Recent exoplanet statistics indicate that photo-evaporation has a great impact on the mass and bulk composition of close-in low-mass planets. While there are many studies addressing photo-evaporation of hydrogen-rich or water-rich atmospheres, no detailed investigation regarding rocky vapor atmospheres (or mineral atmospheres) has been conducted. Here, we develop a new 1-D hydrodynamic model of the UV-irradiated mineral atmosphere composed of Na, Mg, O, Si, their ions and electrons, includin molecular diffusion, thermal conduction, photo-/thermo-chemistry, X--ray and UV heating, and radiative line cooling (i.e., the effects of the optical thickness and non-LTE). The focus of this paper is on describing our methodology but presents some new findings. Our hydrodynamic simulations demonstrate that almost all of the incident X-ray and UV energy from the host-star is converted into and lost by the radiative emission of the coolant gas species such as Na, Mg, Mg$^+$, Si$^{2+}$, Na$^{3+}$ and Si$^{3+}$. For an Earth-size planet orbiting 0.02~AU around a young solar-type star, we find that the X-ray and UV heating efficiency is as small as $1 times 10^{-3}$, which corresponds to 0.3~$Mearth$/Gyr of the mass loss rate simply integrated over all the directions. Because of such efficient cooling, the photo-evaporation of the mineral atmosphere on hot rocky exoplanets with masses of $1Mearth$ is not massive enough to exert a great influence on the planetary mass and bulk composition. This suggests that close-in high-density exoplanets with sizes larger than the Earth radius survive in the high-UV environments.
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