How much do surfaces tilt due to tidal forces?

In summary, according to this article, tides deform the Earth's crust by about 40 cm, which is enough to cause a tilt of 0.013 seconds of arc.
  • #1
Adrian B
21
5
I've read that tides deform the Earth's crust by about 40cm. When I try to visualize the tidal bulge approaching me and then receding away from me, it seems like the local surface under my feet would tilt slightly one way as the bulge approaches, then level out, and then tilt slightly the other way as the bulge recedes. Similar to a surfboard as a wave passes underneath it.

Is this picture correct? If "yes" does anybody have a rough figure for the maximum tilt one would "experience" on Earth due to this? Arc-seconds? Micro arc-seconds?
 
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  • #2
Say the deformation is to approximately an ellipse. Then the ellipse has a ratio of semi-major to semi-minor axes ##\frac{1+\epsilon}{1-\epsilon}## where ##\epsilon## is 40cm / 6371km = ##6\times 10^{-8}##. This can be modeled by the ellipse:

$$\frac{x^2}{(1+\epsilon)^2}+\frac{y^2}{(1-\epsilon)^2}=1$$

The tilt is the angle between the tangent to the ellipse and the line from the point on the ellipse to the origin, which is the centre of mass. That tilt is zero at the x and y intercepts. So it seems reasonable to guess that maximum tilt might be near angles of 45 degrees to the axes. There the angle of the line to COM is 45 degrees. The gradient of the ellipse is:

$$\frac{d}{dx}\left[(1-\epsilon)\sqrt{1-\frac{x^2}{(1+\epsilon)^2}}\right]=-\frac{(1-\epsilon)x}{\sqrt{1-\frac{x^2}{(1+\epsilon)^2}}}$$

##x## is approximately ##\frac{1}{\sqrt{2}}## at that point, so this gives the gradient as ##1-1.3\times 10^{-7}##. Taking the arctan gives an angle that differs from 45 degrees by ##3\times 10^{-6}##. Multiplying that by ##60^2## gives about 0.013 seconds of arc.

E&OE
 
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  • #3
andrewkirk said:
Multiplying that by 60^2 gives about 0.013 seconds of arc.
I agree.

An ellipse can be approximated by a circle of constant radius plus a significant 2'nd harmonic sinewave. One cycle of that 2'nd harmonic covers about 20 Mm, as that is half the Earth circumference, ignoring the flattening. (Remember that Napoleon declared the distance from the North Pole to the Equator, through Paris to be 10Mm, don't you just love the metric system).
If the vertical peak to peak Earth Tide amplitude is 0.4m, half that is the sinewave amplitude = 0.2 m
So scale the 20 Mm by 2π to get 3183100 m. The maximum slope of Sine is at zero = Cos(0) = 1.
Maximum slope of surface is therefore 0.2 in 3183100 = 6.283e-8.
Atan(6.283e-8) = 3.6e–6 deg = 0.01296 arcsec
 
Last edited:
  • #4
Thanks folks!
 

1. How do tidal forces affect the tilt of surfaces?

Tidal forces occur when the gravitational pull of one object, such as a planet or moon, is stronger on one side of another object than the other. This difference in gravitational pull can cause the surface of the object to tilt towards the stronger gravitational force.

2. What determines the amount of tilt caused by tidal forces?

The amount of tilt caused by tidal forces depends on the strength of the gravitational pull, the distance between the two objects, and the composition and structure of the surface being tilted. Objects with weaker gravitational forces, greater distance, and more flexible compositions are more susceptible to tidal forces.

3. Can tidal forces cause significant changes in surface tilt?

Yes, tidal forces can cause significant changes in surface tilt, especially in objects that are closer to the source of the gravitational pull. For example, tidal forces from the moon cause noticeable changes in the tilt of Earth's oceans, resulting in the tides.

4. Are there other factors that can contribute to surface tilt besides tidal forces?

Yes, there are other factors that can contribute to surface tilt, such as seismic activity, plate tectonics, and changes in mass distribution within an object. These factors can also interact with tidal forces to further affect surface tilt.

5. How can scientists measure the tilt caused by tidal forces?

Scientists can measure the tilt caused by tidal forces using instruments such as tiltmeters, which detect changes in the angle of a surface. They can also use satellite imagery and data to track changes in the tilt of larger objects, such as planets and moons.

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