A Rb deficiency by a factor two with respect to the Sun has been found in M dwarfs of solar metallicity. This deficiency is difficult to understand from both the observational and nucleosynthesis point of views. To test the reliability of this Rb deficiency, we study the Rb and Zr abundances in a sample of KM-type giant stars in a similar metallicity range extracted from the AMBRE Project. We derive Rb and Zr abundances in 54 giant stars with metallicity close to solar by spectral synthesis in LTE and NLTE. The impact of the Zeeman broadening in the RbI line is also studied. The LTE analysis results in a Rb deficiency in giant stars smaller than that obtained in M dwarfs, but the NLTE [Rb/Fe] ratios are very close to solar in the full metallicity range. This contrasts with the figure found in M dwarfs. We investigate the effect of gravitational settling and magnetic activity as possible causes of the Rb deficiency found in M dwarfs. While, the former phenomenon has a negligible impact on the surface Rb abundance, the existence of an average magnetic field with intensity typical of that observed in M dwarfs may result in systematic Rb abundance underestimations if the Zeeman broadening is not considered in the spectral synthesis. The new [Rb,Zr/Fe] vs. [Fe/H] relationships can be explained when the Rb production by rotating massive stars and low-and-intermediate mass stars are considered, without the need of any deviation from the standard s-process nucleosynthesis in AGB stars as previously suggested.
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