Monday, August 6, 2012

Star Trek: Tidal Locking

Star Trek the Next Generation (1987-1994) (Picture)
Season 2, Episode 10
The Dauphin

During "The Dauphin" the crew heads to a planet engulfed in a Civil War. This planet rotates once per time it circles its sun, meaning that one side of the planet is always light and the other is always dark. This phenomenon is called tidal locking.

How does tidal locking work, does it actually happen, and what are the consequences?

Why does tidal locking occur?

Tidal locking occurs due to tidal forces. These forces might sound like they are related to the tides, and that's because they are.

The moon is responsible for the tides we experience on the earth. When we go to the beach there are high tides and low tides. Why is the ocean not always at the same level? The answer is that the gravitational force of the moon pulls on the earth's oceans.

Imagine the moon above one location of the earth, as shown in the top picture. The moon pulls on the earth: close to the earth it pulls on the oceans, a middle distance away it pulls on the core/center of the earth, and on the other side it pulls on other oceans.

Tidal forces, showing how the moon causes tides on the earth, for example (Source)

The force of gravity falls of like 1/r2, so the gravitational attraction of the moon is strongest on the close side and weakest on the far side of the earth. The oceans nearest to the moon therefore rise, because they are being pulled toward it. On the far side the oceans are pulled least, meaning they are also at a maximum.

This is a little counter-intuitive. You might believe the oceans on the far side of the earth would be pulled around the earth to the other side (at least a little bit), but the tidal forces act more strongly on the earth between the oceans on the opposite halves of the earth. This means the moon is effectively pulling the earth out from under the ocean on the far side of the earth.

Tidal locking works by the same principles. Let's just think about how gravity can cause the moon to rotate around the earth so that only one side of the moon is ever facing the earth.

The answer has to do with torque. Torque is the reason you use a wrench, not your hands, to unscrew something.


When you go to open a swinging door, where do you push? Near the hinges, or further out? It's easier to open a swinging door by pushing further from the hinges because here you are applying more torque for the same applied force (how hard you push). Try pushing the door open near the hinges and it will be much more difficult.

How a wrench applies a torque. The longer the wrench, the more torque that can be applied and the easier your job becomes (Source)

Torque is defined as τ = r x F, where τ is torque, r is the distance from the axis of rotation, and F is the force applied to the door. The "x" means that it is a cross product (only the force applied perpendicular to the door contributes to the torque).

As much as the moon might seem like a sphere, it isn't. Similar to the way the moon pulls on the earth to produce the tides, the earth pulls on the moon to lock its orbit.

To see how this works, I want to make another analogy. When you spin a top it wobbles a bit and then evens out. This has to do with little corrective torques pulling on the top until it settles into its spinning position. In the same way, when the moon was first rotating around the earth it was spinning. Then the earth's little corrective torques pulled on the moon and it settled into a state where it does not feel net torques from the earth.

Just to be clear...

The moon is tidally locked with the earth only because it is not perfectly homogeneous. If it was a perfect sphere of perfect uniform radial density then it could not be tidally locked, because there would never be net torque from the earth. It is only because the moon is inhomogeneous that there are small bulges here or there on the surface that the earth can effectively pull on. This is analogous to the idea that if you spin a small metal ball it cannot wobble because gravity cannot apply a net torque to it.

Examples of tidal locking

The moon is a good example of tidal locking, but there are others. For example, it is believed that binary star systems are tidally locked, especially when the two stars have about equal masses.

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