Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396

TOI-396 is an F6V star ($V\approx6.4$) orbited by three transiting planets. The orbital periods of the two innermost planets are close to the 5:3 commensurability ($P_b \sim3.6$ d and $P_c \sim6.0$ d). To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS RV observations and retrieved photometric data from TESS. We extracted the RVs via a skew-normal fit onto the HARPS CCFs and performed an MCMC joint analysis of the Doppler measurements and transit photometry, while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a thorough TTV dynamical analysis of the system. Our analysis confirms that the three planets have similar sizes: $R_b=2.004_{-0.047}^{+0.045}R_{\oplus}$; $R_c=1.979_{-0.051}^{+0.054}R_{\oplus}$; $R_d=2.001_{-0.064}^{+0.063}R_{\oplus}$. For the first time, we have determined the RV masses for TOI-396b and d: $M_b=3.55_{-0.96}^{+0.94}M_{\oplus}$ ($\rho_b=2.44_{-0.68}^{+0.69}$ g cm$^{-3}$) and $M_d=7.1\pm1.6M_{\oplus}$ ($\rho_d=4.9_{-1.1}^{+1.2}$ g cm$^{-3}$). Our results suggest a quite unusual system architecture, with the outermost planet being the densest. The Doppler reflex motion induced by TOI-396c remains undetected in our RV time series, likely due to the proximity of $P_c$ to the star's rotation period ($P_{\mathrm{rot}}=6.7\pm1.3$ d). We also discovered that TOI-396b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396c ($M_{c,\mathrm{dyn}}=2.24^{+0.13}_{-0.67}M_{\oplus}$), the result might not be accurate owing to the poor sampling of the TTV phase. We also conclude that TOI-396b and c are close to but out of the 5:3 MMR. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of $\sim$5.2 years.