X-ray and radio data obtained by XMM-Newton and VLA constrain the stellar wind of the magnetic quasi-Wolf-Rayet star in HD45166

Context. Recently, a powerful magnetic field was discovered in the hot helium star classified as a quasi-Wolf-Rayet (qWR) star (∼2 M_⊙), in the HD 45166 system. Upon its explosion as a core-collapse supernova, it is expected to produce a strongly magnetic neutron star - a magnetar. Among the key parameters that govern pre-supernova evolution is the amount of mass lost via stellar wind. However, the magnetic nature of this helium star is expected to affect its stellar wind, which makes the estimation of the wind parameters uncertain. Aims. We report the first observations of HD 45166 in X-rays with the XMM-Newton telescope and in radio with the VLA interferometer array. By placing the observation results in a theoretical framework, we aim to provide a reliable estimate of the wind strength of the magnetic qWR star. Methods. We explain the X-ray properties in the framework of the magnetically confined wind shock scenario, and we apply the semianalytic model of a dynamical magnetosphere (DM) to reproduce the X-ray emission. We compute the thermal radio emission of the wind and its absorption effect on possible gyro-synchrotron emission from the underlying dipolar magnetosphere, sampled in 3D, by integrating the radiative transfer equation. Results. We did not detect radio emissions, which enabled us to set sensitive upper limits on the radio luminosity. The magnetic qWR star is a slow rotator, and comparison with models reveals that the possible acceleration mechanisms that occur within its DM are not as efficient as in fast-rotating magnetic ApBp-type stars. In contrast, the system is detected in X-rays with log(L_X/L_bol)∼-5.6. Using suitable models, we constrain the mass lost from this magnetic quasi-Wolf-Rayet star as Ṁ ≈ 3 ×10^-10 M_⊙ yr^-1. Conclusions. This novel empirical estimate of the mass-loss rate in a ∼2 M_⊙ helium star confirms that it maintains super-Chandrasekhar mass until collapse and can produce a magnetar as its final evolutionary product.