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Astron. Astrophys. 362, 646-654 (2000)

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3. Gamma-ray spectra from ICS cascade

Let us assume that a [FORMULA]-ray photon is injected by the compact object which is in an orbit around the star with the parameters characteristic for Cen X-3 system. As we showed above these photons may create [FORMULA] pair in collision with the soft star photon. We assume that secondary [FORMULA] pairs are locally isotropised by the random component of the magnetic field. These pairs can next produce ICS [FORMULA]-rays which may either escape from the system or collide with the massive star surface or interact with the next soft photon, initiating in this way an ICS cascade in the massive star radiation which is seen anisotropically with respect to the location of the injection place of the primary photons or electrons. The above described procedure is repeated for all cascade [FORMULA] pairs up to the moment of their `complete cooling', i.e. to the moment at which they are not able to produce [FORMULA]-ray photons in ICS process with energies above a certain applied value which in our simulations is chosen as equal to 100 MeV. We also follow all secondary ICS [FORMULA]-rays with energies above the threshold for [FORMULA] pair production in the radiation of a massive star. The conditions for such cascades to occur and the details of the Monte Carlo simulations are presented in Bednarek (1997; note that Eq. (6) of that paper should have the form [FORMULA]).

In the subsections we discuss the results of calculations for different initial injection conditions, i.e. type, distribution, and spectra of primary particles.

3.1. Injection of monodirectional and monoenergetic photons or electrons

It is likely that high energy photons and/or electrons are injected by a neutron star (the compact object in the Cen X-3 system) in the form of highly collimated beams. Such beams may be formed in the outer gaps of pulsar magnetospheres (e.g. Cheng et al. 1986), in collisions of collimated protons escaping from the pulsar magnetospheres with the matter of an accretion disk (Cheng & Ruderman 1989), or they may emerge from regions of the magnetic poles of the neutron star (Kiraly & Meszaros 1988). In our simulations it is assumed that highly collimated beam of [FORMULA]-rays with energy 10 TeV emerge from the distance [FORMULA] at a certain angle [FORMULA], measured with respect to the direction determined by the center of the massive star and the compact object. In fact such a situation corresponds also to the case of highly collimated electron beams with comparable energy since at these energies electrons transfer almost all of their primary energy to a single [FORMULA]-ray photon in ICS process because the scattering occurs in the Klein-Nishina domain.

We are interested in the spectra of secondary photons which escape from the above described type of cascade at a range of angles [FORMULA]. In Figs. 3, we show such secondary spectra for different injection angles of primary monoenergetic photons with energy [FORMULA] MeV [FORMULA] (a), 0.5 (b), 0. (c), -0.5 (d), and -1 (e), observed at the range of angles: [FORMULA] (dotted histograms), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dashed), and [FORMULA] (long-dashed). The cascading effects are the strongest (the highest numbers of secondary [FORMULA]-rays) if the monoenergetic [FORMULA]-ray beam is injected at the intermediate angles [FORMULA] (see thick solid histograms in Figs. 3, which show the secondary photon spectra integrated over all solid angles). This is a consequence of the highest optical depth for photons propagating at directions tangent to the massive star limb (curve for [FORMULA] in Fig. 2a). All secondary spectra are well described by a power law with the index [FORMULA] at energies below [FORMULA] GeV (as expected in ICS cascades with complete cooling of electrons). The photon intensities in these spectra are highest at the direction tangent to the limb of the massive star (dot-dot-dot-dashed histograms in Fig. 3). At higher photon energies ([FORMULA] GeV), the spectra show characteristic cut-offs due to the absorption of secondary photons in the massive star radiation. The spectra recover at TeV energies with the intensities depending on the observation angle. The highest intensities are observed at small angles [FORMULA] for which the optical depth for TeV [FORMULA]-rays is the lowest (see Fig. 2). In Fig. 3f, we compare the spectra escaping at fixed range of angles [FORMULA] but for different angles of injection of primary photons [FORMULA] (dotted histogram), 0.5 (dashed), 0. (dot-dashed), -0.5 (dot-dot-dot-dashed), and [FORMULA] (long dashed). It is clear that primary photons injected at directions of the highest optical depth ([FORMULA]) produce secondary photons with the higher intensities than the ones injected at the directions of the lowest optical depth ([FORMULA]).

[FIGURE] Fig. 3a-f. The secondary spectra of escaping [FORMULA]-rays from the binary system with Cen X-3 parameters, in the case of injection of monodirectional and monoenergetic primary photons with energy [FORMULA] MeV. The spectra are shown for different angles of injection of primary [FORMULA]-rays: [FORMULA] a , 0.5 b , 0. c , -0.5 d , and -1 e , within the range of observation angles: [FORMULA] (dotted histograms), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dashed), and [FORMULA] (long-dashed). The thick solid histogram shows the secondary [FORMULA]-ray spectrum integrated over all sphere. In figure f we compare the spectra escaping at fixed range of angles [FORMULA] in the case of different angles of injection of primary photons [FORMULA] (dotted histogram), 0.5 (dashed), 0. (dot-dashed), -0.5 (dot-dot-dot-dashed), and [FORMULA] (long dashed).

Note that in the type of cascade discussed here, the intensity of secondary photons observed at the direction of injection of primary photons is not completely dominated by these primary photons because of the large optical depths for the considered injection place ([FORMULA]), and the isotropisation of secondary cascade [FORMULA] pairs. Our assumptions on the propagation of photons in massive binaries are different than those applied in e.g. Kirk et al. (1999). These authors consider the binary systems in which the companion stars are largely seperated. In such systems only simple absorption effects of primary [FORMULA]-rays may be considered since the secondary [FORMULA]-rays, produced by secondary [FORMULA] pairs, are not likely to interact again with the soft star photons.

3.2. Isotropic injection of monoenergetic photons or electrons

Monoenergetic photons and electrons can be isotropically injected into the binary system by young pulsars with strong pulsar winds. In fact, it is believed the pulsar winds are composed of relativistic electrons (positrons) which have typical Lorentz factors of the order [FORMULA] (Rees & Gunn 1974). These electrons, if accelerated by the electric fields generated during magnetic reconnection in the pulsar wind zone, can be injected approximately isotropically. Therefore we consider the case of isotropic injection of photons or electrons with energies 10 TeV. If only 10 TeV electrons are injected, they should produce photons with comparable energies in a single scattering as mentioned above. The secondary photon spectra, produced in cascades under such initial conditions, are shown in Fig. 4 for different range of observation angles defined by: [FORMULA] (dotted), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dot-dashed), [FORMULA] (long-dashed). The photon intensities observed at directions defined by very small angles [FORMULA] and directions tangent to the massive star limb behave differently in different energy ranges. The intensities of GeV photons are highest for directions tangent to the stellar limb and lowest for small angles. This is contrary to what is observed at TeV energies. Note that a significant amount of secondary photons emerges also at the range of angles [FORMULA]. This is on the opposite side of the massive star to the location of the compact source of primary photons and/or electrons (directions obscured by the star!). Therefore, the soft radiation of a luminous star may work as a kind of lens for high energy [FORMULA]-ray photons causing the focusing of very high energy photons .

[FIGURE] Fig. 4. The gamma-ray spectra escaping from the Cen X-3 binary system after cascading in the soft radiation field of the massive star in the case of isotropic injection of monoenergetic [FORMULA]-ray photons with energy [FORMULA] MeV. The secondary photon spectra are observed at a range angles: [FORMULA] (dotted), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dot-dashed), [FORMULA] (long-dashed). The full thick histogram shows the secondary spectrum summed over all sphere.

3.3. Isotropic injection of electrons with the power law spectrum

We consider also the case of the isotropic injection of electrons with a power law spectrum. Such electrons can be accelerated at the shock front created by the collision of the pulsar wind with the surrounding matter as proposed in the model by Kennel & Coroniti (1984). It is assumed that electrons have a power law spectrum with index -2 (as expected from the theory of acceleration in strong shocks). They are injected from the discrete source orbiting the massive star in Cen X-3. In the binary system the shock may form relatively close to the compact object because the pulsar in Cen X-3 is relatively slow, and the density of the surrounding plasma inside the binary system is high.

The results of calculations of the secondary photon spectra escaping from ICS cascade, after integration over all solid angles, are shown in Fig 5a for different distances of the discrete source of primary electrons from the massive star. The results show that the integrated spectra in the energy region below [FORMULA] GeV do not depend significantly on the location of the injection distance (in the range from the surface up to [FORMULA]). This effect may result from our assumption on the local capturing of secondary [FORMULA] pairs by the random magnetic field. If the dominant magnetic field has ordered structure inside the binary then the propagation effects of pairs may be important. Such a more complicated cascade scenario is out of the scope of this paper and will be discussed in future work. The photon intensity decreases drastically at TeV energies for injection distances closer to the massive star surface.

[FIGURE] Fig. 5a and b. The [FORMULA]-ray spectra observed from the Cen X-3 system in the case of isotropic injection of primary electrons with the power law spectrum and spectral index -2. The spectra integrated over all solid angle are shown in a for different distances of the point source of primary electrons from the massive star: [FORMULA] (full histogram), 2 (dashed), 3 (dot-dashed), and 5[FORMULA] (dotted). The secondary [FORMULA]-ray spectra escaping at different range of angles of observation: cos[FORMULA] (full histogram), [FORMULA] (dotted), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dot-dashed), [FORMULA] (long-dashed) are shown in b .

We investigate also the dependence of the shape of the escaping spectrum of secondary [FORMULA]-rays on the observation angle [FORMULA], for distance of injection [FORMULA] (see Fig. 5b). The features of these angular dependent spectra are similar to these ones described above for monoenergetic injection of electrons. The highest intensities at TeV energies are observed at small angles [FORMULA] and the lowest intensities at directions behind the massive star. The highest intensities at GeV energies are for directions tangent to the massive star limb and the lowest for small angles and at directions obscured by the massive star.

3.4. Isotropic injection of photons with the power law spectrum

Finally we discuss the case of the isotropic injection of photons with a power law spectrum and spectral index -2. We show in Figs. 6 the spectra of escaping [FORMULA]-rays for different distances of the injection place from the massive star (Fig. 6a) separately, the escaping spectra of primary [FORMULA]-rays and secondary [FORMULA]-rays for the injection place at [FORMULA] (Fig. 6b) and the angular dependence of secondary [FORMULA]-ray spectra on the observation angle [FORMULA] (Fig. 6c). General features of these spectra are very similar to the escaping spectra produced by electrons with a power law spectrum. The significant differences appear at low energies (below a few GeV) and at high energies (above [FORMULA] TeV), and result from the contribution to the escaping spectrum from the primary [FORMULA]-rays which do not cascade in the radiation of the massive star in Cen X-3. The escaping primary [FORMULA]-rays flatten the spectrum at energies below a few [FORMULA] MeV (spectral index close to 1.9), and dominate the angle integrated spectrum at TeV energies.

[FIGURE] Fig. 6a-c. The spectra of [FORMULA]-ray photons escaping from the Cen X-3 system in the case of isotropic injection of primary photons with a power law spectrum and spectral index -2, are shown in a for different distances between the center of the massive star and the injection place: [FORMULA] (full histogram), 2 (dashed), 3 (dot-dashed), and 5[FORMULA] (dotted). The spectrum of photons injected by the compact object is marked by the thin dotted line. In b we show the photon spectra for the injection distance [FORMULA] which: escaped without interaction (dashed curve); are produced in the cascade (dot-dashed curve); and all photons escaping from the system (full curve). In c the gamma-ray spectra of secondary photons, in the case of injection at [FORMULA], are shown for different range of angles: [FORMULA] (full histogram), [FORMULA] (dotted), [FORMULA] (dashed), [FORMULA] (dot-dashed), [FORMULA] (dot-dot-dot-dashed), [FORMULA] (long-dashed).

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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