Measuring titanium isotope ratios in exoplanet atmospheres

[abridged] Measurements of relative isotope abundances can provide unique insights into the formation and evolution histories of celestial bodies. The five stable isotopes of titanium are used to study the early history of the solar system and constrain Galactic chemical models. The minor isotopes of titanium are relatively abundant compared to those of other elements, making them more accessible for challenging observations. We assessed the feasibility of performing titanium isotope measurements in exoplanet atmospheres, and in particular, whether processing techniques used for high-resolution spectroscopy affect the derived isotope ratios. We used an archival high-dispersion CARMENES spectrum of the M-dwarf GJ 1002 as a proxy for an exoplanet observed at very high signal-to-noise. Spectral retrievals using petitRADTRANS models were performed on both narrow (7045-7090 {\AA}) and wide (7045-7500 {\AA}) wavelength regions, resulting in isotope ratios and uncertainties. These retrievals were repeated on the spectrum with its continuum removed to mimic typical high-dispersion exoplanet observations. The relative abundances of all minor Ti isotopes are found to be slightly enhanced compared to terrestrial values. Loss of continuum information from broadband spectral filtering has little effect on the isotope ratios. The CARMENES spectrum was subsequently degraded by adding varying levels of Gaussian noise to estimate the signal-to-noise requirements for future exoplanet atmospheric observations. For the wide wavelength range, a spectrum with signal-to-noise of 5 is required to determine the isotope ratios with relative errors $\lesssim$10%. Super Jupiters at large angular separations from their host star are the most accessible exoplanets, requiring about an hour of observing time on 8-meter-class telescopes, and less than a minute of observing time with the future Extremely Large Telescope.