Chemical Evolution of Odd Elements in an Inhomogeneous Early Galaxy

We investigate the chemical evolution of odd-numbered elements such as sodium(Na) and aluminum (Al) during the early epochs of the Galactic halo with theuse of a model that reproduces the observed box-shaped distribution ofextremely metal-poor stars in the [Na, Al/Mg] versus [Mg/H] plane. Our model isconstructed under the assumptions that those stars retain the elementalabundance patterns produced by individual Type II supernovae (SNe), and thatthe yields of the odd elements depend on the initial metallicity, z, of theirSN progenitors. As a result, recent abundance determinations that clarify howthe [Na, Al/Mg] ratios of field stars have evolved to the solar values enableus to deduce how the yields of these odd elements depend on z. The observedtrends in these abundances, in particular the very large scatter (over 1 dex in[Al/Mg]) requires that the Al yield scales as m_Al proportional to z^0.6 for[Mg/H]<-1.8, while the observed [Na/Mg] trend requires that the Na yield scalesas m_Na proportional to z^-0.4 for [Mg/H]<-1.8 and m_Na proportional to z^0.4for [Mg/H]>-1.8. It is found that the predicted frequency distribution of starsin the [Na/Mg] versus [Mg/H] diagram is very sensitive to the assumed form ofthe primordial IMF, and that its slope is steeper than the Salpeter IMF. Thenecessity to match the observed abundance patterns of odd elements and thefrequency distribution of extremely metal-poor stars should provide usefulconstraints on nucleosynthesis calculations of metal-free massive stars as wellas on theories of their formation.Comment: 6pages including 3 figures, to appear in Ap