Tidal Locking question about the Earth-Moon system

In summary, the section of the Wikipedia article on tidal locking explains that the material of the moon exerts resistance to the tidal forces, causing some delay in reshaping the moon. This results in the bulges being carried forward or lagging behind, depending on the moon's rotation period. The question posed by the conversation is then addressed, discussing the proper length for modeling the moon as a dumbbell and calculating the centroid of the moon's half volume.
  • #1
Swamp Thing
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Why should the moon's tidal bulge lag from the earth-moon axis if the rotation rate is slower than synchronous?
This section of the Wikipedia article on tidal locking https://en.wikipedia.org/wiki/Tidal_locking#Mechanism
states:
(Note : A is the earth; B is the moon)
The material of B exerts resistance to this periodic reshaping caused by the tidal force. In effect, some time is required to reshape B to the gravitational equilibrium shape, by which time the forming bulges have already been carried some distance away from the A–B axis by B's rotation. Seen from a vantage point in space, the points of maximum bulge extension are displaced from the axis oriented toward A. If B's rotation period is shorter than its orbital period, the bulges are carried forward of the axis oriented toward A in the direction of rotation, whereas if B's rotation period is longer, the bulges instead lag behind.

I am having trouble understanding that last "whereas" bit (bold font). It seems to me that if we consider lower and lower rotation speeds (especially sub-synchronous speeds) then what should happen is that the delay caused by "resistance to reshaping" would only become more and more negligible, so that the bulge would align itself more and more closely with the earth-moon axis. Essentially, the bulges "lead angle" with respect to the earth-moon axis should asymptotically approach zero as the rotation slows down.

Is the article wrong on this point, or am I?
 
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  • #2
Okay, I think I get how I was wrong.

Dear Moderator: Please delete the question.
 
  • #3
Don't delete yet.
To study tidal locking (resonance), I wish to model the moon (ellipsoid) as a dumbbell of length, L.
What is a proper length, L, for the dumbbell?
Centroid of half-ellipsoid is 3/8a, half-sphere is 3/8r. A sphere (moon) would produce no torque, so these values seem incorrect.
 
  • #4
I think the primary effect is due to actual density fluctuations (mascons) in the nearly circular moon. I believe the actual shape of the moon is pretty much the usual oblate spheroid slightly flattened at the poles. Post what you find out.
 
  • #5
Subtract the sphere from the ellipsoid - the dumbbell is the remainder.
The equation for the ellipse is ##x^2/a^2 + y^2/b^2 = 1##, ##a>b##, and the equation of a sphere is ##x^2 + y^2 = b^2##, of radius b. We find the centroid, ##\overline{x}##, for half the volume as follows:
\begin{align}
\int{x dV} &= \pi \int{x y^{2}dx} = \int_{0}^{b} [\frac{b^{2}}{a^{2}} (xa^{2} - x^{3}) - (xb^{2} - x^{3})]dx\nonumber \\
&\qquad \qquad \qquad+ \int_{b}^{a}\frac{b^{2}}{a^{2}}(xa^{2} - x^{3})dx \nonumber \\
&\qquad \qquad\qquad\quad \quad= \pi \frac{b^{2}}{4}(a^{2}- b^{2}) \\
\int{dV} &= \frac{\pi}{2}(\frac{4}{3}ab^{2} - \frac{4}{3}b^{3}) \;= \pi \frac{2}{3}b^{2}(a - b) \\
\overline{x} &= \int{x dV} / \int{dV} \:\:= \frac{3}{8} (a + b) = \it{l}
\end{align}
Then, M##_{dumbbell}## = (M - M##_{b}##)/2
 
Last edited:

Related to Tidal Locking question about the Earth-Moon system

1. What is tidal locking?

Tidal locking is a phenomenon where the same side of an astronomical body, such as a planet or moon, always faces the other body due to the gravitational forces between them.

2. How does tidal locking occur in the Earth-Moon system?

In the Earth-Moon system, the Moon's gravitational pull causes tides on Earth. Over time, these tidal forces have slowed down the rotation of the Earth, causing it to become tidally locked with the Moon.

3. How long does it take for the Earth to become tidally locked with the Moon?

It is estimated that it would take around 50 billion years for the Earth to become completely tidally locked with the Moon.

4. What are the effects of tidal locking on the Earth-Moon system?

One of the main effects of tidal locking is that the Moon's rotation has slowed down, causing it to always show the same face to the Earth. This has also caused the Moon's orbit to become slightly elliptical. Additionally, tidal locking has also affected the Earth's rotation, causing the length of a day to increase over time.

5. Can tidal locking occur with other planetary systems?

Yes, tidal locking can occur with other planetary systems as long as there is a significant difference in size and mass between the two bodies and they are in close proximity to each other. Tidal locking has been observed in many other systems, including the Pluto-Charon system and some exoplanets.

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