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These authors have contributed equally to this work and share first authorship

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In addition to gamma-ray binaries which contain a compact object, high-energy and very high–energy gamma rays have also been detected from colliding-wind binaries. The collision of the winds produces two strong shock fronts, one for each wind, both surrounding a shock region of compressed and heated plasma, where particles are accelerated to very high energies. Magnetic field is also amplified in the shocked region on which the acceleration of particles greatly depends. In this work, we performed full three-dimensional magnetohydrodynamic simulations of colliding winds coupled to a code that evolves the kinematics of passive charged test particles subject to the plasma fluctuations. After the run of a large ensemble of test particles with initial thermal distributions, we show that such shocks produce a nonthermal population (nearly 1% of total particles) of few tens of GeVs up to few TeVs, depending on the initial magnetization level of the stellar winds. We were able to determine the loci of fastest acceleration, in the range of MeV/s to GeV/s, to be related to the turbulent plasma with amplified magnetic field of the shock. These results show that colliding-wind binaries are indeed able to produce a significant population of high-energy particles, in relatively short timescales, compared to the dynamical and diffusion timescales.

Star-forming regions are among the sources of high-energy (HE) photons in our galaxy. The reason for that is still a subject of investigation, given that these regions are populated by a mix of magnetized turbulent plasma, as well as a number of stellar objects at different evolutionary stages, given their initial masses, such as collapsing gas cores of stars yet to form, stars of main sequence, as well as more evolved massive ones, which are already at the edge of being subjects to energetic explosive events that end their evolution. Shocks and magnetic fields, basic ingredients for particle acceleration, are present in all the mentioned scenarios and give a hint of possible generation of HE photons from HE particles by synchrotron, inverse Compton, and pion decay processes. While the magnetized plasma of the interstellar medium is understood as important for the particle diffusion and scattering, stellar sources and compact objects have been mentioned as main drivers of particle acceleration in star-forming regions (SFRs).

Supernova (SNe) remnants have been well associated as sources of HE photons and particles for many decades. Among these, the type-II, Ib, and Ic SNe are strongly related to the loci of star formation, given their progenitor’s high mass. Accretion—and associated disks and jets—is also commonly related to the observed HE photons from SFRs, for example, the X-ray binaries. While low-mass X-ray binaries (LMXB) are spatially distributed over the whole galaxy, the high-mass counterpart (HMXB) is associated to SFRs (see, e.g.,

Not only massive binaries with compact objects emit HE photons, strong shocks generated by high-velocity winds of both stars in massive binary systems can also accelerate particles to very high energies, being observed in hard X-rays and gamma-rays bands, the so-called colliding-wind binaries (CWBs). These objects are also strongly related to SFRs. Particle acceleration to high energies from CWBs has been suggested by

The theory of diffusive shock acceleration (DSA) is based on the scattering and change of particle momenta as they interact with plasma waves in a second-order Fermi acceleration fashion (e.g.,

These previous works, although very useful to generally describe spectral features of nonthermal and high-energy sources associated to CWBs, have not been able to provide a proper answer to two major open issues in this field: 1) how the particles are accelerated in CWBs—which includes how fast this acceleration occurs and how they diffuse, and 2) where these particles are mainly accelerated. To answer these two questions, a more complex analysis is needed, based on full magnetohydrodynamics (MHD) combined with a full treatment of particle acceleration either by evolution of momentum distributions or, even better, by studying the kinematics of individual particles.

Full MHD simulations were first employed for the study of nonthermal emission from CWBs by ^{4}–10^{5} s, were given as parametric estimates for that particular MHD setup of eta Carinae, although the energy distribution of particles was not determined.

Given the context described above, a natural improvement of the theoretical modeling of particle acceleration in CWBs would be the development of a full kinetic description of particles as subject to magnetized plasma fluctuations derived from a full MHD simulation of the shocks. This is the main objective of the present work. Here, we provide the first model to integrate a population of thermal hadronic particles, subject to the local distributions of plasma velocity, density, and magnetic fields derived from a full MHD numerical simulation. The organization of the manuscript is given as follows.

In

We model a binary system consisting of primary and secondary stars placed at a distance of approximately 7.2 AU. The stars are described numerically by spherical regions of the radius of about 0.12 AU at which the density, pressure, outflow velocity, and magnetic field are maintained during the whole simulation, and determined by emitted stellar winds. The wind density of the primary star is set to ^{2} times smaller. We assumed the wind speeds of the primary and secondary stars to be 500 km/s and 2500 km/s, respectively. The wind pressure at the star surfaces is determined by the formula

The above setup is evolved by the ideal adiabatic compressible MHD system of equations:

The MHD equations were solved numerically using a high-order shock-capturing adaptive refinement Godunov-type code AMUN^{1}
^{th} order optimized compact monotonicity preserving reconstruction of Riemann states (^{rd} order 4-step Strong Stability Preserving Runge–Kutta (SSPRK) method (

In order to study the behavior of test particles in the binary system evolved as described in the previous subsection, we took the last available domain snapshot (velocity and magnetic field spatial distributions) at ^{5} (if not specified otherwise) protons in random locations near the turbulent region created by colliding winds with random orientations of velocity and the initial particle velocity distribution corresponding to the thermal distribution at temperature T = 10^{6} K.

The trajectories of injected particles were integrated using the relativistic equation of motion for a charged particle

We assumed the particle velocity _{*} = 10^{−5} to 10^{0} G. Both stars are assumed to maintain the same intensity of surface magnetic fields during the whole simulation time. This value is a model parameter and should not be confused with the real stellar magnetic strength. Due to the numerical modeling limitations, it simply represents the magnetic field which is ejected from the star together with the wind. We should also note that in the integration, the unit of time was chosen as 50 h = 1.8×10^{5} s which, assuming the plasma velocity unit of 1000 km/s, results in the unit of length equal to L ≈ 1.2 AU.

The integration of ^{2}
^{th} order embedded Dormand–Prince (DOP853) method (

The model of colliding-wind binary described in the

In ^{−7} c.u. (white, in the middle of the presented color bar). However,

Distribution of magnetic energy in the computational domain of the studied colliding-wind binary system at time

In ^{–1}, indicating high compression efficiency of these interactions. The regions of compressed gas relax the increased pressure converting it into the mechanical energy and producing turbulence and pushing away the turbulence structure. This compression–relaxation process can be maintained in such a system as long as the stars emit their powerful winds.

Distribution of the divergence of velocity in the computational domain at time

In order to allow for comparison of observations with our results on the particle acceleration and maximum energies reached, which are presented in the following sections, we have estimated the mean magnetic field

As we explained in

As an example, we provide a movie demonstrating the evolution of protons during the time of integration for the model with _{*} = 10^{−1} in the _{k}. The integration time elapses in the logarithmic scale.

In

Evolution of distributions of the particle kinetic energy

The situation significantly changes at around ^{3} s and 10^{4} s, and eventually reaching nearly the speed of light between ^{5} s and 10^{6} s (see ^{5} s, one can notice a significant increase of the perpendicular velocity component, reaching the speed of light, as well. Clearly, the increase of the perpendicular velocity results in the increase of the gyroradius, observed in

As seen in ^{4} s, a significant fraction of protons can increase their kinetic energies, defined as

In the left plot of

Left: Distribution of the particle largest kinetic energy for integration done with different values of

The maximum energies shown here were obtained neglecting any process of energy loss. We are basically interested in the acceleration of hadrons, that is, the typical electron energy losses such as synchrotron emission, inverse Compton, Coulomb collisions, and bremsstrahlung emission losses are unimportant and may be disregarded. The protons are expected to suffer energy loss mainly due to hadronic collisions. The process of pion generation, and further decay, is the main energy loss phenomenon at the energy range obtained from our simulations. In order to validate our results, we may estimate the loss rate for proton–proton collisions, as given by

As discussed in the next subsection, the acceleration rates for these particles lie in the range of few to several GeV/s. In order to halt the acceleration process, the hadronic collisions will be dominant, in a mG (miliGauss) shocked region, only for proton number densities above ^{−3}, which is well above any typical post-shock condition expected for the massive systems known. Therefore, the estimates of acceleration and energy distributions obtained here are likely to be independent on these loss terms.

The analysis of the maximum energies of protons produced in the colliding-wind system gives an insight into the possible potential of the system in the production of high or very high energetic protons. In order to analyze how the thermal protons reach higher energies, we have to analyze the particle distributions at given times. In the left plot of

Left: Particle kinetic energy distribution at moments

Observations of nonthermal emission from WR-O binary systems have been very successful in the radio wavelengths, typically associated to the synchrotron emission from relativistic electrons. The nonthermal leptonic emission at radio wavelengths presents spectral indices as steep as

Emission from relativistic hadrons however, such as nonthermal bremsstrahlung and γ-rays from pion decay, has only been systematically observed for the system of η Carinae. Fermi-LAT detected very energetic emission

Besides the basic properties of the acceleration mechanism such as the maximum kinetic energy reached by the protons, their distribution, and the index of the nonthermal tail, it is also interesting to determine what the range of the acceleration rates available in the given system is and the spatial distribution of the sites where the efficient acceleration is favorable.

In

Distribution of the maximum acceleration rate

In

Probability distribution function of the maximum acceleration rate ^{−5} to 10^{0} G. Relatively high fraction of the computational volume is characterized by higher values of the acceleration rates, from keVs to GeVs per second. The maximum acceleration rate is as high as nearly 80 GeV/s for the model with

In this work, we have investigated a colliding-wind binary system as a possible site of high-energy and very high–energy cosmic ray production. We have performed the magnetohydrodynamic simulations of interacting high velocity and high mass loss rate stellar winds from a typical massive binary system. The winds are characterized by supersonic speeds, 2500 km/s and 500 km/s, for the primary and secondary stars, respectively. Collision of such strong magnetized winds causes the production of shocks, turbulence, and magnetic reconnection. These constitute fundamental ingredients for efficient diffusive acceleration. The magnetic reconnection has been proposed as an alternative acceleration mechanism, also represented by the first-order process, gaining more attention recently (see, e.g.,

In order to study the energy gain of nonthermal protons in such system, we injected test particles and integrated their trajectories using the relativistic equation of the charged-particle motion. We present the evolution of distributions of the particle kinetic energy, gyroradia, and parallel and perpendicular, with respect to the local magnetic field, components of velocity. In order to get a deeper insight in the possible energies, the protons can reach in such a system where we estimate how the maximum kinetic energy and nonthermal power-law index depend on the average magnetic field strength in the shock region

Based on our results, we draw the following conclusions:

• The colliding-wind binary studied in this manuscript is able to produce strongly magnetized turbulent region with shock being able to compress the magnetic field to values higher than the magnetic field maintained within the stars. For the limited resolution used in our modeling, the maximum magnetic field strength in the system is about an order of magnitude higher than the magnetic strength,

• The highest available energies achieved by the accelerated protons depend on the strength of magnetic field. For

• For all studied values of magnetic field intensity, we observe the production of nonthermal tail in the energy distribution with the index decreasing from around 0.7 for the weakest

• The regions of the highest acceleration rates

Our results may be compared to those of previous works as follows. There is no record in the literature of full kinematic evolution of thermal test particles interacting with the shocked MHD plasma of a CWB with the objective of understanding the acceleration process of high-energy particles. Estimates of maximum energies and slopes, based on the first-order Fermi acceleration process, as well as DSA, have been posed in the past (e.g.,

Our findings of energies up to tens of TeV are in agreement with the upper limit DSA estimates of

In the forthcoming studies, it is important to test the particle acceleration in CWBs for particles other than proton. Therefore, in the first step, we plan to determine the dependence of the maximum energy limit on the ratio

Visualization of the protons’ motion in the model with

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

DF-G formulated the scientific problem of particle acceleration in CWB. GK prepared and performed numerical simulations of CWB model and the integration of test particle trajectories. Both authors analyzed and interpreted the results and contributed equally to the writing of the article.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

GK acknowledges support from the Brazilian National Council for Scientific and Technological Development (CNPq no. 304891/2016–9) and FAPESP (grants 2013/10559–5 and 2019/03301–8). DF-G thanks the Brazilian agencies CNPq (no. 311128/2017–3) and FAPESP (no. 2013/10559–5) for their financial support. This work has made use of the computing facilities: the cluster of Núcleo de Astrofísica Teórica, Universidade Cruzeiro do Sul, Brazil.

The Supplementary Material for this article can be found online at:

AMUN code is open source and freely available from

GAccel code is open source and freely available from