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The mass of the Mars-sized exoplanet Kepler-138 b from transit timing

Nature volume 522, pages 321–323 (2015)Cite this article

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Abstract

Extrasolar planets that pass in front of their host star (transit) cause a temporary decrease in the apparent brightness of the star, providing a direct measure of the planet’s size and orbital period. In some systems with multiple transiting planets, the times of the transits are measurably affected by the gravitational interactions between neighbouring planets1,2. In favourable cases, the departures from Keplerian orbits (that is, unaffected by gravitational effects) implied by the observed transit times permit the planetary masses to be measured, which is key to determining their bulk densities3. Characterizing rocky planets is particularly difficult, because they are generally smaller and less massive than gaseous planets. Therefore, few exoplanets near the size of Earth have had their masses measured. Here we report the sizes and masses of three planets orbiting Kepler-138, a star much fainter and cooler than the Sun. We determine that the mass of the Mars-sized inner planet, Kepler-138 b, is Earth masses. Its density is grams per cubic centimetre. The middle and outer planets are both slightly larger than Earth. The middle planet’s density ( grams per cubic centimetre) is similar to that of Earth, and the outer planet is less than half as dense at grams per cubic centimetre, implying that it contains a greater portion of low-density components such as water and hydrogen.

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Figure 1: Transit timing variations of the three planets orbiting Kepler-138.
Figure 2: Mass–radius diagram of well characterized planets smaller than 2.1 Earth radii, R⊕.

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Acknowledgements

D.J-H. acknowledges support through the NASA Postdoctoral Program and funding from the Center for Exoplanets and Habitable Worlds. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science and the Pennsylvania Space Grant Consortium. J.F.R. acknowledges NASA grant NNX14AB92G issued through the Kepler Participating Scientist Program. D.C.F. is an Alfred P. Sloane Fellow and was supported by the Kepler Participating Scientist Program award NNX14AB87G. E.B.F. was supported in part by NASA Kepler Participating Scientist Program award NNX14AN76G and NASA Exoplanet Research Program award NNX15AE21G, as well as the Center for Exoplanets and Habitable Worlds.

Author information

Authors and Affiliations

  1. Department of Astronomy, Pennsylvania State University, Davey Laboratory, University Park, 16802, Pennsylvania, USA

    Daniel Jontof-Hutter & Eric B. Ford

  2. NASA Ames Research Center, Moffett Field, 94035, California, USA

    Daniel Jontof-Hutter, Jason F. Rowe & Jack J. Lissauer

  3. SETI Institute, 189 North Bernardo Avenue, Mountain View, 94043, California, USA

    Jason F. Rowe

  4. Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, 60637, Illinois, USA

    Daniel C. Fabrycky

Authors
  1. Daniel Jontof-Hutter
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Contributions

D.J.-H. led the research effort to model the TTV, constrain planetary masses, and wrote the manuscript. J.F.R. measured transit times from the Kepler data set, characterized the host star using spectral follow-up of the target and constraints from the transits and edited the manuscript. J.J.L. led the interpretation effort, assisted in the dynamical study and writing the manuscript. D.C.F. wrote the software to simulate planetary transits, assisted in interpreting results and edited the manuscript. E.B.F. assisted in the development of statistical methodologies and robustness tests for the TTV modelling and edited the manuscript.

Corresponding author

Correspondence to Daniel Jontof-Hutter.

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Extended data figures and tables

Extended Data Figure 1 Folded light curves with corrections for observed TTV for Kepler-138.

The scattered points are photometric relative fluxes and the curves are analytical models of the transit shape described in the text. a, Kepler-138 b; b, Kepler-138 c; c, Kepler-138 d.

Extended Data Figure 2 Stellar mass and radius models using constraints on the stellar mean density inferred from the light curve.

In cyan are models that adopted a uniform prior in eccentricity, and in magenta are constraints found with orbital eccentricities fixed at zero. Grey points with error bars mark stellar parameters found in the literature while the black error bars mark our adopted solution for stellar mass and radius.

Extended Data Figure 3 The distribution of residual normalized deviations from our best fit dynamical model to the raw transit times.

The histogram marks deviations calculated as: (O − S)/σTT, where O is the observed transit time, S is the simulated transit time and σTT is the measurement uncertainty. The green curve marks a Gaussian distribution.

Extended Data Figure 4 Posterior distributions for TTV model parameters.

Data are shown for our nominal model. a, The planet-to-star mass ratio (Mp/M★) for Kepler-138 b; b, Mp/M★ for Kepler-138 c; c, Mp/M★ for Kepler-138 d. d, ecosω for Kepler-138 b; e, ecosω for Kepler-138 c; f, ecosω for Kepler-138 d. g, esinω for Kepler-138 b; h, esinω for Kepler-138 c; i, esinω for Kepler-138 d. The relative frequency for each histogram is scaled to the mode. Mass ratios are scaled to the M⊙/M⊕ mass ratio.

Extended Data Figure 5 Joint posteriors of model parameters and the effects of eccentricity priors.

The dark (light) grey marks the 68.3% (95.4%) credible intervals for each joint posterior. a–c, Mp/M★ and eccentricity vector components for the inner and middle planets. d–f, Mp/M★ and eccentricity vector components for the middle and outer planets. Panels g, h and i compare Kepler-138 b’s Mp/M★ and its orbital eccentricity, for three eccentricity priors: a uniform prior on eccentricity (g), and models with a Rayleigh distribution of scale factor 0.1 (h), and 0.02 (i).

Extended Data Figure 6 Posterior distributions for mass ratios and relative eccentricities between planets.

The mass ratio of the inner and middle planets is shown in a, and relative eccentricity vector components—that is, the difference in ecosω (esinω) in the inner pair—are shown in b (c). The mass ratio of the middle and outer planets are plotted in d, and the relative eccentricity vector components—that is, the difference in ecosω (esinω) in the outer pair—are shown in e (f).

Extended Data Figure 7 Sensitivity tests for the effects of eccentricity prior, outlying transit times and free inclinations on the mass of Kepler-138 b relative to the host star.

Panel a compares a uniform prior (black curve, our nominal posterior for all comparisons) and a Rayleigh distribution with scale factors 0.1 (navy) and 0.02 (cyan). Panel b compares posteriors with 3σ outliers excluded (black), with two alternatives: 4σ outliers (blue) and 2.5σ outliers removed (light green). Panel c compares our nominal model with one with a free ascending node for the inner (purple) or outer (red) planet.

Extended Data Figure 8 Validation of our method with synthetic data sets.

The green curve marks the posterior for a synthetic data set generated with the same parameters as were the medians of our nominal posteriors (in Table 1). The agreement between the green and black curves validates our method and our claim for a positive mass detection for Kepler-138 b. The magenta and purple shades are posteriors for models using data generated with zero mass for Kepler-138 b. These zero-mass synthetic models all reproduced non-detections.

Extended Data Table 1 Transit times of Kepler-138’s planets
Full size table
Extended Data Table 2 Confidence intervals from distributions found with differential evolution Markov Chain Monte Carlo TTV analysis.
Full size table

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Jontof-Hutter, D., Rowe, J., Lissauer, J. et al. The mass of the Mars-sized exoplanet Kepler-138 b from transit timing. Nature 522, 321–323 (2015). https://doi.org/10.1038/nature14494

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