic field is
However, this limit is valid only in the Thomson regime. In general case, we have to compute the ICS losses of electrons in the radiation of a massive star using the full Klein-Nishina cross section. The comparison of these losses with the electron losses on synchrotron process gives the limit on the star magnetic field () which is more restrict for the electrons with very high energies. This critical value of is shown in Fig. 4 as a function of electron's energy, after its normalization to .
The upper limits on the surface magnetic field of massive stars are of the order of a few hundred Gauss, for OB stars (Barker 1986), up to G, for the Wolf-Rayet stars (Maheswaran & Cassinelli 1988). The magnetic field of the stars drops with distance as , for the stars with strong winds, or even faster, e.g. if the star magnetic field can be considered as a dipole (Weber & Davis 1967; Usov & Melrose 1992). So then, if the radiation energy density dominates on the star surface, it has to dominate above the star surface as well. For example, for the parameters of the massive star in Cyg X-3 system (Moffat & Marchenko 1996): cm, K, the ICS losses in the Thomson limit dominates over synchrotron losses if G. This value is above the observational limits on the surface magnetic fields of Wolf-Rayet stars.
It is necessary to check also if for typical parameters of massive companions, the electron energy losses on bremsstrahlung process do not dominate the ICS losses. Since the wind density and the radiation density drops with the distance from the star in this same way (), it is enough to compare these losses at the star surface. It can be simply derived that the ICS losses in the Thomson limit dominate for electrons with the Lorentz factors
where , , and are the radius (in cm) and the surface temperature of the massive star, its wind velocity (in cm ), and the mass loss rate of the star (in ), respectively. Applying the typical parameters of the Wolf-Rayet stars, used above, and assuming the wind velocities of the order of km , and , we obtain that Eq. (4) is fulfilled for electrons with arbitrary Lorentz factors. So then, the bremsstrahlung losses of electrons are negligible in comparison to ICS losses.
In our picture we assume that the secondary cascade electrons are isotropised locally, i.e. in the place of their birth. It is true if the distance scales for the ICS energy losses of secondary electrons are longer than their Larmor orbits in the local magnetic field, . This condition is fulfilled if the local random magnetic field,
For example, for K, (corresponds to the minimal value of , which is on the border between the Klein-Nishina and the Thomson regimes), and for , the magnetic filed has to be Gs. If the scattering of the star photons by secondary electrons occurs in the Klein-Nishina regime, the electron attenuation length and the Larmor radius increases in a similar way with the electron Lorentz factor , and . This means that if the condition for isotropisation of electrons is fulfilled in the Thomson limit (Eq. 5), it has to be valid in the Klein-Nishina regime as well.
To summarise, our picture for the ICS cascade, developing in the radiation field of a massive star, is valid for the massive stars with the surface magnetic field limited by Eq. (3) and (5), and if the secondary pairs have the Lorentz factors above (Eq. 4).
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998