  Post-Noachian Martian paleochannels indicate the existence of liquid water on
the surface of Mars after about 3.5 Gya (Irwin et al., 2015; Palucis et al.,
2016). In order to explore the effects of variations in CO$_{2}$ partial
pressure and obliquity on the possibility of surface water, we created a
zero-dimensional surface energy balance model. We combine this model with
physically consistent orbital histories to track conditions over the last 3.5
Gyr of Martian history. We find that melting is allowed for atmospheric
pressures corresponding to exponential loss rates of $dP/dt \propto t^{-3.73}$
or faster, but this rate is within $0.5 \sigma$ of the rate calculated from
initial measurements made by the Mars Atmosphere and Volatile EvolutioN (MAVEN)
mission, if we assume all the escaping oxygen measured by MAVEN comes from
atmospheric CO$_{2}$ (Lillis et al., 2017; Tu et al., 2015). Melting at this
loss rate matches selected key geologic constraints on the formation of
Hesperian river networks, assuming optimal melt conditions during the warmest
part of each Mars year (Irwin et al., 2015; Stopar et al., 2006; Kite et al.,
2017a,b). The atmospheric pressure has a larger effect on the surface energy
than changes in Mars's mean obliquity. These results show that initial
measurements of atmosphere loss by MAVEN are consistent with atmospheric loss
being the dominant process that switched Mars from a melt-permitting to a
melt-absent climate (Jakosky et al., 2017), but non-CO$_{2}$ warming will be
required if $<2$ Gya paleochannels are confirmed, or if most of the escaping
oxygen measured by MAVEN comes from H$_{2}$O.
