A model for the thermal radio-continuum emission from radiative shocks in colliding stellar winds

Context. In massive-star binary systems, the interaction of the strong stellar winds results in a wind collision region (WCR) between the stars, which is limited by two shock fronts. Besides the nonthermal emission resulting from the shock acceleration, these shocks emit thermal (free-free) radiation detectable at radio frequencies that increase the expected emission from the stellar winds. Observations and theoretical studies of these sources show that the shocked gas is an important, but not dominant, contributor to the total emission in wide binary systems, while it plays a very substantial role in close binaries. Aims: The interaction of two isotropic stellar winds is studied in order to calculate the free-free emission from the WCR. The effects of the binary separation and the wind momentum ratio on the emission from the wind-wind interaction region are investigated. Methods: We developed a semi-analytical model for calculating the thermal emission from colliding stellar winds. Assuming radiative shocks for the compressed layer, which are expected in close binaries, we obtained the emission measure of the thin shell. Then, we computed the total optical depth along each line of sight to obtain the emission from the whole configuration. Results: Here, we present predictions of the free-free emission at radio frequencies from analytic, radiative shock models in colliding wind binaries. It is shown that the emission from the WCR mainly arises from the optically thick region of the compressed layer and scales as \~{}D$^{4/5}$, where D is the binary separation. The predicted flux density S$_{nu}$ from the WCR becomes more important as the frequency {$ν$} increases, showing higher spectral indices than the expected 0.6 value (S$_{nu}$ {\prop} {$ν$}$^{alpha}$, where {$\alpha$} = 0.6) from the unshocked winds. We also investigate the emission from short-period WR+O systems calculated with our analytic formulation. In particular, we apply the model to the binary systems WR 98 and WR 113 and compare our results with the observations. Our theoretical results are in good agreement with the observed thermal spectra from these sources.