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I read somewhere that if Moon's orbital plane would be perpendicular to the Earth's orbital plane - Moon will fall on Earth within 4 years (given today's position)

Can this be shown analytically without computer simulation?

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If the Moon traveled in what was effectively a polar orbit, it would not be stable in the long term. We might expect "problems" in about 8 years. The Moon would either collide with the Earth (although the Roche effect would break it up first) or be slung out of its orbit entirely. This is due to the Kozai-Lidov mechanism.

The greater the inclination of the Moon and the smaller its orbital distance is from the Earth, the more the Earth & Moon would torque on each other, which alters the angular momentum of both of them. This leads to changes in eccentricity and inclination of their orbits, and due to it's lesser mass, the Moon would experience the most drastic orbital changes in less time.

From Wikipedia Kozai Mechanism

The Lidov-Kozai mechanism places restrictions on the orbits possible within a system, for example:

For a regular moon: if the orbit of a planet's moon is highly inclined to the planet's orbit, the eccentricity of the moon's orbit will increase until, at closest approach, the moon is destroyed by tidal forces

For irregular satellites: the growing eccentricity will result in a collision with a regular moon, the planet, or alternatively, the growing apocenter may push the satellite outside the Hill sphere

The mechanism has been invoked in searches for Planet X, hypothetical planets orbiting the Sun beyond the orbit of Neptune.

The basic timescale associated with Kozai oscillations is. $${\displaystyle T_{\mathrm {Kozai} }=2\pi {\frac {\sqrt {GM}}{Gm_{2}}}{\frac {a_{2}^{3}}{a^{3/2}}}\left(1-e_{2}^{2}\right)^{3/2}={\frac {M}{m_{2}}}{\frac {P_{2}^{2}}{P}}\left(1-e_{2}^{2}\right)^{3/2}}$$ where $a$ indicates semimajor axis, $P$ is orbital period, $e$ is eccentricity and $m$ is mass; variables with subscript $2$ refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner (satellite) orbit; $M$ is the mass of the primary. The period of oscillation of all three variables ($e$, $i, $\omega$) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the separatrix orbit that separates librating (Kozai) orbits from oscillating orbit

 

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    $\begingroup$ What is the distinction between a "regular moon" and "irregular satellite"? $\endgroup$ – uhoh Oct 9 '16 at 13:38
  • $\begingroup$ What's the perturber in the case of the moon? $\endgroup$ – EL_DON Oct 9 '16 at 18:25
  • $\begingroup$ An irregular satellite has been captured by the primary and usually has a weird orbit: high eccentricity, high inclination, or retrograde. A regular moon forms around the primary and tends to have a more circular prograde orbit. $\endgroup$ – EL_DON Oct 9 '16 at 18:27
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    $\begingroup$ @EL_DON The sun. $\endgroup$ – Robin Ekman Oct 9 '16 at 18:40

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