We explore the kinematics and orbital properties of a sample of 323 very metal-poor stars in the halo system of the Milky Way, selected from the high-resolution spectroscopic follow-up studies of Aoki et al. and Yong et al. The combined sample contains a significant fraction of carbon-enhanced metal-poor (CEMP) stars (22% or 29%, depending on whether a strict or relaxed criterion is applied for this definition). Barium abundances (or upper limits) are available for the great majority of the CEMP stars, allowing for their separation into the CEMP-$s$ and CEMP-no sub-classes. A new method to assign membership to the inner- and outer-halo populations of the Milky Way is developed, making use of the integrals of motion, and applied to determine the relative fractions of CEMP stars in these two sub-classes for each halo component. Although limited by small-number statistics, the data suggest that the inner halo of the Milky Way exhibits a somewhat higher relative number of CEMP-$s$ stars than CEMP-no stars (57% vs. 43%), while the outer halo possesses a clearly higher fraction of CEMP-no stars than CEMP-$s$ stars (70% vs. 30%). Although larger samples of CEMP stars with known Ba abundances are required, this result suggests that the dominant progenitors of CEMP stars in the two halo components were different; massive stars for the outer halo, and intermediate-mass stars in the case of the inner halo.
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