摘要：K2-146 is a cool, 0.358${M}_{\odot }$ dwarf that was found to host a mini-Neptune with a 2.67 day period.The planet exhibited strong transit timing variations (TTVs) of greater than 30 minutes, indicative of the presence of an additional object in the system.Here we report the discovery of the previously undetected outer planet in the system, K2-146 c, using additional photometric data.K2-146 c was found to have a grazing transit geometry and a 3.97 day period.The outer planet was only significantly detected in the latter K2 campaigns presumably because of precession of its orbital plane.The TTVs of K2-146 b and c were measured using observations spanning a baseline of almost 1200 days.We found strong anti-correlation in the TTVs, suggesting the two planets are gravitationally interacting.Our TTV and transit model analyses revealed that K2-146 b has a radius of 2.25 ± 0.10 ${R}_{\oplus }$ and a mass of 5.6 ± 0.7 ${M}_{\oplus }$, whereas K2-146 c has a radius of ${2.59}_{-0.39}^{+1.81}$ ${R}_{\oplus }$ and a mass of 7.1 ± 0.9 ${M}_{\oplus }$.The inner and outer planets likely have moderate eccentricities of e = 0.14 ± 0.07 and 0.16 ± 0.07, respectively.Long-term numerical integrations of the two-planet orbital solution show that it can be dynamically stable for at least 2 Myr.We show that the resonance angles of the planet pair are librating, which may be an indication that K2-146 b and c are in a 3:2 mean motion resonance.The orbital architecture of the system points to a possible convergent migration origin.
