Aims: We aim at constraining the conditions of the wind and high-energy emission of the host star reproducing the non-detection of Ly$alpha$ planetary absorption. Methods: We model the escaping planetary atmosphere, the stellar wind, and their interaction employing a multi-fluid, three-dimensional hydrodynamic code. We assume a planetary atmosphere composed of hydrogen and helium. We run models varying the stellar high-energy emission and stellar mass-loss rate, further computing for each case the Ly$alpha$ synthetic planetary atmospheric absorption and comparing it with the observations. Results: We find that a non-detection of Ly$alpha$ in absorption employing the stellar high-energy emission estimated from far-ultraviolet and X-ray data requires a stellar wind with a stellar mass-loss rate about six times lower than solar. This result is a consequence of the fact that, for $pi$ Men c, detectable Ly$alpha$ absorption can be caused exclusively by energetic neutral atoms, which become more abundant with increasing the velocity and/or the density of the stellar wind. By considering, instead, that the star has a solar-like wind, the non-detection requires a stellar ionising radiation about four times higher than estimated. This is because, despite the fact that a stronger stellar high-energy emission ionises hydrogen more rapidly, it also increases the upper atmosphere heating and expansion, pushing the interaction region with the stellar wind farther away from the planet, where the planet atmospheric density that remains neutral becomes smaller and the production of energetic neutral atoms less efficient. Conclusions: Comparing the results of our grid of models with what is expected and estimated for the stellar wind and high-energy emission, respectively, we support the idea that the atmosphere of $pi$ Men c is likely not hydrogen-dominated.
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