The carbon isotope ratio is a powerful tool for studying the evolution of stellar systems due to its sensitivity to the local chemical environment. Recent detections of CO isotopologs in disks and exoplanet atmospheres revealed a high variability in the isotope abundance, ponting towards significant fractionation in these systems. In order to fully understand the evolution of this quantity in stellar and planetary systems, however, it is crucial to trace the isotope abundance from stellar nurseries to the time of planet formation. During the protostellar stage, the multiple vibrational modes of CO_2 and CO ice, which peak in the near- and mid-infrared, provide a unique opportunity to examine the carbon isotope ratio in the solid state. With the current sensitivity and wide spectral coverage of the James Webb Space Telescope, the multiple weak and strong absorption features of CO_2 and CO have become accessible at a high signal-to-noise ratio in solar-mass systems. We aim to study the carbon isotope ratio during the protostellar stage by deriving column densities and ratios from the various absorption bands of CO_2 and CO ice, and by comparing our results with the ratios derived in other astronomical environments. We quantify the CO_2 CO_2 and the CO CO isotope ratios in 17 class 0/I low-mass protostars from the CO_2 nu_1 nu_2 and 2 nu_1 nu_2 combination modes (2.70 mu m and 2.77 mu m), the CO_2 nu_3 stretching mode (4.27 mu m), the CO_2 nu_3 stretching mode (4.39 mu m), the CO_2 nu_2 bending mode (15.2 mu m), the CO 1-0 stretching mode (4.67 mu m), and the CO 1-0 stretching mode (4.78 mu m) using the James Webb Space Telescope NIRSpec and MIRI observations. We also report a detection of the 2-0 overtone mode of CO at 2.35 mu m. The column densities and CO_2 CO_2 ratios derived from the various CO_2 vibrational modes agree within the reported uncertainties, and we find mean ratios of 85 pm 23, 76 pm 12, and 97 pm 17 for the 2.70 mu m band, the 4.27 mu m band, and the 15.2 mu m band, respectively. The main source of uncertainty on the derived values stems from the error on the band strengths; the observational errors are negligible in comparison. Variation of the CO_2 CO_2 ratio is observed from one source to the next, which indicates that the chemical conditions of their envelopes might be genuinely different. The CO CO ratios derived from the 4.67 mu m band are consistent, albeit elevated with respect to the CO_2 CO_2 ratios, and we find a mean ratio of 165 pm 52. These findings indicate that ices leave the prestellar stage with elevated carbon isotope ratios relative to the overall values found in the interstellar medium, and that fractionation becomes a significant factor during the later stages of star and planet formation.
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