Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory

The tidal stress at the surface of a satellite is derived from the gravitational potential of the satelliteʹs parent planet, assuming that the satellite is fully differentiated into a silicate core, a global subsurface ocean, and a decoupled, viscoelastic lithospheric shell. We consider two types of time variability for the tidal force acting on the shell: one caused by the satelliteʹs eccentric orbit within the planetʹs gravitational field (diurnal tides), and one due to nonsynchronous rotation (NSR) of the shell relative to the satelliteʹs core, which is presumed to be tidally locked. In calculating surface stresses, this method allows the Love numbers h and ℓ, describing the satelliteʹs tidal response, to be specified independently; it allows the use of frequency-dependent viscoelastic rheologies (e.g. a Maxwell solid); and its mathematical form is amenable to the inclusion of stresses due to individual tides. The lithosphere can respond to NSR forcing either viscously or elastically depending on the value of the parameter Δ ≡ μ η ω , where μ and η are the shear modulus and viscosity of the shell respectively, and ω is the NSR forcing frequency. Δ is proportional to the ratio of the forcing period to the viscous relaxation time. When Δ ≫ 1 the response is nearly fluid; when Δ ≪ 1 it is nearly elastic. In the elastic case, tensile stresses due to NSR on Europa can be as large as ∼ 3.3 MPa , which dominate the ∼ 50   kPa stresses predicted to result from Europaʹs diurnal tides. The faster the viscous relaxation the smaller the NSR stresses, such that diurnal stresses dominate when Δ ≳ 100 . Given the uncertainty in current estimates of the NSR period and of the viscosity of Europaʹs ice shell, it is unclear which tide should be dominant. For Europa, tidal stresses are relatively insensitive both to the rheological structure beneath the ice layer and to the thickness of the icy shell. The phase shift between the tidal potential and the resulting stresses increases with Δ. This shift can displace the NSR stresses longitudinally by as much as 45° in the direction opposite of the satelliteʹs rotation.