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Retention of water in terrestrial magma oceans and carbon-rich early atmospheres

Massive steam and CO$_2$ atmospheres have been proposed for magma ocean outgassing of Earth and terrestrial planets. Yet formation of such atmospheres depends on volatile exchange with the molten interior, governed by volatile solubilities and redox reactions. We determine the evolution of magma ocean--atmosphere systems for a range of oxygen fugacities, C/H ratios and hydrogen budgets that include redox reactions for hydrogen (H$_2$--H$_2$O), carbon (CO--CO$_2$), methane (CH$_4$), and solubility laws for H$_2$O and CO$_2$. We find that small initial budgets of hydrogen, high C/H ratios, and oxidizing conditions, suppress outgassing of hydrogen until the late stage of magma ocean crystallization. Hence early atmospheres in equilibrium with magma oceans are dominantly carbon-rich, and specifically CO-rich except at the most oxidizing conditions. The high solubility of H$_2$O limits its outgassing to melt fractions below $\sim$30\%, the fraction at which the mantle transitions from vigorous to sluggish convection with melt percolation. Sluggish melt percolation could enable a surface lid to form, trapping water in the interior and thereby maintaining a carbon-rich atmosphere (equilibrium crystallization). Alternatively, efficient crystal settling could maintain a molten surface, promoting a transition to a water-rich atmosphere (fractional crystallization). However, additional processes, including melt trapping and H dissolution in crystallizing minerals, further conspire to limit the extent of H outgassing, even for fractional crystallization. Hence, much of the water delivered to planets during their accretion can be safely harbored in their interiors during the magma ocean stage, particularly at oxidizing conditions.