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The origin and tidal evolution of hot Jupiters constrained by a broken age–frequency relation

Nature Astronomy volume 10, pages 92–104 (2026)Cite this article

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Abstract

The discovery of hot Jupiters has challenged classical planet formation theories. Although various formation mechanisms have been proposed, their relative contributions remain unclear. Furthermore, hot Jupiters offer a unique opportunity to test tidal theory and measure the fundamental tidal quality factor \({Q}_{* }^{{\prime} }\), which is yet to be well constrained. Here we use a sample of 123 hot Jupiters around single Sun-like stars and find that the slope of the decline in frequency with age abruptly changes at around 2 Gyr, indicative of the presence of two populations of hot Jupiters that formed at different timescales. We use a tidal evolution model to infer a value of \(\log {Q}_{* }^{{\prime} } \approx5.{7}_{-0.3}^{+0.4}\) for Sun-like stars, which reproduces well the number of observed hot Jupiters undergoing orbital decay. We also constrain the relative importance of the two formation channels: most hot Jupiters form within a few hundred million years through ‘early’ models (for example, in situ formation, disk migration, planet–planet scattering and Kozai–Lidov interactions), whereas a substantial portion (\(3{8}_{-14}^{+16} \%\)) forms late with a timescale of several billion years, mainly thorough secular chaotic migration. This result is supported by the observed obliquity distribution of ‘late-arriving’ hot Jupiters. Our findings provide a unified framework that reconciles hot Jupiter demographics and long-term evolution with multichannel formation.

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Fig. 1: Observed age–frequency relation of hot Jupiters.
Fig. 2: Fitting the observed age–frequency relation of hot Jupiters with a hybrid model.
Fig. 3: Supporting evidence for secular chaos in the late model.
Fig. 4: Evaluation of the constraints on \(\log {Q}_{* }^{{\prime} }\) from our work and previous studies.

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Data availability

The datasets generated and analysed in this study are available via Zenodo at https://doi.org/10.5281/zenodo.16907928 (ref. 95). Other data were collected and processed from publicly available datasets, and the selection procedures are described in the main text and Methods.

Code availability

The simulation procedures are described in detail in Methods. The code supporting the findings of this study is available from the corresponding author upon reasonable request.

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Acknowledgements

We thank S. Dong and W. Zhu for helpful discussions and suggestions. LAMOST is operated and managed by the National Astronomical Observatories, CAS, and supported by the Chinese NDRC. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper makes use of data from the first public release of the WASP data as provided by the WASP consortium. This work is supported by the National Key R&D Program of China (Grant No. 2024YFA1611803) and the National Natural Science Foundation of China (Grant Nos. 12273011, 12150009 and 12403071). We acknowledge science research grants from the China Manned Space Project (Grant No. CMS-CSST-2021-B12). J.-W.X. acknowledges the support from the National Youth Talent Support Program. D.-C.C. acknowledges the Cultivation project for LAMOST Scientific Payoff, the Research Achievement of CAMS-CAS and the fellowship of Chinese postdoctoral science foundation (Grant No. 2022M711566).

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Authors and Affiliations

  1. School of Astronomy and Space Science, Nanjing University, Nanjing, China

    Di-Chang Chen, Ji-Wei Xie & Ji-Lin Zhou

  2. Key Laboratory of Modern Astronomy and Astrophysics, Ministry of Education, Nanjing, China

    Di-Chang Chen, Ji-Wei Xie & Ji-Lin Zhou

  3. School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, China

    Di-Chang Chen & Bo Ma

  4. Center of CSST in the Great Bay Area, Sun Yat-sen University, Zhuhai, China

    Di-Chang Chen & Bo Ma

  5. Institute for Astronomy, University of Hawai’i, Honolulu, HI, USA

    Fei Dai

  6. Department of Astronomy, Indiana University, Bloomington, IN, USA

    Songhu Wang

  7. National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China

    Chao Liu

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Contributions

J.-W.X. conceived the project and designed the research. D.-C.C. led the data analyses and numerical simulations. D.-C.C. and J.-W.X. analysed the results and drafted the paper. All authors contributed to discussing the results and to editing and revising the paper.

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Correspondence to Ji-Wei Xie.

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Extended data

Extended Data Fig. 1 The probability density distribution for the typical timescale of planet-star Kozai cycles.

To overcome general relativity pericenter precession, we only keep these Kozai-Lidov oscillations with timescales shorter than the pericenter precession timescales.

Extended Data Fig. 2 The cumulative distribution of the difference between the kinematic ages and tidal evolution timescales t − ttide.

The solid orange line denotes the arrival timescale of hot Jupiters via the secular models derived from simulations Hamers et al. 2017. The results from the observed and synthetic hot Jupiter samples that can be classified as ‘Late-arrived’ are plotted as solid black line and dashed red line, respectively. We also print the two-sample KS test p − value between the observed and and synthetic hot Jupiter sample that can be classified as ‘Late-arrived’.

Extended Data Fig. 3 The evolutionary patterns of hot Jupiters formed via different origin models.

Panel a: The number ratio of the formed hot Jupiters before t over the formed hot Jupiters of all times as a function of age for different origin models. The green and origin lines represent ‘Early ’model and ‘Late’ model, respectively. Panel b: The number ratio of the left (formed – tidally disrupted) hot Jupiters before t over the formed hot Jupiters of all times as a function of age for different origin models. The initial conditions are set as the standard case, that is, the a-distribution of hot Jupiters is set as the results inferred from Kepler data and the initial planetary mass is set as that of cold Jupiters. The modified stellar tidal quality factor \({Q}_{* }^{{\prime} }=1{0}^{6}\).

Extended Data Fig. 4 Fitting the observed age-frequency relation of hot Jupiters with the ‘Early’ model for the standard case.

The comparison between FHJ obtained from observation data and theoretical simulation. Here the hot Jupiters are formed/migrated all by ‘Early’ model (that is, fLate = 0). Panel a: The observation data is plotted as solid black points and line segments denote the 1-σ interval. The solid line denotes the best match. The initial conditions of simulations are as standard case. The modified stellar tidal quality factor \({Q}_{* }^{{\prime} }\) ranges from 104 to 109. Panel b: The residual of the best match of numerical simulation to the observational results. Panel c: Relative likelihood in logarithm as a function of \({Q}_{* }^{{\prime} }\). The blue, green, and red hatched regions indicate the 1-σ, 2-σ, and 3-σ confidence levels.

Extended Data Fig. 5 Fitting the observed age-frequency relation of hot Jupiters with the ‘Late’ model for the standard case.

Similar to Extended Data Fig. 4 but here the hot Jupiters are formed/migrated all by ‘Late’ model (that is, fLate = 1). Panel a: The observation data is plotted as solid black points and line segments denote the 1-σ interval. Panel b: The residual of the best match of numerical simulation to the observational results. Panel c: Relative likelihood in logarithm as a function of Q∗′. The blue, green, and red hatched regions indicate the 1-σ, 2-σ, and 3-σ confidence levels.

Extended Data Fig. 6 Fitting the observed age-frequency relation of hot Jupiters with the Hybrid model for the standard case.

Similar to Extended Data Fig. 4 but here the hot Jupiters are formed/migrated by two origin mechanisms: ‘Early’ model plus ‘Late’ model. FLate is the fraction of hot Jupiters formed by the ‘Late’ model. Panel a: The observation data is plotted as solid black points and line segments denote the 1-σ interval. Panel b: The residual of the best match of numerical simulation to the observational results. The panel c displays the relative likelihood in logarithm as a function of \({Q}_{* }^{{\prime} }\) and fLate.

Extended Data Fig. 7 The cumulative distribution Functions (CDF) of stellar metallicity [Fe/H].

The red and blue lines denote the ‘late-arrived’ hot Jupiter hosts and the ‘early-arrived’ hot Jupiter hosts neighboring to the ‘late-arrived’ hot Jupiter hosts in stellar mass, radius and TD/D. The average metallicities and corresponding uncertainties from bootstrapping are printed in the top-right corner.

Extended Data Table 1 The fitting parameters for theoretical models under various conditions for different observational data
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Extended Data Table 2 The fitting parameters for theoretical models under various conditions for different observational data
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Extended Data Table 3 The KS p − values of the obliquity distributions between the observed hot Jupiter systems and theoretical predictions (Fig. 3)
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Supplementary Discussion and Figs. 1–18.

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Chen, DC., Xie, JW., Zhou, JL. et al. The origin and tidal evolution of hot Jupiters constrained by a broken age–frequency relation. Nat Astron 10, 92–104 (2026). https://doi.org/10.1038/s41550-025-02693-6

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