Do similar ideas about impact craters and geological activity also apply to ice-rich worlds?

Good question! The answer is yes in general, but with at least two major differences.

The first major difference is in what constitutes “geological activity” on ice-rich worlds. In our solar system, even the largest ice-rich worlds (Jupiter’s moon Ganymede and Saturn’s moon Titan) are much smaller than the planet Mars, which means that if they were rocky, we’d put them all in the “small” category along with Mercury, the Moon, and asteroids. In that sense, you might expect their interiors all to be too cool to drive any volcanoes or other geological activity. However, remember that they contain large amounts of ice, and that ice can melt at much lower temperatures than rock. As a result, some of these worlds have enough internal to heat to drive active “ice geology” — for example, volcanoes that erupt with liquid water rather than molten lava. Because of this fact, we in general see fewer impact craters and more evidence of past or present geological activity on the larger ice-rich worlds (such as large moons and Pluto).

The second major difference arises from a source of internal heating that affects numerous ice-rich worlds but not the rocky worlds (at least in our solar system). Scientists call this “tidal heating,” because it arises from tidal effects similar to the effects that cause our tides on Earth. We won’t go into the details of tidal heating here (you can watch this video about it if you are interested), but it can make a world much hotter inside than it would be based on its size alone. For example, tidal heating explains the extreme volcanic activity on Jupiter’s moon Io and the heat that is thought to melt ice into a subsurface ocean on Jupiter’s moon Europa.

To sum up, we do indeed find that larger ice-rich worlds generally have few impact craters and more geological activity than smaller ones. However, the geological activity generally involves flows of melted ice rather than melted rock, and the pattern is not quite as closely tied to size because some of these worlds are also affected by tidal heating.