atmospheres and to understand the loss rates of hydrogen, deuterium, and oxygen. In Section \[sect:model\], I describe the physical upper atmosphere model used here, in Section \[sect:results\], I present the modelling results, in Section \[sect:evo\], I use these results to study the long term evolution of water losses, and in Section \[sect:conclusions\], I discuss the significance of the results.

![ Evolution of stellar X-ray flux at 1 AU for solar mass stars with different initial rotation rates as calculated by @Tu15. The red, green, and blue tracks correspond to the cases of slow, median, and fast rotators and the dashed black line shows the case of a star that remains at the saturation threshold for its entire lifetime. The grey shades area shows the range of activity levels that I study in this paper. []{data-label="fig:XUVtracks"}](XrayTracks.pdf){width="45.00000%"}

Model {#sect:model}
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![ Hydrodynamic simulation of a rapidly outflowing water vapor atmosphere. The panels show as functions of altitude the outflow speed (*upper-panel*) and the densities of selected neutral (*middle-panel*) and ion (*lower-panel*) species. In the upper panel, the sound and escape speeds are shown and all three lines cross at the same location, as required for a pressure driven Parker wind. []{data-label="fig:Case5"}](VelCase5.pdf "fig:"){width="45.00000%"} 