I can understand a temporary heating, from the tital forces exerted on the moon but wouldn't there be cooling as well eventually when particles "give in" to contraction? Wouldn't they eventually net a unchanging whole body temperature? i.e. How can Europa's oceans be warmed by that and how can Io's crust be melted by that?
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I think you've misunderstood both the nature of tidal stresses and the nature of the heating.
Concerning the forces there is tension in the radial direction and compression in both non-radial direction simultaneously present. If these forces were static they would do nothing, but they can change two different ways. In a case like Io or Europa the magnitude of both sets of forces grow and shrink as the moon moves around it's slightly eccentric orbit (thereby getting closer to and farther from the planet). In a case like the tides induced on the Earth by the moon, the primarily change is the steady progression of the tidal direction relative the body linked frame of reference that changes the stress on any given part of the body. In principle both effects can be present.
Concerning the heating, it arises from dissipative rather than elastic processes: think friction not rubber bands. That is to the degree that the body acts like a spring (simply storing the energy and gives it up again) there is no heating. To the extent that there is friction or fracture--irreversible processes--the energy involved becomes thermalized. The thermal energy can't go back into ordered bulk motion because of the usual thermodynamic constraints.
answered Jan 25, 2013 at 1:00
dmckee --- ex-moderator kittendmckee --- ex-moderator kitten
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$\begingroup$ Where does the energy come from? Does it actually drop slightly from orbit? $\endgroup$– j rivJan 25, 2013 at 13:17
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1$\begingroup$ The energy must come from the system, but the details matter: in the Earth-moon system if comes from the rotational kinetic energy of the Earth and some of that also goes into lifting the moon into a higher orbit. I'm uncertain of the dynamics of the Jupiter-Io system. It's a good question. $\endgroup$ Jan 25, 2013 at 20:17
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$\begingroup$ So should it actually drop orbit to produce energy on it (or give in in internal structure)? $\endgroup$– j rivJan 25, 2013 at 21:23
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2$\begingroup$ @LelaDax In the case of tidal heating due to a highly eccentric orbit, the energy often does come from the orbital energy of the smaller body being heated. The size and eccentricity of the orbit decrease with time. This is one of the proposed mechanisms for hot Jupiters to move inward toward their stars. $\endgroup$– user10851Jan 29, 2013 at 18:40