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As the Moon loops around the Earth, on one side of the Earth it moves toward the Sun (let's call it downward side) and on the other side it moves away from the Sun (let's call it the upward side).

Now, I imagine the Sun's gravity attraction accelerate the Moon on the downward side and slows it on the upward side. This should raise the Moon's apoapsis on the upward side and lowers its periapsis on the downward side, gradually putting the Moon on a collision course with Earth.

Yet, this does not happen. I am curious to understand why. I suppose even a very very slim effect should build up upon billions of years, so is there a fault in my reasoning or some tidal effect correcting the Moon's trajectory?

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  • $\begingroup$ Not a full answer, but consider that for each time the Moon passes closest to the sun (new moon) the Earth moves about 27 degrees, so the apogee and perigee don't line up with the tugging. It's off by 27 more degrees with each lunar pass. $\endgroup$ – userLTK Feb 18 '19 at 21:45
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The Sun's gravity does perturb the Moon's orbit but more subtly than you imagined. The Moon's perigee and apogee migrate eastward in an 8.9 year cycle called apsidal precession. Also the plane of the lunar orbit, inclined about 5.1 degrees to Earth's orbit, shifts westward in an 18.6 year cycle called nodal precession. Lunar theory also addresses various effects on shorter timescales.

If the Moon were 4 times farther from the Earth (outside the Earth's Hill sphere), then the Sun would destabilize the lunar orbit. Even so, the Moon would most likely drift into an independent orbit around the Sun, with little risk of collision with the Earth.

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  • $\begingroup$ Since what we're talking about here is the Three-Body Problem, the pedantic answer is that we cannot predict with absolute accuracy the future trajectories of either the moon or the Earth with respect to the Sun. BTW, the moon is slowly drifting away, so at some point it will be outside the Hill Sphere. $\endgroup$ – Carl Witthoft Feb 18 '19 at 18:54
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    $\begingroup$ Thanks to the Wikipedia's lunar theory page I understood my mistake. This one sentence sums it up well : "Newton concluded that it is only the difference between the Sun's accelerative attraction on the Moon and the Sun's attraction on the Earth that perturbs the motion of the Moon relative to the Earth". I forgot that the Sun acts on the Earth as well. This difference is tiny and, what is more, symmetric, which explains quite well why the Moon is not going to fall anytime soon. $\endgroup$ – armand Feb 19 '19 at 3:52

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