The ion escape of Mars CO$_2$ atmosphere caused by its dissociation products C and O atoms is simulated from present time to $approx$4.1 billion years ago (Ga) by numerical models of the upper atmosphere and its interaction with the solar wind. The planetward-scattered pick-up ions are used for sputtering estimates of exospheric particles including $^{36}$Ar and $^{38}$Ar isotopes. Total ion escape, sputtering and photochemical escape rates are compared. For solar EUV fluxes $geq$3 times that of todays Sun (earlier than $approx$2.6 Ga) ion escape becomes the dominant atmospheric non-thermal loss process until thermal escape takes over during the pre-Noachian eon (earlier than $approx$4.0-4.1 Ga). If we extrapolate the total escape of CO$_2$-related dissociation products back in time until $approx$4.1 Ga we obtain a theoretical equivalent to CO$_2$ partial pressure of more than $approx$3 bar, but this amount did not necessarily have to be present. The fractionation of $^{36}$Ar/$^{38}$Ar isotopes through sputtering and volcanic outgassing from its initial chondritic value of 5.3, as measured in the 4.1 billion years old Mars meteorite ALH 84001, until the present day can be reproduced for assumed CO$_2$ partial pressures between $approx$0.2-3.0 bar, depending on the cessation time of the Martian dynamo (assumed between 3.6-4.0 Ga) - if atmospheric sputtering of Ar started afterwards.
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