| Summary: | The chemical evolution of the Universe and several phases of stellar life are regulated by minute nuclear reactions. The key point for each of these reactions is the value of cross-sections at the energies at which they take place in stellar environments. Direct cross-section measurements are mainly hampered by the very low counting rate and by cosmic background; nevertheless, they have become possible by combining the best experimental techniques with the cosmic silence of an underground laboratory. In the nineties, the LUNA (Laboratory for Underground Nuclear Astrophysics) collaboration opened the era of underground nuclear astrophysics, installing first a homemade 50 kV and, later on, a second 400 kV accelerator under the Gran Sasso mountain in Italy: in 25 years of experimental activity, important reactions responsible for hydrogen burning could have been studied down to the relevant energies thanks to the high current proton and helium beams provided by the machines. The interest in the next and warmer stages of star evolution (i.e., post-main sequence and helium and carbon burning) drove a new project based on an ion accelerator in the MV range called LUNA-MV, able to deliver proton, helium, and carbon beams. The present contribution is aimed to discuss the state of the art for some selected key processes of post-main sequence stellar phases: 12C(α,γ)16O and 12C+12C are fundamental for helium and carbon burning phases, and 13C(α,n)16O and 22Ne(α,n)25Mg are relevant to the synthesis of heavy elements in AGB stars. The perspectives opened by an underground MV facility will be highlighted.
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