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A resonant chain of four transiting, sub-Neptune planets

Nature volume 533, pages 509–512 (2016)Cite this article

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Abstract

Surveys have revealed many multi-planet systems containing super-Earths and Neptunes in orbits of a few days to a few months1. There is debate whether in situ assembly2 or inward migration is the dominant mechanism of the formation of such planetary systems. Simulations suggest that migration creates tightly packed systems with planets whose orbital periods may be expressed as ratios of small integers (resonances)3,4,5, often in a many-planet series (chain)6. In the hundreds of multi-planet systems of sub-Neptunes, more planet pairs are observed near resonances than would generally be expected7, but no individual system has hitherto been identified that must have been formed by migration. Proximity to resonance enables the detection of planets perturbing each other8. Here we report transit timing variations of the four planets in the Kepler-223 system, model these variations as resonant-angle librations, and compute the long-term stability of the resonant chain. The architecture of Kepler-223 is too finely tuned to have been formed by scattering, and our numerical simulations demonstrate that its properties are natural outcomes of the migration hypothesis. Similar systems could be destabilized by any of several mechanisms5,9,10,11, contributing to the observed orbital-period distribution, where many planets are not in resonances. Planetesimal interactions in particular are thought to be responsible for establishing the current orbits of the four giant planets in the Solar System by disrupting a theoretical initial resonant chain12 similar to that observed in Kepler-223.

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Figure 1: Transit timing variations (TTVs) for all four planets with respect to a linear ephemeris.
Figure 2: Binned planet transits.
Figure 3: A migration simulation that ends in a configuration matching the observed semi-major axis ratios, and libration angle centres and amplitudes.

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Acknowledgements

We thank A. Howard and G. Marcy for their role in obtaining spectra, and E. Agol, J. Lissauer, and J. Bean for comments on the manuscript. This material is based on work supported by NASA under grant numbers NNX14AB87G (D.C.F.), NNX12AF73G (E.B.F.) and NNX14AN76G (E.B.F.) issued through the Kepler Participating Scientist Program. E.B.F. received support from NASA Exoplanet Research Program award NNX15AE21G. D.C.F. received support from the Alfred P. Sloan Foundation. C.M. was supported by the Polish National Science Centre MAESTRO grant DEC-2012/06/A/ST9/00276.

Author information

Authors and Affiliations

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

    Sean M. Mills & Daniel C. Fabrycky

  2. Institute of Physics and CASA*, University of Szczecin, Wielkopolska 15, 70-451, Szczecin, Poland

    Cezary Migaszewski

  3. Torun Centre for Astronomy, Nicolaus Copernicus University, Gagarina 11, 87-100, Torun, Poland

    Cezary Migaszewski

  4. Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA

    Eric B. Ford

  5. Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA

    Eric B. Ford

  6. Center for Astrostatistics, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA

    Eric B. Ford

  7. University of California at Berkeley, Berkeley, 94720, California, USA

    Erik Petigura & Howard Isaacson

  8. California Institute of Technology, Pasadena, 91125, California, USA

    Erik Petigura

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  1. Sean M. Mills
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Contributions

S.M.M. performed the photodynamic, stability, tidal dissipation and spectral evolution analyses and led the paper authorship. D.C.F. designed the study, performed TTV and Laplace-angle libration analysis, and assisted writing the paper. C.M. performed the migration analysis, assisted in initial data fitting and contributed to the writing of the paper. E.B.F. advised on the DEMCMC analysis and paper direction. E.P. and H.I. obtained and analysed the spectra. All authors read and edited the manuscript.

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Correspondence to Sean M. Mills.

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The authors declare no competing financial interests.

Additional information

Kepler data are publicly available at http://archive.stsci.edu/kepler/.

Extended data figures and tables

Extended Data Figure 1 Spectroscopic fit of the Kepler-223 star.

A fit to Yonsei–Yale (Y2) evolution tracks (coloured lines) with 0.01-Gyr increments marked with filled circles. Colours correspond to mass with increments of 0.01 from 1.0 (orange) to 1.4 (darkest blue). Isochrones (grey lines) are over-plotted in 2-Gyr increments from 4 Gyr (darkest grey) to 10 Gyr (lightest grey) with filled circles every 0.01 increment. One point is labelled for reference (Msun = ). The best-fit (Teff, log(g)) value (black cross) and an ellipse (black) whose semi-major axes indicate 1σ uncertainties of each parameter found from spectral matching are indicated. The stars in this area of parameter space have evolved off the main sequence.

Extended Data Figure 2 Long-cadence light curve for each planet, broken down by quarter (Q).

Data (black filled circles) are binned via a moving average to give the blue curve, to reduce the scatter relative to the horizontal red line indicating no signal. Each panel is centred on the transit times predicted using the linear ephemeris (T0 and P) of ref. 52 (vertical black lines), with the horizontal axis the time in days from the Eth predicted transit time. The box-and-whisker error bars indicate the best-fit mid-transit time and 1σ and 3σ uncertainties based on Δχ2 = 1 and Δχ2 = 9. χ2 values are computed by sliding an overall fit to the transit horizontally across the data and interpolating. Their offset relative to the linear ephemeris lines indicates the magnitudes of the TTVs.

Source data

Extended Data Figure 3 Laplace-angle librations detected by binning transits into quarters and assuming zero eccentricity.

a–c, Error bars show 1σ uncertainties based on Δχ2 = 1. Almost a full libration cycle of all angles is observed in the ~1,500-day observing window. The amplitude of oscillation in the four-body Laplace angle (ϕ3; c) is similar in amplitude to each of the individual Laplace angles (ϕ1, a; ϕ2, b). Because ϕ3 = −3ϕ1 + 2ϕ2, this amplitude could naively be expected to be much larger; however, ϕ1 and ϕ2 are closely related, owing to the four-body resonance of the Kepler-223 system, in contrast to two independent three-body resonances.

Source data

Extended Data Figure 4 Variation in Laplace angles for two 107-year-stable solutions.

a, The librating Laplace angles (ϕ1, red; ϕ2, blue) for a solution from the DEMCMC posterior. Laplace angles librate over the entire 107 years. The orbital-period distribution in Extended Data Fig. 5 uses this model. b, Another solution from , in which the inner Laplace angle (ϕ1; red) librates near the observed value initially, but begins switching chaotically between three different libration centres. This is not uncommon in the DEMCMC posterior. Despite the initial constraint on the outer Laplace angle (ϕ2; blue), there are long periods of circulation with intermittent libration.

Source data

Extended Data Figure 5 Orbital-period ratios of librating and non-librating solutions fitted to data.

a, c, e, The distribution of osculating period ratios for each neighbouring planet pair (Pc/Pb, a; Pd/Pc, c; Pe/Pd, e) over a randomly selected 4-year window in the first 104 years for two 107-year-stable parameter sets from the DEMCMC posterior solution. The dotted histogram represents a solution that showed substantial periods of Laplace-angle circulation. The solid histogram represents a solution in which both ϕ1 and ϕ2 librate for 107 years. The blue vertical line indicates the empirical mean period; blue dashed vertical lines represent the highest and lowest quarter-to-quarter period measured. b, d, f, The same as in a, c, e, but over the entire 107-year interval.

Source data

Extended Data Figure 6 System stability as a function of mean planetary eccentricity.

The fraction of 500 random draws from the posterior that survive for 107 years (crosses) and 106 years (squares) as a function of four-planet-mean eccentricity in bins of width 0.01. 1σ statistical uncertainties are included as vertical error bars on the crosses. Dotted lines indicate the two eccentricity limits for the planets used in : 0.175 (planets c and e) and 0.212 (planets b and d). Numbers represent the total number of draws in each eccentricity bin. The fraction of 107-year-stable systems falls sharply and is consistent with zero well below the eccentricity cuts imposed by .

Source data

Extended Data Table 1 Mean Kepler-223 quarterly TTVs
Full size table
Extended Data Table 2 Complete Kepler-223 parameters
Full size table
Extended Data Table 3 Best-fit Kepler-223 initial conditions
Full size table

Supplementary information

Supplementary Data

This file contains Supplementary Table Dataset 1, a tsv file of predicted transmit times for Kepler-223. Columns: [Transit_Number] \t [Time_(BJD-2454900)] \t [1-Sigma_Uncertainty_(d)]. Description: Transits times and errors are estimated from integrations of the randomly selected 107 year-stable chains from posterior. Transit times are listed quarterly (the nearest transit time every 3 months) and extend for 10 years past the end of the Kepler mission. The transits of each planet are given sequentially and indexed from 0 at BJD 2455700, the data epoch used in our fits. (TXT 10 kb)

Supplementary Data

This file contains the source code compressed into zip format. (ZIP 8 kb)

Supplementary Data

This file contains source data for Extended Data Table 1. (TXT 2 kb)

Supplementary Data

This file contains source data for Extended Data Table 2. (TXT 4 kb)

Supplementary Data

This file contains source data for Extended Data Table 3. (TXT 1 kb)

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Mills, S., Fabrycky, D., Migaszewski, C. et al. A resonant chain of four transiting, sub-Neptune planets. Nature 533, 509–512 (2016). https://doi.org/10.1038/nature17445

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