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In an academic study titled 'The Moon and the Origin of Life on Earth' it is said that

'If the Moon did not exist, the orientation of the Earth’s axis would not be stable, and would be subject to large chaotic variations over the ages. The resulting climatic changes very likely would have markedly disturbed the development of organized life'.

Since I don't have enough physics and mathematics knowledge in order to understand why the Moon stabilizes the Earth's axis, can anybody explain in simple words this phenomenon?

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One way to look at this problem is to consider angular momentum. The Earth spins around its axis, and has therefore some angular momentum by itself. The angular momentum is proportional to the mass of the Earth, to its squared radius, and to its angular velocity. But the Earth is not alone; it has the Moon rotating around it that adds angular momentum to the Earth-Moon system. And, even though the Moon is not as massive as the Earth (it is about 100 times less massive), nor rotates very fast around the Earth (and therefore a lesser angular velocity), it has a large orbit (about 300 000 km) and overall, it adds to the system an amount of angular momentum comparable to that of the Earth itself.

Now, think of a spinning-top: the faster it rotates (and thus the larger the angular momentum), the more stable it is. It is the same for the Earth-Moon system: without the Moon, the angular momentum of the Earth itself would be such that gravitational perturbations could be sufficient, in the long run, to significantly perturb its axis (exactly like the spinning-top: if it does not rotate very fast, a small perturbation will rapidly increase and the spinning-top axis will start to oscillate more and more). But with the Moon, the global angular momentum of the system is larger, and it is therefore harder to sufficiently perturb the system to get it to oscillate strongly.

Sources:

For those who wants the dirty details, you can have a look at Laskar et al. 1993.

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    $\begingroup$ It's a pity the dirty details are behind a paywall. I have a hard time understanding how the moon's angular momentum will bring the earth's angular momentum back into line if it drifts. $\endgroup$
    – Josiah Yoder
    Commented Jul 20, 2022 at 20:27
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    $\begingroup$ @JosiahYoder The moon speeds up the precession of the earth's axial tilt, giving it sort of a second-order angular momentum about that precession axis. This is caused by the moon tugging on the equatorial bulge of the earth, attempting to bring the bulge in line with the moon's orbital plane. You can think of a slowing top that precesses wider and wider until it falls over due to the fact that the torque force (gravity) is always pulling down. Here, the torque points both up and down as the moon drifts above/below the equator, keeping the precession going but not letting the top fall over. $\endgroup$
    – Nuclear Hoagie
    Commented Jul 21, 2022 at 18:30
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    $\begingroup$ The cited article doesn't back up this answer. They just plugged in random values into a model to see what effects they had, there was no analysis of why the numbers had the effect they did. At no point do they claim the moon's angular momentum added to the system stabilizes the Earth. $\endgroup$
    – Greg Miller
    Commented Jul 22, 2022 at 2:19
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    $\begingroup$ @GregMiller The article may not make claims about the moon's momentum, but it does make claims about the moon stabilizing the earth. That abstract explicitly states that the earth's chaotic region is reduced from 0 through 85 degrees to the range 60 through 90 degrees. Thus, the earth's orbit would not be stable at 23.3 degrees without the moon. I agree that the mechanics of how this achieved may not be discussed in that article. $\endgroup$
    – Josiah Yoder
    Commented Jul 22, 2022 at 19:33
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    $\begingroup$ @JosiahYoder this question was about why the moon stabalizes the Earth, the answer claims it has something to do with angular momentum, and to go to that article for the "dirty details". It does not have those. $\endgroup$
    – Greg Miller
    Commented Jul 24, 2022 at 18:58
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Without the Moon to stabilize our earth the wobble and angular tilt would be erradic, and would result in extreme seasons, weather and incredible high and low tides. Our Moon is tidally locked to benefit further its gravitational tug and stability of earth's seasons weather and tides. A good example of a planet without a moon is Mars which really wobbles from time to time and does not have any real moon to speak of other than tiny Phobos and Deimos which are moot because they cannot stabilize its wobble.

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    $\begingroup$ Hello Robert. This answer seems to restate the facts that the OP already knows (The moon stabilises the tilt of the Earth) without explaining how it does this. The question was not "what would happen if there was no moon", but "explain in simple words why the moon prevents erratic variation in angular tilt". $\endgroup$
    – James K
    Commented Jan 28 at 8:10

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