A&A 507, 12171224 (2009)
Onedimensional pair cascade emission in gammaray binaries
An upperlimit to cascade emission at superior conjunction in LS 5039
B. Cerutti  G. Dubus  G. Henri
Laboratoire d'Astrophysique de Grenoble, UMR 5571 CNRS, Université Joseph Fourier, BP 53, 38041 Grenoble, France
Received 14 May 2009 / Accepted 17 September 2009
Abstract
Context. In gammaray binaries such as LS 5039, a large
number of electronpositron pairs are created by the annihilation of
primary very highenergy (VHE) gamma rays with photons from the massive
star. The radiation from these particles contributes to the total
highenergy gammaray flux and can initiate a cascade, decreasing the
effective gammaray opacity in the system.
Aims. The aim of this paper is to model the cascade emission and
investigate whether it can account for the VHE gammaray flux detected
by HESS from LS 5039 at superior conjunction, where the primary
gamma rays are expected to be fully absorbed.
Methods. A onedimensional cascade develops along the
lineofsight if the deflections of pairs induced by the surrounding
magnetic field can be neglected. A semianalytical approach can then be
adopted, including the effects of the anisotropic seed radiation field
from the companion star.
Results. Cascade equations are numerically solved, yielding the
density of pairs and photons. In LS 5039, the cascade contribution
to the total flux is large and anticorrelated with the orbital
modulation of the primary VHE gamma rays. The cascade emission
dominates close to superior conjunction but is too strong to be
compatible with HESS measurements. Positron annihilation does not
produce detectable 511 keV emission.
Conclusions. This study provides an upper limit to cascade
emission in gammaray binaries at orbital phases where absorption is
strong. The pairs are likely to be deflected or isotropized by the
ambient magnetic field, which will reduce the resulting emission seen
by the observer. Cascade emission remains a viable explanation for the
detected gamma rays at superior conjunction in LS 5039.
Key words: radiation mechanisms: nonthermal  stars: individual: LS 5039  gamma rays: theory  Xrays: binaries
1 Introduction
The massive star in gammaray binaries plays a key role in the formation of very highenergy (VHE, >100 GeV) radiation. The large seedphoton density provided by the O or Be companion star, contributes to the production of gamma rays via inverse Compton scattering on ultrarelativistic electrons accelerated in the system (e.g. in a pulsar wind or a jet). The same photons annihilate with gamma rays, leading to electronpositron pairs production . In some tight binaries such as LS 5039, this gammaray absorption mechanism is strong if the VHE emission occurs close to the compact object. Gammaray absorption can account for an orbital modulation in the VHE gammaray flux from LS 5039, as observed by HESS (Dubus 2006; Bednarek 2006; Böttcher & Dermer 2005).
A copious number of pairs may be produced in the surrounding medium as a byproduct of the VHE gammaray absorption. If the number of pairs created is large enough and if they have enough time to radiate VHE photons before escaping, a sizeable electromagnetic cascade can be initiated. New generations of pairs and gamma rays are produced as long as the secondary particles have enough energy to boost stellar photons beyond the pair production threshold energy. Because of the anisotropic stellar photon field in the system, the inverse Compton radiation produced in the cascade has a strong angular dependence. The cascade contribution depends on the position of the primary gammaray source with respect to the massive star and a distant observer.
The VHE modulation in LS 5039 was explained in Dubus et al. (2008) using phasedependent absorption and inverse Compton emission, ignoring the effect of pair cascading. This model did not predict any flux close to superior conjunction, i.e. where the massive star lies between the compact object and the observer. This is contradicted by HESS observations (Aharonian et al. 2006a). Interestingly, this mismatch intervenes at phases where opacity is known to be high . The development of a cascade could contribute to the residual flux observed in the system, with secondary gammaray emission filling in for the highly absorbed primary gamma rays. This possibility has been proposed to explain this discrepancy (Aharonian et al. 2006a) and is quantitatively investigated in this article.
The ambient magnetic field strength has a critical impact on the development of pair cascading. If the magnetic field strength is low enough to neglect the induced deflections on pair trajectories then the cascade develops along the line of sight joining the primary source of gamma rays and a distant observer. The particles do not radiate synchrotron radiation. Cascade calculations are then reduced to a onedimension problem. Such a situation would apply in an unshocked pulsar wind where the pairs are cold relative to the magnetic field carried in the wind. This paper explores the development of an onedimensional pair cascade in a binary and its implications.
Previous computations of cascade emission in binary environment were carried out by Bednarek (2007); Zdziarski et al. (2009); Bednarek (2006,1997); SierpowskaBartosik & Torres (2008); Aharonian et al. (2006b); Orellana et al. (2007); Sierpowska & Bednarek (2005); Khangulyan et al. (2008). Except for Aharonian et al. (2006b), all these works are based on Monte Carlo methods. One peculiarity of the gammaray binary environment is that the source of seedphotons for pair production and inverse Compton emission is the high luminosity companion star. This study proposes a semianalytical model for onedimensional cascades calculations, taking into account the anisotropy in the seedphoton field. The aim of the paper is to investigate and compute the total contribution from pair cascading in the system LS 5039, and see if it can account for the measured flux close to superior conjunction. The next section presents the main assumptions and equations for cascade computations. The development and the anisotropic effects of pair cascading in compact binaries are investigated. The density of escaping pairs and their rate of annihilation are also calculated in this part. The cascade contribution along the orbit in LS 5039 is computed and compared with the available observations in Sect. 3. The last section concludes on the implications of onedimensional cascades in gammaray binaries. More details about pair production are available in the appendices.
2 Anisotropic pair cascading in compact binaries
2.1 Assumptions
This part examines onedimensional cascading in the context of binary systems. The massive star sets the seedphoton radiation field for the cascade. For simplicity, the massive star is assumed pointlike and monoenergetic. This is a reasonable approximation as previous studies on absorption (Dubus 2006) and emission (Dubus et al. 2008) have shown. The effects of the magnetic field and pair annihilation are neglected (see Sect. 2.5). Triplet pair production (TPP) due to the highenergy electrons or positrons propagating in a soft photon field ( , Mastichiadis 1991) is not taken into account here. The cross section for this process becomes comparable to inverse Compton scattering when that is for electron energies TeV interacting with eV stellar photons. With a scattering rate of about 10^{2} , only a few pairs can be created via TPP by each VHE electron, before it escapes or loses its energy in a Compton scattering. The created pairs have much lower energy than the primary electrons. TPP cooling remains inefficient compared to inverse Compton for VHE electrons with energy PeV. HESS observations of LS 5039 show a break in the spectrum at a few TeV so few electrons are expected to interact by TPP in the cascade. Observations of other gammaray binaries also show steep spectra but this assumption will have to be revised if there is significant primary emission beyond 10 TeV. Pair production due to highenergy gamma rays interacting with the surrounding material is also neglected. This occurs for rays >1 MeV and the crosssection is of order (see e.g. Longair 1992), with the Thomson crosssection. Since the measured is at most 10^{22} cm^{2} in gammaray binaries, pair production on matter will not affect the propagation of gamma rays towards the observer.
Due to the high velocity of the centerofmass (CM) frame in the observer frame, the direction of propagation of pairs created by absorption is boosted in the direction of the initial gamma ray. For a gamma ray of energy TeV, the Lorentz factor of the CM to the observer frame transform is (see the Appendix, Eq. (A.2)). Pairs produced in the cascade are ultrarelativistic with typical Lorentz factor . Their emission is forward boosted within a cone of semiaperture angle in the direction of electrons. The deviations on the electron trajectory due to scattering in the Thomson regime are . In the KleinNishina regime most of the electron energy is given to the photon. It is assumed here that electrons and photons produced in the cascade remain on the same line, a good approximation since and . This line joins the primary gammaray source to a distant observer (Fig. 1).
Pair cascading is onedimensional as long as magnetic deviations of pairs trajectories along the Compton interaction length remain within the cone of emission of the electrons. This condition holds if , with the Larmor radius. For a typical interaction length cm for TeV pairs in LS 5039, the ambient magnetic field must be lower than G. If the magnetic field strength is much greater, pairs locally isotropize and radiate in all directions. In between, pairs follow the magnetic field lines and the dynamics of each pairs must be followed as treated in Sierpowska & Bednarek (2005). The above limit may appear unrealistically stringent. However, since deviations and isotropization will dilute the cascade flux, the onedimensional approach can be seen as maximizing the cascade emission. More exactly, this redistribution induced by magnetic deflections would decrease the cascade flux at orbital phases where many pairs are produced to the benefit of phases where only a few are created. Hence, the onedimensional approach gives an upper limit to the cascade contribution at phases where absorption is strong. If the flux calculated here using this assumption is lower than required by observations then cascading will be unlikely to play a role. Finally, onedimensional cascading should hold in the free pulsar wind as long as the pairs move strictly along the magnetic field. In Sierpowska & Bednarek (2005) and SierpowskaBartosik & Torres (2008), the cascade radiation is computed up to the termination shock using a Monte Carlo approach. Sierpowska & Bednarek (2005) also include a contribution from the region beyond the shock. The cascade electrons in this region are assumed to follow the magnetic field lines (in contrast with the pulsar wind zone where the propagation is radial). There is no reacceleration at the shock and synchrotron losses are neglected. In the method expounded here, the cascade radiation is calculated semianalytically from a pointlike gammaray source at the compact object location up to infinity, providing the maximum possible contribution of the onedimensional cascade in gammaray binaries.
Figure 1: This diagram describes the system geometry. A gammaray photon of energy from the primary source (compact object) interacts with a soft photon of energy at a distance r from the source and R from the massive star (assumed pointlike and monoenergetic), producing a pair boosted toward a distant observer. The system is seen at an angle . 

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2.2 Cascade equations
In order to compute the contribution from the cascade, the radiative transfer equation and the kinetic equation of the pairs have to be solved simultaneously.
The radiative transfer equation for the gammaray density
at a distance r from the source is
where is the electrons distribution, the seedphoton density from the massive star and the Compton kernel. The kernel is normalized to the soft photon density and depends on the energy of the electron and the angle between the photon and the direction of motion of the electron (Dubus et al. 2008). In the monoenergetic and pointlike star approximation the stellar photon density can be estimated as , where is the stellar luminosity, the mean thermal photon energy and R the distance to the massive star (see Fig. 1). The absorption rate is given by Eq. (B.8), convoluted to the soft photon density.
The kinetic equation for the pairs is given by the following integrodifferential equation for
(D'Avezac et al. 2007; Blumenthal & Gould 1970; Zdziarski 1988)
where is the transition rate for an electron of energy downscattered at an energy at r. The first two terms on the right side of the equation describe the inverse Compton cooling of pairs, taking into account catastrophic losses in the deep KleinNishina regime. In this case, most of the electron energy is lost in the interaction and the scattered photon carries away most of its energy since . A continuous losses equation inadequately describes sizeable stochastic losses in a single interaction (Zdziarski 1989; Blumenthal & Gould 1970).
Since the inverse Compton kernel gives the probability per electron of energy
to produce a gamma ray of energy
,
the scattering rate can be rewritten as
The expression of is the same as the Compton kernel as described before but gives the spectrum of scattered electrons instead of the outcoming photon. The first integral in Eq. (2) is the inverse Compton scattering rate and can be analytically expressed as
where is the electron velocity in the observer frame and is the total inverse Compton crosssection (for the full expression see e.g. Rybicki & Lightman 1979, Eq. (7.5)). The last term in the kinetic equation is a source of pairs from absorption coupled with the photon density (see the appendices). The pair production kernel is normalized to the soft photon density.
The anisotropic cascade can be computed by inserting the anisotropic kernels for inverse Compton scattering (see Eq. (A.6) in Dubus et al. 2008) and for pair production obtained in Eq. (B.5) in Eqs. (1), (2). The following sections present cascade calculations applied to compact binaries, using a simple RungeKutta 4 integration method. It is more convenient to perform integrations over an angular variable rather than r. Here, calculations are carried out using , the angle between the line joining the massive star to the observation point and the line of sight (see Fig. 1).
2.3 Cascade growth along the line of sight
Figure 2 presents cascade calculations for different distances r from the primary gammaray source. For illustrative purpose, the source is assumed isotropic and pointlike, injecting a powerlaw distribution of photons with an index 2 at r=0 but no electrons. The calculations were carried out for a system like LS 5039 and for a viewing angle . In this geometric configuration, absorption is known to be strong ( for 200 GeV photons) and a significant fraction of the total absorbed energy is expected to be reprocessed in the cascade, inverse Compton scattering being also very efficient in this configuration.
Close to the source ( with d the orbital separation), absorption produces a sharp and deep dip in the spectrum (light dashed line) but the cascade starts to fill the gap (black solid line). The angle increases with the distance r to the primary source. Hence, the threshold energy for pair production increases as well. Cascading adds more flux to higher energy gamma rays where absorption is maximum. The cascade produces an excess of low energy gamma rays below the minimum threshold energy GeV. Because these new photons do not suffer from absorption, they accumulate at lower energies. This is a wellknown feature of cascading.
2.4 Anisotropic effects
Figure 2: Cascade development along the path to the observer. The primary source of photons, situated at the location of the compact object, has a power law spectral distribution with photon index 2 (dotted line). Spectra are computed using the parameters appropriate for LS 5039 at superior conjunction ( , , K) for . The transmitted spectrum, including cascade emission, is shown at various distances from the primary source: (black dashed line), , , (solid lines) and (dotteddashed line). Pure absorbed spectra are shown for comparison (light dashed line). 

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Figure 3: Spectra as seen by an observer at infinity, taking into account the effect of cascading. Calculations are applied to LS 5039 at periastron for different viewing angle , , , and . Left panel: complete spectra (solid line) are compared to the pure absorbed (light dashed line) and injected (dotted line) spectra. The contribution from the cascade is presented in the right panel. 

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The left panel in Fig. 3 shows the complete spectrum taking into account cascading (solid line) compared to the pure absorbed powerlaw (dashed line). Due to the angular dependence in the pair production process, higher viewing angles shift the cascade contribution to higher energies and decrease its amplitude (Fig. 3, right panel). The cascade flux is low enough to be ignored for .
Three different zones can be distinguished in the cascade spectra. First, below the pair production threshold energy, photons accumulate in a low energy tail (photon index 1.5) produced by inverse Compton cooling of pairs. For , a low energy cutoff is observed due to the pairs escaping the system (Cerutti et al. 2008; Ball & Kirk 2000). This low energy cutoff is at about 0.1 GeV for . The cutoff occurs when the cascade reaches a distance from the primary source corresponding to . Then, the electrons cannot cool effectively because the inverse Compton interaction angle diminishes and the stellar photon density decreases as they propagate. For , particles escape right away from the vicinity of the companion star and no tail is produced. Second, above the threshold energy, there is a competition between absorption and gammaray production by reprocessed pairs, particularly for low angles where both effects are strong. Even if cascading increases the transparency for gamma rays, absorption still creates a dip in the spectrum. Third, well beyond the threshold energy, absorption becomes inefficient. Fewer pairs are created, producing a highenergy cutoff (10 TeV, for ). KleinNishina effects also contribute to the decrease of the highenergy gammarays production.
2.5 Escaping pairs
The spectrum of pairs produced in the cascade as seen at infinity is shown in Fig. 4. The density depends strongly on the viewing angle as expected, but the mean energy of pairs lies at very high energies ( GeV, see Table 1). The accumulation of very highenergy particles can be explained by two concurrent effects. Far from the massive star (), most of the pairs are created at very high energy due to the high threshold energy (almost rearend collision). The second effect is that inverse Compton losses are in deep KleinNishina regime for highenergy electrons. The cooling timescale increases and becomes longer than the propagation timescale of electrons close to the companion star, producing an accumulation of pairs at very high energies.
The distribution of pairs allows to assess the fraction of the total absorbed energy escaping the system in the form of kinetic energy in the pairs. This nonradiated power can be compared to the radiated power released in the cascade . Energy conservation yields the total absorbed power .
Figure 4: Distribution of escaping pairs seen by a distant observer, depending on the viewing angle (dashed line), , , and (dotted line). The binary parameters are the same as in Fig. 3. 

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Table 1: Mean energy of escaping pairs and radiated power efficiency of the cascade.
The asymptotic radiated power reached by the cascade is compared to the total absorbed power integrated over energy in Table 1. The fraction of lost energy increases with the viewing angle. In fact, for most of the power remains in kinetic energy. Once the electrons are created, only a few have time to radiate through inverse Compton interaction. Below ( ), the radiative power dominates and the cascade is very efficient (recycling efficiency up to 80% for ). The cascade is fully linear, since the power reradiated remains much lower than the star luminosity (Svensson 1987). Selfinteractions in the cascade are then negligible. This is also a consequence of KleinNishina cascading (Zdziarski 1988). In addition, interactions between particles in the cascade would be forcedly rearend, hence highly inefficient.
The created positrons will annihilate and form a 511 keV line. However, the expected signal is very weak. The annihilation crosssection is (see e.g. Longair 1992). The escaping positrons have a very high average Lorentz factor (Table 1) so they are unlikely to annihilate within the system. They will thermalize and annihilate in the interstellar medium. Escaping positrons from gammaray binaries are unlikely to contribute much to the diffuse 511 keV emission. The average number of pairs created along the orbit in LS 5039 (based on the results to be discussed in the following section) is . This estimate does not take into account contributions from triplet pair production or from the pulsar wind (for a pulsar injecting pairs with and a luminosity of , about pairs are produced). Gammaray binaries have short lifetimes and it is unlikely there is more than a few hundred currently active in the Galaxy. Hence, the expected contribution is ordersofmagnitude below the positron flux required to explain the diffuse 511 keV emission ( , Knödlseder et al. 2005). Even if the positrons thermalize close to or within the system (because magnetic fields contain them, see Sect. 5) then, following Guessoum et al. (2006), the expected contribution from a single source at 2 kpc would be at most 10^{9} ph cm^{2} s^{1}, which is currently well below detectability.
3 Cascading in LS 5039
Figure 5: Orbitaveraged spectra in LS 5039 at INFC ( , grey lines) and SUPC ( or , black lines) and comparisons with EGRET (dark) and HESS (light) bowties (Aharonian et al. 2006a; Hartman et al. 1999). Dotteddashed lines represent the primary source of gamma rays with pure absorption, injected at , computed with the model described in Dubus et al. (2008) for a monoenergetic and pointlike star. Dashed lines show the contribution from the cascade and thick solid lines the sum of the primary absorbed source and the cascade contributions. 

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Figure 6: Computed lightcurves along the orbit in LS 5039, in the HESS energy band (flux 100 GeV). The cascade contribution (dashed line) is compared to the primary pure absorbed source (dotteddashed line) and HESS observations. The thick solid line shows the sum of both components. 

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LS 5039 was detected by HESS (Aharonian et al. 2005) and the orbital modulation of the TeV gammaray flux was later on reported in Aharonian et al. (2006a). Most of the temporal and spectral features can be understood as a result of anisotropic gammaray absorption and emission from relativistic electrons accelerated in the immediate vicinity of the compact object, e.g. in the pulsar wind termination shock (Dubus et al. 2008). However, this description fails to explain the residual flux observed close to superior conjunction where a significant excess has been detected (6.1 at phase ). The primary gamma rays should be completely attenuated. The aim of this part is to find if cascading can account for this observed flux. The cascade is assumed to develop freely from the primary gammaray source up to the observer. The contribution of the cascade as a function of the orbital phase is also investigated.
The primary source of gamma rays now considered is the spectrum calculated in Dubus et al. (2008). Figure 5 shows phaseaveraged spectra along the orbit at INFC (orbital phase ) and SUPC ( or ) for the primary source, the cascade and the sum of both components. The orbital parameters and the distance (2.5 kpc) are taken from Casares et al. (2005) for an inclination so varies between . The cascade contribution is highly variable along the orbit and dominates at SUPC for GeV, where a high pairproduction rate is expected. At INFC, cascading is negligible compared with the primary flux. With pair cascading the spectral differences between INFC and SUPC are very small at VHE, contrary to what is observed by HESS. In the GeV band, cascades contribute to a spectral hardening at SUPC close to 1030 GeV.
Orbital lightcurves in the HESS energy band give a better appreciation of the contribution from both components (Fig. 6). The contribution from cascading is anticorrelated with the primary absorbed flux. The cascade lightcurve is minimum at inferior conjunction ( ). The non trivial double peaked structure of the lightcurve at phases 0.850.35 is due to competition in the cascade between absorption and inverse Compton emission. Absorption has a slight edge at superior conjunction ( ), producing a dip at this phase. Elsewhere, the primary contribution dominates over the cascade emission. At lower energies ( GeV), the cascade contribution is undistinguishable from the primary source.
In this configuration, the cascade does add VHE gammaray emission close to superior conjunction but the expected contribution overestimates HESS observations. Decreasing the inclination of the system does not help: the cascade flux in the TeV energy band increases, since the primary source is on average more absorbed along the orbit (see Sect. 3 in Dubus 2006). For , the cascade contribution dominates the primary flux at every orbital phases in the VHE band. Onedimension cascades can be ruled out by the current HESS observations of LS 5039.
4 Conclusion
This paper explored the impact of onedimensional pair cascading on the formation of the very highenergy radiation from gammaray binaries in general, LS 5039 specifically. A significant fraction of the total absorbed energy can be reprocessed at lower energy by the cascade, decreasing the global opacity of the primary source. Anisotropic effects also play a major role on the cascade radiation spectrum seen by a distant observer.
A large contribution from cascading is expected in LS 5039, large enough that it significantly overestimates the flux observed by HESS. Onedimensional cascading is too efficient in redistributing the absorbed primary flux and can be ruled out. However, the fact that it overestimates the observed flux means a more general cascade cannot be ruled out (it would have been if the HESS flux had been underestimated). If the ambient magnetic field is high enough ( G) the pairs will be deflected from the lineofsight. For G the Larmor radius of a TeV electron becomes smaller than the LS 5039 orbital separation and the pairs will be more and more isotropized locally. All of this will tend to dilute cascade emission compared to the onedimensional case, which should therefore be seen as an upper limit to the cascade contribution at orbital phases where absorption is strong, particularly at superior conjunction. The initiated cascade will be threedimensional as pointed out by Bednarek (1997). Each point in the binary system becomes a potential secondary source able to contribute to the total gammaray flux at every orbital phases. Cascade emission can still be sizeable all along the orbit in LS 5039, yet form a more weakly modulated background in the lightcurve on account of the cascade radiation redistribution at other phases. The strength and structure of the surrounding magnetic field (from both stars) has a strong influence on the cascade (BoschRamon et al. 2008b; Sierpowska & Bednarek 2005; BoschRamon et al. 2008a). More realistic pair cascading calculations cannot be treated with the semianalytical approach exposed here. Complementary investigations using a Monte Carlo approach are needed to better appreciate the cascade contribution in gammaray binaries.
Finally, the cascade will be quenched if the created pairs lose energy to synchrotron rather than inverse Compton scattering. This requires ambient magnetic fields G, as found by equating the radiative timescales for a 1 TeV electron at periastron in LS 5039. Such ambient magnetic field strengths could be reached close to the companion star. In this case an alternative explanation is needed to account for the flux at superior conjunction. A natural one to consider is that the primary gammaray source is farther from the massive star. The VHE source would not be coincident with the compact object location anymore and would suffer less from absorption. In the microquasar scenario, Bednarek (2007) can account for consistent flux with HESS observations at superior conjunction if some electrons are injected well above the orbital plane (jet altitude ). In addition to LS 5039, this possibility was also considered for the system Cyg X1 by BoschRamon et al. (2008b) and Zdziarski et al. (2009).
In practice, reality may consist of a complex threedimensional cascade partly diluted and partly quenched depending upon position, angle and magnetic field configuration.
AcknowledgementsG.D. thanks A. Mastichiadis for discussions of triplet pair production. This work was supported by the European Community via contract ERCStG200911.
Appendix A: Pair production
The main equations for the pair production process are briefly presented here. Detailed calculations can be found in Gould & Schréder (1967), Bonometto & Rees (1971) and Böttcher & Schlickeiser (1997).
A.1 Kinematics and crosssections
The interaction of a gammaray photon of energy
and a soft photon of energy
in
the observer frame leads to the production of an electronpositron pair
if the total available energy in the centerofmass (CM) frame is
greater than the rest mass energy of the pair
where is the electron mass and the angle between the two incoming photons in the observer frame. It is useful to define the Lorentz invariant . Pairs are produced if and the velocity of the electronpositron pair in the CM frame is .
The differential crosssection in the CM frame depends on and the angle between the outcoming electronpositron pair and the incoming photons. The full expression can be found in e.g. Bonometto & Rees (1971), Eq. (2.7). The differential crosssection presents a symmetric structure, peaked at and minimum for . Electrons are mostly created in the same and opposite direction with respect to the incoming hard photon direction in the CM frame. The double peaked structure is enhanced with increasing energy ( ) and becomes less pronounced close to the threshold ( ). The integration over the angles gives the total pair production crosssection , maximum close to the threshold (see Eq. (1) in Gould & Schréder 1967).
The construction of the CM frame with respect to the observer
frame can be simplified if one of the incoming photons carries most of
the energy. This case is appropriate in the present context. For
,
the CM frame can be considered as propagating along the same direction
as the highenergy photon. The velocity of the CM frame in the observer
frame can be expressed as
The total energy of say the electron in the observer frame can then be formulated using the Lorentz transform from the CM to the observer frames
providing a relation between and .
A.2 Rate of absorption and pair spectrum kernels
A gammaray photon going through a soft photon gas of density
is absorbed at a rate per unit of path length l
The absorption rate gives the probability for a gamma ray of energy to be absorbed but does not give the energy of the pair created in the interaction.
Following Bonometto & Rees (1971), the probability for a gamma ray of energy
to be absorbed between l and
yielding an electron of energy between
and
(with a positron of energy
for
)
is
As with anisotropic inverse Compton scattering (Dubus et al. 2008), it is useful to consider the case of a monoenergetic beam of soft photons. The normalized soft photon density in the observer frame is
where is the Dirac distribution. Injecting Eqs. (A.6) into (A.5) gives the anisotropic pair production kernel, a convenient tool for spectral computations. The detailed calculation is presented in Appendix B and the complete expression given in Eq. (B.5). The pair production kernel has a strong angular dependence and a symmetric structure, centered at and peaked at (see Appendix B, Fig. B.1). The effect of the angle is reduced close to the threshold where the particles share equally the energy of the primary gammaray photon . Far from the threshold, one particle carries away almost all the available energy .
The anisotropic kernel integrated over all the pitch angles, in the case of an isotropic gas of photons, is consistent with the kernel found by Aharonian et al. (1983). Note that a general expression for the anisotropic kernel valid beyond the approximation is presented in Böttcher & Schlickeiser (1997).
A.3 Pair density
The number of pair created per unit of length path and electron energy
depends on the probability to create a pair and on the probability for
the incoming gamma ray to remain unabsorbed up to the point of
observation so that
Because of the symmetry in and since electrons and positrons cannot be distinguished here, . The integration over electron energy yields
The total number of pairs produced by a single gamma ray bathed in a soft radiation along the path l up to the distance r is then
For low opacity , pair production is inefficient and the number of particles produced tends to . For high opacity , a pair is always created.
Appendix B: Anisotropic pair production kernel
Figure B.1: Anisotropic pair production kernel with set at 1 eV for a headon collision ( ). The kernel is computed for GeV (dotted line), 300 GeV, 500 GeV, 1 TeV and 10 TeV (dashed line). The yielding of pairs occurs for GeV. 

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Combining the expression of with the equations Eqs. (A.2)(A.3) and defining , the differential crosssection variables can be written as
The differential crosssection can then be expressed as
The complete general formula to compute the spectrum of the pair for a nonspecified soft radiation field is
corresponding to Eq. (2.14) in Bonometto & Rees (1971). The injection of a monoenergetic and unidirectional soft photon density (Eq. (A.6)) in this last equation yields
where and
This expression is valid for and . The minimum E_{} and maximum E_{+} energy reached by the particles is set by the kinematics of the reaction and given by
Figure B.1 presents the pair production kernel for different incoming gammaray energy .
Note that a kernel can be calculated as well for the absorption rate. Injecting Eqs. (A.6) into (A.4) is straightforward and gives
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All Tables
Table 1: Mean energy of escaping pairs and radiated power efficiency of the cascade.
All Figures
Figure 1: This diagram describes the system geometry. A gammaray photon of energy from the primary source (compact object) interacts with a soft photon of energy at a distance r from the source and R from the massive star (assumed pointlike and monoenergetic), producing a pair boosted toward a distant observer. The system is seen at an angle . 

Open with DEXTER  
In the text 
Figure 2: Cascade development along the path to the observer. The primary source of photons, situated at the location of the compact object, has a power law spectral distribution with photon index 2 (dotted line). Spectra are computed using the parameters appropriate for LS 5039 at superior conjunction ( , , K) for . The transmitted spectrum, including cascade emission, is shown at various distances from the primary source: (black dashed line), , , (solid lines) and (dotteddashed line). Pure absorbed spectra are shown for comparison (light dashed line). 

Open with DEXTER  
In the text 
Figure 3: Spectra as seen by an observer at infinity, taking into account the effect of cascading. Calculations are applied to LS 5039 at periastron for different viewing angle , , , and . Left panel: complete spectra (solid line) are compared to the pure absorbed (light dashed line) and injected (dotted line) spectra. The contribution from the cascade is presented in the right panel. 

Open with DEXTER  
In the text 
Figure 4: Distribution of escaping pairs seen by a distant observer, depending on the viewing angle (dashed line), , , and (dotted line). The binary parameters are the same as in Fig. 3. 

Open with DEXTER  
In the text 
Figure 5: Orbitaveraged spectra in LS 5039 at INFC ( , grey lines) and SUPC ( or , black lines) and comparisons with EGRET (dark) and HESS (light) bowties (Aharonian et al. 2006a; Hartman et al. 1999). Dotteddashed lines represent the primary source of gamma rays with pure absorption, injected at , computed with the model described in Dubus et al. (2008) for a monoenergetic and pointlike star. Dashed lines show the contribution from the cascade and thick solid lines the sum of the primary absorbed source and the cascade contributions. 

Open with DEXTER  
In the text 
Figure 6: Computed lightcurves along the orbit in LS 5039, in the HESS energy band (flux 100 GeV). The cascade contribution (dashed line) is compared to the primary pure absorbed source (dotteddashed line) and HESS observations. The thick solid line shows the sum of both components. 

Open with DEXTER  
In the text 
Figure B.1: Anisotropic pair production kernel with set at 1 eV for a headon collision ( ). The kernel is computed for GeV (dotted line), 300 GeV, 500 GeV, 1 TeV and 10 TeV (dashed line). The yielding of pairs occurs for GeV. 

Open with DEXTER  
In the text 
Copyright ESO 2009