Tidal Evolution of Rubble Piles

Many small bodies in the solar system are believed to be rubble piles, a collection of smaller elements separated by voids. We propose a model for the structure of a self-gravitating rubble pile. Static friction prevents its elements from sliding relative to each other. Stresses are concentrated around points of contact between individual elements. The effective dimensionless rigidity, $\tilde\mu_{rubble}$, is related to that of a monolithic body of similar composition and size, $\tilde\mu$ by $\tilde \mu_{rubble} \sim \tilde \mu^{1/2} \epsilon_Y^{-1/2}$, where $\epsilon_Y \sim 10^{-2}$ is the yield strain. This represents a reduction in effective rigidity below the maximum radius, $R_{max}\sim [\mu\epsilon_Y/(G\rho^2)]^{1/2}\sim 10^3\km$, at which a rubble pile can exist. Densities derived for binary near-Earth asteroids imply that they are rubble piles. As a consequence, their tidal evolution proceeds $10^3$ to $10^4$ times faster than it would if they were monoliths. This accounts for both the sizes of their semimajor axes and their small orbital eccentricities. We show that our model for the rigidity of rubble piles is compatible with laboratory experiment in sand.