significantly higher atmospheric losses. In and near the range of input X-ray fluxes considered, the mass flux at the upper boundary of the simulation domain (at an altitude of $10^5$ km) can be described by $$\label{eqn:Mdot}
\log F_\mathrm{mass} = 
-0.239 \left( \log F_\mathrm{X} \right)^2 
+ 1.982 \log  F_\mathrm{X}
- 3.681 ,$$ where the units of $F_\mathrm{X}$ are erg s$^{-1}$ cm$^{-2}$ and $F_\mathrm{mass}$ are . As a note of caution, it is not trivial to estimate total atmospheric mass loss rates from these fluxes and this problem is discussed in more detail in Section \[sect:evo\]. Since I calculate 1D hydrodynamic models for the direction pointing in the direction of the star, simply multiplying the mass flux by , where $R$ is the Earth’s radius plus $10^5$ km, would overestimate the mass loss rates by a factor of a few. The outward particle fluxes for hydrogen and oxygen and the hydrogen-oxygen and hydrogen-deuterium fractionation factors are also shown in Fig. \[fig:MdotgridFx\] and discussed below.

For Case 2 and Case 14, representing moderate and high activity cases, information about the chemical structures of the upper atmospheres is shown in Fig. \[fig:chem\] for five neutral and two ion species. At the base of the 