Tidal Disruptions: Applications of an Analytical Theory for Solid Bodies

Introduction: Although the theory of Roche [1] for the tidal disruption limits of orbiting satellites assumes a fluid body, a length to diameter of exactly 2.07:1, and a particular body orientation, the theory is commonly applied to the satellites of the solar system and to small asteroids and comets passing nearby a planet. Clearly these bodies are neither fluid nor generally that elongated, so a more appropriate theory is needed. Here we present selected exact analytical results for the distortion and disruption limits of solid spinning ellipsoidal bodies subjected to tidal forces. A detailed presentation is given in Holsapple and Michel [2]. This study uses the same approach as the studies of spin limits for solid ellipsoidal bodies given in Holsapple [3] and Holsapple [4]. Strength of Rocks, Soils and Ices: We use the Drucker-Prager strength model with zero cohesion. It is the appropriate model for dry granular materials such as sands and rocks, for rubble-pile asteroids and comets, and for all larger satellites, asteroids and comets where the cohesion is small compared to gravity pressures. The strength is characterized by an angle of friction φ, which can range from zero to 90°. The case with zero friction angle has no shear strength whatsoever, so it is the model for a fluid or gas. Typical dry soils have angles of friction of 30°40°. Since the fluid case is included in the theory as a special case, the classical results of Roche [1] and Jeans [5] are included and re-derived in their entirety; but the general solid case has much more variety and applicability. Stress states at Failure: The distribution of the stresses at a limit failure state are given (Holsapple [1]), for an ellipsoidal body with a, b and c semiaxes, in body-fixed coordinates by