Radii, masses, and TTVs of TOI-396

Our analysis confirms that the three planets have similar sizes (R_b_=2.004_-0.047_^+0.045^R_{sun}_; R_c_=1.979_-0.051_^+0.054^R_{sun}_; R_d_=2.001_-0.064_^+0.063^R_{sun}_), in agreement with previous findings. Notably, our measurements are ~1.4 times more precise, thanks to the use of two additional TESS sectors. For the first time, we determine RV masses for TOI-396 b and d, and find them to be M_b_=3.55_-0.96_^+0.94^M_{sun}_ and M_d_=7.1+/-1.6M_{sun}_, implying bulk densities of rho_b_=2.44_-0.68_^+0.69^g/cm^3^ and rho_d_=4.9_-1.1_^+1.2^g/cm^3^, respectively.

Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be Prot=6.7+/-1.3d, in agreement with the value predicted from logR'_HK_-based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period.

We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of M_c,dyn_=2.24^+0.13^_-0.67_M_{sun}_, the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to-, but out of- the 5:3 MMR.

A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most effective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ~5.2 years, which should be duly taken into account when scheduling future observations of TOI-396.