Gyroscopes and Conservation of Angular Momentum follow-up

Question

I'd like to follow up on a question I asked two years ago under the same title (Sept. 22, 2019).

I am trying to understand just exactly how angular momentum around the vertical axis is conserved during precession of a gyroscope. The moment we release the gyroscope, it begins to rotate around the vertical (z) axis that has its origin at the pivot point. Shouldn't something else have to start rotating in the opposite direction? If so, what is that exactly?

Last time, we got as far as saying that, in this open system, gravity exerts a torque on the gyroscope, causing it to fall slightly and give up a little potential energy to the kinetic energy of the precession. The torque induces a rotation around z. This leaves unanswered the question of conservation however.

An earlier answer indicated, in response to a follow-up question, that the earth itself would be induced to rotate (imperceptibly of course) in the opposite direction, so that angular momentum would be conserved, assuming the earth and the apparatus comprise a closed system.

How does the gyroscope induce the earth to do this? Of course the gyroscope exerts its own weak gravitational pull on the earth but wouldn't that simply drag the earth along with it - in the same direction? But we need a rotation in the opposite direction. How does that happen?

Also, the gravitation forces are acting vertically from gyroscope to earth and back, but the induced motion would need to be at right angles to them.

Sorry to ask the question again, but upon contemplation, the earlier answer didn't seem complete to me.

Answer 1

You conjecture exchange of momentum with the Earth. While it is possible to create a setup such that the exchange of angular momentum is (ultimately) exchange between the gyroscope and the Earth, the case of accounting for the onset of gyroscopic precesssion does not require that.

One way of showing that is to point out that it is customary to demonstrate onset of gyroscopic precession using a setup where the gyroscope is suspended with a single (thin) string. This string can be twisted several turns with negligable counter-torque developing along the string. That excludes the idea that onset of gyroscopic precession requires exchange of angular momentum with the Earth.

See the following paper by Svilen Kostov and Daniel Hammer: It has to go down a little, in order to go around

Once the gyro wheel has settled in steady gyroscopic precession (nutation damped out) it is seen that the center of mass of the gyro wheel is lower than in the starting position.

Kostov and Hammer provide a diagram.
The normal vector operations such as vector addition/subtraction are applicable for the angular momentum vector. The original angular momentum vector was pointing in a more upwards direction then the angular momentum vector of the steady precession state. That difference in height of the two angular momentum vectors gives a resultant vector that is parallel to the vertical. That vertical angular momentum corresponds to the angular momentum of the steady precessing motion.