Rotational Evolution of Asteroids: The Confined Tumbling State as the Origin of the Excess of Slowly Rotating Asteroids

1. Introduction

It was initially assumed that asteroid rotation distribution had to take a Maxwellian form, peaking at about four revolutions per day as the result of the collision-induced random distribution of their spin vectors in three-dimensional velocity space (Harris & Burns, 1979). However, as observations progressed, an excess of slow rotators was noted for small asteroids. The abnormal excess of slow rotators is not captured by the prevailing model involving the YORP effect and collisions (Pravec et al, 2008). Moreover, recent observations (Durech & Hanus, 2023) reveal an obvious drop in the number density in specific size-dependent periods, forming a visible ``gap'' that separates the slow rotators from the faster rotators.

Another puzzle relates to the asteroids in non-principal rotation states, termed “tumblers” (Harris, 1994). Observation shows that nearly all observed tumblers are distributed in the slow rotation zone. The distribution of these tumblers is constrained by a transition line fitting a power-law on period-diameter diagram, which coincidentally matches the newly discovered ``gap''. A plausible explanation for the distribution of tumblers is still lacking, especially for its correlation with the visible gap in the spin distribution of asteroids. 

In this work, we constructed a self-consistent rotational evolution model that takes into account collisional excitation, internal friction damping, and the YORP effect on tumblers. We also developed a semi-supervised machine learning method to identify the gap in the period-diameter diagram.

2. Results

Our model successfully produced the observational features in the period-diameter diagram of main belt asteroids, including the excess of slow rotators, the gap separating slow rotators from fast rotators, and the distribution of tumblers. Our model suggests that the slow rotator group is mainly populated by “tumblers”, i.e., asteroids with unstable rotation vectors, which are less affected by radiative torques than principal-axis rotators and therefore get stranded in a size-dependent region of long rotation periods. The distribution of these bodies is, hence, determined by the competition between the effects of collisions and internal friction damping, the latter depending on the body's viscosity.

By fitting the gap in the simulation results to that in Gaia data, our best-fit model suggests Q/k₂ ~ 5× 10⁸ (D/km)^-2 or μQ ~ 4 × 10⁹ Pa. This is much smaller than usually assumed  μQ > 10¹¹ Pa for monolithic boulders or 10¹³ Pa for cold imporous solid minerals, indicating that rubble piles are weaker (e.g. have a high porosity or a thick regolith layer) and more susceptible to the tidal effect which leads to a faster evolution and a larger equilibrium separation of binary asteroids.

References

Harris, A. W., & Burns, J. A. (1979). Asteroid rotation: I. Tabulation and analysis of rates, pole positions and shapes. Icarus, 40(1), 115-144.

Pravec, P., Harris, A. W., Vokrouhlický, D., Warner, B. D., Kušnirák, P., Hornoch, K., ... & Goncalves, R. (2008). Spin rate distribution of small asteroids. Icarus, 197(2), 497-504.

Ďurech, J., & Hanuš, J. (2023). Reconstruction of asteroid spin states from Gaia DR3 photometry. Astronomy & Astrophysics, 675, A24.

Harris, A. W. (1994). Tumbling asteroids. Icarus, 107(1), 209-211.