Re: Tidal influence on Earth's interior temperature

There are two main mechanisms by which the Earth is heated: the first was the accretion and differentiation of the Earth, and the second is the decay of radioactive elements. Accretion heated the proto-Earth as it was forming by the energy released from objects impacting into it. Differentiation released heat very early in the Earth’s history when the iron which forms the Earth’s core sank to the center of the Earth (the heat comes from loss of gravitational potential energy of the dense iron). The decay of radioactive elements distributed throughout the Earth also provides heat, much like a nuclear reactor boils water. This mechanism accounts for much of the present day heat escaping from the Earth (the contributions to present day heat flow are (very) roughly 2/3 from radioactive elements and 1/3 from leftover "primordial" heat of the Earth, including accretion and differentiation). This heat source has declined through time because the short-lived elements (technically, isotopes of elements) decayed rapidly, leaving only long-lived elements like Uranium to heat the Earth.

In very special circumstances, tidal friction can be an important source of heat for a planet. This is the case with Jupiter’s moon Io, and possibly Europa. Io is tidally locked with Jupiter, meaning one side always faces Jupiter. Io’s orbit is locked into a resonance with Europa, which means it always gets pulled on by Europa at the same points in its orbit. This forces Io’s orbit to be somewhat eccentric, so the tidal bulge on Io keeps increasing and decreasing and getting shifted back and forth slightly across the surface. Basically, the friction from this flexing creates the heat that drives Io’s spectacular volcanoes. If Europa was not there, the dissipation of energy from this flexing would quickly circularize Io’s orbit, and that would be the end of major tidal heating. Tidal flexing is not a major source of heat on the Earth. The gravitational influence of Jupiter on Io is about 20,000 times greater than that of the Moon on the Earth, and thus the tidal bulge raised on the Earth by the Moon is much, much smaller than the bulge raised on Io by Jupiter (which, if it stayed in one place, is estimated to be about 8 km high!).

The main reason why the Earth is still hot and geologically active and Mars is not is size. Earth is just bigger and can’t lose its heat as fast, basically because the Earth has a smaller surface area to volume ratio. If you’ve ever baked bread or cookies, you know that bigger loaves of bread or fatter cookies take longer to cook. This is because the heat of the oven only affects the surfaces of the baked goodies and then must diffuse to the (comparatively) cold interior. Bakers of cake and quick breads know that the depth to the cold, gooey interior (as measured by sticking a toothpick into the cake to see if it’s done) increases with time as the heat of the oven slowly conducts into the interior and cooks it. Similarly, the only way for heat to escape from a planet is through its surface. The heat must somehow get to the surface through conduction, convection (plate tectonics, mantle flow), or advection (volcanoes, hot springs), where the heat then radiates into space.

Mars is about half the radius of the Earth, which means it has roughly one- quarter of the surface area and one-eighth the volume of the Earth. Thus, it has about double the surface area to volume ratio. This means, in a very basic sense, that the heat from the warm interior of Mars can escape more rapidly than the Earth’s heat can escape. This is why we observe that the heat-driven geologic activity (such as volcanoes) waned quickly on Mars, while the Earth is still going strong.