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Less effective hydrodynamic escape of H$_2$-H$_2$O atmospheres on terrestrial planets orbiting pre-main sequence M dwarfs

Terrestrial planets currently in the habitable zones around M dwarfs likely experienced a long-term runaway greenhouse condition because of a slow decline in host-stellar luminosity in its pre-main sequence phase. Accordingly, they might have lost significant portions of their atmospheres including water vapor at high concentration by hydrodynamic escape induced by the strong stellar XUV irradiation. However, the atmospheric escape rates remain highly uncertain due partly to a lack of understanding of the effect of radiative cooling in the escape outflows. Here we carry out 1-D hydrodynamic escape simulations for an H$_{2}$-H$_{2}$O atmosphere on a planet with mass of $1M_{\oplus}$ considering radiative and chemical processes to estimate the atmospheric escape rate and follow the atmospheric evolution during the early runaway greenhouse phase. We find that the atmospheric escape rate decreases with the basal H$_{2}$O/H$_{2}$ ratio due to the energy loss by the radiative cooling of H$_{2}$O and chemical products such as OH and H$_{3}^{+}$: the escape rate of H$_{2}$ becomes one order of magnitude smaller when the basal H$_{2}$O/H$_{2}=0.1$ than that of the pure hydrogen atmosphere. The timescale for H$_{2}$ escape exceeds the duration of the early runaway greenhouse phase, depending on the initial atmospheric amount and composition, indicating that H$_{2}$ and H$_{2}$O could be left behind after the end of the runaway greenhouse phase. Our results suggest that temperate and reducing environments with oceans could be formed on some terrestrial planets around M dwarfs.