Explaining the $^{12}\text{C}/^{13}\text{C}$ ratio in the Galactic halo: the contribution from shell mergers in primordial massive stars

Recent campaigns of observations have provided new measurements of the carbon isotopes in the most metal-poor stars of the Galaxy. These stars are so metal-poor that they could only have been enriched by one or few generations of massive progenitors. However, explaining the primary production of $^{13}$C and the low $^{12}$C/$^{13}$C ratio measured in these stars is challenging. Making use of the most up-to-date models for zero-metal and low-metallicity stars, we investigate the possible sources of $^{13}$C at low metallicity and verify whether massive stars could be the sole responsible for the $^{12}$C/$^{13}$C ratio observed in halo stars. We employ the stochastic model for Galactic chemical evolution GEMS to reproduce the evolution of CNO elements and $^{12}$C/$^{13}$C ratio, including the enrichment from rotating massive stars, some of which show the occurrence of H-He shell mergers. We find that stars without H-He shell mergers do not produce enough $^{13}$C to be compatible with the observations. Instead, the primary production by shell mergers and later ejection during the supernova explosion can explain 30 < $^{12}$C/$^{13}$C < 100. The observations are best reproduced assuming a large frequency of shell mergers. The $^{12}$C/$^{13}$C < 30 can only be reproduced assuming an outer layer ejection and no explosion, but requiring a larger production of $^{12}$C and $^{13}$C. Zero-metal and low-metallicity spinstars with H-He shell mergers appear as the most plausible scenario to explain the low $^{12}$C/$^{13}$C ratio in CEMP-no stars. The entire range of $^{12}$C/$^{13}$C values can be explained by assuming that some stars fully explode while others only eject their outer layers. Shell mergers should be also more frequent and productive, which is allowed by the current uncertainties in the treatment of convection in stellar modelling.