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According to the current cosmological cold dark matter paradigm, the Galactichalo could have been the result of the assemblage of smaller structures. Herewe explore the hypothesis that the classical and ultra-faint dwarf spheroidalsatellites of the Milky Way have been the building blocks of the Galactic haloby comparing their [$\alpha$/Fe] and [Ba/Fe] versus [Fe/H] patterns with theones observed in Galactic halo stars. The $\alpha$ elements deviatesubstantially from the observed abundances in the Galactic halo stars for[Fe/H] values larger than -2 dex, while they overlap for lower metallicities.On the other hand, for the [Ba/Fe] ratio the discrepancy is extended at all[Fe/H] values, suggesting that the majority of stars in the halo are likely tohave been formed in situ. Therefore, we suggest that [Ba/Fe] ratios are abetter diagnostic than [$\alpha$/Fe] ratios. Moreover, for the first time weconsider the effects of an enriched infall of gas with the same chemicalabundances as the matter ejected and/or stripped from dwarf satellites of theMilky Way on the chemical evolution of the Galactic halo. We find that theresulting chemical abundances of the halo stars depend on the assumed infalltime scale, and the presence of a threshold in the gas for star formation. Inparticular, in models with an infall timescale for the halo around 0.8 Gyrcoupled with a threshold in the surface gas density for the star formation (4$\mathrm{M}_{\odot}\,\mathrm{pc}^{-2}$), and the enriched infall from dwarfspheroidal satellites, the first halo stars formed show [Fe/H]$>$-2.4 dex. Inthis case, to explain [$\alpha$/Fe] data for stars with [Fe/H]$<$-2.4 dex weneed stars formed in dSph systems.

Are ancient dwarf satellites the building blocks of the Galactic halo?