Evidence for the volatile-rich composition of a 1.5-Earth-radius planet

a,b,c Same as Fig. 1a-c, for a photodynamical fit including only the three previously-known planets Kepler-138 b,c, and d. This illustrates the extent to which the timescale over which the predicted transit times of Kepler-138d are modulated (the super-period) is underestimated by the three-planet solution. This discrepancy was already hinted at by the mismatch with the Kepler transit times but revealed at high significance by the now longer baseline over which we obtained transits with HST and Spitzer, at times beyond BJD=2457000.

 

The four panels show the corrected light curve of Kepler-138 (open circles) folded in a 2 day window around the expected transit epochs of Kepler-138 b, c, d, and e from the photodynamical fit (see Methods). Transit models corresponding to the median retrieved planet parameters are superimposed to the data (solid colored lines), conservatively assuming an Earth-like composition to estimate the radius of Kepler-138 e. The transits of Kepler-138 b, c, and d are detected in the Kepler light curve, but while Kepler-138 e should be larger than Kepler-138 b, its transit is not detected. We interpret this as originating from a likely non-transiting configuration of Kepler-138 e’s orbit, with an inclination of ≲ 89^∘ consistent with the photodynamical solution, too low to occult the stellar disk from our perspective.

 

From top to bottom, Lomb-Scargle periodogram of the RV dataset, the Kepler light curve, the activity indicator (S-index) and the window function of the RVs. The orbital periods of the four planets, the rotational period of the star and its first harmonic are shown. False-alarm probability levels of 0.1, 1 and 10% are indicated by dashed gray lines in the top two panels. Significant signals are detected at the stellar period and its first harmonic in the light curve. No significant periodicity is detected in the RV and S-index time series.

Zoom on the last 200 days of the Kepler photometric observations (black points) and the best-fitting stellar activity model using a GP (gray shading). The mean is the solid line and the variance is shown as the shaded region. The lower panel shows residuals around the best-fit model divided by the single-point scatter. Posterior constraints on the stellar rotation period from rotational brightness modulations are shown on the right. The GP model reproduces the photometric variability and provides tight constraints on the covariance structure of the stellar signal.

A trained GP model was used to account for stellar activity in the RV fit. The model corresponding to the median retrieved parameters is plotted in purple while the corresponding parameters are annotated in each panel. We add in quadrature the RV jitter term (Supplementary Table 2) with the measurement uncertainties for all RVs. a, Full HIRES time series. b, Residuals to the best fit model. c, RVs phase-folded to the ephemeris of planet b. The phase-folded model for planet b is shown (purple line), while Keplerian orbital models for all other planets have been subtracted. d,e,f, Same as c for Kepler-138 c, d, and e.

a, Temperature-pressure and b, temperature-radius profiles computed to generate a complete planet model for a mass of 2.36 M_⊕, a H₂/He mass fraction of 3%, and no water layer. Self-consistent atmosphere models are shown down to the radiative–convective boundary (dotted, black), for the irradiation of Kepler-138 d, but varying the reference radius at a pressure of 1 kbar. Interior models are displayed for the same composition but different specific entropies (solid, colors). For consistency, full-planet models with a given planet mass and composition are obtained from the combination of interior and atmosphere model that have both matching temperatures and radii at the radiative–convective boundary (bold profiles show the closest match in this example).

We show the joint and marginalized posterior distributions of the planet structure fit for Kepler-138 d in the case of a hydrosphere lying on top of a rock/iron core. The 1, 2 and 3σ probability contours are shown. As expected, the water mass fraction is strongly correlated with the relative amount of rock and iron. The correlation between irradiance temperature and water mass fraction is weak across the considered temperature range.

Transmission spectrum of Kepler-138 d (black points) superimposed with three scenarios for the level of stellar contamination: spot covering fractions of 0.1, 3 or 10% (colored lines, colored filled circles show bandpass-integrated values). The potential impact of unocculted stellar spots on the radius in the Kepler bandpass is small compared to its measurement uncertainty.

 

Same as Extended Data Figure 7, for Kepler-138 c.