2022AJ....164..242S

Kepler-33 hosts five validated transiting planets ranging in period from 5 to 41 days. The planets are in nearly coplanar orbits and exhibit remarkably similar (appropriately scaled) transit durations indicative of similar impact parameters. The outer three planets have a radius of 3.5 ≲ R_p/R_⊕ ≲ 4.7 and are closely packed dynamically, and thus transit timing variations can be observed. Photodynamical analysis of transit timing variations provide 2σ upper bounds on the eccentricity of the orbiting planets (ranging from <0.02 to <0.2) and the mean density of the host star (${0.39}_{-0.02}^{+0.01}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$). We combine Gaia Early Data Release 3 parallax observations, the previously reported host-star effective temperature and metallicity, and our photodynamical model to refine properties of the host star and the transiting planets. Our analysis yields well-constrained masses for Kepler-33 e (${6.6}_{-1.0}^{+1.1}\,{M}_{\oplus }$) and f (${8.2}_{-1.2}^{+1.6}\,{M}_{\oplus }$) along with 2σ upper limits for planets c (<19 M_⊕) and d (<8.2 M_⊕). We confirm the reported low bulk densities of planet d (<0.4 g cm^–3), e (0.8 ± 0.1 g cm^–3), and f (0.7 ± 0.1 g cm^–3). Based on comparisons with planetary evolution models, we find that Kepler-33 e and f exhibit relatively high envelope mass fractions of ${f}_{\mathrm{env}}={7.0}_{-0.5}^{+0.6} \% $ and f_env = 10.3% ± 0.6%, respectively. Assuming a mass for planet d ∼4 M_⊕ suggests that it has f_env ≳ 12%.