188 17. Tidal Dissipation and Evolution What is the Purpose of this Chapter? There are numerous texts and reviews on the general problem of tides and tidal evolution. For satellites, see for example, Peale, Ann. Rev. Astron. Astrophys. 37 533-602 (1999). This chapter focuses on the narrower but important issue (not well covered in existing literature): The actual mechanism of tidal heating and the resulting heat flow. Since the applications of greatest interest are Io, Europa and Enceladus (but maybe also Titan and Ganymede), the emphasis here is on eccentricity tides in synchronously rotating bodies. However, the ideas can be applied more generally. The one exception is tides in giant planets, where the mechanism is more mysterious and probably different from anything discussed here. Why does tidal dissipation happen? As with any mechanical system, energy is converted from macroscopic motion into heat to the extent that there is a phase lag between forcing and response. This means there is hysteresis. The tide has some period, and the work done per period in steady state will be converted into heat. This can be most readily understood by thinking about a stress-strain diagram: A well-known example of the RHS case is a viscous fluid where stress is linearly related to strain rate. So if stress behaved like sinωt then strain would behave like cosωt and the hysteresis loop would be maximal (stress and strain exactly π/2 out of phase). strain stress strain stress area is heat generated per tidal period perfectly elastic: No heat generated189 A non-synchronously rotating satellite is despun to synchroneity in a geologically short time if it is big enough and /or close enough to the planet. All the cases of interest to us are in this category*. Synchroneity is preserved if there is a long-lived non-hydrostatic bulge that stabilizes the orientation of the body (and the bulge will be along the line joining the satellite to the planet). A very small permanent bulge suffices. _____________________________________________________________ *There is, however, an interesting issue of slightly asynchronous rotation that may be of relevance for understanding some of these bodies. For example, the most prominent long linear features on Europa may have arisen through slight migration of the body orientation away from synchroneity over a long time. This is not discussed further here because it does not contribute significantly to heating even though it may be geologically important). ______________________________________________________ In the case of a synchronously rotating satellite, there is typically a large permanent tidal deformation but it dissipates no heat because it is steady when the orbit is circular. The only source of time-dependent stressing and therefore dissipation is the time-dependent part of the tidal forcing and resultant deformation, arising because of the eccentricity of the orbit. As viewed from the perspective of an observer on the satellite standing at the point on the equator with the planet directly overhead on average, two things are observed: One is the changing distance to the planet as the satellite orbits and the other is the changing position of the planet in the sky because the satellite is rotating almost exactly uniformly* (once per orbit) whereas the angular velocity of the satellite’s orbital motion is varying in accordance with Keplers’ laws. _____________________________________________________________*The reason for the phrase “almost exactly” is that the permanent bulge is alternately carried ahead of and lags behind the satellite-planet line during the orbit in accordance with the difference between almost constant satellite rotation and non-constant satellite orbital motion. The action of planet gravity on this bulge creates an oscillatory torque that causes the satellite to librate. This libration amplitude (expressed as displacement of a point on the surface) can be tens to hundreds of meters and could be detectable. (It has been detected for Mercury). It is a very small modulation of the satellite rotation. It should not be confused with the rocking motion of the eccentricity tide, which is a movement of material relative to rigid body rotation, not a change in rotation.190 The “rocking” motion of the tidal bulge is roughly as important as the up-down motion for the tidal heating, but since we are only seeking to understand general principles here, we will proceed as if it were only the latter that is present. For that part of the response, the picture we have is as shown above. The “unstressed” case is the state corresponding to the mean tidal stress; the eccentricity of the orbit causes oscillations about this mean and results in a tidal amplitude h of the surface. To determine tidal dissipation, we need to estimate the strains (or equivalently tidal amplitude), the stresses associated with that strain, and the phase lag between them. We first seek to estimate h. Homogeneous Satellites: The Perfect Fluid Response “Homogeneous” means uniform properties throughout, but in practice a radially layered structure will not behave much differently except for a constant of order unity. By “perfect fluid”, we mean that the surface will adjust hydrostatically to the tidal potential. This is idealized in multiple ways but it serves as an extremely useful benchmark. Let hf be the fluid tidal amplitude. (Here and everywhere, this should be thought of as varying as sin(nt), where n is the man orbital angular velocity, but the time dependence is implicit rather than explicitly stated). The time-dependent part of the tidal potential is ~ eGMR2/a3 and this is balanced by the change in potential due to satellite self gravity: ghf ~ eGMR2/a3 (17.1) “unstressed” body stressed eccentricity tidal amplitude h191 where g=satellite gravity, M= planet mass, R=satellite radius, a =orbital radius. Equivalently, hf ~ en2R/g. Notice that this says that the tidal bulge is smaller than the hydrostatic rotational bulge (or the permanent tidal bulge) by ~e. (For a synchronously rotating body, the rotational and average tidal distortions are the same to within a factor of order unity since n is both the rotational angular velocity and orbital angular velocity.) Here are some values (remember that hf is not the actual tide, merely what it would be if the body were behaving like a non-resistive