Detection of the YORP Effect for Small Asteroids in the Karin Cluster

The Karin cluster is a young asteroid family thought to have formed only $\simeq 5.75$~My ago. The young age can be demonstrated by numerically integrating the orbits of Karin cluster members backward in time and showing the convergence of the perihelion and nodal longitudes (as well as other orbital elements). Previous work has pointed out that the convergence is not ideal if the backward integration only accounts for the gravitational perturbations from the Solar System planets. It improves when the thermal radiation force known as the Yarkovsky effect it is accounted for. This argument can be used to estimate the spin obliquities of the Karin cluster members. Here we take advantage of the fast growing membership of the Karin cluster and show that the obliquity distribution of diameter $D\simeq 1-2$ km Karin asteroids is bimodal, as expected if the YORP effect acted to move obliquities toward the extreme values ($0^\circ$ or $180^\circ$). The measured magnitude of the effect is consistent with the standard YORP model. The surface thermal conductivity is inferred to be $0.07$-0.2 W m$^{-1}$ K$^{-1}$ (thermal inertia $\simeq 300-500$ J m$^{-2}$ K$^{-1}$s$^{-1/2}$). We find that the strength of the YORP effect is roughly $\simeq 0.7$ of the nominal strength obtained for a collection of random Gaussian spheroids. These results are consistent with a surface composed of rough, rocky regolith. The obliquity values predicted here for 480 members of the Karin cluster can be validated by the light-curve inversion method.