  The relatively warm temperatures required on early Earth and Mars have been
difficult to account for via warming from greenhouse gases. We tested whether
this problem can be resolved for both Earth and Mars by a young Sun that is
brighter than predicted by the standard solar model. We computed high-precision
solar evolutionary models with slightly increased initial masses of M_i = 1.01
to 1.07 M_sun; for each mass, we considered three different mass loss
scenarios. We then tested whether these models were consistent with the current
high-precision helioseismic observations. The relatively modest mass loss rates
in these models are consistent with observational limits from young stars and
estimates of the past solar wind obtained from lunar rocks, and do not
significantly affect the solar lithium depletion. For appropriate initial
masses, all three mass loss scenarios are capable of yielding a solar flux 3.8
Gyr ago high enough to be consistent with water on ancient Mars. We find that
all of our mass-losing solar models are consistent with the helioseismic
observations. The early solar mass loss of a few percent does indeed leave a
small fingerprint on the Sun's internal structure. However, for helioseismology
to significantly constrain early solar mass loss would require higher accuracy
in the observed solar parameters and input physics, namely, by a factor of
about 3 for the observed solar surface composition, and a factor of 2 for the
solar interior opacities, the pp nuclear reaction rate, and the diffusion
constants for gravitational settling.
