A rotating hollow cone anisotropy of TeV emission from binary systems
aa r X i v : . [ a s t r o - ph ] N ov Draft version December 24, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
A ROTATING HOLLOW CONE ANISOTROPY OF TEV EMISSION FROM BINARY SYSTEMS
A.Neronov
INTEGRAL
Science Data Center, 16 ch. d’Ecogia, CH-1290, Versoix, Switzerland andM.Chernyakova
Dublin Institute for Advanced Studies,31 Fitzwilliam Place, Dublin 2, Ireland
Draft version December 24, 2018
ABSTRACTWe show that TeV γ -ray emission produced via interactions of high-energy particles with anisotropicradiation field of a massive star in binary systems should have a characteristic rotating hollow coneanisotropy pattern. The hollow cone, whose axis is directed away from the massive star, rotates withthe period equal to the orbital period of the system. We note that the two maxima pattern of the TeVenergy band lightcurve of the γ -ray loud binary LS 5039 can be interpreted in terms of this rotatinghollow cone model. Adopting such an interpretation, we are able to constrain the geometry of thesystem – either the inclination angle of the binary orbit, or the elevation of the γ -ray emission regionabove the orbital plane. Subject headings: gamma rays: theory — radiation mechanisms: non-thermal — binaries: general
Introduction. γ -ray-loud binary systems are a newlyidentified class of sources in which either accretion ontothe compact object (a neutron star, or a black hole),or interaction of an outflow from the compact objectwith the wind and radiation from a massive companionstar leads to the production of very-high energy (VHE) γ -ray emission. Three such systems, PSR B1259-63,LS 5039 and LSI +61 303, have been firmly detectedas persistent or regularly variable TeV γ -ray emitters(Aharonian et al. 2005, 2006; Albert et al. 2006). TheVHE γ -ray emission from the γ -ray-loud binaries is vari-able on the orbital period (or shorter) time scale. Thisimplies that the emission region is located close to the bi-nary system, in a highly inhomogeneous and anisotropicparticle and photon background produced by massivecompanion star.In what follows we show that if the γ -ray emission fromsuch a region is produced in interactions of isotropicallydistributed VHE particles with photons from the mas-sive star, it should have a characteristic “rotating hollowcone” anisotropy, i.e. most of the photons are emittedat a certain angle ζ with respect to a symmetry axisdirected radially away from the massive star. Orbitalmotion of the emission region around the massive starleads to the rotation of the emission cone. Rotation of thehollow cone on the orbital time scale leads to the appear-ance of 0, 1, or 2 maxima in the orbit-folded lightcurvein the VHE band, occurring at the phases when the lineof sight is inclined at an angle ζ with respect to thecone axis, i.e. at the moments of passage of the of thecone through the line of sight (similarly to the hollowcone models of period-folded lightcurves of pulsars, seee.g. (Lyne & Graham-Smith 2005)).The orbital modulation of the γ -ray flux, related tothe passage of the hollow cone through the line of sightcould be most clearly detected if there are no additionalsources of the modulation, related e.g. to the ellipticity Electronic address: [email protected] address: [email protected] of the binary orbit, absence of spherical symmetry of thewind/radiation from the companion star etc. Among thethree γ -ray-loud binary systems mentioned above, thesystem LS 5039 is characterized by the lowest ellipticityof the orbit. In this system the compact object orbits aO6.5V star which emits isotropic stellar wind (contraryto the other two systems in which the massive star is ofthe Be type).The influence of the anisotropy of the photon field ofthe massive star on the properties of the γ -ray emissionin binaries in general, and in LS 5039 in particular, wasfirst studied by Khangulyan et al. (2005, 2007). Herewe calculate the angular brightness profile of the hollowcone in LS 5039, and find that the observation of the twomaxima of the orbit-folded lightcurve constrains the in-clination of the binary orbit to be i > ◦ , if the emissionis produced in the vicinity of the compact object. Thisresult can be stated also in an opposite way: if the in-clination of the binary orbit is i < ◦ , the two maximastructure of the orbit-folded lightcurve can be explainedonly if the VHE γ -ray emission region is displaced fromthe position of the compact object. This can be the caseif the emission is produced in a jet. In this latter case,we show that the existence of the two-maxima of thelightcurve constrains the elevation of the emission pointabove the orbital plane. Anisotropy of VHE γ -ray emission in a central photonfield. Consider the γ -ray emission produced by inter-actions of VHE particles X (e.g. protons or electrons)with the soft photon field in the vicinity of a massivestar. Assume for simplicity that the size of the emissionregion is much less than the distance from the region tothe center of the star and that the VHE particles in theregion have isotropic velocity distribution. In spite ofthe isotropy of the VHE particle distribution, the γ -rayemission will be anisotropic. The anisotropy arises be-cause of the Doppler effect which leads to the decrease(increase) of the rate of interaction of the VHE particlesco-moving with (moving oppositely to) the soft photon Neronov & Chernyakovafield of the massive star.The interaction rate of particles X with momenta P X with soft photons with momenta p ∗ is given by(Landau & Lifshitz 1980) R = Z n X n ∗ σ P X · p ∗ E X ǫ ∗ dE X dǫ ∗ d Ω , (1)where σ is the interaction cross-section, n X ( P X ) is theparticle distribution, n ∗ ( p ∗ ) is the soft photon distribu-tion and P X · p ∗ is the scalar product of the 4-momentaof the interacting particles, P X · p ∗ ≃ E x ǫ ∗ (1 − cos ζ ) , (2)( ζ is the angle between the particle velocities) which con-tains the Doppler factor, (1 − β cos ζ ) (we assume thatparticle velocity is β ≃ − cos ζ ), so that the γ -rayemission intensity, which is proportional to the interac-tion rate, would be isotropic. However, since all the softphotons crossing the emission region move in the samedirection (away from the massive star), the interactionrate depends on the angle ζ between the direction ofemission and the direction ”from the massive star”. Par-ticles X comoving with the soft photon field (moving atthe angles ζ → ◦ ) interact more rarely than the particlesmoving opposite to the photon field (at ζ → ◦ ).The reduction of the interaction rate has two-fold con-sequences. On one hand, it leads to the reduction of thepower of the γ -ray emission by the particles moving inthe direction away from the star. On the other hand, thereduction of the interaction rate of the soft photons withthe emitted VHE γ -rays facilitates the escape of γ -raysmoving in the direction away from the star. A competi-tion between the decrease of the γ -ray production rate, R prod ( ζ ), and the increase of the γ -ray “survival prob-ability” (i.e. of the exp ( − τ abs ( ζ )), where τ abs ( ζ ) is theoptical depth with respect to the pair production) leadsto the appearance of a maximum of the γ -ray flux F γ ( ζ ) ∼ R prod ( ζ ) e − τ abs ( ζ ) (3)at an angle 0 < ζ < π , i.e. to the appearance of a hollowcone anisotropy pattern.At large distances from the massive star one can ap-proximate the angular distribution of the soft photons bythat of a point source. Under this simplifying assump-tion, one finds that the integration over the angular dis-tribution of the UV photons is easily performed and theresulting expression for the rate of production of γ -raystakes the form R prod ( ζ ) ≃ n X n ∗ σ prod (1 − cos ζ ) , (4)where σ prod is the cross-section of production of γ -rays ininteraction of the particles X with the soft photon field.An estimate of the optical depth for the γ -rays, es-caping from the production region, can be obtained bymultiplying the absorption rate per γ -ray on the size ofthe absorbing region. The absorption rate is given bythe expression (1) in which the particle X is a γ -ray andthe cross-section σ is the pair production cross section, σ abs . Estimating the size of the absorption region to be Fig. 1.—
Thick solid curve: angular brightness profile, Eq. (6),in the approximation of constant cross-sections. Thin dotted anddashed curves show the γ -ray production rate and absorption co-efficients as functions of the angle ζ . of the order of the distance D of the emission point fromthe massive star, one finds τ abs ≃ R abs D/n γ ∼ n ∗ σ abs D (1 − cos ζ ) . (5)Substituting Eqs. (4),(5) into Eq. (3) one finds F γ ( ζ ) ∼ σ prod (1 − cos ζ ) e − n ∗ Dσ abs (1 − cos ζ ) . (6)Two effects affect the anisotropy pattern of the γ -rayemission. First, the explicit dependence of F γ on cos ζ is introduced in Eq. (6) by the Doppler effect. An ad-ditional implicit dependence on ζ is introduced throughthe energy dependence of the interaction cross-sections σ prod , σ abs . Indeed, in general σ = σ ( E CM ), where the E CM if the center-of-mass energy, which, in the case E CM ≫ m X depends on ζ as E CM ∼ √ − cos ζ .The anisotropy pattern resulting from the Doppler ef-fect can be found if one ignores the energy dependenceof σ prod , σ abs . This is done in Fig. 1 (the constant τ = n ∗ Dσ abs is taken to be τ = 3). From this figureone can see that most of the γ -ray flux is emitted alonga ”thick hollow cone” with the opening angle ζ and thethickness comparable to the opening angle, ∆ ζ ∼ ζ .The energy dependence of the interaction cross-sectionsleads to the energy-dependent angular brightness profileof the thick hollow cone. The cone becomes wider athigher energies, where the absorption is less efficient, seeFig. 5 below. The non-zero brightness at ζ = 0 is due tothe finite radius of the star. Geometrical model of variability of γ -ray emission. Ifthe location of the emission region is determined by theposition of the compact object (e.g. the γ -ray emission isproduced in the vicinity of the compact object, or in thejet emitted by the compact object), the orientation ofthe hollow cone changes when the compact object movesaround the star, as it is shown in Fig. 2 (where theemission region is supposed to be situated at an elevation h above the compact object, so that the hollow cone axisis inclined at an angle χ = atan( h/D ), where D is thebinary separation distance).Changes of the orientation of the hollow cone with re-spect to the line of sight should lead to the orbital mod-ulation of the observed γ -ray flux. Depending on the re-lation between the inclination angle i of the binary orbitplane, and the opening angle of the hollow cone, ζ , twocharacteristic patterns of the orbital modulation are pos-sible. If the inclination of the orbit is i > π/ − ζ − χ , thedirection of the line of sight passes through the ”wall” ofthe cone two times per orbit. This should lead to the ap-pearance of two maxima in the orbit-folded lightcurve. Ifthe binary orbit is circular, the maxima of the lightcurveoccur when the true anomaly of the orbit (angular or-eV emission from γ -ray-loud binaies 3 χ Dh ∆Φ t o ob s e r v e r i ζ Fig. 2.—
Model of the anisotropy of the γ -ray emission. Maximaof the lightcurve are expected during the passage of the line ofsight through the hollow cone, whose axis is directed along the lineconnecting the emission region. bital phase 0 < Φ < ◦ , counted from the focal pointof the orbit, Φ = 0 ◦ at the periastron) takes the valuesΦ , = Φ inf ± ∆Φ (7)where Φ inf is the anomaly of the inferior conjunction and∆Φ = acos [1 − (sin( i + χ ) − cos ζ ) / (cos χ sin i )] (8)If the binary orbit is elliptical, an additional orbital mod-ulation of the γ -ray flux can occur because of the varia-tion of the distance of the emission region from the mas-sive star (which leads to the modulation of the γ -rayproduction/absorption rates) with the orbital phase. Asthe inclination of the orbit decreases to i ≤ π/ − ζ − χ ,the two maxima at Φ and Φ merge at the phase Φ inf .Since the opening angle of the hollow cone, ζ dependson the γ -ray energy, the condition i > π/ − ζ ( E γ ) − χ can be satisfied only in a certain interval of energies, sothat a merger of the two maxima Φ , = Φ inf ± ∆Φ( E γ )at the phase of inferior conjunction can be observable ata particular energy E at which ζ ( E ) = π/ − i − χ . The case of LS 5039.
LS 5039 is one of the sev-eral X-ray binaries detected as sources of the VHE γ -ray emission (Aharonian et al. 2006). In this binary sys-tem the compact object rotates with the period P =3 . ± . e = 0 . ◦ < i < ◦ (Casares et al. 2005).The orbit-folded lightcurve of the source at the energies E > φ ≃ . , φ ≃ .
85. The two maximaare apparently not symmetric: the first maximum around φ spans a broader range of the orbital phase, while thesecond maximum is more sharp. The second maximumhappens closer to the phase of the inferior conjunction, φ inf ≃ . γ -ray lightcurve as a function of the trueanomaly, Φ, (to produce Fig. 3, we have calculated thetrue anomalies of each data point of the top panel of Fig.5 of Aharonian et al. (2006) and rebinned the data intobins of the width d Φ = 15 ◦ ) rather than as a function of φ , a symmetry in the positions of the two maxima canbe found.Namely, the lightcurve can be satisfactory fitted witha phenomenological model which is a sum of a constant(weakly modulated emission which can be produced e.g.at larger distances) plus two gaussians, whose centers areequally spaced from the phase of the inferior conjunction,Φ inf ≃ ◦ (see Fig. 3). The positions of the centersof the gaussians, found by the fit, are Φ , ≃ Φ inf ± ◦ , while the widths of the gaussians are nearly equal, Fig. 3.—
A phenomenological model fit to the lightcurve of LS5039. The model consists of a constant plus two gaussians of equalwidth at equal distance from the inferior conjunction (shown by avertical line).
Fig. 4.—
Dependence of the opening angle of the hollow cone onthe inclination of the binary orbit in LS 5039. Grey shaded regionindicates the range of ζ found from numerical calculations. δ Φ , ≃ . ◦ (if the phase of the inferior conjunctionis left free, while fitting, the fit finds the phase Φ inf ≃ . ◦ , consistent with the value Φ inf ≃ . ◦ ± . ◦ found by Casares et al. (2005)). The gaussian centeredat the phase Φ is found to have ≃ . .The observed symmetry of the positions of the max-ima of the orbit folded lightcurve can be interpreted, ina straightforward way, in terms of the ”rotating hollowcone” model, discussed in the previous sections. In par-ticular the phase shift of Eq. (7) is ∆Φ ≃ ◦ . From Eq.(8) (which can be used, for a low-eccentricity orbit, as afirst approximation) one can find a relation between i and ζ , shown in Fig. 4. Taking into account the constrainton the inclination angle, 20 < i <
60 (Casares et al.2005), we find that the opening angle of the hollow coneanisotropy pattern is constrained to 40 ◦ < ζ < ◦ , ifthe emission is assumed to come from the vicinity of thecompact object ( χ = 0 ◦ in Fig. 2). A constraint on the geometry of LS 5039.
The openingangle of the hollow cone, ζ ( E γ ), can be found, once theemission process leading to the γ -ray production and thelocation of the emission region are known. Comparingthe theoretically predicted value of ζ ( E γ ) to the oneimplied by the data one can, in principle, constrain thegeometry of the system. In particular, assuming thatthe location of the emission region is known, one canconstrain the inclination of the binary orbit with respectto the line of sight. Otherwise, if the inclination of theorbit would be known, one would be able to constrain thelocation of the emission region, in particular, its distancefrom the star and/or elevation above the orbital plane.Different locations of the VHE γ -ray emission regionare assumed in different models of activity of LS 5039.In the model of ”compact pulsar wind nebula” (see e.g.(Dubus et al. 2007)) the emission region is assumed tosurround the compact object ( h = 0 in notations of Fig.2). In a ”microquasar” model the TeV emission is as- Neronov & Chernyakova Fig. 5.—
Angular brightness profile of the hollow cone in LS5039, found assuming that emission is produced at the distance ofperiastron (black) or apastron (grey) of the orbit. Thin dotted anddashed curves show the production rate and absorption coefficientfor an emission region at the periastron distance. sumed to be produced in a jet, so that the emission re-gion is displaced from the position of the compact object(it is situated above or below the orbital plane, h = 0, seee.g. (Khangulyan et al. 2007)). In both types of modelsthe VHE γ -rays are produced via the inverse Comptonscattering of the soft photons by the VHE electrons.To find the rate of production of γ -rays via the inverseCompton scattering, one has to numerically integrate theEq. (1), with σ prod being the Klein-Nishina cross section,over the angular and energy distributions of soft photonsat a distance D from the massive star. The result ofsuch integration is shown by the dotted curves in Fig.5 for electron energies, E e = 1 and 10 TeV for the casewhen the emission region is situated at the distance ofthe periastron of the binary orbit.To find the optical depth τ abs ( ζ ) for the γ -rays emittedat different angles ζ (see Eq. 3), one first calculates the γ -ray absorption rate, given by Eq. (1) with the crosssection σ being the pair production cross-section, at eachpoint of the γ -ray trajectory. This is done similarly tothe calculation of the inverse Compton scattering rate,via an integration over the soft photon angular distri-bution. Next, one has to integrate the absorption ratealong the γ -ray trajectory, from the emission point to in-finity. The result of such numerical integration is shownby the dashed curves in Fig. 5 for γ -ray energies E γ = 1and 10 TeV, assuming that emission is produced at thedistance of the periastron of the binary orbit.Substituting the numerically found R prod ( ζ, E γ ) and τ abs ( ζ, E γ ) into Eq. (3) one can find the angular bright-ness profile of the hollow cone for different orbital phasesand different photon energies. The angular brightnessprofile, calculated for the periastron/apastron of the bi-nary orbit, is shown by the thick, solid, black/grey curvein Fig. 5. We assume that the energies of γ -rays areapproximately equal to the energies of the primary elec-trons, which is true in the Klein-Nishina regime of theinverse Compton scattering. The first maximum of the TeV band lightcurve of LS5039 takes place at the orbital phase Φ ≃ ◦ − ◦ ≃ ◦ , close to the apastron of the orbit. From Fig. 5one can see that at this phase the opening angle of thecone changes in the range 30 ◦ < ζ ( E γ ) < ◦ when the γ -ray energy changes from 1 to 10 TeV. Comparing thisnumerically calculated range of values (shaded region inFig. 4) to the one implied by the observational data, onecan find from Fig. 4, that if the VHE γ -ray emission isproduced close to the compact object (i.e. at χ = 0 ◦ ) theinclination angle of the binary orbit should be i > ◦ .The constraint on i, χ can be re-formulated in a differ-ent way: if the inclination angle of the orbit is small, e.g. i ∼ ◦ (see Casares et al. (2005)), the γ -ray emissionis not produced close to the compact object. Instead,from Fig. 4 one can find that in this case the elevation χ of the emission region above the orbital plane shouldbe χ ≥ ◦ . If the emission region is located in a jet-likeoutflow orthogonal to the orbital plane, the emission re-gion should be situated at the height h = D tan χ > . D above the orbital plane, where D is the binary separationdistance (see Fig. 2 for notations). Summary.
We have shown that VHE γ -ray emis-sion from γ -ray-loud binaries is expected to have a ro-tating hollow cone anisotropy pattern, determined bythe Doppler effect in the anisotropic radiation field ofa massive star (Fig. 1). This anisotropy leads to the ap-pearance of a double-peak structure of the orbit-foldedlightcurve, with the two peaks situated at equal distance∆Φ (Eq. 8) from the phase of the inferior conjunction.We have demonstrated that such a symmetric double-peak structure is observed in the particular case of LS5039. In this case, a measurement of the phase shift ∆Φenables to find a relation between the opening angle ofthe hollow cone, ζ , and the inclination of the binary or-bit, i (see Fig. 4). Comparing the value of ζ , inferredfrom the data, to the one found from numerical calcula-tion of the angular brightness profile of the cone (Fig. 5)we were able to constrain the inclination of the binaryorbit and/or the elevation of the VHE γ -ray emissionregion above the orbital plane in this particular source.In the particular case of LS 5039, the rotating hollowcone model discussed above can be tested if the statisticsof the signal from the source becomes high enough toallow a splitting of the source lightcurve at the energies E > < E <
10 TeV and
E >
10 TeV). In this case the predictedshifts of the two maxima of the lightcurve toward eachother (or even a merger of them) at the higher energiesshould be observable. If observed, such an effect wouldbe a clear evidence in favor of the proposed model.We thank F.Aharonian and A.Zdziarsky for the discus-sions of the subject.
REFERENCESAharonian F.A. et al., 2005, A&A, 442, 1.Aharonian F.A. et al., 2006, A&A, 460, 743.Albert J., et al., 2006, Science, 312, 1771. Dubus G., Cerutti B., Henri G., 2007, A&A, accepted(arXiv:0710.0968).Khangulyan D., Aharonian F., AIP Conf. Proc., 2005, 745, 359. eV emission from γ -ray-loud binaies 5 Khangulyan D., Aharonian F., Bosch-Ramon V., 2007, MNRAS,accepted (arXiv:0707.1689).Landau L.D., Lifshitz E.M., 1980,
The Classical Theory of Fields ,Elsevier.Lyne A., Graham-Smith F., 2005,