To make the tidal forces easier to understand, I made some animations.
These are visualizations of the images that came to my mind when I saw the equations.
Pictures may be easier to understand than equations or words.

(Caution) Please note that watching the rotating images for a long time may make you feel sick.

**1. Translational acceleration system**

Let us look at the forces that appear on the earth in translational motion along a circle.
Since we consider only the translational motion of the earth, we assume that the orientation of the earth does not change.
Figure B-1 shows the translational earth viewed from the inertial system.
The + symbol represents the coordinate system fixed to the translating earth, i.e., the translational acceleration system.
When the earth is viewed from the inertial system, only gravity is visible as shown in Figure B-1.
You may sometimes see animations with inertial force drawn even though the animation is based on an inertial system, but that is not true.
In an inertial system, inertial force cannot be seen.
In the figure, the force acting on the moon is omitted.
In fact, the force acting on the moon (per unit mass) is tens of times larger than that on the earth.

Figure B-2 is a view of the moon in a coordinate system fixed to the translational motion of the earth.
The translational acceleration motion produces a uniform inertial force, which almost cancels out the gravitational force.
What remains is the tidal force.

In translational motion, if the background is a distant starry sky, it is difficult to know whether we are looking at an inertial system or a translational acceleration system, because the background does not move. Figures B-3 and B-4 show an imaginary stationary grid just behind the earth-moon system so that the coordinate system we are looking at can be easily recognized.

**2. Rotating system**

Assuming that the earth is on a coordinate system rotating with its orbital motion, let us consider the forces that appear in a rotating system. Fig. B-5 is the earth on the rotating system viewed from the inertial system. The + symbol represents the coordinate system fixed to the rotating earth. When the earth is viewed from the inertial system, only gravity is visible as shown in Fig. B-5. Fig. B-6 is the view from the coordinate system fixed to the earth, i.e., the rotating system. Centrifugal force around the common center of mass appears. Fig. B-7 shows the centrifugal force centered on the common center of mass, divided into uniform inertial force and earth-centered centrifugal force. Of these, the uniform component almost cancels out gravity. The earth-centered centrifugal force still remains. From the viewpoint of the earth, it is easier to divide the force into two components in this way.

For reference, assuming that the earth is on the coordinate system rotating with the orbital motion, and let us look at it from the common center of mass.
The + symbol represents a coordinate system revolving around the common center of mass.
Fig. B-8 is the earth revolving around the common center of mass viewed from the inertial system.
Fig. B-9 is the earth revolving around the common center of mass viewed from the rotating system centered on the common center of mass.
Centrifugal force around the common center of mass appears.

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Takao FUJIWARA, June 2021