Our Sun. V. A Bright Young Sun Consistent with Helioseismology and Warm   Temperatures on Ancient Earth and Mars
  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. 
I.-Juliana Sackmann and Arnold I. Boothroyd