# Can tidal locking increase rotational kinetic energy? Where does the energy come from then?

I was thinking about the explanation for how the Moon gets tidally locked with the Earth. We are working in the non-rotating reference frame of the Earth, and assume it is inertial (to an approximate degree). I was looking at the explanation given in this link (see 2nd and 3rd paragraphs).

Because tidal forces of gravitation cause bulges of the moon (in the direction of the Earth-Moon axis), any rotation of the Moon that does not match the orbit of the Moon around Earth will lead to a torque on the Moon, causing the rotation period to gradually become closer to the orbital period.

If the Moon rotates too fast, the usual explanation tell us that the rotational kinetic energy dissipates into heat via tidal friction. This makes complete sense. However, there seems to be an obvious follow-up question to this: What if the rotation of the Moon is too little? Where does the energy come from?

I can see how rotational kinetic energy of the Moon is dissipated into heat by tidal forces, but I don't understand where the energy comes from in the case where the Moon is rotating too slowly (from the point-of-view of Earth non-rotating reference frame). Is tidal friction the right terminology here?

If there are any answers, are there any calculations to support them?

The referenced link is not that good:

The planet Mercury is tidally locked to the Sun so that the same scorchingly hot side always faces the Sun and the other side is perpetually cold.

This was what was thought to be the case a century ago. That Mercury does not rotate at the same rate that it orbits the Sun was overturned over half of a century ago. Mercury is in a 3:2 spin-orbit resonance.

Where does the energy come from?

It of course comes from gravitation potential energy, the same source as the torques that keep the Moon tidally locked.

The ultimate fate of a small, non-spherical, semi-rigid body in the two body problem is for the small body to be orbiting circularly, rotating at the same rate that it is orbiting, and rotating such that the rotation is about the axis with the largest moment of inertia and such that the axis with the smallest moment of inertia always points at the central body.

Without dissipative forces, gravity gradient torques would either make the small body oscillate forever about this stable position or would make the small body rotate chaotically. The presence of dissipative forces (e.g. tidal friction) make the orbiting body seek a minimum energy configuration, and that minimum energy configuration is the configuration described in the previous paragraph.

• Thank you so much for the correction. I am astonished at how much misinformation exists on the internet. Jul 22, 2020 at 1:25
• To be completely clear: you're saying that in the ideal, non-dissipative case, tidal forces (causing the moon to oscillate or rotate chaotically) still conserve energy, because it is converted between gravitational potential energy and kinetic energy. Then additional friction forces on the moon cause its oscillations to be damped, and the energy is then conserved to heat. Do I understand this accurately? Jul 22, 2020 at 1:27
• @MaximalIdeal - You understand correctly. Jul 22, 2020 at 1:49