The binary stellar system HD 93129A is one of the most massive known binaries in our Galaxy. This system presents non-thermal emission in the radio band, which can be used to infer its physical conditions and predict its emission in the high-energy band. We intend to constrain some of the unknown parameters of HD 93129A through modelling the non-thermal emitter, and also to analyse the detectability of this source in hard X-rays and $gamma$-rays. We develop a broadband radiative model for the wind-collision region taking into account the evolution of the accelerated particles streaming along the shocked region, the emission by different radiative processes, and the attenuation of the emission propagating through the local matter and radiation fields. From the analysis of the radio emission, we find that the binary HD~93129A is more likely to have a low inclination and a high eccentricity. The minimum energy of the non-thermal electrons seems to be between $sim 20 - 100$MeV, depending on the intensity of the magnetic field in the wind-collision region. The latter can be in the range $sim 20 - 1500$ mG. Our model is able to reproduce the observed radio emission, and predicts that the non-thermal radiation from HD~93129A will increase in the near future. With instruments such as textit{NuSTAR}, textit{Fermi}, and CTA, it will be possible to constrain the relativistic particle content of the source, and other parameters such as the magnetic field strength in the wind collision zone, which in turn can be used to obtain upper-limits of the magnetic field on the surface of the very massive stars, thereby inferring whether magnetic field amplification is taking place in the particle acceleration region.
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