The GAPS programme at TNG XLIX. TOI-5398, the youngest compact multi-planet system composed of an inner sub-Neptune and an outer warm Saturn

Short-period giant planets are frequently found to be solitary compared to other classes of exoplanets. Small inner companions to giant planets with $P \lesssim$ 15 days are known only in five compact systems: WASP-47, Kepler-730, WASP-132, TOI-1130, and TOI-2000. Here, we report the confirmation of TOI-5398, the youngest compact multi-planet system composed of a hot sub-Neptune (TOI-5398 c, $P_{\rm c}$ = 4.77271 days) orbiting interior to a short-period Saturn (TOI-5398 b, $P_{\rm b}$ = 10.590547 days) planet, both transiting around a 650 $\pm$ 150 Myr G-type star. As part of the GAPS Young Object project, we confirmed and characterised this compact system, measuring the radius and mass of both planets, thus constraining their bulk composition. Using multidimensional Gaussian processes, we simultaneously modelled stellar activity and planetary signals from TESS Sector 48 light curve and our HARPS-N radial velocity time series. We have confirmed the planetary nature of both planets, TOI-5398 b and TOI-5398 c, alongside a precise estimation of stellar parameters. Through the use of astrometric, photometric, and spectroscopic observations, our findings indicate that TOI-5398 is a young, active G dwarf star (650 $\pm$ 150 Myr), with a rotational period of $P_{\rm rot}$ = 7.34 days. The transit photometry and radial velocity measurements enabled us to measure both the radius and mass of planets b, $R_b = 10.30\pm0.40 R_{\oplus}$, $M_b = 58.7\pm5.7 M_{\oplus}$, and c, $R_c = 3.52 \pm 0.19 R_{\oplus}$, $M_c = 11.8\pm4.8 M_{\oplus}$. TESS observed TOI-5398 during sector 48 and no further observations are planned in the current Extended Mission, making our ground-based light curves crucial for ephemeris improvement. With a Transmission Spectroscopy Metric value of around 300, TOI-5398 b is the most amenable warm giant (10 < $P$ < 100 days) for JWST atmospheric characterisation.