How exactly does the Moon stabilizes Earth axial tilt?

Question

There are many references regarding the Moon stabilizing the tilt of the Earth's rotational axis. I'd like to see some support for that claim, more than non-sequitur handwaving "Moon causes tides, therefore Earth axis doesn't wobble" that a google search yields.

EDIT: based on comments, I should mention that the question is not about precession, which doesn't change axial tilt and therefore doesn't have any effect on average annual illumination of any place on Earth. I was taking about axial tilt variation.

From the above reference: "Indeed, Laskar and Robutel also showed that the axial tilt of Mars, which has only two tiny moons, has varied between 10° and 60° in the past, which caused huge climate variations that in turn could have contributed to the loss of most of the planet’s atmosphere, leaving Mars the bone-dry desert world that it is now. Since then, most astrobiologists have assumed that Earth-like planets in other solar systems would need a comparatively large moon to support complex life over long periods of time."

The latter part, how a large moon would stabilize axial tilt, is what I am trying to understand.

Answer 1

The original 1993 article is available as file with scanned pages:
The chaotic obliquity of the planets

(Unfortunately the scans are of poor quality; very dark, low contrast, and the images were saved as .jpg files)

As I understand it: the approach was to throw massive computational power at it. I surmise that in the model every planet was affecting the motion of each other planet. The model, I surmise, was so close to being an exhaustive model that Laskar and Robutel could run simulations corresponding to several billions of years of solar system evolution, and reasonably expect the outcomes of the simulation runs to be physically realistic.

To obtain statistical data Laskar and Robutal started simulation runs with a range of different starting conditions.

For Earth they started runs both with the Moon present, and with an Earth without Moon.

In the simulation runs with an Earth without moon they obtained as statistical result that in many runs the Earth had periods of significant change of the angle of inclination.

Conversely, in the runs with the Moon present significant change of angle of inclination did not occur, or far less often.

The thing is: that model of the solar system is so complex that it is not humanly possible to trace what is going on.

There is another article, also available on the website that goes to the underlying mechanism:
The Moon and the Origin of Life on Earth

This article was first published in french, in the magazine 'Pour la Science, which is the french franchise of the Scientific American magazine.

In the section: 'Variations in the Earth’s Obliquity'

Laskar describes: the various other planets in the solar system cause changes in the orbit of the Earth that go very slowly. Given how slow this type of rate of change is: it is customary to express it in terms of seconds of arc per year.

[...] the principal frequencies of the motion of the Earth’s orbit range from 26.33 seconds per year down to nearly 0.67 seconds per year, with the main frequencies at 18.85 and 17.75 seconds per year.

In the case of the Earth: the presence of the Moon induces a relatively fast rate of precession: 50.47 seconds per year.

Since the Earth rate of precession is significantly faster than most of the periodic changes in Earth orbit the probability of a significant resonance is low.

(The contributions of Sun and Moon to rate of precession are in the same ratio as the contributions to tidal effect. The Moon contributes about 2/3 of the total effect.)

Without The Moon the rate of precession is significantly slower, which puts the Earth at higher probability of a resonance with some periodic influence from another planet. When such a resonance occurs the resonance will affect the inclination of the orbit.

(You can try that with a gimbal mounted gyroscope. If that gyroscope is precessing, and you give a bit of a nudge to make that precessing motion go a bit faster then in response the inclination of the gyro wheel will change.)

I'm just relaying here what Jacques Laskar writes in that article. The explanation that Jacques Laskar offers may not be correct.

I can confirm that a spinning gyroscope does respond in the way that Jacques Laskar describes.

(About the PDF with scanned pages: I extracted the images, and I adjusted the brightness and the contrast. That way the article is much more readable. If you are interested in the processed images: contact information for me is on the contact page of my webste; link to my website is on my stackexchange profile page.)