We test the hypothesis that the observed first-peak (Sr, Y, Zr) and second-peak (Ba) s-process elemental abundances in low metallicity Milky Way stars ($text{[Fe/H]} lesssim -0.5$), and the abundances of the intervening elements Mo and Ru, can be explained by a pervasive r-process contribution that originates in neutrino-driven winds from highly-magnetic and rapidly rotating proto-neutron stars (proto-NSs). To this end, we construct chemical evolution models that incorporate recent calculations of proto-NS yields in addition to contributions from AGB stars, Type Ia supernovae, and two alternative sets of yields for massive star winds and core collapse supernovae. For non-rotating massive star yields from either set, models without proto-NS winds underpredict the observed s-process peak abundances by $0.3$-$1,text{dex}$ at low metallicity, and they severely underpredict Mo and Ru at all metallicities. Models that include the additional wind yields predicted for proto-NSs with spin periods $P sim 2$-$5,text{ms}$ fit the observed trends for all these elements well. Alternatively, models that omit proto-NS winds but adopt yields of rapidly rotating massive stars, with $v_{rm rot}$ between $150$ and $300,text{km},text{s}^{-1}$, can explain the observed abundance levels reasonably well for $text{[Fe/H]}<-2$. These models overpredict [Sr/Fe] and [Mo/Fe] at higher metallicities, but with a tuned dependence of $v_{rm rot}$ on stellar metallicity they might achieve an acceptable fit at all [Fe/H]. If many proto-NSs are born with strong magnetic fields and short spin periods, then their neutrino-driven winds provide a natural source for Sr, Y, Zr, Mo, Ru, and Ba in low metallicity stellar populations. Spherical winds from unmagnetized proto-NSs, on the other hand, overproduce the observed Sr, Y, and Zr abundances by a large factor.
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