JWST observations of $^{13}$CO$_{2}$ ice: Tracing the chemical environment and thermal history of ices in protostellar envelopes

The structure and composition of simple ices can be modified during stellar evolution by protostellar heating. Key to understanding the involved processes are thermal and chemical tracers that can diagnose the history and environment of the ice. The 15.2 $\mu$m bending mode of $^{12}$CO$_2$ has proven to be a valuable tracer of ice heating events but suffers from grain shape and size effects. A viable alternative tracer is the weaker $^{13}$CO$_2$ isotopologue band at 4.39 $\mu$m which has now become accessible at high S/N with the $\textit{James Webb}$ Space Telescope (JWST). We present JWST NIRSpec observations of $^{13}$CO$_2$ ice in five deeply embedded Class 0 sources spanning a wide range in luminosities (0.2 - 10$^4$ L$_{\odot}$ ) taken as part of the Investigating Protostellar Accretion Across the Mass Spectrum (IPA) program. The band profiles vary significantly, with the most luminous sources showing a distinct narrow peak at 4.38 $\mu$m. We first apply a phenomenological approach and show that a minimum of 3-4 Gaussian profiles are needed to fit the $^{13}$CO$_2$ absorption feature. We then combine these findings with laboratory data and show that a 15.2 $\mu$m $^{12}$CO$_2$ band inspired five-component decomposition can be applied for the isotopologue band where each component is representative of CO$_2$ ice in a specific molecular environment. The final solution consists of cold mixtures of CO$_2$ with CH$_3$OH, H$_2$O and CO as well as segregated heated pure CO$_2$ ice. Our results are in agreement with previous studies of the $^{12}$CO$_2$ ice band, further confirming that $^{13}$CO$_{2}$ is a useful alternative tracer of protostellar heating events. We also propose an alternative solution consisting only of heated CO$_2$:CH$_3$OH and CO$_2$:H$_2$O ices and warm pure CO$_2$ ice for decomposing the ice profiles of the two most luminous sources in our sample.