### When did the Moon become tidally locked to Earth?

• SF.

8 years ago

According to various theories the Moon was created around 4.5 billion years ago. About all of these theories suggest that it was rotating around its axis at that time though. Currently, Moon is at tidal lock with Earth, despite some monthly "wiggling" a flat zero on the long-term rotation speed relative to it.

I wanted to ask when did that stop occur - relative to Moon's age, how long was the period of rotating Moon?

The answer would shed some light on my other question - Why are most lunar maria on the visible side? as Earth tends to catch or deflect many bodies heading for Moon surface from "our" direction - still, there is no erosion on the Moon, so craters once formed are extremely slow to vanish - if that period was relatively long, Earth's "protection" wouldn't explain the maria, as rotating Moon would get 'cratered' uniformly all over its surface.

• 8 years ago

"Protection" isn't the only effect of Earth. Here is a different POV: Earth may have accelerated impactors by gravity assist.

A different approch is the thinner crust, as suggested for the near side, which may have allowed asteroids to penetrate Moon's crust, such that lava could flow into the basins, or which may have favoured volcanism on the near side (see "Lunar interior" on this site).

A third approach is the protective property of Earth preventing the near side to be covered with many new craters, hence leave the maria visible.

According to Wikipedia the time to lock tidally is about
$$t_{\mbox{lock}}=\frac{wa^6IQ}{3G{m_p}^2k_2R^5},$$
with $$I=0.4m_sR^2.$$
For Moon $k_2/Q = 0.0011$, hence
$$t_{\mbox{lock,Moon}}=121\frac{wa^6m_s}{G{m_p}^2R^3}.$$
With Earth's mass $m_p=5.97219\cdot 10^{24}\mbox{ kg}$, Moon's mass $m_s=7.3477\cdot 10^{22}\mbox{ kg}$, Moon's mean radius of $R=1737.10\mbox{ km}$, $G=6.672\cdot 10^{-11}\frac{\mbox{Nm}^2}{\mbox{kg}^2}$we get
$$t_{\mbox{lock,Moon}}=121\frac{wa^67.3477\cdot 10^{22}\mbox{ kg}}{6.672\cdot 10^{-11}\frac{\mbox{Nm}^2}{\mbox{kg}^2}\cdot{(5.97219\cdot 10^{24}\mbox{ kg})}^2(1737.10\mbox{ km})^3},$$
or
$$t_{\mbox{lock,Moon}}=7.12753\cdot 10^{-25}wa^6 \frac{\mbox{kg}}{\mbox{Nm}^2 \mbox{km}^3}.$$
Parameters are $w$ the spin rate in radians per second, and $a$ the semi-major axis of the moon orbit.

If we take the the current simi-major axis of the moon orbit of 384399 km
and a maximum possible spin rate of
$$w=v/(2\pi R)=\frac{2.38 \mbox{ km}/\mbox{s}}{2\pi\cdot 1737.10\mbox{ km}}=\frac{1}{4586 \mbox{ s}},$$
with $v=2.38 \mbox{ km}/\mbox{s}$, Moon's escape velocity, 1737.1 km Moon's radius,
we get
$$t_{\mbox{lock,Moon}}=7.12753\cdot 10^{-25}\cdot \frac{1}{4586 \mbox{ s}}\cdot (384399\mbox{ km})^6 \frac{\mbox{kg}}{\mbox{Nm}^2 \mbox{km}^3}\\ =501416\mbox{ s}^{-1}\cdot \mbox{ km}^6 \frac{\mbox{kg}}{\mbox{Nm}^2 \mbox{km}^3}= 5.01416\cdot 10^{14} \mbox{ s}.$$
That's about 16 million years, as an upper bound.

If we assume a higher Love number for the early moon, or slower initial rotation, the time may have been shorter.

The time for getting locked is very sensitive to the distance Earth-Moon (6th power). Hence if tidal locking occurred closer to Earth, the time will have been shorter, too. That's likely, because Moon is spiraling away from Earth. "Earth may have accelerated impactors by gravity assist." - Yes, like the one that created Tycho. Fewer - but stronger. Another theory for the near side difference is that the far side of the moon was slowly impacted by another, smaller moon in the same orbit, such that it just went splat! instead of blasting the moon apart. This left the far side with a much thicker crust, such that few impactors could penetrate to the mantle. Yet another is that the intense heat from the molten Earth when the moon was much closer ablated away some of the crust on the near side, which condensed on the far side. @DanHanson Maybe. There's a fairly large mascon in the south on the far side. See https://en.wikipedia.org/wiki/Gravitation_of_the_Moon