A while back my old college roommate called me out of the blue - he and his sister were arguing over why we only see one side of the moon, and they were hoping their space nerd friend could deliver an explanation! Unfortunately, I only knew the general concept of tidal locking, I couldn't give more than that. So I figured I'd lay it all out here in simplest layman's terms first, then go one step deeper with a pretty cool mathematical formula I found!

#### Basics of tidal locking - our one-sided relationship with the Moon

Mankind has looked up at the moon since the dawn of antiquity, but due to a gravitational phenomenon called tidal locking, no one had ever seen the far side of the moon until the coming of the Space Age when unmanned probes returned the first images

A satellite being tidally locked means it rotates on its axis at the same time as it takes to revolve around its central body. Said plainly, its "day" is the same as its "year". In the case of the moon and the Earth, the moon rotates once and orbits the Earth once every ~28 days - see the below gif to visualize why this synchronization means we only see one side of the moon

So how does this happen? Billions of years ago when the Earth and moon were newly formed, the moon's orbit was almost certainly asynchronous. However moon actually isn't a perfect sphere - the tidal force caused by Earth's gravity deformed the moon into a slight oval shape (slightly wider than tall), and as the early moon rotated out of sync with its orbit, the Earth's gravity would pull more heavily on its equatorial bulges:

Confused? Don't be - it's a concept that's hard to explain in words but easy to visualize in motion. Check out this great video here to see!

It's actually an exceptionally difficult phenomenon to predict (almost as hard as projecting future cash flows at my job in investment banking, haha!). But here's the formula that describes the timescale of tidal locking:

I know the formula's crazy, I have no idea how it's derived, but don't let that scare you away because there are just a few simple takeaways I'm trying to highlight - notice how the terms $a$ and $R$ are raised to the 6th and 5th powers! Two enormous factors that determine tidal locking timescale are 1) distance between the satellite and the central body and 2) the size of the satellite. All else equal, smaller and more distant satellites take longer to lock. Earth is larger than the moon, which should make it lock faster, but it's so much further from the Sun than the moon is from the Earth that the distance factor significantly outweighs all others. That explains why the moon locked so quickly but the Earth will take an eternity

Finally, while all the other terms are relatively commonplace units from math and physics, $Q$ and $k_2$ are unique and very interesting:

I found $Q$ to be super complicated, but here's my best crack at explaining it: it's a known fact that the moon is gradually accelerating and drifting away from the Earth, due to the same forces that cause high tide and low tide in our oceans. Basically, angular momentum is being transferred from Earth's rotation to the moon's revolution, and as with everything in physics, angular momentum must be conserved. But the transfer isn't perfect, some of angular moment ends up being dissipated as heat due to friction from the movement of the Earth's oceans. $Q$ is a measure of that dissipation

$k_2$ has nothing do with romance, it's named after the mathematician Augustus E.H. Love, whose work covered the mathematical theory of elasticity. A satellite's Love number is a function that relates a satellite's density, surface gravity, and rigidity to its susceptibility to tidal forces

Other than for the moon, $Q$ and $k_2$ are extremely poorly known for other satellites in the Solar System, which explains our difficulty predicting tidal locking in satellites we can't directly observe. Like we often say in banking about our financial models, "garbage in, garbage out" - our projections are only as reliable as our inputs!

So how does this happen? Billions of years ago when the Earth and moon were newly formed, the moon's orbit was almost certainly asynchronous. However moon actually isn't a perfect sphere - the tidal force caused by Earth's gravity deformed the moon into a slight oval shape (slightly wider than tall), and as the early moon rotated out of sync with its orbit, the Earth's gravity would pull more heavily on its equatorial bulges:

- If the moon were spinning too fast, the Earth would pull at an angle slightly against its direction of rotation, slowing it down until it eventually synced up with its orbit
- If the moon were spinning too slow, the Earth would pull the moon's rotation forward, speeding its rotation up until it eventually synced up with its orbit

Confused? Don't be - it's a concept that's hard to explain in words but easy to visualize in motion. Check out this great video here to see!

#### One level deeper - what about the Earth and the Sun?

So then the next logical question to ask is, why are some satellites tidally locked but not others? After all, the Earth isn't tidally locked to the Sun, otherwise one hemisphere would see perpetual day while the other experiences endless night (what a strange world that would be!) Here's how it works - any imperfectly spherical satellite will eventually become tidally locked by the gravitational lasso described above. But it's a question of time - while the moon probably took only about a thousand years to become tidally locked (a blink of an eye in astronomical time scales), the Sun will probably expand and die (destroying the Earth with it) long before our planet becomes tidally locked. How do we know this?It's actually an exceptionally difficult phenomenon to predict (almost as hard as projecting future cash flows at my job in investment banking, haha!). But here's the formula that describes the timescale of tidal locking:

$t_{lock} \approx \frac{\omega \thinspace a^6 \thinspace I \thinspace Q}{3G\thinspace m_{central}^2 \thinspace k_2 \thinspace R^5}$

$\omega$ = initial spin rate in radians per second

$a$ = semi-major axis of the satellite's orbit (basically distance between satellite and central body)

$I$ = moment of inertia of the satellite (determines the torque needed for a desired angular acceleration)

$Q$ = dissipation function of the satellite

$G$ = Newton's gravitational constant

$m_{central}$ = mass of the central body

$k_2$ = tidal Love number of the satellite

$R$ = satellite mean radius

Finally, while all the other terms are relatively commonplace units from math and physics, $Q$ and $k_2$ are unique and very interesting:

I found $Q$ to be super complicated, but here's my best crack at explaining it: it's a known fact that the moon is gradually accelerating and drifting away from the Earth, due to the same forces that cause high tide and low tide in our oceans. Basically, angular momentum is being transferred from Earth's rotation to the moon's revolution, and as with everything in physics, angular momentum must be conserved. But the transfer isn't perfect, some of angular moment ends up being dissipated as heat due to friction from the movement of the Earth's oceans. $Q$ is a measure of that dissipation

$k_2$ has nothing do with romance, it's named after the mathematician Augustus E.H. Love, whose work covered the mathematical theory of elasticity. A satellite's Love number is a function that relates a satellite's density, surface gravity, and rigidity to its susceptibility to tidal forces

Other than for the moon, $Q$ and $k_2$ are extremely poorly known for other satellites in the Solar System, which explains our difficulty predicting tidal locking in satellites we can't directly observe. Like we often say in banking about our financial models, "garbage in, garbage out" - our projections are only as reliable as our inputs!

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*Correction: an earlier version of this article incorrectly stated that the moon's equatorial bulge was caused by the centrifugal force of its rotation about its axis, but this is not the case; it's caused by the tidal force of Earth's gravity. A reader on Facebook was kind enough to point out the error for me. I'm just an investment banker, I'm pretty sure that guy is a legit theoretical physicist, so I'm inclined to believe him! (And upon my own further research, I concluded he was correct). Learn something new every day!*

Haha! Thanks for a great explanation. This is a question that my husband and I have argued about vociferously for forty years! Me: The rotation of the moon differs from that of the earth, because it always falls back and we never see the "far side". He: the moon is just like any other celestial body, but its spin on its own axis coincides with its (monthly) rotation around the earth.

ReplyDeleteForty years, that's a long argument hahaha! Glad I could help clarify - it's a really interesting phenomenon. And sometimes it's not a perfect 1:1 match. Mercury is tidally locked to the Sun in a 3:2 resonance. https://en.wikipedia.org/wiki/Mercury_(planet)#Spin%E2%80%93orbit_resonance

DeleteI like the gif a lot!

ReplyDeleteBlogger is pretty jank, but somehow it handles gifs well lol

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