Pulsar wind model of close massive γ-ray binaries · PDF fileWe present and derive the...

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Mon. Not. R. Astron. Soc.000, 1–27 (2008) Printed 10 March 2009 (MN LATEX style file v2.2)

Pulsar wind model of close massiveγ-ray binaries:The influence of geometry in the pulsar wind zone processes

Agnieszka Sierpowska-Bartosik1⋆ & Diego F. Torres2,1†1Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain2Institucio Catalana de Recerca i Estudis Avancats (ICREA)

ABSTRACTSeveralγ-ray binaries have been recently detected by the High-Energy Stereoscopy Array(H.E.S.S.) and the Major Atmospheric Imaging Cerenkov (MAGIC) telescope. In at least twocases, their nature is unknown, since a distinctive, final observational feature for a black holeor a pulsar compact object companion is still missing. In this paper we aim to provide thedetails of a theoretical model of closeγ-ray binaries containing a young energetic pulsar ascompact object. This model includes a detailed account of the system geometry, the angulardependence of processes such as Klein-Nishina inverse Compton andγγ absorption, and aMonte Carlo simulation of cascading. We present and derive the used formulae and give alldetails about their numerical implementation, particularly, on the computation of cascades. Inthis model, emphasis is put in the processes occurring in thepulsar wind zone of the binary,i.e., the region between the pulsar and the shock in between of the two stars, since as we show,opacities can be already important for close systems. We thus provide a detailed study on allrelevant opacities and geometrical dependencies along theorbit of binaries, exemplifying withthe case of LS 5039, for which we have recently presented summary results elsewhere. Thisis used to understand the formation of the very high-energy lightcurve and phase dependentspectrum. For the particular case of LS 5039, we uncover an interesting behavior of the mag-nitude representing the shock position in the direction to the observer along the orbit, andanalyze its impact in the predictions. We show that in the case of LS 5039, the H.E.S.S. phe-nomenology is completely matched by the presented model, and explore the reasons why thishappens while discussing future ways of testing the model.

Key words: γ-rays: theory, X-ray binaries (individual LS 5039),γ-rays: observations

1 INTRODUCTION

Very recently, a few massive binaries have been identified asvari-able very-high-energy (VHE)γ-ray sources. They are PSR B1259-63 (Aharonian et al. 2005a), LS 5039 (Aharonian et al. 2005b,2006), LS I+61 303 (Albert et al. 2006), and Cyg X-1 (Albertet al. 2007). The nature of only two of these binaries is consideredknown: PSR B 1259-63 is formed with a pulsar whereas Cyg X-1is formed with a black hole compact object. The nature of the tworemaining systems is under discussion.

The high-energy phenomenology of Cyg X-1 is different fromthat of the others. It has been detected just once in a flare statefor which a duty cycle is yet unknown. The three other sources,instead, present a behavior that is fully correlated with the orbitalperiod. The latter varies from about 4 days in the case of LS 5039to several years in the case of PSR B1259-63: this span of orbital

⋆ E-mail: [email protected]† E-mail: [email protected]

periodicities introduces its own complications in analyzing the sim-ilarities among the three systems.

LS I +61 303 shares with LS 5039 the quality of being theonly two known microquasars/γ-ray binaries that are spatially co-incident with sources above 100 MeV listed in the Third EnergeticGamma-Ray Experiment (EGRET) catalog (Hartman et al. 1999).These sources both show low X-ray emission and variability,andno signs of emission lines or disk accretion. For LS I+61 303,extended, apparently precessing, radio emitting structures at an-gular extensions of 0.01-0.05 arcsec have been reported by Massiet al. (2001, 2004); this discovery has earlier supported its micro-quasar interpretation. But the uncertainty as to what kind of com-pact object, a black hole or a neutron star, is part of the system(e.g., Casares et al. 2005a), seems settled for many after the re-sults presented by Dhawan et al. (2006). These authors have pre-sented observations from a July 2006 VLBI campaign in whichrapid changes are seen in the orientation of what seems to be acometary tail at periastron. This tail is consistent with itbeing theresult of a pulsar wind. No large features or high-velocity flowswere noted on any of the observing days, which implies at least its

http://arXiv.org/abs/0801.3427v1

2 Agnieszka Sierpowska-Bartosik& Diego F. Torres

non-permanent nature. The changes within 3 hours were foundtobe insignificant, so the velocity can not be much over 0.05c. Sitll,discussion is still on-going (e.g. see Romero et al. 2007). New cam-paigns with similar radio resolution, as well as new observations intheγ-ray domain have been obtained since (Albert et al. 2008). Akey aspect to analyze in these high-angular-resolution campaignsis the maintenance of the morphology of the radio emission ofthe system in time: the changing morphology of the radio emis-sion along the orbit (Dhawan et al. 2006) would require a highlyunstable jet, which detailed morphology is not expected to be re-produced orbit after orbit as current results indicate (Albert et al.2008). The absence of accretion signatures in X-rays in Chandraand XMM-Newton observations (as reported by Sidoli et al. 2006,Chernyakova et al. 2006, and Paredes et al. 2007) is another rel-evant aspect of the discussion about the compact object compan-ion. Finally, it is interesting to note that neutrino detection or non-detection with ICECUBE will shed light on the nature of theγ-rayemission irrespective of the system composition (e.g., Aharonian etal. 2006b, Torres and Halzen 2007).

For LS 5039, a periodicity in theγ-ray flux, consistent withthe orbital timescale as determined by Casares et al. (2005b),was found with amazing precision (Aharonian et al. 2006). Shorttimescale variability displayed on top of this periodic behavior,both in flux and spectrum, was also reported. For instance, itwasfound that the parameters of power-law fits to theγ-ray data ob-tained in 0.1 phase binning already displayed significant variabil-ity. Current H.E.S.S. observations of LS 5039 (∼ 70 hours dis-tributed over many orbital cycles, Aharonian et al. 2006) consti-tute one of the most detailed datasets of high-energy astrophysics.Accordingly, it has been subject of intense theoretical studies (e.g.,Bednarek 2006, 2007; Bosch-Ramon et al. 2005; Bottcher 2007;Bottcher and Dermer 2005; Dermer and Bottcher 2006; Dubus2006a,b; Paredes et al. 2006; Khangulyan et al. 2007; Dubus etal. 2007). Similarly to LS I+61 303, the discovery of a jet-like radio structure in LS 5039 and the fact of it being the onlyradio/X-ray source co-localized with a mildly variable (Torres etal. 2001a,b) EGRET detection, prompted a microquasar interpreta-tion (advanced already by Paredes et al. 2000). However, thecur-rent mentioned findings at radio and VHEγ-rays in the cases ofLS I +61 303 (Dhawan et al. 2006, Albert et al. 2006, 2008) orPSR B1259-63 (Aharonian et al. 2005), gave the perspective thatall three systems are different realizations of the same scenario: apulsar-massive star binary. Dubus (2006a,b) has studied these sim-ilarities. He provided simulations of the extended radio emissionof LS 5039 showing that the features found in high resolutionradioobservations could also be interpreted as the result of a pulsar wind.Recently, Ribo et al. (2008) provided VLBA radio observations ofLS 5039 with morphological and astrometric information at mil-liarcsecond scales. They showed that a microquasar scenario can-not easily explain the observed changes in morphology. All theseresults, together with the assessment of the low X-ray state(Mar-tocchia et al. 2005) made the pulsar hypothesis tenable, andthe pos-sibility of explaining the H.E.S.S. phenomenology in such acase,an interesting working hypothesis.

In recent works (Sierpowska-Bartosik and Torres 2007, 2008)under the assumption that LS 5039 is composed by a pulsar rotatingaround an O6.5V star in the∼ 3.9 day orbit, presented the resultsof a leptonic (for a generic hadronic model see Romero et al. 2003)theoretical modeling for the high-energy phenomenology observedby H.E.S.S. These works studied the lightcurve, the spectral orbitalvariability in both broad orbital phases and in shorter (0.1phasebinning) timescales and have found a complete agreement between

H.E.S.S. observations and our predictions. They have also analyzedhow this model could be tested by Gamma-ray Large Area SpaceTelescope (GLAST), and how much time would be needed for thissatellite in order to rule the model out in case theory significantlydeparts from reality. But many details of implementation which arenot only useful for the case of LS 5039 but for all others closemas-siveγ-ray binaries, as well as many interesting results concerninggeometry, wind termination, opacities to different processes alongthe orbit of the system, and further testing at the highest energyγ-ray domain were left without discussion in our previous works.Here, we provide all these details, an in-depth step-by-step discus-sion of the model and the methods used, together with benchmarkcases (e.g., mono-energetic primaries) that are useful to understandthe formation of the very high-energy lightcurve and phase depen-dent spectra in this pulsar wind model of closeγ-ray binaries.

High energy emission from pulsar binaries have been subjectof study for a long time (just to quote a non-exhaustive list of ref-erences note the works of Maraschi and Treves 1981; Protheroeand Stanev 1987, Arons and Tavani 1993, 1994; Moskalenko etal. 1993; Bednarek 1997, Kirk et al. 1999, Ball and Kirk 2000,Romero et al. 2001, Anchordoqui et al. 2003, and others alreadycited above). In most of these studies, although not in all, high-energy primaries are assumed to be accelerated at the shock formedby the collision of the pulsar and the massive star winds. In our re-cent works (Sierpowska-Bartosik and Torres 2007a, 2008), also inSierpowska-Bartosik and Bednarek (2005), initial injection couldcome directly from the pulsar (the interacting particle populationcan of course be a result of equilibrium between this injected dis-tribution and the losses to which it is subject, just as in thecase ofshock-provided electron primaries). The closer the binary, the morethese two settings (shock injection plus losses, and pulsarinjectionplus losses) are similar to each other: for those cases in which theshock is close to the pulsar, similar results would be obtained ifwe assume that the shock accelerates the electrons incidentuponit and injects them into the downstream region such that whenin-tegrated over the shock surface, the injection rate is isotropic inthe rest frame of the pulsar. Given the directionality of thehighLorentz factor inverse Compton process, photons directed towardsthe observer are generally those coming from electrons moving inthe same direction. Opacities to processes such as inverse Comptonandγγ absorption are high in closeγ-ray binaries, cascades can de-velop, and dominating high-energy processes can already happen,as we explicitly show below, in the pre-shock region, from the pul-sar side, i.e., in the pulsar wind zone (PWZ).

The rest of this paper is organized as follows. Next Sectionintroduces the model concept and its main properties. It provides adiscussion of geometry, wind termination, and opacities along theorbit of the system (we focus on LS 5039 to make use of our earlierresults). It also deals with its numerical implementation,and de-scribes in detail the Monte Carlo simulation of the cascading pro-cesses. An accompanying Appendix provides mathematical deriva-tions of the formulae used and useful intermediate results that arekey for the model, but too cumbersome to include them as part ofthe main text. This Section and the accompanying Appendix arethe computational core of this paper. The results follow: Section3 deals with a mono-energetic interacting particle population, andSection 4, with power-law primary distributions. Comparison withH.E.S.S. results is made in these Sections and details aboutaddi-tional tests are given. Final concluding remarks are provided at theend.

Pulsar wind model of close massiveγ-ray binaries 3

2 DESCRIPTION OF THE MODEL AND ITSIMPLEMENTATION

In this Section we provide a detailed description of some of thekey issues of the theoretical model presented, its concept,andits numerical implementation. In an accompanying Appendix, wepresent an account of needed formulae, some of their cumbersomederivation, and further details on the approach.

2.1 Concept

Under the assumption that the pulsar in the binary is energeticenough to prevent matter from the massive companion from ac-creting, a termination shock is created in the interaction region ofthe pulsar and donor star winds. This is represented in Fig. 1. Wefocus on the specific case of the binary LS 5039, which we use asatestbed all along this paper. The volume of the system is separatedby the termination shock, which structure depends on features ofthe colliding winds: it may be influenced by the anisotropy ofthewinds themselves, the motion of the pulsar along the orbit, turbu-lences in the shock flow, etc. For simplicity it is assumed here thatthe winds are radial and spherically symmetric, and that theter-mination of the pulsar wind is an axial symmetric structure withnegligible thickness. In this general picture there are three regionsof different properties in the binary: the pulsar wind zone (PWZ),the shock (SR), and the massive star wind zone (MSWZ).

The PWZ is considered as an unshocked pulsar wind. Weassume that this wind is spherically symmetric and that particleswithin it propagate radially from the pulsar. The relativistic e+e−

pairs are assumed to be injected not far from the light cylinder, andtheir Lorentz factors, as high asΓpw ∼ 106 − 107 (e.g., Kennel andCoroniti 1984a,b). The magnetic field is frozen in this relativisticwind so that pairs do not produce synchrotron radiation therein.Synchrotron emission from the system is to come from electronsbeyond the PWZ, as is typically considered in models where theγ-emission mainly comes from the shock region, especially appli-cable in the case of binaries with more eccentric orbits (e.g., Kirk,Ball and Skjaeraasen 1999). Concomitantly, in such very eccentricmassive binaries with large separations (e.g., PSR B1259-63) theradiation field of the massive star may not dominate over the wholeorbit and IC scattering may not be effective in the PWZ.

The energy content in the interaction population of parti-cles is assumed as a fraction of pulsar spin-down powerLsd. Incase of young energetic pulsars, this power is typically∼ 1036 −1037 erg s−1. Propagating pairs up-scatter thermal photons from themassive star due to inverse Compton process. For close binaries,the radiation field of hot massive stars (type O, Be or WR, havingtypical surface temperatures in the rangeTs ∼ 104 − 105 K and lin-ear dimensionRs ∼ 10R⊙) dominates along the whole orbit overother possible fields (e.g., the magnetic field or the thermalfield ofthe neutron star). This thermal radiation field is anisotropic, partic-ularly for e+e− injected close to the pulsar (the radiation source ismisplaced with respect to the electron injection place).

The high-energy photons produced by pairs can initiate cas-cades due to subsequent pair production in absorption (γγ) processwith the same radiation field (as sketched in Fig. 1). We assumethat these cascades develop along the primary injection direction,i.e., in a one-dimensional way, which is certainly justifiedbased onthe relativistic velocity of the interacting electrons. This process isfollowed up to the termination shock unless leptons lose their en-ergy before reaching it. Those leptons which propagate to the shockregion are trapped there by its magnetic field. Radiation from them

isisotropisedd. The photons produced in cascades which reach theshock can get through it and finally escape from the binary or beabsorbed in the radiation field close to the massive star, some mayeven reach the stellar surface.

In the shock region leptons move along it with velocity∼ c/3.They could be re-accelerated and produce radiation via synchrotron(local magnetic field from the pulsar side) or inverse Comptonscattering (ICS, thermal radiation field from the massive star) pro-cesses. However, as they are isotropised in the local magnetic field,photons are produced in different directions and their directionalitytowards the observer is lost. It was already shown by Sierpowskaand Bednarek (2005) that in compact binary systems (as an exam-ple, the parameters of Cyg X-3 were taken by these authors) theradiation processes in the shock region do not dominate: theen-ergy carried bye+e− reaching the shock is a small fraction of totalinjected power. Furthermore, we will show that for the parametersrelevant to the LS 5039 scenario, the PWZ is relatively largewithrespect to the whole volume of the system for the significant rangeof the binary orbit.

2.2 Hydrodynamic balance

Assuming that both, the pulsar and the massive star winds arespherically symmetric, and based on the hydrodynamic equilibriumof the flows, the geometry of the termination shock is described byparameter

η =MiVi

MoVo, (1)

whereMiVi andMoVo are the loss mass rates and velocities of thetwo winds (Girard and Wilson, 1987). The shock will be symmetricwith respect to the line joining two stars, with a shock frontat adistancers from the one of the stars:

rs = D

√η

(1+√η). (2)

The surface of the shock front can be approximated then by a cone-like structure with opening angle given by

ψ = 2.1

(

1−η2/5

4

)

η1/3, (3)

whereη = min(η, η−1) takes the smaller value between the twomagnitudes quoted. This last expression was achieved undertheassumption of non-relativistic winds in the simulations ofthe ter-mination shock structure in Girard and Wilson (1987), albeit it isalso in agreement with the relativistic winds case (e.g., Eichler andUsov, 1993; Bogovalov et al., 2007). If one of the stars is a pulsarof a spin down luminosityLsd and the power of the massive star isMsVs the parameterη can be calculated from the formula

η =Lsd

c(MsVs)(4)

(e.g., Ball and Kirk 2000). Note that forη < 1, the star wind dom-inates over the pulsar’s and the termination shock wraps around it.Note also that forη = 1, the shock is at equal distance,d/2, betweenthe stars. In the case of LS 5039, for the assumed (nominal) spin-down luminosityLsd discussed below, the value ofη is between 0.5(periastron) and 0.3 (apastron).

The massive star in the LS 5039 binary system is of O type,which wind is radiation driven. The velocity of the wind at a certaindistance can be described by classical velocity law


A

SUPC

MS

α obs

P

φ

INFC

to observer

PSR

α

γ

γ

γ

γ

γ

γ

obs e e

e e

e e

e ee e

e e+ −

+ −

+ −

+ −

+ −+ −

ε

ε

ε

ε

εε

εε

ε

ε

ε

ε

e e+ −

MS

Figure 1. Left: Sketch of a close binary, such as the LS 5039 system. P stands for the orbital position of periastron, A for apastron,INFC for inferiorconjunction, and SUPC for superior conjunction. The orbital phase,φ, and the angle to the observer,αobs, are marked. The orbital plane is inclined withrespect to the direction of the observer (this angle is not marked). The termination shock created in the interaction of the pulsar and the massive star winds isalso marked for two opposite phases –periastron and apastron– together with the direction to the observer for both phases. Right: The physical scenario forhigh-energy photon production in the PWZ terminated by the shock.e+e− are injected by the pulsar or by a close-to-the-pulsar shockand travel towards theobserver, producing Inverse Compton photons,γ, via up-scattering thermal photons from the massive star,ε. γ-photons can initiate IC cascade due to absorptionin the same thermal field. The cascade is developing up to the termination shock. Electron reaching the shock are trapped there in the local magnetic field,while photons propagate further and escape from the binary or are absorbed in massive star radiation field. The cascade develops radially following the initialinjection direction given byαobs.

Vs(r) = V0 + (V∞ − V0)(

1−Rs

r

)β

, (5)

whereV∞ is the wind velocity at infinity,Rs the hydrostatic radiusof the star,V0 is the velocity close to the stellar surface, andβ =0.8−1.5 (e.g., Cassinelli 1979, Lamers and Cassinelli 1999) and weassumeβ = 1.5. As can be seen by plotting Eq. (5), the influence ofthis parameter is minor. Typical wind velocities for O/Be type starsareV∞ ≈ (1−3)× 103 km s−1. For LS 5039 we haveV∞ = 2.4×103

km s−1 andV0 = 4 km s−1 (Casares et al. 2005). The assumed valuefor the star mass-loss rate is 10−7 M⊙ yr−1, while the typical valuesfor O/Be stars are 10−6 − 10−7 M⊙ yr−1.

2.3 Interacting electron population

The starting point of the model is the assumption that a fractionof the pulsar power is injected as relativistice+e− pairs. Once thepulsar injects relativistic electrons (what could itself be subject toorbital variability), or alternatively, a close-to-the-pulsar shock ac-celerates a primary population assumed to be isotropic in the pulsarrest frame (e.g., Kirk et al. 1999), the interacting primaryleptonpopulation arises from the equilibrium between injection and thelosses operating in the system along the orbital evolution.Of thisinteracting population, we only care on those electrons directed to-wards the observer, since the opening angle of the Inverse Comptonprocess is very small at high Lorentz factors.

Whereas the spin-down power for young energetic pulsars isan indirectly measurable quantity, the fraction of it that is trans-ferred into relativistic electrons is not. To proceed forward, and for

our benchmark case of LS 5039, we assumeLsd = 1037 erg s−1 andwill fix the parameterβ according to the injection model. We dis-cuss these values below. Yet, it is less known whether, if pulsar-injected, pairs would follow a mono-energetic or rather show agiven energy distribution, since the processes of acceleration ontop of the light-cylinder are not well known. We will here discusstwo models for the interactinge+e− distribution; in both cases weassume that the interacting population of electrons is isotropic (pul-sar centered), what in close binaries should be a sensitive enoughapproach even for shock acceleration. First, a benchmark case ofa pulsar-provided mono-energetic lepton injection which remainsessentially unchanged by losses, and where the energy of thepri-maries is fixed atE0 = 10 TeV, i.e., an injection rate given by aDirac Delta functionNe+e− (E) ∝ δ(E − E0). Second, we particular-ize on the simplest case in which leptons are described, after beingreprocessed by losses which dominance can be a function of orbitalphase, by a power-law in energy that may be constant or vary alongthe orbit: Ne+e− (E) ∝ E−αi . The fraction of the pulsar spin-downpower ending in thee+e− interacting pairs can then be written as:

βLsd =

∫

Ne+e− (E)EdE. (6)

Assuming that the distance to the source isd = 2.5 kpc (d2 =

5.6 × 1043 cm2, Casares et al. 2005), the normalization factor forelectrons traveling towards Earth is

A = Ne+e−/4πd2. (7)

The specific normalization factors in the expression of the injection


Figure 2. Dependence of the distance to the shock form the pulsar side,rs,as a function of the spin-down power,Lsd, for fixed values of star mass lossrate,M = 10−7 M⊙ yr−1, and terminal star wind velocity,V∞ = 2.4× 103

km s−1.

rateNe+e− (E) will be given together with the results for two modelsin the corresponding sections below. In the models presented here,only a small fraction (∼1%) of the pulsar’sLSD ends up in relativis-tic leptons. This is consistent with ions carrying much of the windenergetics.

2.4 On parameter interdependencies

The η expression represents the realization of a specific scenariofixed by different values of the pulsar and massive star parameters.For instance, assumingLsd = 1037 erg s−1 and MsVs as given inthe Table 1 below, we getη ≈ 0.5 for the periastron phase. Butwe would achieve the same value ofη with a smallerLsd, say 1036

erg s−1, if within uncertainties we adopt a smaller value ofMsVs.Clearly, if only one of the quantities,Lsd or MsVs changes, theshock position moves, what results in a correspondingly smaller orlarger PWZ in which cascading processes are set. A key parameteris then given byrs, instead ofη, since this is what defines the realsize of the PWZ. The mild dependence ofrs on η makes possibleto accommodate a large variation in the latter maintaining similarresults. Fig. 2 shows this dependence for a relevant range ofLsd.

In addition, it is obvious thatLsd andβ are linearly (inversely)related. As a first normalization of the simulation results below, wewill fix β = 10−2 for a spin-down luminosityLsd = 1037 erg s−1. Wenote that a fixed value ofη does not imply a specific value forLsd

as it is combined with parameters of the massive star wind, sub-ject to uncertainty in their measurements for the specific case ofbinary treated (e.g., for LS 5039), if at all known. In the end, thesame results can be obtained for different sets of parameters. Thereis a degeneracy between the shock position, which determines theextent of the PWZ, and the injected power in relativistic electrons.For a smaller PWZ (e.g., if we assume a smallerLsd for the sameproduct ofMsVs) a larger amount of injected power compensatesthe reduced interacting region. We found that to get the similar re-sults when theη parameter is smaller, theβ parameter have to beincreased roughly by the same factor.

2.5 Monte-Carlo implementation for the cascading process

A Monte-Carlo procedure is applied to calculate the place ofelec-tron interaction,xe, and the energy of the resulting photon in the

ICS, Eeγ; as well as the place of photon interaction,xp, and the

energy of the produced electron/positron,Epe (see, e.g., Bednarek

1997). The following summarizes the procedure.When the primary electron is injected, the computational pro-

cedure for IC scattering is invoked first in order to get the initialplace of interaction (the radial distance from the injection position)and the energy of the up-scattered photon if the interactiontakesplace. For this same electron, the procedure is repeated afterwardsup to the moment of electron cooling or when reaching the shockregion. The photons produced along the electron path switchon aparallel computational procedure for pair production. This in turngives the place of photon absorption and the energy of thee+e−

pair if the process occurs. Because the photons can get trough thetermination shock if not absorbed in the PWZ, the numerical pro-cedure is not limited to the termination shock radius. If ae+e− pairis created within the PWZ, then the procedure for IC scattering isinitiated for it, from which a next generation of photons canbe pro-duced. When photons cross the shock, information about themisseparately saved in order to get account of the level of flux absorp-tion in the MSWZ. The emission spectra are produced from pho-tons which finally escape from the binary, so the photons absorbedin the MSWZ are not included in it.

The place of lepton interaction in IC scattering, that is, theproduction of a relativistic photon atxe1, after being injected at adistancexi from the star, at an angleα (we discuss further detailsof geometry in the next Section and in the Appendix, especiallysee Fig. 20), is calculated from an inverse method. For an specificrandom numberP1 in the range (0,1), the interaction placexe1 isgiven by the formula:

P1 ≡ e−τICS = exp

(

−∫ xe1

0λ−1

ICS(Ee, xi , α, xe) dxe

)

, (8)

whereλ−1ICS(Ee, xi , α, xe) is the rate of electron interaction to IC pro-

cess (see Eq. 96 of the Appendix). From this it follows:− ln P1 =∫ xe1

0λ−1

ICS(Ee, xi , α, xe) dxe, where the integration is over the propa-gation path of the electron,xe. Knowingλ−1

ICS(Ee, xi , α, xe), we keepintegrating forward until we find numerical equality of the inte-gral result with the random-generated quantity− ln P1, what definesthe position of interactionxe1. For a large number of simulations,the distribution of interaction distances is presented in Fig. (3, leftpanel).

Once we get the place of electron interaction,xe1, simulatedin the way described, we get the energy of the scattered photonfrom an inverse cumulative distribution. The energy of the photonEeγ produced in IC is then given by the spectrum of photons after

scatteringdN/dEγdt (see Appendix, Eq. 86). For a random numberP2 we define:

P2 ≡[∫ Ee

γ

0

dNdEγdt

dEγ

]

/

[ ∫ Emaxγ

0

dNdEγdt

dEγ

]

, (9)

whereEmaxγ is the maximal scattered photon energy (see Appendix

Eq. 73), while the energyEeγ is the Monte-Carlo result for the simu-

lated photon energy. Note that the denominator of the latterexpres-sion is the normalization needed for the Monte-Carlo associationto succeed. The statistics for the energies of up-scatteredphotons isshown in Fig. 3, right panel. All relevant formulae and more detailsof implementation are given in the Appendix.

In a similar way we randomize by Monte-Carlo the neededmagnitudes forγγ absorption. The probability of interaction for aphoton with energyEγ at a given distance, sayxp3, from an injec-tion place (i.e., providedxi andα are known) during the propaga-tion in an anisotropic radiation field is given by expression:


Figure 3. Left: The distribution of the place of electron interactiondue to IC scaterring in the anisotropic radiation field of a massive star. The initial electronenergy is here assumed asE = 1 TeV, and the place of its injection is given by the distance to the massive star in the LS 5039 system,ds = 2.25 Rs, beinginjected at the angleα = 150o with respect to the massive star. The number of injected electrons (interactions) is N= 3360. Right: Distribution for the scatterredphoton energies for the same simulations.

P3 ≡ e−τγγ = exp

(

−∫ xp3

0λ−1γγ (Eγ , xi , α, xγ) dxγ

)

. (10)

From this it follows,∫ xp3

0λ−1γγ (Eγ , xi , α, xγ) dxγ = − ln P3, thus, sim-

ilarly to the process just described, for the random numberP3 wecan get the specific place of interactionxp3. The results of suchMonte-Carlo simulations are presented in Fig. 4, left panel.

The energy of the lepton (e+e−) produced in the photon ab-sorption process (with the photon having an energyEγ) is calcu-lated again from an inverse function. The latter is obtainedfromthe integration of thee+e− spectra produced in the process (see Ap-pendix, Eq. 68):

P4 ≡

∫ Epe

0.5Eγ

dW(Eγ ,x0,α,xp)dEedxγ

dEe

∫ Emaxe

0.5Eγ

dW(Eγ ,x0,α,xp)dEedxγ

dEe

, (11)

whereP4 is a new random number. The integration is over the elec-tron energyEe. For normalizing, as the produced lepton spectraare symmetric with respect to the energyEe = Eγ/2, the lowerlimit of this integral is fixed to this latter energy, while the up-per limit is equal to maximum energy available in this processEmax

e = Eγ − mec2. From the energy of the electron,Epe , the en-

ergy of the associated positron is also obtained asEpe′= Eγ − Ep

e .Again, note that the denominator of the latter expression isthe nor-malization needed for the Monte-Carlo association to succeed.

2.6 Checks for the Monte-Carlo distributions

Here we discuss the rightness of the simulated random distributionof electrons and photons resulting from the cascading process. InFig. 5 we show the comparison between the event statistics, i.e.,N(x < xe)/Ntotal with Ntotal being the total number of simulationsrun in these examples, and the analytical computed probability ofinteraction 1− exp(−x/λ), whereλ is, correspondingly, the onecorresponding toγγ absorption and ICS. We see total agreementof the Monte-Carlo and analytical probabilities, i.e., whereas theposition of interaction of a single photon or electron, fromwhichthe subsequent cascading process is followed, is obtained throughMonte-Carlo and thus is random, the overall distribution maintainsthe shape provided by the physical scenario: given the target and

injection energy, the mean free path defines the distribution for asufficiently high number of runs.

2.7 Basic geometry

Parameters such as the viewing angle towards the observer,αobs,the separation of the binary,d, and the distance to the shock regionalong the orbit,rs (Eq. 2), are directly connected toγ-ray observa-tional results and are thus crucial for detailed model discussion. Inthis Section, the basis sketch of the model depicted in Fig. 1has tobe kept in mind to follow the discussion. Table 1 gives the setofLS 5039 orbital and binary parameters that are relevant for model-ing. These parameters come from the recent work by Casares etal.(2005b), either directly from their measurements or from the valuescompiled by them.

The inferior conjunction (INFC) is defined as the phase whenthe binary is viewed from the pulsar side (the smallestαobs). Ex-actly in the opposite side of the orbit we find superior conjunction(SUPC), when the pulsar is behind the massive star (the largestαobs), see also Fig. 1. One can see that for LS 5039, INFC (φ ≈0.72) is close to apastron phase, while SUPC (φ ≈ 0.06) is close toperiastron.

Figs. 7 and 6 shows the angle to the observer,αobs, as a func-tion of orbital phase for two considered inclination of the binary,i = 30o and i = 60o. In addition, the separation of the binary ismarked. The viewing angle changes within the limits (900−i,900+i)and depends also on the longitude of the periastronωp. The deriva-tion of the angle to the observer is cumbersome and we give thedetails in the accompanying Appendix. The comparison for the twoinclinations shows already thatαobs varies significantly in the caseof the larger angle, as the difference isδαobs = 120o, what is cru-cial for all angle-dependent processes discussed in this paper. Theseparation of the binary is given by

r =p

1+ ε cosθ, (12)

whereε is the eccentricity of the orbit,

ε2 = (a2 − b2)/a2, (13)

the value ofp is defined as


Figure 4. Left: Distribution of the position of theγ-photon interaction due to absorption in the anisotropic radiation field. The initial photon energy in thisexample is assumed asE = 1 TeV and the place of its injection is given by the distance tothe massive star in the LS 5039 system,ds = 2.25 Rs at the angleα = 150o with respect to the massive star. The number of injected photons (interaction) is N= 3358. Right: Distribution of the produced electron energy forthe same simulations.

Figure 5. Comparison between the event statistics, i.e.,N(x < xe)/Ntotal with Ntotal being the total number of simulations run, and the analytical computedprobability of interaction 1− exp(−x/λ), whereλ is both, the one corresponding toγγ absorption (left) and ICS (right).

Table 1.LS 5039: system parameters

Meaning Symbol Adopted value

Radius of star R⋆ 9.3R⊙Mass of star M⋆ 23M⊙Temperature of star T⋆ 3.9× 104 KMass loss rate of star M 10−7 M⊙ yr−1

Wind termination velocity V∞ 2400 km s−1

Wind initial velocity V0 4 km s−1

Distance to the system D 2.5 kpcEccentricity of the orbit ε 0.35Semimajor axis a 0.15 AU ∼ 3.5R⋆Longitude of periastron ωp 226o


PSR

α

o

β

MS

tφ,

φ t i

obs

iN

observer

Figure 6. The geometry of the orbit of the inclinationi in the binary system(with the MS - massive star in the center) where the angle to the observer,αobs, is defined.No is the normal to the orbital plane,φt is the orbital phaseandφ′t corresponds to the orbital phase in the plane of the observer. Theangleβ gives the height of the pulsar for given phase above the observerplane.

p = a(1− ε2) (14)

and whereb is the semi-minor axis. As the semi-major axis of LS5039 isa = 0.15 AU = 3.5Rs the assumed pulsar in the orbit is atperiastron only 2.25Rs from the massive star, whereas at apastron,it lies atd = 4.7Rs, about a factor of 2 farther.

To get the relation for the angle to the observer we use thegeometry shown in Fig. 6 where starting from the spherical trianglecontaining the inclination anglei, we have the relations:

sinφt cosi = cosβ sinφ′t (15)

and

cosφt = cosβ cosφ′t (16)

which is the cosine theorem for a rectangular spherical triangle.Then, the angle to the observer defined in Fig. 6, is calculated fromthe formula:

cos2 β = sin2 φt cos2 i + cos2 φt, (17)

from which:

αobs = π/2± β. (18)

The sign in this relation depends on the orientation to the observer,i.e., if the pulsar is below or above the orbital plane. Note that Eq.(17) is not fulfilled all around the orbit as in here the phaseφt = 0 isa common point for the orbital and observer plane. The relation canthen be used after defining the orientation of the orbital plane withrespect to the observer given by the inclinationi and the periastronpositionωper. The latter is done when we tilt the inclined orbitalplane withωper + π/2. To find the internal phaseφt dependent onthe orbital phaseφ (with φ = 0 at periastron), we have to find thespherical triangle from Fig. 6 in the system of the orbital plane.Including the tilting of the systems we have the relation:

φt = φ + ωper + π/2. (19)

Then we have to check if the calculated value ofφt is within oneorbital phase 0< φt < 2π. If not the phase have to be replacedby φt + 2π (if φt < 0) or φ − 2π (φt > 2π). To use the Eq. (17)and calculate the corresponding angle to the observer we have torecalculate the phaseφt once again as it is only valid in the range0 < φt < π/2. We can divide the orbit in four quarters in which thefollowing transformations are needed:

(i) π/2− φt if 0 < φt < π/2,(ii) φt − π/2 if π/2 < φt < π,(iii) 3π/2− φt if π < φt < 3π/2,(iv) φt − 3π/2 if 3π/2 < φt < 2π.

Inserting such phase into the Eqs. (17) and (18) we get the angle toobserver corresponding to phaseφ.

2.8 Wind termination

Very interesting in the context of LS 5039 model properties is thedependence of the distance from the pulsar to the termination shockin the direction to the observer as a function of the orbital phase.This is shown in Fig. 8. We find that for both inclinations, thewindis unterminated for a specific range of phases along the orbit, i.e.,the electron propagating in the direction of the observer would findno shock. The PWZ would always be limited in the observer’s di-rection only if the inclination of the system is close to zero, i.e., thesmaller the binary inclination the narrower the region of the windnon-termination viewed by the observer. In the discussed scenar-ios for LS 5039, the termination of the pulsar wind is limitedto thephasesφ ∼ 0.36−0.92 for inclinationi = 60o and toφ ∼ 0.45−0.89for i = 300, what coincides with the phases where the emissionmaximum is detected in the very high-energy photon range (seebelow the observed lightcurve obtained by H.E.S.S.). The untermi-nated wind is viewed by the observer at the phase range betweenapastron and INFC, while a strongly limited wind appears from pe-riastron and SUPC. Note that the important differences in the rangeof phases in which the unterminated wind appears for distinct in-clinations (e.g., the observer begins to see the unterminated pulsarwind at∼ 0.36 for i = 60o compared to∼ 0.45 for i = 300) producea distinguishable feature of any model with a fixed orbital inclina-tion angle.

From the geometrical properties of the shock surface, therecan also be specific phases for which the termination shock isdi-rectededge-onto the observer. These are phases close to the non-terminated wind viewing conditions: slightly before and after thosespecific phases for which the wind becomes non-terminated, e.g.,for the case of LS 5039 andi = 300, φ ∼ 0.4 and 0.93; whereasfor i = 600, it appears atφ ∼ 0.32 and 0.93. Thus, we find thatthe condition for lepton propagation change significantly in a rel-atively short phase period, when the termination shock is gettingfurther and closer to the pulsar (phases periods∼ 0.2 − 0.45 and∼ 0.9− 0.96). In Table 2 we show the differences in these geomet-rical parameters for a few characteristic orbital phases. Note thateven when they are essential to understand the formation of thelightcurve and phase dependent highenergyy energy photon spec-tra, all the above features are based on the simple approximation ofthe geometry of the termination shock. In a more realistic scenario,the transition between the terminated and unterminated wind, andits connection with orbital phases and inclination are expected tobe more complicated yet.

2.9 Opacities along the LS 5039 orbit

The conditions for leptonic processes for this specific binary can bediscussed based on optical depths to ICS andγγ absorption. Thetarget photons for IC scattering of injected electrons and for e+e−

pair production for secondary photons are low energy photons ofblack-body spectrum with temperatureTs = 3.9× 104, which is thesurface temperature of the massive star. This is an anisotropic fieldas the source of thermal photons differ from the place of injection


Table 2.Geometrical properties for specificphasess along the orbit. p1 andp2 stand for the phases at which the angle to the observer is 90o; r1 andr2 stand forthe phases between which the shock in the observer directionis non-terminated. The separationd and the shock distancers are given in units of star radiiRs.

i = 300 i = 600

φ d αobs rs φ d αobs rs

Periastron 0.0 2.25 1110 0.8 0.0 2.25 1300 0.6SUPC 0.06 2.45 1200 0.8 0.06 2.45 1500 0.6p1 0.27 4.06 900 3.2 0.27 4.06 900 3.2r1 0.45 4.7 720 . . . 0.36 4.5 730 . . .Apastron 0.5 4.72 690 . . . 0.5 4.72 430 . . .INFC 0.72 4.1 600 . . . 0.72 4.1 300 . . .r2 0.89 2.8 760 . . . 0.92 2.6 800 . . .p2 0.94 2.47 900 2.4 0.94 2.47 900 2.4

Figure 9. Opacities to ICS for electrons and pair production for photons as a function of electron/photon energy. Opacities were calculated up to infinity forinjection at the periastron (left panel) and apastron (right panel) pulsar distance for different angles of propagation with respect to the massive star.

Figure 7. The LS 5039 pulsar angle to the observer as a function of phasealong the orbit for two different values of the binary inclination anglei.INFC, SUPC, periastron, and apastron phases are marked.

of relativistic electrons, which for simplicity is assumedto be atthe pulsar location for any given orbital phase. For fixed geometryparameters, the optical depths change with the separation of thebinary (in general with the distance to the massive star), the angle ofinjection (the direction of propagation with respect to themassivestar), and the energy of injected particle (the electrons orphotonsfor γ absorption).

Figure 8. The distance from the pulsar to the termination shock in the di-rection to the observer (in units of stellar radiusRs), for the two differentinclination angles,i, analyzed in this paper. INFC, SUPC, periastron, andapastron phases are marked. Additionally, the gray line shows the separa-tion of the binary (also in units ofRs) as a function of phase along the orbit.

To have a first handle on opacities, we have calculated the op-tical depths adopting the binary separation at periastron and apas-tron. In case of the optical depths for photons we had to do an addi-tional simplifying assumption to allow for a direct comparison withthe optical depth for electrons. In the discussed model of this pa-


per, photons are secondary particles and they do not have thesameplace of injection as electrons, but because the path of electrons upto their interaction is much smaller than the system separation, weassume now the same place of injection for photons and electrons:for a first generation of photons, this is sufficient for comparison.

Results are shown in Fig. 9 and the needed formulae for itscomputation are given in the Appendix. The optical depths for ICScan be as high as∼ few 100 for significant fractions of the orbit, andare still above unity for energies close to 10 TeV. With respect tothe injection angle, the opacities are above∼ 1 for all propagationangles, except the outward directions at the apastron separation.(Recall that outward –inward– directions refer toα-angles close to0o and 180o, respectively, as shown in Fig. 1). In the case ofγγ

pair production, the interactions are limited to specific energies ofthe photons and are strongly angle-dependent. The most favorablecase for pair creation is represented by those photons with energy inthe range 0.1 to few TeV, propagating at& 50o. The highest opticaldepths to pair production are found for directions tangent to themassive star surface: for propagation directly towards themassivestar pair production processes are limited by the star surface itself(for periastron the tangent angle is at∼ 154o; for apastron, it isgiven at an angle of∼ 168o). This is better shown in Fig. 10.1

The opacities for pair production are characterized by maximawhich change with respect to the angle of photon injection. Forinward directions (in general for angles larger then 90o), the peakenergy is higher,∼ 102 GeV − 3 × 102 GeV, than for outwarddirections, where we find∼ 3×102 GeV−5 TeV. For energies up tothepeakfor pair production, the IC process dominate for the sameangle of particle injection, but for higher energies the opacities forγ-absorption become slightly higher than for ICS.

Cascading processes are effective when the interaction pathfor electrons and photons are short enough so that a cascade can beinitiated already inside the pulsar wind zone. As we can see in Fig.9, the opacities at high-energy range (Klein-Nishina) are compara-ble to the maximal opacities forγγ absorption. This means simi-lar probability of interaction for both photons and electrons. If theelectrons scatter in the Klein-Nishina regime, the photon produced,with comparable energy to that carried by the initiating electron, islikely to be absorbed, and ae+e− cascade is produced.

The dependence of the optical depths upon the orbital phasefor specific parameters of the LS 5039 is shown in Fig. 11. Theseopacities are calculated up to the termination shock in the observerdirection. The presence of the shock (at a distancers) limits the op-tical depths. Since this parameter is highly variable alongthe orbit(see Fig. 8), its influence on the optical depth values is not minor.As the angle to the observer, which defines the primary injectionangle, vary in the range (90− i,90+ i) (for INFC and SUPC, respec-tively), we found that there is a range of orbital phases for which thePWZ is non-terminated. In that case the cascading process –whichdevelops linearly– is followed up to electron’s complete cooling(defined by the energyEmin = 0.5 GeV). The optical depths in thenon-terminated pulsar wind are thus comparable to those presentedin Fig. 9, taking into account the differences between the separationand angle of injection.

Apart from the condition for electron cooling, in the case of

1 The range of parameters depicted in this figure is just to emphasize thatthere could also be propagation and interactions outside the innermost partsof the system, in general, although of course with a reduced opacity. Theactual range of angles towards the observer allowed in a particular systemdepend on the orbit inclination.

the terminated wind the cascade is followed up to the moment whenthe electron reaches the shock region, whatever happens first. Ascan be seen from comparison of Fig. 9 and 11, the opacities aresignificantly smaller when the propagation is limited. For instance,as a comparison we can choose the electron injection at perias-tron, the angleαobs = 110o (130o for i = 60o), and fix the energyto ∼ 1 TeV. Then, the opacities along the orbit up to the termi-nated shock are∼ 2 − 3 while for the unterminated processes theopacity are∼ 10− 20. A direct comparison for specific energiesof injected particles and the two inclinations considered is shownin Fig. 12. Moreover, the opacities for both processes decreasesalong the propagation path, see Fig. 13, as the angle to the observeralso decreases. This indicate that even if the photons propagate veryclose to the massive star, the probability for absorption can be muchsmaller than that at its injection place, and finally become less then1 at the distance from the pulsar∼ 3−4Rs, for an energy of 10 TeVand 1 TeV respectively.

3 MONO-ENERGETIC ELECTRON INJECTION

As a benchmark case, we first consider a mono-energetic injectionof primary leptons at the pulsar position. The energy of the mono-energetic electron is fixed at 10 TeV, what is to be understoodas anexample from where to extract the physics underlying more com-plex results that are shown in the next Section. A comparisonwithother values of lepton energy is also made. It is to be noted thatthe 10 TeV assumption for the energy of leptons could be in agree-ment with models of pulsar winds, which give estimations fortheLorentz factors of the wind as high as 106 − 107 (e.g. Kennel andCoroniti 1984). Indeed, the observation of the Crab pulsar in thehigh-energy range shows that emitted photons has maximal energy∼ 80 TeV. Observational data for LS 5039 binary shows that thephotons escaping from the system have energy up to∼ few TeVand possibly beyond, implying a higher energy for primary parti-cles.

Since we continue working with the case of LS 5039, we adoptthe geometry and orbital parameters based on the work by Casareset al. (2005b), shown in Table 1. Specific values for the separa-tion and angle to the observer at different orbital positions, anddistance from the pulsar to the shock front, for both inclinationsdiscussed herein, are gathered in Table 2. Following the normal-ization formula given by Eq. (6), for the primary lepton energyE = 10 TeV and the nominal value of pulsar spin-down powerLsd = 1037erg s−1, and assuming a distance to the sourced = 2.5kpc (d2 = 5.6× 1043 cm2) we get:

Ne+e− = β × 6.24× 1035 δ(E − 10TeV) s−1. (20)

Then the normalization factor for electrons traveling towards Earthis A = Ne/4πd2 = β × 8.83× 10−10 s−1 cm−2.

The opacities for ICS andγγ absorption along the orbit of thesystem for an electron with energy 10 TeV are less than unity,sothese processes are not very efficient. We already showed this inFig. 12. The statistics for the simulations have been fixed accord-ingly – the number of injected leptons have to be high, and depend-ing on the computational time involved it was fixed from 103 to5 × 102 for different orbital phases. The produced photon spectraare normalized then to one injected particle, and the overall resultsshown, with the normalization factorA.


Figure 10. Opacities to ICS for electrons and pair production for photons as a function of the injection angle,α, in the LS 5039 system, for different initialenergies of the particles (left panelE = 1 TeV, right panelE = 10 TeV) and distance to the massive star (including periastron and apastron separation).Opacities were calculated up to infinity. The injection angle is provided in the plot and for pair production process theyare given in the same order as for ICS.

Figure 11.Opacities to ICS for electrons and pair production for photons as a function of the orbital phase,φ in the LS 5039 system, for different inclinationof the binary orbit (left paneli = 300, right paneli = 600) and energy of injected particle. Opacities are calculatedfor specific phases up to the terminationshock in direction to the observer.

3.1 Lightcurve and broad-phase spectra

Based on the simulations for the specific phases along the orbit, thephoton spectra and lightcurves in different energy ranges (10− 100GeV, 10− 103 GeV, and above 1 TeV) were calculated for twoassumed inclinations of the orbit of the system,i = 30o and i =60o. The highest energy range corresponds to the data presentedbyH.E.S.S. (Aharonian et. al. 2006). To allow for a comparison, thephoton spectra were calculated also for a primary electron energyof 1 TeV for specific orbital phases (periastron, apastron, SUPCand INFC).

The photon spectra (SED) are presented in Fig. 14. All thespectra for primary electron energyE = 10 TeV are hard, present-ing a photon index< 2, and they have significantly higher flux atthe highest energy range, close to the initial primaries’∼ 10 TeV, asthe ICS of primary pairs mainly occur in Klein-Nishina range. Be-low this peak the spectra are well represent by the power-lawwithphoton index close to∼ −1, albeit one can also notice the changein the photon index from SUPC to INFC (harder spectrum) andthe notably different energy cutoffs. These features are discussedbelow.

While for phases around SUPC the PWZ is limited in the di-rection to the observer, for the opposite phases the PWZ is unter-minated. On the other hand, for INFC the angle to the observeris at its minimum (and also the separation of the system is larger)what causes the decreasing of optical depth to both processes con-sidered. In contrary to the INFC phases, those at SUPC presentstrong absorption features at an energy range from∼ 0.1 to fewTeV, which cause the spectra to be cut at lower energies. Thisis dueto the dependence of the optical depth toγγ absorption on the en-ergy of photon and the influence of geometry in defining the photonpath towards the observer. The larger the angle to the observer, thestronger the effects of absorption are, what can be seen from com-parison of the results for two inclination angles (SUPC and perias-tron phases). Notice, that fori = 60o the angle to the observer at-tains the largest value of the discussed examples,αobs = 150o. Theabsorption for SUPC phases takes place already in the PWZ, whatcan initiate the cascading processes. Photons which pass throughthe shock region can be also absorbed in the MSWZ. The opacitiesfor γγ absorption also strongly depend on photon energy (see Fig.12). High energy photons, with energy from few GeV up to fewTeV, are the most likely to be absorbed, as the opacities decrease


Figure 12.Comparison of the opacities to ICS for electrons and pair production for photons along the orbit of LS 5039, for fixed particle energy but differentinclination angles (i = 300, i = 600).

Figure 13. Opacities to ICS for electrons and pair production for photons as a function of the distance from the injection place along the propagation pathfrom the pulsar side,dr, in the LS 5039 system. Opacities are calculated for two values of the particles initial energies,E = 1 TeV (left panel), andE = 10TeV (right panel), and different angles of injection.

with energy for all shown angles of the photon injection. Indeed,the number of the photons absorbed in MSWZ is about∼ 10− 20% of the number of photons reaching the shock region and the effectis significant in the final spectra. On the other hand, the cascadingin PWZ produce lower energy leptons and photons (in the cooling

process of secondary pairs) and this causes a higher photon fluxat energies up to few 10 GeV. The processes in PWZ are limitedby the presence of the shock, what influence the photon productionrate. For INFC phases the absorption ofγ-rays is minor, as the pho-tons propagate outwards of the massive star; once produced,most


Figure 14.VHE photon spectra for specific phases, at periastron, apastron (thin lines), INFC, and SUPC (bold lines), for two inclination angles,i = 30o (top)and i = 60o (bottom). The spectra were calculated for two different primary energy of electrons,E = 10 TeV (left) andE = 1 TeV (right). The few freeparameters involved in the mono-energetic model and their assumed values are given in Table 3. For direct comparison, the same normalization factor wasused in case of each injection energy. Discussion on the non-uniqueness of model parameters is given in Section 2.

Table 3.Model parameters for the example of mono-energetic injection considered


Spin-down power of assumed pulsar Lsd 1037 erg s−1

Fraction of spin-down power injected in relativistic electrons β 10−2

The energy of injected pairs E0 10 TeV

photons can escape from the system (also because of the thresholdto pair production). Higher energy photons are produced mostly inthe first IC interaction of the primary pairs, while further electroncooling, not limited by the termination shock, supply the spectrumin lower energy photons. This flux is not as high as in the case ofSUPC phases, where most of the cascading takes place.

Similardependenciess, both for INFC and SUPC phases, arepresent in the spectra obtained from the simulations for primaryenergy of pairsE = 1 TeV. As the optical depths to IC scattering arehigher in that case, the processes are more efficient and the numberof produced photons are higher. To give an example, most photonsof energy> 20-30 GeV at SUPC-periastron phase are absorbed inMSWZ.

Comparing the photon spectra produced at SUPC and INFC,

an anticorrelation between GeV and TeV photon fluxes is evi-dent, at least comparing the spectra at energies below and above∼ 10 GeV. All these effects play a role in the formation of theγ-ray lightcurve, shown in Fig. 15. The lightcurve in the high-est energy range (> 1 TeV) has a broad minimum around SUPC,0.96< φ < 0.25. From the comparison of the position of the termi-nation shock along the orbit, the opacities to ICS andγγ absorptionwe see that this minimum is formed despite being at phases with thehighest opacities to both processes considered and so having an ef-fective cascading in the PWZ region. This minimum is mainly dueto the absorption of the photons which get through the shock andpropagate into the massive star wind region. The broad maximumin the high-energy lightcurve is formed at opposite phases aroundINFC, 0.4 < φ < 0.9. The maximum for higher inclination is also


Figure 15.Theoretical lightcurves for mono-energetic injection andtwo different inclination angles. As before: i=600, solid; i=300, dot-dashed.

characterized by a local minimum close to the INFC phase, what isthe result of the IC opacity dependence on the propagation angle.The range of INFC phases corresponds to the unterminated pulsarwind in the observer direction. The opacities for both inclinationshave local peaks in this range, what was discussed in the previ-ous paragraph. The local minimum in photon flux within the INFCrange reflects a similar behavior of the opacities. So, as faras thepropagation of the particles in the PWZ is unterminated, thehigh-energy lightcurve formation is in agreement with the dependenceof the optical depths. Additionally, we can also see that thefirst lo-cal peak in this lightcurve is formed earlier in phase for inclinationi = 60o (φ ∼ 0.35), what is also in agreement with the depen-dence shown by the distance to the termination shock. Note thatthe second peak in the broad TeV lightcurve maximum, at∼ 0.85,is higher than the first one, at∼ 0.4 what is the result of a change inthe separation of the system together with the angle to the observer.

3.2 Understanding peaks and dips in the lightcurve

Let us summarize the connections between model features andtheγ-ray lightcurve formation. The geometrical conditions forlep-tonic processes change significantly along the orbit, and they aremore efficient for phases around SUPC. However, these processes,as we can already see from the opacities dependence on the or-bital phases, is limited by presence of the termination shock. Thiscauses that the best conditions for very high-energy photonpro-

duction, and finally escaping from the system, occur for an specificcombination of the angle to the observer, the separation of the sys-tem, and the distance to the shock from the pulsar side. From Figs.11 and 8 we can see that this happens at the phases∼ 0.3 − 0.4and∼ 0.9 what reflects in the TeV lightcurve. At phases close toINFC photons are produced mainly in the primarye+e− pairs cool-ing when the propagating electron undergo frequent scatterings. To-gether with the fact that there is no efficient absorption of photonsonce they are produced, they finally escape from the system, whatyields to the broad maximum in the TeV lightcurve. On the otherhand, at phases around SUPC, high-energy photons are absorbedwhen propagating through the system in the MSWZ, what causesa dip in the lightcurve. For these phases many more lower energyphotons are produced in cascades in the PWZ, what yields to amaximum in the GeV lightcurve for SUPC, anticorrelated withthebehavior at TeV energies.

A condition which enlarges the photon population is a highopacity to ICS (depending on the angle of propagation and dis-tance to the massive star, and is best for SUPC phases). The con-ditions which limits the photon production are high opacities tophoton absorption (which peaks at the same phases than ICS, andis also angle dependent, as related to the photon field density), andthe presence of the termination shock in the direction to theob-server (which results in a limitation for the cascade developmentand possible subsequent absorption in the massive star windre-


gion, affecting especially SUPC phases). The interrelationship ofthese processes anddependenciess can not be over emphasized.

4 A POWER-LAW INTERACTING ELECTRONDISTRIBUTION

Albeit useful for understanding the interrelationship of the com-ponents of our model, a mono-energetic injection failed to repro-duce the high-energy spectra from LS 5039. Thus, for furtherex-ploration of theγ-ray production model we set a new assumption:the energy distribution of the interactinge+e− pairs is given by apower-law spectrum. We will additionally assume that the power-law may be constant or vary along the orbit. Indeed, this latter casemay be the result when the steady distribution of pairs is managedby equilibrium between injection and losses (radiative andadia-batic losses which relative dominance can depend on the orbitalphase), and/or if the injection itself on top of the light cylinder orthe acceleration efficiency in a close-to-the-pulsar shock is phasedependent. To simplify the problem and motivated by the differentobservational behavior found, we have assumed that two differentspectral indices correspond to the two broad orbital intervals pro-posed by HESS (Aharonian et. al., 2006). For direct comparisonwe have specified the interval around the inferior conjunction (withthe apastron phase): 0.45 < φ < 0.90, and around superior con-junction (including the periastron phase):φ < 45 andφ > 0.90as being bathed by different electron distributions. The results forpower-laws distribution were already summarily presentedin ourearlier work Sierpowska-Bartosik and Torres (2007a,b) andwe re-fer to these works for further details. The agreement in bothspectraand lightcurve is notable (particularly for the case of a variable lep-ton spectrum along the orbit), as can be seen in Figs. 16 and 17discussed below.

4.1 Lightcurve and broad-phase spectra

To construct the lightcurve, spectra for over 20 orbital phases werecalculated to cover the whole orbit of LS 5039. The specific av-eraged spectra were obtained based on the same orbital intervalsas presented in H.E.S.S. data. The averaged spectra were obtainedsumming up individual contributions from orbital spectra in givenphases, each with a weight (δt/T) corresponding to the fraction oforbital time that the system spends in the corresponding phase bin.The phase of the adopted spectrum is assumed valid in that smallbin, and is obtained at a central phase in the middle of the bin. Then,comparing the observational and theoretical averaged spectra, theparameterβwas estimated. Both for constant and variable injectionmodel, we get that the fraction of the spin-down power in the pri-mary leptons has to be at the level of∼ 1 %. With this parameter inhand each of the single spectra can be equally normalized.

It is worth noticing how well these lightcurves compare withthose in the work by Bednarek (2007), at least for some of the spe-cific phases considered by him. Bednarek also included cascadingin his simulations, and the physical input of his model (althoughin the case of a microquasar scenario) is similar to ours. As are-sult, the anti-correlation phenomena (from GeV to TeV energies) isalso a result of his work. The spectrum along the orbit with respectto H.E.S.S. datapoints and the possible short-timescale variability(see below) was not provided by Bednarek, so that a comparisonwith these results is not possible. A detailed differentiation betweenthese two models is a key input for distinguishing (microquasar orγ-ray binary) scenarios, even when some assumptions are intrinsic

to each of the models, isolating the contribution of an equalphys-ical input can help decide on what object constitute the system LS5039. As an example: Bednarek did find in his model that a fixedinclination angle (i = 600) was needed in order to reproduce theshape of the H.E.S.S. lightcurve results, whereas in our case, as wesee in Figure 17 the influence of inclination is minor.

For completenesss, we mention that as noted above, H.E.S.S.has also provided the evolution of the normalization and slope of apower-law fit to the 0.2–5 TeV data in 0.1 phase-binning alongtheorbit (Aharonian et al. 2006). The use of a power-law fit was limitedby low statistics in such shorter sub-orbital intervals, i.e., higher-order functional fittings such as a power-law with exponential cut-offwere reported to provide a no better fit and were not justified.Todirectly compare with these results, we have applied the same ap-proach to treat the model predictions, i.e., we fit a power-law in thesame energy range and phase binning. For completeness we showhere this comparison in Fig. 18 (taken from Sierpowska-Bartosikand Torres 2008) in the case of a variable lepton distribution alongthe orbit. We find a rather good agreement between model predic-tions and data. Results for constant lepton distribution along theorbit can be seen in Sierpowska-Bartosik and Torres (2007a).

In Fig. 16 the SED in H.E.S.S. energy range are shown forboth, constant and variable lepton distributions along theorbit, andtwo inclination of the systemi = 30o andi = 60o. It was shown inthe previous Section that in case of the mono-energetic injection,there are photon flux differences in INFC phase due to the changein the angle to the observer. In the power-law distribution modelthis effect is less significant due to the contributions of differentenergies, and the dependence on energy of the processes involved.Based for instance on the injection of constant spectrum with Γe =

−2.0 we can see that there is no significant differences involvingthe inclination.

Exploring the more evolved model of the variable injection wesee that the steeper the primary spectrum is, the higher the photonflux at lower energy range (SED for lower energies were shownby Sierpowska-Bartosik and Torres 2008). For constant lepton dis-tribution along the orbit, the differences in photon flux below 100GeV are not so significant, but the difference gets important in thecase of a variable lepton distribution (the difference between bothmodels in the lower energy range is about one order of magnitude).Overall, the variable lepton distribution provides a better agree-ment with all data (note that it has only two extra free parameterswhen compared with the constant distribution case, see Table 4, butmatches more than 10 data points that were missed in the previouscase). In particular, it is worth noticing that there is goodagree-ment with the H.E.S.S. spectra at an energy∼ 200 GeV where bothSUPC and INFC spectra coincide and above which the photon fluxfor INFC is higher then the flux for SUPC dominating in the lowerenergy range; i.e., the energy where the anticorrelation begins (seeSierpowska-Bartosik and Torres 2008). It is also interesting to re-mind that absorption alone would produce strict modulation(zeroflux) in the energy range 0.2 to 2 TeV whereas the observationsshow that the flux at∼ 0.2 TeV is stable. Additional processes,i.e. cascading, must be considered to explain the spectral modula-tion. Our model have these processes consistently includedand thelightcurve details arise then as the interplay of the absorption ofγ-rays with the cascading process, in the framework of a varyinggeometry along the orbit of the system.


Table 4.Model parameters for interacting electrons described by power-laws


Spin-down power of assumed pulsar Lsd 1037 erg s−1

VHE cutoff of the injection spectra Emax 50 TeV

Constant lepton spectrum along the orbit

Fraction ofLsd in leptons β 10−2

Slope of the power-law Γe −2.0

Variable lepton spectrum along the orbit

Fraction ofLsd in leptons at INFC interval β 8.0× 10−3

Slope of the power-law at INFC interval Γe −1.9Fraction ofLsd in leptons at SUPC interval β 2.4× 10−2

Slope of the power-law at SUPC interval Γe −2.4

Figure 16. Very high-energy spectra of LS 5039 around INFC and SUPC, together with the theoretical predictions in equal phase intervals for power-lawselectron distribution and two different inclination angles. The free parameters involved in our model and their assumed values are given in Table 4. AfterSierpowska-Bartosik & Torres (2008).

4.2 Testing with future data

Apart from possible testing with GLAST (at the level of lightcurve,spectra, hardness ratios, and differentiation between constant andvariable electron distribution, see Sierpowska-Bartosikand Torres2008), we can provide further possible tests at highγ-ray ener-gies. It was already said that power-laws do not always present thebest fit to the specific spectra along the orbit. Even when fittingsuch power-laws to the theoretical predictions provide an excellentagreement with data (see Fig. 18), observations with largerstatistics(with H.E.S.S., H.E.S.S. II, or CTA) could directly test themodel inspecific phases. A model failure in specific phases would allow fur-ther illumination about the physics of the system. Figure 19showsthe evolution of individual spectrum in the best fitting caseof vari-able lepton distribution along the orbit, at individual phase bins,from 0.1 to 0.9, for testing with future quality of data.

5 CONCLUDING REMARKS

We presented the details of a theoretical model of closeγ-ray bi-naries, and applied it to the particular case of LS 5039. The modelassumes a pulsar scenario, where either the pulsar or a close-to-the-pulsar shock injects leptons that after being reprocessed by lossesto constitute an steady population, are assumed to interactwith thetarget photon field provided by the companion star. The modelac-counts for the highly variable system geometry with respectto theobserver, and radiative processes; essentially, anisotropic Klein-Nishina ICS andγγ absorption, put together with a Monte Carlocomputation of cascading. The formation of lightcurve and spec-tra in this model was discussed in detail for the case where theinteracting leptons are assumed mono-energetic consideredd as abenchmark model) and described by power-laws. The influenceofgeometry in the pulsar wind zone processes is large and was spelledin detail in this work.

Comparing the interacting models of mono-energetic lep-tons and power-law distributions we can see similar dependen-


Figure 17. H.E.S.S. run-by-run folded (ephemeris of Casares et al. 2005) observations (each point corresponds to 28 min of observations) of LS 5039with the results of the theoretical model for power-law distribution and two different inclination angles for i=300 (left) and i=600 (right). Light (green) linesstand for results obtained with a constant interacting lepton spectrum along the orbit, whereas dark lines (black) correspond to a variable spectrum. AfterSierpowska-Bartosik & Torres (2008).

Figure 18. Shaded areas in the left (right) panel show the change in the normalization (photon index) of a power-law photon spectra fitted to the theoreticalprediction for each of the 0.1 bins of phase in the case of a variable lepton distribution along the orbit. The two different colors of the shading stand for thetwo inclination angles considered. The size of the shading gives account of the error in the fitting parameters. Data points represent the H.E.S.S. results for asimilar procedure: a power-law fit to the observational spectra obtained in the same phase binning. From Sierpowska-Bartosik & Torres (2008).

cies for the spectral changes along the orbit and the GeV toTeV lightcurves. Thus, the simplified benchmark case of a mono-energetic population is useful to understand some of the aspects re-garding the formation of the observational features. For the mono-energetic case, we have shown the spectra produced at characteris-tic phases (INFC, SUPC, periastron, apastron) for primary energiesof 1 TeV and 10 TeV. In the power-law distribution model, the spe-cific features of the photon spectra and the lightcurves producedfor an specific primary electron energy overlap. We can stillnoticethe absorption features in the spectra produced at SUPC. We havediscussed effects solely based on the optical depths and the generalgeometrical dependence along the orbit, as well as on the presenceof the shock which terminates the pulsar wind in the direction tothe observer at SUPC phases.

We have provided full details of our finding: a power-law lep-ton distribution describes very well the phenomenology found inthe LS 5039 system at all timescales, both flux and spectrum-wise(two power-laws corresponding to INFC and SUPC broad phase-

intervals provide a complete match to the phenomenology observedby H.E.S.S.). However, this model certainly has room for improve-ment. The assumption of power-laws is an approximation to a morecomplex scenario where the real interacting lepton population is theresult of a full escape-loss equation. In addition, we are not consid-ering yet the multiwavelength emission at lower energies, since weleft out of our description, e.g., the synchrotron emissionof elec-trons accelerated at the shock and the morphology of the shockalong the orbit, what we expect to discuss elsewhere. Eventhough,after the a posteriori success of the model in the reproduction ofsuch a complex observed phenomenology, its predictive power isimpressive. Other than system scalings that are fixed by multiwave-length observations, the model is based on just a handful of freeparameters, and it is subject to tests at high and very high-energyγ-ray observations with both GLAST (described in Sierpowska-Bartosik & Torres 2008) and future samples of data at higher ener-gies, where more statistics at finer phase bins can determinebetterthe spectral evolution along the orbit of this interesting system.


Figure 19.Evolution of individual spectrum in the case of variable lepton distribution along the orbit, at individual phase bins,from 0.1 to 0.9. The shadowrepresents the GLAST energy coverage; the rest of the span ofthe x-axis can be observed by ground-based facilities.

APPENDIX

This Appendix introduces further essential details concerning ge-ometry and formulae for the implemented process of InverseCompton scattering ande+e− pair production in the anisotropic ra-diation field of the massive star.

Thermal radiation

The source of the thermal radiation field is the massive star ofearly type (O, Be, WR). The spectrum is described by Planck’slaw, which differential energy spectrum (the number of photons ofgiven energyǫ per unit energydǫ, per unit solid angleΩ, per unitvolumeV) is given by:

n(ǫ) =dn(ǫ,Ω)dǫdΩdV

=4π

(hc)3

ǫ2

eǫ/kTs − 1, (21)

whereǫ is thermal photon energy,h is the Planck constant, andk isBoltzmann constant.

αγ

θ

x

x

x

x

Rβ S S

ε

e

i

0

Figure 20. Basic parameters for photon and electron propagation and theirinteraction in the anisotropic radiation field of the massive star. The injec-tion is produced at a distancexi and angleα. Rs denotes the massive starradius, and the tangent direction to its surface is given by angle βs. Thepropagation path is denoted byxγ andxe, while the interaction occur at dis-tancex0. The scattering angle for electron-photon (ǫ) or photon-photon (ǫ)interactions isθ.

Gamma ray absorption ande+e− production: opacity andgeometry

The optical depth toγ-photon absorption in the radiation field of themassive star up to infinity can then be calculated from the integral:


τγγ(Eγ , xi , α) =∫ ∞

0λ−1γγ(Eγ , xi , α, xγ) dxγ, (22)

whereλ−1γγ is a photon interaction rate toe+e− production in an

anisotropic radiation field andxγ is its propagation length. Whenthe propagation occur toward the massive star surface the integra-tion is performed up to the stellar surface. The photon interactionrate,λ−1

γγ , is related to a photon of energyEγ injected at a distancexi from the massive star, at angleα (see Fig. 20), and is given bythe formula:

λγγ−1(Eγ , xi , α, xγ) =

∫

(1+ µ)dµ∫

dφ∫

n(ǫ)σγγ(β)dǫ, (23)

wherexγ is the distance to the interacting photon from the injec-tion place along its propagating path. The integration limits will bediscussed in a different part of the Appendix. For now we focus onthe variableµ in the first integration. It is the cosine of the photon-photon scattering angleµ = cosθ (see Fig. 20). The angleφ is theazimuthal angle between the photon (γ-ray) propagation directionand the direction to the massive star, whileφs gives its limit valuefor the direction tangent to the massive star surface (see Fig. 22).The cross section toe+e− production is denoted asσγγ(β), wherethe parameterβ in the center of mass system is

β =√ω2 −m2/ω, (24)

with

ω2 =12

Eγǫ(1+ cosθ) (25)

being the photon energy squared (in this notation it is assumed alsothatc = 1). From this we get

β2 = 1− 2m2/Eγǫ(1+ µ). (26)

The kinematic condition for the angleθ which defines the thresholdfor thee+e− creation is given by expression:

µ > µlim =2m2

Eγǫ− 1. (27)

To simplify the equations we rewrite the internal integration in Eq.(23) making use ofI1(µ), such that,

I1(µ) =∫

n(ǫ)σγγ(β)dǫ. (28)

With the replacement

β2 = 1− a/ǫ, (29)

where

a = 2m2/Eγ(1+ µ), (30)

and defining the constantS = 8π/(hc)3, the spectrum of thermalphotons is now given by the formula:

n(ǫ) = Sa2

(1− β2)2

1

e(a/[(1−β2)kTs)] − 1. (31)

Substituting in the internal integralI1(µ), and introducing the inte-gration variable toβ, via

dǫ = 2aβ/(1− β2)2 dβ, (32)

yields to the integral:

I1(a) = 2S∫ 1

0

a3

(1− β2)2

1

e(a/(1−β2)kTs) − 1σγγ(β)dβ. (33)

The lower limit of integration is from the energy condition for theprocessγ + γ → e+e−, i.e., it follows from the threshold condition

Figure 21.The plot of the functionC(b) = I1(β)/C1, where the integralI1

is given by Eq. (36).

Eγǫ(1 + µ) = 2m2, whereω = m. . The upper integration limitcomes from the relativistic limitω ≫ m, where we getβ ≈ 1. Toproceed forward, we introduce a dumb variable,b, by a = kTsb, toget:

I1(b) = 2S∫ 1

0(kTs)3b3σγγ(β)

β

(1− β2)4

1

e(b/(1−β2)) − 1dβ. (34)

The cross section fore+e− pair production is (Jauch andRohrlich, 1980):

σγγ(β) =12

r20π(1− β2)[(3 − β4) ln

1+ β1− β

− 2β(2− β2)], (35)

where r0 is the classical electron radius, andσT =83πr2

0 is theThomson cross section. When putting this expression into the in-tegralI1(b) (Eq. 34) we get finally,

I1(b) = C1316

b3

∫ 1

0

[

(3− β4) ln1+ β1− β

− 2β(2− β2)

]

×

β

(1− β2)3

1

e(b/(1−β2)) − 1dβ, (36)

with C1 = 16π(kTs/hc)3σT .Weparametrizee the internal integral and write it as a function

of b,

C(b) ≡ I1(β)/C1. (37)

The plot for this function is shown in Fig. (21). Then, the integralwe are after can be written asI1(b) = C1C(b) and

λγγ−1(Eγ , xi , α, xγ) = C1

∫

(1+ µ)dµ∫

dφC(b). (38)

In the second internal integral, we have to fix the limits (havingin mind that the angleφ depends on the angleθ, then also on thevariableµ),


I2 =

∫ φs

−φs

C(b)dφ = 2φsC(b) = C(b)Φ(µ). (39)

The angleφs determines the maximal azimuthal angle ofγ-ray pho-ton propagation with respect to the direction of the thermalphoton,so that it gives the directions tangent to the star surface. This con-dition for φs can be determined from the spherical triangle (shownin Fig. 22). From the cosine theorem of spherical trigonometry

cosβs = cosθ cos (π − α) + sinθ sin (π − α) cosφs, (40)

whereβs is defined as in Fig. (22). After some algebra we get:

cosφs =cosβs + µ cosα√

1− µ2 sinα, (41)

where sinβs = Rs/xi , while 0 6 φs 6 π. The angleα determinesthe direction of incomingγ-ray photon (see Fig. 20).

The limits of integration with respect to the parameterµ (seeFig. 22), are the range of angles for soft photons coming fromthestar,

λγγ−1(Eγ , xi , α, xγ) = C1

∫ µ2

µ1

(1+ µ)Φ(µ) dµC(b), (42)

whereµ1 = cosθs1, µ2 = cosθs

2. Using basic trigonometrical depen-dencies between the angles,θs

1 = π− α− β andθs2 = π− α+ β (Fig.

22, left), we get two expressions:

µ1 = − cosα cosβs + sinα sinβs,

µ2 = − cosα cosβs − sinα sinβs. (43)

Depending on the angle ofγ-photon propagationα we canprecise the conditions for the above integration limits. Wedistin-guish three regimes:

(i) the range forα towards the star surface: (π − βs 6 α 6 π),(ii) the “middle” range: (βs < α < π − βs),(iii) the range outwards of the massive star surface: (06 α 6

βs).

Integration overµ can be simplified by separating the main inte-gral, Eq. (42), in the sum of the corresponding integrals fortheseregimes, each depending on their limitsµ1, µ2, µlim (see Eq. 43 and27). A simplification occurs for the specific range of the angleθ for which the functionΦ(µ) = 2φs (Eq. 39) is maximal, i.e.,Φ(µ) = 2π, what corresponds to the scenario when the photon prop-agate toward the star. Another simplification happens whenΦ(µ) isminimal, i.e.,Φ(µ) = 0, for an outgoing photon. To simplify theformulae for numerical implementation we introduce the functionf (µ,b) = C1C(b)(1+ µ) such that


∫ µ2

µ1

f (µ,b)Φ(µ) dµ, (44)

where we introduced internal functionf (µ,b) = C1(1+ µ)C(b). Tofasten the simulations the integration limits can be fixed followingthe rules (note that relation (µ1 > µ2) is always fulfilled):

(i) for “inward” propagation :

• if (µlim 6 µ2 6 µ1):λγγ−1(Eγ , xi , α, xγ) =

∫ µ1

µ2f (µ,b)Φ(µ) dµ +

∫ 1

µ1f (µ,b)2π dµ,

• if (µ2 6 µlim 6 µ1):λγγ−1(Eγ , xi , α, xγ) =

∫ µ1

µlimf (µ,b)Φ(µ) dµ +

∫ 1

µ1f (µ,b)2π dµ,

• if (µ2 6 µ1 6 µlim):λγγ−1(Eγ , xi , α, xγ) =

∫ 1

µlimf (µ,b)2π dµ.

Apart from these, the case ofα = π can also be separately defined

(incoming at a central spot of the star) when (µ1 = µ2) and


∫ 1

µminf (µ,b)2π dµ, whereµmin is the smaller

value from (µ1, µlim).

(ii) for the “middle” range of propagation:


∫ µ1

µ2f (µ,b)Φ(µ) dµ,


∫ µ1

µlimf (µ,b)Φ(µ) dµ,

• if (µ2 6 µ1 6 µlim):λγγ−1(Eγ , xi , α, xγ) = 0.

(iii) for “outwards” directions:


∫ µ1

µ2f (µ,b)Φ(µ) dµ +

∫ µ2

µlimf (µ,b)2π dµ,


∫ µ1

µlimf (µ,b)Φ(µ) dµ,

• if (µ2 6 µ1 6 µlim):λγγ−1(Eγ , xi , α, xγ) = 0.

Apart from these, as before, the case ofα = 0 can also be defined(escaping radially from a central spot of the star) when (µ1 = µ2) iif (µ1 > µlim), and we haveλγγ−1(Eγ , xi , α, xγ) =

∫ µ1

µlimf (µ,b)2π dµ,

otherwiseλγγ−1(Eγ , xi , α, xγ) = 0.

The energy spectrum fore+e− pairs produced byγγabsorption process

To simulate the energy of an electron (positron) produced intheprocess of gamma absorption we follow the energydistributionn ofe+e− pairs (see Eq. 11). Thee+e− pair spectrum produced by a pho-ton of energyEγ at a specific distancexp is given by the expression:

dW(Eγ , xi , α, xγ)

dEedxγ=

∫

(1+ µ)dµ∫

dφ∫

n(ǫ)σ(Eγ ,Ee, µ)dǫ, (45)

whereEγ is the energy of the interacting photon,xγ is the propaga-tion length,xi the distance of the injection place form the massivestar,α is the angle of propagation, andEe is the energy of producedlepton (e+e−). The limits of integration are discussed in the follow-ing paragraph.

The energy spectrum of thermal photonsǫ is again given bythe Planck’s spectrumn(ǫ) (Eq. 21). The cross section for pair pro-ductionσ(Eγ ,Ee, µ) depends on the photon energyEγ and energyof electronEe. The variableµ = cosθ, anglesθ andφs are definedin Fig. 22).

The cross section is given by the formula (Akhieser &Berestezki, 1965):

σ(Eγ,Ee, µ) =r2

0

2m2

Eγ

1εcβcγc

× [2ε2

c −m2 + (ε2c −m2)A2(Ee, εc)

m2B2(Ee, εc) + ε2cA2(Ee, εc)

−2(ε2

c −m2)2A4(Ee, εc)(m2B2(Ee, εc) + ε2

cA2(Ee, εc))2], (46)

whereεc is the photon energy in centrum of momentum (CM) ofthe interacting photonsε2

c =12 Eγǫ(1 + µ). The parameterγc =

(Eγ + ǫ)/2εc is Lorentz factor in CM system andm = mec2 is theelectron mass. In Eq. (46), the following functions are introduced:

B2(Ee, εc) =(Ee − γcεc)2

(γcβc)2(ε2c −m2)

, (47)

A2(Ee, εc) = 1− B2(Ee, εc), (48)


ε

γ

θ

β s

π−αφs

Figure 22. Left: Spherical geometry for a photon absorption process inthe radiation field of the star (a thermal photon is denoted with ǫ). The angleβs isdefined by the line to the star center and tangent to its surface. The anglesα andθ define the direction of gamma-photon incoming towards the star. The angleα is measured from the central line (joining the star and injection place) whereas the angleθ, from the line of gamma photon propagation. The directionsof gamma-photon propagationdefinedd as “inward” and “outward” with respect to the star are shaded. Right: two dimensional geometry for definition of theparameterµ = cosθ, whereµ1 = cosθs

1 andµ2 = cosθs2 are given by Eq. (43). The azimuthal angleφs is the angle between the direction of gamma propagation

and propagation ofǫ..

whereβc is the velocity of the CM system in units of the speedof light, γcβc =

√

γ2c − 1. The internal integration in Eq. (45), i.e.,

that performed over the thermal photons spectrum, according to Eq.(46), is:

I1(Eγ ,Ee, µ) =8π

(hc)3

∫ ∞

ǫmin

ǫ

(eǫ/kTs − 1)1

εcβcγc×

[

2ε2c −m2 + (ε2

c −m2)A2(Ee, εc)m2B2(Ee, εc) + ε2

cA2(Ee, εc)−

2(ε2c −m2)2A4(Ee, εc)

(m2B2(Ee, εc) + ε2cA2(Ee, εc))2

]

dǫ. (49)

The lower integration limit is given by the condition:

ǫmin =m2Eγ

2Ee(Eγ − Ee)(1+ µ)=

Fκ(1− κ)

. (50)

In the last expression, we have introduced the new variablesκ =

Ee/Eγ andF = m2/2Eγ(1+ µ), whereκ ∈ (0, 1).For a photon energyEγ ≫ ǫ, we haveγc ≫ 1, β ≃ 1, and

εcβcγc = Eγ/2. After rewriting Eq. (49) we get


(hc)3

2Eγ

∫ ∞

ǫmin

ǫ

(eǫ/kTs − 1)f (ǫ) dǫ, (51)

where, to simplify the formula, we have defined

f (ǫ) =2ε2

c −m2 + (ε2c −m2)A2(Ee, εc)

m2B2(Ee, εc) + ε2cA2(Ee, εc)

−

2(ε2c −m2)2A4(Ee, εc)

(m2B2(Ee, εc) + ε2cA2(Ee, εc))2

. (52)

To proceed forward, we introduce the variablez = ǫ/kTs, sothatkTsdz= dǫ, and the integral is now given by the expression:


(hc)3

2(kTs)2

Eγ

∫ ∞

zmin

zez − 1

f (z) dz, (53)

where the lower limit of the integral is given byzmin = ǫmin/kTs,

zmin = m2/(2EγkTs(1+ µ)κ(1− κ)) = m2/χκ(1− κ) (54)

and a new parameter,

χ = 2EγkTs(1+ µ), (55)

was introduced. The parameterχ is limited by the conditionε2c >

m2, so then

χ > 4m2/zmax= 4kTsm2/ǫmax, (56)

whereǫmax is the maximal thermal photon energy from the black-body spectrum (we assumeǫmax= 30kTs). By using the variable

= ε2c =

12

Eγǫ(1+ µ) = zχ/4, (57)

the additional needed functions are given by expressions:

B2(χ, κ) =(2κ − 1)2

−m2, (58)

A2(χ, κ) = 1− B2(χ, κ). (59)

Function (52) can now be rewritten as:

f (z) =2 −m2 + ( −m2)A2(χ, κ)

m2B2(χ, κ) + A2(χ, κ)−

2( −m2)2A4(χ, κ)[

m2B2(χ, κ) + A2(χ, κ)]2. (60)

We can further conveniently replace the parameterχ with adimensionless formχ = 2kTsEγ(1+ µ)/m2 to get finally the full setof needed magnitudes in a useful way for integrating:

I1(Eγ, Ee, µ) =16π(kTs)2

(hc)3

1Eγ

D(χ, κ), (61)

D(χ, κ) =∫ ∞

zmin

zez − 1

f (z) dz, (62)

f (z) =2 − 1+ ( − 1)A2(χ, κ)

B2(χ, κ) + A2(χ, κ)−

2( − 1)2A4(χ, κ)[

B2(χ, κ) + A2(χ, κ)]2, (63)

B2(χ, κ) =(2κ − 1)2

− 1, (64)

A2(χ, κ) = 1− B2(χ, κ). (65)

To produce the electron of the energyκ the process is limited by theconditionz> zκ, where

zκ =1

χκ(1− κ). (66)

On the other hand, from the kinematics of the process we havelimitation z> zχ, where

zχ =4χ=

2m2

EγkTs(1+ µ). (67)


Figure 23. The integral functionD(χ, κ) (Eq. 61), whereχ = 2kTsEγ(1 +µ)/m2 andκ = Ee/Eγ are the parameters.

The lower limit of the sought integral is the greater value ofthetwo expressions quoted above. The integral given in Eq. (62)canbe tabulated with respect to the parametersχ andκ. The plot of thefunctionD(χ, κ) (Eq. 62) is presented in Fig. (23).

The remaining internal integral in Eq. (45) is given by thefunctionΦ(µ) =

∫ φs

−φsdφ = 2φs, where the angleφs is defined by

Eq. (41). With this in mind, the formula for the spectrum of pairs isgiven by:

dW(Eγ , xi , α, xγ)

dEedxγ=

C2

E2γ

∫

(1+ µ)Φ(µ)D(χ, κ) dµ, (68)

where the constantC2 = 3σTm2(kTs)2/(hc)3. The integral limits,(µ1, µ2) can be calculated according to Eq. (43), following the rulesdescribed in previous paragraphs.

Thee+e− pair spectra as a function of initial parameters for theprocess,xi , α, (see Fig. 20), and the energy of interacting photon,Eγ, are shown in Fig. (24). The producede+e− pair energy spec-

α

θ1

1

eγ

Figure 25.The geometry for the Inverse Compton scattering in the observer(LAB) frame. The Lorentz factor of an electron isγe, the energy of a photonbefore scattering isα1, and angle of scattering isθ1 (the angle betweenthe directions of propagation of the low energy photon and the relativisticelectron).

β

α

θ

θ

χ

’’

’

α ’

φ’

1

1’

2

2

Figure 26. The kinematics of IC scattering in the center of mass system(CMS). The photon before scattering is denoted byα′1 = ǫ/mec2, and afterit, by α′2 = Eγ/mec2. The propagation of the photon before and after inter-action are given by anglesθ′1 andθ′2, both are polar angles measured with

respect to the velocity vector of the electron~β = ~v/c. The scattering anglesare the polar angleχ′, and the azimuthal angleφ′.

tra are symmetric with respect to the electron energyEe/Eγ = 0.5,what is characteristic for the gamma absorption process. Figuresshow also that the process of pair production strongly depends onthe assumed geometry of interacting particles so that the efficiencyis decreasing for gamma photons propagating in the outward di-rection and if injected farther from the star surface. Note that thepresented spectra are calculated for specific parameters ofthe mas-sive star, but serve as a general example.

The production of high-energy photons in Inverse Comptonscattering

The electron propagates in the radiation field of densityn(ǫ), andthe electron injection place is defined by the distancexi and angleαwith respect to the massive star (see Fig. 20). The thermal photonsare again described by a blackbody energy spectrum (Eq. 21).

The rate of interaction of a high-energy electron of energyEe,at the distancexe from its injection place, is given by the integral:

λ−1ICS(Ee, xi , α, xe) =

1c

∫ Emaxγ

0

dN(Ee, xi , α, xe)dEγdt

dEγ . (69)

The integration of the scattered photon spectradN(Ee, xi , α, xe)/dEγdt is taken over the energies of the producedphotons,Eγ, and is limited by the maximal photon energy scatteredEmaxγ .

We calculate the optical depth to IC scattering, at the propa-gation pathxe, integrating the rate of electron interactions in theanisotropic radiation field:

τ(Ee, xi , α, xe) =∫ ∞

0λ−1

ICS(Ee, xi , α, xe) dxe. (70)


Figure 24. The differential spectra fore+e− pairs produced in the absorption process of a photon with energy Eγ . The massive star has here a surfacetemperatureTs = 105 K, and radiusRs = 10R⊙. The spectra for a fixed place of photon injection, defined byxi = 2Rs, andα = 170o for different energiesof relativistic photon are shown at the left in the top panels. The energies of gamma photon areEγ = 104,105,106, 107 MeV, from top to bottom curves. Thesame dependence but forxi = 10Rs is shown to the right in the top panels. Thee+e− spectra from differentangleaα at the same injection distancexi = 2Rs andenergy of photonEγ = 105 MeV are presented in the left, bottom panels. The spectra arecalculated forα = 30o, 60o,90o,120o, 150o. The dependency on thedistance of the photon injection place, forxi = 2, 5, 10, 15, 20, 30Rs, and fixedEγ = 106 MeV andα = 170o are shown in figure to the right, bottom panels.The energy of leptons are normalized to the injected photon energy.

If the electron propagates directly towards the massive star, the pathis terminated by the massive star surface.

The kinematics of Inverse Compton scattering

We adapt here the formulas from Jones (1968) (therein Eq. 1through 10) and Blumenthal and Gould (1970) (their chapters2.6-2.7), wherein the scattered photon spectra are calculated in theisotropic radiation case. In our work, we integrate over thean-gles of incoming low energy photons (the solid angle which defines


the surface of the massive star) what differs from integration overwhole solid angle in the isotropic radiation field.

The energy of photon before scattering in the LAB system (seeFig. 25) is denoted byα1 = ǫ/mec2, while that after scattering, byα2 = Eγ/mec2. In the CMS (see Fig. 26) all quantities are markedwith prime variables, so that the energy of the photon beforeandafter scattering are written asα′1 andα′2. The angle between the di-rection of the electron propagation and the incoming stellar photonis denoted asθ1 (θ′1 in the CMS), while the angle between the elec-tron direction and the photon after scattering, isθ2 (θ′2 in the CMS).These are polar angles measured with respect to the electronve-locity vector~β = ~v/c. The angles of scattering in the electron restframe generate a spherical triangle, so that

cosθ′2 = cosθ′1 cosχ′ + sinθ′1 sinχ′ cosφ′ (71)

(see Fig. 26). The energy of the scattered photon in the CMS isgiven by the formula:

α′2 =α′1

1+ α′1(1− cosχ′), (72)

whereχ′ is the photon scattering angle, defined as the azimuthalangle between the propagation direction of the photon afterscatter-ing,α′2, and the photon propagation direction before it,α′1.

The energy of photon in the LAB system can be found froma Lorentz transformation. In a simplified case, when the photonpropagate head-on with respect to the electron direction, and afterinteraction, it has the energy:

α2 = γeα′2(1+ β cos(π − χ′)) ≈ γeα

′2(1− cosχ′). (73)

In the electron’s rest frame (see Fig. 26), for the polar angle θ′1 = 0we haveχ′ = θ′2 (see Eq. 73), what means that the photon energyafter scattering (see Eq. 73) does not depend on the azimuthal scat-tering angleφ′.

We can calculate the maximal energy of the scattered pho-ton from relations (72) and (73), such thatαmax = 2γeα

′2. In the

Thomson limit (α′1 ≪ 1), the maximal photon energy depends onthe electron energy and equalsαT

max ≈ 4γ2eα1. In the Klein-Nishina

limit, which is the highly relativistic case, this energy ispropor-tional to the initial electron energy and it isαKN

max ≈ γeα1.

The IC scattered photon spectrum in an anisotropic targetfield

In the case of mono-energetic isotropic photon background (in theLAB system) the angular spectrum of incoming low energy photonsin the electron rest frame is given by the formula (Jones 1968):

n′(θ′1)d cosθ′1 =d cosθ′1

2γe(1− β cosθ′1)2. (74)

For relativistic electrons (γe≫ 1), the incoming target photon polarangles are in the range 06 θ′1 6 θ

′1/2, whereθ1/2 ≈ 1/γe is the angle

of a photon ‘cone’ apex.The photon energy spectrum before scattering in the electron

rest frame is:

n′(α′1)dα′1 =

α′1

2γeα21

Θ(α′1;α1

2γe, α12γe)dα

′1, (75)

for γe ≫ 1, where functionΘ(x; a,b) is the characteristic functiondefined in a range (a,b) and takes the values:

Θ(x; a, b) =

1 a 6 x 6 b,0 b < x < a.

Figure 27.The photon spectra (Eq. 96) for fixed initial parameters (theelec-tron injection place): the distance to the massive starxi = 10Rs and angleα = 180o (see Fig. (20)), as a function of electron energyEe. The temper-ature of the star is herein taken asTs = 105 K and the radiusRs = 10R⊙.The electron Lorentz factors shown are :γe = 103, 104,105,106,107, 108.

The cross section to IC scattering in the Klein-Nishina range isgiven by the formula:

σ(α′2, α′1, y′) =

r20(1+ y′2)

2[1+ α′1(1− y′)]2×

1+α′21 (1− y′)2

(1+ y′2)[1 + α′1(1− y′)]

×

δ(α′2 − f (α′1, y′)), (76)

wherey′ = cosχ′, r0 = e2/mc2 is the classical electron radius andthe internal function is given by

f (α′1, y′) = α′1/[1 + α

′1(1− y′)]. (77)

The number of scatterings per unit timet′ is n′cσ. If dN/dt =γ−1

e dN/dt′, then after integration overφ′ one gets:

d4Ndtdα′1dα′2dy′

=πr2

0c

2α21γ

2e

1+ y′2

[1 + α′1(1− y′)]2×

(

1+α′21 (1− y′)2

(1+ y′2)[1 + α′1(1− y′)]

)

×

α′1δ(α′2 − f (α′1, y

′))Θ(α′1;α1

2γe, α2

12γe). (78)

From the relation

dα′2dα′1dy′ = [1 + α′1(1− y′)]2dα′2dy′d f (79)

and after integration overf we get:

d3Ndtdα′2dy′

=πr2

0c

2α21γ

2e

[

(1+ y′2) +(α′2)

2(1− y′)2

1− α′2(1− y′)

]

×

α′21− α′2(1− y′)

×


Θ(α′2

1− α′2(1− y′);α1

2γe, α2

12γe). (80)

Applying the Doppler-shift formula to the energyα′2 and LABframe energyα2 yields to relation

α′2 = α2/γe(1− βy′). (81)

Then, by means of the replacementρ = 1− βy′:

d3Ndtdα2dρ

=πr2

0cα2

2α21γ

4e(1− α2/γe)

×[

ρ2 − 2ρ + 2+(α2/γe)2

1− α2/γe

]

Θ(ρ; ρ1, ρ2)ρ2

, (82)

which is obtained under the assumption that (1− y′) ≈ ρ. The lim-iting values in theargumentt of the functionΘ(ρ; ρ1, ρ2) are:

ρ1 =α2

2α1γ2e(1− α2/γe)

, (83)

ρ2 =2α2

α1(1− α2/γe), (84)

where the parameterρ is limited by conditionsρl 6 ρ 6 ρp, with

ρl =1

2γ2e

,

ρp = 2 (85)

To calculate the spectra of scattered photons fromthe spectrum (Eq. 82) as a function of our parameters,dN/dtdEγ(Ee, xi , α, xe), we replaceEγ = α2mec2 andǫ = α1mec2

and integrate over the background photon spectran(ǫ) andparameterρ,

dNdtdEγ

(Ee, xi , α, xe) =r2

0cEγ2γ4

e(mc2)3(1− Eγ/Ee)

∫

n(ǫ1)

ǫ21

dǫ1 ×

∫

Φ(µ(ρ))

[

ρ2 − 2ρ + 2+(Eγ/Ee)2

1− Eγ/Ee

]

×

Θ(ρ; ρ1, ρ2)ρ2

dρ, (86)

where

µ = Eγmc2/(ǫγe(Ee − Eγ)ρ) − 1, (87)

and the functionΦ(µ) has been already defined in previous para-graphs asΦ(µ) =

∫ φs

−φsdφ = 2φs.

For convenience, let us rewrite the internal integral onρ asfollows:

G(Eγ , ǫ, µ) =∫ ρmax

ρmin

Φ(µ)

[

ρ2 − 2ρ + 2+(Eγ/Ee)2

1− Eγ/Ee

]

×

Θ(ρ; ρ1, ρ2)ρ2

dρ. (88)

If the range of parameter forρ from Eq. (88) is consistent withthe range (ρl , ρp) (Eq. 85), then the integration limits (ρmin, ρmax)correspond to the common part: (ρ1, ρ2) ∩ (ρl , ρp), and the integraloverρ is given by function:

G(Eγ , ǫ, µ) = 2π

[

ρ − 2 logρ − 2ρ−

(Eγ/Ee)2

(1− Eγ/Ee)ρ

]ρmax

ρmin

. (89)

From the Doppler formula and transformations between the sys-tems we get,

ρ = Eγ/γ2eǫ(1+ β cosθ2)(1− Eγ/Ee), (90)

so that the integration limit overρ, for anglesθs1 andθs

2 (see Fig.22) are given by the formulas:

ρ(θ1) =Eγ

ǫγe(1+ β cosθs1)(1− Eγ/Ee)

, (91)

ρ(θ2) =Eγ

ǫγe(1+ β cosθs2)(1− Eγ/Ee)

, (92)

where the anglesθs1, θ

s2 are the angles which defined the directions

of low energy photons coming from the star. In addition, the pa-rameterρ is limited by kinematic condition for the scattering tohappen,

cosθkin > 2m2e/Eγǫ − 1. (93)

The integration in respect toρ can be separated to get the sum ofintegrals, as forρ > ρ(θkin) the internal functionΦ(µ) = 2π, andthe functionG(Eγ, ǫ, µ) is given by (89). If the functionΦ(µ) = 2πwe can integrate the functionG(Eγ, ǫ, µ) (Eq. 88). Otherwise, ifΦ(µ) = 2φs we can rewrite the functionG(Eγ , ǫ, µ) in a compactway as follow. Thus, these cases are summarized by

G(Eγ , ǫ, µ) =∫ ρmax

ρmin

2φsGρ(Eγ , ǫ, µ) dρ, (94)

G(Eγ , ǫ, µ) = 2π[

Gint(Eγ , ǫ, µ)]ρmax

ρmin. (95)

where Gρ(Eγ, ǫ, µ) =[

ρ2 − 2ρ + 2+ (Eγ/Ee)2/(1− Eγ/Ee)]

×Θ(ρ; ρ1, ρ2)/ρ2 and Gint(Eγ , ǫ, µ) = ρ − 2 logρ − 2/ρ −(Eγ/Ee)2/((1− Eγ/Ee)ρ). The limits of integration depend on com-binations of (ρ(θ1), ρ(θ2)) and (ρl , ρp), and respecting the con-straints put byρ(θkin). The final integrationG(Eγ , ǫ, µ) is given bythe formulas:

(i) for “outwards” propagation :

• if (ρmax < ρ1): G(Eγ , ǫ, µ) = 0,

• if (ρmax > ρ1) and (ρmin < ρ1) :

and (ρmax< ρ2) :G(Eγ, ǫ, µ) =

∫ ρmax

ρ12φsGρ(Eγ , ǫ, µ) dρ,

or (ρmax > ρ2) :G(Eγ, ǫ, µ) =

∫ ρ2

ρ12φsGρ(Eγ , ǫ, µ) dρ + 2π

[


ρ2,

• if (ρmax > ρ1) and (ρmin > ρ1):

if (ρmin < ρ2) and (ρmax < ρ2):G(Eγ, ǫ, µ) =

∫ ρmax

ρmin2φsGρ(Eγ , ǫ, µ) dρ,

if (ρmin < ρ2) and (ρmax > ρ2) :G(Eγ, ǫ, µ) =

∫ ρ2

ρmin2φsGρ(Eγ , ǫ, µ) dρ + 2π

[


ρ2,

if (ρmin > ρ2):G(Eγ, ǫ, µ) = 2π

[


ρ2.

(ii) for the “middle” range of propagation:

• if (ρmax < ρ1) or (ρmin > ρ2):G(Eγ, ǫ, µ) = 0,• if (ρmax > ρ1) and (ρmax< ρ2):

G(Eγ, ǫ, µ) =∫ ρmax

MAX(ρmin,ρ1)2φsGρ(Eγ , ǫ, µ) dρ,

• if (ρmax > ρ1) and (ρmax> ρ2):G(Eγ, ǫ, µ) =

∫ ρ2

MAX(ρmin,ρ1)2φsGρ(Eγ , ǫ, µ) dρ.

whereMAX(a,b) gives the larger number from the brackets.(iii) for “inward” directions:


Figure 28.Left: The photon spectra (Eq. (97)) for a specific electron energyγe = 106 and fixed distance to the massive starxi = 10Rs (xe = 0), calculated fordifferent electron propagation angles:α = 0o, 30o, 60o, 90o,120o, 150o, 180o. Right: The photon spectra depending on the electron energyEe and the initialinjection parameters. The spectra are calculated forγe = 106, the angle of propagationα = 180o and subsequent distances to the massive star:xi = 5, 10, 20,30, 50 Rs (rs = xi/Rs, xe = 0) (right panel). The temperature of the star is herein takenasTs = 105 K and its radius isRs = 10R⊙.

• if (ρmin > ρ2) : G(Eγ , ǫ, µ) = 0,• if (ρmin < ρ2) and (ρmin > ρ1):

if (ρmax> ρ2) :G(Eγ , ǫ, µ) =

∫ ρ2


if (ρmax< ρ2) :G(Eγ , ǫ, µ) =

∫ ρmax


• if (ρmin < ρ2) and (ρmin < ρ1):

if (ρmax> ρ2) :G(Eγ , ǫ, µ) =

∫ ρ2


[

Gint(Eγ, ǫ, µ)]ρ1

ρmin,

if (ρmax< ρ2) and (ρmax> ρ1) :G(Eγ , ǫ, µ) =

∫ ρmax


[

Gint(Eγ , ǫ, µ)]ρ1

ρmin,

if (ρmax< ρ2) and (ρmax< ρ1) :G(Eγ , ǫ, µ) = 2π

[

Gint(Eγ, ǫ, µ)]ρmax

ρmin.

Finally the rate of electron scattering on its propagation wayto IC process is given by expression:

λ−1ICS(Ee, xi , α, xe) =

1c

∫ Emaxγ

0

dNdEγdt

(Ee, xi , α, xe) dEγ , (96)

where,

dNdtdEγ

(Ee, xi , α, xe) = C1γ4

e

Eγ

(1− Eγ/Ee)×

∫ ǫmax

ǫmin

dǫeǫ/kTs − 1

G(Eγ, ǫ, µ), (97)

with the constantC = r20c/2(mc2)3. The functionG(Eγ , ǫ, µ) is de-

scribed by Eqs. (88) and (89), which have to be chosen with respectto the value of the functionΦ(µ) and the limits to the parameterρ, as

explained. The lower limit for the integral in Eq. (97) correspondsto the lowest energy of the scattered photon

ǫmin = Eγ/4γ2e, (98)

while the upper one in the integral of Eq. (96) is the maximal scat-tered photon energy

Emaxγ = 4γ2

eǫ. (99)

The photon spectra calculated from Eq. (96) are presented inFigs.(27) and (28). They are calculated with respect to the electron en-ergy Ee and its initial parameters of injection close to the massivestar (see Fig. 20). The photon energy spectra reflect the features ofthe cross section for IC scattering, with characteristic peaks at thehigh energy range (forEγ/Ee ∼ 1) – Klein-Nishina regime if theenergy of the incoming electron is relativistic. These plots showsalso the dependence on the propagation angle (with the highest pro-duction rate for direction toward the massive star) and place of theelectron injection.

ACKNOWLEDGMENTS

We acknowledge extended use of IEEC-CSIC parallel computerscluster. We acknowledge W. Bednarek for help and encouragement.This work was supported by Spanish grants AYA 2006-00530 andCSIC-PIE 200750I029.

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