A direct numerical verification of tidal locking mechanism using the discrete element method

We use a discrete element method to simulate the tidal evolution of the spin of a viscoelastic circular body (a secondary body) moving in a circular orbit under the attraction of a large point-mass (a primary body) located at the centre, where the secondary body can have general elasticity (e.g. variable Poisson’s ratio). The model consists of a group of rigid particles linked by elastic and dissipative springs and allows for translational and bending degrees of freedom and rotation of particles. The tidal deformation of the secondary body when it is orbiting around the primary body under the gravitational attraction, and a small lag angle between the direction of the bulge and a line that connects the two bodies have been reproduced. We measure the angular velocity evolution of the secondary body for different initial angular velocities. It is found that if the initial angular velocity is set as the special value (the locked angular velocity) such that the spin period equals its orbital period, the angular velocity of this body remains constant, indicating a stable “locked state”. However, if the initial angular velocity is smaller/larger than the locked angular velocity, the body will spin up/down (i.e. its angular velocity will increase/decrease) due to the effect of tidal torque. Therefore, the spin velocity of an orbiting body (moon) will finally lock onto the orbiting period. Parameters which determine how rapidly the tidal locking occurs have been identified. These parameters include damping coefficient, the gravitational constant, the mass of the primary body, the distance between the primary and the secondary body, the rigidity parameter and Poisson’s ratio of the secondary body, the radius of the secondary body and self-gravitation parameters. Tidal torques obtained from our simulations are compared with the one from the existing tidal theories and a good agreement is found. We demonstrate that the discrete element method is capable of directly simulating the deformation, spinning and tidal evolution of a viscoelastic object under tidal stress.