Four ages of rotating stars in the rotation–activity relationship and gyrochronology

¹ Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
² Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China
³ INAF-Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy
⁴ School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
⁵ Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia
⁶ Università di Catania, Dipartimento di Fisica e Astronomia, Via S. Sofia 78, 95123 Catania, Italy

^⋆ Corresponding author: yhq@nao.cas.cn

Received: 5 March 2025
Accepted: 8 June 2025

Abstract

Context. Gyrochronology and the rotation–activity relationship are standard techniques used to determine the evolution phase and dynamo process of low-mass stars based on their slowing down. Gyrochronology identifies two tracks in color–period diagrams: the convective (C) phase (younger, faster rotating) and the interface (I) phase (older, slower rotating). These phases are separated by a transition, or “gap” (g phase), and there is no precise estimate of its duration. The rotation–activity relation also identifies two stages: the saturated regime (faster, with higher activity) and unsaturated regime (slower, with declining activity). The mismatch in the definition of the evolutionary phases has so far raised many issues in physics and mathematics and has hampered the understanding of how the internal dynamo processes affect the observable properties.

Aims. To address this problem, we seek a unified scheme that shows a one-to-one mapping from gyrochronology to the rotation–activity relationship.

Methods. We combined LAMOST spectra, the Kepler mission, and two open clusters to obtain the chromospheric activity R^″_HK of 6846 stars and their rotation periods. We used R^′_HK and the rotation period to investigate the rotation–activity relationship. Instead of the traditional two-interval model, we applied a three-interval model to fit the rotation–activity relationship in the range of the Rossby number Ro < 0.7. We also used the X-ray data to verify our new model.

Results. We find that the rotation–activity relationship is best fit by three intervals in the rang of Ro < 0.7. We associate those intervals with the convective, gap, and interface phases of gyrochronology. The mean Ro of the C-to-g and g-to-I transition is ≈0.022 and ≈0.15, respectively. The g-to-I transition is on the edge of the intermediate period gap, indicating that the transition of surface brightness from being spot dominated to facula dominated can be associated with the transition from the gap to the I sequence. Furthermore, based on previous studies, we suggest an additional epoch at late times of the I phase (Ro > 0.7; weakened magnetic breaking phase) from the perspective of activity. We further used the three-interval models to fit the period–activity relationship in temperature bins and determine the duration of the transition phase as a function of effective temperature. By comparing the critical temperature and period of the g-to-I transition with the slowly rotating sequence of ten young open clusters whose ages range from 1 Myr to 2.5 Gyr, we conclude that our new model finds the pure I sequence without fast rotating outliers, which defines the zero-age I sequence (ZAIS). We propose that there is an ambiguous consensus on when the I sequence begins. This ambiguity is from the visually convergent sequence of the color–period diagrams in open clusters. This visually convergent sequence is younger than the ZAIS and is actually the pre-I sequence that can be associated with the stall of the spin-down. Our results unify the rotation–activity relationship and gyrochronology for the stellar evolution of low-mass stars, which we denote as the “CgIW” scenario.