Spatial and Temporal Patterns of Tidal Dissipation in Synchronous Satellites

Introduction: Tidal heating is an important energy source for several solar system bodies, and there is a wide-spread perception that the pattern of surface heat flow is diagnostic of internal structure. We wish to clarify that situation. Our analysis depends upon two important assumptions: First, that heat transport is dominated by conduction. Second, that the body can be modeled by a sequence of spherically symmetric layers, each with a linear visco-elastic rheology. Under these assumptions, surface heat flow patterns in tidally dominated satellites will reflect radially integrated dissipation patterns. For synchronously rotating satellites with zero obliquity, this pattern depends quite strongly on orbital eccentricity but relatively little on purely radial variations in internal structure. The total amount of heat generated within the body does depend sensitively on internal structure, but the spatial pattern is rather insensitive to structure, especially at low orbital eccentricities. Finite obliquity can complicate the spatial and temporal pattern of dissipation. Motion of the tidal bulge north and south of the equator causes an additional strain field which interacts with the patterns due to radial and longitudinal tidal components. The radial and longitudinal patterns repeat once per anomalistic cycle and the latitudinal pattern follows the nodical cycle. As the nodal line advances and the apsidal line regresses, these tidal components form changing patterns of dissipation. Though tidal evolution will tend to damp initially large obliquity values, they are not expected to vanish entirely, and thus temporal variations in tidal heat generation might be important. Calculation Strategies: Calculating the rate of tidal dissipation for a specified internal structure model can be done in at least two different ways. The clearest approach, conceptually, is to calculate stress, strain, and strain rate at each point within the body, and average the product of stress and strain-rate over the period of tidal forcing [1,2]. That averaged product yields the volumetric dissipation rate. This approach is very general, and allows radial dissection of the dissipation pattern. However, the expressions obtained can be quite complex in form, and the global properties are not evident by inspection. If the radial integral of this local dissipation rate is sought, an alternative approach [3,4] is to take the product of the imposed tidal potential and the time derivative of the induced potential, due to deformation, and average that quantity over the forcing period. With appropriate scaling, this potential product also yields the expected surface heat flow and is much easier to implement. The principal advantage of this approach, in the current context, is that it makes the dependence on internal structure much easier to see. The degree 2 tidal potentials, both imposed and induced, can each be written as a sum of 3 spherical harmonic basis functions. The dissipation pattern, which is constructed from the time average of the product of the potentials, has 9 separate terms in the initial product, but only 4 of them survive the time averaging. Each of the surviving terms can be written as a sum of spherical harmonics of even degree l (0, 2, or 4), even order m (0, 2, or 4), and even parity (cosine in longitude). The resulting dissipation pattern has 3 orthogonal planes of reflection symmetry, with the equator and prime meridian delineating two of them.