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

Context. TOI-396 is an F6 V bright naked-eye star (V ≈ 6.4) orbited by three small (Rp ≈ 2 R) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability (Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR). Aims. 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 radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors. Methods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo 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 transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration. Results. Our analysis confirms that the three planets have similar sizes (Rb = 2.004−+00047045 R; Rc = 1.979−+00051054 R; Rd = 2.001−+00064063 R) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3.55+−009496 M and Md = 7.1 ± 1.6 M, which implies bulk densities of ρb = 2.44−+006869 g cm−3 and ρd = 4.9−+1112 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 ± 1.3 d, in agreement with the value predicted from log R′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 Mc,dyn = 2.24−+006713 M, 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. Conclusions. 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 five hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396.