Do centrifugal forces contribute to the Earth's tidal bulges?

Centrifugal force.

Elementary physics textbooks usually mention centrifugal force, and give a few simple examples. Their treatment often leads to student confusion and misconceptions.

Centrifugal forces are classed as "fictitious" forces because they do not arise from material objects such as those with mass. They are a mathematical trick to simplify calculations carried out in non-intertial (accelerating) frames of reference. In a reference frame rotating with angular velocity ω this requires modifying Newton's law F = ma by adding fictitious force terms to ma:

Feffmarot = mafixedm ω × (ω × rrot ) − 2mω × vrot

The term − mω × (ω × r) is the centrifugal force where ω is the angular velocity. Angular velocity is a vector along the rotation axis of the reference frame. This reduces to mω2r when all the motion lies in a common plane. The last term is called the Coriolis force.

The subscript "eff" labels effective force as observed in the rotating reference frame. The label "rot" refers a measurement made in this rotating reference frame. The label "fixed" refers to a measurement made in the fixed (inertial) reference frame.

Students should be a bit disturbed by this conceptual trick. How does it square with Newton's third law? That law is, "If body A exerts a force on body B, then B exerts an equal and oppositely directed force on A." If a centrifugal force acts on a body A in a system, what does that body exert an equal oppositely directed force upon? Where is body B? Real forces always have a material body source. What physical body is the source of a centrifugal force? Where is it located?

One "solution" to this dilemma is to note that centrifugal forces are always directed perpendicular to and away from the reference frame's rotation axis. Suppose we invent a "fictitious mass" along that axis. That mass would have to be negative, repulsing other masssive objects in the physical system with a force that gets stronger linearly with distance from the axis. Newton's third law is now satisfied. We thus create a fictitious "centrifugal force field" with the force law Fc = − mω2r. The minus sign indicates that it is directed outward from the axis of the reference frame's rotation.

Can such a force field deform an elastic body? As with gravitational fields, deformation results from the elastic properties of the body, and the gradient of the force field. In the gravitational field we found this derivative to be proportional to 1/r3 and we called it the "tide raising force".

What is the gradient of a field whose strength is linearly dependent on distance from a fixed axis? It is constant! It alone cannot deform an elastic body.

This tells us that it is not centrifugal force that causes Earth's tidal bulges, though many textbooks would have you believe that, and students uncritically accept it as "common sense". Fictitious forces are never the cause of anything.

Earth's equatorial bulge.

Physics students suppose they know the mechanics of the Earth's equatorial bulge. They say, "It is caused by centrifugal force pulling earth and water outward all around the equatorial belt." There's much more to it than that. The earth and water at the equator is not simply stretched outward. That material to fatten the Earth's midriff must come from somewhere. It comes at the expense of material near Earth's poles. In fact, historically this distortion of the Earth from sphericity was called "polar flattening". What forces nudge Earth's material to favor the equator?

How do we account for the Earth's equatorial bulge without invoking centrifugal force? Rotation and elasticity of materials are responsible. We have shown that a body of mass m moving in a circle must be acted on by a radial force 2r where ω is the angular velocity of the mass. For a chunk of matter at the equator that r is the radius of the Earth, R. At Earth's poles it is zero. At other latitudes it is between R and 0. In all cases, it acts perpendicular to the Earth's rotation axis. The Earth's own gravity is nearly constant anywhere on Earth's surface and in a radial direction toward Earth's center. Of course similar considerations apply throughout Earth's volume.

Now imagine a spherical rotating sphere made of Jello™ with gravity negligible. Suppose the tensile strength of Jello is sufficient to maintain all of it rotating at the same angular velocity ω about a fixed axis. It is kept from flying apart by its own tensile force. (Think of this sphere in the language we use for the Earth, with equator, latitude and poles.) Material at the equator has the greatest orbital speed around the axis. At higher latitudes, the material has a smaller orbital radius and smaller speed. Remember Fcentripetal = mω2raxial. At a latitude where raxial = R/2 the axial force required to keep a chunk of Jello in circular orbit is half that at the equator. The tensile stress on Jello there is also smaller. The original spherical shape is distorted into an oblate spheroid.

This is not the whole story. To stress a chunk of matter to change its shape we must apply differential force across its volume. If forces were constant over its volume there would be no stress to stretch or compress or in any way distort its shape. The answer lies in the previous paragraph. The radial tensile force at a given latitude varies proportional to distance from the rotation axis. So for two chunks of matter of the same size, the stress at distance R/2 is half that at R.

In addition to the axial component of stress (acting away from the rotation axis) there's also another component of stress that acts in a direction toward the equator. A rotating globe of water in space would deform to an oblate spheroid by physically relocating some water toward the equator. Water does not stretch or compress much. This is especially important when considering the Earth's ocean waters.

Earth's gravity alone produces a spherical shape. Add rotation to this and you get the modification of that spherical shape to an oblate spheroid. This shape is entirely due to gravitational and tensile forces. No centrifugal forces are necessary to account for it.

Centrifugal forces due to motion around Earth's barycenter?

Some textbooks try to account for lunar tides as due to centrifugal forces produced by Earth's monthly movement around the Earth-Moon center of mass (the Barycenter). This revolution has an angular velocity of about (1/28)2 = 1/734 that of the Earth's daily axial spin. The barycenter is 4,670 km from Earth's center.

Earth orbits around barycenter (O)
with rotation about its
center (C) suppressed.

To isolate this possible effect, we temporarily suppress Earth's axial rotation, and consider only its monthly motion around the barycenter. This has an unexpected geometric result. Earth's center moves in a circle of radius 4,670 km. and each point on the Earth moves around its own circle of the same radius. This is an uncommon sort of motion, seldom seen in textbook physics problems. But we all have experienced this if we ever took a ride in a Ferris Wheel. The passenger seats or cars on the wheel do not rotate with respect to the ground.

In this motion, adjacent chunks of the Earth experience the same centrifugal force, equal in size and direction. This force field has no gradiant across Earth's volume. Therefore it cannot cause deformations of the Earth, and no tides.

Ferris wheel in Vienna.

Why did the Earth's axial spin cause deformation of the Earth, but this movement did not?

The centrifugal forces acting on each piece of the axially spinning Earth had different sizes (and directions). Adjacent chunks of earth therefore experienced different size forces and this meant the compression or expansion of them was not zero. This difference (the force gradient) deformed the Earth.

Turning to the Earth/Moon system, we suppressed Earth's axial spin and changed our refererence frame to one that rotates around the Earth/Moon barycenter, matching the motion of the moon.

Then we considered whether there might be an additional deformation due to the Earth's movement around the barycenter. As with any rotating reference we must include centrifugal forces. But they are constant in size and direction at any time and have no gradient. So they can not deform the Earth's shape.

This little exercise was futile and redundant. We already established that choice of reference frame and its fictitious forces cannot ever be the cause or reason for any physical effect.

Summary.

I've looked into the history of textbook treatments of centrifugal force, and journal papers on that subject. Controversy and confusion on these topics go back to Newton, Mach, and Einstein. I found the history of this subject in books and papers on philosophy, mathematics, physics and oceanography. I knew the common mistakes before, but now I know better how to resolve them. Eugene Butikov's paper on tides gets it right. But the interesting thing (like the study of perpetual motion proposals) is to examine how and why so many get it wrong and even propose bogus "explanations" that suggest centrifugal force causes (or contributes to) the tidal bulges as the Earth rotates around the Earth-Moon barycenter. In fact, contrary to common assumptions, centrifugal and Coriolis forces generate no tidal forces to deform the Earth. Earth's axial rotation is the reason for the equatorial bulge, but that is not a tide.

A look at the formula clarifies this. The centrifugal force perceived on any body in the rotating system is 2r. This depends on the angular rotation of the chosen reference frame. If we choose a reference frame rotating around the Earth/Moon baycenter, all points on Earth have the same radius of rotation at any instant. So this results in a rotating centrifugal field of constant strength.

The Coriolis force is perceived in our system as 2mωv where v is the velocity of a mass m moving in the rotating system. In this coordinate system at any instant all points on or within the Earth have the same angular velocity, so the Coriolis force on each and every one has the same size, and therefore they also cannot stress or deform the Earth's shape.

In short, the gravity gradient alone causes the tidal bulges. Earth's axial rotation causes the equatorial bulge (and polar flattening), which is not a tide. Earth's wobbling around the Earth-Moon barycenter does not generate additional stress effects on the Earth's shape, and no additional contribution to tides.

The only force field that can stress or deform a material object must have a field gradient. Uniform force fields cannot stress or deform material objects.

With earth rotation suppressed, using a reference frame rotating at the same rate as the Moon, each chunk of matter on Earth experiences the same centrifugal force (constant in size and direction), therefore this cannot act to deform the shape of the Earth. I.e., this field exerts the same force on adjacent chunks of earth material, so there's no differential force to squeeze, stretch or displace materials. The same is true for Coriolis forces. [1]

If we consider a frame of reference rotating about the Earth-Sun center of mass, we reach the same conclusion.

Tidal bulges are solely due to the gradient of gravitational forces acting aginst tensile forces of earth and water. Centrifugal and Coriolis forces contribute nothing to them.

Endnotes.

  1. An interesting fact is that even in a rotating reference frame, fictitious forces exert a net zero effect in this case. The Coriolis force exactly cancels the centrifugal force. See:

    Takuya Matsuda et. al.Confusion around the tidal force and the centrifugal force. Jein Institute for Fundamental Science, Research Note. 2015.

    From the abstract of this paper: The tidal force arises because the gravitational force exerted on the extended body by the secondary mass is not uniform across the primary. In the derivation of the tidal force, the non-uniformity of the gravity is essential, and inertial forces such as the centrifugal force are not needed. Nevertheless it is often asserted that the tidal force can be explained by the cenrifugal force. If we literally take into account the centrifugal force, it would mislead us.

    Donald E. Simanek, May 24, 2020.

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