Gamma-rays, neutrinos and cosmic rays from dense regions in open clusters

Gamma-rays, neutrinos and cosmic rays from dense regions in open clusters

W. Bednarek, J. Pabich & T. Sobczak

Department of Astrophysics, The University of Łodz, 90-236 Łodz, ul. Pomorska 149/153, Poland

Abstract

We analyse the high energy processes, occurring within open clusters containing massive binary systems, which turnto the production of high energy γ-rays and neutrinos. Nuclei, accelerated within the binary systems, inject protonsand neutrons as a result of their fragmentation in collisions with stellar radiation and matter of the winds. We calculatethe radiation produced by these protons and neutrons during their interaction with the matter of the stellar wind andthe open cluster. The detectability of γ-ray emission and neutrino emission by the present and future Cherenkovtelescopes and the neutrino telescopes is discussed.

Keywords: stars: binaries: close — open clusters — radiation mechanisms: non-thermal — gamma-rays: theory —neutrinos

1. Introduction

Up to now, TeV γ-ray emission has been detected fromthe direction of 3 open clusters (i.e. Cyg OB 2 [1], West-erlund 2[2], and Westerlund 1[3]). Also the supermas-sive binary system Eta Carina, within the Carina Nebulacomplex, has been detected up to ∼100 GeV[4,5,6,7,8]but not in the TeV energy range[9]. This emissioncan originate in collisions of strong winds producedby massive stars in open clusters (e.g.[10,11,12,13,14]),at the shocks formed within massive binary systems(e.g.[15,16,17,18,19,20]), in the interaction of SNRsshock waves with dense clouds[21], within PWNe, or asa result of the interaction of PWNe with dense matter ofthe open cluster[22,23,24,25,26]. It is likely that in spe-cific open cluster a few processes can be important sincedifferent types of objects (massive star winds, SNRs,PWNe) might inject relativistic particles with compa-rable power, i.e. of the order of 1050 erg.

In the present paper we investigate in detail the highenergy radiation expected in hadronic processes withinand around the massive binary systems surrounded bythe large concentration of matter. It is assumed thatnuclei are accelerated in the region of colliding windswithin the binary stars (e.g. see[27]). These nuclei can

severely disintegrate, in the interaction with the radia-tion field of massive stars and with the matter of the stel-lar winds, injecting neutrons and protons. Charged pro-tons diffuse through the open cluster producing γ-raysand neutrinos in collisions with the matter of the stellarwind cavity and dense environment of the open cluster.On the other hand, neutrons move balistically throughthe wind cavity and decay on protons at some distancefrom the binary system which can be still within thewind cavity or already within the open cluster. Protons,from their decay, can also contribute to the high energyγ-ray and neutrino spectrum. As an example, we per-form calculations of the γ-ray and neutrino fluxes pro-duced in the clusters surrounding the Eta Carina super-massive binary system and WR 20a binary system in theopen cluster Westerlund 2. The details of this scenarioare discussed in Bednarek et al.[28].

2. Binary systems within the open cluster

We consider a massive binary system in which oneor both companion stars belongs to the class of theWolf-Rayet (WR) stars. WR type star produces fastand dense wind, due to the huge mass loss rate, which

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can be of the order of MWR = 10−5M5 M� yr−1.The winds propagate with the characteristic velocitiesof the order of vw = 103v3 km s−1. The densityof the wind drops with the distance from the star ac-cording to, nw(r) ≈ 3.2 × 1011M−5/v3R2

12r2 cm−3,where RWR = 1012R12 cm is the radius of the star, andr = R/RWR is the distance from the star in units of thestellar radius. The massive binary systems are usuallyimmersed itself within a relatively dense open clusters(OCs). Typical densities of the OCs are of the order ofnoc = 10n10 cm−3 and temperatures of the gas of theorder of Toc = 104T4 K. At certain distance from thebinary system, the pressure of the stellar wind is bal-anced by the pressure of thermal gas within the OC. Weestimate the dimension of such stellar wind cavity on,Rcav ≈ 1.1 × 1019[M−5v3/(n10T4)]1/2 cm. The radii ofthe wind cavities around WR type binary systems withinthe open clusters are typically of the order of a few par-secs for the density of surrounding matter of the orderof ∼ 10 cm−3 and its temperature Toc ∼ 104K.

We consider consequences of acceleration of nucleiwithin the binary system located in the open cluster. Nu-clei (from the stellar winds) can be accelerated withinthe collision region of the winds (e.g.[14]). The recon-nection of the magnetic field and the diffusive shockacceleration process can play an important role in thisplace. We show that nuclei can efficiently disintegratein the dense radiation and matter of the winds from mas-sive stars. As a result, neutrons are injected. They decayat the distance from the binary system which is deter-mined by their Lorentz factors. Protons, extracted fromnuclei, lose energy on interaction with the dense windclose to the binary system. They also suffer adiabaticenergy losses in the expanding wind within the windcavity. We consider high energy processes in which γ-rays and neutrinos are produced in hadronic collisionsin the above described scenario.

Massive stars produce fast and dense winds with themass loss rates of the order of a few 10−6 − 10−4 M�yr−1 (characteristic for the Wolf-Rayet (WR) and O typestars). During the main sequence stage, the outer partsof stars are completely lost and only inner parts, com-posed of heavy nuclei, are left. Therefore, the windsof early type stars are expected to be mainly composedfrom nuclei heavier than hydrogen such as helium tooxygen. Massive stars are frequently found within themassive binary systems in which strong winds collideproviding conditions for acceleration of nuclei to largeenergies. The propagation and interaction of relativis-tic nuclei with the stellar radiation field results in theirphoto-disintegration to neutrons, protons and secondarynuclei. We perform numerical simulations of propaga-

tion of nuclei in the radiation field of the massive starsin order to determine the rate of injection of differentnuclei. Note that density of stellar photons in the vicin-ity of massive star, nph ≈ 2 × 1016T 3

5 ph. cm−3 (whereT = 105T5 K is the surface temperature of the mas-sive star), is a few orders of magnitude larger than den-sity of matter in the stellar wind. Therefore, it is ex-pected that nuclei with sufficiently large energies inter-act at first in collisions with stellar photons rather thanwith the matter of the stellar wind. After that, nucleican suffer significant fragmentation in collisions withthe matter of the stellar wind. This second process is in-dependent on energy of nuclei. Therefore, lower energynuclei can also suffer strong disintegration process incollisions with dense matter of the stellar winds in thecase of stars with exceptionally strong winds such asconsidered in this paper. Neutrons released from nucleiin collisions with the matter have the spectrum similarto the spectrum of primary nuclei. Therefore, neutronswith low energies can decay relatively close to the bi-nary system where the density of matter is still high.On the other hand, high energy neutrons can even reachdense regions outside the wind cavity of the massive bi-nary system. In this paper we calculate γ-ray and neu-trino spectra produced by protons extracted from nucleiand also from protons from neutrons decaying at somedistance from the binary system. These neutrons decaywithin the wind cavity and also within the dense opencluster in which binary system is immersed.

We estimate the maximum energies of hadrons accel-erated in the shock region of colliding winds followingthe conditions considered e.g. in[20,27]. The maximumenergies of hadrons are determined by comparing theiracceleration time scale and their escape time scale fromthe acceleration region or collision time scale with thematter of the stellar wind. The acceleration time scale isgiven by, τacc = Eh/Pacc ≈ 0.02γh/(χ−5B3) s, where Ehand γh are the energy and the Lorentz factor of particles,Pacc = χcEh/RL is the acceleration rate, χ = 10−5χ−5is the acceleration coefficient, RL = Acγh/ZeBsh is theLarmor radius of hadron in the magnetic field at theshock Bsh, c is the velocity of light, e is the elemen-tal charge, and A and Z are the mass and charge num-bers of nuclei. We apply A/Z = 2. Bsh, is related tothe surface magnetic field of the massive star by as-suming its dipolar structure close to the surface up to∼ 1.2R� and radial structure at larger distances[29].For the distance of the collision region from the starequal to Rsh = 2R�, Bsh drops to ∼ 0.25B�. The ad-vection time scale of hadrons from the collision regionis estimated from, τadv ≈ Rsh/vw ≈ 104R12/v3 s,where Rsh = 1012R12rsh cm is the distance of the col-

W. Bednarek et al. / Nuclear Physics B (Proc. Suppl.) 256–257 (2014) 107–116108

lision region from the star and rsh = Rsh/RWR. Themaximum energy of hadrons accelerated at the regionof colliding stellar winds is obtained from comparisonof above estimated time scales. It is then given by,γadv

max ≈ 5 × 105B3v3R12rsh GeV, where the accelera-tion coefficient is estimated on χ = (vw/c)2 ≈ 10−5v2

3and Bsh = 103B3 G.

On the other hand, hadrons lose also energy on col-lisions with the matter of the stellar wind. The hadron-hadron energy loss time scale can be estimated from,τhh = (cnw(r)kσpp)−1 ≈ 3.5×103R2

12r2v3/M−5 s,whereσpp = 3 × 10−26 cm2 is the cross section for proton-proton collision, and k = 0.5 is the in-elasticity coef-ficient. The maximum energies of hadrons allowed bycollisional energy losses are,γhh

max ≈ 3.5 × 105R212r2v3

3B3/M−5. We have calculatedthe maximum energies of hadrons due to these two pro-cesses for two example massive stars within the binarysystems. The results are shown in Table 1. In the caseof the binary system containing WR 20a stars, the max-imum energies of hadrons are comparable in the case ofadvection process and collisional energy losses. But, inthe case of Eta Carina the maximum energies of hadronsare limited by collisional energy losses. In all consid-ered cases hadrons can reach energies of the order of∼106 GeV.

We calculate the optical depths for fragmentation ofnuclei in the radiation field, which are injected at dif-ferent directions than those defined by the stars, for twomass numbers corresponding to the helium and the oxy-gen nuclei. The approximation of the cross section forphoto-disintegration process of nuclei is applied as de-scribed in Karakuła & Tkaczyk[30]. Nuclei are con-sidered to propagate in the vicinity of two representa-tive massive stars with different parameters, WR starin the WR 20a binary system and supermassive star inthe Eta Carina binary system. We conclude that in thecase of WR 20a binary system the process of photo-disintegration of nuclei can be neglected. In the caseof the star with parameters observed in Eta Carina, sig-nificant number of nuclei with energies in the range∼ 10(5−6) will lose nucleons in collisions with stellarradiation.

In order to conclude whether nuclei can be effi-ciently disintegrated in collisions with the matter ofthe stellar wind, we estimate the optical depth for rel-ativistic hadrons on collisions with the matter of thewind. This optical depth can be estimated from, τhp =∫ Rc

RBSσppnwc/vwdR ≈ 2.9 × 1012M−5v−2

3 (1/RBS − 1/Rc),where RBS (in cm) is the radius of the binary systemat which hadrons are injected within the wind. It is

assumed to be equal to the shock radius. The opticaldepths for protons on the interaction with the matter ofthe wind are shown in Table 2. Note that the opticaldepth for nuclei with specific mass number A1 in colli-sions with nuclei with the mass number A2 scales as,τA1A2 ∝ A1/3

1 A1/32 . For the case of the helium nuclei

(A1 = A2 = 4), the optical depths will be a factor of∼ 2.5 larger and for the oxygen nuclei a factor of ∼ 6.3larger. We conclude that in the case of WR 20a andEta Carina binary systems, nuclei should be completelydisintegrated. Large values of the optical depth for colli-sions of protons with the matter of the stellar winds indi-cate that these relativistic hadrons should efficiently loseenergy on production of high energy γ-rays and neutri-nos in the close vicinity of the massive binary system.On the other hand, neutrons, released from these nuclei,should move realistically and decay at large distancesfrom the binary system within the wind cavity or, in thecase of the most energetic neutrons, directly in the re-gion of the open cluster. Protons, from decay of theseneutrons, can also produce high energy radiation at largedistances from the binary systems, i.e in the region withlow level of the soft radiation field and low density ofmatter in respect to the density of the wind within thebinary system.

3. Radiation from hadrons escaping from the binarysystem

Accelerated nuclei initiate a sequence of processeswithin the binary system and its surrounding. As wehave shown above nuclei suffer complete disintegra-tion if injected within massive binary systems con-taining luminous stars characterising by dense stellarwinds (WR type stars). Unstable neutrons, from photo-disintegration of these nuclei, decay at some distancefrom the binary system. Depending on their energy, theycan decay within the stellar wind region (the wind cav-ity) or outside the stellar wind shock, i.e. within theopen cluster. Protons, from neutrons decaying withinthe stellar wind, are expected to suffer adiabatic energylosses during the fast expansion of the wind. Theseprotons are also partially advected with the wind to theopen cluster. Neutrons, with large enough energies, canalso decay outside the wind cavity in the volume of theopen cluster. Protons, from their decay, diffuse gradu-ally through the open cluster suffering some collisionswith the distributed matter. We conclude that the sim-ple scenario which postulates acceleration of hadronswithin the massive binary system immersed in the opencluster provides variety of conditions for production of

W. Bednarek et al. / Nuclear Physics B (Proc. Suppl.) 256–257 (2014) 107–116 109

Table 1: Parameters of massive stars in binary systems, Lorentz factors of accelerated nuclei and the optical depths for protons in the matterof the stellar wind

Name B RWR vw Rsh MWR Lw τhp γadvmax γcol

maxunit G cm km/s RWR K erg s−1

WR 20a 103 1.4 × 1012 103 2 3 × 10−5 1037 ∼2.4 1.4 × 106 106

Eta Car 200 1.2 × 1013 700 1.4 2.5 × 10−4 4 × 1037 ∼4.1 1.2 × 106 2.7 × 105

Figure 1: The average γ-ray reduction factors due to absorption of γ-rays in the stellar radiation (γ + γ → e±), as a function of the energy of γ-rayphoton. The results are shown for the massive star in WR 20a (on the left) and Eta Carina (on the right) binary systems. The optical depths arecalculated assuming isotropic injection of γ-rays at a specific distance from the massive star in units of the stellar radius equal to r = 3 (dashed), 10(solid), 30 (dotted), 100 (dot-dashed), 300 (dot-dot-dashed), and 103 (thin dashed).

high energy radiation in different parts of the open clus-ter characterised by different conditions, e.g the windcavity of the binary system and dense medium of theopen cluster. We are interested in the γ-ray and neutrinoemission produced in such scenario. Let us at first es-timate whether charged hadrons can be captured in thestellar wind. We compare the Larmor radius of chargedhadrons with a specific energy with the characteristicdistance scale which is the distance at which hadrons arelocated from the binary system, R. The Larmor radius ofhadrons with the Lorentz factor γn is RL = 3×106γn/BGcm, where B = 1BG G is the magnetic field strength inthe wind at the distance R from the binary system. Themagnetic field in the stellar wind is expected to havecomplicated structure as a function of the distance fromthe star (dipolar, radial and at farther distances toroidal).This structure becomes even more complicated in thecase of stars within the compact binary system whenboth stars have large velocities. We assume that themagnetic field is radial at distances below ∼ 10RWR,i.e. in this region B ∝ B�(10RWR/R)2. At largerdistances the magnetic field becomes toroidal. Then,

for distances greater than ∼ 10RWR, the magnetic fieldstrength can be approximated by B(R) ≈ 100B3(RWR/R)G, where B = 103B3 G is the surface magnetic field ofthe WR type star. Applying this simple scaling, the con-dition for capturing of protons, RL < R, is fulfilled forγn < 3×106. Protons with energies fulfilling this condi-tion are expected to suffer strong adiabatic energy lossesduring gradual expansion of the wind from the star. Wecan determine the Lorentz factor of hadrons at a specificdistance from the binary system by taking into accountadiabatic and collisional energy losses,γh(R) = γh(RBS)RBSkτhp/R, where k ≈ 0.5 is the in-elasticity coefficient in proton-proton collisions. De-pending on the parameters of the considered scenario,either adiabatic losses or collision losses determine theLorentz factors of hadrons at a specific distance of thewind from the binary system. We conclude that the adi-abatic energy losses plays an important role in the pro-cess of production of high energy radiation in the openclusters.

We calculate the spectra of γ-rays and neutrinos pro-duced within the wind cavity. Protons are advected


from the binary system with the velocity of the wind.They produce high energy radiation in collisions withthe matter of the stellar wind during their propagationwithin the wind cavity. On the other hand, neutrons, ex-tracted from nuclei, move realistically through the windcavity. They decay at distances from the binary systemwhich depend on their Lorentz factors. Secondary pro-tons, from decaying neutrons, are also captured in thestellar wind. All these secondary protons suffer colli-sional and adiabatic energy loses due to the expansionof the wind.

Significant amount of γ-rays produced in hadroniccollisions originate close enough to the massive star thattheir absorption in the stellar radiation can play impor-tant role. The γ − γ absorption, in the case of binarysystems without precise orbital parameters, can be ap-proximately evaluated by calculating the average opti-cal depths for γ-ray photons. We calculate the opticaldepths for γ-rays injected isotropically at the distance Rfrom the massive star (see e.g.[31,32,33,34]). The ex-act values of the optical depths depend on the propaga-tion angle of γ-rays in respect to the direction towardsthe star. Therefore, we show their values averaged overthe injection angles assuming isotropic injection of γ-rays from a point source at the shock distance from thestar (see Fig. 1). It is clear that γ-rays with energiesabove several GeV are efficiently absorbed even if theyare produced at relatively large distances from the bi-nary system.

We consider production of radiation by protons fromnuclei accelerated within the binary system. These pro-tons are advected outside the binary system with the ve-locity of the stellar wind. The number of hadrons whichinteract during the advection process with the stellarwind, on the range of distances from the binary sys-tem between RBS (considered as the injection place) andR, is determined by the factor [1 − exp(−τpp)], whereτpp = A(1/RBS − 1/R), and A = Mcσpp/(4πv2

w) ≈2.9 × 1012M−5/v2

3 cm. The spectrum of hadrons, whichinteract at the distance R from the binary system, is

Nh =dNh(γh,R)dγhdRdt

= NhJAR2 eA(1/R−1/RBS), (1)

where Nh = dN(γh)/dγhdt is the injection rate ofhadrons with the Lorentz factors γh from the binary sys-tem, J = dγh(RBS)/dγh(R) = R/(RBSKτhp ) is the Jaco-bian of transformation of hadron energy. In order tocalculate the γ-ray and neutrino spectra produced byprotons, we have to integrate the above injection rateof protons over the whole dimension of the wind cavity

and over the spectrum protons,

dNγ,νdEγ,νdt

=

∫ Rc

RBS

∫ γmax

γmin

NhdNγ,ν(γh)

dEγ,νe−τγγdγhdR (2)

The spectra of γ-rays and neutrinos are calculated by ap-plying the scaling break model for hadronic collisionsdeveloped by Wdowczyk & Wolfendale[35], which issuitable in the considered energy range of relativistichadrons. The spectra are calculated for the range of en-ergies of hadrons γmin = 10 and γmax as reported in Ta-ble 1. Eγ and Eν are the energies of γ-ray photons andneutrinos, respectively.

Neutrons, extracted from nuclei within the binarysystem, propagate along the straight lines and gradu-ally decay into protons in the dense medium surround-ing the binary system. Depending on the Lorentz factorsof neutrons, these protons appear within the wind cavityor outside the wind cavity (i.e. within the open cluster).We calculate the spectra of protons from neutrons de-caying within the wind cavity. The neutron decay rateat the distance, R′, from the binary system (equal to therate of creation of protons from their decay) is given by,

dNp(γp,R′)dγpdR′dt

=Nn

cγnτne−R′/(cγnτn), (3)

where Nn = dN/dγndt is the neutron injection rate fromthe binary system (equal to the acceleration rate of nu-clei), τn = 900 s is the neutron decay time, and theLorentz factor of secondary protons (γp) is assumed tobe equal to the Lorentz factor of neutrons (γn). Protons,from decaying neutrons, are confined by the magneticfield of the stellar wind. Therefore, they suffer adiabaticand collisional energy losses as considered for protonsfrom fragmentation of nuclei in the previous section.The collision rate of protons at the distance, R > R′,from the binary system, is given by the formula,

dNp(γp,R′,R)dγpdR′dRdt

=dNp

dγpdR′dtAR2 JeA(1/R′−1/R), (4)

where J = dγp(R′)/dγp(R) = R/(R′kτpp ), and τpp ≈ 2.9×1012(M−5/v2

3)(1/R′ − 1/R) = A(1/R′ − 1/R).The γ-ray and neutrino spectra, produced by these se-

condary protons in the wind cavity, can be calculatedby integrating the above formula over the spectrum ofprotons, their creation distance in the cavity measuredfrom binary system, R′, and their interaction distance R,

dNγ,νdEγ,νdt

=

∫ Rc

RBS

dR∫ Rc

RdR′∫ γmax

γmin

dγp

× dNp

dγpdRdR′dtdNγ,ν(γp(R))

dEγ,ν. (5)


Figure 2: Gamma-ray spectra produced by protons from disintegrated nuclei (dashed curves), and by protons (solid curves) decaying neutrons(solid curves). These protons interact with the matter within the wind cavity in the case of two example binary systems the WR 20a (on the left)and Eta Carina (on the right). The absorption effects of γ-rays in the stellar radiation are included. The γ-ray spectrum observed by HESS fromthe direction of WR 20a binary system is shown by the dot-dashed line[2]. In the case of Eta Carina binary system, the Fermi-LAT spectrum isshown by triangles[5] and the HESS upper limits[9] are shown by the thin dot-dashed line. The γ-ray spectra have been calculated for the spectraof primary nuclei which have been normalized to the stellar wind powers (see Table 1). The normalization coefficients are equal to η = 5× 10−3, inorder to be consistent with the γ-ray flux reported by HESS from the open cluster Westerlund 2 (see thin dot-dashed line), and η = 10−2, in orderto be consistent with the Fermi observations of Eta Carina below 10 GeV. The broken thin dotted line show the level of sensitivity of the CTA[36].

We have performed calculations of the γ-ray spectraproduced by protons from neutrons decaying within thewind cavity for the parameters of two considered binarysystems. These spectra are clearly below the γ-ray spec-tra produced by protons directly extracted from nuclei.This effect is due to the appearance of protons (fromdecaying neutrons) at a relatively large distances fromthe binary system where the density of the stellar windis already low. On the other hand, these γ-ray spectrashow much smaller absorption effects in the stellar ra-diation due to their production at larger distances fromthe massive star.

The γ-ray spectra, produced in the wind cavity by twoconsidered above populations of protons, are comparedwith the available observations of the two consideredbinary systems in the GeV-TeV energy range and withthe sensitivity of the planned CTA. In fact, the TeV γ-ray source has been reported in the direction of the opencluster Westerlund 2 which contains binary system WR20a[2,24]. The γ-ray spectrum of the source in the di-rection of this open cluster has the spectral index 2.58 inthe energy range ∼1-10 TeV. The nature of this sourceis at present unknown. It is supposed that this emis-sion can be related to the binary system WR 20a, thepulsar wind nebula (PWNe) around PSR J1022-5746,or maybe also to the dense molecular clouds present inthis open cluster. We compare the γ-ray spectrum ex-pected from the wind cavity around the binary systemWR 20a with the above mentioned observations. The

results are shown in Fig. 2. In order to be consistentwith the TeV observations, reasonable value for the en-ergy conversion efficiency from the stellar wind to rela-tivistic nuclei is required (∼ 5×10−3). According to ourcalculations, γ-rays produced in the wind cavity shouldbe mainly responsible for a part of this emission at thehighest observed energies, i.e. ∼ 10 TeV. As we showbelow, the lower energy part of the γ-ray emission fromWesterlund 2 could be either produced by protons whichescape from the wind cavity into dense regions of theopen cluster, or it comes from the other sources presentwithin the Westerlund 2.

We also compare our calculations with the observa-tions of the binary system Eta Carina and the surround-ing dense complex Carina Nebula. The source in thisdirection has been detected in the GeV energies by theAgile[4] and Fermi telescopes[5]. γ-ray emission fromthis source shows two component spectrum, first one ex-tending to a few GeV and the second one extending upto ∼ 100 GeV[7]. The highest energy component showevidences of variability with the orbital period of the bi-nary system[6,8]. This component is expected to be pro-duced within the binary system. On the other hand, thelower component seems to be steady. We compare theγ-ray spectrum expected in terms of our model from thewind cavity around the binary system with the observa-tions (see Fig. 2). The spectrum is normalized to the ob-served lower energy component extending to a few GeV.The γ-ray emission, produced within the wind cavity, is


strongly absorbed in the stellar radiation. Therefore, itis steep in the TeV energy range (spectral index close to-4, see Fig. 2). Therefore, this emission is clearly belowthe present upper limit reported by the HESS Collabo-ration in the TeV energy range[9].

Next, we calculate the γ-ray (and neutrino) spectraproduced by protons which originate as a decay prod-ucts of neutrons within the wind cavity, but they are ad-vected with the stellar wind into the surrounding opencluster. We also calculate the γ-ray emission from rela-tivistic protons which originate from the most energeticneutrons decaying directly outside the wind cavity, i.e.within the open cluster. It is assumed that the open clus-ter has the following parameters, density of matter 10cm−3, magnetic field strength 10−4 G, and radius 20 pc.

A part of protons, which is advected with the stellarwind, can arrive up to the wind cavity border. They canbe injected into the dense open cluster surrounding thebinary system. However, only protons from decayingneutrons can still have large enough energies allowingthem to produce the TeV γ-rays and neutrinos. Protons,from direct disintegration of nuclei, appear close to thebinary system and therefore suffer huge adiabatic en-ergy losses. The spectrum of protons injected from thewind cavity (with the radius Rc) into the open clustercan be calculated from,

dNp(γp)dγpdt

=

∫ Rc

RBS

dNp

dγpdR′dtJdR′, (6)

where the Jacobian is J = dγp(R′)/dγp(Rc). We calcu-late the spectra of protons advected into the open clus-ter for two considered binary systems (WR 20a and EtaCarina). Two different values for the density of mat-ter within the open cluster, which determine the outerradius of the wind cavity, are assumed. The resultsare shown in Fig. 3. As expected, for denser matterwithin the open cluster (corresponding to smaller radiusof the wind cavity) the spectra have larger intensitiesand the maximum in the spectra is shifted to lower ener-gies. These effects are due to the lower adiabatic energylosses in the case of the wind cavities with smaller radii.

In order to calculate the γ-ray and neutrino spectraproduced by these protons in the open cluster, we haveto estimate their residency time within the open clus-ter, due to the diffusion process, and compare it withthe collisional time scale. We assume that protons dif-fuse in the turbulent medium of the open cluster withthe rate well determined by the Bohm diffusion coeffi-cient, DB = RLc/3 ≈ 3 × 1026γ6/B−4 cm2s−1, wherethe magnetic field strength within the open cluster isBoc = 10−4B−4 G. Then, the average diffusion time

scale of protons within the open cluster, with the char-acteristic dimension Roc = 20R20 pc, is estimated on,τdif = R2

oc/2DB ≈ 7 × 1012R220B−4/γ6 s.

The collision time of these protons can be estimatedfrom tout

p = (cnocσpp)−1 ≈ 1014/n10 s, where the densityof surrounding matter is noc = 10n10 cm−3. The inter-action process of these protons within the open clusterdepends on the lifetime of the massive binary systemand the diffusion time scale of protons from the opencluster.

By comparing the diffusion time scale with collisiontime scale, we estimate critical energy of relativisticprotons, γp < γ

intp = 7 × 104R2

20B−4n10, below whichthey can interact efficiently within the open cluster. Weassume that protons with such energies reaches steadystate equilibrium in which the rate of proton injectionequals the rate of proton interaction.

The γ-ray and neutrino spectra produced by protons,advected from the wind cavity, are calculated by inte-gration of the above derived proton spectrum,

dNγ,νdEγ,νdt

=

∫ γintp

γmin

dNp(γp)dγpdt

dNγ,ν(γh)dEγ,ν

dγp. (7)

The most energetic neutrons, extracted from nucleiwithin the binary system, can decay directly outside thewind cavity, since their propagation distance Ln = γnτncbecomes comparable to the radius of the wind cavity.The spectrum of protons from decay of these neutronsis given by,

dNp,cl(γp)dγpdt

=dNn,cl(γn)

dγndt= Nne

−Rcav(γnτnc) , (8)

where Nn = dN(γn)/dγndt is the spectrum of neutronsextracted from nuclei, Rcav is the radius of the wind cav-ity. This neutron spectrum has exactly the same shapeas the spectrum of accelerated nuclei, i.e. the power lawtype with the spectral index equal to 2. It is assumed thatLorentz factors of neutrons are equal to Lorentz factorsof parent nuclei.

We calculate the γ-ray spectra produced by protonsin hadronic collisions with the matter of the open clus-ter, applying their diffusion and interaction model asdescribed above. As before, we consider the openclusters which contain the WR 20a and Eta Carina bi-nary systems. The calculations of the γ-ray spectraare performed for the same parameters of the acceler-ation scenario as discussed for the γ-ray production inthe wind cavity regions of these two binaries. The re-sults are shown for two different values of the param-eter R2

20B−4n10 which determines the conditions withinthe open cluster.


Figure 3: Gamma-ray spectra produced by protons (from decaying neutrons) in the dense cluster around WR 20a and Eta Carina binary systems.The escape of protons from the open cluster is described by assuming the Bohm diffusion prescription and the parameter R2

20B−4n10 = 1.5 (solidcurve) and 0.15 (dashed) for WR 20a binary, and 0.2 (solid) and 0.04 (dashed) for Eta Carina binary. The HESS spectrum observed from thedirection of the open cluster around WR 20a is marked by the thin dot-dashed line and the CTA sensitivity is marked by the thin dotted curve. Theefficiency of acceleration of nuclei is equal to η = 5 × 10−3 for WR 20a and 10−2 for Eta Carina.

In the case of the open cluster containing the binarysystem WR 20a, the γ-ray spectrum expected from theopen cluster Westerlund 2 is consistent with the positivedetection of this open cluster by the HESS Collabora-tion[3], provided that the parameter R2

20B−4n10 ≤ 1.5(see Fig. 3). Note that γ-rays produced in this caseare expected to contribute mainly to the energy rangearound ∼ 1 TeV. This is clearly below the energy rangeat which the γ-ray emission from the wind cavity isexpected. γ-rays, expected from the open cluster interms of our model for Westerlund 2, are predicted tobe still detected by the CTA provided that the param-eter R2

20B−4n10 ≥ 0.15. We have also compared thepredictions of our model with the observations of theCarina Complex containing the Eta Carina binary sys-tem. HESS Collaboration has derived an upper limit onthe γ-ray flux from the extended source from this di-rection[9], which is about a factor of five higher thanthe upper limit on the point source towards the Eta Ca-rina itself. We have compared the calculated γ-ray spec-tra with the HESS upper limit in Fig. 3. These spectraare still consistent with the upper limit provided that thevalue of the parameter R2

20B−4n10 ≤ 0.2. γ-rays pro-duced in terms of our model should contribute only tothe sub-TeV energy range. We conclude that the γ-rayemission, produced by relativistic protons in the CarinaComplex, is expected to be still detectable by the CTAprovided that the open cluster is characterised by the pa-rameter R2

20B−4n10 ≥ 0.04. However, this γ-ray compo-nent is expected to extend only through the low energyrange of the CTA sensitivity, i.e. at ∼100 GeV.

Detection of neutrinos, produced in hadronic inter-actions between relativistic protons and the matter ofthe stellar wind and/or open cluster, could provide ad-ditional constraints on the high energy processes in theconsidered scenario. Therefore, we calculate the neu-trino spectra from the wind cavity and the open clus-ter applying derived above spectra of relativistic pro-tons in these regions. The neutrino spectra from thewind cavity (on the left) and the open cluster (on theright) are shown in Fig. 4 for the WR 20a binary system(solid) and Eta Carina binary system (dashed). Theyare obtained for the parameters of the model as usedfor the corresponding γ-ray spectra shown above, i.e.for the energy conversion efficiencies as described inFig.2 and the distances to these binary systems equalto ∼ 2 kpc. These spectra are compared with the atmo-spheric neutrino background (ANB) and the upper limiton the neutrino flux from discrete sources reported bythe ANTARES Collaboration[37] and the 5 yr sensitiv-ity of the IceTop + IceCube[38]. The neutrino spec-tra, expected from these two binary systems immersedwithin the open clusters, are on the level of the atmo-spheric neutrino background. However, they are aboutan order of magnitude below the present upper limit ofthe flux from discrete sources by ANTARES collabo-ration. It looks that nuclei are not energetic enough inorder to produce neutrino fluxes which could be poten-tially detected by the Ice Cube telescope. We have alsocalculated the expected neutrino flux produced by rela-tivistic protons within the open cluster applying the nor-malization obtained from the comparison of the γ-ray


Figure 4: Spectra of neutrinos (SED) produced by hadrons in the above discussed scenarios from the wind cavity (left) and from the open cluster(right). The minimum energy of escaping protons is equal to 105 GeV, corresponding to R2

20B−4n10 = 1.5, for WR 20a (solid) and 1.5 × 104 GeV,corresponding to R2

20B−4n10 = 0.2 for Eta Carina (dashed). The neutrino spectra are produced by protons which are the decay products of neutronsextracted from nuclei in their collisions with stellar wind. The atmospheric background (ANB) in a viewcone of 1◦ radius around the source isshown by the thin dashed curves[39], the 5 yr sensitivity of the IceTop + IceCube is shown by the thin dotted line[38], and the ANTARES upperlimit on the point sources is shown by dot-dashed line[37]. The proton spectra are normalized in this same way is described for the γ-ray spectra.

fluxes with observations of these two open clusters (seeFig.3). These neutrino fluxes are predicted on a muchlower level than those expected from the wind cavitiesof these binary systems. We conclude that consideredbinary systems are not expected to produce detectableneutrino fluxes from the directions of the open clustersWesterlund 2 and Carina Complex. Note however, thatneutrino fluxes should be larger if the specific open clus-ter contains many massive binary systems. Therefore,our results do not completely exclude future detectionof neutrinos from massive stellar clusters at distances ofa few kpc from the vicinity of the Sun.

4. Conclusions

In this paper we concentrate on the processes due tothe presence of massive binary systems in dense regions(open clusters). We have formulated a model for theinteraction of the binary system with the matter of theopen cluster. It is postulated that nuclei (from heliumto oxygen) can be efficiently accelerated within the bi-nary system. We show that these nuclei are disintegratedin the dense stellar radiation and the matter of the stel-lar wind injecting relativistic protons and neutrons. Wecalculate the expected γ-ray and neutrino emission pro-duced in the interaction of these particles with the mat-ter of the stellar wind and surrounding open cluster.

The results of calculations are shown for the case ofthe two well known massive binary systems (WR 20aand Eta Carina), which have been recently reported asa γ-ray sources in the GeV-TeV energy range. Our

calculations show that the largest fluxes of γ-rays areproduced by protons close to the binary system wherethe density of stellar wind is the largest. However, theabsorption of gamma-rays in the stellar radiation fieldhas the important effect on the gamma-ray spectrum,produced by protons extracted directly from the nuclei.Therefore, the γ-ray spectra produced in these regionsare clearly steeper than the injection spectra of protons(equal to the spectrum of accelerated nuclei). On theother hand, γ-ray spectra, from hadronic collisions ofprotons which are decay products of neutrons extractedfrom nuclei, are mainly produced at larger distancesfrom the binary systems since neutrons move ballisti-caly through the wind cavity and decay at relativelylarge distances. They interact with the matter of thewind with lower density but do not suffer strong absorp-tion in the stellar radiation field. In total, these γ-rayspectra have clearly lower level than spectra producedin previous process.

Significant amount of protons is advected with thestellar wind to the open cluster. Protons, directly ex-tracted from nuclei, cannot be advected to the open clus-ter with large energies due to the huge adiabatic energylosses. Therefore the radiation produced by them can besafely neglected. Another population of relativistic pro-tons is provided by the most energetic neutrons whichcan decay directly into the open cluster, i.e. outsidethe wind cavity. The fate of relativistic protons in theopen cluster depends on the cluster parameters. Protonsdiffuse outside the cluster and interact with the clustermatter. We show, that in the case of Bohm diffusion ap-


proximation, the escape/interaction conditions of pro-tons depend on the parameter R2

20B−4n10, which is thecombination of the parameters characterising the opencluster, its radius, magnetic field strength and density ofmatter. We show that for likely parameters of the opencluster, protons with largest expected energies escapefrom the cluster practically without interaction with thematter. Only lower energy protons are captured in theopen cluster and lose energy on the production of γ-rays and neutrinos. Note that protons escaping from theclusters are expected to have Lorentz factors in the range∼ 10(4−5). Therefore, we predict that the open clusters,containing massive binary systems, might become inter-esting sources of relativistic protons in the Galaxy.

We confronted the γ-ray emission, expected in termsof this model, with the observations of the open clus-ters containing binary systems WR 20a (Westerlund 2)and Eta Carina (Carina Complex). It is concluded thatprotons within the specific open cluster can contributeto the observed TeV γ-ray spectrum (mainly at its lowerenergy part) observed from Westerlund 2. This γ-rayemission is consistent with the upper limits on the TeVflux from Carina Complex. We have also calculated theneutrino spectra expected in this model for these twobinary systems. Unfortunately, these neutrino fluxes areabout two orders of magnitude below the present upperlimit on neutrino emission from discrete sources pro-vided by ANTARES Collaboration [37]. These neutri-nos will be also difficult to observe with the IceCubetelescope since their fluxes, produced in the wind cav-ity regions, are comparable to the atmospheric neutrinobackground.

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Gamma-rays, neutrinos and cosmic rays from dense regions in open clusters