Refining the Masses and Radii of the Star Kepler-33 and its Five Transiting Planets

Kepler-33 hosts five validated transiting planets ranging in period from 5 to 41 days. The planets are in nearly co-planar orbits and exhibit remarkably similar (appropriately scaled) transit durations indicative of similar impact parameters. The outer three planets have radii of $3.5\lesssim R_{\mathrm{p}}\mathrm{/}{R}_{\oplus }\lesssim 4.7$ and are closely-packed dynamically, and thus transit timing variations can be observed. Photodynamical analysis of transit timing variations provide $2\sigma$ upper bounds on the eccentricity of the orbiting planets (ranging from $<0.02$ to $<0.2$) and the mean density of the host-star ($0.39_{-0.02}^{+0.01}\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^3$). We combine \emph{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 $2\sigma$ upper limits for planets c ($<19M_{\oplus }$) and d ($<8.2M_{\oplus }$). We confirm the reported low bulk densities of planet d ($<0.4\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^3$), e ($0.8\pm 0.1\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^3$), and f ($0.7\pm 0.1\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^3$). Based on comparisons with planetary evolution models, we find that Kepler-33~e and f exhibit relatively high envelope mass fractions of $f_{\mathrm{e}\mathrm{n}\mathrm{v}}=7.0_{-0.5}^{+0.6}\mathrm{\%}$ and $f_{\mathrm{e}\mathrm{n}\mathrm{v}}=10.3\pm 0.6\mathrm{\%}$, respectively. Assuming a mass for planet d $\sim 4M_{\oplus }$ suggests that it has $f_{\mathrm{e}\mathrm{n}\mathrm{v}}\gtrsim 12\mathrm{\%}$.