  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.
