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$a We aim to interpret the radio emission of stellar bow shocks and assess under which conditions it could be either thermal or non-thermal, and how to use the observational data to infer their physical properties. We used an extended non-thermal emission model for stellar bow shocks for which we incorporated a consistent calculation of the thermal emission from the forward shock. We fitted this model to the available radio data, including largely unexplored data at low frequencies. In addition, we used a simplified one-zone model to estimate the gamma-ray emission from particles escaping the bow shocks. We can only explain the radio data from (BD+43°3654 and BD+60°2522) assuming a hard electron energy distribution below $\sim$1 GeV, a high efficiency of conversion of (shocked) wind kinetic power into relativistic electrons, and a relatively high magnetic-to-thermal pressure ratio. In the other systems, the interpretation of the observed flux density is ambiguous, although a non-thermal scenario is also favoured. We also show how complementary observations at other frequencies can allow us to place stronger constraints in the model. We also estimated the gamma-ray fluxes from the HII regions around the bow shocks of BD+43°3654 and BD+60°2522, and obtained luminosities at GeV energies of $\sim 10[elevado a la]{33}$[aprox.]erg\,s$[elevado a la]{-1}$ and $10[elevado a la]{32}$[aprox.]erg\,s$[elevado a la]{-1}$, respectively. Stellar bow shocks can potentially be very efficient particle accelerators. This work provides multi-wavelength predictions of their emission and demonstrates the key role of low-frequency radio observations in unveiling particle acceleration processes. The prospects of detections with next-generation observatories such as SKA and ngVLA are promising. Finally, BD+43°3654 may be detected in GeV in the near future, while bow shocks in general may turn out to be non-negligible sources of low-energy cosmic rays. |
