Tidal Lock | Why the Moon Always Faces Us
By Bu Quan, 17th Jun 2022
Translated by Yifan Wang, 30th Jul 2023
As most people interested in science would know, all celestial objects have their own rotational velocity, such is the case with the Moon, which rotates on its axis as it orbits Earth. This fact also leads to an interesting question, why do we only ever see one side of the Moon? And why does it seem to have its back against us? This phenomenon is known as “tidal locking” and is caused by the gravitational interaction between the Earth and the Moon, which is able to permanently keeps one side of the Moon hidden from our view.
PART.1
To better picture the effects of tidal locking, imagine a dance with a partner. With your hands tightly intertwined, despite any moving and spinning, you and your partner will always be facing each other. The force of attraction keeps the two of you together, just like the gravity between the Earth and the Moon.
Tidal forces refer to the gravitational gradient cause by the gravitational fields of two celestial objects, producing tides and various effects on the Earth’s surface. Interestingly, tidal forces predominantly effect the ocean’s tides. When the Earth’s oceans are pulled by the Sun or the Moon, water rises towards the celestial objects and create high and ebb tides dependent on their orientation. This periodic phenomenon causes constant changes in the water level of the ocean, with two high tides and two low tides occurring per day.
The strength of the tidal force is inversely proportional to the cube of the distance between the two celestial bodies and proportional to their mass. For the oceans of Earth, the Moon has the largest influence, this is as although the Moon has a low mass, it is relatively close to the Earth. The mass of the sun is much bigger than that of the Moon (by a factor of 20700000 times), but it is also 389 times further away from Earth, thus it has weaker effects on the oceans, with the Moon having an effect approximately 2.17 times greater than that of the sun. (However, due to the slightly elliptical orbit of the Moon, tidal forces caused by the Moon differ in respect to their distance to the Moon.)
Tidal locking occurs when the gravitational gradient causes a celestial body to permanently face another, “locking” it in place. The most example of this is obviously the Moon and the Earth, and this mechanism can be explained in four steps, using an example system of objects A and B, with B orbiting A:
1. Tidal forces exert forces of different magnitudes on the near and far ends of the celestial body, causing the shape of the celestial body to begin to distort and be elongated on the axis facing towards the other celestial body. Since the total volume remains the same, the diameter perpendicular to the axis must then decrease. This phenomenon is known as a tidal bulge, which moves across a body's surface when it is not tidally locked.
2. In reality, this process takes some time to happen, and during this period, the original A-B axis along the bulge has changed because of the rotation of B, so the budge formed will be distanced from the current axis between objects A and B. From a lookout point in space, the direction of the peak of the bulge deviates from the direction pointing to A. If B's rotation period is shorter than its orbital period, the bulge will lead the direction of the A-B axis; conversely, if B's rotation period is longer, the bulge will instead lag behind.
3. As the bulge deviates from the direction of the axis connecting the two bodies, moments of force will be formed at both the near and far ends of object B, causing both ends to try to rotate towards object A. However, as the near end of object B experience a greater force, it will cause the object to rotate and align the peak of the near end along the A-B axis, this rotation causes changes in the rotational speeds of object, which increases if the bulge lags behind the A-B axis, and decreases if the bulge leads the A-B axis.
4. Lastly, because of the conservation of angular momentum, when the angular momentum of object B changes due to the tidal locking, its orbital angular momentum will also change (In this example, the change of angular momentum of object A is ignored because A has a large mass). For example, when B slows down its rotation, its orbit relative to A will speed up. In the opposite case, when B's rotation speed is too slow, the effect of tidal locking will speed its rotation up, and at the same time, also slowing down B’s orbit.
PART.2
(The following section is mostly comprised of physics calculations, a firm foundation in mathematics and physics is recommended to achieve better understanding of the content below.)
Now we have explained the mechanisms behind tidal locking, another interesting question may be raises. Why are the effects of tidal locking between the Earth and the Sun so minimal despite being a similar system of a high and low mass objects? And under what conditions can a stable tidal lock between the two be achieved? Both questions can be answered by the following calculations.
Imagine again celestial bodies A and B, with A being significantly more massive. Object B’s angular momentum relative to A (excluding B’s rotation) can be derived:
Differentiating both sides of the expression to get:
Using the circular motion of B around A and the gravitational force formula, we obtain:
Differentiate both sides again to get:
Combining the above two differential equations then yields:
Which is the relationship between A's rotation angular velocity and B's orbital revolution angular velocity. Then we differentiate both sides of the system’s mechanical energy conservation equation to obtain:
We then bring all the previous calculations into the following energy equation:
When the system reaches constant energy and achieves equilibrium, the derivative of energy with respect to angular velocity is now 0, . At the same time, in order to achieve a stable balance, its second-order derivative is greater than 0: . Hence:
We're just two steps away! By substituting our previous results, the above complex equation can be simplified into:
Finally, by substituting the moment of inertia of a ball into the final result, we obtain:
When masses of objects A and B satisfy this inequality, object B will be successfully tidal locked to object A stably.
If you are still interested, you can also query the relevant data and calculate the value of the Earth-Moon and Earth-Sun systems yourself. It is not difficult to find that the Earth-Moon system is three to four orders of magnitude larger, which is the reason why the Moon only faces us with one side. Even with just high school knowledge and some college knowledge, we can explain many of the wonderful phenomena we find in everyday life.