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The formation of the TRAPPIST-1 system in two steps during the recession of the disk inner edge

Nature Astronomy volume 8, pages 1408–1415 (2024)Cite this article

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Abstract

TRAPPIST-1 hosts seven planets. The period ratios of neighbouring pairs are close to the 8:5, 5:3, 3:2, 3:2, 4:3 and 3:2 ratios in increasing distance from the star. The Laplace angles associated with neighbouring triplets are observed to be librating, proving the resonant nature of the system. This compact, resonant configuration is a manifest sign of disk-driven migration; however, the preferred outcome of such evolution is the establishment of first-order resonances, not the high-order resonances observed in the inner system. Here, we explain the observed orbital configuration with a model that is largely independent of the specific disk migration and orbital circularization efficiencies. Together with migration, the two key elements of our model are that the inner border of the protoplanetary disk receded with time and that the system was initially separated into two subsystems. Specifically, the inner b, c, d and e planets were initially placed in a 3:2 resonance chain and then evolved to the 8:5–5:3 commensurability between planets b, c and d due to the recession of the inner edge of the disk, whereas the outer planets migrated to the inner edge at a later time and established the remaining resonances. Our results pivot on the dynamical role of the presently unobservable recession of the inner edge of protoplanetary disks. They also reveal the role of recurring phases of convergent migration followed by resonant repulsion with associated orbital circularization when resonant chains interact with migration barriers.

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Fig. 1: Evolution of a 3:2 resonant chain under dissipation.
Fig. 2: Joining of the inner and outer subsystems.
Fig. 3: Post-nebular evolution under tidal dissipation of the TRAPPIST-1 system.

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Data availability

The time series of the simulations displayed in the manuscript and the data used to plot the analytical curves in Fig. 1 are available at https://github.com/GabrielePichierri/FormingTrappist-1. Data corresponding to the observed physical and orbital state of the TRAPPIST-1 system were taken from https://github.com/ericagol/TRAPPIST1_Spitzer.

Code availability

The N-body integrations were run using the publicly available SWIFT subroutine package (https://www.boulder.swri.edu/~hal/swift.html), which has been modified with the extra forces needed. The other subroutines are available upon request from the corresponding author (G.P.). The analytical calculations were performed using the computational software Mathematica.

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Acknowledgements

G.P. is grateful for support from the European Research Council (Starting Grant No. 757448-PAMDORA) and from the Barr Foundation for their financial support. A.M. is grateful for support from the European Research Council (Advanced Grant No. HolyEarth 101019380) and to Caltech for the visiting professor programme, which he could benefit from. K.B. is grateful to Caltech’s Center for Comparative Planetology, the David and Lucile Packard Foundation and the National Science Foundation (Grant No. AST 2109276) for their generous support. This study is supported by the Research Council of Norway through its Centres of Excellence funding scheme (Project No. 332523 PHAB to R.B.). G.P. thanks M. Goldberg and B. Bitsch for helpful discussions.

Author information

Authors and Affiliations

  1. Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA

    Gabriele Pichierri & Konstantin Batygin

  2. Max Planck Institute for Astronomy, Heidelberg, Germany

    Gabriele Pichierri

  3. Laboratoire Lagrange, Université Cote d’Azur, CNRS, Observatoire de la Cote d’Azur, Nice, France

    Alessandro Morbidelli

  4. Collège de France, CNRS, PSL Univ., Sorbonne Univ., Paris, France

    Alessandro Morbidelli

  5. Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences, MTA Centre of Excellence, Budapest, Hungary

    Ramon Brasser

  6. Centre for Planetary Habitability, University of Oslo, Oslo, Norway

    Ramon Brasser

Authors
  1. Gabriele Pichierri
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  2. Alessandro Morbidelli
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  3. Konstantin Batygin
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  4. Ramon Brasser
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Contributions

G.P. conceived the model, designed the N-body simulations, modified the N-body code with extra routines to model more effects (migration, the inner edge of the disk and its evolution, and the OLT implementation), performed the numerical experiments, analysed the data, produced the figures and wrote the paper. A.M. conceived of and designed the model, suggested using resonant repulsion as a mechanism for circularizing the orbits at the inner edge, supplied part of the code for the model migration and wrote the paper. K.B. contributed the part of the N-body code that models the planetary tides and helped in designing the N-body simulations and in writing the paper. R.B. originated the discussion on the formation of the TRAPPIST-1 system, contributed data on the planets’ physical and orbital properties and helped in writing the paper.

Corresponding author

Correspondence to Gabriele Pichierri.

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Extended data

Extended Data Fig. 1 Purely tidal evolution of a Trappist-1b,c,d,e 3:2 – 3:2 – 3:2 chain.

This evolution is similar to the one in Fig. 1, with the sole difference that NAM increase is instead provided by a dissipative force onto the planets inside the cavity. The evolution of period ratios (panel (a)) and eccentricities (panel (b)) is equivalent to that of Fig. 1 as expected, until the crossing of the double 8:5 – 5:3 resonance. At this point, unlike the case of NAM increase provided by a receding inner edge (Fig. 1), the dissipative force quickly re-establishes a 3:2 – 3:2 commensurability between planets b,c and d by efficiently lowering their eccentricities, thus restoring the resonant repulsion mechanism. Thus, Tc/Tb and Td/Tc continue to grow past the observed 8:5 – 5:3 ratios, while planet e jumps out of resonance. This shows that direct dissipation onto the planets alone is not as robust a mechanism to explain the assembly of the inner-most 8:5 – 5:3 chain.

Extended Data Fig. 2 Assembly of a primordial 3:2 chain among planets Trappist-1b,c,d,e.

Planets b,c,d,e form a 3:2 – 3:2 – 3:2 chain inside the disc (top shade area), followed by planets b, c and d entering the inner disc cavity, with planet e migrating and reaching the inner edge (the blue line enclosing the top shaded area). The evolution of the semi-major axes is plotted in panel (a). Each planet is indicated by a coloured circle whose size reflects the observed size of the planet. This N-body simulation mimics the entry of planets b, c and d inside the inner cavity as a removal of the inner portion of the disc surrounding these planets. The corresponding evolution of the surface density in this phase is sketched in the panel (b), where the arrow indicates the drop of surface density in time (see also the dashed region in panel (a) and the shift in the initial and final position of the inner edge).

Extended Data Fig. 3 The planets’ period ratios while the inner and outer subsystems are joined.

Two examples of the joining of the inner system (with b, c and d in their 8:5 – 5:3 resonance) with the outer system are shown (see also the left panel of Fig. 2). Panel (a): planets f and g already in their 4:3 resonance. Panel (b): planets f and g close, but not yet inside, their 4:3 resonance. The evolution in both cases is very similar: planet e interacts with planet f via their 3:2 resonance (red curve) and starts to migrate inward with respect to planet d (green curve). Note in particular that, when planet e crosses high-order resonances with planet d, the period ratio Tf/Te increases slightly, which is associated with very efficient ee-damping (see Fig. 1a on the structure of 3:2 resonances). This efficient damping helps in preventing spurious captures in unwanted high-order resonances.

Extended Data Table 1 Statistical study of the successful joining of the inner and outer system
Full size table

Supplementary information

Supplementary Information

Supplementary Figs. 1–3, Table 1 and discussion (with references).

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Pichierri, G., Morbidelli, A., Batygin, K. et al. The formation of the TRAPPIST-1 system in two steps during the recession of the disk inner edge. Nat Astron 8, 1408–1415 (2024). https://doi.org/10.1038/s41550-024-02342-4

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